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Widely used approaches include +mechanical scanning with antenna apertures and phase switching +in arrays. Both of those realizations have severe limitations, +related to scanning speeds and implementation costs. Here we +demonstrate a solution, where the antenna pattern is switched with +optical signals. The system encompasses an active element, +surrounded by a set of cylindrically arranged passive dipolar +directors, functionalized with tunable impedances. The control +circuit is realized as a bipolar transistor, driven by a photodiode. +Light illumination in this case serves as a trigger, capable of either +closing or opening the transistor, switching the impedance +between two values. Following this approach, a compact half-a- +wavelength footprint antenna, capable to switch between 6 dBi +directional patterns within a few milliseconds’ latency was +demonstrated. The developed light activation approach allows +constructing devices with multiple almost non-interacting degrees +of freedom, as brunched feeding network is not required. The +capability of MHz and faster switching between multiple +electromagnetic degrees of freedom open pathways to new wireless +applications, where fast beam steering and beamforming +performances are required. + +Index Terms—steerable antenna, electro-optical control, dual- +band, compact antenna, latency. + +I. INTRODUCTION +HE ABILITY to control the radiation pattern with high +accuracy allows for establishing efficient point-to-point +communication, where one or more participants can change +their locations during the process. A radar, tracking a moving +target in both azimuth and elevation, is one notable example. +Recently, the automotive industry raised a demand for high- +resolution short-range radar-based imaging systems, where +high-quality fast scanning small aperture antennas are essential +components [1]–[3]. Another realm is 5G communications, +where beamforming with millisecond-scale latency is the +enabling technology to support fast-speed broadband wireless +communication [4], [5]. In all the beforehand mentioned +applications, antenna devices are subject to engineering +tradeoffs where high scanning speed and low cost are +contradictory requirements. There are several traditional +approaches to beam steering. The first one is a mechanical scan, + +(Corresponding +author: +Dmytro +Vovchuk +e-mail: +dimavovchuk@gmail.com). +Dmytro Vovchuk, Anna Mikhailovskaya, Dmitry Dobrykh, Pavel Ginzburg +School of Electrical Engineering, Tel Aviv University, Ramat Aviv, Tel Aviv, +69978, Israel (e-mail: dimavovchuk@gmail.com, pginzburg@tauex.tau.ac.il ). +where a motor controls the angular position of a highly directive +antenna. This technique is frequently used for implementing +marine and airport tracking radars, where scanning speeds are +not the main factor to consider. Another approach to beam +steering is based on antenna phased arrays. Here multiple +elements are phased-locked and radiate simultaneously. While +this architecture allows achieving fast all-electronic scanning, +the realization of high-quality and directive beams requires +employing tens or even hundreds of phase-shifting elements. +This approach is used e.g., in airborne applications, where the +speed and scan quality requirements predominate over- +involved costs of realizations. Recently, several approaches, +complementary to traditional phased arrays have been proposed +and demonstrated. The ability to tailor and control the laws of +refraction with the help of artificially structured media +(metamaterials [6]–[9]) opened a range of new capabilities in +beam shaping and control. Carefully designed surfaces +(metasurfaces) can provide capabilities to tailor properties of +transmitted and reflected waves [6], [10]–[12]. While many +metasurface studies concentrate on static configurations (e.g., +[13]–[15]), introducing fast real-time tunability is the +demanded feature. Several realizations of dynamically +reconfigurable metasurfaces and metasurface-based antennas +have been demonstrated (e.g. [11], [16]–[20]). The key +underlining concept is typically based on controlling individual +resonant elements within an array with electronics. For +example, tunable capacitance allows shifting resonant +responses of individual elements, and as the result, either +amplitude or phase switchable screens are achieved [11], [21], +[22]. While this type of realization does not rely on expensive +phase shifters, it still requires using numerous (yet simple and +cheap) electronic elements, and, even more critically, a +branched set of wires to drive them. While reflect array +configurations allow hiding wires behind a ground plane [23], +[24], electric circuitry can significantly affect electromagnetic +performances in other realizations. For example, a mesh of thin +wires with subwavelength spacing will have a predominating +undesired electromagnetic response. A probable solution to this +problem has been demonstrated in the case of volumetric +metamaterial-based scatterers [25]. It relies on driving +individual meta-atoms with light. Light and light guiding +materials do not interact with cm and mm waves, which enables +uncoupling of these two phenomena in the design. The +interaction happens directly within an individual antenna +Dmytro Vovchuk Department of Radio Engineering and Information +Security, Yuriy Fedkovych Fernivtsi National University, Chernivtsi, 58012, +Ukraine (e-mail: dimavovchuk@gmail.com) +Dmytro Vovchuk and Anna Mikhailovskaya contributed equally to this +work. +Dual-band electro-optically steerable antenna +Dmytro Vovchuk, Anna Mikhailovskaya, Dmitry Dobrykh, and Pavel Ginzburg +T + +element, where optical energy is rectified within a +photoelement to drive electronics. Here we develop the concept +of electro-optically driven beamforming, which allows fast +manipulation over radiation patterns by arranging arrays of +auxiliary optically switchable reflectors and directors around a +radiating element. The optical link allows for obtaining both +high switching speeds and modularity, i.e., almost any radiating +element can be granted with scanning capabilities, as the +constraints, related to a wired feeding network are relaxed. +The manuscript is organized as follows – the design and +implementation of a single reflector are introduced first and +then followed by its integration into an antenna device. Beam +steering performances are assessed next along with +investigating of other antenna characteristics. Measurements of +the beam steering rates are demonstrated next. The capability to +grant steering capabilities for several commercial and custom- +made antennas radiating elements is discussed before the +Conclusion. +II. ELECTRO-OPTICALLY DRIVEN ELEMENT +Quite a few designs of directive antennas are based on +interference phenomena between several elements. A +representative example here is Yagi-Uda antenna, where a set +of passive elements – reflectors and directors, are responsible +for a narrow beam formation. Each of them introduces a +different phase lag, which is tuned by controlling lengths of +elements within the architecture. While physical size of a +resonant element cannot be controlled dynamically on a +reasonably fast timescale, electric length can be governed by +introducing a tunable lumped element. As the first step, we will +demonstrate a design of wirelessly tunable single element, +which will be subsequently integrated within a beam steering +array. Two states – ‘on’ and ‘off’ correspond to either presence +or absence of the illumination. Our basic component is a half- +wavelength element (λ/2), formed by a pair of λ/4-lenght wires +with a gap in between (Fig. 1(a)). The driving circuit consists +of two photodiodes (BPW34) and a bipolar transistor +(BFU730F115 NPN-type BJT) as in Fig. 1(b). Two +photoelements are used to elevate the voltage drop to open the +transistor. If the illumination power on the circuit is insufficient, +the element acts as a cut. After passing a threshold, the diode +become a shortage. +Figure 1(c) demonstrates the forward scattering spectra of the +system at its two states. Wire dimensions are length l = 72 mm +and radius r = 0.5 mm. The gap at the middle is 1mm. Those +parameters were tuned to make the device complying with +IEEE 802.11 communication standard (in terms of radiation +bands). It is worth mentioning that the transistor impedance is +also considered for both open and short operation states. Fig. +1(c) demonstrates the capability to tune the scattering peak from +2.2GHz to 1.9 and vice versa upon light illumination. Full-wave +numerical analysis, including an introduction of lumped +elements, was done with CST Microwave Studio. The surface +current distribution on the element strongly depends on the light +state (insets to Fig. 1(c)), demonstrating the switching between +dipolar and quadrupolar operation modes. 0.5 pF of the lumped +element C was found to provide a reliable model to switching +for the state ‘off’ and a solid λ/2 wire – for the state ‘on’. Slight +differences between numerical and experimental data come +from nonvanishing formfactors of active elements, which were +not considered in simulations. + +Fig. 1. Optically-switchable passive element – photograph (a) and the +schematics (b) of the photo-activated driving circuit (BJT – bipolar junction +transistor, C – collector, B – base and E – emitter). (c) Numerical analysis and +experimental forward scattering spectra of the device at light ‘on’ and ‘off’ +states. Color lines – responses of individual elements. Inset – current +distributions along the elements (numerical results). + +A choice of elements for implementing the driving circuit +worth a discussion. Among possible architectures (i) varactor + +photodiode; (ii) PIN-photodiode and (iii) phototransistor or +transistor + photodiode can be considered. While varactors are +commonly used in related designs [10], [11], those are not the +best candidates for the current implementation as they demand +quite high voltage to provide a pF-scale capacitance tunability. +0.7 V for Si and 0.35 V for Ge implementations are requited. +Phototransistors are typically designed for low-frequency +applications, e.g., fire protection or motion detection. +Therefore, we will investigate a combination of a low-cost +high-frequency BFU730F115 npn-type BJT and BPW34 +photodiodes. The photodiode’s anode is connected to the +transistor’s base and cathode to the emitter (Fig. 1(b)). The +collector and emitter of the transistor are the outputs of the +driving circuit and are soldered to the λ/4 wires. This +arrangement allows shifting the scattering resonance to higher +frequencies. + +driving +(a) +(b) +circuit +light += +Front view +Back view +Photodiodes +on +off +(c) +on +off +Forward scattering, a.u. +A/m, a.u. +0 +0.5 +dashed-simulations +solid-experiment +0 +1 +1.5 +2 +2.5 +3 +Frequency, GHzIII. OPTICALLY STEERABLE ANTENNA +After designing single elements, those will be assembled to +form a larger-scale system, which aims on providing beam +steering capabilities. Six passive director elements were chosen +to form the geometry. This number, being found beneficial to +optimize wire bundle scatterers [26]–[29], was chosen as a +tradeoff between design simplicity and functionality. While this +configuration fits demands of 6-sector 4G wireless network, it +can be further tuned per application, i.e., the number of +scanning +lobes +can +be +increased, +and +various +5G +communication protocols can be implemented. +The antenna consists of seven elements in overall: one active +(marked with ‘#’) placed exactly at the center and six passives +(1-6) are equidistantly placed on an imaginary cylindrical +surface (Fig. 2). A broadband monopole antenna (W1096), +covering the investigated frequency range and providing rather +flat frequency response, was chosen as a feed [30]. This +commercial element can be replaced by a custom-made +monopole, tuned per frequency. Before assembling the +structure, each of six passive elements was calibrated to provide +the identical response (as in Fig. 1(c)). Here both scattering +parameters and optical activation power are adjusted. Each +individual element was checked separately by performing a +forward scattering experiment. As the element acts as a dipole, +this parameter almost completely characterizes its response. +The manual adjustment was done by cutting the wire’s length. +It is also worth noting that nominals of lumped elements can +vary from item to item. Hence, an individual calibration is +required. Fig. 1(c) demonstrates the calibration curves, the +averaged parameters of which was used as in antenna modeling. + +Fig. 2. (a) Schematic layout and (b) photograph of the optically steerable +antenna. On the insets (c) photograph of the top view. (d) S11 parameters of +antennas – standalone monopole, steering antenna with light ‘on and ‘off’, as +in the legends. + +Without a light activation, all six passive elements are +identical and, as a result, the radiation pattern has no directivity +in-plane (end-fire). To break the symmetry, several elements +can be triggered with light. For an initial approximate analysis, +the elements can be considered as present for 2.2GHz wave if +the light is “on” and absent if there is no direct illumination on +them. For 1.8GHz the scenario is reversed. As a result, several +elements form a directive pattern. A more accurate analysis +suggests considering impact of non-resonant inactivated +elements. This was done numerically, and the system +parameters were additionally optimized. The optimization is +applied to maximize directivity and gain of the antenna, +constraining its overall size [31]. While a directivity in a Yagi- +Uda antenna relies on interference phenomena between several +directors and reflectors, the proposed realization involves +multipolar interaction and near-field coupling between +elements [26]–[28]. The radius of the imaginary cylindrical +surface (taking into account the cylinder radius R = 20 mm), +containing optically switchable passive elements, was chosen +to be 41 mm ≈ 0.26λ. 1.8 and 2.2GHz were chosen quite +arbitrary within the wireless band and can be tuned per a +specific application. +Fig. 3 summarizes the patterns, obtained both numerically +and experimentally at an anechoic camber. ‘1’ and ‘0’ in the +figure captions indicate whenever the element was illuminated +or not, respectively. Antenna matching conditions (S11 +parameters) appear in Fig. 2(d). While the initial design was +made for a single-element activation (Fig. 3a-d), different +combinations can be considered as well. Theoretically the +system has 2N independent degrees of freedom, where N is the +number of elements. Potentially, 2N antenna patterns can be +achieved, nevertheless not all of them can be considered as +practically relevant. Several reports have demonstrated N +patterns with N tunable elements [24], [32], [33]. While our +structure was not designed to maximize the number of patterns, +we found that activating pairs of adjacent elements leads to +formation of directional beams, shifted by 30° in respect to the +single-element case (Fig. 3e-h). As a result, we have +demonstrated 12 directional beams, i.e., 2N useful patterns. +Furthermore, the device shows a dual band performance – both +1.8 and 2.2 GHz with a 10% fractional bandwidth. Activating +other combination of elements didn’t lead to formation of +patterns with reasonable directivity. +Directivity (D) and gain (G) of the antenna will be +characterized next. As the pattern is formed primarily in-plane, +the following relation will be used to process the experimental +data [34]: +𝐷(φ, θ = const) = +𝑃𝑚𝑎𝑥 +1 +2𝜋 ∫ +𝑃(φ)𝑑φ +2𝜋 +0 + , (1) +where Pmax is the maximal radiated power of the antenna. The +assessment is made for a constant elevation angle (θ = 0) and +for the entire 2π of the azimuth φ. The realized gain GTx is +extracted by comparing the device with an etalon antenna +(IDPH-2018 S/N-0807202 horn) with a known gain GRx. Eq. 2 +is used for the analysis [34]. +𝐺𝑇𝑥 = ( +4𝜋𝑎 +𝜆 ) +2 𝑃𝑅𝑥 +𝑃𝑇𝑥 +1 +𝐺𝑅𝑥 , (2) +where ‘a’ is the distance between the apertures of the transmit +Tx and the receive Rx antenas, λ is the operational wavelength +and PRx/PTx = |S21|2 is the power transmission coefficient. +To assess the switching parameter, we calculated the +differential gain values between the ‘on’ and ‘off’ states (Gon +and Goff), as following: +𝐺𝑑𝑖𝑓𝑓 = 𝐺𝑜𝑛 − 𝐺𝑜𝑓𝑓. (3) +The results are summarized in Table I. The numerical results + +(a) 16 +2 # 5 +(b) +Activeelement-# +Passive elements-1...6 +C +Topview +0 +S11, +20 +Monopole +Light'OFF +(d) +Light ON +30 +1.4 +1.8 +2.2 +2.4 +GHzon directivity are presented for the 2D (φ,θ=0) and 3D (φ,θ) +cases, while the experiments are shown only for 2D case. One +can see the difference between the directivity of numerical and +experimental values, especially at 2.2 GHz. The results can be +assessed by comparing patterns in Fig. 3. The most pronounced +difference was found for the data on panels (c) and (d). A +significant back lobe, being predicted numerically (imperfect +optimization), was not found in the measurements. The +opposite behavior was found for the two-element illumination +at 2.2 GHz – here back lobes were found in the experiment, +while the numerical prediction suggests rather minor back +radiation. The reason for this can be several-fold: (i) +imperfection in elements, affecting the interference phenomena +and (ii) a parasitic illumination due to the ambient illumination +and the pollution from nearby light sources – the driving LED +(as will be discussed hereinafter). Nevertheless, the back lobe +suppression effect is not dramatic. (iii) Nevertheless, the +feeding monopole connector has an orientation, perpendicular +to the antenna axis, it breaks the symmetry between different +radiation patterns (e.g., yellow, and purple lines in Fig. 3). +It is worth mentioning that the system cannot perform an +independent simultaneous beam steering at two different +frequency bands, as the same photodiodes are in use. + + +Fig. 3. Radiation patterns – numerical and experimental results. Single (a-d) and double-element (e-h) illumination at the frequencies 1.8 (director case) and +2.2 GHz (reflector case). Antenna 3D radiation patterns (numerical results) are in left insets. + + + + + + +Single-element +[100000] +[000100] +[010000] +[000010] +Illumination +[001000] +[000001] +Radiation patterns +simulations +measurements +90 +90 +(a) +120 +60 +1.8 GHz +120 +60 +(b) +150 +30 +150 +30 +180 +0 +180 +0 +210 +330 +210 +330 +240 +300 +240 +300 +270 +270 +90 +90 +(c) +120 +60 +120 +60 +(d) +2.2 GHz +150 +30 +150 +30 +180 +0 +180 +0 +210 +330 +210 +330 +240 +300 +240 +300 +270 +270 +0 +W/m, a.u. +[110000] +[000110] +Two-element +[011000] +[000011] +Illumination +[001100] +[100001] +Radiation patterns +simulations +measurements +90 +90 +(e) +120 +60 +1.8 GHz +120 +60 +(f) +150 +30 +150 +30 +180 +180 +0 +210 +330 +210 +330 +240 +300 +240 +300 +270 +270 +90 +90 +(g) +120 +60 +2.2 GHz +120 +1 +60 +(h) +150 +30 +150 +30 +180 +0 +180 +0 +210 +330 +210 +330 +240 +300 +240 +300 +270 +270TABLE I +The directivity D and differential gain Gdiff. + +f, GHz +Numerical +Experimental +2D +3D +2D +Single-element +illumination +D, dBi +1.8 +2.68 +5.21 +3.31 +2.2 +2.48 +5.17 +4.11 +Gdiff, dBi +1.8 + +2.06 +2.65 +2.2 + +5.68 +5.56 +Two-element +illumination +D, dBi +1.8 +3.36 +6.01 +3.37 +2.2 +6.17 +9.27 +3.85 +Gdiff, dBi +1.8 + +2.49 +2.2 +2.2 + +7.25 +4.62 + +Free-space illumination of photodiodes requires an extra- +consideration. The first factor is an ambient radiation, which +can accidentally bring the system to a threshold. For an +assessment, we compared chamber conditions with an office +space and outdoors (direct summer sunlight). In last two cases, +a light concealment arrangement is required to maintain the +correct operation of the device. The second factor is undesired +light from a nearby illuminated element. The distance between +the LED and the photodiode is 1cm (inset to Fig. 4(a), thus the +light leakage was found to play no role. In both cases the +voltage on the diode was measured and compared with 0.7V +threshold. It is worth noting that introducing integrated optics +arrangements (e.g., waveguiding devices) are capable to solve +issues of the undesired overexposure to light. +One of the main advantages of the proposed design is its +potentially fast switching rates. 5G standards demand latencies +as a small as a milli-second. It implies having capabilities of +sub-MHz beam steering rates. To assess this parameter, the +following setup have been constructed – a signal from a high- +frequency generator (N5173B EXG X-Series Microwave +Analog Signal Generator) is split via ZX10-2-852-S+ Splitter +into two channels: the first feeds the active element of the +antenna and the second provides the synchronization signal and +feeds the local oscillator (LO) input of a mixer ZX05-C24-S+ +at the receiver (Fig. 4(a)). The LF pulse sequences generator +(81160A Pulse Function Arbitrary Noise Generator) feeds a +LED SMD5630, which is located close to the antenna +photodiodes and performs the on/off-switching with a period T += 1 ms. 50% duty cycle (τ) was chosen. The receiver includes +Rx antenna, feeding the RF input of the mixer. The output, after +a low-pass filter (LPF) BLP-100-75+, is displaced on a scope. +The digitalized scope’s output allows investigating switching +properties of the device (antenna under test – AUT). The results +show that f0 = 1/T at 1 kHz can be obtained (Fig. 4(b)). To +determine rise (tr) and fall (tf) times, the received signal was +smoothed and fitted with a sine series (Fig. 4(c)). The extracted +rise and fall times for the system are ~ 0.1 ms. + + +Fig. 4. (a) Schematics of the setup for measuring the switching rate. (b) +Zoomed IF signal on the scope. (c) Post-processed signal - period T = 1 ms +(50% duty cycle for τ = 0.5 ms), rise tr, and fall tf time. +IV. BEAM STEERING WITH OTHER ANTENNAS + +To demonstrate the flexibility of the proposed method, 3 +different antennas have been considered, namely the +commercial monopole from the previous studies, symmetric +dipole antenna and a monopole above a ground plane (panels a, +d, and f in Fig. 5, respectively). Each of those has an omni- +directional pattern in-plane. Two switching elements has been +used do demonstrate the concept. As the structures have +reflection symmetry, only one directional pattern per frequency +was demonstrated. Yellow and green lines correspond to 2.2 +and 1.8 GHz, respectively. Illuminating one side of the structure +leads to a creation of directional patterns, which are oppositely +oriented for both of those frequencies. Switching between the +illumination side will case the flip in the patters. The +commercial monopole antenna has slightly better performances +owing as it underwent a significant optimization by the vendor. +The dipole demonstrates less directive pattern at 1.8GHz owing +to the frequency-dependent balun. This aspect does not affect +the monopole configuration, which also demonstrates good +switching capabilities. + +AUT +87..0 +Radiation +direction +on/off (T) +top view +Rx +1cm +LED'S +RF +HFsignal +(f1,2) +LO +IFJ +(a) +LPF +ch. 1 +(b) +ch. 1 +(c) +zoomed in ch. 1 +n'e +tr +tf +t.ms> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < + +6 + + +Fig. 5. The concept of granting a radiating element with beam steering +capabilities. (a), (c), and (e) – photographs of antenna devices. (b), (d), and (f) +- experimentally obtained (in-plane) radiation patterns. Switching between 2 +sectors has been considered. +V. CONCLUSION +A scanning antenna with optical control is demonstrated +experimentally. The device consists of six passive resonators, +arranged around the feed. Electromagnetic properties of passive +elements, serving as either directors or reflectors, are tuned with +light. The driving circuit, containing photodiodes and bipolar +transistor, is activated remotely with light. This approach +allows tuning electromagnetic properties of the system without +a need of a brunched network of metal wires. The demonstrated +design provides steering capabilities of directional beams with +~5 dBi of the directivity and 6 dBi of the differential gain with +a switching rate around at sub-MHz rate. The demonstrated +antenna belongs to the class of compact (2r/λ ≈ 0.5-0.6, where +r is the radius of an imaginary sphere that surrounds the whole +antenna [31], [35]) low-cost devices (the active element + six +passive elements with driving circuits cost around 20$). +Furthermore, it was shown to provide a dual-band operation at +frequencies, relevant to wireless communications. Further +optimization of the electromagnetic design and introduction of +fast elements (transistors and fast photodiodes) can elevate the +switching rates towards MHz and higher opening pathways to +new applications, where fast beam steering and beamforming +performances are required (e.g., radars and 5G). Frequency +bands in 5G protocols are quite broad and utilized per +application, though a capability of fast beam control remains +essential. Light activation approach allows constructing devices +with multiple almost non-interacting degrees of freedom, as +brunched feeding network is not required and, in principle, +almost any radiating element can be granted with beam steering +capabilities. + +ACKNOWLEDGEMENTS +The work was supported by ERC POC, grant 101061890 +“DeepSight”. +REFERENCES +[1] +S. M. Patole, M. Torlak, D. Wang, and M. Ali, +“Automotive Radars: A review of signal processing +techniques,” IEEE Signal Process. 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Antennas Propag., vol. 51, no. 8, pp. 2116– +2123, Aug. 2003, doi: 10.1109/TAP.2003.814754. + + + diff --git a/-dAyT4oBgHgl3EQf3fmN/content/tmp_files/load_file.txt b/-dAyT4oBgHgl3EQf3fmN/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8c310f4b15ad6d1010b5dfa5e99b95d16d0d350f --- /dev/null +++ b/-dAyT4oBgHgl3EQf3fmN/content/tmp_files/load_file.txt @@ -0,0 +1,759 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf,len=758 +page_content='Abstract— The ability to obtain dynamic control over an antenna radiation pattern is one of the main functions, desired in a vast range of applications, including wireless communications, radars, and many others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Widely used approaches include mechanical scanning with antenna apertures and phase switching in arrays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Both of those realizations have severe limitations, related to scanning speeds and implementation costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Here we demonstrate a solution, where the antenna pattern is switched with optical signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' The system encompasses an active element, surrounded by a set of cylindrically arranged passive dipolar directors, functionalized with tunable impedances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' The control circuit is realized as a bipolar transistor, driven by a photodiode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Light illumination in this case serves as a trigger, capable of either closing or opening the transistor, switching the impedance between two values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Following this approach, a compact half-a- wavelength footprint antenna, capable to switch between 6 dBi directional patterns within a few milliseconds’ latency was demonstrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' The developed light activation approach allows constructing devices with multiple almost non-interacting degrees of freedom, as brunched feeding network is not required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' The capability of MHz and faster switching between multiple electromagnetic degrees of freedom open pathways to new wireless applications, where fast beam steering and beamforming performances are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Index Terms—steerable antenna, electro-optical control, dual- band, compact antenna, latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' INTRODUCTION HE ABILITY to control the radiation pattern with high accuracy allows for establishing efficient point-to-point communication, where one or more participants can change their locations during the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' A radar, tracking a moving target in both azimuth and elevation, is one notable example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Recently, the automotive industry raised a demand for high- resolution short-range radar-based imaging systems, where high-quality fast scanning small aperture antennas are essential components [1]–[3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Another realm is 5G communications, where beamforming with millisecond-scale latency is the enabling technology to support fast-speed broadband wireless communication [4], [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' In all the beforehand mentioned applications, antenna devices are subject to engineering tradeoffs where high scanning speed and low cost are contradictory requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' There are several traditional approaches to beam steering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' The first one is a mechanical scan, (Corresponding author: Dmytro Vovchuk e-mail: dimavovchuk@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='com).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Dmytro Vovchuk, Anna Mikhailovskaya, Dmitry Dobrykh, Pavel Ginzburg School of Electrical Engineering, Tel Aviv University, Ramat Aviv, Tel Aviv, 69978, Israel (e-mail: dimavovchuk@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='com, pginzburg@tauex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='tau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='il ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' where a motor controls the angular position of a highly directive antenna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' This technique is frequently used for implementing marine and airport tracking radars, where scanning speeds are not the main factor to consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Another approach to beam steering is based on antenna phased arrays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Here multiple elements are phased-locked and radiate simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' While this architecture allows achieving fast all-electronic scanning, the realization of high-quality and directive beams requires employing tens or even hundreds of phase-shifting elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' This approach is used e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=', in airborne applications, where the speed and scan quality requirements predominate over- involved costs of realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Recently, several approaches, complementary to traditional phased arrays have been proposed and demonstrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' The ability to tailor and control the laws of refraction with the help of artificially structured media (metamaterials [6]–[9]) opened a range of new capabilities in beam shaping and control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Carefully designed surfaces (metasurfaces) can provide capabilities to tailor properties of transmitted and reflected waves [6], [10]–[12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' While many metasurface studies concentrate on static configurations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=', [13]–[15]), introducing fast real-time tunability is the demanded feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Several realizations of dynamically reconfigurable metasurfaces and metasurface-based antennas have been demonstrated (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' [11], [16]–[20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' The key underlining concept is typically based on controlling individual resonant elements within an array with electronics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' For example, tunable capacitance allows shifting resonant responses of individual elements, and as the result, either amplitude or phase switchable screens are achieved [11], [21], [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' While this type of realization does not rely on expensive phase shifters, it still requires using numerous (yet simple and cheap) electronic elements, and, even more critically, a branched set of wires to drive them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' While reflect array configurations allow hiding wires behind a ground plane [23], [24], electric circuitry can significantly affect electromagnetic performances in other realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' For example, a mesh of thin wires with subwavelength spacing will have a predominating undesired electromagnetic response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' A probable solution to this problem has been demonstrated in the case of volumetric metamaterial-based scatterers [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' It relies on driving individual meta-atoms with light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Light and light guiding materials do not interact with cm and mm waves, which enables uncoupling of these two phenomena in the design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' The interaction happens directly within an individual antenna Dmytro Vovchuk Department of Radio Engineering and Information Security, Yuriy Fedkovych Fernivtsi National University, Chernivtsi, 58012, Ukraine (e-mail: dimavovchuk@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='com) Dmytro Vovchuk and Anna Mikhailovskaya contributed equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Dual-band electro-optically steerable antenna Dmytro Vovchuk, Anna Mikhailovskaya, Dmitry Dobrykh, and Pavel Ginzburg T element, where optical energy is rectified within a photoelement to drive electronics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Here we develop the concept of electro-optically driven beamforming, which allows fast manipulation over radiation patterns by arranging arrays of auxiliary optically switchable reflectors and directors around a radiating element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' The optical link allows for obtaining both high switching speeds and modularity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=', almost any radiating element can be granted with scanning capabilities, as the constraints, related to a wired feeding network are relaxed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' The manuscript is organized as follows – the design and implementation of a single reflector are introduced first and then followed by its integration into an antenna device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Beam steering performances are assessed next along with investigating of other antenna characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Measurements of the beam steering rates are demonstrated next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' The capability to grant steering capabilities for several commercial and custom- made antennas radiating elements is discussed before the Conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' ELECTRO-OPTICALLY DRIVEN ELEMENT Quite a few designs of directive antennas are based on interference phenomena between several elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' A representative example here is Yagi-Uda antenna, where a set of passive elements – reflectors and directors, are responsible for a narrow beam formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Each of them introduces a different phase lag, which is tuned by controlling lengths of elements within the architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' While physical size of a resonant element cannot be controlled dynamically on a reasonably fast timescale, electric length can be governed by introducing a tunable lumped element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' As the first step, we will demonstrate a design of wirelessly tunable single element, which will be subsequently integrated within a beam steering array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Two states – ‘on’ and ‘off’ correspond to either presence or absence of the illumination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Our basic component is a half- wavelength element (λ/2), formed by a pair of λ/4-lenght wires with a gap in between (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' 1(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' The driving circuit consists of two photodiodes (BPW34) and a bipolar transistor (BFU730F115 NPN-type BJT) as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Two photoelements are used to elevate the voltage drop to open the transistor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' If the illumination power on the circuit is insufficient, the element acts as a cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' After passing a threshold, the diode become a shortage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Figure 1(c) demonstrates the forward scattering spectra of the system at its two states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Wire dimensions are length l = 72 mm and radius r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='5 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' The gap at the middle is 1mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Those parameters were tuned to make the device complying with IEEE 802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='11 communication standard (in terms of radiation bands).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' It is worth mentioning that the transistor impedance is also considered for both open and short operation states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' 1(c) demonstrates the capability to tune the scattering peak from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='2GHz to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='9 and vice versa upon light illumination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Full-wave numerical analysis, including an introduction of lumped elements, was done with CST Microwave Studio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' The surface current distribution on the element strongly depends on the light state (insets to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' 1(c)), demonstrating the switching between dipolar and quadrupolar operation modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='5 pF of the lumped element C was found to provide a reliable model to switching for the state ‘off’ and a solid λ/2 wire – for the state ‘on’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Slight differences between numerical and experimental data come from nonvanishing formfactors of active elements, which were not considered in simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Optically-switchable passive element – photograph (a) and the schematics (b) of the photo-activated driving circuit (BJT – bipolar junction transistor, C – collector, B – base and E – emitter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' (c) Numerical analysis and experimental forward scattering spectra of the device at light ‘on’ and ‘off’ states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Color lines – responses of individual elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Inset – current distributions along the elements (numerical results).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' A choice of elements for implementing the driving circuit worth a discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Among possible architectures (i) varactor + photodiode;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' (ii) PIN-photodiode and (iii) phototransistor or transistor + photodiode can be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' While varactors are commonly used in related designs [10], [11], those are not the best candidates for the current implementation as they demand quite high voltage to provide a pF-scale capacitance tunability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='7 V for Si and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='35 V for Ge implementations are requited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Phototransistors are typically designed for low-frequency applications, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=', fire protection or motion detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Therefore, we will investigate a combination of a low-cost high-frequency BFU730F115 npn-type BJT and BPW34 photodiodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' The photodiode’s anode is connected to the transistor’s base and cathode to the emitter (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' 1(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' The collector and emitter of the transistor are the outputs of the driving circuit and are soldered to the λ/4 wires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' This arrangement allows shifting the scattering resonance to higher frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' driving (a) (b) circuit light = Front view Back view Photodiodes on off (c) on off Forward scattering, a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' A/m, a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='5 dashed-simulations solid-experiment 0 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='5 3 Frequency, GHzIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' OPTICALLY STEERABLE ANTENNA After designing single elements, those will be assembled to form a larger-scale system, which aims on providing beam steering capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Six passive director elements were chosen to form the geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' This number, being found beneficial to optimize wire bundle scatterers [26]–[29], was chosen as a tradeoff between design simplicity and functionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' While this configuration fits demands of 6-sector 4G wireless network, it can be further tuned per application, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=', the number of scanning lobes can be increased, and various 5G communication protocols can be implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' The antenna consists of seven elements in overall: one active (marked with ‘#’) placed exactly at the center and six passives (1-6) are equidistantly placed on an imaginary cylindrical surface (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' A broadband monopole antenna (W1096), covering the investigated frequency range and providing rather flat frequency response, was chosen as a feed [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' This commercial element can be replaced by a custom-made monopole, tuned per frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Before assembling the structure, each of six passive elements was calibrated to provide the identical response (as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' 1(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Here both scattering parameters and optical activation power are adjusted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Each individual element was checked separately by performing a forward scattering experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' As the element acts as a dipole, this parameter almost completely characterizes its response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' The manual adjustment was done by cutting the wire’s length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' It is also worth noting that nominals of lumped elements can vary from item to item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Hence, an individual calibration is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' 1(c) demonstrates the calibration curves, the averaged parameters of which was used as in antenna modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' (a) Schematic layout and (b) photograph of the optically steerable antenna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' On the insets (c) photograph of the top view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' (d) S11 parameters of antennas – standalone monopole, steering antenna with light ‘on and ‘off’, as in the legends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Without a light activation, all six passive elements are identical and, as a result, the radiation pattern has no directivity in-plane (end-fire).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' To break the symmetry, several elements can be triggered with light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' For an initial approximate analysis, the elements can be considered as present for 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='2GHz wave if the light is “on” and absent if there is no direct illumination on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' For 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='8GHz the scenario is reversed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' As a result, several elements form a directive pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' A more accurate analysis suggests considering impact of non-resonant inactivated elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' This was done numerically, and the system parameters were additionally optimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' The optimization is applied to maximize directivity and gain of the antenna, constraining its overall size [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' While a directivity in a Yagi- Uda antenna relies on interference phenomena between several directors and reflectors, the proposed realization involves multipolar interaction and near-field coupling between elements [26]–[28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' The radius of the imaginary cylindrical surface (taking into account the cylinder radius R = 20 mm), containing optically switchable passive elements, was chosen to be 41 mm ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='26λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='8 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='2GHz were chosen quite arbitrary within the wireless band and can be tuned per a specific application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' 3 summarizes the patterns, obtained both numerically and experimentally at an anechoic camber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' ‘1’ and ‘0’ in the figure captions indicate whenever the element was illuminated or not, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Antenna matching conditions (S11 parameters) appear in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' 2(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' While the initial design was made for a single-element activation (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' 3a-d), different combinations can be considered as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Theoretically the system has 2N independent degrees of freedom, where N is the number of elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Potentially, 2N antenna patterns can be achieved, nevertheless not all of them can be considered as practically relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Several reports have demonstrated N patterns with N tunable elements [24], [32], [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' While our structure was not designed to maximize the number of patterns, we found that activating pairs of adjacent elements leads to formation of directional beams, shifted by 30° in respect to the single-element case (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' 3e-h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' As a result, we have demonstrated 12 directional beams, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=', 2N useful patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Furthermore, the device shows a dual band performance – both 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='8 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='2 GHz with a 10% fractional bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Activating other combination of elements didn’t lead to formation of patterns with reasonable directivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Directivity (D) and gain (G) of the antenna will be characterized next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' As the pattern is formed primarily in-plane, the following relation will be used to process the experimental data [34]: 𝐷(φ, θ = const) = 𝑃𝑚𝑎𝑥 1 2𝜋 ∫ 𝑃(φ)𝑑φ 2𝜋 0 , (1) where Pmax is the maximal radiated power of the antenna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' The assessment is made for a constant elevation angle (θ = 0) and for the entire 2π of the azimuth φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' The realized gain GTx is extracted by comparing the device with an etalon antenna (IDPH-2018 S/N-0807202 horn) with a known gain GRx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' 2 is used for the analysis [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' 𝐺𝑇𝑥 = ( 4𝜋𝑎 𝜆 ) 2 𝑃𝑅𝑥 𝑃𝑇𝑥 1 𝐺𝑅𝑥 , (2) where ‘a’ is the distance between the apertures of the transmit Tx and the receive Rx antenas, λ is the operational wavelength and PRx/PTx = |S21|2 is the power transmission coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' To assess the switching parameter, we calculated the differential gain values between the ‘on’ and ‘off’ states (Gon and Goff), as following: 𝐺𝑑𝑖𝑓𝑓 = 𝐺𝑜𝑛 − 𝐺𝑜𝑓𝑓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' (3) The results are summarized in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' The numerical results (a) 16 2 # 5 (b) Activeelement-# Passive elements-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content="6 C Topview 0 S11, 20 Monopole Light'OFF (d) Light ON 30 1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='4 GHzon directivity are presented for the 2D (φ,θ=0) and 3D (φ,θ) cases, while the experiments are shown only for 2D case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' One can see the difference between the directivity of numerical and experimental values, especially at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='2 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' The results can be assessed by comparing patterns in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' The most pronounced difference was found for the data on panels (c) and (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' A significant back lobe, being predicted numerically (imperfect optimization), was not found in the measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' The opposite behavior was found for the two-element illumination at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='2 GHz – here back lobes were found in the experiment, while the numerical prediction suggests rather minor back radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' The reason for this can be several-fold: (i) imperfection in elements, affecting the interference phenomena and (ii) a parasitic illumination due to the ambient illumination and the pollution from nearby light sources – the driving LED (as will be discussed hereinafter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Nevertheless, the back lobe suppression effect is not dramatic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' (iii) Nevertheless, the feeding monopole connector has an orientation, perpendicular to the antenna axis, it breaks the symmetry between different radiation patterns (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=', yellow, and purple lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' It is worth mentioning that the system cannot perform an independent simultaneous beam steering at two different frequency bands, as the same photodiodes are in use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Radiation patterns – numerical and experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Single (a-d) and double-element (e-h) illumination at the frequencies 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='8 (director case) and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='2 GHz (reflector case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Antenna 3D radiation patterns (numerical results) are in left insets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Single-element [100000] [000100] [010000] [000010] Illumination [001000] [000001] Radiation patterns simulations measurements 90 90 (a) 120 60 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='8 GHz 120 60 (b) 150 30 150 30 180 0 180 0 210 330 210 330 240 300 240 300 270 270 90 90 (c) 120 60 120 60 (d) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='2 GHz 150 30 150 30 180 0 180 0 210 330 210 330 240 300 240 300 270 270 0 W/m, a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' [110000] [000110] Two-element [011000] [000011] Illumination [001100] [100001] Radiation patterns simulations measurements 90 90 (e) 120 60 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='8 GHz 120 60 (f) 150 30 150 30 180 180 0 210 330 210 330 240 300 240 300 270 270 90 90 (g) 120 60 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='2 GHz 120 1 60 (h) 150 30 150 30 180 0 180 0 210 330 210 330 240 300 240 300 270 270TABLE I The directivity D and differential gain Gdiff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' f, GHz Numerical Experimental 2D 3D 2D Single-element illumination D, dBi 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='68 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='21 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='31 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='48 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='17 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='11 Gdiff, dBi 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='06 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='65 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='68 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='56 Two-element illumination D, dBi 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='36 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='01 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='37 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='17 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='27 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='85 Gdiff, dBi 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='49 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='25 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='62 Free-space illumination of photodiodes requires an extra- consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' The first factor is an ambient radiation, which can accidentally bring the system to a threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' For an assessment, we compared chamber conditions with an office space and outdoors (direct summer sunlight).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' In last two cases, a light concealment arrangement is required to maintain the correct operation of the device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' The second factor is undesired light from a nearby illuminated element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' The distance between the LED and the photodiode is 1cm (inset to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' 4(a), thus the light leakage was found to play no role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' In both cases the voltage on the diode was measured and compared with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='7V threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' It is worth noting that introducing integrated optics arrangements (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=', waveguiding devices) are capable to solve issues of the undesired overexposure to light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' One of the main advantages of the proposed design is its potentially fast switching rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' 5G standards demand latencies as a small as a milli-second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' It implies having capabilities of sub-MHz beam steering rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' To assess this parameter, the following setup have been constructed – a signal from a high- frequency generator (N5173B EXG X-Series Microwave Analog Signal Generator) is split via ZX10-2-852-S+ Splitter into two channels: the first feeds the active element of the antenna and the second provides the synchronization signal and feeds the local oscillator (LO) input of a mixer ZX05-C24-S+ at the receiver (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' 4(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' The LF pulse sequences generator (81160A Pulse Function Arbitrary Noise Generator) feeds a LED SMD5630, which is located close to the antenna photodiodes and performs the on/off-switching with a period T = 1 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' 50% duty cycle (τ) was chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' The receiver includes Rx antenna, feeding the RF input of the mixer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' The output, after a low-pass filter (LPF) BLP-100-75+, is displaced on a scope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' The digitalized scope’s output allows investigating switching properties of the device (antenna under test – AUT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' The results show that f0 = 1/T at 1 kHz can be obtained (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' 4(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' To determine rise (tr) and fall (tf) times, the received signal was smoothed and fitted with a sine series (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' 4(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' The extracted rise and fall times for the system are ~ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='1 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' (a) Schematics of the setup for measuring the switching rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' (b) Zoomed IF signal on the scope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' (c) Post-processed signal - period T = 1 ms (50% duty cycle for τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='5 ms), rise tr, and fall tf time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' BEAM STEERING WITH OTHER ANTENNAS To demonstrate the flexibility of the proposed method, 3 different antennas have been considered, namely the commercial monopole from the previous studies, symmetric dipole antenna and a monopole above a ground plane (panels a, d, and f in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' 5, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Each of those has an omni- directional pattern in-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Two switching elements has been used do demonstrate the concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' As the structures have reflection symmetry, only one directional pattern per frequency was demonstrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Yellow and green lines correspond to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='2 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='8 GHz, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Illuminating one side of the structure leads to a creation of directional patterns, which are oppositely oriented for both of those frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Switching between the illumination side will case the flip in the patters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' The commercial monopole antenna has slightly better performances owing as it underwent a significant optimization by the vendor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' The dipole demonstrates less directive pattern at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='8GHz owing to the frequency-dependent balun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' This aspect does not affect the monopole configuration, which also demonstrates good switching capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' AUT 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=".0 Radiation direction on/off (T) top view Rx 1cm LED'S RF HFsignal (f1,2) LO IFJ (a) LPF ch." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' 1 (b) ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' 1 (c) zoomed in ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=" 1 n'e tr tf t." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='ms> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 6 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' The concept of granting a radiating element with beam steering capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' (a), (c), and (e) – photographs of antenna devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' (b), (d), and (f) experimentally obtained (in-plane) radiation patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Switching between 2 sectors has been considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' CONCLUSION A scanning antenna with optical control is demonstrated experimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' The device consists of six passive resonators, arranged around the feed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Electromagnetic properties of passive elements, serving as either directors or reflectors, are tuned with light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' The driving circuit, containing photodiodes and bipolar transistor, is activated remotely with light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' This approach allows tuning electromagnetic properties of the system without a need of a brunched network of metal wires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' The demonstrated design provides steering capabilities of directional beams with ~5 dBi of the directivity and 6 dBi of the differential gain with a switching rate around at sub-MHz rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' The demonstrated antenna belongs to the class of compact (2r/λ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='5-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='6, where r is the radius of an imaginary sphere that surrounds the whole antenna [31], [35]) low-cost devices (the active element + six passive elements with driving circuits cost around 20$).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Furthermore, it was shown to provide a dual-band operation at frequencies, relevant to wireless communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Further optimization of the electromagnetic design and introduction of fast elements (transistors and fast photodiodes) can elevate the switching rates towards MHz and higher opening pathways to new applications, where fast beam steering and beamforming performances are required (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=', radars and 5G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Frequency bands in 5G protocols are quite broad and utilized per application, though a capability of fast beam control remains essential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Light activation approach allows constructing devices with multiple almost non-interacting degrees of freedom, as brunched feeding network is not required and, in principle, almost any radiating element can be granted with beam steering capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' ACKNOWLEDGEMENTS The work was supported by ERC POC, grant 101061890 “DeepSight”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' REFERENCES [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Rudant, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Belmkaddem, “Miniature directive antennas,” Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Microw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Wirel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' 6, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' 45–50, Feb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' 2014, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='1017/S1759078713001098.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' [32] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Lee and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Choi, “Five-element ESPAR antenna using the annular ring slot active element,” Microw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' 58, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' 12, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' 2800– 2804, Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' 2016, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='1002/MOP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='30155.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' [33] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Liang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Sun, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Zhou, “Multiple Beam Parasitic Array Radiator Antenna for 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='4 GHz WLAN Applications,” IEEE Antennas Wirel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Propag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' 17, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' 12, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' 2513–2516, Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' 2018, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='1109/LAWP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='2880208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' [34] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Balanis, Antenna Theory: Analysis and Design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Wiley-Interscience;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' 3 edition, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' [35] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Geyi, “Physical limitations of antenna,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' Antennas Propag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' 51, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' 8, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' 2116– 2123, Aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content=' 2003, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='1109/TAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} +page_content='814754.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQf3fmN/content/2301.00770v1.pdf'} diff --git a/.gitattributes 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--git a/3NE0T4oBgHgl3EQfeABh/content/tmp_files/2301.02384v1.pdf.txt b/3NE0T4oBgHgl3EQfeABh/content/tmp_files/2301.02384v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..e667b5d5a3c01996872384a4999b22163c050bde --- /dev/null +++ b/3NE0T4oBgHgl3EQfeABh/content/tmp_files/2301.02384v1.pdf.txt @@ -0,0 +1,742 @@ +arXiv:2301.02384v1 [cond-mat.mes-hall] 6 Jan 2023 +Universal optical polarizability for plasmonic nanostructures +Tigran V. Shahbazyan +Department of Physics, Jackson State University, Jackson, Mississippi 39217 USA +We develop an analytical model for calculation of optical spectra for metal nanostructures of +arbitrary shape supporting localized surface plasmons (LSPs). For plasmonic nanostructures with +characteristic size below the diffraction limit, we obtain the explicit expression for optical polariz- +ability that describes the lineshape of optical spectra solely in terms of the metal dielectric function +and LSP frequency. The amplitude of LSP spectral band is determined by the effective system +volume that, for long wavelength LSPs, can significantly exceed the physical volume of metal nanos- +tructure. These results can be used to model or interpret the experimental spectra of plasmonic +nanostructures and to tune their optical properties for various applications. +Localized surface plasmons (LSPs) are collective elec- +tron excitation resonantly excited by incident light in +metal nanostructures with characteristic size below the +diffraction limit [1–3]. Optical interactions between LSPs +and excitons in dye molecules or semiconductors un- +derpin numerous phenomena in plasmon-enhanced spec- +troscopy, such as surface-enhanced Raman scattering [4], +plasmon-enhanced fluorescence and luminescence [5–11], +strong exciton-plasmon coupling [12–25], and plasmonic +laser [26–29]. Optical properties of metal nanostructures +of various sizes and shapes are of critical importance for +numerous plasmonics applications [30–32] and were ex- +tensively studied experimentally and theoretically [33– +39]. The generic optical characteristics that defines the +response of plasmonic nanostructure to an incident elec- +tromagnetic (EM) field Eine−iωt as well as the interac- +tions between LSPs and excitons is the optical polar- +izability tensor α(ω), where ω is the incident field fre- +quency. If the characteristic system size is much smaller +than the radiation wavelength, so that Ein is nearly uni- +form on the system scale, the induced dipole moment of +plasmonic structure has the form p(ω) = α(ω)Ein, where +α(ω) can be calculated, with a good accuracy, within the +quasistatic approach [3]. Analytical models for α(ω) have +been available only for highly symmetric systems, such as +spherical, spheroidal or cylindrical geometries [34]. For +example, a metal nanosphere of radius a placed in the air +is characterized by the scalar polarizability +α(ω) = a3 ε(ω) − 1 +ε(ω) + 2, +(1) +where ε(ω) = ε′(ω) + iε′′(ω) is complex dielectric func- +tion of the metal. However, for more complicated shapes +used in the experiment, calculations of optical spectra are +performed using various numerical techniques [34–38]. +On the other hand, for actual structures explored in +the experiment, analytical or numerical models attempt- +ing to determine both the LSP frequency and the line- +shape of optical spectra, as Eq. (1) does, are not even +necessary due to inevitable uncertainties in the nanos- +tructure shapes and sizes. Typically, the spectral posi- +tion of LSP resonance is measured with a reasonably high +precision, and so the main challenge is to describe or in- +terpret the lineshape of optical spectra. Therefore, an +analytical model describing accurately the optical spec- +tra of plasmonic nanostructures at given LSP frequencies +would be highly useful. Here, we present such a model. +Specifically, we show that the optical polarizability of a +small metal structure of arbitrary shape associated with +the LSP resonance at a frequency ωp has the form +αp(ω) = Vp +ε(ω) − 1 +ε(ω) − ε′(ωp), +(2) +where Vp = Vm|χ′(ωp)|sp is the effective volume. Here, +Vm is the metal volume, χ = (ε−1)/4π is the susceptibil- +ity (we use Gaussian units), while the parameter sp ≤ 1 +reflects the system geometry but, for dipole LSP modes, +is independent of its volume. Remarkably, for any system +geometry, the lineshape of optical spectra is determined +solely by the metal dielectric function and LSP frequency. +The polarization (2) can be extended to larger systems +by including the LSP radiation damping as well. +LSP modes and the Green function.—We consider +metal nanostructures supporting LSPs that are localized +at the length scale much smaller than the radiation wave- +length. In the absence of retardation effects, each region +of the structure, metallic or dielectric, is characterized +by the dielectric function εi(ω), so that the full dielectric +function is ε(ω, r) = � +i θi(r)εi(ω), where θi(r) is unit +step function that vanishes outside the region volume Vi. +We assume that dielectric regions’ permittivities are con- +stant and adopt ε(ω) for the metal one. The LSP modes +are defined by the lossless Gauss’s equation [3], +∇ · [ε′(ωp, r)∇Φp(r)] = 0, +(3) +where Φp(r) and Ep(r) = −∇Φp(r), which we chose +real, are the mode’s potential and electric field. +Note +that the eigenmodes of Eq. (3) are orthogonal in each +region [40]: +� +dViEp(r)·Ep′(r) = δpp′ � +dViE2 +p(r). +The EM dyadic Green function D(ω; r, r′) satisfies (in +the operator form) ∇×∇×D−(ω2/c2)εD = (4πω2/c2)I, +while equation for the longitudinal part of D is obtained +by applying ∇ to both sides. In the near field, we switch + +2 +to the scalar Green function for the potentials D(ω; r, r′), +defined as D(ω; r, r′) = ∇∇′D(ω; r, r′), satisfying +∇ · [ε(ω, r)∇D(ω; r, r′)] = 4πδ(r − r′). +(4) +We now adopt the decomposition D = D0 +DLSP, where +D0 = −|r − r′|−1 is the free-space Green function and +DLSP is the LSP contribution, satisfying +∇· +� +ε(ω, r)∇DLSP(ω; r, r′) +� += −∇· +� +[ε(ω, r) − 1]∇D0(ω; r, r′) +� +. +(5) +Consider first the lossless case and set ε′′ = 0 for now. +For real dielectric function ε(ω, r), we can expand DLSP +over the eigenmodes of Eq. (3) as +DLSP(ω; r, r′) = +� +p +Dp(ω)Φp(r)Φp(r′), +(6) +with real coefficients Dp(ω) [41, 42]. In the next step, +we apply the operator ∆′ to both sides of Eq. (5), +and integrate the result over V ′ with the factor Φp(r′). +Using the LSP Green function expansion (6) and the +modes’ orthogonality, one can prove the following rela- +tions, +� +dV ′Φp(r′)∆′DLSP(ω; r, r′) = −DpΦp(r) +� +dV E2 +p +and � dV ′Φp(r′)∆′D0(ω; r, r′) = 4πΦp(r), to use in the +left-hand side and right-hand side of Eq. (5), yielding +Dp∇· +� +ε(ω, r)Ep(r) +� += 4π ∇· +� +[ε(ω, r) − 1]Ep(r) +� +� +dVE2p +. (7) +Multiplying Eq. (7) by Φp(r) and integrating the result +over the system volume, we obtain the coefficients Dp as +Dp(ω) = +4π +� +dV E2n(r) − +4π +� +dV ε(ω, r)E2p(r), +(8) +where the first term ensures the boundary condition for +ε = 1, and will be omitted in the following. Accordingly, +the LSP dyadic Green function for the electric fields takes +the form DLSP(ω; r, r′) = � +p Dp(ω)Ep(r)Ep(r′). +Note that even though the LSP Green function is ex- +pressed in terms of real eigenmodes Ep, it is valid for plas- +monic systems with complex dielectric function ε(ω, r) = +ε′(ω, r) + iε′′(ω, r) as well. Indeed, if ε′′ is included as +perturbation, then the first-order (diagonal) correction +leads to Eq. (8), while the high-order corrections include +non-diagonal terms of the form ε′′(ω) +� +dVmEp(r)Ep′(r), +which vanish due to the modes’ orthogonality, and so the +LSP Green function DLSP with complex coefficients (8) +is, in fact, exact in all orders. +We now note that, in the quasistatic approximation, +the frequency and coordinate dependencies in the LSP +Green function can be separated out. Using the Gauss’s +equation in the form � dV ε′(ωp, r)E2 +p(r) = 0, the integral +in Eq. (8) over the system volume can be presented as +� +dV ε(ω, r)E2 +p(r) = [ε(ω) − ε′(ωp)] +� +dVmE2 +p(r), +(9) +where the last integration is now carried over the metal +volume Vm, while contributions from the dielectric re- +gions, characterized by constant permittivities, cancel +out. Finally, the LSP Green function takes the form +DLSP(ω; r, r′) = − +� +p +4π +� +dVmE2p +Ep(r)Ep(r′) +ε(ω) − ε′(ωp), +(10) +which is the basis for our analysis of the optical properties +of metal nanostructures that follows. +Plasmon LDOS, DOS, and mode volume.—Using rep- +resentation (10) for the LSP Green function, we can es- +tablish some general spectral properties of LSPs. In the +following, we consider metal nanostructures of arbitrary +shape in dielectric medium with permittivity εd (we set +εd = 1 for now). An important quantity that is criti- +cal in numerous applications is the local density of states +(LDOS), which describes the number of LSP states in +unit volume and frequency interval: +ρ(ω, r) = +1 +2π2ω Im TrDLSP(ω; r, r) = +� +p +ρp(ω, r). (11) +Here, ρp(ω, r) is LDOS for the individual LSP mode +which, using the Green function (10), has the form +ρp(ω, r) = 2 +πω +E2 +p(r) +� +dVmE2p +Im +� +−1 +ε(ω) − ε′(ωp) +� +. +(12) +Integration of LDOS over the volume yields the LSP den- +sity of states (DOS) ρp(ω) = +� +dV ρp(ω, r), describing the +number of LSP states per unit frequency interval. To elu- +cidate the LSP states’ distribution, let us compare the +DOS inside the metal ρm +p (ω) = +� +dVmρp(ω, r) and in the +dielectric region ρd +p(ω) = +� +dVdρp(ω, r). From Eq. (12), +ρm +p (ω) is readily obtained as +ρm +p (ω) = 2 +πω Im +� +−1 +ε(ω) − ε′(ωp) +� +. +(13) +To evaluate ρd +p(ω), we note that, using the Gauss’s equa- +tion, the integral over the dielectric region Vd can be pre- +sented as +� +dVdE2 +p = −ε′(ωp) +� +dVmE2 +p. Since ε′(ωp) < 0, +we obtain ρd +p(ω) = |ε′(ωp)|ρm +p (ω), implying that the LSP +states are predominantly distributed outside the metal. +The full LSP DOS ρp(ω) = ρm +p (ω) + ρd +p(ω) has the form +ρp(ω) = 2 +πω Im +� +ε′(ωp) − 1 +ε(ω) − ε′(ωp) +� +, +(14) +which is independent of the nanostructure shape. +Let us now evaluate the number of LSP states per +mode, Np = � dωρp(ω). +Expanding Eq. (14) near the +LSP pole and evaluating the integral, we obtain +Np = +2|ε′(ωp) − 1| +ωp∂ε′(ωp)/∂ωp +, +(15) + +3 +where we disregarded the corrections ∼ |ε′′/ε′|2 ≪ 1. +For the Drude form of ε(ω), Eq. (15) yields Np ≈ 1, +implying that LSP states saturate the oscillator strength. +However, for the experimental dielectric function, Np can +be substantially below that value, which has implications +for the optical spectra (see below). +Another important quantity that characterizes the lo- +cal field confinement is LSP mode volume Vp, which is +related to the LDOS as V−1 +p += +� +dωρp(ω, r) = ρp(r), +where ρp(r) is the LSP spatial density [41, 42]. Perform- +ing the frequency integration of Eq. (12), we obtain +Vp(r) = ωp +∂ε′(ωp) +∂ωp +� +dVmE2 +p +2E2p(r) . +(16) +Note that the LSP mode volume is a local quantity that +can be very small [i.e., the density ρp(r) is large] at hot +spots characterized by large field intensities, but it is +bound by the relation +� +dV/Vp = Np ≤ 1. Finally, de- +spite suggestions in the literature to the contrary [43], +the LSP mode volume (16) is independent of ε′′. +Optical polarizability.—Consider now a metal nanos- +tructure in the incident EM field Eine−iωt that is nearly +uniform on the system scale. The induced dipole mo- +ment of plasmonic structure is obtained by integrating +the electric polarization vector over the system volume, +p(ω) = +� +dV χ(ω, r)E(ω, r), where E(ω, r) is the local +field inside the nanostructure, given by +E(ω, r) = Ein + +� +dV ′χ(ω, r′)DLSP(ω; r, r′)Ein. (17) +Using the LSP Green function (10), we obtain +E(ω, r) = Ein − +� +p +4πEp(r) +� +dVmE2p +pp(ω)·Ein +ε(ω) − ε′(ωp), +(18) +where pp(ω) = +� +dV χ(ω, r)Ep(r) is dipole moment of the +LSP mode. Noting that pp(ω) = χ(ω) +� +dVmEp, the local +field takes the form +E(ω, r) = Ein − +� +p +cpEp(r) +ε(ω) − 1 +ε(ω) − ε′(ωp), +(19) +where cp = +� +dVmEp·Ein/ +� +dVmE2 +p. Inside the metal, the +incident field Ein can be expanded over the LSP eigen- +modes as Ein = � +p cpEp(r), and we obtain +E(ω, r) = − +� +p +cpEp(r) +ε′(ωp) − 1 +ε(ω) − ε′(ωp). +(20) +Multiplying Eq. (20) by χ(ω, r) and integrating over the +system volume, we obtain the system’s induced dipole +moment as p(ω) = � +p αp(ω)Ein, where +αp(ω) = npnp|χ′(ωp)| +�� +dVmEp +�2 +� +dVmE2p +ε(ω) − 1 +ε(ω) − ε′(ωp) +(21) +is polarizability tensor of the LSP mode, while unit vector +np = +� +dVmEp/| +� +dVmEp| describes the mode’s polariza- +tion. Finally, introducing the effective volume Vp as +Vp = Vm|χ′(ωp)|sp, +sp = +�� +dVmEp +�2 +Vm +� +dVmE2p +, +(22) +we obtain αp(ω) = αp(ω)npnp, where αp(ω) is given +by Eq. (2). +The parameter sp is independent of the +overall field amplitude and, for the dipole LSP modes, +of the nanostructure volume as well. +For spherical or +spheroidal shape, its exact value is sp = 1, while smaller +values sp ≲ 1 should be expected for other shapes. For +a nanosphere of radius a, we have sp = 1, ε′(ωp) = −2, +and we recover Vp = a3, which is significantly smaller +than the system volume. However, for long-wavelength +LSPs characterized by large values of |χ′(ωp)|, the effec- +tive volume can exceed the metal volume Vm (see below). +The above expression for the polarizability (2) is valid +for small nanostructures characterized by weak LSP radi- +ation damping as compared to the Ohmic losses in metal. +For larger systems, to satisfy the optical theorem, the +LSP radiation damping must be included by considering +the system’s interaction with the radiation field, which +leads to the replacement αp → αp[1 − (2iω3/3c3)αp]−1, +where c is the speed of light [44, 45]. Finally, after restor- +ing the permittivity of surrounding medium εd, the po- +larizability takes the form +αp(ω) = Vp +ε(ω) − εd +ε(ω) − ε′(ωp) − 2i +3 k3Vp[ε(ω) − εd], +(23) +where k = √εdω/c is the light wave vector, while the +system effective volume now has the form +Vp = Vm|ε′(ωp)/εd − 1|sp/4π. +(24) +The optical polarizability (23) is the central result of this +work which permits accurate description of optical spec- +tra for diverse plasmonic structures, including those of +irregular shape, using, as input, only the basic system pa- +rameters and the LSP frequency. In terms of αp, the ex- +tinction and scattering cross-sections have the form [45] +σext(ω) = 4πω +c |ep|2α′′ +p(ω), σsc(ω) = 8πω4 +3c4 |ep|2 |αp(ω)|2 , +(25) +where ep = np · Ein/|Ein| is projection of LSP polariza- +tion on that of incident light. For metal structures with +multiple LSP resonances, including porous structures +[46], the polarizability tensor is α(ω) = � +p αp(ω)npnp, +where Vp can now be considered as fitting parameters. +Numerical results.—Below we present the results of nu- +merical calculations for small gold nanostructures to il- +lustrate some general features of the LSP optical spectra +that are common for any system geometry (we use the ex- +perimental gold dielectric function). In Fig. 1, we plot the + +4 +600 +700 +800 +900 +1000 +1100 +1200 +0.2 +0.4 +0.6 +0.8 +1.0 +600 +700 +800 +900 1000 1100 1200 +0 +5 +10 +15 +20 +25 +Qp +lp (nm) +Np +lp (nm) +(a) +600 +700 +800 +900 +1000 +1100 +1200 +0 +1 +2 +3 +4 +5 +Vp / Vm +lp (nm) +(b) +FIG. 1. (a) Number of LSP states for Au nanostructures is +plotted against the LSP wavelength. Inset: LSP quality factor +wavelength dependence. (b) Normalized effective volume is +plotted against the LSP wavelength. +number of plasmon states per mode Np and the effective +volume Vp against the LSP wavelength λp in the interval +from 550 nm to 1200 nm, i.e., for energies below the inter- +band transitions onset in gold. With increasing λp, as the +the system enters the Drude regime, Np increases, albeit +slowly, towards its maximal value [see Fig. 1(a)]. How- +ever, for typical LSP wavelengths from 550 nm to 800 nm, +Np remains substantially below its maximal value, imply- +ing that the interband transitions can influence the LSP +states even at frequencies well below the transitions on- +set; due to the Kramers-Kronig relations, the real part of +dielectric function ε′(ωp), which defines Np via Eq. (15), +is determined by ε′′(ω) at all frequencies. Notably, the +frequency dependence of Np does not follow that of the +LSP quality factor [3] Qp = ωp[∂ε′(ωp)/∂ωp]/2ε′′(ωp), +shown in the inset, which peaks at λp ≈ 700 nm due +to the minimum of ε′′ at this wavelength. In Fig. 1(b), +we plot the effective volume Vp normalized by the metal +volume Vm in the same LSP wavelength interval. The +normalized effective volume increases about tenfold from +the LSP wavelength value 550 nm, roughly corresponding +to LSP in gold nanosphere, to the value 1200 nm corre- +sponding to elongated particles with large aspect ratio, +implying that the optical spectra of metal nanostructures +can be tuned in a wide range by altering the system shape +at the same metal volume. +500 +550 +600 +650 +700 +750 +800 +850 +900 +4 +8 +12 +16 +20 + L = 10 nm + L = 20 nm + L = 30 nm + L = 40 nm +Im ap / Vm +l (nm) +L +(a) +500 +550 +600 +650 +700 +750 +800 +850 +900 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 + Ext + Scatt +Normalized spectrum +l (nm) +(b) +FIG. 2. (a) Imaginary part of polarizability for Au structures +of various sizes in water is shown at various LSP wavelengths. +(b) Normalized extinction and scattering spectra are shown +for L = 30 nm nanostructures at the same LSP wavelengths. +In Fig. 2, we show the optical spectra of gold nanos- +tructures in water (εd = 1.77) for different values of +characteristic size L and, accordingly, of metal volume +Vm = L3, calculated using Eqs. (23)-(25) at the LSP +wavelength values 550 nm, 610 nm, 670 nm, 730 nm, +and 790 nm (we set sp = ep = 1). +The imaginary +part of polarizability normalized by the metal volume +increases dramatically in amplitude with the LSP wave- +length [see Fig. 2(a)], consistent with similar effective +volume increase in Fig. 1(b). For larger structures, the +LSP peak amplitudes of α′′ +p(ω)/Vm decrease due to radi- +ation damping. Although for full α′′ +p(ω), such a decrease +would be masked by larger Vm values, it is clear that, for +the same metal volume, radiation damping is stronger +for long wavelength LSPs since it is also determined by +the effective volume Vp [see Eq. (23)]. +In Fig. 2(b), we plot the extinction and scattering spec- +tra, normalized by their respective maxima, for L = 30 +nm gold nanostructures at the same LSP wavelengths +(for such system size, the extinction is dominated by the +absorption). For shorter wavelengths (< 700 nm), the +scattering spectra exhibit apparent redshift relative to +the extinction spectra despite both are calculated at the +same LSP wavelength. This redshift is not related to the +LSP since, at such wavelengths, the LSP states carry only +about 50% of the full oscillator strength [see Fig. 1(a)]. + +UA5 +In summary, we have developed the analytical model +for optical polarization of plasmonic nanostructures with +characteristic size below the diffraction limit. For such +systems, the lineshape of optical spectra is defined explic- +itly by the metal dielectric function and LSP frequency +while their amplitude depends on the system effective +volume which increases with the LSP wavelength. We +have also established some general features of LSP opti- +cal spectroscopy independent of the system geometry. +Note finally, that the universal form (23) for optical po- +larizability is valid for metal nanostructures in a dielectric +medium. For more complex layered systems, including +core-shell structures, the corresponding expressions for +polarizability are more cumbersome and, importantly, no +longer universal, and therefore are not presented here. +This work was supported in part by the National Sci- +ence Foundation Grants No. DMR-2000170, No. DMR- +1856515, and No. DMR-1826886. +[1] S. A. Maier and H. A. Atwater, J. Appl. Phys. 98, 011101 +(2005). +[2] E. Ozbay, Science 311, 189 (2006). +[3] M. I. 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Klar, Nano Lett. 18, 1269 (2018). + diff --git a/3NE0T4oBgHgl3EQfeABh/content/tmp_files/load_file.txt b/3NE0T4oBgHgl3EQfeABh/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..782e1ba00fe283ed20e959c43b02f946d6daa72f --- /dev/null +++ b/3NE0T4oBgHgl3EQfeABh/content/tmp_files/load_file.txt @@ -0,0 +1,575 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf,len=574 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content='02384v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content='mes-hall] 6 Jan 2023 Universal optical polarizability for plasmonic nanostructures Tigran V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' Shahbazyan Department of Physics, Jackson State University, Jackson, Mississippi 39217 USA We develop an analytical model for calculation of optical spectra for metal nanostructures of arbitrary shape supporting localized surface plasmons (LSPs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' For plasmonic nanostructures with characteristic size below the diffraction limit, we obtain the explicit expression for optical polariz- ability that describes the lineshape of optical spectra solely in terms of the metal dielectric function and LSP frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' The amplitude of LSP spectral band is determined by the effective system volume that, for long wavelength LSPs, can significantly exceed the physical volume of metal nanos- tructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' These results can be used to model or interpret the experimental spectra of plasmonic nanostructures and to tune their optical properties for various applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' Localized surface plasmons (LSPs) are collective elec- tron excitation resonantly excited by incident light in metal nanostructures with characteristic size below the diffraction limit [1–3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' Optical interactions between LSPs and excitons in dye molecules or semiconductors un- derpin numerous phenomena in plasmon-enhanced spec- troscopy, such as surface-enhanced Raman scattering [4], plasmon-enhanced fluorescence and luminescence [5–11], strong exciton-plasmon coupling [12–25], and plasmonic laser [26–29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' Optical properties of metal nanostructures of various sizes and shapes are of critical importance for numerous plasmonics applications [30–32] and were ex- tensively studied experimentally and theoretically [33– 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' The generic optical characteristics that defines the response of plasmonic nanostructure to an incident elec- tromagnetic (EM) field Eine−iωt as well as the interac- tions between LSPs and excitons is the optical polar- izability tensor α(ω), where ω is the incident field fre- quency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' If the characteristic system size is much smaller than the radiation wavelength, so that Ein is nearly uni- form on the system scale, the induced dipole moment of plasmonic structure has the form p(ω) = α(ω)Ein, where α(ω) can be calculated, with a good accuracy, within the quasistatic approach [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' Analytical models for α(ω) have been available only for highly symmetric systems, such as spherical, spheroidal or cylindrical geometries [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' For example, a metal nanosphere of radius a placed in the air is characterized by the scalar polarizability α(ω) = a3 ε(ω) − 1 ε(ω) + 2, (1) where ε(ω) = ε′(ω) + iε′′(ω) is complex dielectric func- tion of the metal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' However, for more complicated shapes used in the experiment, calculations of optical spectra are performed using various numerical techniques [34–38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' On the other hand, for actual structures explored in the experiment, analytical or numerical models attempt- ing to determine both the LSP frequency and the line- shape of optical spectra, as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' (1) does, are not even necessary due to inevitable uncertainties in the nanos- tructure shapes and sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' Typically, the spectral posi- tion of LSP resonance is measured with a reasonably high precision, and so the main challenge is to describe or in- terpret the lineshape of optical spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' Therefore, an analytical model describing accurately the optical spec- tra of plasmonic nanostructures at given LSP frequencies would be highly useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' Here, we present such a model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' Specifically, we show that the optical polarizability of a small metal structure of arbitrary shape associated with the LSP resonance at a frequency ωp has the form αp(ω) = Vp ε(ω) − 1 ε(ω) − ε′(ωp), (2) where Vp = Vm|χ′(ωp)|sp is the effective volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' Here, Vm is the metal volume, χ = (ε−1)/4π is the susceptibil- ity (we use Gaussian units), while the parameter sp ≤ 1 reflects the system geometry but, for dipole LSP modes, is independent of its volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' Remarkably, for any system geometry, the lineshape of optical spectra is determined solely by the metal dielectric function and LSP frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' The polarization (2) can be extended to larger systems by including the LSP radiation damping as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' LSP modes and the Green function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content='—We consider metal nanostructures supporting LSPs that are localized at the length scale much smaller than the radiation wave- length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' In the absence of retardation effects, each region of the structure, metallic or dielectric, is characterized by the dielectric function εi(ω), so that the full dielectric function is ε(ω, r) = � i θi(r)εi(ω), where θi(r) is unit step function that vanishes outside the region volume Vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' We assume that dielectric regions’ permittivities are con- stant and adopt ε(ω) for the metal one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' The LSP modes are defined by the lossless Gauss’s equation [3], ∇ · [ε′(ωp, r)∇Φp(r)] = 0, (3) where Φp(r) and Ep(r) = −∇Φp(r), which we chose real, are the mode’s potential and electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' Note that the eigenmodes of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' (3) are orthogonal in each region [40]: � dViEp(r)·Ep′(r) = δpp′ � dViE2 p(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' The EM dyadic Green function D(ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' r, r′) satisfies (in the operator form) ∇×∇×D−(ω2/c2)εD = (4πω2/c2)I, while equation for the longitudinal part of D is obtained by applying ∇ to both sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' In the near field, we switch 2 to the scalar Green function for the potentials D(ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' r, r′), defined as D(ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' r, r′) = ∇∇′D(ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' r, r′), satisfying ∇ · [ε(ω, r)∇D(ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' r, r′)] = 4πδ(r − r′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' (4) We now adopt the decomposition D = D0 +DLSP, where D0 = −|r − r′|−1 is the free-space Green function and DLSP is the LSP contribution, satisfying ∇· � ε(ω, r)∇DLSP(ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' r, r′) � = −∇· � [ε(ω, r) − 1]∇D0(ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' r, r′) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' (5) Consider first the lossless case and set ε′′ = 0 for now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' For real dielectric function ε(ω, r), we can expand DLSP over the eigenmodes of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' (3) as DLSP(ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' r, r′) = � p Dp(ω)Φp(r)Φp(r′), (6) with real coefficients Dp(ω) [41, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' In the next step, we apply the operator ∆′ to both sides of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' (5), and integrate the result over V ′ with the factor Φp(r′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' Using the LSP Green function expansion (6) and the modes’ orthogonality, one can prove the following rela- tions, � dV ′Φp(r′)∆′DLSP(ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' r, r′) = −DpΦp(r) � dV E2 p and � dV ′Φp(r′)∆′D0(ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' r, r′) = 4πΦp(r), to use in the left-hand side and right-hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' (5), yielding Dp∇· � ε(ω, r)Ep(r) � = 4π ∇· � [ε(ω, r) − 1]Ep(r) � � dVE2p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' (7) Multiplying Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' (7) by Φp(r) and integrating the result over the system volume, we obtain the coefficients Dp as Dp(ω) = 4π � dV E2n(r) − 4π � dV ε(ω, r)E2p(r), (8) where the first term ensures the boundary condition for ε = 1, and will be omitted in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' Accordingly, the LSP dyadic Green function for the electric fields takes the form DLSP(ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' r, r′) = � p Dp(ω)Ep(r)Ep(r′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' Note that even though the LSP Green function is ex- pressed in terms of real eigenmodes Ep, it is valid for plas- monic systems with complex dielectric function ε(ω, r) = ε′(ω, r) + iε′′(ω, r) as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' Indeed, if ε′′ is included as perturbation, then the first-order (diagonal) correction leads to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' (8), while the high-order corrections include non-diagonal terms of the form ε′′(ω) � dVmEp(r)Ep′(r), which vanish due to the modes’ orthogonality, and so the LSP Green function DLSP with complex coefficients (8) is, in fact, exact in all orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' We now note that, in the quasistatic approximation, the frequency and coordinate dependencies in the LSP Green function can be separated out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' Using the Gauss’s equation in the form � dV ε′(ωp, r)E2 p(r) = 0, the integral in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' (8) over the system volume can be presented as � dV ε(ω, r)E2 p(r) = [ε(ω) − ε′(ωp)] � dVmE2 p(r), (9) where the last integration is now carried over the metal volume Vm, while contributions from the dielectric re- gions, characterized by constant permittivities, cancel out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' Finally, the LSP Green function takes the form DLSP(ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' r, r′) = − � p 4π � dVmE2p Ep(r)Ep(r′) ε(ω) − ε′(ωp), (10) which is the basis for our analysis of the optical properties of metal nanostructures that follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' Plasmon LDOS, DOS, and mode volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content='—Using rep- resentation (10) for the LSP Green function, we can es- tablish some general spectral properties of LSPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' In the following, we consider metal nanostructures of arbitrary shape in dielectric medium with permittivity εd (we set εd = 1 for now).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' An important quantity that is criti- cal in numerous applications is the local density of states (LDOS), which describes the number of LSP states in unit volume and frequency interval: ρ(ω, r) = 1 2π2ω Im TrDLSP(ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' r, r) = � p ρp(ω, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' (11) Here, ρp(ω, r) is LDOS for the individual LSP mode which, using the Green function (10), has the form ρp(ω, r) = 2 πω E2 p(r) � dVmE2p Im � −1 ε(ω) − ε′(ωp) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' (12) Integration of LDOS over the volume yields the LSP den- sity of states (DOS) ρp(ω) = � dV ρp(ω, r), describing the number of LSP states per unit frequency interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' To elu- cidate the LSP states’ distribution, let us compare the DOS inside the metal ρm p (ω) = � dVmρp(ω, r) and in the dielectric region ρd p(ω) = � dVdρp(ω, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' (12), ρm p (ω) is readily obtained as ρm p (ω) = 2 πω Im � −1 ε(ω) − ε′(ωp) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' (13) To evaluate ρd p(ω), we note that, using the Gauss’s equa- tion, the integral over the dielectric region Vd can be pre- sented as � dVdE2 p = −ε′(ωp) � dVmE2 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' Since ε′(ωp) < 0, we obtain ρd p(ω) = |ε′(ωp)|ρm p (ω), implying that the LSP states are predominantly distributed outside the metal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' The full LSP DOS ρp(ω) = ρm p (ω) + ρd p(ω) has the form ρp(ω) = 2 πω Im � ε′(ωp) − 1 ε(ω) − ε′(ωp) � , (14) which is independent of the nanostructure shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' Let us now evaluate the number of LSP states per mode, Np = � dωρp(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' Expanding Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' (14) near the LSP pole and evaluating the integral, we obtain Np = 2|ε′(ωp) − 1| ωp∂ε′(ωp)/∂ωp , (15) 3 where we disregarded the corrections ∼ |ε′′/ε′|2 ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' For the Drude form of ε(ω), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' (15) yields Np ≈ 1, implying that LSP states saturate the oscillator strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' However, for the experimental dielectric function, Np can be substantially below that value, which has implications for the optical spectra (see below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' Another important quantity that characterizes the lo- cal field confinement is LSP mode volume Vp, which is related to the LDOS as V−1 p = � dωρp(ω, r) = ρp(r), where ρp(r) is the LSP spatial density [41, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' Perform- ing the frequency integration of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' (12), we obtain Vp(r) = ωp ∂ε′(ωp) ∂ωp � dVmE2 p 2E2p(r) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' (16) Note that the LSP mode volume is a local quantity that can be very small [i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=', the density ρp(r) is large] at hot spots characterized by large field intensities, but it is bound by the relation � dV/Vp = Np ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' Finally, de- spite suggestions in the literature to the contrary [43], the LSP mode volume (16) is independent of ε′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' Optical polarizability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content='—Consider now a metal nanos- tructure in the incident EM field Eine−iωt that is nearly uniform on the system scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' The induced dipole mo- ment of plasmonic structure is obtained by integrating the electric polarization vector over the system volume, p(ω) = � dV χ(ω, r)E(ω, r), where E(ω, r) is the local field inside the nanostructure, given by E(ω, r) = Ein + � dV ′χ(ω, r′)DLSP(ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' r, r′)Ein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' (17) Using the LSP Green function (10), we obtain E(ω, r) = Ein − � p 4πEp(r) � dVmE2p pp(ω)·Ein ε(ω) − ε′(ωp), (18) where pp(ω) = � dV χ(ω, r)Ep(r) is dipole moment of the LSP mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' Noting that pp(ω) = χ(ω) � dVmEp, the local field takes the form E(ω, r) = Ein − � p cpEp(r) ε(ω) − 1 ε(ω) − ε′(ωp), (19) where cp = � dVmEp·Ein/ � dVmE2 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' Inside the metal, the incident field Ein can be expanded over the LSP eigen- modes as Ein = � p cpEp(r), and we obtain E(ω, r) = − � p cpEp(r) ε′(ωp) − 1 ε(ω) − ε′(ωp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' (20) Multiplying Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' (20) by χ(ω, r) and integrating over the system volume, we obtain the system’s induced dipole moment as p(ω) = � p αp(ω)Ein, where αp(ω) = npnp|χ′(ωp)| �� dVmEp �2 � dVmE2p ε(ω) − 1 ε(ω) − ε′(ωp) (21) is polarizability tensor of the LSP mode, while unit vector np = � dVmEp/| � dVmEp| describes the mode’s polariza- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' Finally, introducing the effective volume Vp as Vp = Vm|χ′(ωp)|sp, sp = �� dVmEp �2 Vm � dVmE2p , (22) we obtain αp(ω) = αp(ω)npnp, where αp(ω) is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' The parameter sp is independent of the overall field amplitude and, for the dipole LSP modes, of the nanostructure volume as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' For spherical or spheroidal shape, its exact value is sp = 1, while smaller values sp ≲ 1 should be expected for other shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' For a nanosphere of radius a, we have sp = 1, ε′(ωp) = −2, and we recover Vp = a3, which is significantly smaller than the system volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' However, for long-wavelength LSPs characterized by large values of |χ′(ωp)|, the effec- tive volume can exceed the metal volume Vm (see below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' The above expression for the polarizability (2) is valid for small nanostructures characterized by weak LSP radi- ation damping as compared to the Ohmic losses in metal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' For larger systems, to satisfy the optical theorem, the LSP radiation damping must be included by considering the system’s interaction with the radiation field, which leads to the replacement αp → αp[1 − (2iω3/3c3)αp]−1, where c is the speed of light [44, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' Finally, after restor- ing the permittivity of surrounding medium εd, the po- larizability takes the form αp(ω) = Vp ε(ω) − εd ε(ω) − ε′(ωp) − 2i 3 k3Vp[ε(ω) − εd], (23) where k = √εdω/c is the light wave vector, while the system effective volume now has the form Vp = Vm|ε′(ωp)/εd − 1|sp/4π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' (24) The optical polarizability (23) is the central result of this work which permits accurate description of optical spec- tra for diverse plasmonic structures, including those of irregular shape, using, as input, only the basic system pa- rameters and the LSP frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' In terms of αp, the ex- tinction and scattering cross-sections have the form [45] σext(ω) = 4πω c |ep|2α′′ p(ω), σsc(ω) = 8πω4 3c4 |ep|2 |αp(ω)|2 , (25) where ep = np · Ein/|Ein| is projection of LSP polariza- tion on that of incident light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' For metal structures with multiple LSP resonances, including porous structures [46], the polarizability tensor is α(ω) = � p αp(ω)npnp, where Vp can now be considered as fitting parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' Numerical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content='—Below we present the results of nu- merical calculations for small gold nanostructures to il- lustrate some general features of the LSP optical spectra that are common for any system geometry (we use the ex- perimental gold dielectric function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' 1, we plot the 4 600 700 800 900 1000 1100 1200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content='0 600 700 800 900 1000 1100 1200 0 5 10 15 20 25 Qp lp (nm) Np lp (nm) (a) 600 700 800 900 1000 1100 1200 0 1 2 3 4 5 Vp / Vm lp (nm) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' (a) Number of LSP states for Au nanostructures is plotted against the LSP wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' Inset: LSP quality factor wavelength dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' (b) Normalized effective volume is plotted against the LSP wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' number of plasmon states per mode Np and the effective volume Vp against the LSP wavelength λp in the interval from 550 nm to 1200 nm, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=', for energies below the inter- band transitions onset in gold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' With increasing λp, as the the system enters the Drude regime, Np increases, albeit slowly, towards its maximal value [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' 1(a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' How- ever, for typical LSP wavelengths from 550 nm to 800 nm, Np remains substantially below its maximal value, imply- ing that the interband transitions can influence the LSP states even at frequencies well below the transitions on- set;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' due to the Kramers-Kronig relations, the real part of dielectric function ε′(ωp), which defines Np via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' (15), is determined by ε′′(ω) at all frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' Notably, the frequency dependence of Np does not follow that of the LSP quality factor [3] Qp = ωp[∂ε′(ωp)/∂ωp]/2ε′′(ωp), shown in the inset, which peaks at λp ≈ 700 nm due to the minimum of ε′′ at this wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' 1(b), we plot the effective volume Vp normalized by the metal volume Vm in the same LSP wavelength interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' The normalized effective volume increases about tenfold from the LSP wavelength value 550 nm, roughly corresponding to LSP in gold nanosphere, to the value 1200 nm corre- sponding to elongated particles with large aspect ratio, implying that the optical spectra of metal nanostructures can be tuned in a wide range by altering the system shape at the same metal volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' 500 550 600 650 700 750 800 850 900 4 8 12 16 20 L = 10 nm L = 20 nm L = 30 nm L = 40 nm Im ap / Vm l (nm) L (a) 500 550 600 650 700 750 800 850 900 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content='0 Ext Scatt Normalized spectrum l (nm) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' (a) Imaginary part of polarizability for Au structures of various sizes in water is shown at various LSP wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' (b) Normalized extinction and scattering spectra are shown for L = 30 nm nanostructures at the same LSP wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' 2, we show the optical spectra of gold nanos- tructures in water (εd = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content='77) for different values of characteristic size L and, accordingly, of metal volume Vm = L3, calculated using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' (23)-(25) at the LSP wavelength values 550 nm, 610 nm, 670 nm, 730 nm, and 790 nm (we set sp = ep = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' The imaginary part of polarizability normalized by the metal volume increases dramatically in amplitude with the LSP wave- length [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' 2(a)], consistent with similar effective volume increase in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' For larger structures, the LSP peak amplitudes of α′′ p(ω)/Vm decrease due to radi- ation damping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' Although for full α′′ p(ω), such a decrease would be masked by larger Vm values, it is clear that, for the same metal volume, radiation damping is stronger for long wavelength LSPs since it is also determined by the effective volume Vp [see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' (23)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' 2(b), we plot the extinction and scattering spec- tra, normalized by their respective maxima, for L = 30 nm gold nanostructures at the same LSP wavelengths (for such system size, the extinction is dominated by the absorption).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' For shorter wavelengths (< 700 nm), the scattering spectra exhibit apparent redshift relative to the extinction spectra despite both are calculated at the same LSP wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' This redshift is not related to the LSP since, at such wavelengths, the LSP states carry only about 50% of the full oscillator strength [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' 1(a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' UA5 In summary, we have developed the analytical model for optical polarization of plasmonic nanostructures with characteristic size below the diffraction limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' For such systems, the lineshape of optical spectra is defined explic- itly by the metal dielectric function and LSP frequency while their amplitude depends on the system effective volume which increases with the LSP wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' We have also established some general features of LSP opti- cal spectroscopy independent of the system geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' Note finally, that the universal form (23) for optical po- larizability is valid for metal nanostructures in a dielectric medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' For more complex layered systems, including core-shell structures, the corresponding expressions for polarizability are more cumbersome and, importantly, no longer universal, and therefore are not presented here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' This work was supported in part by the National Sci- ence Foundation Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' DMR-2000170, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' DMR- 1856515, and No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' DMR-1826886.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3NE0T4oBgHgl3EQfeABh/content/2301.02384v1.pdf'} +page_content=' A.' metadata={'source': 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b/3tFQT4oBgHgl3EQfHDWI/content/tmp_files/2301.13247v1.pdf.txt @@ -0,0 +1,2357 @@ +Online Loss Function Learning +Christian Raymond 1 Qi Chen 1 Bing Xue 1 Mengjie Zhang 1 +Abstract +Loss function learning is a new meta-learning +paradigm that aims to automate the essential task +of designing a loss function for a machine learn- +ing model. Existing techniques for loss function +learning have shown promising results, often im- +proving a model’s training dynamics and final in- +ference performance. However, a significant limi- +tation of these techniques is that the loss functions +are meta-learned in an offline fashion, where the +meta-objective only considers the very first few +steps of training, which is a significantly shorter +time horizon than the one typically used for train- +ing deep neural networks. This causes significant +bias towards loss functions that perform well at +the very start of training but perform poorly at the +end of training. To address this issue we propose +a new loss function learning technique for adap- +tively updating the loss function online after each +update to the base model parameters. The exper- +imental results show that our proposed method +consistently outperforms the cross-entropy loss +and offline loss function learning techniques on a +diverse range of neural network architectures and +datasets. +1 +Introduction +When applying deep neural networks to a given learning +task, a significant amount of time is typically allocated to- +wards performing manual tuning of the hyper-parameters to +achieve competitive learning performances (Bengio, 2012). +Selection of the appropriate hyper-parameters is critical +for embedding the relevant inductive biases into the learn- +ing algorithm (Gordon & Desjardins, 1995). The induc- +tive biases control both the set of searchable models and +the learning rules used to find the final model parameters. +Therefore, the field of meta-learning (Schmidhuber, 1987; +1School of Engineering and Computer Science, Victoria Uni- +versity of Wellington, New Zealand. Correspondence to: Christian +Raymond . +Proceedings of the X th Conference on Machine Learning, City, +State, Country, Publisher, 2022. Copyright 2022 by the author(s). +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Predicted Probability (y = 1) +60 +40 +20 +0 +20 +40 +60 +Learned Loss +100000 +80000 +60000 +40000 +20000 +0 +Figure 1: An example of an adaptive meta-learned loss +function generated by AdaLFL on the CIFAR-10 dataset, +where color represents the current gradient step. +Vanschoren, 2018; Peng, 2020; Hospedales et al., 2022), as +well as the closely related field of hyper-parameter optimiza- +tion (Bergstra et al., 2011; Feurer & Hutter, 2019), aim to +automate the design and selection of a suitable set of induc- +tive biases (or a subset of them) and have been long-standing +areas of interest to the machine learning community. +One component that has only very recently been receiving +attention in the meta-learning context is the loss function. +The loss function (Wang et al., 2022) is one of the most +central components of any gradient-based supervised learn- +ing system, as it determines the base learning algorithm’s +learning path and the selection of the final model (Reed +& MarksII, 1999). Furthermore, in deep learning, neural +networks are typically trained through the backpropagation +of gradients that originate from the loss function (Rumelhart +et al., 1986; Goodfellow et al., 2016). Given this importance, +a new and emerging subfield of meta-learning referred to as +Loss Function Learning (Gonzalez & Miikkulainen, 2020; +Bechtle et al., 2021; Raymond et al., 2022; Collet et al., +2022) aims to attempt the difficult task of inferring a highly +performant loss function directly from the given data. +Loss function learning aims to meta-learn a specialized +task-specific loss function, which yields improved perfor- +mance capabilities when utilized in training compared to +handcrafted loss functions on one or many related tasks, +i.e., a task distribution (Hospedales et al., 2022). Initial +approaches to loss function learning have shown promise at +enhancing various aspects of deep neural network training, +arXiv:2301.13247v1 [cs.LG] 30 Jan 2023 + +Online Loss Function Learning +2 +such as improving the convergence and sample efficiency +(Gonzalez & Miikkulainen, 2020; Bechtle et al., 2021), as +well as the generalization (Gonzalez & Miikkulainen, 2021; +Liu et al., 2020; Li et al., 2022; Leng et al., 2022), and model +robustness (Gao et al., 2021; 2022). However, one prevail- +ing limitation of the existing approaches to loss function +learning is that they have thus far exclusively focused on +learning a loss function in the offline meta-learning settings. +In offline loss function learning, training is prototypically +partitioned into two phases. In the first phase, the base loss +function is meta-learned via iteratively updating the loss +function by performing one or a few base training steps +to approximate the performance. Second, the base model +is trained using the learned loss function, which is now +fixed, and is used in place of the conventional handcrafted +loss function. Unfortunately, this methodology is prone +to a severe short-horizon bias (Wu et al., 2018) towards +loss functions which are performant in the early stages of +training but often have poor performance in the later stages. +To address the limitation of offline loss function learning, +we propose a new technique for online loss function learn- +ing called Adaptive Loss Function Learning (AdaLFL). In +the proposed technique, the learned loss function is repre- +sented as a small feed-forward neural network that is trained +simultaneously with the base learning model. Unlike prior +methods, AdaLFL can adaptively transform both the shape +and scale of the loss function throughout the learning pro- +cess to adapt to what is required at each stage of the learning +process, as shown in Figure 1. In offline loss function learn- +ing, the central goal is to improve the performance of a +model by specializing the loss function to a small set of +related tasks. Online loss function learning naturally ex- +tends this general philosophy to instead specialize the loss +function to each individual gradient step on a single task. +1.1 +Contributions +• We propose a method for efficiently learning adaptive +loss functions via online meta-learning by utilizing online +unrolled differentiation to update the meta-learned loss +function after each update to the base model. +• We address shortcomings in the design of neural network- +based loss function parameterizations, which previously +caused learned loss functions to be biased toward overly +flat shapes resulting in poor training dynamics. +• Empirically, we demonstrate that models trained with our +method has enhanced convergence capabilities and infer- +ence performance compared to handcrafted loss functions +and offline loss function learning methods. +• Finally, we analyze the meta-learned loss functions, high- +lighting several key trends to explore why our adaptive +meta-learned loss functions are so performant in contrast +to traditional handcrafted loss functions. +2 +Online Loss Function Learning +In this work, we aim to automate the design and selection +of the loss function and improve upon the performance of +supervised machine learning systems. This is achieved via +meta-learning an adaptive loss function that transforms both +its shape and scale throughout the learning process. To +achieve this, we propose Adaptive Loss Function Learning +(AdaLFL), an efficient task and model-agnostic approach +for online adaptation of the base loss function. +2.1 +Problem Setup +In a prototypical supervised learning setup, we are given +a set of N independently and identically distributed (i.i.d.) +examples of form D = {(x1, y1), . . . , (xN, yN)}, where +xi ∈ X is the ith instance’s feature vector and yi ∈ Y is +its corresponding class label. We want to learn a mapping +between X and Y using some base learning model, e.g., a +classifier or regressor, fθ : X → Y, where θ is the base +model parameters. In this paper, similar to others (Finn et al., +2017; Bechtle et al., 2021; Raymond et al., 2022), we con- +strain the selection of the base models to those amenable to a +stochastic gradient descent (SGD) style training procedures +such that optimization of model parameters θ occurs via +optimizing some task-specific loss function LT as follows: +θt+1 = θt − α∇θtLT (y, fθt(x)) +(1) +where LT is a handcrafted loss function, typically the cross +entropy between the predicted label and the ground truth +label in classification or the squared error in regression. The +principal goal of AdaLFL is to replace this conventional +handcrafted loss function LT with a meta-learned adaptive +loss function Mφ, where the meta-parameters φ are learned +simultaneously with the base parameters θ, allowing for +online adaptation of the loss function. We formulate the task +of learning φ and θ as a non-stationary bilevel optimization +problem, where t is the current time step +φt+1 = arg min +φ +LT (y, fθt+1(x)) +s.t. +θt+1(φt) = arg min +θ +Mφt(y, fθt(x)). +(2) +The outer optimization problem aims to meta-learn a per- +formant loss function Mφ that minimizes the error on the +given task. The inner optimization problem directly mini- +mizes the learned loss value produced by Mφ to learn the +base model parameters θ. +2.2 +Loss Function Representation +In AdaLFL, the choice of loss function parameterization is a +small feedforward neural network, which is chosen due to its +high expressiveness and design flexibility. Our meta-learned +loss function parameterization inspired by (Bechtle et al., +2021) is a small feedforward neural network denoted by ℓφ + +Online Loss Function Learning +3 +Algorithm 1 Loss Function Initialization (Offline) +Input: LT ← Task loss function (meta-objective) +1: Mφ0 ← Initialize parameters of meta learner +2: for i ∈ {0, ..., Sinit} do +3: +θ0 ← Reset parameters of base learner +4: +for j ∈ {0, ..., Sinner} do +5: +X, y ← Sample from Dtrain +6: +Mlearned ← Mφi(y, fθj(X)) +7: +θj+1 ← θj − α∇θjMlearned +8: +end for +9: +X, y ← Sample from Dvalid +10: +Ltask ← LT (y, fθj+1(X)) +11: +φi+1 ← φi − η∇φiLtask +12: end for +with two hidden layers and 40 hidden units each, which is +applied class/output-wise. +Mφ(y, fθ(x)) = 1 +C +�C +i=0 ℓφ(yi, fθ(x)i) +(3) +Crucially, key design decisions are made regarding the ac- +tivation functions used in ℓφ to enforce desirable behavior. +In (Bechtle et al., 2021), ReLU activations are used in the +hidden layers, and the smooth Softplus activation is used +in the output layer to constrain the loss to be non-negative, +i.e., ℓφ : R2 → R+ +0 . Unfortunately, this network architec- +ture is prone to unintentionally encouraging overly flat loss +functions, see Appendix A.1. Generally, flat regions in the +loss function are very detrimental to training as uniform loss +is given to non-uniform errors. Removal of the Softplus +activation in the output can partially resolve this flatness +issue; however, without it, the learned loss functions would +violate the typical constraint that a loss function should be at +least C1, i.e., continuous in the zeroth and first derivatives. +Alternative smooth activations, such as Sigmoid, TanH, Soft- +Plus, ELU, etc., can be used in the hidden layers instead; +however, due to their range-bounded limits, they are also +prone to encouraging loss functions that have large flat re- +gions when their activations saturate. Therefore, to inhibit +this behavior, the unbounded leaky ReLU (Maas et al., 2013) +is combined with the smooth ReLU, i.e., SoftPlus (Dugas +et al., 2000), as follows: +ϕhidden(x) = 1 +β log(eβx + 1) · (1 − γ) + γx +(4) +This smooth leaky ReLU activation function with leak pa- +rameter γ and smoothness parameter β has desirable char- +acteristics for representing a loss function. It is smooth and +has linear asymptotic behavior necessary for tasks such as +regression, where extrapolation of the learned loss function +can often occur. Furthermore, as its output is not bounded, +it does not encourage flatness in the learned loss function. +See Appendix A.2 for more details. +Algorithm 2 Loss Function Adaptation (Online) +Input: Mφ ← Learned loss function (base-objective) +Input: LT ← Task loss function (meta-objective) +1: θ0 ← Initialize parameters of base learner +2: for i ∈ {0, ..., Strain} do +3: +X, y ← Sample from Dtrain +4: +Mlearned ← Mφi(y, fθi(X)) +5: +θi+1 ← θi − α∇θiMlearned +6: +X, y ← Sample from Dvalid +7: +Ltask ← LT (y, fθi+1(X)) +8: +φi+1 ← φi − η∇φiLtask +9: end for +2.3 +Loss Function Initialization +One challenge for online loss function learning is achiev- +ing a stable and performant initial set of parameters for the +learned loss function. If φ is initialized poorly, too much +time is spent on fixing φ in the early stages of the learn- +ing process, resulting in poor base convergence, or in the +worst case, fθ to diverge. To address this, offline loss func- +tion learning using Meta-learning via Learned Loss (ML3) +(Bechtle et al., 2021) is utilized to fine-tune the initial loss +function to the base model prior to performing online learn- +ing. The initialization process is summarized in Algorithm +1, where Sinit = 2500. In AdaLFL’s initialization process +one step on θ is taken for each step in φ, i.e., inner gradient +steps Sinner = 1. However, if Sinner < 1, implicit gra- +dients (Lorraine et al., 2020; Gao et al., 2022) can instead +be utilized to reduce the initialization process’s memory +footprint and computational overhead. +2.4 +Online Meta-Optimization +To optimize φ, unrolled differentiation is utilized in the outer +loop to update the learned loss function after each update +to the base model parameters θ in the inner loop, which +occurs via vanilla backpropagation. This is conceptually +the simplest way to optimise φ as all the intermediate it- +erates generated by the optimizer in the inner loop can be +stored and then backpropagate through in the outer loop +(Maclaurin et al., 2015). The full iterative learning pro- +cess is summarized in Algorithm 2 and proceeds as follows: +perform a forward pass fθt(x) to obtain an initial set of +predictions. The learned loss function Mφ is then used to +produce a base loss value +Mlearned = Mφt(y, fθt(x)). +(5) +Using Mlearned, the current weights θt are updated by +taking a step in the opposite direction of the gradient of the +loss with respect to θt, where α is the base learning rate. +θt+1 = θt − α∇θtMφt(y, fθt(x)) += θt − α∇θtEX,y +� +Mφt(y, fθt(x)) +� +(6) + +Online Loss Function Learning +4 +Meta Update +Base Update +Inner Optimization +Outer Optimization +Figure 2: Computational graph of AdaLFL, where θ is updated using Mφ in the inner loop (Base Update). The +optimization path is tracked in the computational graph and then used to update φ based on the meta-objective in the +outer loop (Meta Update). The dashed lines show the gradients for θ and φ with respect to their given objectives. +which can be further decomposed via the chain rule as +shown in Equation (7). Importantly, all the intermediate +iterates generated by the (base) optimizer at the tth time- +step when updating θ are stored in memory. +θt+1 = θt − α∇fMφt(y, fθt(x))∇θtfθt(x) +(7) +φt can now be updated to φt+1 based on the learning pro- +gression made by θ. Using θt+1 as a function of φt, compute +a forward pass using the updated base weights fθt+1(x) to +obtain a new set of predictions. The instances can either be +sampled from the training set or a held-out validation set. +The new set of predictions is used to compute the task loss +LT to optimize φt through θt+1 +Ltask = LT (y, fθt+1(x)) +(8) +where LT is selected based on the respective application. +For example, the squared error loss for the task of regression +or the cross-entropy loss for classification. The task loss is a +crucial component for embedding the end goal task into the +learned loss function. Optimization of the current meta-loss +network loss weights φt now occurs by taking the gradient +of LT , where η is the meta learning rate. +φt+1 = φt − η∇φtLT (y, fθt+1(x)) += φt − η∇φtEX,y +� +LT (y, fθt+1(x)) +� +(9) +where the gradient computation is decomposed by applying +the chain rule as shown in Equation (10) where the gradient +with respect to the meta-loss network weights φt requires +the updated model parameters θt+1 from Equation (6). +φt+1 = φt − η∇fLT ∇θt+1fθt+1∇φtθt+1 +(10) +This process is repeated for a fixed number of gradient steps +Strain, which is identical to what would typically be used +for training fθ. An overview and summary of the full asso- +ciated data flow between the inner and outer optimization +of θ and φ, respectively, is given in Figure 2. +2.5 +Implicit Tuning of Learning Rate Schedule +In offline loss function learning, it is known from (Gonzalez +& Miikkulainen, 2021; Raymond et al., 2022) that there is +implicit initial learning rate tuning of α when meta-learning +a loss function since +∃α∃φ : θ − α∇θLT ≈ θ − ∇θMφ. +(11) +Consequently, an emergent behavior, unique to online loss +function learning, is that the adaptive loss function generated +by AdaLFL implicitly embodies multiple different learning +rates throughout the learning process hence often causing a +fine-tuning of the fixed learning rate or of a predetermined +learning rate schedule. +3 +Related Work +The method that we propose in this paper addresses the +general problem of meta-learning a (base) loss function, +i.e. loss function learning. Existing loss function learn- +ing methods can be categorized along two key axes, loss +function representation and meta-optimization. Frequently +used representations in loss function learning include para- +metric (Gonzalez & Miikkulainen, 2020; Raymond et al., +2022) and nonparametric (Liu et al., 2020; Li et al., 2022) +genetic programming expression trees. In addition to this, +alternative representations such as truncated Taylor polyno- +mials (Gonzalez & Miikkulainen, 2021; Gao et al., 2021; +2022) and small feed-forward neural networks (Bechtle +et al., 2021) have also been recently explored. Regard- +ing meta-optimization, loss function learning methods have +heavily utilized computationally expensive evolution-based +methods such as evolutionary algorithms (Koza et al., 1994) +and evolutionary strategies (Hansen & Ostermeier, 2001). +While more recent approaches have made use of gradient- +based approaches unrolled differentiation (Maclaurin et al., +2015), and implicit differentiation (Lorraine et al., 2020). + +Online Loss Function Learning +5 +A common trait among these methods is that, in contrast to +AdaLFL, they perform offline loss function learning, result- +ing in a severe short-horizon bias and sub-optimal perfor- +mance at the end of training. This short-horizon bias arises +from how the various approaches compute their respective +meta-objectives. In offline evolution-based approaches, the +fitness, i.e., meta-objective, is typically calculated by com- +puting the performance at the end of a partial training ses- +sion, e.g., ≤ 1000 gradient steps (Gonzalez & Miikkulainen, +2021; Raymond et al., 2022). A truncated number of gradi- +ent steps are required to be used as evolution-based meth- +ods have to evaluate the performance of a large number of +candidate solutions, typically L loss function over K iter- +ations/generations, where 25 ≤ L, K ≤ 100. Therefore, +performing full training sessions, which can be hundreds +of thousands or even millions of gradient steps for each +candidate solution, is infeasible. +Regarding the existing gradient-based approaches, offline +unrolled optimization requires the whole optimization path +to be stored in memory; in practice, this significantly re- +stricts the number of inner gradient steps before computing +the meta-objective to only a small number of steps. Methods +such as implicit differentiation can obviate these memory +issues; however, it would still require a full training session +in the inner loop, which is a prohibitive number of forward +passes to perform in tractable time. Furthermore, the de- +pendence of the model-parameters on the meta-parameters +increasingly shrinks and eventually vanishes as the number +of steps increases (Rajeswaran et al., 2019). +3.1 +Online vs Offline Loss Function Learning +The key algorithmic difference of AdaLFL from prior of- +fline gradient-based methods (Bechtle et al., 2021; Gao +et al., 2022) is that φ is updated after each update to θ in +lockstep in a single phase as opposed to learning θ and φ +in separate phases. This is achieved by not resetting θ after +each update to φ (Algorithm 1, line 3), and consequently, +φ has to adapt to each newly updated timestep such that +φ = (φ0, φ1, . . . , φStrain). In offline loss function learning, +φ is learned separately at meta-training time and then is +fixed for the full duration of the meta-testing phase where θ +is learned and φ = (φ0). Another crucial difference is that +in online loss function learning, there is implicit tuning of +the learning rate schedule, as mentioned in Section 2.5. +3.2 +Alternative Paradigms +Although online loss function learning has not been explored +in the meta-learning context, some existing research outside +the subfield has previously explored the possibility of adap- +tive loss functions, such as in (Li et al., 2019) and (Wang +et al., 2020). However, we emphasize that these approaches +are categorically different in that they do not learn the loss +function from scratch; instead, they interpolate between a +small subset of handcrafted loss functions, updating the loss +function after each epoch. Furthermore, in contrast to loss +function learning which is both task and model-agnostic, +these techniques are restricted to being task-specific, e.g., +face recognition only. Finally, this class of approaches does +not implicitly tune the base learning rate α, as is the case in +loss function learning. +4 +Experimental Evaluation +In this section, the experimental setup for evaluating +AdaLFL is presented. In summary, experiments are con- +ducted across four open-access datasets and multiple well- +established network architectures. The performance of the +proposed method is contrasted against the handcrafted cross- +entropy loss and AdaLFL’s offline counterpart ML3 Super- +vised (Bechtle et al., 2021). The experiments were imple- +mented in PyTorch (Paszke et al., 2017), and Higher +(Grefenstette et al., 2019), and the code for reproducing the +experiments can be found at github.com/*redacted*. +4.1 +Benchmark Tasks +Following the established literature on loss function learn- +ing (Gonzalez & Miikkulainen, 2021; Bechtle et al., 2021; +Raymond et al., 2022), MNIST (LeCun et al., 1998) is ini- +tially used as a simple domain to illustrate the capabilities +of the proposed method. Following this, the more challeng- +ing tasks of CIFAR-10, CIFAR-100 (Krizhevsky & Hinton, +2009), and SVHN (Netzer et al., 2011), are employed to +assess the performance of AdaLFL to determine whether +the results can generalize to larger, more challenging tasks. +The original training-testing partitioning is used for all four +datasets, with 10% of the training instances allocated for +validation. In addition, standard data augmentation tech- +niques consisting of normalization, random horizontal flips, +and cropping are applied to the training data of CIFAR-10, +CIFAR-100, and SVHN during meta and base training. +4.2 +Benchmark Models +A diverse set of commonly used and well-established bench- +mark architectures are utilized to evaluate the performance +of AdaLFL. For MNIST, logistic regression (McCullagh +et al., 1989), a simple two hidden layer multi-layer per- +ceptron (MLP) taken from (Baydin et al., 2018), and the +LeNet-5 (LeCun et al., 1998) architecture is used. Follow- +ing this experiments are conducted on CIFAR-10, VGG-16 +(Simonyan et al., 2014), AllCNN-C (Springenberg et al., +2014), ResNet-18 (He et al., 2016), and SqueezeNet (Ian- +dola et al., 2016) are used. For the remaining datasets, +CIFAR-100 and SVHN, WideResNet 28-10 and WideRes- +Net 16-8 (Zagoruyko et al., 2016) is employed, respectively. +5 +Results and Analysis +The results in Figure 3 show the average training learning +curves of AdaLFL compared with the baseline cross-entropy + +Online Loss Function Learning +6 +0 +5000 +10000 +15000 +20000 +25000 +0.0 +0.1 +0.2 +0.3 +0.4 +Error Rate +(a) MNIST + Logistic +0 +5000 +10000 +15000 +20000 +25000 +0.00 +0.05 +0.10 +0.15 +0.20 +Error Rate +(b) MNIST + MLP +0 +5000 +10000 +15000 +20000 +25000 +0.00 +0.05 +0.10 +0.15 +0.20 +Error Rate +(c) MNIST + LeNet-5 +0 +20000 +40000 +60000 +80000 +100000 +0.00 +0.05 +0.10 +0.15 +0.20 +Error Rate +(d) CIFAR-10 + VGG-16 +0 +20000 +40000 +60000 +80000 +100000 +0.00 +0.05 +0.10 +0.15 +0.20 +Error Rate +(e) CIFAR-10 + AllCNN-C +0 +20000 +40000 +60000 +80000 +100000 +0.00 +0.05 +0.10 +0.15 +0.20 +Error Rate +(f) CIFAR-10 + ResNet-18 +0 +20000 +40000 +60000 +80000 +100000 +0.00 +0.05 +0.10 +0.15 +0.20 +Error Rate +(g) CIFAR-10 + SqueezeNet +0 +25000 +50000 +75000 +100000 125000 150000 +0.00 +0.05 +0.10 +0.15 +0.20 +Error Rate +(h) CIFAR-100 + WRN 28-10 +0 +25000 +50000 +75000 +100000 125000 150000 +0.00 +0.05 +0.10 +0.15 +0.20 +Error Rate +(i) SVHN + WRN 16-8 +Baseline +ML3 (Offline) +AdaLFL (Online) +Figure 3: Mean learning curves across 10 independent executions of each algorithm on each task + model pair, showing the +training error rate (y-axis) against gradient steps (x-axis). Best viewed in color. +loss and ML3 across 10 executions of each method on each +dataset + model pair. The results show that AdaLFL makes +clear and consistent gains in convergence speed compared +to the baseline and offline loss function learning method +ML3, except on CIFAR-100 where there was difficulty in +achieving a stable initialization. Furthermore, the errors ob- +tained by AdaLFL at the end of training are typically better +(lower) than both of the compared methods, suggesting that +performance gains are being made in addition to enhanced +convergence and training speeds. +Another key observation is that AdaLFL improves upon +the performance of the baseline on the more challenging +tasks of CIFAR-10, CIFAR-100, and SVHN, where offline +loss functions learning method ML3 consistently performs +poorly. Improved performance on these datasets is achieved +via AdaLFL adaptively updating the learned loss function +throughout the learning process to the changes in the train- +ing dynamics. This is in contrast to ML3, where the loss +function remains static, resulting in poor performance on +tasks where the training dynamics at the beginning of train- +ing vary significantly from those at the end of training. +5.1 +Final Inference Testing Performance +The corresponding final inference testing results reporting +the average error rate across 10 independent executions of +each method are shown in Table 1. The results show that +AdaLFL’s meta-learned loss functions produce superior in- +ference performance when used in training compared to +the baseline on all the tested problems. A further observa- +tion is that the gains achieved by AdaLFL are consistent +and stable. Notably, in most cases, lower variability than +the baseline is observed, as shown by the relatively small +standard deviation in error rate across the independent runs. +Contrasting the performance of AdaLFL to ML3, similar per- +formance is obtained on the MNIST experiments, suggest- +ing that the training dynamics at the beginning of training +are similar to those at the end; hence the modest difference +in results. While on the more challenging tasks of CIFAR- +10, CIFAR-100, and SVHN, AdaLFL produced significantly +better results than ML3, demonstrating the scalability of the +newly proposed loss function learning approach. +The results attained by AdaLFL are are promising given +that the base models tested were designed and optimized +around the cross-entropy loss. We hypothesize that larger +performance gains may be attained using networks designed +specifically around meta-learned loss function, similar to +the results shown in (Kim et al., 2018; Elsken et al., 2020; +Ding et al., 2022). Thus future work will explore learning +the loss function in tandem with the network architecture. + +Online Loss Function Learning +7 +Table 1: Results reporting the mean ± standard deviation of final inference testing error rates across 10 independent +executions of each algorithm on each task + model pair (using no base learning rate scheduler). +Task +Model +Baseline +ML3 (Offline) +AdaLFL (Online) +MNIST +Logistic (McCullagh et al., 1989) +0.0766±0.0009 +0.0710±0.0010 +0.0697±0.0010 +MLP (Baydin et al., 2018) +0.0203±0.0006 +0.0185±0.0004 +0.0184±0.0006 +LeNet-5 (LeCun et al., 1998) +0.0125±0.0007 +0.0094±0.0005 +0.0091±0.0004 +CIFAR-10 +VGG-16 (Simonyan et al., 2014) +0.1036±0.0049 +0.1027±0.0062 +0.0903±0.0032 +AllCNN-C (Springenberg et al., 2014) +0.1030±0.0062 +0.1015±0.0055 +0.0835±0.0050 +ResNet-18 (He et al., 2016) +0.0871±0.0057 +0.0883±0.0041 +0.0788±0.0035 +SqueezeNet (Iandola et al., 2016) +0.1226±0.0080 +0.1162±0.0052 +0.1083±0.0049 +CIFAR-100 +WRN 28-10 (Zagoruyko et al., 2016) +0.3046±0.0087 +0.3108±0.0075 +0.2668±0.0283 +SVHN +WRN 16-8 (Zagoruyko et al., 2016) +0.0512±0.0043 +0.0500±0.0034 +0.0441±0.0014 +Table 2: Results reporting the mean ± standard deviation +of testing error rates when using an increasing number of +inner gradient steps Sinner with ML3. +Method +CIFAR-10 + AllCNN-C +ML3 (Sinner = 1) +0.1015±0.0055 +ML3 (Sinner = 5) +0.0978±0.0052 +ML3 (Sinner = 10) +0.0985±0.0050 +ML3 (Sinner = 15) +0.0989±0.0049 +ML3 (Sinner = 20) +0.0974±0.0061 +AdaLFL (Online) +0.0835±0.0050 +5.2 +Inner Gradient Steps +In ML3, (Bechtle et al., 2021) suggested taking only one +inner step, i.e., setting Sinner = 1 in Algorithm 1. A rea- +sonable question to ask is whether increasing the number +of inner steps to extend the horizon of the meta-objective +past the first step will reduce the disparity in performance +between ML3 and AdaLFL. To answer this question, exper- +iments are performed on CIFAR-10 AllCNN-C with ML3 +setting Sinner = {1, 5, 10, 15, 20}. The results reported +in Table 2 show that increasing the number of inner steps +in ML3 up to the limit of what is feasible in memory on +a consumer GPU does not resolve the short horizon bias +present in offline loss function learning. Furthermore, the +results show that increasing the number of inner steps only +results in marginal improvements in the performance over +Sinner = 1. Hence, offline loss function learning methods +that seek to obviate the memory issues of unrolled differen- +tiation to allow for an increased number of inner steps, such +as (Gao et al., 2022), which uses implicit differentiation, are +still prone to a kind of short-horizon bias. +Table 3: Average run-time of the entire learning process for +each benchmark method. Each algorithm is run on a single +Nvidia RTX A5000, and results are reported in hours. +Task and Model +Baseline +Offline +Online +MNIST + Logistic +0.06 +0.31 +0.55 +MNIST + MLP +0.06 +0.32 +0.56 +MNIST + LeNet-5 +0.10 +0.38 +0.67 +CIFAR-10 + VGG-16 +1.50 +1.85 +5.56 +CIFAR-10 + AllCNN-C +1.41 +1.72 +5.53 +CIFAR-10 + ResNet-18 +1.81 +2.18 +8.38 +CIFAR-10 + SqueezeNet +1.72 +2.02 +7.88 +CIFAR-100 + WRN 28-10 +8.81 +10.3 +50.49 +SVHN + WRN 16-8 +7.32 +7.61 +24.75 +5.3 +Run-time Analysis +The average run-time of the entire learning process of all +benchmark methods on all tasks is reported in Table 3. No- +tably, there are two key reasons why the computational +overhead of AdaLFL is not as bad as it may at first seem. +First, the time reported for the baseline does not include +the implicit cost of manual hyper-parameter selection and +tuning of the loss function, as well as the initial learning +rate and learning rate schedule, which is needed prior to +training in order to attain reasonable performance (Goodfel- +low et al., 2016). Second, a large proportion of the cost of +AdaLFL comes from storing a large number of intermediate +iterates needed for the outer loop. However, the intermedi- +ate iterates stored in this process are identical to those used +in other popular meta-learning paradigms (Andrychowicz +et al., 2016; Finn et al., 2017). Consequently, future work + +Online Loss Function Learning +8 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Predicted Probability (y = 1) +15.0 +12.5 +10.0 +7.5 +5.0 +2.5 +0.0 +Learned Loss +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Predicted Probability (y = 0) +0.0 +2.5 +5.0 +7.5 +10.0 +12.5 +15.0 +17.5 +Learned Loss +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Predicted Probability (y = 1) +10 +20 +30 +40 +50 +60 +Learned Loss +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Predicted Probability (y = 0) +80 +60 +40 +20 +0 +20 +Learned Loss +0 +20000 +40000 +60000 +80000 +100000 +Figure 4: Loss functions generated by AdaLFL on the CIFAR-10 dataset, where each row represents +a loss function, and the color represents the current gradient step. +can explore combining AdaLFL with other optimization- +based meta-learning methods with minimal overhead cost, +as is the case in methods such as MetaSGD (Li et al., 2017), +where both the initial parameters and a parameter-wise ma- +trix of learning rate terms are learned simultaneously. +5.4 +Visualizing Learned Loss Functions +To better understand why the meta-learned loss functions +produced by AdaLFL are so performant, two of the learned +loss functions are highlighted in Figure 4, where the learned +loss function is plotted at equispaced intervals throughout +the training. See Appendix C for further examples of the +diverse and creative loss function meta-learned by AdaLFL. +Analyzing the learned loss functions, it can be observed +that the loss functions change significantly in their shape +throughout the learning process. In both cases, the learned +loss functions attributed strong penalties for severe mis- +classification at the start of the learning process, and than +gradually pivoted to a more moderate or minor penalty as +learning progressed. This behavior enables fast and effi- +cient learning early on, and reduces the sensitivity of the +base model to outliers in the later stages of the learning pro- +cess. A further observation is that the scale of the learned +loss function changes, confirming that implicit learning rate +tuning, as noted in Section 3.1, is occurring. +5.5 +Implicit Early Stopping +A unique property observed in the loss functions generated +by AdaLFL is that often once base convergence is achieved +the learned loss function will intentionally start to flatten or +take on a parabolic form, see Figures 7 and 12 in Appendix. +This is implicitly a type of early stopping, also observed in +related paradigms such as in hypergradient descent (Baydin +et al., 2018), which meta-learns base learning rates. In hy- +pergradient descent the learned learning rate has previously +been observed to oscillate around 0 near the end of training, +at times becoming negative, essentially terminating training. +Implicit early stopping is beneficial as it is known to have a +regularizing effect on model training (Yao et al., 2007); how- +ever, if not performed carefully it can also be detrimental to +training due to terminating training prematurely. Therefore, +in future work, we aim to further investigate and explore +regulating this behavior, as a potential avenue for further +improving performance. +6 +Conclusion +In this work, the first fully online approach to loss function +learning is proposed. The proposed technique, AdaLFL, +infers the base loss function directly from the data and adap- +tively trains it with the base model parameters simultane- +ously using unrolled differentiation. The results showed that +models trained with our method have enhanced convergence +capabilities and inference performance compared with the +de facto standard cross-entropy loss and offline loss func- +tion learning method ML3. Further analysis on the learned +loss functions identified common patterns in the shape of +the learned loss function, as well revealed unique emergent +behavior present only in adaptively learned loss functions. +Namely, implicit tuning of the learning rate schedule as +well as implicit early stopping. While this work has solely +set focus on meta-learning the loss function in isolation to +better understand and analyze its properties, we believe that +further benefits can be realized upon being combined with +existing optimization-based meta-learning techniques. + +Online Loss Function Learning +9 +References +Andrychowicz, M., Denil, M., Gomez, S., Hoffman, M. 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Constructive Approximation, +26(2):289–315, 2007. +Zagoruyko, S., Komodakis, N., et al. Wide residual net- +works. arXiv preprint arXiv:1605.07146, 2016. + +Online Loss Function Learning +11 +0 +20000 +40000 +60000 +80000 +100000 +Figure 5: Four example loss functions generated by AdaLFL using the network architecture proposed in (Bechtle et al., +2021), which uses a softplus activation in the output layer, causing flattening behavior degrading learning performance. +A +Loss Function Representation +The representation of the learned loss function under consideration in AdaLFL is a simple feed-forward neural network. We +consider the general case of a feed-forward neural network with one input layer, L hidden layers, and one output layer. A +hidden layer refers to a feed-forward mapping between two adjacent layers hφ(l) such that +hφ(l) +� +hφ(l−1) +� += ϕ(l)� +φ(l)Thφ(l−1) +� +, ∀l = 1, . . . L, +(12) +where ϕ(·)(l) refers to the element-wise activation function of the lth layer, and φ(l) is the matrix of interconnecting +weights between hφ(l−1) and hφ(l). For the input layer, the mapping is defined as hφ(0)(yi, fθ(x)i), and for the output +layer as hφ(out)(hφ(L)). Subsequently, the meta-learned loss function ℓφ parameterized by the set of meta-parameters +φ = {φ0, . . . , φl, φout} can be defined as a composition of feed-forward mappings such that +ℓφ +� +yi, fθ(x)i +� += hφ(out) +� +hφ(L) +� +. . . +� +hφ(0) +� +yi, fθ(x)i +�� +. . . +�� +(13) +which is applied output-wise across the C output channels of the ground truth and predicted labels, e.g., applied to each index +of the one-hot encoded class vector in classification, or to each continuous output in regression. The loss value produced by +ℓφ is then summed across the output channel to reduce the loss vector into its final scalar form. +Mφ(y, fθ(x)) = 1 +C +C +� +i=0 +ℓφ(yi, fθ(x)i) +(14) +A.1 +Network Architecture +The learned loss function used in our experiments has L = 2 hidden layers and 40 hidden units in each layer, inspired by the +network configuration utilized in Meta-Learning via Learned Loss (ML3 Supervised) (Bechtle et al., 2021). We found no + +600 +500 +Learned Loss +400 +300 +200 +100 +0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Predicted Probability (y = 1)80 +Learned Loss +60 +40 +20 +0 +0.2 +0.4 +0.0 +0.6 +0.8 +1.0 +Predicted Probability (y = 1)100 +Learned Loss +80 +60 +40 +20 +0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Predicted Probability (y = 1)600 +500 +Learned Loss +400 +300 +200 +100 +0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Predicted Probability (y = 1)Online Loss Function Learning +12 +consistent improvement in performance across our experiments by increasing or decreasing the number of hidden layers +or nodes. However, it was found that the choice of non-linear activations used in ML3, was highly prone to encouraging +poor-performing loss functions with large flat regions, as shown in Figure 5. +In ML3, rectified linear units, ϕReLU(x) = max(0, x), are used in the hidden layers and the smooth SoftPlus ϕsoftplus = +log(eβx + 1) is used in the output layer to enforce the optional constraint that Mlearned should be non-negative, i.e., +∀y∀fθ(x)Mφt(y, fθ(x)) ≥ 0. An adverse side-effect of using the softplus activation in the output is that all negative inputs +to the output layer go to 0, resulting in flat regions in the learned loss. Furthermore, removal of the output activation does +not resolve this issue, as ReLU, as well as other common activations such as Sigmoid, TanH, and ELU, are also bounded +and are prone to causing flatness when their activations saturate, a common occurrence when taking gradients through long +unrolled optimization paths (Antoniou et al., 2018). +A.2 +Smooth Leaky ReLU +To inhibit the flattening behavior of learned loss functions, a range unbounded activation function should be used. A popular +activation function that is unbounded (when the leak parameter γ < 0) is the Leaky ReLU (Maas et al., 2013) +ϕleaky(x) = max(γ · x, x) +(15) += max(0, x) · (1 − γ) + γx +(16) +However, it is typically assumed that a loss function should be at least C1, i.e., continuous in the zeroth and first derivatives. +Fortunately, there is a smooth approximation to the ReLU, commonly referred to as the SoftPlus activation function (Dugas +et al., 2000), where β (typically set to 1) controls the smoothness. +ϕsmooth(x) = 1 +β · log(eβx + 1) +(17) +The leaky ReLU is combined with the smooth ReLU by taking the term max(0, x) from Equation (16) and substituting it +with the smooth SoftPlus defined in Equation (17) to construct a smooth approximation to the leaky ReLU +ϕhidden(x) = 1 +β log(eβx + 1) · (1 − γ) + γx +(18) +where the derivative of the smooth leaky ReLU with respect to the input x is +ϕ′ +hidden(x) = d +dx +�log(eβx + 1) · (1 − y) +β ++ γx +� +(19) += +d +dx[log(eβx + 1)] · (1 − y) +β ++ γ +(20) += +d +dx[eβx + 1] · (1 − y) +β · eβx + 1 ++ γ +(21) += eβx · β · (1 − y) +β · eβx + 1 ++ γ +(22) += eβx(1 − γ) +eβx + 1 ++ γ +(23) += eβx(1 − γ) +eβx + 1 ++ γ(eβx + 1) +eβx + 1 +(24) += eβx + γ +eβx + 1 +(25) +The smooth leaky ReLU and its corresponding derivatives are shown in Figure 6. Early iterations of AdaLFL learned γ +and β simultaneously with the network weights φ, however; empirically, we found that setting γ = 0.01 and β = 10 gave +adequate inference performance across our experiments. +B +Experimental Setup +To initialize Mφ, Sinit = 2500 steps are taken in offline mode with a meta learning rate of η = 1e − 3. In contrast, in +online mode, a meta learning rate of η = 1e − 5 is used (note, a high meta learning rate in online mode can cause a jittering + +Online Loss Function Learning +13 +4 +2 +0 +2 +4 +4 +2 +0 +2 +4 + = 0.0 + = 0.25 + = 0.5 + = 0.75 + = 1.0 +(a) +4 +2 +0 +2 +4 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 + = 0.0 + = 0.25 + = 0.5 + = 0.75 + = 1.0 +(b) +4 +2 +0 +2 +4 +2 +0 +2 +4 + = 1 + = 2 + = 3 + = 4 + = 5 +(c) +4 +2 +0 +2 +4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 + = 1 + = 2 + = 3 + = 4 + = 5 +(d) +2 +0 +2 +4 +2 +0 +2 +4 +ReLU +Leaky ReLU +Smooth ReLU +Smooth Leaky ReLU +(e) +Figure 6: The proposed activation function and its corresponding derivatives when shifting γ are shown in (a) and (b), +respectively. In (c) and (d) the activation function and its derivatives when shifting β are shown. Finally, in (c), the smooth +leaky ReLU is contrasted with the original smooth and leaky variants ReLU. +effect in the loss function, which can cause training instability). The popular Adam optimizer is used in the outer loop for +both initialization and online adaptation. +In the inner-loop, all base models are trained using stochastic gradient descent (SGD) with a base learning rate α = 0.01, +and on CIFAR-10, CIFAR-100, and SVHN, Nesterov momentum 0.9, and weight decay 0.0005 are used. The remaining +base-model hyper-parameters are selected using their respective values from the literature in an identical setup to (Gonzalez +& Miikkulainen, 2021). +All experimental results reported show the average results across 10 independent executions on different seeds for the +purpose of analysing algorithmic consistency. Importantly, our experiments control for the base initializations such that all +methods get identical initial parameters across the same random seed; thus, any difference in variance between the methods +can be attributed to the respective algorithms and their loss functions. Furthermore, the choice of hyper-parameters between +ML3 and AdaLFL has been standardized to allow for a fair comparison. +C +Learned Loss Functions (Extended) + +Online Loss Function Learning +14 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Predicted Probability (y = 1) +5.0 +7.5 +10.0 +12.5 +15.0 +17.5 +20.0 +22.5 +Learned Loss +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Predicted Probability (y = 0) +10 +15 +20 +25 +30 +Learned Loss +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Predicted Probability (y = 1) +6 +8 +10 +12 +14 +16 +18 +Learned Loss +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Predicted Probability (y = 0) +5 +10 +15 +20 +25 +Learned Loss +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Predicted Probability (y = 1) +4 +6 +8 +10 +12 +14 +16 +18 +Learned Loss +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Predicted Probability (y = 0) +5 +10 +15 +20 +25 +Learned Loss +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Predicted Probability (y = 1) +10.0 +12.5 +15.0 +17.5 +20.0 +22.5 +25.0 +Learned Loss +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Predicted Probability (y = 0) +10 +15 +20 +25 +30 +Learned Loss +0 +5000 +10000 +15000 +20000 +25000 +Figure 7: Loss functions generated by AdaLFL on the MNIST dataset, where each row represents +a loss function, and the color represents the current gradient step. + +Online Loss Function Learning +15 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Predicted Probability (y = 1) +17.5 +15.0 +12.5 +10.0 +7.5 +5.0 +2.5 +Learned Loss +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Predicted Probability (y = 0) +5 +0 +5 +10 +Learned Loss +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Predicted Probability (y = 1) +0 +2 +4 +6 +8 +10 +Learned Loss +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Predicted Probability (y = 0) +0 +5 +10 +15 +20 +Learned Loss +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Predicted Probability (y = 1) +0 +2 +4 +6 +8 +10 +12 +14 +16 +Learned Loss +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Predicted Probability (y = 0) +0 +5 +10 +15 +20 +Learned Loss +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Predicted Probability (y = 1) +8 +10 +12 +14 +16 +18 +20 +Learned Loss +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Predicted Probability (y = 0) +0 +5 +10 +15 +20 +Learned Loss +0 +5000 +10000 +15000 +20000 +25000 +Figure 8: Loss functions generated by AdaLFL on the MNIST dataset, where each row represents +a loss function, and the color represents the current gradient step. + +Online Loss Function Learning +16 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Predicted Probability (y = 1) +10 +0 +10 +20 +30 +40 +50 +60 +Learned Loss +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Predicted Probability (y = 0) +20 +40 +60 +80 +100 +Learned Loss +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Predicted Probability (y = 1) +10 +20 +30 +40 +50 +Learned Loss +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Predicted Probability (y = 0) +80 +60 +40 +20 +0 +20 +40 +60 +80 +Learned Loss +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Predicted Probability (y = 1) +10 +0 +10 +20 +30 +40 +Learned Loss +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Predicted Probability (y = 0) +20 +40 +60 +80 +100 +120 +Learned Loss +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Predicted Probability (y = 1) +20 +40 +60 +80 +100 +Learned Loss +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Predicted Probability (y = 0) +70 +60 +50 +40 +30 +20 +10 +0 +Learned Loss +0 +5000 +10000 +15000 +20000 +25000 +Figure 9: Loss functions generated by AdaLFL on the MNIST dataset, where each row represents +a loss function, and the color represents the current gradient step. + +Online Loss Function Learning +17 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Predicted Probability (y = 1) +100 +80 +60 +40 +20 +0 +20 +Learned Loss +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Predicted Probability (y = 0) +50 +25 +0 +25 +50 +75 +100 +Learned Loss +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Predicted Probability (y = 1) +0 +25 +50 +75 +100 +125 +150 +Learned Loss +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Predicted Probability (y = 0) +80 +60 +40 +20 +0 +20 +Learned Loss +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Predicted Probability (y = 1) +20 +0 +20 +40 +60 +80 +Learned Loss +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Predicted Probability (y = 0) +80 +60 +40 +20 +0 +20 +40 +Learned Loss +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Predicted Probability (y = 1) +20 +40 +60 +80 +100 +120 +Learned Loss +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Predicted Probability (y = 0) +100 +75 +50 +25 +0 +25 +50 +75 +Learned Loss +0 +20000 +40000 +60000 +80000 +100000 +Figure 10: Loss functions generated by AdaLFL on the CIFAR-10 dataset, where each row represents +a loss function, and the color represents the current gradient step. + +Online Loss Function Learning +18 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Predicted Probability (y = 1) +20 +0 +20 +40 +60 +80 +100 +120 +Learned Loss +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Predicted Probability (y = 0) +0 +20 +40 +60 +80 +Learned Loss +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Predicted Probability (y = 1) +60 +40 +20 +0 +20 +40 +60 +Learned Loss +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Predicted Probability (y = 0) +60 +40 +20 +0 +20 +40 +60 +80 +100 +Learned Loss +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Predicted Probability (y = 1) +20 +0 +20 +40 +60 +Learned Loss +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Predicted Probability (y = 0) +20 +40 +60 +80 +100 +120 +Learned Loss +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Predicted Probability (y = 1) +20 +40 +60 +80 +100 +120 +Learned Loss +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Predicted Probability (y = 0) +150 +125 +100 +75 +50 +25 +0 +25 +50 +Learned Loss +0 +20000 +40000 +60000 +80000 +100000 +Figure 11: Loss functions generated by AdaLFL on the CIFAR-10 dataset, where each row represents +a loss function, and the color represents the current gradient step. + +Online Loss Function Learning +19 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Predicted Probability (y = 1) +50 +40 +30 +20 +10 +0 +10 +Learned Loss +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Predicted Probability (y = 0) +40 +30 +20 +10 +0 +10 +20 +Learned Loss +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Predicted Probability (y = 1) +30 +20 +10 +0 +10 +20 +30 +Learned Loss +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Predicted Probability (y = 0) +30 +20 +10 +0 +10 +20 +30 +Learned Loss +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Predicted Probability (y = 1) +30 +20 +10 +0 +10 +20 +30 +Learned Loss +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Predicted Probability (y = 0) +20 +10 +0 +10 +20 +30 +Learned Loss +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Predicted Probability (y = 1) +10 +20 +30 +40 +50 +60 +Learned Loss +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Predicted Probability (y = 0) +30 +20 +10 +0 +10 +20 +30 +40 +Learned Loss +0 +20000 +40000 +60000 +80000 +100000 +Figure 12: Loss functions generated by AdaLFL on the CIFAR-10 dataset, where each row represents +a loss function, and the color represents the current gradient step. + +Online Loss Function Learning +20 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Predicted Probability (y = 1) +20 +10 +0 +10 +20 +30 +Learned Loss +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Predicted Probability (y = 0) +10 +0 +10 +20 +30 +40 +50 +Learned Loss +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Predicted Probability (y = 1) +40 +30 +20 +10 +0 +10 +20 +Learned Loss +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Predicted Probability (y = 0) +30 +20 +10 +0 +10 +20 +30 +40 +Learned Loss +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Predicted Probability (y = 1) +20 +10 +0 +10 +20 +30 +Learned Loss +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Predicted Probability (y = 0) +20 +10 +0 +10 +20 +30 +40 +50 +60 +Learned Loss +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Predicted Probability (y = 1) +40 +30 +20 +10 +0 +10 +20 +Learned Loss +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Predicted Probability (y = 0) +20 +10 +0 +10 +20 +30 +40 +50 +Learned Loss +0 +20000 +40000 +60000 +80000 +100000 +Figure 13: Loss functions generated by AdaLFL on the CIFAR-10 dataset, where each row represents +a loss function, and the color represents the current gradient step. + diff --git a/3tFQT4oBgHgl3EQfHDWI/content/tmp_files/load_file.txt b/3tFQT4oBgHgl3EQfHDWI/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e2c3c30680f6046c4aad34c465f9069623531995 --- /dev/null +++ b/3tFQT4oBgHgl3EQfHDWI/content/tmp_files/load_file.txt @@ -0,0 +1,1432 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFQT4oBgHgl3EQfHDWI/content/2301.13247v1.pdf,len=1431 +page_content='Online Loss Function Learning Christian Raymond 1 Qi Chen 1 Bing Xue 1 Mengjie Zhang 1 Abstract Loss function learning is a new meta-learning paradigm that aims to automate the essential task of designing a loss function for a machine learn- ing model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFQT4oBgHgl3EQfHDWI/content/2301.13247v1.pdf'} +page_content=' Existing techniques for loss function learning have shown promising results, often im- proving a model’s training dynamics and final in- ference performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFQT4oBgHgl3EQfHDWI/content/2301.13247v1.pdf'} +page_content=' However, a significant limi- tation of these techniques is that the loss functions are meta-learned in an offline fashion, where the meta-objective only considers the very first few steps of training, which is a significantly shorter time horizon than the one typically used for train- ing deep neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFQT4oBgHgl3EQfHDWI/content/2301.13247v1.pdf'} +page_content=' This causes significant bias towards loss functions that perform well at the very start of training but perform poorly at the end of training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFQT4oBgHgl3EQfHDWI/content/2301.13247v1.pdf'} +page_content=' To address this issue we propose a new loss function learning technique for adap- tively updating the loss function online after each update to the base model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFQT4oBgHgl3EQfHDWI/content/2301.13247v1.pdf'} +page_content=' The exper- imental results show that our proposed method consistently outperforms the cross-entropy loss and offline loss function learning techniques on a diverse range of neural network architectures and datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFQT4oBgHgl3EQfHDWI/content/2301.13247v1.pdf'} +page_content=' 1 Introduction When applying deep neural networks to a given learning task, a significant amount of time is typically allocated to- wards performing manual tuning of the hyper-parameters to achieve competitive learning performances (Bengio, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFQT4oBgHgl3EQfHDWI/content/2301.13247v1.pdf'} +page_content=' Selection of the appropriate hyper-parameters is critical for embedding the relevant inductive biases into the learn- ing algorithm (Gordon & Desjardins, 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFQT4oBgHgl3EQfHDWI/content/2301.13247v1.pdf'} +page_content=' The induc- tive biases control both the set of searchable models and the learning rules used to find the final model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFQT4oBgHgl3EQfHDWI/content/2301.13247v1.pdf'} +page_content=' Therefore, the field of meta-learning (Schmidhuber, 1987;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFQT4oBgHgl3EQfHDWI/content/2301.13247v1.pdf'} +page_content=' 1School of Engineering and Computer Science, Victoria Uni- versity of Wellington, New Zealand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFQT4oBgHgl3EQfHDWI/content/2301.13247v1.pdf'} +page_content=' Correspondence to: Christian Raymond p2 > 1). +(4) +Under the above assumptions, there holds +∥�fc∥L1(R+) ≲ ∥f ′∥Op(R+), +(5) +provided that the right-hand side is finite for some p > 1. In fact, a different notation is convenient +for the case where the Op norm is calculated for the derivative: ∥f∥Vp := ∥f ′∥Op. Just this notation +is appropriate for further generalization. On the one hand, (5) follows from (3) and (4). On the +other hand, a direct proof for (5) is given in [7], where the main ingredient is the Hausdorff-Young +inequality. To provide similar reasoning for measures µf generated by functions of bounded variation +f rather than functions (however, we shall write df rather than dµf), we need a corresponding +extension of the Hausdorff-Young inequality. And here is the point where our special harmonic +analysis comes into play. We do not restrict ourselves to finding immediate tools for the above +problem but try to establish a kind of general and multivariate theory. A variety of relevant issues +will be introduced and studied. +1.1. Basic notions. +We define an analog of Lp spaces for measures by means of an associated norm. For a given +p ∈ [1, ∞], we use the notation ∥ · ∥p to denote the standard norm in Lp(Rn) = Lp(Rn, dx), where +by dx we mean the Lebesgue measure. + +Lp SIMULATION FOR MEASURES +3 +We denote by S(Rn) the Schwartz space of rapidly decreasing C∞ functions, and either by F(f) +or by �f the Fourier transform of a function f ∈ S(Rn), written +�f(y) = +� +Rn f(x)e−2πix·y dx, +where x·y = x1y1+...+xnyn. Recall that F : S(Rn) → S(Rn) is one-to-one, and the inverse Fourier +transform is ˇf(y) = �f(−y). In this paper, we will not distinguish between Fourier transform and +inverse Fourier transform, unless it becomes necessary. +For p ∈ [1, 2], the operator F : Lp(Rn) → Lp′(Rn), with 1 +p+ 1 +p′ = 1, is bounded, with ∥ �f∥p′ ≤ ∥f∥p +and equality if p = 2. For Lp(Rn), p > 2, the Fourier transform can be defined in the distributional +sense as +⟨ �f, ψ⟩ = +� +Rn f(x) �ψ(x) dx, +ψ ∈ S(Rn); +clearly, �f is a function if and only if f = �g for some g ∈ Lp′(Rn). With this observation in mind, +we give the following definition. +For a given p ∈ [1, ∞], we let +�Lp(Rn) = {f ∈ Lp(Rn) : f = �g for some g ∈ Lp′(Rn)}. +(6) +In a natural way, we endow �Lp(Rn) with the norm +∥f∥�Lp = ∥ �f ∥p′. +(7) +With this definition, the Fourier transform +F : Lp(Rn) → (�Lp′(Rn), ∥ · ∥�Lp′) +is a one-to-one isometry. When p ∈ [1, 2], the Hausdorff-Young inequality yields ∥f∥�Lp ≤ ∥f∥p, +with equality if p = 2. +We denote by M the space of sigma-finite Borel measures on Rn. For every p ∈ [1, ∞], we +define the functional ∥ · ∥∗ +p : M → [0, ∞] as +∥µ∥∗ +p = +sup +h∈� +Lp′ (Rn) : +∥h∥� +Lp′ ≤1 +���� +� +Rn h(t)dµ(t) +���� ; +(8) +we let +Mp = {µ ∈ M : ∥µ∥∗ +p < ∞}. +(9) +Note that, for every µ ∈ Mp and every h ∈ �Lp′(Rn), we have that +���� +� +Rn h(x)dµ(x) +���� ≤ ∥h∥�Lp′∥µ∥∗ +p. +(10) +We do not assume that our measures are positive, or even real-valued. +For definition and +properties of non-positive measure see e.g. [9]. With this assumption, the spaces Mp are vector +spaces, and we will prove in Section 2 that the functional ∥µ∥∗ +p is a norm on Mp. + +4 +L. DE CARLI AND E. LIFLYAND +1.2. Structure of the paper. +With ∥µ∥∗ +p and Mp denoted by similarity to Lp, we then establish basic properties of these +measure spaces. +We will prove in Section 2 that the spaces Mp have many properties in common with Lp +spaces. We establish the properties of measures in Mp spaces and the properties of functions in +spaces �Lp(Rn). +Discussing then the Fourier transform of a measure, we establish a Hausdorff- +Young type inequality. Further, for the convolution of a function and a measure, we prove a Young +type inequality for our setting. We mention that the results in Section 2 are supplemented with +examples. +Section 3 is devoted to applications of the introduced machinery. One of them is a development +of an uncertainty principle for measures. The uncertainty principle in Fourier analysis quantifies +the intuition that a function and its Fourier transform cannot both be concentrated on small sets. +Many examples of this principle can be found, e.g., in the book by Havin and J¨oricke [10] and in +an article by Folland and Sitaram [6]. In Subsection 3.1, using a quantitative version of a result +in [1], we prove that a finite measure and its Fourier transform cannot both be supported on sets +of finite Lebesgue measure. Recall that a measure µ is supported in a set E ⊂ Rn if µ(F) = 0 +whenever F is a measurable set that does not intersect E. +In conclusion, we formulate and prove an analog of (5) for functions of bounded variation without +assuming absolute continuity. This is Theorem 10. In order to formulate and prove it, as an analog +of Vp spaces for functions, we introduce the notion f ∈ V ∗ +p for measures, with +∥f∥V ∗ +p = +� ∞ +0 +x− 1 +p ∥χ(x,2x)µf∥∗ +p dx +where χE denotes the characteristic function of E. The product of a measure µ and a measurable +function f is the measure defined by (fµ)(F) = +� +F f dµ for every measurable set F. For 1 < p ≤ 2, +our new Hausdorff-Young inequality will be helpful, while for p > 2, we prove an analog of (4) and +use an embedding argument. +2. Lp properties of measures +In this section we establish basic properties of measures in the spaces Mp defined in the intro- +duction, with p ∈ [1, ∞], that mimic those of functions in Lp spaces. We also establish properties +of the spaces �Lp(Rn) defined in (6). +If E is a measurable subset of Rn, with |E| ̸= 0, we let +∥µ∥∗ +p,E = ∥χEµ∥∗ +p = +sup +h∈� +Lp′ (Rn) : +∥h∥� +Lp′ ≤1 +���� +� +E +h(t) dµ(t) +���� , +(11) +and Mp,E = {µ : ∥µ∥∗ +p,E < ∞}. +We can also define +M1,loc = {µ : ∥µ∥∗ +1,E < ∞ for every measurable bounded set E}. +(12) +The standard Lebesgue measure and the Delta measures are notable examples of Mp measures. +In the rest of this paper we will use L (or dx in integration) to denote the standard Lebesgue +measure. +For a given a ∈ Rn, we let δa be the measure defined as +� +Rn f(x) dδa = f(a). + +Lp SIMULATION FOR MEASURES +5 +Example 1. We show that the standard Lebesgue measure is in M∞ and ∥L∥∗ +∞ = 1. +Indeed, +∥L∥∗ +∞ = +sup +h∈� +L1(Rn) : +∥h∥� +L1=∥�h∥∞ +≤1 +���� +� +Rn h(t) dt +���� ≤ +sup +h∈L1(Rn) : +∥h∥1≤1 +���� +� +Rn h(t) dt +���� ≤ 1. +To prove that equality holds, we can consider g = e−π|x|2. It is easy to verify that �g(x) = g(x), and +so g ∈ �L1(Rn) and ∥g∥�L1 = ∥�g∥∞ = 1. Since 1 = �g(0) = +� +Rn g(t) dt = ∥g∥1, we have that +∥L∥∗ +∞ ≥ +� +Rn g(t) dt = 1, +as desired. +Example 2. We show that δa ∈ Mp only for p = 1 and ∥δa∥1 = 1. Indeed, assuming a = 0 for +simplicity, we can easily see that +∥δ0∥∗ +1 = +sup +h∈S(Rn): +∥h∥� +L∞ ≤1 +���� +� +Rn h(t) dδ0 +���� = +sup +h∈S(Rn): +∥�h ∥1≤1 +|h(0)| += +sup +h∈S(Rn): +∥�h∥1≤1 +���� +� +Rn +�h(x) dx +���� ≤ +sup +h∈S(Rn): +∥�h ∥1≤1 +∥�h ∥1 = 1. +To prove that equality holds, we can consider the function g = e−π|x|2 in the previous example +and verify that ∥δ0∥∗ +1 ≥ ∥ˆg∥1 = 1. An easy variation of this argument shows that δ0 ̸∈ Mp if p > 1. +2.1. H¨older type inequalities. +We prove the following +Theorem 1. If µ ∈ Mp and f ∈ �Lq(Rn), and 1 +r = 1 +q + 1 +p, then +∥fµ∥∗ +r ≤ ∥f∥�Lq∥µ∥∗ +p. +(13) +Proof. Assume ∥f∥�Lq = 1, or else replace f with ˜f = +f +∥f∥� +Lq . With the notation previously +introduced, +∥fµ∥∗ +r = +sup +h∈� +Lr′ (Rn): +∥h∥� +Lr′ ≤1 +���� +� +Rn h(y)f(y) dµ(y) +����. +Let us show that hf ∈ �Lp′ and ∥hf∥�Lp′ ≤ 1. Indeed, � +hf = �h ∗ �f (standard convolution). Since +1 +r + 1 +q′ = 1 + 1 +p, by Young’s inequality for convolution and the Hausdorff-Young inequality, +∥hf∥�Lp′ = ∥� +hf∥p = ∥�h ∗ �f∥p ≤ ∥�h∥r∥ �f∥q′ = ∥h∥�Lr′∥f∥�Lq ≤ 1. +Thus, ∥hf∥�Lp′ ≤ 1, and so +∥fµ∥∗ +r ≤ +sup +k∈� +Lp′ (Rn): +∥k∥� +Lp′ ≤1 +���� +� +Rn k(y) dµ(y) +���� = ∥µ∥∗ +p = ∥µ∥∗ +p ∥f∥�Lq, +as required. +□ +Remark 2. When r = 1, for every µ ∈ Mp and f ∈ �Lp′(Rn) we have that +∥fµ∥∗ +1 ≤ ∥f∥�Lp′∥µ∥∗ +p. +This is the case of (13) that most closely resembles the standard H¨older’s inequality. + +6 +L. DE CARLI AND E. LIFLYAND +Corollary 3. Let E be a bounded subset of Rn. Then Mr,E ⊂ Mp,E whenever 1 ≤ p ≤ r ≤ ∞. +Proof. Assume p < r, since the case p = r is trivial. Assume also E ⊂ QR = [−R, R]n for +some R > 0. By (11), +∥µ∥∗ +r,E = +sup +∥h∥� +Lr′ ≤1 +���� +� +E +h(y) dµ(y) +���� = +sup +∥�h∥Lr ≤1 +���� +� +Rn χQ(y)h(y)χE(y) dµ(y) +����. +Let q = +rp +r−p. Since r ̸= p, we have q < ∞ and q′ > 1. The Fourier transform of the characteristic +function of QR is +�χQR(x) = +n +� +j=1 +sin(πRxj) +πxj +, +and so �χQR(x) ∈ Ls(Rn) for every s > 1. We have ∥�χQR∥s = Cn +s R +n +s′ , where +Cs = +���� +sin(π·) +π· +���� +s += + + + + + + + + + +� +2s′ +π +� 1 +s, +1 < s < 2, +� +2 +s +� 1 +2s, +2 ≤ s < ∞ +1, +s = ∞, +is independent of R. In fact, Cs can be taken +� +2s′ +π +� 1 +s for all s < ∞. This is calculated by minimal +means: split the integral +� +R +���sin(πt) +πt +��� +s +dt = 1 +π +� +R +���sin(t) +t +��� +s +dt +(14) +into two, over |t| ≤ 1 and over |t| > 1, and replace +��� sin(t) +t +��� in the first by 1 and in the second by +1 +|t|. +However, it is known (see [2, Lemma 3] or [14, Ch.VI, 7.5]) that for s ≥ 2, the sharp bound for +(14) is +� +2 +s. +Applying Proposition 1 with f = χQR and χEµ in place of µ, we obtain +∥µ∥∗ +p,E ≤ ∥χQR∥�Lq∥µ∥∗ +r,E = Cn +q R +n +q′ ∥µ∥∗ +r,E, +(15) +and so ∥µ∥∗ +p,E < ∞ whenever ∥µ∥∗ +r,E < ∞, as required. +□ +Corollary 4. For every p ∈ [1, ∞], we have that Mp ⊂ M1,loc. +Proof. Follows from Corollary 3 and (12). +□ +Corollary 5. The functional ∥ ∥∗ +p is a norm on Mp for every p ∈ [1, ∞]. +Proof. It is trivial to verify that for every µ, σ ∈ Mp and every λ ∈ C, +∥µ + σ∥∗ +p ≤ ∥µ∥∗ +p + ∥σ∥∗ +p, +∥λµ∥∗ +p = |λ| ∥µ∥∗ +p. +We now prove that ∥µ∥∗ +p = 0 if and only if µ ≡ 0, in the sense that µ(E) = 0 for every µ−measurable +set E. + +Lp SIMULATION FOR MEASURES +7 +In order to show that µ ≡ 0, it is enough to verify that µ(E) = 0 for every bounded set E. +Let E be bounded and µ−measurable. Assume that E ⊂ QR for some R > 0. Using (15) and +Proposition 8, we can see at once that +µ(E) = +� +E +dµ(x) = +� +QR +χEdµ(x) ≤ ∥χQR∥�L∞∥µ∥∗ +p,E ≤ ∥µ∥∗ +p = 0 +and so µ(E) = 0 for every µ−measurable bounded set E. +□ +2.2. Properties of �Lp spaces. +In this sub-section we will establish properties of the spaces �Lp(Rn) defined in (6). We first +shows how measures of the form dµ = fdx behave with respect to the norms introduced when +f ∈ �Lp, +Theorem 6. Let dµ = fdx, with f ∈ �Lp(Rn) for some p ∈ [1, ∞]; then µ ∈ Mp and +∥µ∥∗ +p = ∥f∥�Lp. +Before discussing Theorem 6, we prove the following +Lemma 1. S(Rn) is dense in �Lp(Rn) for every p ∈ [1, ∞]. +Proof. Since S(Rn) ⊂ �Lp(Rn) ⊂ Lp(Rn) and S(Rn) is dense in Lp(Rn) for every p ∈ [1, ∞), +we can see at once that S(Rn) is also dense in �Lp(Rn). To see that S(Rn) is dense also in �L∞(Rn), +we observe that every f ∈ �L∞(Rn) is the image of g ∈ L1(Rn) via the Fourier transform. We can +find functions ψn ∈ S(Rn) such that lim +n→∞ ∥ψn − g∥1 = 0. But +∥ψn − g∥1 = ∥ � +�ψn − �f ∥1 = ∥ �ψn − f∥�L∞, +and so lim +n→∞ ∥ �ψn−f∥�L∞ = 0. Since �ψn ∈ S(Rn), we have proved that S(Rn) is dense in �L∞(Rn). +□ +Proof of Theorem 6. Since S(Rn) is dense in Lp(Rn) and in �Lp′(Rn), and the Fourier trans- +form is one-to-one in S(Rn), we can see at once that +∥f∥�Lp = ∥ �f∥p′ = +sup +g∈S(Rn): +∥g∥p≤1 +���� +� +Rn g(t) �f(t) dt +���� = +sup +g∈S(Rn): +∥g∥p≤1 +���� +� +Rn �g(t)f(t) dt +���� += +sup +h∈S(Rn): +∥�h∥p≤1 +���� +� +Rn h(t)f(t) dt +���� = +sup +h∈S(Rn): +∥h∥� +Lp′ ≤1 +���� +� +Rn h(t)f(t) dt +���� = ∥µ∥∗ +p, +which completes the proof. +□ +Remark 7. If dµ = fdx is as in Theorem 6 and p ∈ [1, 2], then there holds +∥µf∥∗ +p = ∥f∥�Lp = ∥ �f∥p′ ≤ ∥f∥p. +Corollary 8. Let E ⊂ Rn be a (Lebesgue) measurable set. +a) For every p ∈ [1, ∞], we have +∥χE∥�Lp ≤ |E| +1 +p. +(16) +b) For every 1 ≤ p ≤ q ≤ ∞ and every µ ∈ Mq, we have +∥µ∥∗ +p,E ≤ ∥µ∥∗ +q|E| +1 +r , +where 1 +r = 1 +p − 1 +q. +We have used the standard convention +1 +∞ = 0. Thus, (16) yields ∥χE∥�L∞ ≤ 1, for every set E. + +8 +L. DE CARLI AND E. LIFLYAND +Proof. We first prove a). When p ∈ [1, 2], Remark 7 yields +∥χE∥�Lp ≤ ∥χE∥p = |E| +1 +p. +Assume now p ∈ (2, ∞). By the Hausdorff-Young inequality, we can see at once that +{f ∈ �Lp′ : ∥f∥�Lp′ = ∥ �f∥p ≤ 1} ⊂ {f ∈ Lp′(Rn) : ∥f∥p′ ≤ 1}. +In view of this observation and Theorem 6, we can let dσ = χEdx and write the following chain of +inequalities: +∥χE∥�Lp = ∥σ∥∗ +p,E = +sup +f∈� +Lp′(Rn): +∥f∥� +Lp′ ≤1 +���� +� +Rn χE(x)f(x)dx +���� +≤ +sup +f∈Lp′ (Rn): +∥f∥p′ ≤1 +���� +� +Rn χE(x)f(x)dx +���� +(17) +≤ |E| +1 +p∥f∥p′ ≤ |E| +1 +p. +We have used H¨older’s inequality in the last step. +When p = ∞, it follows from (17) that +sup +f∈L1(Rn): +∥f∥1≤1 +���� +� +Rn χE(x)f(x)dx +���� ≤ +sup +f∈L1(Rn): +∥f∥1≤1 +� +Rn χE(x)|f(x)|dx ≤ 1. +Part b) follows from H¨older’s inequality (1) and part a). Indeed, letting r = +pq +q−p, we have +∥µ∥∗ +p,E = ∥χEµ∥∗ +p ≤ ∥χE∥�Lr∥µ∥∗ +q ≤ |E| +1 +r ∥µ∥∗ +q. +The proof of the corollary is complete. +□ +We use Theorem 6 to prove inclusion relations of the �Lp spaces and their duals. Recall that the +dual of a normed space X, denoted by (X)′, is the set of linear functionals L : V → C such that +sup∥f∥X≤1 |L(f)| < ∞. +By definition, �Lp(Rn) = Lp(Rn) when p ∈ [1, 2] but in general �Lp(Rn) is a proper subspace of +Lp(Rn). For example, the Riemann-Lebesgue Lemma yields that �L∞(Rn) is a space of uniformly +continuous functions that go to zero at infinity. +Even though Lp(Rn) = �Lp(Rn) when p ∈ [1, 2], the norms on these spaces are different and so +the duals of these spaces are different too. When p ≤ 2, the Hausdorff-Young inequality yields, +∥f∥�Lp = ∥ ˆf∥p′ ≤ ∥f∥p. +When p = 2 we have ∥f∥2 = ∥f∥�L2 but when p > 2 the inequality above can be strict. +We prove the following +Proposition 1. For every p ∈ [1, ∞], we have +�Lp′(Rn) ⊂ (�Lp(Rn))′. +When p ∈ [1, 2], we have �Lp′(Rn) ⊂ (�Lp(Rn))′ ⊂ Lp′(Rn). + +Lp SIMULATION FOR MEASURES +9 +Proof. For a given g ∈ �Lp′(Rn), we let dµ = gdx and we let Lg : �Lp(Rn) → C, +Lg(f) = +� +Rn f(x)g(x)dx. +By H¨older’s inequality (13) and Theorem 6 +|Lg(f)| = +���� +� +Rn f(x)g(x)dx +���� ≤ ∥f∥�Lp∥µ∥∗ +p′ = ∥f∥�Lp∥g∥�Lp′ +and so L ∈ (�Lp(Rn))′. +When p ≤ 2, for every L ∈ (�Lp(Rn))′, we have that +|L(f)| ≤ C∥f∥�Lp = C∥ ˆf∥p′ ≤ C∥f∥p +and so L ∈ (Lp(Rn))′ = Lp′(Rn). +□ +2.3. Fourier transform of finite measures. The Fourier transform of a finite Borel measure +µ is the function defined as +�µ(y) = +� +Rn e−2πix·ydµ(x). +(18) +To distinguish it from the Fourier transform for functions, it is sometimes called the Fourier-Stieltjes +transform. It is well-known (see, e.g., [3, §5.3] or [15, §4.4]) that the function �µ is continuous and +bounded. By the Riemann-Lebesgue Lemma, the Fourier transform of an L1 function vanishes at +infinity, but the Fourier transform of a M1 measure does not need to do so. For example, we have +shown in Example 2 that the Delta measure µ = δa is in M1; its Fourier transform is �µ(x) = e2πia·x, +and |�µ(x)| ≡ 1. +We prove the following analog of the Hausdorff-Young inequality. +Proposition 2. Let µ ∈ Mp, with 1 ≤ p ≤ 2. Then, �µ ∈ Lp′(Rn), and +∥�µ∥p′ ≤ ∥µ∥∗ +p. +(19) +Proof. We have observed that the Fourier transform of a finite measure is always bounded, +so the proposition is trivial for p = 1. When p ∈ (1, 2], we have +∥�µ∥p′ = +sup +h∈S(Rn): +∥h∥p≤1 +� +Rn h(y)�µ(y) dy. +By Fubini’s theorem, +� +Rn h(y)�µ(y) dy = +� +Rn +� +Rn h(y)e−2πix·y dµ(x) dy = +� +Rn +�h(x) dµ(x). +(20) +In view of (20) and the fact that ∥�h ∥p′ ≤ ∥h∥p ≤ 1, we can see at once that +∥�µ∥p′ = +sup +h∈S(Rn): +∥h∥p≤1 +� +Rn +�h(y) µ(y) ≤ +sup +k∈S(Rn): +∥�k ∥p′ ≤1 +���� +� +Rn k(x) dµ(x) +���� = ∥µ∥∗ +p, +which completes the proof. +□ + +10 +L. DE CARLI AND E. LIFLYAND +Example 3. If µf is generated by the singular function f in [16], we have +|� +µf(x)| = O +� 1 +|x|δ +� +for |x| large, with 0 < δ < 1 +2. Then � +µf ∈ Lp′(R), with 1 +δ < p′ < ∞, and correspondingly, 2 < p′ < ∞. +By this, ∥µf∥∗ +p < ∞, since +∥µf∥∗ +p = +sup +∥h∥� +Lp′ ≤1 +���� +� +R +h(t) df(t) +���� = +sup +∥g∥Lp≤1 +���� +� +R +�g(x) df(x) +���� += +sup +∥g∥Lp≤1 +���� +� +R +g(x) �df(x) dx +���� < ∞, +because of g ∈ Lp and � +µf ∈ Lp′, with 1 < p < +1 +1−δ < 2. +We have used the pioneer example of a singular function in [16] but there are more subtle ones. +However, for all of them there is a barrier to L2, like 0 < δ < 1 +2 above; see, e.g., [11] and references +therein. +2.4. Convolution of a function and a measure. Let µ be a sigma-finite Borel measure, +and let f : Rn → R be a measurable function such that the function +x → +� +Rn f(x − y)dµ(y) +(21) +is finite for a.e. x ∈ Rn. The convolution of f and µ, denoted by f ∗ µ, is the function defined in +(21). We prove the following analog of the Young inequality for convolution. +Proposition 3. If µ ∈ Mp and f ∈ �Lq(Rn) with 1 +p + 1 +q = 1 +r, then f ∗ µ ∈ �Lr(Rn) and +∥f ∗ µ∥�Lr ≤ ∥f∥�Lq∥µ∥∗ +p. +Proof. In view of Proposition 6, +∥f ∗ µ∥�Lr = ∥f ∗ µ∥∗ +r = +sup +h∈S(Rn): +∥h∥� +Lr′ ≤1 +� +Rn h(x)(f ∗ µ)(x) dx. +(22) +For every h ∈ S(Rn), +� +Rn h(x)(f ∗ µ)(x) dx = +� +Rn h(x) +� +Rn f(x − y) dµ(y) dx, = +� +Rn h ∗ ˜f(y) dµ(y) +(23) +where ˜g(t) = g(−t). By (23) and (10), +� +Rn h ∗ ˜f(y) dµ(y) ≤ ∥h ∗ ˜f∥�Lp′∥µ∥∗ +p = ∥�h �f∥p ∥µ∥∗ +p. +(24) +Recalling that 1 + 1 +r = 1 +p + 1 +q, we have 1 +p = 1 +r + 1 +q′. By H¨older’s inequality, +∥�h �f∥p ≤ ∥�h ∥r∥ �f ∥q′ = ∥h∥�Lr′∥f∥�Lq +By (22) and (24), we conclude that +∥f ∗ µ∥�Lr ≤ ∥f∥�Lq∥µ∥∗ +p, +as required. +□ + +Lp SIMULATION FOR MEASURES +11 +3. Applications +As mentioned, in this section we present applications of the obtained results. +3.1. Uncertainty principle. +In this subsection, we show that the uncertainty principle has its embodiment also for measures. +We prove the following +Theorem 9. A finite nonzero measure µ ∈ M2 and its Fourier transform �µ cannot both be +supported in sets of finite Lebesgue measure. +The proof of the theorem relies on the following +Lemma 2. Let E, F ⊂ Rn be sets of finite Lebesgue measure. There exists a constant C > 0 +such that for every measure µ ∈ M2, we have +∥dµ∥∗ +2,F ≤ C∥ �µ ∥L2(Ec). +Proof. Recall the following quantitative form of an uncertainty principle result obtained by +Amrein and Berthier in [1]: Let E, F ⊂ Rn be sets of finite measure. There exists a constant C > 0 +such that for every function f ∈ L2(Rn), +∥ �f ∥L2(F ) ≤ C∥f∥L2(Ec). +(25) +Let h ∈ L2(Rn). By (25), the inequality ∥h∥L2(Ec) ≤ 1 yields ∥�h ∥L2(F ) ≤ C. In view of (20), we +can write the following chain of inequalities: +∥ �µ ∥L2(Ec) = +sup +h∈L2(Rn): +∥h∥L2(Ec)≤1 +� +Rn h(x)�µ(x) += +sup +h∈L2(Rn): +∥h∥L2(Ec)≤1 +� +Rn +�h(y) dµ(y) ≥ +sup +h∈L2(Rn): +∥�h ∥L2(F )≤C +� +Rn +�h(x) dµ(x) += 1 +C +sup +k∈L2(Rn): +∥ �k ∥2≤1 +� +F +k(x)dµ(x) = 1 +C ∥µ∥∗ +2,F, +obtaining the required result. +□ +Proof of Theorem 9. Assume by contradiction that µ is supported in F and �µ is supported +in E, where E, F ⊂ Rn are both of finite measure. By Lemma 2, we have ∥µ∥∗ +2,F = ∥χFµ∥∗ +2 = 0, and +Corollary 5 yields χFµ ≡ 0. Since χF cµ ≡ 0 is assumed, we have µ = 0, which is a contradiction. +□ +3.2. The Fourier transform theorem. +In order to reveal an analogy to the case of absolutely continuous f, we prove a counterpart of +corresponding embeddings in (4). +Proposition 4. For p1 > p2 > 1, there holds +V ∗ +p1 ֒→ V ∗ +p2. + +12 +L. DE CARLI AND E. LIFLYAND +Proof. We are going to apply Corollary 3. Since for E = (x, 2x), we have in (15) that by (16), +there holds +∥χQR∥�Lq ≲ x +1 +q , +and it follows that +∥µf∥∗ +p2,(x,2x) ≲ x +1 +q ∥µf∥∗ +p1,(x,2x). +The corresponding relation +1 +p2 = 1 +q + 1 +p1 yields 1 +q = p1−p2 +p1p2 . It remains to observe that +x− 1 +p2 x +p1−p2 +p1p2 = x− 1 +p1 , +which leads to the needed embedding. +□ +With these embeddings and the tools elaborated before, we study, for γ = 0 or 1 +4, the Fourier +transforms +�fγ(x) = +� ∞ +0 +f(t) cos 2π(xt − γ) dt. +(26) +It is clear that �fγ represents the cosine Fourier transform in the case γ = 0, while taking γ = 1 +4 +gives the sine Fourier transform. +Theorem 10. Let f be of bounded variation on R+ and vanishing at infinity, that is, lim +t→∞ f(t) = +0. If f ∈ V ∗ +p , then for x > 0, we have +�fγ(x) = +1 +2πxf +�1 +x +� +sin 2πγ + Γ(x), +where γ = 0 or 1 +4, and ∥Γ∥L1(R+) ≲ ∥f∥V ∗ +p provided that the last value is finite for some p, 1 < p ≤ +∞. +Proof. Splitting the integral in (26) and integrating by parts, we obtain +�fγ(x) = − 1 +2πxf +�1 +x +� +sin 2π(1 − γ) ++ +� +1 +x +0 +f(t) cos 2π(xt − γ) dt − +1 +2πx +� ∞ +1 +x +sin 2π(xt − γ) df(t). +Further, +� +1 +x +0 +f(t) cos 2π(xt − γ) dt += +� +1 +x +0 +[f(t) − f +�1 +x +� +] cos 2π(xt − γ) dt + +� +1 +x +0 +f +�1 +x +� +cos 2π(xt − γ) dt += − +� +1 +x +0 +� � +1 +x +t +df(s) +� +cos 2π(xt − γ) dt ++ +1 +2πxf +�1 +x +� +sin 2π(1 − γ) + +1 +2πxf +�1 +x +� +sin 2πγ += +1 +2πxf +�1 +x +� +sin 2πγ + +1 +2πxf +�1 +x +� +sin 2π(1 − γ) + O +�� +1 +x +0 +s|df(s)| +� +. +To continue the proof, we need the following + +Lp SIMULATION FOR MEASURES +13 +Lemma 3. We have the inequality +� ∞ +0 +|df(s)| ≲ ∥f∥V ∗ +p . +(27) +Proof. There holds +ln 2 +� ∞ +0 +|df(s)| = +� ∞ +0 +1 +x +� 2x +x +|df(s)| dx += +� ∞ +0 +x− 1 +p +���� +� 2x +x +h(s) df(s) +����dx, +where h(s) = x− 1 +p′ sign df(s) if x < s < 2x and zero otherwise. This h is not necessarily of bounded +variation; however, since it will always be under the integral sign, we can take an equivalent function +that is of bounded variation. This is possible because the number of jumps of f is of measure zero. +We will continue to use notation h for such a function. +It is easy to see that ∥h∥p′ = 1. Let +g(u) = �h(u). We have +� ∞ +0 +|g(u)|p du = +�� +1 +x +0 ++ +� ∞ +1 +x +� +|g(u)|p du +≲ 1 +x +� x +x +1 +p′ +�p ++ x− p +p′ +� ∞ +1 +x +� ���� h(s)e−ius +−iu +��� +2x +x +���� + 1 +u +� 2x +x +|dh(s)| +�p +du. +The first term on the right is bounded. Since +� ∞ +1 +x +du +up ≲ x +1 +p′ , +the definition of h leads to the boundedness of the second term as well. +Therefore, h is the Fourier transform of an Lp function g. This leads to the needed right-hand +side in (27). +□ +We return to the proof of the theorem. Since +� ∞ +0 +� +1 +x +0 +s|df(s)| dx = +� ∞ +0 +|df(s)|, +it follows from (27) that to prove the theorem it remains to estimate +� ∞ +0 +1 +x +����� +� ∞ +1 +x +sin 2π(xt − γ) df(t) +����� dx. +We have +ln 2 +� ∞ +0 +1 +x +���� +� ∞ +1 +x +sin 2π(xt − γ) df(t) +����dx +≤ +� ∞ +0 +1 +u +� ∞ +1 +u +1 +x +���� +� 2u +u +sin 2π(xt − γ) df(t) +���� dx du + ln 2 +� ∞ +0 +1 +x +� +2 +x +1 +x +|df(t)| dx. + +14 +L. DE CARLI AND E. LIFLYAND +The latter summand on the right is controlled by +� ∞ +0 |df(t)|. Applying H¨older’s inequality to the +integral in x of the first summand, we have to estimate +� ∞ +0 +1 +u +�� ∞ +1 +u +x−pdx +� 1 +p �� ∞ +0 +���� +� 2u +u +sin 2π(xt − γ) df(t) +���� +p′ +dx +� 1 +p′ +du += +� ∞ +0 +u− 1 +pI(u) du. +(28) +where by I(u) the term in the second parenthesis is denoted. We can see that +I = 1 +2 +�� ∞ +0 +���� +� +R +� +e2πi(xt−γ) − e−2πi(xt−γ)� +χ(u,2u)(t) df(t) +���� +p′ +dx +� 1 +p′ += 1 +2 +�� ∞ +0 +�� e2πiγ � +χ(u,2u)µf(x) − e−2πiγ � +χ(u,2u)µf(−x) +��p′ +dx +� 1 +p′ +≤ ∥ � +χ(u,2u)µf∥p′. +For 1 < p ≤ 2, applying the Hausdorff-Young inequality (19), we obtain +I ≤ ∥ � +χ(u,2u)µf∥p′ ≤ ∥χ(u,2u)µf∥∗ +p, +from which we derive that (28) is bounded by +� ∞ +0 +u− 1 +p∥µf∥∗ +p,(u,2u) du, +as desired. For p > 2, Proposition 4 completes the proof. +□ +Remark 11. There exist analogs of (5) for the multivariate setting; see, e.g., [12] or [13]. +However, the above one-dimensional result is more transparent and illustrative in the sense that +extending it to several dimensions is a plain business with awkward notation and technicalities. +References +[1] W.O. Amrein and A.M. Berthier, On support properties of Lp-functions and their Fourier transforms, J. Funct. +Anal. 24 (1977), 258–267. +[2] K. Ball, Cube slicing in Rn, Proc. Amer. Math. Soc. 97 (1986), 465–473. +[3] P. L. Butzer and R. J. Nessel, Fourier Analysis and Approximation. Volume 1. One-Dimensional Theory, +Academic Press, New York and London, 1971. +[4] J.A. Cima, A.L. Matheson and W.T. Ross, The Cauchy transform, Mathematical Surveys and Monographs, +125, Amer. Math. Soc., Providence, RI, 2006. +[5] S. Fridli, Hardy Spaces Generated by an Integrability Condition, J. Approx. Theory 113 (2001), 91–109. +[6] G. Folland and A. Sitaram, The Uncertainty Principle, J. Fourier Anal. Appl. 3 (1997), 207–238. +[7] D.V. Giang and F. M´oricz, On the L1 theory of Fourier transforms and multipliers, Acta Sci. Math. (Szeged) +61 (1995), 293–304. +[8] A. Iosevich and E. Liflyand, Decay of the Fourier transform: analytic and geometric aspects, Birkhauser, 2014. +[9] P.R. Halmos, Measure Theory, Van Nostrand, New York, 1950 +[10] V.P. Havin and B. J¨oricke, The Uncertainty Principle in Harmonic Analysis, Springer-Verlag, Berlin, 1994. +[11] T.W. K¨orner, Fourier transforms of distributions and Hausdorff measures, 20 (2014), 547–565. +[12] E. Liflyand, Fourier transforms of functions from certain classes, Anal. Math. 19 (1993), 151–168. +[13] E. Liflyand, Functions of Bounded Variation and their Fourier Transforms, Birkh¨auser, 2019. +[14] B. Makarov and A. Podkorytov, Real Analysis: Measures, Integrals and Applications, Springer, 2013. + +Lp SIMULATION FOR MEASURES +15 +[15] H. Reiter and J.D. Stegeman, Classical harmonic analysis and locally compact groups. Second edition, London +Mathematical Society Monographs. New Series, 22. The Clarendon Press, Oxford University Press, New York, +2000. +[16] N. Wiener and A. Wintner, Fourier-Stieltjes Transforms and Singular Infinite Convolutions, Amer. J. Math. +60 (1938), 513–522. +Department of Mathematics and Statistics, Florida International University, Miami, FL, 33199, +USA +Email address: decarlil@fiu.edu +Department of Mathematics, Bar-Ilan University, 52900 Ramat-Gan, Israel +Email address: liflyand@gmail.com + diff --git a/7NE1T4oBgHgl3EQfTgOa/content/tmp_files/load_file.txt b/7NE1T4oBgHgl3EQfTgOa/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3fc7777e4f150ba66e812fd2593bf100e30de4d9 --- /dev/null +++ b/7NE1T4oBgHgl3EQfTgOa/content/tmp_files/load_file.txt @@ -0,0 +1,444 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf,len=443 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content='03079v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content='FA] 8 Jan 2023 Lp simulation for measures L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' De Carli and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Liflyand Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Being motivated by general interest as well as by certain concrete problems of Fourier Analysis, we construct analogs of the Lp spaces for measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' It turns out that most of standard properties of the usual Lp spaces for functions are extended to the measure setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' We illustrate the obtained results by examples and apply them to obtain a version of the uncertainty principle and an integrability result for the Fourier transform of a function of bounded variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Introduction Looking through any book devoted to Fourier analysis or just the table of contents, one will see that the L1 theory of the Fourier transform or the Hilbert transform goes with the corresponding Lp theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' This is not the case for the theories of the corresponding transforms for measures, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=', [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' A simple curiosity may force one to wonder where the analogs for measures are hidden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' We have not succeeded to find such a machinery in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' However, we have a more concrete reason to be interested in the depository of such treasures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Let us consider the following example, somewhat sketchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' The cosine Fourier transform of a function of bounded variation on the half-axis, to wit f ∈ BV (R+), is �fc(x) = � ∞ 0 f(t) cos(2πxt) dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' (1) Let f be locally absolutely continuous on (0, ∞);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' note that here we use not R+ = [0, ∞) but (0, ∞) since it is of considerable importance and generality that we can avoid claiming absolute continuity at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Let in addition, lim t→∞ f(t) = 0 and Hof ′ ∈ L1(R+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Here, for any integrable function g on R+, Hog(x) = 2 π � ∞ 0 tg(t) x2 − t2 dt (2) is the Hilbert transform applied to the odd extension of g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' of course, understood in the principle value sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' When it is integrable, we will denote the corresponding Hardy space of such functions g by H1 0(R+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Then the cosine Fourier transform of f in (1) is Lebesgue integrable on R+, with ∥�fc∥L1(R+) ≲ ∥f ′∥L1(R+) + ∥Hof ′∥L1(R+) = ∥f ′∥H1 0(R+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' (3) 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Primary: 28A33;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Secondary: 42A38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Measure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Fourier transform;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Hausdorff-Young inequality;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Young inequality;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' uncertainty principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' 1 2 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' DE CARLI AND E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' LIFLYAND For this result as well as many other more advanced ones, see [12] (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' [5] and [7];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' see also [8, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content='3] or more recent [13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Recall that the derivative of a function of bounded variation exists almost everywhere and is Lebesgue integrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Here and in what follows ϕ ≲ ψ means that ϕ ≤ Cψ with C being an absolute constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' A natural question arises whether we can relax the assumption of absolute continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' The first step in an eventual proof is obvious: we integrate by parts in the Stieltjes sense in (1) and arrive at �fs(x) = − 1 2πx � ∞ 0 sin(2πxt) df(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' However, if we try to follow the lines of the proof of (3) and arrive at a version of Hardy’s space with integrable Hilbert transform of df, we will fail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' The point is that the Hilbert transform of df does exist almost everywhere (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=', [3, §8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content='1 ]) but its integrability leads to absolute continuity, the property that we aimed to remove (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=', [4] and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' On the other hand, there is a scale of handy subspaces of H1 0(R+), for which the integrability of the cosine Fourier transform is valid, with the norm of f ′ in one of such spaces on the right-hand side of (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' More precisely, for 1 < p < ∞, set ∥g∥Op = � ∞ 0 �1 x � x≤t≤2x |g(t)|pdt � 1 p dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Further, for p = ∞, let ∥g∥O∞ = � ∞ 0 ess sup x≤t≤2x |g(t)| dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Known are (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=', the above sources) the following relations: O∞ ֒→ Op1 ֒→ Op2 ֒→ H1 0 ֒→ L1 (p1 > p2 > 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' (4) Under the above assumptions, there holds ∥�fc∥L1(R+) ≲ ∥f ′∥Op(R+), (5) provided that the right-hand side is finite for some p > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' In fact, a different notation is convenient for the case where the Op norm is calculated for the derivative: ∥f∥Vp := ∥f ′∥Op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Just this notation is appropriate for further generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' On the one hand, (5) follows from (3) and (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' On the other hand, a direct proof for (5) is given in [7], where the main ingredient is the Hausdorff-Young inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' To provide similar reasoning for measures µf generated by functions of bounded variation f rather than functions (however, we shall write df rather than dµf), we need a corresponding extension of the Hausdorff-Young inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' And here is the point where our special harmonic analysis comes into play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' We do not restrict ourselves to finding immediate tools for the above problem but try to establish a kind of general and multivariate theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' A variety of relevant issues will be introduced and studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Basic notions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' We define an analog of Lp spaces for measures by means of an associated norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' For a given p ∈ [1, ∞], we use the notation ∥ · ∥p to denote the standard norm in Lp(Rn) = Lp(Rn, dx), where by dx we mean the Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Lp SIMULATION FOR MEASURES 3 We denote by S(Rn) the Schwartz space of rapidly decreasing C∞ functions, and either by F(f) or by �f the Fourier transform of a function f ∈ S(Rn), written �f(y) = � Rn f(x)e−2πix·y dx, where x·y = x1y1+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content='+xnyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Recall that F : S(Rn) → S(Rn) is one-to-one, and the inverse Fourier transform is ˇf(y) = �f(−y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' In this paper, we will not distinguish between Fourier transform and inverse Fourier transform, unless it becomes necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' For p ∈ [1, 2], the operator F : Lp(Rn) → Lp′(Rn), with 1 p+ 1 p′ = 1, is bounded, with ∥ �f∥p′ ≤ ∥f∥p and equality if p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' For Lp(Rn), p > 2, the Fourier transform can be defined in the distributional sense as ⟨ �f, ψ⟩ = � Rn f(x) �ψ(x) dx, ψ ∈ S(Rn);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' clearly, �f is a function if and only if f = �g for some g ∈ Lp′(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' With this observation in mind, we give the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' For a given p ∈ [1, ∞], we let �Lp(Rn) = {f ∈ Lp(Rn) : f = �g for some g ∈ Lp′(Rn)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' (6) In a natural way, we endow �Lp(Rn) with the norm ∥f∥�Lp = ∥ �f ∥p′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' (7) With this definition, the Fourier transform F : Lp(Rn) → (�Lp′(Rn), ∥ · ∥�Lp′) is a one-to-one isometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' When p ∈ [1, 2], the Hausdorff-Young inequality yields ∥f∥�Lp ≤ ∥f∥p, with equality if p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' We denote by M the space of sigma-finite Borel measures on Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' For every p ∈ [1, ∞], we define the functional ∥ · ∥∗ p : M → [0, ∞] as ∥µ∥∗ p = sup h∈� Lp′ (Rn) : ∥h∥� Lp′ ≤1 ���� � Rn h(t)dµ(t) ���� ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' (8) we let Mp = {µ ∈ M : ∥µ∥∗ p < ∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' (9) Note that, for every µ ∈ Mp and every h ∈ �Lp′(Rn), we have that ���� � Rn h(x)dµ(x) ���� ≤ ∥h∥�Lp′∥µ∥∗ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' (10) We do not assume that our measures are positive, or even real-valued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' For definition and properties of non-positive measure see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' With this assumption, the spaces Mp are vector spaces, and we will prove in Section 2 that the functional ∥µ∥∗ p is a norm on Mp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' 4 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' DE CARLI AND E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' LIFLYAND 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Structure of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' With ∥µ∥∗ p and Mp denoted by similarity to Lp, we then establish basic properties of these measure spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' We will prove in Section 2 that the spaces Mp have many properties in common with Lp spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' We establish the properties of measures in Mp spaces and the properties of functions in spaces �Lp(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Discussing then the Fourier transform of a measure, we establish a Hausdorff- Young type inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Further, for the convolution of a function and a measure, we prove a Young type inequality for our setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' We mention that the results in Section 2 are supplemented with examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Section 3 is devoted to applications of the introduced machinery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' One of them is a development of an uncertainty principle for measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' The uncertainty principle in Fourier analysis quantifies the intuition that a function and its Fourier transform cannot both be concentrated on small sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Many examples of this principle can be found, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=', in the book by Havin and J¨oricke [10] and in an article by Folland and Sitaram [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' In Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content='1, using a quantitative version of a result in [1], we prove that a finite measure and its Fourier transform cannot both be supported on sets of finite Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Recall that a measure µ is supported in a set E ⊂ Rn if µ(F) = 0 whenever F is a measurable set that does not intersect E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' In conclusion, we formulate and prove an analog of (5) for functions of bounded variation without assuming absolute continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' This is Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' In order to formulate and prove it, as an analog of Vp spaces for functions, we introduce the notion f ∈ V ∗ p for measures, with ∥f∥V ∗ p = � ∞ 0 x− 1 p ∥χ(x,2x)µf∥∗ p dx where χE denotes the characteristic function of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' The product of a measure µ and a measurable function f is the measure defined by (fµ)(F) = � F f dµ for every measurable set F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' For 1 < p ≤ 2, our new Hausdorff-Young inequality will be helpful, while for p > 2, we prove an analog of (4) and use an embedding argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Lp properties of measures In this section we establish basic properties of measures in the spaces Mp defined in the intro- duction, with p ∈ [1, ∞], that mimic those of functions in Lp spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' We also establish properties of the spaces �Lp(Rn) defined in (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' If E is a measurable subset of Rn, with |E| ̸= 0, we let ∥µ∥∗ p,E = ∥χEµ∥∗ p = sup h∈� Lp′ (Rn) : ∥h∥� Lp′ ≤1 ���� � E h(t) dµ(t) ���� , (11) and Mp,E = {µ : ∥µ∥∗ p,E < ∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' We can also define M1,loc = {µ : ∥µ∥∗ 1,E < ∞ for every measurable bounded set E}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' (12) The standard Lebesgue measure and the Delta measures are notable examples of Mp measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' In the rest of this paper we will use L (or dx in integration) to denote the standard Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' For a given a ∈ Rn, we let δa be the measure defined as � Rn f(x) dδa = f(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Lp SIMULATION FOR MEASURES 5 Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' We show that the standard Lebesgue measure is in M∞ and ∥L∥∗ ∞ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Indeed, ∥L∥∗ ∞ = sup h∈� L1(Rn) : ∥h∥� L1=∥�h∥∞ ≤1 ���� � Rn h(t) dt ���� ≤ sup h∈L1(Rn) : ∥h∥1≤1 ���� � Rn h(t) dt ���� ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' To prove that equality holds, we can consider g = e−π|x|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' It is easy to verify that �g(x) = g(x), and so g ∈ �L1(Rn) and ∥g∥�L1 = ∥�g∥∞ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Since 1 = �g(0) = � Rn g(t) dt = ∥g∥1, we have that ∥L∥∗ ∞ ≥ � Rn g(t) dt = 1, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' We show that δa ∈ Mp only for p = 1 and ∥δa∥1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Indeed, assuming a = 0 for simplicity, we can easily see that ∥δ0∥∗ 1 = sup h∈S(Rn): ∥h∥� L∞ ≤1 ���� � Rn h(t) dδ0 ���� = sup h∈S(Rn): ∥�h ∥1≤1 |h(0)| = sup h∈S(Rn): ∥�h∥1≤1 ���� � Rn �h(x) dx ���� ≤ sup h∈S(Rn): ∥�h ∥1≤1 ∥�h ∥1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' To prove that equality holds, we can consider the function g = e−π|x|2 in the previous example and verify that ∥δ0∥∗ 1 ≥ ∥ˆg∥1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' An easy variation of this argument shows that δ0 ̸∈ Mp if p > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' H¨older type inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' We prove the following Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' If µ ∈ Mp and f ∈ �Lq(Rn), and 1 r = 1 q + 1 p, then ∥fµ∥∗ r ≤ ∥f∥�Lq∥µ∥∗ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' (13) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Assume ∥f∥�Lq = 1, or else replace f with ˜f = f ∥f∥� Lq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' With the notation previously introduced, ∥fµ∥∗ r = sup h∈� Lr′ (Rn): ∥h∥� Lr′ ≤1 ���� � Rn h(y)f(y) dµ(y) ����.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Let us show that hf ∈ �Lp′ and ∥hf∥�Lp′ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Indeed, � hf = �h ∗ �f (standard convolution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Since 1 r + 1 q′ = 1 + 1 p, by Young’s inequality for convolution and the Hausdorff-Young inequality, ∥hf∥�Lp′ = ∥� hf∥p = ∥�h ∗ �f∥p ≤ ∥�h∥r∥ �f∥q′ = ∥h∥�Lr′∥f∥�Lq ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Thus, ∥hf∥�Lp′ ≤ 1, and so ∥fµ∥∗ r ≤ sup k∈� Lp′ (Rn): ∥k∥� Lp′ ≤1 ���� � Rn k(y) dµ(y) ���� = ∥µ∥∗ p = ∥µ∥∗ p ∥f∥�Lq, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' □ Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' When r = 1, for every µ ∈ Mp and f ∈ �Lp′(Rn) we have that ∥fµ∥∗ 1 ≤ ∥f∥�Lp′∥µ∥∗ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' This is the case of (13) that most closely resembles the standard H¨older’s inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' 6 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' DE CARLI AND E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' LIFLYAND Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Let E be a bounded subset of Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Then Mr,E ⊂ Mp,E whenever 1 ≤ p ≤ r ≤ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Assume p < r, since the case p = r is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Assume also E ⊂ QR = [−R, R]n for some R > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' By (11), ∥µ∥∗ r,E = sup ∥h∥� Lr′ ≤1 ���� � E h(y) dµ(y) ���� = sup ∥�h∥Lr ≤1 ���� � Rn χQ(y)h(y)χE(y) dµ(y) ����.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Let q = rp r−p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Since r ̸= p, we have q < ∞ and q′ > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' The Fourier transform of the characteristic function of QR is �χQR(x) = n � j=1 sin(πRxj) πxj , and so �χQR(x) ∈ Ls(Rn) for every s > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' We have ∥�χQR∥s = Cn s R n s′ , where Cs = ���� sin(π·) π· ���� s = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f3 � 2s′ π � 1 s, 1 < s < 2, � 2 s � 1 2s, 2 ≤ s < ∞ 1, s = ∞, is independent of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' In fact, Cs can be taken � 2s′ π � 1 s for all s < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' This is calculated by minimal means: split the integral � R ���sin(πt) πt ��� s dt = 1 π � R ���sin(t) t ��� s dt (14) into two, over |t| ≤ 1 and over |t| > 1, and replace ��� sin(t) t ��� in the first by 1 and in the second by 1 |t|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' However, it is known (see [2, Lemma 3] or [14, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content='VI, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content='5]) that for s ≥ 2, the sharp bound for (14) is � 2 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Applying Proposition 1 with f = χQR and χEµ in place of µ, we obtain ∥µ∥∗ p,E ≤ ∥χQR∥�Lq∥µ∥∗ r,E = Cn q R n q′ ∥µ∥∗ r,E, (15) and so ∥µ∥∗ p,E < ∞ whenever ∥µ∥∗ r,E < ∞, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' □ Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' For every p ∈ [1, ∞], we have that Mp ⊂ M1,loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Follows from Corollary 3 and (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' □ Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' The functional ∥ ∥∗ p is a norm on Mp for every p ∈ [1, ∞].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' It is trivial to verify that for every µ, σ ∈ Mp and every λ ∈ C, ∥µ + σ∥∗ p ≤ ∥µ∥∗ p + ∥σ∥∗ p, ∥λµ∥∗ p = |λ| ∥µ∥∗ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' We now prove that ∥µ∥∗ p = 0 if and only if µ ≡ 0, in the sense that µ(E) = 0 for every µ−measurable set E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Lp SIMULATION FOR MEASURES 7 In order to show that µ ≡ 0, it is enough to verify that µ(E) = 0 for every bounded set E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Let E be bounded and µ−measurable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Assume that E ⊂ QR for some R > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Using (15) and Proposition 8, we can see at once that µ(E) = � E dµ(x) = � QR χEdµ(x) ≤ ∥χQR∥�L∞∥µ∥∗ p,E ≤ ∥µ∥∗ p = 0 and so µ(E) = 0 for every µ−measurable bounded set E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Properties of �Lp spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' In this sub-section we will establish properties of the spaces �Lp(Rn) defined in (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' We first shows how measures of the form dµ = fdx behave with respect to the norms introduced when f ∈ �Lp, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Let dµ = fdx, with f ∈ �Lp(Rn) for some p ∈ [1, ∞];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' then µ ∈ Mp and ∥µ∥∗ p = ∥f∥�Lp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Before discussing Theorem 6, we prove the following Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' S(Rn) is dense in �Lp(Rn) for every p ∈ [1, ∞].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Since S(Rn) ⊂ �Lp(Rn) ⊂ Lp(Rn) and S(Rn) is dense in Lp(Rn) for every p ∈ [1, ∞), we can see at once that S(Rn) is also dense in �Lp(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' To see that S(Rn) is dense also in �L∞(Rn), we observe that every f ∈ �L∞(Rn) is the image of g ∈ L1(Rn) via the Fourier transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' We can find functions ψn ∈ S(Rn) such that lim n→∞ ∥ψn − g∥1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' But ∥ψn − g∥1 = ∥ � �ψn − �f ∥1 = ∥ �ψn − f∥�L∞, and so lim n→∞ ∥ �ψn−f∥�L∞ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Since �ψn ∈ S(Rn), we have proved that S(Rn) is dense in �L∞(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' □ Proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Since S(Rn) is dense in Lp(Rn) and in �Lp′(Rn), and the Fourier trans- form is one-to-one in S(Rn), we can see at once that ∥f∥�Lp = ∥ �f∥p′ = sup g∈S(Rn): ∥g∥p≤1 ���� � Rn g(t) �f(t) dt ���� = sup g∈S(Rn): ∥g∥p≤1 ���� � Rn �g(t)f(t) dt ���� = sup h∈S(Rn): ∥�h∥p≤1 ���� � Rn h(t)f(t) dt ���� = sup h∈S(Rn): ∥h∥� Lp′ ≤1 ���� � Rn h(t)f(t) dt ���� = ∥µ∥∗ p, which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' □ Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' If dµ = fdx is as in Theorem 6 and p ∈ [1, 2], then there holds ∥µf∥∗ p = ∥f∥�Lp = ∥ �f∥p′ ≤ ∥f∥p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Let E ⊂ Rn be a (Lebesgue) measurable set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' a) For every p ∈ [1, ∞], we have ∥χE∥�Lp ≤ |E| 1 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' (16) b) For every 1 ≤ p ≤ q ≤ ∞ and every µ ∈ Mq, we have ∥µ∥∗ p,E ≤ ∥µ∥∗ q|E| 1 r , where 1 r = 1 p − 1 q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' We have used the standard convention 1 ∞ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Thus, (16) yields ∥χE∥�L∞ ≤ 1, for every set E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' 8 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' DE CARLI AND E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' LIFLYAND Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' We first prove a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' When p ∈ [1, 2], Remark 7 yields ∥χE∥�Lp ≤ ∥χE∥p = |E| 1 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Assume now p ∈ (2, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' By the Hausdorff-Young inequality, we can see at once that {f ∈ �Lp′ : ∥f∥�Lp′ = ∥ �f∥p ≤ 1} ⊂ {f ∈ Lp′(Rn) : ∥f∥p′ ≤ 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' In view of this observation and Theorem 6, we can let dσ = χEdx and write the following chain of inequalities: ∥χE∥�Lp = ∥σ∥∗ p,E = sup f∈� Lp′(Rn): ∥f∥� Lp′ ≤1 ���� � Rn χE(x)f(x)dx ���� ≤ sup f∈Lp′ (Rn): ∥f∥p′ ≤1 ���� � Rn χE(x)f(x)dx ���� (17) ≤ |E| 1 p∥f∥p′ ≤ |E| 1 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' We have used H¨older’s inequality in the last step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' When p = ∞, it follows from (17) that sup f∈L1(Rn): ∥f∥1≤1 ���� � Rn χE(x)f(x)dx ���� ≤ sup f∈L1(Rn): ∥f∥1≤1 � Rn χE(x)|f(x)|dx ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Part b) follows from H¨older’s inequality (1) and part a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Indeed, letting r = pq q−p, we have ∥µ∥∗ p,E = ∥χEµ∥∗ p ≤ ∥χE∥�Lr∥µ∥∗ q ≤ |E| 1 r ∥µ∥∗ q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' The proof of the corollary is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' □ We use Theorem 6 to prove inclusion relations of the �Lp spaces and their duals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Recall that the dual of a normed space X, denoted by (X)′, is the set of linear functionals L : V → C such that sup∥f∥X≤1 |L(f)| < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' By definition, �Lp(Rn) = Lp(Rn) when p ∈ [1, 2] but in general �Lp(Rn) is a proper subspace of Lp(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' For example, the Riemann-Lebesgue Lemma yields that �L∞(Rn) is a space of uniformly continuous functions that go to zero at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Even though Lp(Rn) = �Lp(Rn) when p ∈ [1, 2], the norms on these spaces are different and so the duals of these spaces are different too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' When p ≤ 2, the Hausdorff-Young inequality yields, ∥f∥�Lp = ∥ ˆf∥p′ ≤ ∥f∥p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' When p = 2 we have ∥f∥2 = ∥f∥�L2 but when p > 2 the inequality above can be strict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' We prove the following Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' For every p ∈ [1, ∞], we have �Lp′(Rn) ⊂ (�Lp(Rn))′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' When p ∈ [1, 2], we have �Lp′(Rn) ⊂ (�Lp(Rn))′ ⊂ Lp′(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Lp SIMULATION FOR MEASURES 9 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' For a given g ∈ �Lp′(Rn), we let dµ = gdx and we let Lg : �Lp(Rn) → C, Lg(f) = � Rn f(x)g(x)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' By H¨older’s inequality (13) and Theorem 6 |Lg(f)| = ���� � Rn f(x)g(x)dx ���� ≤ ∥f∥�Lp∥µ∥∗ p′ = ∥f∥�Lp∥g∥�Lp′ and so L ∈ (�Lp(Rn))′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' When p ≤ 2, for every L ∈ (�Lp(Rn))′, we have that |L(f)| ≤ C∥f∥�Lp = C∥ ˆf∥p′ ≤ C∥f∥p and so L ∈ (Lp(Rn))′ = Lp′(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Fourier transform of finite measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' The Fourier transform of a finite Borel measure µ is the function defined as �µ(y) = � Rn e−2πix·ydµ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' (18) To distinguish it from the Fourier transform for functions, it is sometimes called the Fourier-Stieltjes transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' It is well-known (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=', [3, §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content='3] or [15, §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content='4]) that the function �µ is continuous and bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' By the Riemann-Lebesgue Lemma, the Fourier transform of an L1 function vanishes at infinity, but the Fourier transform of a M1 measure does not need to do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' For example, we have shown in Example 2 that the Delta measure µ = δa is in M1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' its Fourier transform is �µ(x) = e2πia·x, and |�µ(x)| ≡ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' We prove the following analog of the Hausdorff-Young inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Let µ ∈ Mp, with 1 ≤ p ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Then, �µ ∈ Lp′(Rn), and ∥�µ∥p′ ≤ ∥µ∥∗ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' (19) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' We have observed that the Fourier transform of a finite measure is always bounded, so the proposition is trivial for p = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' When p ∈ (1, 2], we have ∥�µ∥p′ = sup h∈S(Rn): ∥h∥p≤1 � Rn h(y)�µ(y) dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' By Fubini’s theorem, � Rn h(y)�µ(y) dy = � Rn � Rn h(y)e−2πix·y dµ(x) dy = � Rn �h(x) dµ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' (20) In view of (20) and the fact that ∥�h ∥p′ ≤ ∥h∥p ≤ 1, we can see at once that ∥�µ∥p′ = sup h∈S(Rn): ∥h∥p≤1 � Rn �h(y) µ(y) ≤ sup k∈S(Rn): ∥�k ∥p′ ≤1 ���� � Rn k(x) dµ(x) ���� = ∥µ∥∗ p, which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' □ 10 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' DE CARLI AND E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' LIFLYAND Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' If µf is generated by the singular function f in [16], we have |� µf(x)| = O � 1 |x|δ � for |x| large, with 0 < δ < 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Then � µf ∈ Lp′(R), with 1 δ < p′ < ∞, and correspondingly, 2 < p′ < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' By this, ∥µf∥∗ p < ∞, since ∥µf∥∗ p = sup ∥h∥� Lp′ ≤1 ���� � R h(t) df(t) ���� = sup ∥g∥Lp≤1 ���� � R �g(x) df(x) ���� = sup ∥g∥Lp≤1 ���� � R g(x) �df(x) dx ���� < ∞, because of g ∈ Lp and � µf ∈ Lp′, with 1 < p < 1 1−δ < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' We have used the pioneer example of a singular function in [16] but there are more subtle ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' However, for all of them there is a barrier to L2, like 0 < δ < 1 2 above;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=', [11] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Convolution of a function and a measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Let µ be a sigma-finite Borel measure, and let f : Rn → R be a measurable function such that the function x → � Rn f(x − y)dµ(y) (21) is finite for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' x ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' The convolution of f and µ, denoted by f ∗ µ, is the function defined in (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' We prove the following analog of the Young inequality for convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' If µ ∈ Mp and f ∈ �Lq(Rn) with 1 p + 1 q = 1 r, then f ∗ µ ∈ �Lr(Rn) and ∥f ∗ µ∥�Lr ≤ ∥f∥�Lq∥µ∥∗ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' In view of Proposition 6, ∥f ∗ µ∥�Lr = ∥f ∗ µ∥∗ r = sup h∈S(Rn): ∥h∥� Lr′ ≤1 � Rn h(x)(f ∗ µ)(x) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' (22) For every h ∈ S(Rn), � Rn h(x)(f ∗ µ)(x) dx = � Rn h(x) � Rn f(x − y) dµ(y) dx, = � Rn h ∗ ˜f(y) dµ(y) (23) where ˜g(t) = g(−t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' By (23) and (10), � Rn h ∗ ˜f(y) dµ(y) ≤ ∥h ∗ ˜f∥�Lp′∥µ∥∗ p = ∥�h �f∥p ∥µ∥∗ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' (24) Recalling that 1 + 1 r = 1 p + 1 q, we have 1 p = 1 r + 1 q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' By H¨older’s inequality, ∥�h �f∥p ≤ ∥�h ∥r∥ �f ∥q′ = ∥h∥�Lr′∥f∥�Lq By (22) and (24), we conclude that ∥f ∗ µ∥�Lr ≤ ∥f∥�Lq∥µ∥∗ p, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' □ Lp SIMULATION FOR MEASURES 11 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Applications As mentioned, in this section we present applications of the obtained results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Uncertainty principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' In this subsection, we show that the uncertainty principle has its embodiment also for measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' We prove the following Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' A finite nonzero measure µ ∈ M2 and its Fourier transform �µ cannot both be supported in sets of finite Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' The proof of the theorem relies on the following Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Let E, F ⊂ Rn be sets of finite Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' There exists a constant C > 0 such that for every measure µ ∈ M2, we have ∥dµ∥∗ 2,F ≤ C∥ �µ ∥L2(Ec).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Recall the following quantitative form of an uncertainty principle result obtained by Amrein and Berthier in [1]: Let E, F ⊂ Rn be sets of finite measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' There exists a constant C > 0 such that for every function f ∈ L2(Rn), ∥ �f ∥L2(F ) ≤ C∥f∥L2(Ec).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' (25) Let h ∈ L2(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' By (25), the inequality ∥h∥L2(Ec) ≤ 1 yields ∥�h ∥L2(F ) ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' In view of (20), we can write the following chain of inequalities: ∥ �µ ∥L2(Ec) = sup h∈L2(Rn): ∥h∥L2(Ec)≤1 � Rn h(x)�µ(x) = sup h∈L2(Rn): ∥h∥L2(Ec)≤1 � Rn �h(y) dµ(y) ≥ sup h∈L2(Rn): ∥�h ∥L2(F )≤C � Rn �h(x) dµ(x) = 1 C sup k∈L2(Rn): ∥ �k ∥2≤1 � F k(x)dµ(x) = 1 C ∥µ∥∗ 2,F, obtaining the required result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' □ Proof of Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Assume by contradiction that µ is supported in F and �µ is supported in E, where E, F ⊂ Rn are both of finite measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' By Lemma 2, we have ∥µ∥∗ 2,F = ∥χFµ∥∗ 2 = 0, and Corollary 5 yields χFµ ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Since χF cµ ≡ 0 is assumed, we have µ = 0, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' The Fourier transform theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' In order to reveal an analogy to the case of absolutely continuous f, we prove a counterpart of corresponding embeddings in (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' For p1 > p2 > 1, there holds V ∗ p1 ֒→ V ∗ p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' 12 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' DE CARLI AND E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' LIFLYAND Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' We are going to apply Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Since for E = (x, 2x), we have in (15) that by (16), there holds ∥χQR∥�Lq ≲ x 1 q , and it follows that ∥µf∥∗ p2,(x,2x) ≲ x 1 q ∥µf∥∗ p1,(x,2x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' The corresponding relation 1 p2 = 1 q + 1 p1 yields 1 q = p1−p2 p1p2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' It remains to observe that x− 1 p2 x p1−p2 p1p2 = x− 1 p1 , which leads to the needed embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' □ With these embeddings and the tools elaborated before, we study, for γ = 0 or 1 4, the Fourier transforms �fγ(x) = � ∞ 0 f(t) cos 2π(xt − γ) dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' (26) It is clear that �fγ represents the cosine Fourier transform in the case γ = 0, while taking γ = 1 4 gives the sine Fourier transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Let f be of bounded variation on R+ and vanishing at infinity, that is, lim t→∞ f(t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' If f ∈ V ∗ p , then for x > 0, we have �fγ(x) = 1 2πxf �1 x � sin 2πγ + Γ(x), where γ = 0 or 1 4, and ∥Γ∥L1(R+) ≲ ∥f∥V ∗ p provided that the last value is finite for some p, 1 < p ≤ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Splitting the integral in (26) and integrating by parts, we obtain �fγ(x) = − 1 2πxf �1 x � sin 2π(1 − γ) + � 1 x 0 f(t) cos 2π(xt − γ) dt − 1 2πx � ∞ 1 x sin 2π(xt − γ) df(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Further, � 1 x 0 f(t) cos 2π(xt − γ) dt = � 1 x 0 [f(t) − f �1 x � ] cos 2π(xt − γ) dt + � 1 x 0 f �1 x � cos 2π(xt − γ) dt = − � 1 x 0 � � 1 x t df(s) � cos 2π(xt − γ) dt + 1 2πxf �1 x � sin 2π(1 − γ) + 1 2πxf �1 x � sin 2πγ = 1 2πxf �1 x � sin 2πγ + 1 2πxf �1 x � sin 2π(1 − γ) + O �� 1 x 0 s|df(s)| � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' To continue the proof, we need the following Lp SIMULATION FOR MEASURES 13 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' We have the inequality � ∞ 0 |df(s)| ≲ ∥f∥V ∗ p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' (27) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' There holds ln 2 � ∞ 0 |df(s)| = � ∞ 0 1 x � 2x x |df(s)| dx = � ∞ 0 x− 1 p ���� � 2x x h(s) df(s) ����dx, where h(s) = x− 1 p′ sign df(s) if x < s < 2x and zero otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' This h is not necessarily of bounded variation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' however, since it will always be under the integral sign, we can take an equivalent function that is of bounded variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' This is possible because the number of jumps of f is of measure zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' We will continue to use notation h for such a function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' It is easy to see that ∥h∥p′ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Let g(u) = �h(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' We have � ∞ 0 |g(u)|p du = �� 1 x 0 + � ∞ 1 x � |g(u)|p du ≲ 1 x � x x 1 p′ �p + x− p p′ � ∞ 1 x � ���� h(s)e−ius −iu ��� 2x x ���� + 1 u � 2x x |dh(s)| �p du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' The first term on the right is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Since � ∞ 1 x du up ≲ x 1 p′ , the definition of h leads to the boundedness of the second term as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Therefore, h is the Fourier transform of an Lp function g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' This leads to the needed right-hand side in (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' □ We return to the proof of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Since � ∞ 0 � 1 x 0 s|df(s)| dx = � ∞ 0 |df(s)|, it follows from (27) that to prove the theorem it remains to estimate � ∞ 0 1 x ����� � ∞ 1 x sin 2π(xt − γ) df(t) ����� dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' We have ln 2 � ∞ 0 1 x ���� � ∞ 1 x sin 2π(xt − γ) df(t) ����dx ≤ � ∞ 0 1 u � ∞ 1 u 1 x ���� � 2u u sin 2π(xt − γ) df(t) ���� dx du + ln 2 � ∞ 0 1 x � 2 x 1 x |df(t)| dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' 14 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' DE CARLI AND E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' LIFLYAND The latter summand on the right is controlled by � ∞ 0 |df(t)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Applying H¨older’s inequality to the integral in x of the first summand, we have to estimate � ∞ 0 1 u �� ∞ 1 u x−pdx � 1 p �� ∞ 0 ���� � 2u u sin 2π(xt − γ) df(t) ���� p′ dx � 1 p′ du = � ∞ 0 u− 1 pI(u) du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' (28) where by I(u) the term in the second parenthesis is denoted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' We can see that I = 1 2 �� ∞ 0 ���� � R � e2πi(xt−γ) − e−2πi(xt−γ)� χ(u,2u)(t) df(t) ���� p′ dx � 1 p′ = 1 2 �� ∞ 0 �� e2πiγ � χ(u,2u)µf(x) − e−2πiγ � χ(u,2u)µf(−x) ��p′ dx � 1 p′ ≤ ∥ � χ(u,2u)µf∥p′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' For 1 < p ≤ 2, applying the Hausdorff-Young inequality (19), we obtain I ≤ ∥ � χ(u,2u)µf∥p′ ≤ ∥χ(u,2u)µf∥∗ p, from which we derive that (28) is bounded by � ∞ 0 u− 1 p∥µf∥∗ p,(u,2u) du, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' For p > 2, Proposition 4 completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' □ Remark 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' There exist analogs of (5) for the multivariate setting;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=', [12] or [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' However, the above one-dimensional result is more transparent and illustrative in the sense that extending it to several dimensions is a plain business with awkward notation and technicalities.' metadata={'source': 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(1938), 513–522.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content=' Department of Mathematics and Statistics, Florida International University, Miami, FL, 33199, USA Email address: decarlil@fiu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content='edu Department of Mathematics, Bar-Ilan University, 52900 Ramat-Gan, Israel Email address: liflyand@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} +page_content='com' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'} diff --git a/7dAyT4oBgHgl3EQfcvf_/content/tmp_files/2301.00291v1.pdf.txt b/7dAyT4oBgHgl3EQfcvf_/content/tmp_files/2301.00291v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..06d5d28490f873b7eac939f2cd81e4f8ee96302a --- /dev/null +++ b/7dAyT4oBgHgl3EQfcvf_/content/tmp_files/2301.00291v1.pdf.txt @@ -0,0 +1,1640 @@ +The Functional Wiener Filter + +Benjamin Colburn, Luis G. Sanchez Giraldo, Jose C. Principe + +Abstract +This paper presents a close form solution in Reproducing Kernel Hilbert Space (RKHS) for the +famed Wiener filter, which we called the functional Wiener filter (FWF). Instead of using the +Wiener-Hopf factorization theory, here we define a new lagged RKHS that embeds signal statistics +based on the correntropy function. In essence, we extend Parzen’s work on the autocorrelation +function RKHS to nonlinear functional spaces. The FWF derivation is also quite different from +kernel adaptive filtering (KAF) algorithms, which utilize a search approach. The analytic FWF +solution is derived in the Gaussian kernel RKHS with a constant computational complexity similar +to the Wiener solution, and never composes nor employs the error as in conventional optimal +modeling. Because of the lack of congruence between the Gaussian RKHS and the space of time +series, we compare performance of two pre-imaging algorithms: a fixed-point optimization +(FWFFP) that finds and approximate solution in the RKHS, and a local model implementation +named FWFLM. The experimental results show that the FWF performance is on par with the KAF +for time series modeling, and it requires far less computation. + +Introduction + +Norbert Wiener’s 1949 work on minimum mean square error estimation opened the door +for the theory of optimum filtering [1]. The mathematics to solve integral equations, the Wiener- +Hopf method [2], were crucial to arrive at the optimal parameter function, however, the +methodology is rather complex. In digital signal processing using finite impulse response filters, +the Wiener solution coincides with least squares, as proven by the Wiener-Kinchin theorem [3]. +Therefore, the solution still belongs to the span of the input data i.e., the corresponding filter is +linear in the parameters and therefore it is not a universal functional approximator. +In the late 50’s, Emmanuel Parzen [4] presented an alternative approach to solve the +minimum mean square estimation (MMSE) problem in a Reproducing Kernel Hilbert space +(RKHS) defined by the autocorrelation function of the data. Since RKHS theory will be +extensively employed, we define here a RKHS. Let 𝐸 be a non-empty set, and 𝜅(𝑢, 𝑣) a function +defined on 𝐸 × 𝐸 that is nonnegative definite. Due to the Moore-Aronzsajn theorem [5], 𝜅(𝑢, 𝑣) +defines uniquely a RKHS, ℋ𝜅, such that 𝜅(⋅, 𝑣) ∈ ℋ𝜅 and for any 𝑔 ∈ ℋ𝜅, 〈𝑔, 𝜅(⋅,𝑣)〉ℋ𝜅 = 𝑔(𝑣). +Therefore, a RKHS is a special Hilbert vector space associated with a kernel such that it reproduces +(via the inner product) in the space i.e., 〈 𝜅(⋅,𝑢), 𝜅(⋅,𝑣)〉ℋ𝜅 = 𝜅(𝑢, 𝑣); or equivalently, a space +where every point evaluation functional is bounded. The history of RKHS applications started in +physics [6], statistics [7], signal processing [8] and machine learning [12]. Here, it will also be +clear that the RKHS framework provides a natural link between stochastic processes and +deterministic functional analysis. +Parzen introduced for the first time the RKHS methodology in statistical signal-processing +and time-series analysis in [4]. His essential idea is that there exists a congruence map between +the set of random variables spanned by the random process {𝑋(𝑡), 𝑡 ∈ 𝑇} with covariance function +𝑅(𝑡, 𝑠) = 𝐸[𝑋(𝑡)𝑋(𝑠)] and the RKHS of vectors spanned by the set {𝑅(⋅,𝑡), 𝑡 ∈ 𝑇} denoted as +ℋ𝑅. Note that the kernel expresses the second-order statistics of the data through the expected +value (a data-dependent kernel) and Parzen clearly stated that this RKHS offers an elegant + +functional analysis framework for minimum mean square error (MMSE) solutions such as +regression coefficients, least squares estimation of random variables, detection of signals in +Gaussian noise, and others [9],[10],[11]. Unfortunately, ℋ𝑅 is defined in the input data space, so +yields only linear solutions to the regression problem. Parzen beautiful interpretation did not +provide any practical improvement, so it was quickly forgotten in signal processing. +More recent work by Vapnik on support vector machines brought back a lot of interest to +RKHS theory for pattern recognition [12], where the RKHS is used primarily as a high- +dimensional feature space and the inner product is efficiently computed by means of the kernel +trick. A nonnegative definite kernel function (e.g., Gaussian, Laplacian, polynomial, and others +[13]) nonlinearly projects the data sample-by-sample into a high-dimensional RKHS. This +development was included in adaptive filtering, yielding the class of kernel adaptive filters (KAF) +[14], which allows the design of convex universal learning machines (CULMs) [15]. KAFs +estimate a functional model that approximates the MMSE solution using search techniques in the +RKHS defined by the Gaussian kernel [14], and the order grows linearly with the number of +samples, if no sparsification is considered. Another branch of RKHS theory important for this +paper is kernel Principal Component Analysis (KPCA) [16]. When the kernel function is infinite +dimensional as the Gaussian, denoted as 𝐺(𝑥𝑖,. ), the eigen decomposition of the empirical +covariance operator 𝐶 = 1 𝑁 ∑ +𝐺(𝑥𝑖, . )𝐺(𝑥𝑖,. )𝑇 +𝑁 +𝑖=1 +⁄ + needs to be truncated (we assume 𝐺(𝑥𝑖,. ) are +centered in the RKHS). In such cases, a more efficient approach uses only inner products of +functionals centered at the projected samples, which can be computed in the input space using the +reproducing property of the kernel (also called the kernel trick). The goal is to rewrite the eigen +decomposition of the empirical covariance operator 𝐶 through a functional eigenvalue equation as +𝐶𝑉 = 𝜆𝑉, where 𝑉 is the eigenfunction 𝑉 = 1 𝑁 ∑ +𝛼𝑖𝐺(𝑥𝑖,.) +𝑁 +𝑖=1 +⁄ + and 𝜆 is a vector of scalars that +correspond to the eigenvalues. For any nonzero 𝜆, the eigenfunction exists in the span of the RKHS +defined by the kernel. Since the number of samples is finite this methodology is very appealing +and efficient. However, the span of the functional space defined by the kernel is much larger than +the mappings of single mapped samples into the RKHS, which means that the inverse mapping of +RKHS functionals to the input space cannot be necessarily expressed as the image of a single input +pattern i.e., given a function 𝜁 in the RKHS span, there is no guarantee that there is exist a 𝑧 ∈ ℝ𝑁 +such that 𝐺(𝑧, . ) = 𝜁. This has been called the preimage problem [17]. We call 𝑧̂ an approximate +preimage of 𝜁 if ‖𝐺(𝑧, . ) − 𝜁‖2 is small, according to the application. We will see that this pre- +imaging will be important in our approach. +This paper takes Parzen’s work one step further, combining it with KAF concepts to yield +a RKHS defined by the covariance function of the projected data in a Gaussian RKHS, which is +nonlinearly related to the data space. More specifically, we define a data dependent kernel based +on the correntropy function [16] that incorporates full data statistics and defines a RKHS of +deterministic functions, even when the input data is a random variable (r.v.). Correntropy has been +heavily used for robust cost functions in adaptive signal processing [17], but here its functional +extension [16] will be employed as a methodology to solve the famous Wiener filter in the space +of nonlinear functions, without using the Wiener-Hopf spectral factorization. Previous attempts by +others e.g., the kernel Wiener filter [18], approximate the Wiener solution employing subspace +projections in RKHS. An early attempt to solve the Wiener-Hopf equations in RKHS was not +successful [19]. This paper shows how to pose the optimum filtering problem, derive a solution, +and present a methodology to implement the filter directly from samples, which effectively extends +MMSE for nonlinear universal approximators. The framework is named the functional Wiener +filter (FWF) and amazingly, it does not require the use of the error signal as in the traditional + +Wiener solution to adapt parameters. It takes advantage of the geometry of the RKHS and finds, +just like Least Squares, the orthogonal projection of the desired response in the space spanned by +the correntropy function, and in this sense, it is model agnostic. Preliminary results show that +performance is on par with KLMS but it is worse than KRLS. The major advantage is the simplicity +in implementation that is similar to the Wiener solution. + +Review of Linear Prediction of Continuous Time Series in RKHS + +A stochastic process 𝑋(𝑡, 𝜔) is broadly defined as a collection of random variables on a +measurable sample space (Ω, ℬΩ), indexed by a set 𝑇. Here, we restrict 𝑋(𝑡,𝜔) to random variables +taking values in ℝ, 𝑇 ⊂ ℝ, which we call a time-series, {𝑋(𝑡), 𝑡 ∈ 𝑇} and omit the dependence on +𝜔. For a time-series with finite second order moments, let 𝐿2(𝑋(𝑡), 𝑡 ∈ 𝑇) denote the space of all +real valued random variables spanned by the time series, that is, this space consists of all r.v. 𝑈 +that are either linear combinations of finite number of 𝑋(𝑡𝑖) or are limits of such linear +combinations. The time structure is quantified by the joint probability density 𝑝𝑡,𝑠(𝑥𝑡,𝑥𝑠) of the +pair of random variables 𝑋(𝑡),𝑋(𝑠) at two points in time 𝑡 and 𝑠. Assuming a strictly stationary +stochastic model for 𝑋(𝑡), the marginal density 𝑝𝑡(𝑥) is the same for any 𝑡. Normally, the joint +density 𝑝𝑡,𝑠(𝑥𝑡,𝑥𝑠) is quantified by its mean value, called the autocorrelation function. To simplify +notation, let us define the time autocorrelation of the finite second order moment time series as: + +𝑅(𝑠, 𝑡) = 𝐸[𝑋(𝑠)𝑋(𝑡)] +(1) + +This kernel on time sample pairs is positive semi definite, hence by Moore-Aronzsajn +theorem [5] it defines a RKHS space of functions on 𝑇 × 𝑇, denoted ℋ𝑅. Notice that the functions +in ℋ𝑅 are deterministic because of the 𝐸[. ] operator, while the inner product in ℋ𝑅 depends on +the statistics of the data through 𝑋(𝑡). + +For any r.v. 𝑈, 𝐴 ∈ 𝐿2(𝑋(𝑡),𝑡 ∈ 𝑇), define the inner product between the two as + +〈𝑈, 𝐴〉 = 𝐸[𝑈𝐴], +(2) + +and the norm of 𝑈 in 𝐿2(𝑋(𝑡), 𝑡 ∈ 𝑇) by the inner product 〈𝑈, 𝑈〉 = 𝐸[𝑈2]. Obviously, this inner +product coincides with the autocorrelation function if 𝑈 is 𝑋(𝑠) and 𝐴 is 𝑋(𝑡). However, notice +that 𝐿2(𝑋(𝑡), 𝑡 ∈ 𝑇) is not an RKHS. + +Explicit Expression for MMSE +One of the important problems in time series analysis is the representation of an +unobservable r.v. 𝑍. Let {𝑋(𝑡), 𝑡 ∈ 𝑇} be an observable time series assumed stationary. The goal +is to create a linear combination of the observable time series that has the smallest mean square +distance to 𝑍. By the Hilbert projection theorem, there is a unique minimum norm projection +between the abstract Hilbert space ℋ and any subspace 𝑀 of ℋ. Then, there exists a unique vector +𝐴∗ in 𝑀, given by 𝐴∗ = 𝐸∗[𝐴|𝑀], which projects orthogonally a vector 𝐴 in ℋ to 𝑀. For a family +of vectors {𝑋(𝑡), 𝑡 ∈ 𝑇} the projection becomes + +𝐴∗ = 𝐸∗[𝐴|𝑋(𝑡), 𝑡 ∈ 𝑇]. +(3) + + +Then with 𝐴 = 𝑍, the optimum linear predictor is the unique r.v. in 𝐿2(𝑋(𝑡), 𝑡 ∈ 𝑇) that satisfies + +𝐸[𝐸∗[𝑍|𝑋(𝑡),𝑡 ∈ 𝑇]𝑋(𝑠)] = 𝐸[𝑍𝑋(𝑠)] +(4) + +This result gives immediately rise to the famous Wiener equation. Indeed, if T is a finite interval +𝑇 = {𝑡: 𝑎 ≤ 𝑡 ≤ 𝑏} and w(t) a weighting function in 𝐿2, the integral +∫ 𝑋(𝑡)𝑤(𝑡)𝑑𝑡 +𝑏 +𝑎 + +(5) + +represents a r.v. in 𝐿2(𝑋(𝑡), 𝑡 ∈ 𝑇), then the weighting function of the best linear predictor can be +written as + +𝐸∗[𝑍|𝑋(𝑡), 𝑡 ∈ 𝑇] = ∫ 𝑋(𝑡)𝑤∗(𝑡)𝑑𝑡 +𝑏 +𝑎 + + +(6) + +and must satisfy the generalized Wiener equation: + +∫ 𝑤∗(𝑡)𝑅(𝑠, 𝑡)𝑑𝑡 = 𝜌𝑧(𝑠) +𝑏 +𝑎 + (7) + +in 𝑎 ≤ 𝑠 ≤ 𝑏, with 𝑅(𝑠, 𝑡) = 𝐸[𝑋(𝑠), 𝑋(𝑡)], 𝜌𝑧(𝑠) = 𝐸[𝑍𝑋(𝑠)]. + +This equation states that one can always find a representation for the function 𝜌𝑧(𝑠) in terms of +the functions {𝑅(𝑠, 𝑡), 𝑡 ∈ 𝑇} such that the minimum mean square error linear predictor +𝐸∗[𝑍|𝑋(𝑡), 𝑡 ∈ 𝑇] can be written in terms of the corresponding linear operator on the time series +{𝑋(𝑡), 𝑡 ∈ 𝑇}. + +Hilbert Space Representation of Time Series +First, let us state an important theorem that is very important for this line of work [4]. + +Basic Congruence Theorem. Let ℋ1 and ℋ2 be two abstract Hilbert spaces. Let 𝑇 be an index set +and let {𝑢𝑡,𝑡 ∈ 𝑇} be a family of vectors spanning ℋ1, and similarly {𝑎𝑡,𝑡 ∈ 𝑇} a family of vectors +spanning ℋ2. Suppose that for every 𝑠, 𝑡 in 𝑇, + +〈𝑢𝑠, 𝑢𝑡〉 ℋ1 = 〈𝑎𝑠, 𝑎𝑡〉 ℋ2 +(8) + +then there is a congruence (a one-to-one inner product preserving linear mapping) 𝜓 from ℋ1 to +ℋ2 such that 𝜓(𝑢𝑡) = 𝑎𝑡 for any 𝑡 ∈ 𝑇. + +Definition: A family of vectors {𝑢𝑡,𝑡 ∈ 𝑇} in a Hilbert space ℋ𝑅 is a representation of a wide sense +stationary time series {𝑋(𝑡), 𝑡 ∈ 𝑇} if for every s, t in T + +〈𝑢𝑠, 𝑢𝑡〉ℋ𝑅 = 𝑅(𝑠,𝑡) = 𝐸[𝑋(𝑠), 𝑋(𝑡)] +(9) + +Then there is a congruence 𝜓 between the Hilbert space spanned by {𝑢𝑡,𝑡 ∈ 𝑇} and +denoted as 𝐿2(𝑢𝑡,𝑡 ∈ 𝑇), onto 𝐿2(𝑋(𝑡), 𝑡 ∈ 𝑇) satisfying 𝜓(𝑢𝑡) = 𝑋(𝑡), and every r.v. 𝑈 in +𝐿2(𝑋(𝑡), 𝑡 ∈ 𝑇) may be written 𝑈 = 𝜓(𝑔) for some unique vector 𝑔 in 𝐿2(𝑢𝑡, 𝑡 ∈ 𝑇). + + +The natural representation of a time series is obtained in the RKHS ℋ𝑅 i.e., a Hilbert space +where the kernel has two properties: +𝑅(⋅,𝑡) ∈ ℋ𝑅 +〈𝑔, 𝑅(⋅, 𝑡)〉ℋ𝑅 = 𝑔(𝑡) +(10) + +This result is the well-known Riez representation theorem, which yields for our +discussion + +𝑅(𝑠, 𝑡) = 〈𝑅(⋅,𝑠), 𝑅(⋅,𝑡)〉ℋ𝑅 = 𝐸[𝑋(𝑠), 𝑋(𝑡)] + +(11) + +It can be further shown that for any time series {𝑋(𝑡), 𝑡 ∈ 𝑇} with covariance kernel 𝑅, the family +of functions {𝑅(⋅, 𝑡),𝑡 ∈ 𝑇} in ℋ𝑅 is a representation of 𝐿2(𝑋(𝑡), 𝑡 ∈ 𝑇). Indeed, for any two +vectors 𝑈,𝐴 ∈ 𝐿2(𝑋(𝑡), 𝑡 ∈ 𝑇) such that the congruence is denoted by 𝑈 = 𝜓(𝑔) and 𝐴 = 𝜓(ℎ), +and 𝐴 = 𝜓(ℎ) = 〈𝑋, ℎ〉ℋ𝑅, + +〈𝑋, 𝑅(⋅,𝑡)〉ℋ𝑅 = 𝑋(𝑡) +𝐸[〈𝑋, ℎ〉ℋ𝑅〈𝑋, 𝑔〉ℋ𝑅] = 〈ℎ, 𝑔〉ℋ𝑅 + + + + +It is easy to see that if the two vectors ℎ, 𝑔 ∈ ℋ𝑅 correspond to random variables +𝑈, 𝐴 ∈ 𝐿2(𝑋(𝑡), 𝑡 ∈ 𝑇) +〈ℎ, 𝑔〉 ℋ𝑅 = ∫ +∫ +ℎ(𝑠)𝑅−1(𝑠, 𝑡)𝑔(𝑡)𝑑𝑠 + +𝑠∈𝑇 +𝑑𝑡, + +𝑡∈𝑇 + +where 𝑅−1(𝑠, 𝑡) is the kernel of the inverse of the covariance operator 𝑅𝑔 = ∫ +𝑔(𝑡)𝑅(𝑠, 𝑡)𝑑𝑡 + +𝑡∈𝑇 +. +Moreover, if 𝑈 = ∑ +𝑤𝑔(𝑡𝑖)𝑋(𝑡𝑖) +𝑁𝑔 +𝑖=1 + and 𝐴 = ∑ +𝑤ℎ(𝑠𝑗)𝑋(𝑠𝑗) +𝑁ℎ +𝑗=1 +, their inner product in the RKHS +can be computed in the input space from vectors {𝑤ℎ(𝑠𝑗)}𝑗=1 +𝑁ℎ and {𝑤𝑔(𝑡𝑖)}𝑖=1 +𝑁𝑔 (what is now called +the kernel trick) as + +〈ℎ, 𝑔〉 ℋ𝑅 = ∑ +∑ +𝑤ℎ(𝑠𝑗)𝑅(𝑠𝑗,𝑡𝑖) +𝑁𝑔 +𝑖=1 +𝑁ℎ +𝑗=1 +𝑤𝑔(𝑡𝑖) = ∑ +∑ +ℎ(𝑠𝑗)𝑟𝑠𝑗,𝑡𝑖 +−1 +𝑁𝑔 +𝑖=1 +𝑁ℎ +𝑗=1 +𝑔(𝑡𝑖) (13) + +where 𝑟𝑠𝑗,𝑡𝑖 +−1 is the 𝑠𝑗, 𝑡𝑖 element of the inverse of the covariance kernel 𝑅(𝑠𝑖, 𝑡𝑖) i.e., the kernel +modifies the traditional inner product of vectors in the input space. This explains the nature of +ℋ𝑅 quite well: because of the mapping 𝑅(𝑠,. ), which contains the statistics of the data, the inner +product in ℋ𝑅 takes advantage of the data statistics over time instances. Hence, in the input space +the solution must be a quadratic form employing 𝑅−1 as shown in (13) to meet the congruence. + +Theorem (from [4]): Let {𝑋(𝑡),𝑡 ∈ 𝑇} be a time series with covariance kernel 𝑅(𝑠, 𝑡), and let ℋ𝑅 +be the corresponding RKHS. Between 𝐿2(𝑋(𝑡), 𝑡 ∈ 𝑇) and ℋ𝑅 there exists a one-to-one inner +product preserving linear mapping under which a vector ℎ ∈ {𝑅(⋅,𝑡), 𝑡 ∈ 𝑇} and 𝑈 ∈ 𝐿2(𝑋(𝑡), 𝑡 ∈ +𝑇) are mapped into one another. Denote by 〈ℎ, 𝑋〉ℋ𝑅 the r.v. in 𝐿2(𝑥𝑡,𝑡 ∈ 𝑇) which corresponds +to the function ℎ ∈ ℋ𝑅 under the mapping. Then the solution of the prediction problem may be +written as follows. If 𝑍 is a r.v. with finite second moments and 𝜌𝑍(𝑡) = 𝐸[𝑍𝑋(𝑡)] then +𝜌𝑍 ∈ ℋ𝑅, and + (14) + + +𝐸∗[𝑍|𝑋(𝑡), 𝑡 ∈ 𝑇] = 〈𝜌𝑧,𝑋〉ℋ𝑅 + +(15) + +with prediction mean square error given by + +𝐸[(𝑍 − 𝐸∗[𝑍 |𝑋(𝑡), 𝑡 ∈ 𝑇])2] = 𝐸[𝑍2] − 〈𝜌𝑧, 𝜌𝑧〉ℋ𝑅 +(16) + +The equivalent minimum mean square error solution (15) in the data space, because of +(13), becomes + +𝑌 = 𝐸∗[𝑍|𝑋(𝑡),𝑡 ∈ 𝑇] = 〈𝜌𝑧, 𝑋〉ℋ𝑅 = ∫ +∫ +𝑅−1(𝑠, 𝑡)𝜌𝑍(𝑠)𝑋(𝑡)𝑑𝑠 + +𝑠∈𝑇 +𝑑𝑡 + +𝑡∈𝑇 + (17) + +which is exactly the Wiener solution ∫ +𝑋(𝑡)𝑤∗(𝑡)𝑑𝑡 + +𝑡∈𝑇 +. Note that the effective role of this inverse +operator is to decorrelate the input space data and it is a steppingstone for finding the orthogonal +projection as demonstrated by Wiener. However, in ℋ𝑅 this solution for the prediction problem is +coordinate free, does not use the approximation error, and directly uses the structure of ℋ𝑅. In +fact, it is sufficient to compute the linear projection of 𝜌𝑧(𝑠) with the input data because the +covariance kernel 𝑅(𝑠, 𝑡) provides its statistics, unlike Wiener-Hopf method that requires spectral +factorization. This coordinate free property of RKHS solutions with the covariance kernel was first +noted by Loeve [20] who suggested that instead of finding a set of functional projections (e.g. +Karuhnen Loeve transform [21]) it is sufficient to employ the statistics of 𝑋(𝑡) embedded in the +structure of the RKHS. Parzen [4] further states that for this reason “RKHS defined by the +covariance kernel is the natural setting in which to solve problems of statistical inference on time +series”. These are fundamental results that will be very useful when seeking an extension of the +theory to nonlinear solutions. +The fundamental issue with Parzen approach is twofold: first, it does not elucidate efficient +alternatives to implement the conditional mean operator. Moreover, from (13) we can see that the +inverse may not always exist, needs to be accurate, and it is computationally expensive because it +needs to be applied to every test sample. Despite approximations for the inverse, this is +cumbersome but a necessity for continuous time models. Second, for discrete time signal +processing, this approach is computationally not competitive with the famous Wiener solution +𝑤∗ = 𝑅−1𝜌, where 𝑅 is the autocorrelation matrix (the kernel 𝑅(𝑠, 𝑡) evaluated at a finite set of +times), which finds the optimal weighting 𝑤∗ only once in the training set using the error, and does +an inner product in the data space of two vectors in the test set. Hence, we conclude that the +advantage of the RKHS theory is on the mathematical tools of congruence and representation of +time series in RKHS, which open the door to seek more general solutions such as the nonlinear +prediction case. In fact, the advantage of the RKHS theory is that the operations defined in the +RKHS are independent of the kernel utilized, hence the key goal is to concentrate on designing +proper kernels when the goal is nonlinear extensions. + +The Nonlinear Prediction Case +A. Kernel Adaptive Filtering +The goal is to construct a function 𝑓: ℝ𝐿 → ℝ based on a real sequence {(𝒙𝑖,𝑑𝑖)}𝑖=1 +𝑁 of +examples (𝑥𝑖,𝑑𝑖) ∈ 𝑆 × 𝐷, where 𝐷 is a compact subset of ℝ and 𝑆 a compact subspace of ℝ𝐿. +As described below, the function 𝑓 ∈ ℋ𝑘 is obtained based on a positive definite kernel 𝜅: 𝑆 × 𝑆 → +ℝ that defines a RKHS ℋ𝑘. A commonly employed kernel is the Gaussian kernel 𝐺(𝒙, 𝒙𝑖) = + +exp (− +‖𝒙 −𝒙𝑖‖2 +2𝜎2 +), where 𝜎 is the kernel size or bandwidth. Kernel adaptive filtering (KAF) [14] +implements nonlinear filtering on discrete time series by mapping the input sampled data {𝒙𝑖}𝑖=1 +𝑁 +to ℋ𝐺 using a positive definite kernel 𝐺, and using search techniques based on the gradient and or +Hessian information to adapt functional parameters. +The Gaussian kernel maps each embedding vector 𝒙𝑖 of size 𝐿, to a function in ℋ𝐺, which +we will also denote as 𝐺(𝒙𝑖,⋅), where the “⋅” in the second argument means that a data point is +represented by a Gaussian function centered at 𝒙𝑖. The inner product in the RKHS of two such +functions centered at 𝒙𝑖 and 𝒙𝑗 can be easily computed in the input space as a Gaussian kernel +evaluation i.e., 〈𝐺(𝒙𝑖,⋅),𝐺(𝒙𝑗,⋅)〉ℋ𝐺 = 𝐺(𝒙𝑖,𝒙𝑗). The ℋ𝐺 defined by the Gaussian is infinite +dimensional and nonlinearly related to the input data space 𝑆 [22]. For the case of samples from a +stochastic process {𝑋(𝑡), 𝑡 ∈ 𝑇}, 𝐺(𝑋(𝑡),⋅) is a random function. One notable example of KAF is +the kernel least mean square (KLMS) algorithm, for which the non-linear filter output is simply +given by + +𝑦𝑛 = ∑ +𝜂𝑒𝑖𝐺(𝒙𝑛 − 𝒙𝑖) +𝑛−1 +𝑖=1 + + +(18) + +where  is the stepsize, 𝑒𝑖 is the error at iteration 𝑖, and {𝒙𝑖}𝑖=1 +𝑛−1 are the past samples in the training +set that constitute the “dictionary” to construct the output. This algorithm uses gradient search to +construct the optimal function Ω∗, such that 𝑓∗(𝒙) = 〈𝐺(𝒙,⋅),Ω∗〉ℋ𝐺, and converges in the mean +to the optimal least minimum square solution in ℋ𝐺 for small step sizes and large number of data +samples. The appeal of the KLMS is that it is an online algorithm, does not need explicit +regularization [23], and is a CULM (convex and universal learning machine) [15]. However, +because of the nonlinearity of the kernel mapping there is no congruence between the input space +defined by the span of the time series and the RKHS ℋ𝐺. The solution needs to be expressed in +terms of observations from the time series, which means that the order of the filter grows linearly +in time, if no sparsification is included [14]. This is a shortcoming of this class of algorithms +because it affects the computation complexity in the test set. In KAF, since the kernel evaluations +are weighted by the error, the algorithm has an automatic way to preserve the scale of the +representations when applying the kernel trick. +The ℋ𝐺 defined by the Gaussian kernel differs from the ℋ𝑅 defined by Parzen’s covariance +kernel in four fundamental ways. +• First, Parzen used a “linear” kernel ℋ𝑅 yielding a close form optimal linear model in 𝐿2 as +mentioned above. +• Second, the Parzen kernel is computed by employing the expected value over data lags 𝑠 = +𝑡 − 𝜏 to take advantage of the wide sense stationarity of the time series, unlike the pairwise +sample set as ℋ𝐺. +• Third, the map to ℋ𝐺 is stochastic because samples are mapped from a random process rather +than mapping the elements of the index set 𝑇, directly. In contrast, the map to ℋ𝑅 is +deterministic because of the congruence. +• Fourth, ℋ𝐺 is infinite dimensional, while in ℋ𝑅 is a finite dimensional RKHS space defined +by the number of lags required for the covariance kernel, which is dictated by the input data +dynamics (normally small). +Our goal now is to define a new RKHS that preserves the correlation structure defined by +the data as ℋ𝑅, but also maps the time series by a nonlinear kernel to achieve CULM properties. +To be practical, this approach uses the kernel trick to perform the computation in the input space. + + +B. Definition of the Correntropy RKHS + +Let {𝑋(𝑡),𝑡 ∈ 𝑇} be a strictly stationary stochastic process (i.e., the joint PDF {𝑝𝑠,𝑡(𝑥𝑠,𝑥𝑡) } is +unaffected by a change of the time origin, that is 𝑝𝑠,𝑡(𝑥𝑠,𝑥𝑡) = 𝑝𝑠−𝜏,𝑡−𝜏(𝑥𝑠,𝑥𝑡) ) with T being an +index set and 𝑥𝑡 ∈ ℝ𝐿. The autocorrentropy function 𝑣(𝑠, 𝑡) is defined as a function from 𝑇 × 𝑇 → +ℝ given by +𝑣𝜎(𝑠,𝑡) = 𝐸𝑠,𝑡[𝐺𝜎(𝑋(𝑠), 𝑋(𝑡))] = ∬ 𝐺𝜎(𝑥𝑠,𝑥𝑡)𝑝𝑠,𝑡(𝑥𝑠,𝑥𝑡)𝑑𝑥𝑠𝑑𝑥𝑡 (19) +where 𝐸𝑠,𝑡[⋅] denotes mathematical expectation over a pair of r.v. in the time series {𝑋(𝑡),𝑡 ∈ 𝑇} . +While it is true that any symmetric positive definite kernel (i.e., Mercer kernel) 𝜅(𝑥𝑠,𝑥𝑡) can be +employed instead of the Gaussian kernel 𝐺𝜎, the symmetry, scaling, and translation invariant +properties of 𝐺𝜎, confer additional properties and interpretation to correntropy, which are reviewed +in the appendix. The autocorrentropy function defined in (19) is a reproducing kernel on the index +set 𝑇 × 𝑇. We will denote its corresponding RKHS by ℋ𝑣. The functions 𝑣𝜎(𝑠,⋅) are in ℋ𝑣 and +𝑣𝜎(𝑠,𝑡) = 〈𝑣𝜎(𝑠,⋅),𝑣𝜎(𝑡,⋅) 〉ℋ𝑣. + +Another space that can be defined by the composition of the random variable 𝑋(𝑡) and the positive +definite Gaussian kernel 𝐺𝜎(⋅,⋅) is the span of the set of random elements {𝐺𝜎(𝑋(𝑡),⋅),𝑡 ∈ 𝑇} +taking values in ℋ𝐺. We will denote this space by ℋ𝑅𝐺 and the inner product between two elements +𝑈 = ∑ 𝛼𝑖𝐺𝜎(𝑋(𝑡𝑖),⋅) +𝑖 + and 𝐴 = ∑ 𝛽𝑗𝐺𝜎(𝑋(𝑠𝑗),⋅) +𝑗 + is given by + + ⟨𝑈, 𝐴⟩ℋ𝑅𝐺 = 𝐸[〈∑ 𝛼𝑖𝐺𝜎(𝑋(𝑡𝑖),⋅) +𝑖 +, ∑ 𝛽𝑗𝐺𝜎(𝑋(𝑠𝑗),⋅) +𝑗 +〉ℋ𝑅𝐺] = ∑ 𝛼𝑖𝛽𝑗𝐸[𝐺𝜎(𝑋(𝑡𝑖), 𝑋(𝑠𝑗))] +𝑖𝑗 +. + +There is a congruence between ℋ𝑅𝐺 and ℋ𝑣. Moreover, we see that for strictly stationary time +series making 𝑠 = 𝑡 − 𝜏, the function 𝑣𝜎 can also be written as a function of 𝜏 only as follows: + +𝑣𝜎(𝜏) = 𝐸𝑡,𝑡−𝜏[𝐺𝜎(𝑋(𝑡), 𝑋(𝑡 − 𝜏))], (20) + +where any 𝑡 ∈ 𝑇can be used. This shows its similarity with the Parzen covariance kernel of (11), +except that 𝑣𝜎(𝜏) is computed in ℋ𝑣, a space nonlinearly related to the original time series. +The autocorrentropy functional can then be interpreted in two vastly different feature +spaces. One is the RKHS ℋ𝐺 induced by the Gaussian kernel on pairs of observations 𝐺𝜎(⋅,⋅), which +is widely used in kernel learning. The elements of this RKHS are infinite-dimensional vectors +expressed by the eigenfunctions of the Gaussian kernel, and they lie on the positive hyperoctant of +a sphere because ‖𝐺𝜎(𝑥, . )‖2 = 𝐺𝜎(0) = 1/√2𝜋𝜎. The correntropy functional performs statistical +averages on the functionals in this sphere. +The second feature space is the RKHS ℋ𝑣 induced by the correntropy kernel 𝑣(𝑠, 𝑡), which +is defined on the index set of the random variables in the time series. This inner product is defined +by the correlation of the kernel at two different lags and the mapping produces a single +deterministic scalar for each element on the index set, that is, the practical dimension of ℋ𝑣 is the +size of the index set. ℋ𝑣 has very nice properties for statistical signal processing: +• +ℋ𝑣 provides a straightforward way to apply optimal projection algorithms based on mean +statistical embeddings that are expressed by inner products. +• +The effective dimension of ℋ𝑣 is under the control of the designer by selecting the number + +of lags (just like with the RKHS defined by the autocorrelation function). +• +Elements of ℋ𝑣 can be readily manipulated algebraically for statistical inference (i.e. +without taking averages over realizations). +• +ℋ𝑣 is nonlinearly related to the input space, unlike the RKHS defined by the autocorrelation +of the random process. Therefore, it is in principle very appealing for nonlinear statistical +signal processing. + +The table presents the different types of RKHS defined so far that summarize our approach. +Table I +RKHS +Functional Mapping +Hilbert Space Characteristics +ℋ𝑅 Parzen + 𝐸[𝑋(𝑡),. ] +Linear mapping of data, size of lags, +deterministic functions +ℋ𝐺 Gaussian +𝐺(𝑥,⋅) +Nonlinear mappings of data, infinite +dimensional, random functions +ℋ𝑅𝐺 Random Gaussian +𝐺(𝑋(𝑡),⋅) +Nonlinear mapping of data, size of lags, +random functions +ℋ𝑣 Correntropy +𝑣𝜎(𝑡,⋅) +Nonlinear mapping of data, size of lags, +deterministic functions + +Representing an Unobservable Random Variable in ℋ𝑅𝐺 + +Like the original problem where the random variable 𝑍 was approximated by a random variable in +the span of the time series {𝑋(𝑡),𝑡 ∈ 𝑇} by the Hilbert projection theorem, we can define the +approximation in the space of random elements ℋ𝐺 as follows + +𝜉∗ = argmin +𝜉 +𝐸[‖𝐺(𝑍,⋅) − 𝜉‖ℋ𝐺 +2 ], + (21) + +where 𝜉 is a random element in the span of {𝐺(𝑋(𝑡),⋅),𝑡 ∈ 𝑇}. Solving for 𝜉 gives rise to the +following equation: + +𝐸[〈𝐺𝜎(𝑍,⋅), 𝐺𝜎(𝑋(𝑠),⋅)〉ℋ𝐺] = 𝐸[〈𝜉, 𝐺𝜎(𝑋(𝑠),⋅)〉ℋ𝐺], +(22) + +where 𝜉 is expressed as a linear combination of elements in ℋ𝑅𝐺, + +𝜉 = ∫ 𝐺𝜎(𝑋(𝑡),⋅)𝑤(𝑡)𝑑𝑡 + +𝑇 +, +(23) + +Then the weighting function 𝑤∗ of the best predictor must satisfy: + +𝐸[∫ 𝑤∗(𝑡)〈𝐺𝜎(𝑋(𝑡),⋅),𝐺𝜎(𝑋(𝑠),⋅)〉ℋ𝐺𝑑𝑡 + +𝑇 +] = 𝐸[〈𝐺𝜎(𝑍,⋅),𝐺𝜎(𝑋(𝑠),⋅)〉ℋ𝐺], + +which gives rise to the functional Wiener equation: + +∫ 𝑤∗(𝑡)𝑣𝜎(𝑡,𝑠)𝑑𝑡 + +𝑇 += 𝐸[〈𝐺𝜎(𝑍,⋅), 𝐺𝜎(𝑋(𝑠),⋅)〉ℋ𝐺] = 𝜌𝑍(𝑠), (24) + + +These equations state that one can always find a representation for the function 𝜌𝑧(𝑠) in terms of +the functions {𝑣𝜎(𝑡,⋅),𝑡 ∈ 𝑇} because the best correntropy predictor is computed in the span of the +set {𝐺𝜎(𝑋(𝑡),⋅), 𝑡 ∈ 𝑇}. Nevertheless, because this computation is carried out in the correntropy +RKHS, the best approximation to 𝑍 cannot be directly obtained since the input space where the +time series lies is nonlinearly related to the correntropy RKHS where we compute the projection. + +Solution of the Representation Problem in ℋ𝑣 +To solve the representation problem in ℋ𝑣, that is, finding 𝑤∗(𝑡), let us consider the +representation 𝜁𝑠 in ℋ𝑣 of the random element 𝐺𝜎(𝑋(𝑠),⋅) that can be obtained by the congruence +between ℋ𝑣 and ℋ𝐺. From equation (24) we have that: + +𝜌𝑧(𝑠) = 〈𝜌𝑧, 𝜁𝑠〉ℋ𝑣 = ∫ 𝑤∗(𝑡)〈𝜁𝑡, 𝜁𝑠〉ℋ𝑣𝑑𝑡 + +𝑇 + + + +(25) +This defines a close form functional Wiener filter solution in ℋ𝑣. Notice that the formulation is +the same as (17), the only difference is the structure of the inner product space. +The relation between ℋ𝑅𝐺 and ℋ𝑣 is rather similar to the relation between ℝ𝐿 and ℋ𝑅 so, for two +elements ℎ and 𝑔 in ℋ𝑣, + +〈ℎ, 𝑔〉ℋ𝑣 = ∫ +∫ +ℎ(𝑠)𝑣𝜎 +−1(𝑠, 𝑡)𝑔(𝑡)𝑑𝑠 + +𝑠∈𝑇 +𝑑𝑡, + +𝑡∈𝑇 + (26) + +where 𝑣𝜎 +−1(𝑠, 𝑡) is the element of the inverse of the correntropy operator defined as (𝑉𝜎𝑔)(𝑠) = + ∫ +𝑔(𝑡)𝑣𝜎(𝑠, 𝑡)𝑑𝑡 + +𝑡∈𝑇 +. The above form can be used to compute a solution to (25) as, + +𝑤∗(𝑡) = ∫ +𝜌𝑧(𝑠)𝑣𝜎 +−1(𝑠, 𝑡)𝑑𝑠 + +𝑠∈𝑇 +. (27) + +In this case the solution is nonlinear in the input space, so this is a very elegant extension of +Wiener theory. A major difference to KAF and the Wiener filter in the data space, is that this +solution never uses the error. The reason is that Parzen’s solution decorrelates implicitly the data +(in this case in ℋ𝑅𝐺) and automatically finds the orthogonal projection on the data manifold. +However, not everything is perfect with the solution (26), since we cannot extend the +congruence in (25) to the original time series {𝑋(𝑡), 𝑡 ∈ 𝑇}, i.e. + +〈𝜁𝑡, 𝜁𝑠〉 ℋ𝑣 = 𝐸[𝐺𝜎(𝑋(𝑡),𝑋(𝑠))] ≠ 𝐸[𝑋(𝑡)𝑋(𝑠)] (28) + +because the kernel mapping does not preserve the inner product, i.e. 〈𝑥𝑛,𝑥𝑖〉 ≠ +〈𝐺(𝑥𝑛,. ), 𝐺(𝑥𝑖,. )〉 ℋ𝐺. + +C. Computation of the Functional Wiener Filter in ℋ𝐺 + +How can the solution in (26) be implemented from a sample data stream? In this case, we +restrict our treatment to discrete-time time series. Let us start by assuming that the time series is +ergodic, such that expected values can be estimated by temporal averages. Second, because of the +congruence (25), 〈𝜁𝑡, 𝜁𝑡−𝜏〉ℋ𝑣 can be substituted by 𝐸[𝐺𝜎(𝑋(𝑡), 𝑋(𝑡 − 𝜏))] and by ergodicity, it +can be estimated from samples {𝑥(𝑡)}𝑡=1 +𝑁 over a window of length 𝑁. + + + +𝑣𝜏 = +1 +𝑁 ∑ +𝐺𝜎(𝑥(𝑡), 𝑥(𝑡 − 𝜏)) +𝑁 +𝑡=1 + +(29) + +For 𝜏 = 0,1,⋯ , 𝐿 − 1, 𝑣𝜏 is the 𝜏th entry of the autocorrentropy vector and can be used to +construct the autocorrentropy matrix of size 𝐿 × 𝐿 as follows: + +𝑉 = [ +𝑣0 +⋯ +𝑣𝑇−1 +⋮ +⋱ +⋮ +𝑣𝑇−1 +⋯ +𝑣0 +] +(30) + +This matrix is unlike anything in kernel adaptive filtering, because it is a matrix of scalar +values that can be computed once from the training set and never changed. This matrix is very +unusual in kernel filtering, where the filters always increase in size with each new sample. The +values of the correntropy matrix can be centered in RKHS if necessary [28]: + +𝑣̅𝜏 = 𝑣𝜏 − +1 +𝑁2 ∑ +∑ +𝐺𝜎(𝑥(𝑡), 𝑥(𝑠)) +𝑁 +𝑠=1 +𝑁 +𝑡=1 + +(31) + +The other major difference is that in KAF, one needs to transfer vectors of samples to the +RKHS, where the size of the vector is an estimate of the embedding dimension of the system that +created the time series, using Takens’ embedding theory. The reason is that the KLMS is a pairwise +instantaneous algorithm, so if it is applied to each sample of the input data the algorithm loses the +local time structure of the signal. For FWF, the data can be mapped to RKHS sample by sample, +just like in the input space, because the formulation uses the correntropy matrix where the lag +structure is included. +Let us now show how to estimate the cross correlation functional 𝜌𝑧 in ℋ𝑣. Using the same +approximations as the ones for the correntropy matrix yields + +𝜌̂𝑧(𝜏) = +1 +𝑁 ∑ +𝐺𝜎(𝑥(𝑡 − 𝜏), 𝑧(𝑡)) +𝑁 +𝑡=1 + +(32) + +This is the only term that relates the target and the input signals, and it only needs to be evaluated +in the training set. The optimal weighting vector in (27), 𝑤∗ (𝜏) for 𝜏 = 0,2, … , 𝐿 − 1, is obtained +by solving the system: + +𝜌𝑧(ℓ) = ∑ +𝑉ℓ+1,𝜏+1 +𝐿−1 +𝜏=0 +𝑤(𝜏). +(33) + +In other words, 𝑤∗ = 𝑉−1𝜌𝑧 . During testing, the output of the filter corresponds to an instance +of the random element ∑ +𝑤∗ (𝜏)𝐺𝜎(𝑋(𝑡 − 𝜏),⋅) +𝐿−1 +𝜏=0 +, which is the best approximation to 𝐺𝜎(𝑍,⋅), +namely, + +𝜉∗ (𝑡) = ∑ +𝑤∗(𝜏)𝐺𝜎(𝑥test(𝑡 − 𝜏),⋅) +𝐿−1 +𝜏=0 +. + +(34) + +where 𝑥test(𝑡) is the test input at time 𝑡. This solution shares the form of (6) in 𝐿2(𝑋(𝑡), 𝑡 ∈ 𝑇) +and (23) in ℋ𝑅𝐺.The big difference is that the autocorrelation function was substituted by the +correntropy function, while the input vector [𝑥(𝑡),𝑥(𝑡 − 1),⋯ , 𝑥(𝑡 − 𝐿 + 1)] was substituted by + +a vector of functions nonlinearly related to the input space (the feature space defined by the +Gaussian kernel). +Notice that this solution is quite different from the KAF in several important ways. First, the +optimal weight vector can be computed in the input space, and it appears as a scale factor to change +the finite range of the Gaussian to span the values of the target response. Notice that this weighting +depends on the actual local L sample history of the current input, but it is nonlinear and so it is +more powerful than the linear weighting in linear Wiener filters. Second, there is no sum over the +training set samples in the optimal solution like in KAFs. The best approximant is a combination +of just L Gaussian functions centered at the current test sample, which is a major simplification in +computation when compared with KAF. This algorithm has the complexity of the Wiener solution, +and should be an universal approximator when the number of delays grows to infinity, but we have +not formally proved this statement. Unfortunately, the output of the functional Wiener filter 𝜉∗ (𝑡) +is still in ℋ𝐺, so the task of implementing a filter in the data space is still not finalized. + + +D. Preimage to Estimate the FWF output in the input space + +Ideally, the output of the FWF in the input space would correspond to the inverse map from +ℋ𝐺 to ℝ𝑑, where 𝑑 = 1 in the simplest. Since (34) expresses the optimal filter solution as a linear +combination of Gaussian function, the goal is just to evaluate the function at a point in the input +space, whose image is closest to the optimal solution. However, there is no guarantee such inverse +map exists, so we must resort to an extra optimization or approximation step to find a pre-image +[17] of the optimal solution in the input space, as will be explained next. + +D.1. Preimage using the optimal filter output in 𝓗𝑮 + +For the FWF, the basic concept is to use an approximate pre-image in the input space of +the optimal filter output in ℋ𝐺 i.e., the approximated FWF output to 𝑦∗(𝑡) will be given by: + +𝑦(𝑡) = argmin +𝑦 ∈ ℝ𝑑 ‖𝐺𝜎(𝑦,⋅) − 𝜉∗ (𝑡)‖ℋ𝐺 +2 + +(35) + +This formulation can be applied in practical settings because in a training set, the optimal +weight vector can be estimated using the 𝑉 matrix from (33) and the cross correntropy from (32). +Therefore, and according to (35) it is only required to find the point to evaluate the optimal weight +function, which is equivalent to find the minimum of + + ∑ +𝑤∗(𝜏)𝐺𝜎(𝑥test(𝑡 − 𝜏), 𝑦) +𝐿−1 +𝜏=0 +. + +(36) + +Making the gradient of (36) with respect to 𝑦 equal to zero yields the fixed-point expression + + +𝑦(𝑖+1) = +∑ +𝑤∗(𝜏)𝐺𝜎(𝑥test(𝑡−𝜏),𝑦(𝑖))𝑥test(𝑡−𝜏) +𝐿−1 +𝜏=0 +∑ +𝑤∗(𝜏)𝐺𝜎(𝑥test(𝑡−𝜏),𝑦(𝑖)) +𝐿−1 +𝜏=0 +, (37) + + +where 𝑦(𝑖) denotes the estimate of the preimage at the 𝑖th iteration of the fixed-point update. Notice +that the nature of the pre-imaging solution involves a search on top of the analytic solution. This +solution will be named FWFFP. + +D.2. Preimage using local models + +Intuitively, the goal is to select training set input samples that, when combined with the +current test sample, provide functional evaluations in RKHS that approximate the targets in the +training set. The difficulty is that during testing there is no information about the target value. +Therefore, one simple option is to use the similarity in the input space to cluster locally the input +samples that provide the best approximation to the target signal during training. This approach was +inspired by [29], where a successful table lookup approach was employed to extend linear model +performance that links input samples to their errors in the training set to create outputs outside the +span of the input space. +Here, the approach is to find an input sample 𝑥(𝑚) that when combined with the current +input 𝑥(𝑖), will produce an output in ℋ𝐺 that is close to its target 𝑧(𝑖). Let us represent 𝑧̂(𝑖) = +∑ +𝑤∗(𝜏)𝐺𝜎(𝑥 (𝑖 − 𝜏), 𝑥(𝑚 − 𝜏)). +𝐿−1 +𝜏=0 + The optimization can be written as + +𝑎𝑟𝑔 min +𝑥(𝑚)∈𝑆 ‖𝑧(𝑖) − 𝑧̂(𝑖)‖ +(38) + +where S is the training set. So, we need to implement a search (done once), where we find the +sample pair (𝑥(𝑖),𝑥(𝑚)), 𝑖 = 1,… 𝑁 that produces the closest approximation to the target sample +𝑧(𝑖). Once in testing, we find the closest sample 𝑥(𝑖) in the training set to 𝑥(𝑡𝑒𝑠𝑡) and use its +neighbor 𝑥(𝑚) to plug in (34) to obtain the FWF output as + +𝑦(𝑡) = +𝑧𝑖 +𝑧̂𝑖 ∑ +𝑤∗(𝜏)𝐺𝜎(𝑥 (𝑚 − 𝜏), 𝑥(𝑡)) +𝐿−1 +𝜏=0 + + (39) + +where the ratio 𝑧𝑖 𝑧̂𝑖 +⁄ enforces the scale. This search needs to be done online for every test +sample, but if we rank the training set, it can be done quickly with a tree search. This process can +be repeated K times for a better approximation, where K is a hyper-parameter. The idea is to +probe the neighborhood of 𝑥(𝑡𝑒𝑠𝑡) with K input samples {𝑥(1),… 𝑥(𝐾)} and use their respective +neighbors using (38) to compute K approximate targets {𝑧̂(1),… . 𝑧̂(𝐾)} and represent their mean +by 𝑧̅. The final FWF output will be + +𝑦(𝑡) = ∑ +𝑧𝑘 +𝑧̅ +𝐾 +𝑘=1 +∑ +𝑤∗(𝜏)𝐺𝜎(𝑥 (𝑘 − 𝜏), 𝑥(𝑡)) +𝐿−1 +𝜏=0 + (40) + + Since the filter computation is so small, this improves performance with a minor increase +in computation. The computational complexity of FWFFP and FWFLM are compared in the +following table (i = iterations, M = fixed point updates). + +Table II +Filter +Complexity +(training/testing) +Memory +(training/ testing) +KLMS +O(i) +O(i) +KRLS +O(i2) +O(i2) + +FWFFP +O(L2N)/ O(L) + O(LM) O(N+L2)/O(L) +FWFLM +O(L2N) + O(N)/ +O(KL) + O(logN) +O(2N+L2)/ +O(2NL+L2) + + +E. Experimental Results + +FWF Implementation Challenges +There are several challenges for the FWF implementation. The first issue is numeric +instability and deals with the inverse of the correntropy matrix 𝑉 in (34). Large condition numbers +will bias the solution and need to be corrected through regularization. The second issue stems from +the fundamental fact that learning models must generalize well outside the training set. Note that +there is no error in the FWF methodology, so this presents a different problem than in conventional +machine learning where the regularization can be controlled by penalty terms in the cost function. +In the FWF, generalization is controlled by the kernel size, and by the model order, the two hyper- +parameters in the design. It is easy to see that small kernel sizes yield a correntropy matrix that +approaches a scaled identity matrix, 𝑎𝐼. This is because when kernel sizes are small, correntropy +will peak when signals exactly match, and become very small when signals do not match. This +increases specificity in the training set and also simplifies conditioning of the 𝑉 matrix, but it +requires a large number of samples in the training set and a huge dynamic range in the computation +to avoid losing information in the higher lags. +Therefore, small kernel sizes limit the number of lags that can be used in practice to +represent the input space data correlations. Hence, in order to capture long time dependencies +amongst the lags in a stationary signal, we must use larger kernel sizes in ℋ𝐺. However, if the +kernel size is too large then the behavior of the correntropy function will approach the behavior of +the auto-correlation function i.e., we lose the specificity provided by the higher order moments of +the data PDF. The other drawback of employing larger number of lags is that the chances of ill- +conditioning in the correntropy matrix increase. Hence, these trade-offs mean that kernel size +selection and regularization of 𝑉 are vital for the performance of the FWF, and the kernel size +becomes the key parameter for generalization. + +Regularization of the Correntropy Matrix +We concluded that larger kernel sizes are needed to preserve information over the lags of +the 𝑉 matrix. This means that 𝑉 will be more ill-conditioned, which can be quantified by the +matrix’s condition number. It is important to note that while regularization is helpful, we also need +to control the number of lags to obtain optimal performance. The regularization of the 𝑉 matrix is +depicted in equation (41). Our goal is to find a  such that the condition number of Vreg is +approximately equal to some desired condition number, which becomes a FWF hyper-parameter. + +𝑉𝑟𝑒𝑔 = 𝑉 + 𝜆𝐼; 𝜆 = 𝛾. min 𝐸𝑖𝑔𝑉𝑎𝑙𝑢𝑒 (𝑉) (41) + +Using this framework, we found that condition numbers below 30 worked well, which is +quite restrictive, but can be expected because we expect tiny errors in prediction to make FWF +competitive with KAF approaches. These low condition numbers require a large amount of +regularization, which unfortunately does not utilize all the information in the 𝑉 matrix affecting +the accuracy of the FWF predictions. + + +Initial FWF Results: The Mackey-Glass Time Series + +The Mackey-Glass (MG) times series is a chaotic time series, generated by + +𝑑𝑥(𝑡) +𝑑𝑡 += −𝑏𝑥(𝑡) + +𝑎𝑥(𝑡 − 𝜏) +1 + 𝑥(𝑡 − 𝜏)10 +The MG times series used in the following experiments was generated with b = 0.1, a = 0.2, and + = 30. Experiments testing the KLMS and KRLS kernel adaptive filters with this time series can +be found in [14]. + +One of the hyper-parameters of the FWF is number of lags (L). This defines the length of the +correlation time used to represent each sample, very similar to the Wiener model. Each sample is +represented by a vector of length L with the form [𝑥(𝑖), … 𝑥(𝑖 − 𝐿 − 1)] 𝑇. This is standard practice +for time series prediction. The second hyper-parameter is the kernel size of ℋ𝐺. To estimate the +dependence of performance on hyperparameters, the parameters are scanned and plotted with +training set data to obtain the performance surface of the FWFLM, with two different local model +orders (K = 5 and 15). We can see in Figure 1 that the two local model orders provide basically +the same results. The minimum is obtained around L = 7 delays, and the minimum trough is around + =  which is much larger than the corresponding KAF filters for the same time series. We +also see that the best error is on the order of 10-3 (log 10) which is better than the Wiener filter of +the same order for this data set (MSE = 0.013). + + +Figure 1. Error performance surface over the two FWF hyper-parameters (kernel size and number of +lags), estimated with two different local model orders. + +Experiments with FWFFP and FWFLM +In this section, the performance of the FWF with both pre-imaging methods described +above is compared with two well-known KAF methods, kernel recursive least squares (KRLS) +and KLMS. Figure 2 compares the average test set MSE across 5-folds of cross validation. The +best kernel size from Figure 1 was employed ( =1.5). The figure shows performance with two + +TrainingvsKSandLag,N=2000,K=5 +2.2 +2.4 +Log Error +-2.6 +-2.8 +-3.0 +0.0 +0.5 +1.0 +5 +10 +15 +1.5 +Lags +20 +2.0 +25TrainingvsKSandLag,N=2000,K=15 +-2.0 +-2.2 +Error +-2.4 +Log +-2.6 +-2.8 +-3.0 +0.0 +0.5 +5 +1.0 +10 +15 +1.5 +Lags +20 +25 +2.0different values of K for the FWFLM. We also present results with K=1 for a direct comparison +with the FWFFP. The number of lags considered for FWFLM, was L = 7 the same as embedding for +KLMS, and KRLS. The performance for the FWFFP is the worst, and it improves slightly with the +number of lags, therefore the figures below show results with L=25. Notice that FWFLM with K=1 +is much better than the fixed-point update and here rivals the performance for higher number of +local models. Notice that, as expected, there is no variation with the number of local models in the +FWFFP because the method uses an optimization to find the minimum, so the solution only depends +on L, , and the number of samples in the training set. The FWFLM approaches the performance of +KLMS, but it is far worse than KRL. Remember that the FWF was derived under a strict +stationarity assumption, which is not fulfilled by the MG time series. Therefore, this result is quite +reasonable, taking in consideration the FWF much smaller computation complexity. + +Figure 2. Comparisons of predictions for two different selections of local models (K) as a function of the +number of samples in the training set. Asymptotic performance occurs after 1000 samples. For K=1 +performance is much better than fixed point pre imaging. More models worsen the prediction results on +MG. + +Noisy Mackey-Glass Prediction: +In this experiment the FWF with both pre-imaging methods, KLMS and KRLS are predicting the +MG time series, but with white Gaussian noise added to the input signal. Each algorithm is given +a noisy training and testing input, and the desired signal is the next time point 𝑥(𝑡 + 1) with no +added noise. White Gaussian noise with standard deviations of 0.01, 0.04, 0.1, and 0.2 were tested. +Five-fold cross validation was used for each algorithm at each noise level. The best kernel size is +shown for each algorithm. In general, FWFLM is better than KLMS and KRLS at higher noise +levels. The number of training samples does not have a great effect on the final MSE. Again, the +performance of FWFFP is evaluated at L = 25 while FWFLM use L = 5 and 7. + +MackeyGlass:TestMSE,L=7,K=5 +10-2 +10-3 +TestMSE +10-4 +FWFLM,ks=1.5 +FWFFp,kS=1.0 +KLMS,ks=0.25 +KRLS,kS=0.25 +10~5 +FWFLM,K=1 +250 +500 +750 +1000 +1250 +1500 +1750 +2000 +TrainingSamples (N)MackeyGlass:TestMSE,L=7K=15 +10-2 +下 +10-3 +TestMSE +10-4 +FWFLM,ks=1.5 +FWFFP,kS=1.0 +KLMS,kS=0.25 +KRLS,kS=0.25 +10-5 +FWFLM,K=1 +250 +500 +750 +1000 +1250 +1500 +1750 +2000 +TrainingSamples(N) +Figure 3. FWF has better robustness when noise is added to the time series, as we can expect from the use +of multiple delays. + +Lorenz Prediction: +We decided to test the performance of the FWF in the prediction of a more complex chaotic +dynamical system. The Lorenz system is a well-known system introduced in [32]. We use the x +component of the Lorenz attractor and to make the problem harder, the model predicts 𝑥(𝑡 + 10) +e.g. 10 samples ahead with the last L samples. A version of this experiment can be found in [13]. +Like the previous experiments, the FWFFP was evaluated at L = 30, which is larger than the other +methods. The FWF + +𝐿𝑀 + outperforms KLMS for low number of lags. This difference shrinks as we +consider more lags. As in the other experiments, FWFFP does not perform well when compared to +the other methods. In the Lorenz system, FWFLM performs at the level or better than the KLMS. +Notice that this time series is far from stationary. + + +MackeyGlass:TestMSEvsNoiseVariance,L=7,K=5,N=1oo0 +10-2 +上 +Test MSE +10-3 +FWFLM, kS=1.0 +壬壬壬 +FWFp, ks=1.0 +KLMS, kS=0.25 +KRLS, kS=0.5 +104 +0.025 +0.050 +0.075 +0.100 +0.125 +0.150 +0.175 +0.200 +Noise VarianceMackeyGlass:TestMSEvsNoiseVariance,L=7,K=5,N=2000 +10-2 +Test MSE +103 +王 +FWFLM,kS=1.0 +壬壬壬 +FWFp, ks=1.0 +KLMS,ks=0.25 +10-4 +KRLS, kS=0.5 +0.025 +0.050 +0.075 +0.100 +0.125 +0.150 +0.175 +0.200 +Noise VarianceMackeyGlass:TestMSEvsNoiseVariance,L=5,K=5,N=1oo0 +Test MSE +102 +王 +FWFLM,ks=1.5 +壬壬壬 +103 +FWFfp,ks=1.0 +KLMS,ks=0.25 +KRLS,ks=0.25 +0.025 +0.050 +0.075 +0.100 +0.125 +0.150 +0.175 +0.200 +Noise VarianceMackeyGlass:TestMSEvsNoiseVariance,L=5,K=5,N=2000 +Test MSE +10~2 +FWFLM,ks=1.5 +10-3 +壬壬壬 +FWFfp,ks=1.0 +KLMS,ks=0.25 +KRLS,kS=0.25 +0.025 +0.050 +0.075 +0.100 +0.125 +0.150 +0.175 +0.200 +Noise Variance + +Figure 4. Comparison of performance in the Lorenz time series prediction. For this time series FWFLM +performs better than KLMS but by a small margin. + +Further Analysis on Mackey-Glass sample by sample predictions +Figures 5 shows the training and testing predictions compared to the desired with L = 7, kernel +size of 1.5, and two different local models K = 5 and 100. In both, the prediction is worse in the +parts of the Mackey-Glass series that are more non-stationary (the small ripple across the signal), +but the smoothing effect of using many local models is clearly visible. This explains why K=1 +does such a good job in this signal. This makes sense since when the model is more localized, the +dependency on the stationarity constraint is reduced. + + + +Lorenz:TestMSE.L=10.K=5 +100 +王 +王 +王 +工 +10-1 +TestMSE +10-2 +FWFLM, ks=1.5 +10-3 +壬壬壬 +FWFFp, kS=1.5 +KLMS,ks=0.25 +KRLS,ks=0.25 +250 +500 +750 +1000 +1250 +1500 +1750 +2000 +Training Samples (N)Lorenz:TestMSE,L=15,K=5 +100 +王 +王 +王 +工 +10-1 +TestMSE +10-2 +FWFLM,ks=0.1 +FWFFP,ks=1.5 +10-3 +KLMS,kS=0.25 +KRLS,kS=0.5 +250 +500 +750 +1000 +1250 +1500 +1750 +2000 +TrainingSamples(N)Lorenz: Test MSE, L = 7,K =5 +100 +王 +王 +王 +工 +T +10-1 +Test MSE +工 +工 +10-2 +FWFLM,ks=0.1 +FWFFp, kS=1.5 +10-3 +KLMS,ks=0.1 +KRLS, ks=0.25 +250 +500 +750 +1000 +1250 +1500 +1750 +2000 +Training Samples (N)TestingPredictionsvsDesired +0.4 +0.2 +0.0 +-0.2 +-0.4- +Desired +-0.6 +Predictions +0 +25 +50 +75 +100 +125 +150 +175 +200TestingPredictionsvsDesired +0.4 +0.2 +0.0 +-0.2 +-0.4 +Desired +-0.6 +Predictions +0 +25 +50 +75 +100 +125 +150 +175 +200Figure 5. Sample by sample comparisons of predictions with the FWFLM. Most of the errors occur in the +time varying ripple superimposed in the signal. Notice that less local models perform better. + +Further Prediction Analysis on Lorenz: +Figures 6 shows predictions made by the FWFLM on the Lorenz time series described in the above +section. The hyperparameters here are L = 7,  = 0.1, with two local models, of order K = 5 and K += 100. It is obvious that when the number of local models increases, samples too far away from +the optimal solution will average out the response of the FWF, degrading the prediction. It is also +interesting that the errors at the bottom of the signal ae not smooth, showing that there are not +enough good neighbors in the training set. + +Figure 6. Averaging effect in the quality of the prediction when too many local models are employed +(K=5 left, versus K=100 on the right). + + +F. Conclusions + +The main objective of this paper is to find a principled way to include the input data statistics in +the inner product of a universal RKHS. Recall that KAFs use a data independent kernel (e.g. +Gaussian) to project the data to define in the RKHS, the functional that implements the optimal +model for the application. At test time for online applications, these functionals grow linearly with +the number of samples, which is impractical. In practice, sparcification techniques must be used. +The hypothesis is that a data dependent kernel will substitute the current KAF methodologies and +simplify a lot the functional form to achieve an equally performing model. Parzen inspired this +extension by showing that the ACF of a stationary random process is a positive definite kernel +where optimal statistics models can be implemented. Once in this RKHS, a simple orthogonal +projection is sufficient to find the optimal solution, unlike the incremental solution of KAF. +However, the ACF kernel spans the input data space, so the RKHS solution is still a linear model +with complexity higher than the Wiener filter. With this observation, the goal of this paper can be +stated as extending Parzen’s work to universal models. + +The paper shows how to accomplish this task by defining the positive definite correntropy kernel +as the inner product in a novel RKHS ℋ𝑣. The advantage is that functionals in ℋ𝑣 represent +universal mapping functionals (for infinite number of lags), extending Parzen’s result. The +dimension of ℋ𝑣 is controlled by the number of delays of the autocorrentropy function, so this +space is vastly different from the RKHS created by the Gaussian function, with the promise of + +TestingPredictionsvs Desired +2.0 +1.5 +1.0 +0.5 +0.0 +-0.5 +-1.0 +Desired +-1.5 +Predictions +0 +25 +50 +75 +100 +125 +150 +175 +200TestingPredictionsvsDesired +2.0 +1.5 +1.0 +0.5 +0.0 +0.5 +-1.0 +Desired +1.5 +Predictions +0 +25 +50 +75 +100 +125 +150 +175 +200decreasing the computational complexity of the implementation at test time. The paper presents +the analytical solution of the FWF in ℋ𝐺, but we were unable to find a way to use the kernel trick +to obtain the input space filter. The difficulty is that ℋ𝐺 is not congruent with L2. Two pre-imaging +techniques are proposed to implement the FWF in the input space, which are both approximated +solutions, but they differ in the method and in the computation. FWFFP uses a fixed-point iteration +to find the best solution to evaluate the functional in ℋ𝐺, but since one single Gaussian is unable +to model well a sum of Gaussian at different centers, more sophisticated optimization methods are +needed for good performance. The training set is never used in this pre-imaging technique. The +FWFLM on the other hand uses the training set data to find pairs of samples that approach the best +solution in the training set. This requires a search across the training set to find the best sample +pairs to match the target response, but the method avoids the difficulty of FWFFP fixed-point +iteration by averaging local models obtained in the training set. The simplest of the FWFLM with +K = 1 may be applicable for many nonlinear applications. The FWFLM was found experimentally +more accurate than the linear Wiener filter and is on par with the KLMS performance, but it is still +substantially worse than KRLS. As an advantage, the FWF filter is far more efficient +computationally than KAF implementations and uses less memory. Th FWFFP is of the same +complexity as the Wiener filter but requires a recursive optimization at each iteration, which is not +very expensive computationally. The FWFLM requires a search at the training time to match pairs +of samples to find the pre image, and at test time, a search to find the closest training sample to the +current test input (which is O(log N) if the input training samples are ordered by amplitude). + +Hence, we conclude that this paper does not solve all the issues and should be considered as a first +attempt to develop a new class of universal mappers in RKHS that integrate the data statistics in +the kernel. But the novelty of the technique brings fresh ideas to statistical signal processing that +need also to be further investigated. For instance, the FWF never employs the error, which is +critical in KAFs. Effectively, FWF only works with the minimum norm (orthogonal) projection +in RKHS, so it is “model agnostic”: the important step is to create a RKHS that includes the data +statistics (in the form of its autocorrentropy function) in the inner product. Of course, this +construction requires “parameters” that are the ACF values of the input data and the CCF with the +target, and the number of lags, just like least squares. After this construction, the FWF just finds +the best local projection in the optimal RKHS functional centered at the current test sample. +Therefore, there is no model nor parameters as in conventional optimal filtering and neural +networks, just memory of the training set. In a sense, this approach resembles how brains encode +and react to the physical world; neurons across life encode the structure and similarities given by +the laws of physics, and they react very quickly to implement their response to stimulus, which +means that the response must be very easy to compute. The advantage and disadvantages of the +new approach are not fully understood at this time. Finally, we should focus on ways to avoid the +loss of congruence between the universal RKHS and the input space. The correntropy RKHS has +the very nice property that embeds the statistics of the data in the inner product, but there may be +other kernels that maintain congruence with the input space, exemplified by our work and others +on embedding PDFs in RKHS [33]. Another interesting aspect is that the local linear models seem +to go beyond the strict stationarity assumption that supports theoretically the method. More work +is required to study further this aspect. + +Acknowledgements: This work was partially supported by ONR grants N00014-21-1-2295 and +N00014-21-1-2345 + + +Appendix: Properties of the AutoCorrentropy Function + +The existence of ℋ𝐺 opens new possibilities to extend the work of Parzen on the covariance RKHS +that is defined on the Hilbert space of the data. Recall that the autocorrelation function of a time +series is a similarity measure quantified by the expected value of the product between two random +variables 𝑋(𝑡𝑖), 𝑋(𝑡𝑗) at two different time intervals 𝑡𝑖, 𝑡𝑗 given by their joint distribution. As such +it only measures the first moment (the mean) of the joint PDF over time. The first question is how +to modify the autocorrelation function, as a similarity measure in such a way that it captures all +the statistical information contained in the joint distribution. + +Going Beyond the Autocorrelation Function for Similarity +The most general measure of similarity in the joint space of two r.v. 𝑋, 𝑌 is the cross +covariance operator [26], defined by the bilinear form + +𝒞𝑠,𝑡(𝑓, 𝑔) = 𝐸 [𝑓(𝑋)𝑔(𝑌)] − 𝐸 [𝑓(𝑋)]. 𝐸 [𝑔(𝑌)] (A1) + +The covariance operator has been estimated in RKHS ℋ𝐺 as the matrix Σ𝑥𝑠𝑥𝑡 of size equal to +number of samples such that + +〈𝑓, Σ𝑥𝑡𝑥𝑠𝑔〉 ℋ𝐺 = 𝒞𝑠,𝑡(𝑓, 𝑔) (A2) + +where f and g are functional in RKHS that map the samples from the r.v. 𝑥𝑡 and 𝑥𝑠. But this +treatment might be overly complicated for a stationary random process. Firstly, the marginals have +the same density; secondly, only a scalar similarity over marginals is needed, and the mean +embedding operator (20) can be estimated in ℋ𝑣; and thirdly because time establishes an a priori +order on the r.v. such that a single variable (the delay) can be employed, instead of pairwise +samples. Therefore, we submit that it is not necessary to estimate the full covariance operator for +this application, which is computationally very intensive. + +Measures of Similarity in the Joint Space of Densities + +Definition: Given a strictly stationary time series {𝑋𝑡,𝑡 ∈ 𝑇} the equality in probability density +between two marginals at s and t i.e., 𝑃(|𝑋(𝑠) − 𝑋(𝑡)| < 𝜀) for an infinitesimally small 𝜀, defines +a measure of similarity that can be estimated in ℋ𝐺. +In the joint space of 𝑝𝑠,𝑡(𝑥𝑡,𝑥𝑠) we can define a radial marginal as the bisector of the joint +space. The density over the line 𝑥𝑡 = 𝑥𝑠 approximates +𝑃(|𝑋(𝑠)−𝑋(𝑡)|<𝜀) +𝜀 +, which can be estimated as + +𝐸𝑝𝑠,𝑡[𝛿(𝑋(𝑠) − 𝑋(𝑡))] (A3) + +where 𝛿(. ) is a delta function and we assume that the joint pdf over the lags is smooth along the +bisector of the joint space is non-zero. To simplify, the Dirac calculus is used to illustrate the +concept. +The expected value in (A3) can be written + + +𝐸𝑝𝑠,𝑡[𝛿(𝑋(𝑠) − 𝑋(𝑡))] = ∬ 𝛿(𝑥𝑠 − 𝑥𝑡)𝑝𝑠,𝑡(𝑥𝑠,𝑥𝑡)𝑑𝑥𝑠𝑑𝑥𝑡 (A4) + +The meaning of (A3) is quite clear: it is integrating the area under the joint density along the line +𝑥𝑡 = 𝑥𝑠. Therefore, we can write (A4) as a single integral + +𝐸𝑝𝑠,𝑡[𝛿(𝑋(𝑠) − 𝑋(𝑡))] = ∫ 𝑝𝑠,𝑡(𝑥, 𝑥)𝑑𝑥 +(A5) + +This reduction to a single integral can be expected by the definition of conditional PDF (see +below), and it simplifies the calculation because of the statistical embedding in ℋ𝑣. +Note however, that this procedure needs to be repeated for every lag L of interest i.e., it +should be written as 𝑡 = 𝑠 − 𝑙, 𝑙 = 0,… 𝐿. Fortunately, the maximum lag L is dictated by the +embedding dimension of the real system that produced the time series, which is far smaller than +the number of samples we collect from the world. In engineering applications this order can be +estimated by Takens’ embedding theory [27], or more practically by selecting the first minimum +of the time series autocorrelation function. This computation is much simpler than the covariance +matrix in (A1) because we are reducing the matrix to a vector u of size L. + +Correntropy functional as an approximation to the bisector integral +An empirical estimator of the natural measure of similarity defined above is given by its +inner product (20). It turns out it has been coined in [16] the correntropy functional, which reads + +𝑉𝜎(𝑡, 𝑠) = 𝐸𝑝𝑡,𝑠[𝐺𝜎(𝑥𝑡 − 𝑥𝑠)] (A6) + +where G(.) is the Gaussian function with bandwidth . As discussed above, correntropy is a mean +embedding of the joint pdf of a pair of samples. Rewriting (A6) using the definition of the expected +value over the joint distribution, we obtain + +𝑉𝜎(𝑡, 𝑠) = ∬ 𝐺𝜎(𝑥𝑡 − 𝑥𝑠)𝑝𝑡,𝑠(𝑥𝑡,𝑥𝑠)𝑑𝑥𝑡𝑑𝑥𝑠 = 𝐸[𝐺𝜎(𝑥𝑡 − 𝑥𝑠)] (A7) + +for strictly stationary processes. The best way to interpret this relation is to realize that when 𝑥𝑡 = +𝑥𝑠, i.e. along the bisector of the joint space, the Gaussian kernel function is maximum, i.e. +correntropy weights the joint space of samples with Gaussian kernels placed along the bisector of +the first quadrant [16]. When the kernel size  approaches 0, it approximates a delta function +𝛿(𝑥𝑡 − 𝑥𝑠), so we obtain an approximation to (A3). Moreover, correntropy is easily computed +from samples too. Collect a segment of data of size N from a time series. From (A7) an estimator +of correntropy is simply + +𝑉𝜎(𝜏) = +1 +𝑁−𝜏+1 ∑ +𝐺𝜎(𝑥𝑖 − 𝑥𝑖−𝜏) +𝑁 +𝑖=𝑚 + (A8) + + +Hence, correntropy effectively estimates a radial marginal density obtained by integrating +along the bisector from samples with linear complexity. This is unsuspected, because we are +quantifying similarity in the structure of a time series beyond what we can achieve with the mean +value of the product of samples in the autocorrelation. Note that here the kernel size should be +made small for fine temporal resolution, but there is a compromise, because if we use a very small + +kernel size, the number of samples N must be sufficiently large to get sufficient number of samples +around the bisector of the joint space for accurate statistical estimation. + + +The Relation between 𝑃(𝑥𝑡1 − 𝑥𝑡2) and the Conditional Density in the Joint Space +The Dirac calculus is a short cut and here we provide a more precise derivation of the value of the +radial margin as a conditional distribution. As is well known the definition of conditional +distribution of the r.v. X given Y is +𝑓(𝑥|𝑦) = 𝑓(𝑥, 𝑦) +𝑓(𝑦) = 𝑓(𝑥|𝑌 = 𝑦0) = 𝑓(𝑥, 𝑌 = 𝑦0) +𝑓(𝑌 = 𝑦0) +The meaning of this conditional is that we pick a value for y = y0 and compute the area under the +joint pdf at y0. Here we are interested in a radial marginal, which is the bisector of the joint space +given by the equality in probability i.e., Y=X, and would like to see how to compute it. Let us start +with the distribution function and write the conditional probability as + +𝐹(𝑥|(𝑥 − 𝛿) < 𝑌 ≤ 𝑥) = 𝑃(𝑋 ≤ 𝑥|(𝑥 − 𝛿) < 𝑌 ≤ 𝑥) = +𝑃(𝑋 ≤ 𝑥,(𝑥 − 𝛿) < 𝑌 ≤ 𝑥) +𝑃((𝑥 − 𝛿) < 𝑌 ≤ 𝑥) += lim +𝛿→0 +∫ +∫ +𝑓𝑋,𝑌(𝑢, 𝑣)𝑑𝑢𝑑𝑣 +𝑥 +−∞ +𝑥 +𝑥−𝛿 +∫ +𝑓𝑌(𝑣)𝑑𝑣 +𝑥 +𝑥−𝛿 += 𝑓𝑋,𝑌(𝑥, 𝑥) +𝑓𝑌(𝑥) +So, when the concept of the radial margin is employed as a conditional probability, we see that +there is a normalizing factor that guarantees that the result adds to one as required for probabilities, +but the numerator is exactly what the Dirac calculus quantifies in (A4). + +Approximating 𝑃(𝑥𝑡1 − 𝑥𝑡2) with Correntropy +lim +𝜎→0 𝑣𝜎 (𝑡1, 𝑡2) = ∬ 𝛿(𝑥𝑡1 − 𝑥𝑡2)𝑝𝑝𝑡1𝑝𝑡2(𝑥𝑡1,𝑥𝑡2)𝑑𝑥𝑡1𝑑𝑥𝑡2 = ∫ 𝑝𝑝𝑡1𝑝𝑡2(𝑥𝑡1,𝑥𝑡1)𝑑𝑥𝑡1 (A9) + + + +In practice, the kernel size is always finite so correntropy does not estimate the probability density +over a line in the joint space but the probability on a “Gaussian shaped tunnel” of width  along +the radial direction 𝑥𝑡1 = 𝑥𝑡2, which will be approximated by a parallelepiped of width 2 with  ~ +1.25. We can write + +𝑃(|𝑥𝑡1 − 𝑥𝑡2| < 𝜀) = ∫ +∫ +𝑝𝑝𝑡1𝑝𝑡2(𝑥𝑡1, 𝑥𝑡2)𝑑𝑥𝑡1𝑑𝑥𝑡2 +𝑥𝑡1+𝜀 +𝑥𝑡2=𝑥𝑡1−𝜀 +∞ +𝑥𝑡1=−∞ + (A10) + +If  is small and 𝑝𝑝𝑡1𝑝𝑡2(𝑥𝑡1,𝑥𝑡2) is continuous at every point along the 𝑥𝑡1 = 𝑥𝑡2 line, the function +value does not change a lot along 𝑥(𝑡2) within the interval [𝑥𝑡1 − 𝜀, 𝑥𝑡1 + 𝜀] for any fixed 𝑥(𝑡1). +Thus + +𝑃(|𝑥𝑡1 − 𝑥𝑡2| < 𝜀) ≈ 2𝜀 ∫ +𝑝𝑝𝑡1𝑝𝑡2(𝑥𝑡1,𝑥𝑡1)𝑑𝑥𝑡1 +∞ +𝑥(𝑡1)=−∞ += 2𝜀𝑣𝜎(𝑡1,𝑡2) (A11) + +And finally, we have +𝑣𝜎(𝑡1,𝑡2) = +𝑃(|𝑥𝑡1−𝑥𝑡2|<𝜀) +2𝜀 + (A12) + + +which shows that correntropy estimates indeed the probability density of the event 𝑃(𝑥𝑡1 = 𝑥𝑡2) +in the joint sample space for small kernel sizes. + + +References + +1. Wiener, Norbert (1949). Extrapolation, Interpolation, and Smoothing of Stationary Time +Series. New York: Wiley. +2. N. Wiener, E. Hopf, "Ueber eine Klasse singulärer Integralgleichungen" Sitzungber. +Akad. Wiss. Berlin (1931) pp. 696–706 +3. Wiener, Norbert (1930). "Generalized Harmonic Analysis". Acta Mathematica. 55: 117- +258 +4. E. Parzen, "Statistical inference on time series by Hilbert space methods," Tech. Report +23, Stat. Dept., Stanford Univ., 1959. +5. N. Aronszajn, "The theory of reproducing kernels and their applications," Cambridge +Philos. Soc. Proc., vol. 39, pp. 133-153, 1943. +6. Wahba, Grace, Spline Models for Observational Data, SIAM, 1990 +7. T. Kailath and H. Weinert, “An RKHS approach to detection and estimation problems– +part II: Gaussian signal detection,” IEEE Trans. Inf. Theory, vol. IT-21, no. 1, pp. 15–23, +Jan. 1975. +8. T. Kailath and D. Duttweiler, “An RKHS approach to detection and estimation +problems–part III: Generalized innovations representations and a likelihood-ratio +formula,” IEEE Trans. Inf. Theory, vol. IT-18, no. 6, pp. 730–745, Nov. 1972. +9. D. Duttweiler and T. Kailath, “RKHS approach to detection and estimation problems– +part IV: Non-gaussian detection,” IEEE Trans. Inf. Theory, vol. IT-19, no. 1, pp. 19–28, +Jan. 1973. +10. D. Duttweiler and T. Kailath,, “RKHS approach to detection and estimation problems– +part V: Parameter estimation,” IEEE Trans. Inf. Theory, vol. IT-19, no. 1, pp. 29–37, Jan. +1973. +11. V. N. Vapnik, Statistical Learning Theory. New York: John Wiley & Sons, 1998 +12. M. G. Genton, “Classes of kernels for machine learning: A statistics perspective,” J. +Mach. Learn. Res., vol. 2, pp. 299–312, 2001 +13. Liu W., Haykin S., Principe J., “Kernel Adaptive Filtering”, Wiley 2010 +14. Principe J., Chen B., “Universal Approximation with Convex Optimization: Gimmick or +Reality”, IEEE Computation Intelligent Magazine, vol. 10, no. 2, pp. 68-77, 2015 +15. Santamaria I., Pokharel P., Principe J., “Generalized Correlation Function: Definition, +Properties and Application to Blind Equalization”, IEEE Trans. Signal Proc. vol 54, no 6, +pp 2187- 2186, 2006 +16. Liu W., Pokharel P., Principe J., “Correntropy: Properties and Applications in Non +Gaussian Signal Processing”, IEEE Trans. Sig. Proc., vol 55; # 11, pages 5286-5298, +2007 +17. Schölkopf, Bernhard; Smola, Alex; Müller, Klaus-Robert. "Nonlinear Component +Analysis as a Kernel Eigenvalue Problem". Neural Computation. 10 (5): 1299–1319, +1998. + +18. S. Mika, B. Schölkopf, A. Smola, K. Müller, M. Scholz, and G. Rätsch, “Kernel pca and +de-noising in feature spaces,” in Proceedings of the NIPS II. Cambridge, MA, USA: MIT +Press, 1999, pp. 536–542 +19. I. Constantin, C. Richard, R. Lengelle and L. Soufflet, "Regularized kernel-based Wiener +filtering. Application to magnetoencephalographic signals denoising," Proceedings. +(ICASSP '05). IEEE International Conference on Acoustics, Speech, and Signal +Processing, 2005., 2005, pp. iv/289-iv/292 Vol. 4 +20. Pokharel P., Xu J., Erdogmus D., Principe J., “A Closed Form Solution for a Nonlinear +Wiener Filter”, Proc. IEEE Int. Conf. Acoustics Speech and Signal Processing, Toulose, +France +21. Parzen E. “An approach to time series analysis,” Ann. Math. Stat., vol. 32, no. 4, pp. +951–989, Dec. 1961 +22. Loève, Michel (1955). Probability Theory. Princeton, New Jersey, USA: D Van +Nostrand. +23. Kosambi, D. D. (1943), "Statistics in Function Space", Journal of the Indian +Mathematical Society, 7: 76–88, +24. B. Scholkopf and A. Smola, Learning with kernels. Cambridge, MA: MIT Press, 2002 +25. Liu W., Pokarel P., Principe J., “The Kernel LMS Algorithm”, IEEE Trans. Signal +Processing, Volume 56, Issue 2, Page(s):543 - 554, 2008. +26. Arthur Gretton, Karsten M Borgwardt, Malte J Rasch, Bernhard Schölkopf, and +Alexander Smola. A kernel two-sample test. Journal of Machine Learning Research, +13(Mar):723–773, 2012 +27. F. Takens (1981). "Detecting strange attractors in turbulence". In D. A. Rand and L.-S. +Young (ed.). Dynamical Systems and Turbulence, Lecture Notes in Mathematics, vol. +898. Springer-Verlag. pp. 366–381 +28. Principe J., Information Theoretic Learning: Renyi’s Entropy and Kernel Perspectives, +Springer 2010 +29. Qin Z., Chen B, Zheng N., Principe J., “Augmented Space Linear Models”, IEEE Trans. +Signal Proc., vol 68, 2724 – 2738, 2020 +30. Chen B., Zhao P., Zhu P., Principe J., Quantized Kernel Least Mean Square Algorithm. +IEEE Trans. Neural Netw. Learning Syst. 23(1): 22-32 (2012) +31. Martinsson P., Rokhlin V., Tygert M., “A Fast Algorithm for the Inversion of General +Toeplitz Matrices”, Computers and Mathematics with Applications 50 (2005) 741-752 +32. Lorenz, Edward Norton (1963). "Deterministic nonperiodic flow". Journal of the +Atmospheric Sciences. 20 (2): 130–141. +33. Xu J., Paiva A., Park I., Principe J., “A Reproducing Kernel Hilbert space framework for +information-theoretic learning", IEEE Trans. Signal Processing Volume 56, Issue 12, +Page(s):5891 - 5902, 2008 + + + + diff --git a/7dAyT4oBgHgl3EQfcvf_/content/tmp_files/load_file.txt b/7dAyT4oBgHgl3EQfcvf_/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3235cc8b992eace172632de2284e9151af134a98 --- /dev/null +++ b/7dAyT4oBgHgl3EQfcvf_/content/tmp_files/load_file.txt @@ -0,0 +1,869 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf,len=868 +page_content='The Functional Wiener Filter Benjamin Colburn, Luis G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Sanchez Giraldo, Jose C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Principe Abstract This paper presents a close form solution in Reproducing Kernel Hilbert Space (RKHS) for the famed Wiener filter, which we called the functional Wiener filter (FWF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Instead of using the Wiener-Hopf factorization theory, here we define a new lagged RKHS that embeds signal statistics based on the correntropy function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' In essence, we extend Parzen’s work on the autocorrelation function RKHS to nonlinear functional spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' The FWF derivation is also quite different from kernel adaptive filtering (KAF) algorithms, which utilize a search approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' The analytic FWF solution is derived in the Gaussian kernel RKHS with a constant computational complexity similar to the Wiener solution, and never composes nor employs the error as in conventional optimal modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Because of the lack of congruence between the Gaussian RKHS and the space of time series, we compare performance of two pre-imaging algorithms: a fixed-point optimization (FWFFP) that finds and approximate solution in the RKHS, and a local model implementation named FWFLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' The experimental results show that the FWF performance is on par with the KAF for time series modeling, and it requires far less computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Introduction Norbert Wiener’s 1949 work on minimum mean square error estimation opened the door for the theory of optimum filtering [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' The mathematics to solve integral equations, the Wiener- Hopf method [2], were crucial to arrive at the optimal parameter function, however, the methodology is rather complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' In digital signal processing using finite impulse response filters, the Wiener solution coincides with least squares, as proven by the Wiener-Kinchin theorem [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Therefore, the solution still belongs to the span of the input data i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=', the corresponding filter is linear in the parameters and therefore it is not a universal functional approximator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' In the late 50’s, Emmanuel Parzen [4] presented an alternative approach to solve the minimum mean square estimation (MMSE) problem in a Reproducing Kernel Hilbert space (RKHS) defined by the autocorrelation function of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Since RKHS theory will be extensively employed, we define here a RKHS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Let 𝐸 be a non-empty set, and 𝜅(𝑢, 𝑣) a function defined on 𝐸 × 𝐸 that is nonnegative definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Due to the Moore-Aronzsajn theorem [5], 𝜅(𝑢, 𝑣) defines uniquely a RKHS, ℋ𝜅, such that 𝜅(⋅, 𝑣) ∈ ℋ𝜅 and for any 𝑔 ∈ ℋ𝜅, 〈𝑔, 𝜅(⋅,𝑣)〉ℋ𝜅 = 𝑔(𝑣).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Therefore, a RKHS is a special Hilbert vector space associated with a kernel such that it reproduces (via the inner product) in the space i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=', 〈 𝜅(⋅,𝑢), 𝜅(⋅,𝑣)〉ℋ𝜅 = 𝜅(𝑢, 𝑣);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' or equivalently, a space where every point evaluation functional is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' The history of RKHS applications started in physics [6], statistics [7], signal processing [8] and machine learning [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Here, it will also be clear that the RKHS framework provides a natural link between stochastic processes and deterministic functional analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Parzen introduced for the first time the RKHS methodology in statistical signal-processing and time-series analysis in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' His essential idea is that there exists a congruence map between the set of random variables spanned by the random process {𝑋(𝑡), 𝑡 ∈ 𝑇} with covariance function 𝑅(𝑡, 𝑠) = 𝐸[𝑋(𝑡)𝑋(𝑠)] and the RKHS of vectors spanned by the set {𝑅(⋅,𝑡), 𝑡 ∈ 𝑇} denoted as ℋ𝑅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Note that the kernel expresses the second-order statistics of the data through the expected value (a data-dependent kernel) and Parzen clearly stated that this RKHS offers an elegant functional analysis framework for minimum mean square error (MMSE) solutions such as regression coefficients, least squares estimation of random variables, detection of signals in Gaussian noise, and others [9],[10],[11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Unfortunately, ℋ𝑅 is defined in the input data space, so yields only linear solutions to the regression problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Parzen beautiful interpretation did not provide any practical improvement, so it was quickly forgotten in signal processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' More recent work by Vapnik on support vector machines brought back a lot of interest to RKHS theory for pattern recognition [12], where the RKHS is used primarily as a high- dimensional feature space and the inner product is efficiently computed by means of the kernel trick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' A nonnegative definite kernel function (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=', Gaussian, Laplacian, polynomial, and others [13]) nonlinearly projects the data sample-by-sample into a high-dimensional RKHS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' This development was included in adaptive filtering, yielding the class of kernel adaptive filters (KAF) [14], which allows the design of convex universal learning machines (CULMs) [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' KAFs estimate a functional model that approximates the MMSE solution using search techniques in the RKHS defined by the Gaussian kernel [14], and the order grows linearly with the number of samples, if no sparsification is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Another branch of RKHS theory important for this paper is kernel Principal Component Analysis (KPCA) [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' When the kernel function is infinite dimensional as the Gaussian, denoted as 𝐺(𝑥𝑖,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' ), the eigen decomposition of the empirical covariance operator 𝐶 = 1 𝑁 ∑ 𝐺(𝑥𝑖, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' )𝐺(𝑥𝑖,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' )𝑇 𝑁 𝑖=1 ⁄ needs to be truncated (we assume 𝐺(𝑥𝑖,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' ) are centered in the RKHS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' In such cases, a more efficient approach uses only inner products of functionals centered at the projected samples, which can be computed in the input space using the reproducing property of the kernel (also called the kernel trick).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' The goal is to rewrite the eigen decomposition of the empirical covariance operator 𝐶 through a functional eigenvalue equation as 𝐶𝑉 = 𝜆𝑉, where 𝑉 is the eigenfunction 𝑉 = 1 𝑁 ∑ 𝛼𝑖𝐺(𝑥𝑖,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=') 𝑁 𝑖=1 ⁄ and 𝜆 is a vector of scalars that correspond to the eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' For any nonzero 𝜆, the eigenfunction exists in the span of the RKHS defined by the kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Since the number of samples is finite this methodology is very appealing and efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' However, the span of the functional space defined by the kernel is much larger than the mappings of single mapped samples into the RKHS, which means that the inverse mapping of RKHS functionals to the input space cannot be necessarily expressed as the image of a single input pattern i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=', given a function 𝜁 in the RKHS span, there is no guarantee that there is exist a 𝑧 ∈ ℝ𝑁 such that 𝐺(𝑧, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' ) = 𝜁.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' This has been called the preimage problem [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' We call 𝑧̂ an approximate preimage of 𝜁 if ‖𝐺(𝑧, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' ) − 𝜁‖2 is small, according to the application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' We will see that this pre- imaging will be important in our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' This paper takes Parzen’s work one step further, combining it with KAF concepts to yield a RKHS defined by the covariance function of the projected data in a Gaussian RKHS, which is nonlinearly related to the data space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' More specifically, we define a data dependent kernel based on the correntropy function [16] that incorporates full data statistics and defines a RKHS of deterministic functions, even when the input data is a random variable (r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Correntropy has been heavily used for robust cost functions in adaptive signal processing [17], but here its functional extension [16] will be employed as a methodology to solve the famous Wiener filter in the space of nonlinear functions, without using the Wiener-Hopf spectral factorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Previous attempts by others e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=', the kernel Wiener filter [18], approximate the Wiener solution employing subspace projections in RKHS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' An early attempt to solve the Wiener-Hopf equations in RKHS was not successful [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' This paper shows how to pose the optimum filtering problem, derive a solution, and present a methodology to implement the filter directly from samples, which effectively extends MMSE for nonlinear universal approximators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' The framework is named the functional Wiener filter (FWF) and amazingly, it does not require the use of the error signal as in the traditional Wiener solution to adapt parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' It takes advantage of the geometry of the RKHS and finds, just like Least Squares, the orthogonal projection of the desired response in the space spanned by the correntropy function, and in this sense, it is model agnostic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Preliminary results show that performance is on par with KLMS but it is worse than KRLS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' The major advantage is the simplicity in implementation that is similar to the Wiener solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Review of Linear Prediction of Continuous Time Series in RKHS A stochastic process 𝑋(𝑡, 𝜔) is broadly defined as a collection of random variables on a measurable sample space (Ω, ℬΩ), indexed by a set 𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Here, we restrict 𝑋(𝑡,𝜔) to random variables taking values in ℝ, 𝑇 ⊂ ℝ, which we call a time-series, {𝑋(𝑡), 𝑡 ∈ 𝑇} and omit the dependence on 𝜔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' For a time-series with finite second order moments, let 𝐿2(𝑋(𝑡), 𝑡 ∈ 𝑇) denote the space of all real valued random variables spanned by the time series, that is, this space consists of all r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' 𝑈 that are either linear combinations of finite number of 𝑋(𝑡𝑖) or are limits of such linear combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' The time structure is quantified by the joint probability density 𝑝𝑡,𝑠(𝑥𝑡,𝑥𝑠) of the pair of random variables 𝑋(𝑡),𝑋(𝑠) at two points in time 𝑡 and 𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Assuming a strictly stationary stochastic model for 𝑋(𝑡), the marginal density 𝑝𝑡(𝑥) is the same for any 𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Normally, the joint density 𝑝𝑡,𝑠(𝑥𝑡,𝑥𝑠) is quantified by its mean value, called the autocorrelation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' To simplify notation, let us define the time autocorrelation of the finite second order moment time series as: 𝑅(𝑠, 𝑡) = 𝐸[𝑋(𝑠)𝑋(𝑡)] (1) This kernel on time sample pairs is positive semi definite, hence by Moore-Aronzsajn theorem [5] it defines a RKHS space of functions on 𝑇 × 𝑇, denoted ℋ𝑅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Notice that the functions in ℋ𝑅 are deterministic because of the 𝐸[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' ] operator, while the inner product in ℋ𝑅 depends on the statistics of the data through 𝑋(𝑡).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' For any r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' 𝑈, 𝐴 ∈ 𝐿2(𝑋(𝑡),𝑡 ∈ 𝑇), define the inner product between the two as 〈𝑈, 𝐴〉 = 𝐸[𝑈𝐴], (2) and the norm of 𝑈 in 𝐿2(𝑋(𝑡), 𝑡 ∈ 𝑇) by the inner product 〈𝑈, 𝑈〉 = 𝐸[𝑈2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Obviously, this inner product coincides with the autocorrelation function if 𝑈 is 𝑋(𝑠) and 𝐴 is 𝑋(𝑡).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' However, notice that 𝐿2(𝑋(𝑡), 𝑡 ∈ 𝑇) is not an RKHS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Explicit Expression for MMSE One of the important problems in time series analysis is the representation of an unobservable r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' 𝑍.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Let {𝑋(𝑡), 𝑡 ∈ 𝑇} be an observable time series assumed stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' The goal is to create a linear combination of the observable time series that has the smallest mean square distance to 𝑍.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' By the Hilbert projection theorem, there is a unique minimum norm projection between the abstract Hilbert space ℋ and any subspace 𝑀 of ℋ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Then, there exists a unique vector 𝐴∗ in 𝑀, given by 𝐴∗ = 𝐸∗[𝐴|𝑀], which projects orthogonally a vector 𝐴 in ℋ to 𝑀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' For a family of vectors {𝑋(𝑡), 𝑡 ∈ 𝑇} the projection becomes 𝐴∗ = 𝐸∗[𝐴|𝑋(𝑡), 𝑡 ∈ 𝑇].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' (3) Then with 𝐴 = 𝑍, the optimum linear predictor is the unique r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' in 𝐿2(𝑋(𝑡), 𝑡 ∈ 𝑇) that satisfies 𝐸[𝐸∗[𝑍|𝑋(𝑡),𝑡 ∈ 𝑇]𝑋(𝑠)] = 𝐸[𝑍𝑋(𝑠)] (4) This result gives immediately rise to the famous Wiener equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Indeed, if T is a finite interval 𝑇 = {𝑡: 𝑎 ≤ 𝑡 ≤ 𝑏} and w(t) a weighting function in 𝐿2, the integral ∫ 𝑋(𝑡)𝑤(𝑡)𝑑𝑡 𝑏 𝑎 (5) represents a r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' in 𝐿2(𝑋(𝑡), 𝑡 ∈ 𝑇), then the weighting function of the best linear predictor can be written as 𝐸∗[𝑍|𝑋(𝑡), 𝑡 ∈ 𝑇] = ∫ 𝑋(𝑡)𝑤∗(𝑡)𝑑𝑡 𝑏 𝑎 (6) and must satisfy the generalized Wiener equation: ∫ 𝑤∗(𝑡)𝑅(𝑠, 𝑡)𝑑𝑡 = 𝜌𝑧(𝑠) 𝑏 𝑎 (7) in 𝑎 ≤ 𝑠 ≤ 𝑏, with 𝑅(𝑠, 𝑡) = 𝐸[𝑋(𝑠), 𝑋(𝑡)], 𝜌𝑧(𝑠) = 𝐸[𝑍𝑋(𝑠)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' This equation states that one can always find a representation for the function 𝜌𝑧(𝑠) in terms of the functions {𝑅(𝑠, 𝑡), 𝑡 ∈ 𝑇} such that the minimum mean square error linear predictor 𝐸∗[𝑍|𝑋(𝑡), 𝑡 ∈ 𝑇] can be written in terms of the corresponding linear operator on the time series {𝑋(𝑡), 𝑡 ∈ 𝑇}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Hilbert Space Representation of Time Series First, let us state an important theorem that is very important for this line of work [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Basic Congruence Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Let ℋ1 and ℋ2 be two abstract Hilbert spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Let 𝑇 be an index set and let {𝑢𝑡,𝑡 ∈ 𝑇} be a family of vectors spanning ℋ1, and similarly {𝑎𝑡,𝑡 ∈ 𝑇} a family of vectors spanning ℋ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Suppose that for every 𝑠, 𝑡 in 𝑇, 〈𝑢𝑠, 𝑢𝑡〉 ℋ1 = 〈𝑎𝑠, 𝑎𝑡〉 ℋ2 (8) then there is a congruence (a one-to-one inner product preserving linear mapping) 𝜓 from ℋ1 to ℋ2 such that 𝜓(𝑢𝑡) = 𝑎𝑡 for any 𝑡 ∈ 𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Definition: A family of vectors {𝑢𝑡,𝑡 ∈ 𝑇} in a Hilbert space ℋ𝑅 is a representation of a wide sense stationary time series {𝑋(𝑡), 𝑡 ∈ 𝑇} if for every s, t in T 〈𝑢𝑠, 𝑢𝑡〉ℋ𝑅 = 𝑅(𝑠,𝑡) = 𝐸[𝑋(𝑠), 𝑋(𝑡)] (9) Then there is a congruence 𝜓 between the Hilbert space spanned by {𝑢𝑡,𝑡 ∈ 𝑇} and denoted as 𝐿2(𝑢𝑡,𝑡 ∈ 𝑇), onto 𝐿2(𝑋(𝑡), 𝑡 ∈ 𝑇) satisfying 𝜓(𝑢𝑡) = 𝑋(𝑡), and every r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' 𝑈 in 𝐿2(𝑋(𝑡), 𝑡 ∈ 𝑇) may be written 𝑈 = 𝜓(𝑔) for some unique vector 𝑔 in 𝐿2(𝑢𝑡, 𝑡 ∈ 𝑇).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' The natural representation of a time series is obtained in the RKHS ℋ𝑅 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=', a Hilbert space where the kernel has two properties: 𝑅(⋅,𝑡) ∈ ℋ𝑅 〈𝑔, 𝑅(⋅, 𝑡)〉ℋ𝑅 = 𝑔(𝑡) (10) This result is the well-known Riez representation theorem, which yields for our discussion 𝑅(𝑠, 𝑡) = 〈𝑅(⋅,𝑠), 𝑅(⋅,𝑡)〉ℋ𝑅 = 𝐸[𝑋(𝑠), 𝑋(𝑡)] (11) It can be further shown that for any time series {𝑋(𝑡), 𝑡 ∈ 𝑇} with covariance kernel 𝑅, the family of functions {𝑅(⋅, 𝑡),𝑡 ∈ 𝑇} in ℋ𝑅 is a representation of 𝐿2(𝑋(𝑡), 𝑡 ∈ 𝑇).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Indeed, for any two vectors 𝑈,𝐴 ∈ 𝐿2(𝑋(𝑡), 𝑡 ∈ 𝑇) such that the congruence is denoted by 𝑈 = 𝜓(𝑔) and 𝐴 = 𝜓(ℎ), and 𝐴 = 𝜓(ℎ) = 〈𝑋, ℎ〉ℋ𝑅, 〈𝑋, 𝑅(⋅,𝑡)〉ℋ𝑅 = 𝑋(𝑡) 𝐸[〈𝑋, ℎ〉ℋ𝑅〈𝑋, 𝑔〉ℋ𝑅] = 〈ℎ, 𝑔〉ℋ𝑅 It is easy to see that if the two vectors ℎ, 𝑔 ∈ ℋ𝑅 correspond to random variables 𝑈, 𝐴 ∈ 𝐿2(𝑋(𝑡), 𝑡 ∈ 𝑇) 〈ℎ, 𝑔〉 ℋ𝑅 = ∫ ∫ ℎ(𝑠)𝑅−1(𝑠, 𝑡)𝑔(𝑡)𝑑𝑠 𝑠∈𝑇 𝑑𝑡, 𝑡∈𝑇 where 𝑅−1(𝑠, 𝑡) is the kernel of the inverse of the covariance operator 𝑅𝑔 = ∫ 𝑔(𝑡)𝑅(𝑠, 𝑡)𝑑𝑡 𝑡∈𝑇 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Moreover, if 𝑈 = ∑ 𝑤𝑔(𝑡𝑖)𝑋(𝑡𝑖) 𝑁𝑔 𝑖=1 and 𝐴 = ∑ 𝑤ℎ(𝑠𝑗)𝑋(𝑠𝑗) 𝑁ℎ 𝑗=1 , their inner product in the RKHS can be computed in the input space from vectors {𝑤ℎ(𝑠𝑗)}𝑗=1 𝑁ℎ and {𝑤𝑔(𝑡𝑖)}𝑖=1 𝑁𝑔 (what is now called the kernel trick) as 〈ℎ, 𝑔〉 ℋ𝑅 = ∑ ∑ 𝑤ℎ(𝑠𝑗)𝑅(𝑠𝑗,𝑡𝑖) 𝑁𝑔 𝑖=1 𝑁ℎ 𝑗=1 𝑤𝑔(𝑡𝑖) = ∑ ∑ ℎ(𝑠𝑗)𝑟𝑠𝑗,𝑡𝑖 −1 𝑁𝑔 𝑖=1 𝑁ℎ 𝑗=1 𝑔(𝑡𝑖) (13) where 𝑟𝑠𝑗,𝑡𝑖 −1 is the 𝑠𝑗, 𝑡𝑖 element of the inverse of the covariance kernel 𝑅(𝑠𝑖, 𝑡𝑖) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=', the kernel modifies the traditional inner product of vectors in the input space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' This explains the nature of ℋ𝑅 quite well: because of the mapping 𝑅(𝑠,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' ), which contains the statistics of the data, the inner product in ℋ𝑅 takes advantage of the data statistics over time instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Hence, in the input space the solution must be a quadratic form employing 𝑅−1 as shown in (13) to meet the congruence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Theorem (from [4]): Let {𝑋(𝑡),𝑡 ∈ 𝑇} be a time series with covariance kernel 𝑅(𝑠, 𝑡), and let ℋ𝑅 be the corresponding RKHS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Between 𝐿2(𝑋(𝑡), 𝑡 ∈ 𝑇) and ℋ𝑅 there exists a one-to-one inner product preserving linear mapping under which a vector ℎ ∈ {𝑅(⋅,𝑡), 𝑡 ∈ 𝑇} and 𝑈 ∈ 𝐿2(𝑋(𝑡), 𝑡 ∈ 𝑇) are mapped into one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Denote by 〈ℎ, 𝑋〉ℋ𝑅 the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' in 𝐿2(𝑥𝑡,𝑡 ∈ 𝑇) which corresponds to the function ℎ ∈ ℋ𝑅 under the mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Then the solution of the prediction problem may be written as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' If 𝑍 is a r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' with finite second moments and 𝜌𝑍(𝑡) = 𝐸[𝑍𝑋(𝑡)] then 𝜌𝑍 ∈ ℋ𝑅, and (14) 𝐸∗[𝑍|𝑋(𝑡), 𝑡 ∈ 𝑇] = 〈𝜌𝑧,𝑋〉ℋ𝑅 (15) with prediction mean square error given by 𝐸[(𝑍 − 𝐸∗[𝑍 |𝑋(𝑡), 𝑡 ∈ 𝑇])2] = 𝐸[𝑍2] − 〈𝜌𝑧, 𝜌𝑧〉ℋ𝑅 (16) The equivalent minimum mean square error solution (15) in the data space, because of (13), becomes 𝑌 = 𝐸∗[𝑍|𝑋(𝑡),𝑡 ∈ 𝑇] = 〈𝜌𝑧, 𝑋〉ℋ𝑅 = ∫ ∫ 𝑅−1(𝑠, 𝑡)𝜌𝑍(𝑠)𝑋(𝑡)𝑑𝑠 𝑠∈𝑇 𝑑𝑡 𝑡∈𝑇 (17) which is exactly the Wiener solution ∫ 𝑋(𝑡)𝑤∗(𝑡)𝑑𝑡 𝑡∈𝑇 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Note that the effective role of this inverse operator is to decorrelate the input space data and it is a steppingstone for finding the orthogonal projection as demonstrated by Wiener.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' However, in ℋ𝑅 this solution for the prediction problem is coordinate free, does not use the approximation error, and directly uses the structure of ℋ𝑅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' In fact, it is sufficient to compute the linear projection of 𝜌𝑧(𝑠) with the input data because the covariance kernel 𝑅(𝑠, 𝑡) provides its statistics, unlike Wiener-Hopf method that requires spectral factorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' This coordinate free property of RKHS solutions with the covariance kernel was first noted by Loeve [20] who suggested that instead of finding a set of functional projections (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Karuhnen Loeve transform [21]) it is sufficient to employ the statistics of 𝑋(𝑡) embedded in the structure of the RKHS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Parzen [4] further states that for this reason “RKHS defined by the covariance kernel is the natural setting in which to solve problems of statistical inference on time series”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' These are fundamental results that will be very useful when seeking an extension of the theory to nonlinear solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' The fundamental issue with Parzen approach is twofold: first, it does not elucidate efficient alternatives to implement the conditional mean operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Moreover, from (13) we can see that the inverse may not always exist, needs to be accurate, and it is computationally expensive because it needs to be applied to every test sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Despite approximations for the inverse, this is cumbersome but a necessity for continuous time models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Second, for discrete time signal processing, this approach is computationally not competitive with the famous Wiener solution 𝑤∗ = 𝑅−1𝜌, where 𝑅 is the autocorrelation matrix (the kernel 𝑅(𝑠, 𝑡) evaluated at a finite set of times), which finds the optimal weighting 𝑤∗ only once in the training set using the error, and does an inner product in the data space of two vectors in the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Hence, we conclude that the advantage of the RKHS theory is on the mathematical tools of congruence and representation of time series in RKHS, which open the door to seek more general solutions such as the nonlinear prediction case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' In fact, the advantage of the RKHS theory is that the operations defined in the RKHS are independent of the kernel utilized, hence the key goal is to concentrate on designing proper kernels when the goal is nonlinear extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' The Nonlinear Prediction Case A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Kernel Adaptive Filtering The goal is to construct a function 𝑓: ℝ𝐿 → ℝ based on a real sequence {(𝒙𝑖,𝑑𝑖)}𝑖=1 𝑁 of examples (𝑥𝑖,𝑑𝑖) ∈ 𝑆 × 𝐷, where 𝐷 is a compact subset of ℝ and 𝑆 a compact subspace of ℝ𝐿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' As described below, the function 𝑓 ∈ ℋ𝑘 is obtained based on a positive definite kernel 𝜅: 𝑆 × 𝑆 → ℝ that defines a RKHS ℋ𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' A commonly employed kernel is the Gaussian kernel 𝐺(𝒙, 𝒙𝑖) = exp (− ‖𝒙 −𝒙𝑖‖2 2𝜎2 ), where 𝜎 is the kernel size or bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Kernel adaptive filtering (KAF) [14] implements nonlinear filtering on discrete time series by mapping the input sampled data {𝒙𝑖}𝑖=1 𝑁 to ℋ𝐺 using a positive definite kernel 𝐺, and using search techniques based on the gradient and or Hessian information to adapt functional parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' The Gaussian kernel maps each embedding vector 𝒙𝑖 of size 𝐿, to a function in ℋ𝐺, which we will also denote as 𝐺(𝒙𝑖,⋅), where the “⋅” in the second argument means that a data point is represented by a Gaussian function centered at 𝒙𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' The inner product in the RKHS of two such functions centered at 𝒙𝑖 and 𝒙𝑗 can be easily computed in the input space as a Gaussian kernel evaluation i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=', 〈𝐺(𝒙𝑖,⋅),𝐺(𝒙𝑗,⋅)〉ℋ𝐺 = 𝐺(𝒙𝑖,𝒙𝑗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' The ℋ𝐺 defined by the Gaussian is infinite dimensional and nonlinearly related to the input data space 𝑆 [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' For the case of samples from a stochastic process {𝑋(𝑡), 𝑡 ∈ 𝑇}, 𝐺(𝑋(𝑡),⋅) is a random function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' One notable example of KAF is the kernel least mean square (KLMS) algorithm, for which the non-linear filter output is simply given by 𝑦𝑛 = ∑ 𝜂𝑒𝑖𝐺(𝒙𝑛 − 𝒙𝑖) 𝑛−1 𝑖=1 (18) where \uf068 is the stepsize, 𝑒𝑖 is the error at iteration 𝑖, and {𝒙𝑖}𝑖=1 𝑛−1 are the past samples in the training set that constitute the “dictionary” to construct the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' This algorithm uses gradient search to construct the optimal function Ω∗, such that 𝑓∗(𝒙) = 〈𝐺(𝒙,⋅),Ω∗〉ℋ𝐺, and converges in the mean to the optimal least minimum square solution in ℋ𝐺 for small step sizes and large number of data samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' The appeal of the KLMS is that it is an online algorithm, does not need explicit regularization [23], and is a CULM (convex and universal learning machine) [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' However, because of the nonlinearity of the kernel mapping there is no congruence between the input space defined by the span of the time series and the RKHS ℋ𝐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' The solution needs to be expressed in terms of observations from the time series, which means that the order of the filter grows linearly in time, if no sparsification is included [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' This is a shortcoming of this class of algorithms because it affects the computation complexity in the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' In KAF, since the kernel evaluations are weighted by the error, the algorithm has an automatic way to preserve the scale of the representations when applying the kernel trick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' The ℋ𝐺 defined by the Gaussian kernel differs from the ℋ𝑅 defined by Parzen’s covariance kernel in four fundamental ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' • First, Parzen used a “linear” kernel ℋ𝑅 yielding a close form optimal linear model in 𝐿2 as mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Second, the Parzen kernel is computed by employing the expected value over data lags 𝑠 = 𝑡 − 𝜏 to take advantage of the wide sense stationarity of the time series, unlike the pairwise sample set as ℋ𝐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' • Third, the map to ℋ𝐺 is stochastic because samples are mapped from a random process rather than mapping the elements of the index set 𝑇, directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' In contrast, the map to ℋ𝑅 is deterministic because of the congruence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' • Fourth, ℋ𝐺 is infinite dimensional, while in ℋ𝑅 is a finite dimensional RKHS space defined by the number of lags required for the covariance kernel, which is dictated by the input data dynamics (normally small).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Our goal now is to define a new RKHS that preserves the correlation structure defined by the data as ℋ𝑅, but also maps the time series by a nonlinear kernel to achieve CULM properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' To be practical, this approach uses the kernel trick to perform the computation in the input space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Definition of the Correntropy RKHS Let {𝑋(𝑡),𝑡 ∈ 𝑇} be a strictly stationary stochastic process (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=', the joint PDF {𝑝𝑠,𝑡(𝑥𝑠,𝑥𝑡) } is unaffected by a change of the time origin, that is 𝑝𝑠,𝑡(𝑥𝑠,𝑥𝑡) = 𝑝𝑠−𝜏,𝑡−𝜏(𝑥𝑠,𝑥𝑡) ) with T being an index set and 𝑥𝑡 ∈ ℝ𝐿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' The autocorrentropy function 𝑣(𝑠, 𝑡) is defined as a function from 𝑇 × 𝑇 → ℝ given by 𝑣𝜎(𝑠,𝑡) = 𝐸𝑠,𝑡[𝐺𝜎(𝑋(𝑠), 𝑋(𝑡))] = ∬ 𝐺𝜎(𝑥𝑠,𝑥𝑡)𝑝𝑠,𝑡(𝑥𝑠,𝑥𝑡)𝑑𝑥𝑠𝑑𝑥𝑡 (19) where 𝐸𝑠,𝑡[⋅] denotes mathematical expectation over a pair of r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' in the time series {𝑋(𝑡),𝑡 ∈ 𝑇} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' While it is true that any symmetric positive definite kernel (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=', Mercer kernel) 𝜅(𝑥𝑠,𝑥𝑡) can be employed instead of the Gaussian kernel 𝐺𝜎, the symmetry, scaling, and translation invariant properties of 𝐺𝜎, confer additional properties and interpretation to correntropy, which are reviewed in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' The autocorrentropy function defined in (19) is a reproducing kernel on the index set 𝑇 × 𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' We will denote its corresponding RKHS by ℋ𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' The functions 𝑣𝜎(𝑠,⋅) are in ℋ𝑣 and 𝑣𝜎(𝑠,𝑡) = 〈𝑣𝜎(𝑠,⋅),𝑣𝜎(𝑡,⋅) 〉ℋ𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Another space that can be defined by the composition of the random variable 𝑋(𝑡) and the positive definite Gaussian kernel 𝐺𝜎(⋅,⋅) is the span of the set of random elements {𝐺𝜎(𝑋(𝑡),⋅),𝑡 ∈ 𝑇} taking values in ℋ𝐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' We will denote this space by ℋ𝑅𝐺 and the inner product between two elements 𝑈 = ∑ 𝛼𝑖𝐺𝜎(𝑋(𝑡𝑖),⋅) 𝑖 and 𝐴 = ∑ 𝛽𝑗𝐺𝜎(𝑋(𝑠𝑗),⋅) 𝑗 is given by ⟨𝑈, 𝐴⟩ℋ𝑅𝐺 = 𝐸[〈∑ 𝛼𝑖𝐺𝜎(𝑋(𝑡𝑖),⋅) 𝑖 , ∑ 𝛽𝑗𝐺𝜎(𝑋(𝑠𝑗),⋅) 𝑗 〉ℋ𝑅𝐺] = ∑ 𝛼𝑖𝛽𝑗𝐸[𝐺𝜎(𝑋(𝑡𝑖), 𝑋(𝑠𝑗))] 𝑖𝑗 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' There is a congruence between ℋ𝑅𝐺 and ℋ𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Moreover, we see that for strictly stationary time series making 𝑠 = 𝑡 − 𝜏, the function 𝑣𝜎 can also be written as a function of 𝜏 only as follows: 𝑣𝜎(𝜏) = 𝐸𝑡,𝑡−𝜏[𝐺𝜎(𝑋(𝑡), 𝑋(𝑡 − 𝜏))], (20) where any 𝑡 ∈ 𝑇can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' This shows its similarity with the Parzen covariance kernel of (11), except that 𝑣𝜎(𝜏) is computed in ℋ𝑣, a space nonlinearly related to the original time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' The autocorrentropy functional can then be interpreted in two vastly different feature spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' One is the RKHS ℋ𝐺 induced by the Gaussian kernel on pairs of observations 𝐺𝜎(⋅,⋅), which is widely used in kernel learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' The elements of this RKHS are infinite-dimensional vectors expressed by the eigenfunctions of the Gaussian kernel, and they lie on the positive hyperoctant of a sphere because ‖𝐺𝜎(𝑥, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' )‖2 = 𝐺𝜎(0) = 1/√2𝜋𝜎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' The correntropy functional performs statistical averages on the functionals in this sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' The second feature space is the RKHS ℋ𝑣 induced by the correntropy kernel 𝑣(𝑠, 𝑡), which is defined on the index set of the random variables in the time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' This inner product is defined by the correlation of the kernel at two different lags and the mapping produces a single deterministic scalar for each element on the index set, that is, the practical dimension of ℋ𝑣 is the size of the index set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' ℋ𝑣 has very nice properties for statistical signal processing: • ℋ𝑣 provides a straightforward way to apply optimal projection algorithms based on mean statistical embeddings that are expressed by inner products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' • The effective dimension of ℋ𝑣 is under the control of the designer by selecting the number of lags (just like with the RKHS defined by the autocorrelation function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' • Elements of ℋ𝑣 can be readily manipulated algebraically for statistical inference (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' without taking averages over realizations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' • ℋ𝑣 is nonlinearly related to the input space, unlike the RKHS defined by the autocorrelation of the random process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Therefore, it is in principle very appealing for nonlinear statistical signal processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' The table presents the different types of RKHS defined so far that summarize our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Table I RKHS Functional Mapping Hilbert Space Characteristics ℋ𝑅 Parzen 𝐸[𝑋(𝑡),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' ] Linear mapping of data,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' size of lags,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' deterministic functions ℋ𝐺 Gaussian 𝐺(𝑥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='⋅) Nonlinear mappings of data,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' infinite dimensional,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' random functions ℋ𝑅𝐺 Random Gaussian 𝐺(𝑋(𝑡),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='⋅) Nonlinear mapping of data,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' size of lags,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' random functions ℋ𝑣 Correntropy 𝑣𝜎(𝑡,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='⋅) Nonlinear mapping of data,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' size of lags,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' deterministic functions Representing an Unobservable Random Variable in ℋ𝑅𝐺 Like the original problem where the random variable 𝑍 was approximated by a random variable in the span of the time series {𝑋(𝑡),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='𝑡 ∈ 𝑇} by the Hilbert projection theorem,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' we can define the approximation in the space of random elements ℋ𝐺 as follows 𝜉∗ = argmin 𝜉 𝐸[‖𝐺(𝑍,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='⋅) − 𝜉‖ℋ𝐺 2 ],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' (21) where 𝜉 is a random element in the span of {𝐺(𝑋(𝑡),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='⋅),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='𝑡 ∈ 𝑇}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Solving for 𝜉 gives rise to the following equation: 𝐸[〈𝐺𝜎(𝑍,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='⋅),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' 𝐺𝜎(𝑋(𝑠),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='⋅)〉ℋ𝐺] = 𝐸[〈𝜉,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' 𝐺𝜎(𝑋(𝑠),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='⋅)〉ℋ𝐺],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' (22) where 𝜉 is expressed as a linear combination of elements in ℋ𝑅𝐺,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' 𝜉 = ∫ 𝐺𝜎(𝑋(𝑡),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='⋅)𝑤(𝑡)𝑑𝑡 𝑇 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' (23) Then the weighting function 𝑤∗ of the best predictor must satisfy: 𝐸[∫ 𝑤∗(𝑡)〈𝐺𝜎(𝑋(𝑡),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='⋅),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='𝐺𝜎(𝑋(𝑠),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='⋅)〉ℋ𝐺𝑑𝑡 𝑇 ] = 𝐸[〈𝐺𝜎(𝑍,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='⋅),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='𝐺𝜎(𝑋(𝑠),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='⋅)〉ℋ𝐺],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' which gives rise to the functional Wiener equation: ∫ 𝑤∗(𝑡)𝑣𝜎(𝑡,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='𝑠)𝑑𝑡 𝑇 = 𝐸[〈𝐺𝜎(𝑍,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='⋅),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' 𝐺𝜎(𝑋(𝑠),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='⋅)〉ℋ𝐺] = 𝜌𝑍(𝑠),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' (24) These equations state that one can always find a representation for the function 𝜌𝑧(𝑠) in terms of the functions {𝑣𝜎(𝑡,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='⋅),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='𝑡 ∈ 𝑇} because the best correntropy predictor is computed in the span of the set {𝐺𝜎(𝑋(𝑡),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='⋅),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' 𝑡 ∈ 𝑇}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Nevertheless, because this computation is carried out in the correntropy RKHS, the best approximation to 𝑍 cannot be directly obtained since the input space where the time series lies is nonlinearly related to the correntropy RKHS where we compute the projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Solution of the Representation Problem in ℋ𝑣 To solve the representation problem in ℋ𝑣, that is, finding 𝑤∗(𝑡), let us consider the representation 𝜁𝑠 in ℋ𝑣 of the random element 𝐺𝜎(𝑋(𝑠),⋅) that can be obtained by the congruence between ℋ𝑣 and ℋ𝐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' From equation (24) we have that: 𝜌𝑧(𝑠) = 〈𝜌𝑧, 𝜁𝑠〉ℋ𝑣 = ∫ 𝑤∗(𝑡)〈𝜁𝑡, 𝜁𝑠〉ℋ𝑣𝑑𝑡 𝑇 (25) This defines a close form functional Wiener filter solution in ℋ𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Notice that the formulation is the same as (17), the only difference is the structure of the inner product space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' The relation between ℋ𝑅𝐺 and ℋ𝑣 is rather similar to the relation between ℝ𝐿 and ℋ𝑅 so, for two elements ℎ and 𝑔 in ℋ𝑣, 〈ℎ, 𝑔〉ℋ𝑣 = ∫ ∫ ℎ(𝑠)𝑣𝜎 −1(𝑠, 𝑡)𝑔(𝑡)𝑑𝑠 𝑠∈𝑇 𝑑𝑡, 𝑡∈𝑇 (26) where 𝑣𝜎 −1(𝑠, 𝑡) is the element of the inverse of the correntropy operator defined as (𝑉𝜎𝑔)(𝑠) = ∫ 𝑔(𝑡)𝑣𝜎(𝑠, 𝑡)𝑑𝑡 𝑡∈𝑇 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' The above form can be used to compute a solution to (25) as, 𝑤∗(𝑡) = ∫ 𝜌𝑧(𝑠)𝑣𝜎 −1(𝑠, 𝑡)𝑑𝑠 𝑠∈𝑇 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' (27) In this case the solution is nonlinear in the input space, so this is a very elegant extension of Wiener theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' A major difference to KAF and the Wiener filter in the data space, is that this solution never uses the error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' The reason is that Parzen’s solution decorrelates implicitly the data (in this case in ℋ𝑅𝐺) and automatically finds the orthogonal projection on the data manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' However, not everything is perfect with the solution (26), since we cannot extend the congruence in (25) to the original time series {𝑋(𝑡), 𝑡 ∈ 𝑇}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' 〈𝜁𝑡, 𝜁𝑠〉 ℋ𝑣 = 𝐸[𝐺𝜎(𝑋(𝑡),𝑋(𝑠))] ≠ 𝐸[𝑋(𝑡)𝑋(𝑠)] (28) because the kernel mapping does not preserve the inner product, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' 〈𝑥𝑛,𝑥𝑖〉 ≠ 〈𝐺(𝑥𝑛,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' ), 𝐺(𝑥𝑖,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' )〉 ℋ𝐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Computation of the Functional Wiener Filter in ℋ𝐺 How can the solution in (26) be implemented from a sample data stream?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' In this case, we restrict our treatment to discrete-time time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Let us start by assuming that the time series is ergodic, such that expected values can be estimated by temporal averages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Second, because of the congruence (25), 〈𝜁𝑡, 𝜁𝑡−𝜏〉ℋ𝑣 can be substituted by 𝐸[𝐺𝜎(𝑋(𝑡), 𝑋(𝑡 − 𝜏))] and by ergodicity, it can be estimated from samples {𝑥(𝑡)}𝑡=1 𝑁 over a window of length 𝑁.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' 𝑣𝜏 = 1 𝑁 ∑ 𝐺𝜎(𝑥(𝑡), 𝑥(𝑡 − 𝜏)) 𝑁 𝑡=1 (29) For 𝜏 = 0,1,⋯ , 𝐿 − 1, 𝑣𝜏 is the 𝜏th entry of the autocorrentropy vector and can be used to construct the autocorrentropy matrix of size 𝐿 × 𝐿 as follows: 𝑉 = [ 𝑣0 ⋯ 𝑣𝑇−1 ⋮ ⋱ ⋮ 𝑣𝑇−1 ⋯ 𝑣0 ] (30) This matrix is unlike anything in kernel adaptive filtering, because it is a matrix of scalar values that can be computed once from the training set and never changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' This matrix is very unusual in kernel filtering, where the filters always increase in size with each new sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' The values of the correntropy matrix can be centered in RKHS if necessary [28]: 𝑣̅𝜏 = 𝑣𝜏 − 1 𝑁2 ∑ ∑ 𝐺𝜎(𝑥(𝑡), 𝑥(𝑠)) 𝑁 𝑠=1 𝑁 𝑡=1 (31) The other major difference is that in KAF, one needs to transfer vectors of samples to the RKHS, where the size of the vector is an estimate of the embedding dimension of the system that created the time series, using Takens’ embedding theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' The reason is that the KLMS is a pairwise instantaneous algorithm, so if it is applied to each sample of the input data the algorithm loses the local time structure of the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' For FWF, the data can be mapped to RKHS sample by sample, just like in the input space, because the formulation uses the correntropy matrix where the lag structure is included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Let us now show how to estimate the cross correlation functional 𝜌𝑧 in ℋ𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Using the same approximations as the ones for the correntropy matrix yields 𝜌̂𝑧(𝜏) = 1 𝑁 ∑ 𝐺𝜎(𝑥(𝑡 − 𝜏), 𝑧(𝑡)) 𝑁 𝑡=1 (32) This is the only term that relates the target and the input signals, and it only needs to be evaluated in the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' The optimal weighting vector in (27), 𝑤∗ (𝜏) for 𝜏 = 0,2, … , 𝐿 − 1, is obtained by solving the system: 𝜌𝑧(ℓ) = ∑ 𝑉ℓ+1,𝜏+1 𝐿−1 𝜏=0 𝑤(𝜏).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' (33) In other words, 𝑤∗ = 𝑉−1𝜌𝑧 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' During testing, the output of the filter corresponds to an instance of the random element ∑ 𝑤∗ (𝜏)𝐺𝜎(𝑋(𝑡 − 𝜏),⋅) 𝐿−1 𝜏=0 , which is the best approximation to 𝐺𝜎(𝑍,⋅), namely, 𝜉∗ (𝑡) = ∑ 𝑤∗(𝜏)𝐺𝜎(𝑥test(𝑡 − 𝜏),⋅) 𝐿−1 𝜏=0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' (34) where 𝑥test(𝑡) is the test input at time 𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' This solution shares the form of (6) in 𝐿2(𝑋(𝑡), 𝑡 ∈ 𝑇) and (23) in ℋ𝑅𝐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='The big difference is that the autocorrelation function was substituted by the correntropy function, while the input vector [𝑥(𝑡),𝑥(𝑡 − 1),⋯ , 𝑥(𝑡 − 𝐿 + 1)] was substituted by a vector of functions nonlinearly related to the input space (the feature space defined by the Gaussian kernel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Notice that this solution is quite different from the KAF in several important ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' First, the optimal weight vector can be computed in the input space, and it appears as a scale factor to change the finite range of the Gaussian to span the values of the target response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Notice that this weighting depends on the actual local L sample history of the current input, but it is nonlinear and so it is more powerful than the linear weighting in linear Wiener filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Second, there is no sum over the training set samples in the optimal solution like in KAFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' The best approximant is a combination of just L Gaussian functions centered at the current test sample, which is a major simplification in computation when compared with KAF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' This algorithm has the complexity of the Wiener solution, and should be an universal approximator when the number of delays grows to infinity, but we have not formally proved this statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Unfortunately, the output of the functional Wiener filter 𝜉∗ (𝑡) is still in ℋ𝐺, so the task of implementing a filter in the data space is still not finalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Preimage to Estimate the FWF output in the input space Ideally, the output of the FWF in the input space would correspond to the inverse map from ℋ𝐺 to ℝ𝑑, where 𝑑 = 1 in the simplest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Since (34) expresses the optimal filter solution as a linear combination of Gaussian function, the goal is just to evaluate the function at a point in the input space, whose image is closest to the optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' However, there is no guarantee such inverse map exists, so we must resort to an extra optimization or approximation step to find a pre-image [17] of the optimal solution in the input space, as will be explained next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Preimage using the optimal filter output in 𝓗𝑮 For the FWF, the basic concept is to use an approximate pre-image in the input space of the optimal filter output in ℋ𝐺 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=', the approximated FWF output to 𝑦∗(𝑡) will be given by: 𝑦(𝑡) = argmin 𝑦 ∈ ℝ𝑑 ‖𝐺𝜎(𝑦,⋅) − 𝜉∗ (𝑡)‖ℋ𝐺 2 (35) This formulation can be applied in practical settings because in a training set, the optimal weight vector can be estimated using the 𝑉 matrix from (33) and the cross correntropy from (32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Therefore, and according to (35) it is only required to find the point to evaluate the optimal weight function, which is equivalent to find the minimum of ∑ 𝑤∗(𝜏)𝐺𝜎(𝑥test(𝑡 − 𝜏), 𝑦) 𝐿−1 𝜏=0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' (36) Making the gradient of (36) with respect to 𝑦 equal to zero yields the fixed-point expression 𝑦(𝑖+1) = ∑ 𝑤∗(𝜏)𝐺𝜎(𝑥test(𝑡−𝜏),𝑦(𝑖))𝑥test(𝑡−𝜏) 𝐿−1 𝜏=0 ∑ 𝑤∗(𝜏)𝐺𝜎(𝑥test(𝑡−𝜏),𝑦(𝑖)) 𝐿−1 𝜏=0 , (37) where 𝑦(𝑖) denotes the estimate of the preimage at the 𝑖th iteration of the fixed-point update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Notice that the nature of the pre-imaging solution involves a search on top of the analytic solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' This solution will be named FWFFP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Preimage using local models Intuitively, the goal is to select training set input samples that, when combined with the current test sample, provide functional evaluations in RKHS that approximate the targets in the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' The difficulty is that during testing there is no information about the target value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Therefore, one simple option is to use the similarity in the input space to cluster locally the input samples that provide the best approximation to the target signal during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' This approach was inspired by [29], where a successful table lookup approach was employed to extend linear model performance that links input samples to their errors in the training set to create outputs outside the span of the input space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Here, the approach is to find an input sample 𝑥(𝑚) that when combined with the current input 𝑥(𝑖), will produce an output in ℋ𝐺 that is close to its target 𝑧(𝑖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Let us represent 𝑧̂(𝑖) = ∑ 𝑤∗(𝜏)𝐺𝜎(𝑥 (𝑖 − 𝜏), 𝑥(𝑚 − 𝜏)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' 𝐿−1 𝜏=0 The optimization can be written as 𝑎𝑟𝑔 min 𝑥(𝑚)∈𝑆 ‖𝑧(𝑖) − 𝑧̂(𝑖)‖ (38) where S is the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' So, we need to implement a search (done once), where we find the sample pair (𝑥(𝑖),𝑥(𝑚)), 𝑖 = 1,… 𝑁 that produces the closest approximation to the target sample 𝑧(𝑖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Once in testing, we find the closest sample 𝑥(𝑖) in the training set to 𝑥(𝑡𝑒𝑠𝑡) and use its neighbor 𝑥(𝑚) to plug in (34) to obtain the FWF output as 𝑦(𝑡) = 𝑧𝑖 𝑧̂𝑖 ∑ 𝑤∗(𝜏)𝐺𝜎(𝑥 (𝑚 − 𝜏), 𝑥(𝑡)) 𝐿−1 𝜏=0 (39) where the ratio 𝑧𝑖 𝑧̂𝑖 ⁄ enforces the scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' This search needs to be done online for every test sample, but if we rank the training set, it can be done quickly with a tree search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' This process can be repeated K times for a better approximation, where K is a hyper-parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' The idea is to probe the neighborhood of 𝑥(𝑡𝑒𝑠𝑡) with K input samples {𝑥(1),… 𝑥(𝐾)} and use their respective neighbors using (38) to compute K approximate targets {𝑧̂(1),… .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' 𝑧̂(𝐾)} and represent their mean by 𝑧̅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' The final FWF output will be 𝑦(𝑡) = ∑ 𝑧𝑘 𝑧̅ 𝐾 𝑘=1 ∑ 𝑤∗(𝜏)𝐺𝜎(𝑥 (𝑘 − 𝜏), 𝑥(𝑡)) 𝐿−1 𝜏=0 (40) Since the filter computation is so small, this improves performance with a minor increase in computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' The computational complexity of FWFFP and FWFLM are compared in the following table (i = iterations, M = fixed point updates).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Table II Filter Complexity (training/testing) Memory (training/ testing) KLMS O(i) O(i) KRLS O(i2) O(i2) FWFFP O(L2N)/ O(L) + O(LM) O(N+L2)/O(L) FWFLM O(L2N) + O(N)/ O(KL) + O(logN) O(2N+L2)/ O(2NL+L2) E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Experimental Results FWF Implementation Challenges There are several challenges for the FWF implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' The first issue is numeric instability and deals with the inverse of the correntropy matrix 𝑉 in (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Large condition numbers will bias the solution and need to be corrected through regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' The second issue stems from the fundamental fact that learning models must generalize well outside the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Note that there is no error in the FWF methodology, so this presents a different problem than in conventional machine learning where the regularization can be controlled by penalty terms in the cost function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' In the FWF, generalization is controlled by the kernel size, and by the model order, the two hyper- parameters in the design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' It is easy to see that small kernel sizes yield a correntropy matrix that approaches a scaled identity matrix, 𝑎𝐼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' This is because when kernel sizes are small, correntropy will peak when signals exactly match, and become very small when signals do not match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' This increases specificity in the training set and also simplifies conditioning of the 𝑉 matrix, but it requires a large number of samples in the training set and a huge dynamic range in the computation to avoid losing information in the higher lags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Therefore, small kernel sizes limit the number of lags that can be used in practice to represent the input space data correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Hence, in order to capture long time dependencies amongst the lags in a stationary signal, we must use larger kernel sizes in ℋ𝐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' However, if the kernel size is too large then the behavior of the correntropy function will approach the behavior of the auto-correlation function i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=', we lose the specificity provided by the higher order moments of the data PDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' The other drawback of employing larger number of lags is that the chances of ill- conditioning in the correntropy matrix increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Hence, these trade-offs mean that kernel size selection and regularization of 𝑉 are vital for the performance of the FWF, and the kernel size becomes the key parameter for generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Regularization of the Correntropy Matrix We concluded that larger kernel sizes are needed to preserve information over the lags of the 𝑉 matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' This means that 𝑉 will be more ill-conditioned, which can be quantified by the matrix’s condition number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' It is important to note that while regularization is helpful, we also need to control the number of lags to obtain optimal performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' The regularization of the 𝑉 matrix is depicted in equation (41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Our goal is to find a \uf067 such that the condition number of Vreg is approximately equal to some desired condition number, which becomes a FWF hyper-parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' 𝑉𝑟𝑒𝑔 = 𝑉 + 𝜆𝐼;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' 𝜆 = 𝛾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' min 𝐸𝑖𝑔𝑉𝑎𝑙𝑢𝑒 (𝑉) (41) Using this framework, we found that condition numbers below 30 worked well, which is quite restrictive, but can be expected because we expect tiny errors in prediction to make FWF competitive with KAF approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' These low condition numbers require a large amount of regularization, which unfortunately does not utilize all the information in the 𝑉 matrix affecting the accuracy of the FWF predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Initial FWF Results: The Mackey-Glass Time Series The Mackey-Glass (MG) times series is a chaotic time series, generated by 𝑑𝑥(𝑡) 𝑑𝑡 = −𝑏𝑥(𝑡) + 𝑎𝑥(𝑡 − 𝜏) 1 + 𝑥(𝑡 − 𝜏)10 The MG times series used in the following experiments was generated with b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='1, a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='2, and \uf074 = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Experiments testing the KLMS and KRLS kernel adaptive filters with this time series can be found in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' One of the hyper-parameters of the FWF is number of lags (L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' This defines the length of the correlation time used to represent each sample, very similar to the Wiener model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Each sample is represented by a vector of length L with the form [𝑥(𝑖), … 𝑥(𝑖 − 𝐿 − 1)] 𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' This is standard practice for time series prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' The second hyper-parameter is the kernel size of ℋ𝐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' To estimate the dependence of performance on hyperparameters, the parameters are scanned and plotted with training set data to obtain the performance surface of the FWFLM, with two different local model orders (K = 5 and 15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' We can see in Figure 1 that the two local model orders provide basically the same results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' The minimum is obtained around L = 7 delays, and the minimum trough is around \uf073 = \uf031\uf02e\uf035\uf02c which is much larger than the corresponding KAF filters for the same time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' We also see that the best error is on the order of 10-3 (log 10) which is better than the Wiener filter of the same order for this data set (MSE = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Error performance surface over the two FWF hyper-parameters (kernel size and number of lags), estimated with two different local model orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Experiments with FWFFP and FWFLM In this section, the performance of the FWF with both pre-imaging methods described above is compared with two well-known KAF methods, kernel recursive least squares (KRLS) and KLMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Figure 2 compares the average test set MSE across 5-folds of cross validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' The best kernel size from Figure 1 was employed (\uf073 =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' The figure shows performance with two TrainingvsKSandLag,N=2000,K=5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='4 Log Error -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='6 -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='8 -3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='0 5 10 15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='5 Lags 20 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='0 25TrainingvsKSandLag,N=2000,K=15 -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='0 -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='2 Error -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='4 Log -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='6 -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='8 -3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='5 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='0 10 15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='5 Lags 20 25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='0different values of K for the FWFLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' We also present results with K=1 for a direct comparison with the FWFFP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' The number of lags considered for FWFLM, was L = 7 the same as embedding for KLMS, and KRLS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' The performance for the FWFFP is the worst, and it improves slightly with the number of lags, therefore the figures below show results with L=25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Notice that FWFLM with K=1 is much better than the fixed-point update and here rivals the performance for higher number of local models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Notice that, as expected, there is no variation with the number of local models in the FWFFP because the method uses an optimization to find the minimum, so the solution only depends on L, \uf073, and the number of samples in the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' The FWFLM approaches the performance of KLMS, but it is far worse than KRL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Remember that the FWF was derived under a strict stationarity assumption, which is not fulfilled by the MG time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Therefore, this result is quite reasonable, taking in consideration the FWF much smaller computation complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Comparisons of predictions for two different selections of local models (K) as a function of the number of samples in the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Asymptotic performance occurs after 1000 samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' For K=1 performance is much better than fixed point pre imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' More models worsen the prediction results on MG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Noisy Mackey-Glass Prediction: In this experiment the FWF with both pre-imaging methods, KLMS and KRLS are predicting the MG time series, but with white Gaussian noise added to the input signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Each algorithm is given a noisy training and testing input, and the desired signal is the next time point 𝑥(𝑡 + 1) with no added noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' White Gaussian noise with standard deviations of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='04, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='1, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='2 were tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Five-fold cross validation was used for each algorithm at each noise level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' The best kernel size is shown for each algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' In general, FWFLM is better than KLMS and KRLS at higher noise levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' The number of training samples does not have a great effect on the final MSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Again, the performance of FWFFP is evaluated at L = 25 while FWFLM use L = 5 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' MackeyGlass:TestMSE,L=7,K=5 10-2 10-3 TestMSE 10-4 FWFLM,ks=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='5 FWFFp,kS=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='0 KLMS,ks=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='25 KRLS,kS=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='25 10~5 FWFLM,K=1 250 500 750 1000 1250 1500 1750 2000 TrainingSamples (N)MackeyGlass:TestMSE,L=7K=15 10-2 下 10-3 TestMSE 10-4 FWFLM,ks=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='5 FWFFP,kS=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='0 KLMS,kS=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='25 KRLS,kS=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='25 10-5 FWFLM,K=1 250 500 750 1000 1250 1500 1750 2000 TrainingSamples(N) Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' FWF has better robustness when noise is added to the time series, as we can expect from the use of multiple delays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Lorenz Prediction: We decided to test the performance of the FWF in the prediction of a more complex chaotic dynamical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' The Lorenz system is a well-known system introduced in [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' We use the x component of the Lorenz attractor and to make the problem harder, the model predicts 𝑥(𝑡 + 10) e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' 10 samples ahead with the last L samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' A version of this experiment can be found in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Like the previous experiments, the FWFFP was evaluated at L = 30, which is larger than the other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' The FWF 𝐿𝑀 outperforms KLMS for low number of lags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' This difference shrinks as we consider more lags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' As in the other experiments, FWFFP does not perform well when compared to the other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' In the Lorenz system, FWFLM performs at the level or better than the KLMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Notice that this time series is far from stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' MackeyGlass:TestMSEvsNoiseVariance,L=7,K=5,N=1oo0 10 2 上 Test MSE 10 3 FWFLM, kS=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='0 壬壬壬 FWFp, ks=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='0 KLMS, kS=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='25 KRLS, kS=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='5 104 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='175 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='200 Noise VarianceMackeyGlass:TestMSEvsNoiseVariance,L=7,K=5,N=2000 10 2 Test MSE 103 王 FWFLM,kS=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='0 壬壬壬 FWFp, ks=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='0 KLMS,ks=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='25 10 4 KRLS, kS=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='175 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='200 Noise VarianceMackeyGlass:TestMSEvsNoiseVariance,L=5,K=5,N=1oo0 Test MSE 102 王 FWFLM,ks=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='5 壬壬壬 103 FWFfp,ks=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='0 KLMS,ks=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='25 KRLS,ks=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='175 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='200 Noise VarianceMackeyGlass:TestMSEvsNoiseVariance,L=5,K=5,N=2000 Test MSE 10~2 FWFLM,ks=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='5 10 3 壬壬壬 FWFfp,ks=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='0 KLMS,ks=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='25 KRLS,kS=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='175 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='200 Noise Variance Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Comparison of performance in the Lorenz time series prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' For this time series FWFLM performs better than KLMS but by a small margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Further Analysis on Mackey-Glass sample by sample predictions Figures 5 shows the training and testing predictions compared to the desired with L = 7, kernel size of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='5, and two different local models K = 5 and 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' In both, the prediction is worse in the parts of the Mackey-Glass series that are more non-stationary (the small ripple across the signal), but the smoothing effect of using many local models is clearly visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' This explains why K=1 does such a good job in this signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' This makes sense since when the model is more localized, the dependency on the stationarity constraint is reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Lorenz:TestMSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='L=10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='K=5 100 王 王 王 工 10-1 TestMSE 10-2 FWFLM, ks=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='5 10-3 壬壬壬 FWFFp, kS=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='5 KLMS,ks=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='25 KRLS,ks=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='25 250 500 750 1000 1250 1500 1750 2000 Training Samples (N)Lorenz:TestMSE,L=15,K=5 100 王 王 王 工 10-1 TestMSE 10-2 FWFLM,ks=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='1 FWFFP,ks=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='5 10-3 KLMS,kS=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='25 KRLS,kS=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='5 250 500 750 1000 1250 1500 1750 2000 TrainingSamples(N)Lorenz: Test MSE, L = 7,K =5 100 王 王 王 工 T 10-1 Test MSE 工 工 10-2 FWFLM,ks=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='1 FWFFp, kS=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='5 10-3 KLMS,ks=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='1 KRLS, ks=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='25 250 500 750 1000 1250 1500 1750 2000 Training Samples (N)TestingPredictionsvsDesired 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='0 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='2 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='4- Desired -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='6 Predictions 0 25 50 75 100 125 150 175 200TestingPredictionsvsDesired 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='0 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='2 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='4 Desired -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='6 Predictions 0 25 50 75 100 125 150 175 200Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Sample by sample comparisons of predictions with the FWFLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Most of the errors occur in the time varying ripple superimposed in the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Notice that less local models perform better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Further Prediction Analysis on Lorenz: Figures 6 shows predictions made by the FWFLM on the Lorenz time series described in the above section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' The hyperparameters here are L = 7, \uf073 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='1, with two local models, of order K = 5 and K = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' It is obvious that when the number of local models increases, samples too far away from the optimal solution will average out the response of the FWF, degrading the prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' It is also interesting that the errors at the bottom of the signal ae not smooth, showing that there are not enough good neighbors in the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Averaging effect in the quality of the prediction when too many local models are employed (K=5 left, versus K=100 on the right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Conclusions The main objective of this paper is to find a principled way to include the input data statistics in the inner product of a universal RKHS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Recall that KAFs use a data independent kernel (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Gaussian) to project the data to define in the RKHS, the functional that implements the optimal model for the application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' At test time for online applications, these functionals grow linearly with the number of samples, which is impractical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' In practice, sparcification techniques must be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' The hypothesis is that a data dependent kernel will substitute the current KAF methodologies and simplify a lot the functional form to achieve an equally performing model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Parzen inspired this extension by showing that the ACF of a stationary random process is a positive definite kernel where optimal statistics models can be implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Once in this RKHS, a simple orthogonal projection is sufficient to find the optimal solution, unlike the incremental solution of KAF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' However, the ACF kernel spans the input data space, so the RKHS solution is still a linear model with complexity higher than the Wiener filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' With this observation, the goal of this paper can be stated as extending Parzen’s work to universal models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' The paper shows how to accomplish this task by defining the positive definite correntropy kernel as the inner product in a novel RKHS ℋ𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' The advantage is that functionals in ℋ𝑣 represent universal mapping functionals (for infinite number of lags), extending Parzen’s result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' The dimension of ℋ𝑣 is controlled by the number of delays of the autocorrentropy function, so this space is vastly different from the RKHS created by the Gaussian function, with the promise of TestingPredictionsvs Desired 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='0 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='5 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='0 Desired -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='5 Predictions 0 25 50 75 100 125 150 175 200TestingPredictionsvsDesired 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='5 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='0 Desired 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='5 Predictions 0 25 50 75 100 125 150 175 200decreasing the computational complexity of the implementation at test time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' The paper presents the analytical solution of the FWF in ℋ𝐺, but we were unable to find a way to use the kernel trick to obtain the input space filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' The difficulty is that ℋ𝐺 is not congruent with L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Two pre-imaging techniques are proposed to implement the FWF in the input space, which are both approximated solutions, but they differ in the method and in the computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' FWFFP uses a fixed-point iteration to find the best solution to evaluate the functional in ℋ𝐺, but since one single Gaussian is unable to model well a sum of Gaussian at different centers, more sophisticated optimization methods are needed for good performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' The training set is never used in this pre-imaging technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' The FWFLM on the other hand uses the training set data to find pairs of samples that approach the best solution in the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' This requires a search across the training set to find the best sample pairs to match the target response, but the method avoids the difficulty of FWFFP fixed-point iteration by averaging local models obtained in the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' The simplest of the FWFLM with K = 1 may be applicable for many nonlinear applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' The FWFLM was found experimentally more accurate than the linear Wiener filter and is on par with the KLMS performance, but it is still substantially worse than KRLS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' As an advantage, the FWF filter is far more efficient computationally than KAF implementations and uses less memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Th FWFFP is of the same complexity as the Wiener filter but requires a recursive optimization at each iteration, which is not very expensive computationally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' The FWFLM requires a search at the training time to match pairs of samples to find the pre image, and at test time, a search to find the closest training sample to the current test input (which is O(log N) if the input training samples are ordered by amplitude).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Hence, we conclude that this paper does not solve all the issues and should be considered as a first attempt to develop a new class of universal mappers in RKHS that integrate the data statistics in the kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' But the novelty of the technique brings fresh ideas to statistical signal processing that need also to be further investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' For instance, the FWF never employs the error, which is critical in KAFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Effectively, FWF only works with the minimum norm (orthogonal) projection in RKHS, so it is “model agnostic”: the important step is to create a RKHS that includes the data statistics (in the form of its autocorrentropy function) in the inner product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Of course, this construction requires “parameters” that are the ACF values of the input data and the CCF with the target, and the number of lags, just like least squares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' After this construction, the FWF just finds the best local projection in the optimal RKHS functional centered at the current test sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Therefore, there is no model nor parameters as in conventional optimal filtering and neural networks, just memory of the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' In a sense, this approach resembles how brains encode and react to the physical world;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' neurons across life encode the structure and similarities given by the laws of physics, and they react very quickly to implement their response to stimulus, which means that the response must be very easy to compute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' The advantage and disadvantages of the new approach are not fully understood at this time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Finally, we should focus on ways to avoid the loss of congruence between the universal RKHS and the input space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' The correntropy RKHS has the very nice property that embeds the statistics of the data in the inner product, but there may be other kernels that maintain congruence with the input space, exemplified by our work and others on embedding PDFs in RKHS [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Another interesting aspect is that the local linear models seem to go beyond the strict stationarity assumption that supports theoretically the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' More work is required to study further this aspect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Acknowledgements: This work was partially supported by ONR grants N00014-21-1-2295 and N00014-21-1-2345 Appendix: Properties of the AutoCorrentropy Function The existence of ℋ𝐺 opens new possibilities to extend the work of Parzen on the covariance RKHS that is defined on the Hilbert space of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Recall that the autocorrelation function of a time series is a similarity measure quantified by the expected value of the product between two random variables 𝑋(𝑡𝑖), 𝑋(𝑡𝑗) at two different time intervals 𝑡𝑖, 𝑡𝑗 given by their joint distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' As such it only measures the first moment (the mean) of the joint PDF over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' The first question is how to modify the autocorrelation function, as a similarity measure in such a way that it captures all the statistical information contained in the joint distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Going Beyond the Autocorrelation Function for Similarity The most general measure of similarity in the joint space of two r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' 𝑋, 𝑌 is the cross covariance operator [26], defined by the bilinear form 𝒞𝑠,𝑡(𝑓, 𝑔) = 𝐸 [𝑓(𝑋)𝑔(𝑌)] − 𝐸 [𝑓(𝑋)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' 𝐸 [𝑔(𝑌)] (A1) The covariance operator has been estimated in RKHS ℋ𝐺 as the matrix Σ𝑥𝑠𝑥𝑡 of size equal to number of samples such that 〈𝑓, Σ𝑥𝑡𝑥𝑠𝑔〉 ℋ𝐺 = 𝒞𝑠,𝑡(𝑓, 𝑔) (A2) where f and g are functional in RKHS that map the samples from the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' 𝑥𝑡 and 𝑥𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' But this treatment might be overly complicated for a stationary random process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Firstly, the marginals have the same density;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' secondly, only a scalar similarity over marginals is needed, and the mean embedding operator (20) can be estimated in ℋ𝑣;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' and thirdly because time establishes an a priori order on the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' such that a single variable (the delay) can be employed, instead of pairwise samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Therefore, we submit that it is not necessary to estimate the full covariance operator for this application, which is computationally very intensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Measures of Similarity in the Joint Space of Densities Definition: Given a strictly stationary time series {𝑋𝑡,𝑡 ∈ 𝑇} the equality in probability density between two marginals at s and t i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=', 𝑃(|𝑋(𝑠) − 𝑋(𝑡)| < 𝜀) for an infinitesimally small 𝜀, defines a measure of similarity that can be estimated in ℋ𝐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' In the joint space of 𝑝𝑠,𝑡(𝑥𝑡,𝑥𝑠) we can define a radial marginal as the bisector of the joint space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' The density over the line 𝑥𝑡 = 𝑥𝑠 approximates 𝑃(|𝑋(𝑠)−𝑋(𝑡)|<𝜀) 𝜀 , which can be estimated as 𝐸𝑝𝑠,𝑡[𝛿(𝑋(𝑠) − 𝑋(𝑡))] (A3) where 𝛿(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' ) is a delta function and we assume that the joint pdf over the lags is smooth along the bisector of the joint space is non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' To simplify, the Dirac calculus is used to illustrate the concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' The expected value in (A3) can be written 𝐸𝑝𝑠,𝑡[𝛿(𝑋(𝑠) − 𝑋(𝑡))] = ∬ 𝛿(𝑥𝑠 − 𝑥𝑡)𝑝𝑠,𝑡(𝑥𝑠,𝑥𝑡)𝑑𝑥𝑠𝑑𝑥𝑡 (A4) The meaning of (A3) is quite clear: it is integrating the area under the joint density along the line 𝑥𝑡 = 𝑥𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Therefore, we can write (A4) as a single integral 𝐸𝑝𝑠,𝑡[𝛿(𝑋(𝑠) − 𝑋(𝑡))] = ∫ 𝑝𝑠,𝑡(𝑥, 𝑥)𝑑𝑥 (A5) This reduction to a single integral can be expected by the definition of conditional PDF (see below), and it simplifies the calculation because of the statistical embedding in ℋ𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Note however, that this procedure needs to be repeated for every lag L of interest i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=', it should be written as 𝑡 = 𝑠 − 𝑙, 𝑙 = 0,… 𝐿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Fortunately, the maximum lag L is dictated by the embedding dimension of the real system that produced the time series, which is far smaller than the number of samples we collect from the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' In engineering applications this order can be estimated by Takens’ embedding theory [27], or more practically by selecting the first minimum of the time series autocorrelation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' This computation is much simpler than the covariance matrix in (A1) because we are reducing the matrix to a vector u of size L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Correntropy functional as an approximation to the bisector integral An empirical estimator of the natural measure of similarity defined above is given by its inner product (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' It turns out it has been coined in [16] the correntropy functional, which reads 𝑉𝜎(𝑡, 𝑠) = 𝐸𝑝𝑡,𝑠[𝐺𝜎(𝑥𝑡 − 𝑥𝑠)] (A6) where G(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=') is the Gaussian function with bandwidth \uf073.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' As discussed above, correntropy is a mean embedding of the joint pdf of a pair of samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Rewriting (A6) using the definition of the expected value over the joint distribution, we obtain 𝑉𝜎(𝑡, 𝑠) = ∬ 𝐺𝜎(𝑥𝑡 − 𝑥𝑠)𝑝𝑡,𝑠(𝑥𝑡,𝑥𝑠)𝑑𝑥𝑡𝑑𝑥𝑠 = 𝐸[𝐺𝜎(𝑥𝑡 − 𝑥𝑠)] (A7) for strictly stationary processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' The best way to interpret this relation is to realize that when 𝑥𝑡 = 𝑥𝑠, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' along the bisector of the joint space, the Gaussian kernel function is maximum, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' correntropy weights the joint space of samples with Gaussian kernels placed along the bisector of the first quadrant [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' When the kernel size \uf073 approaches 0, it approximates a delta function 𝛿(𝑥𝑡 − 𝑥𝑠), so we obtain an approximation to (A3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Moreover, correntropy is easily computed from samples too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Collect a segment of data of size N from a time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' From (A7) an estimator of correntropy is simply 𝑉𝜎(𝜏) = 1 𝑁−𝜏+1 ∑ 𝐺𝜎(𝑥𝑖 − 𝑥𝑖−𝜏) 𝑁 𝑖=𝑚 (A8) Hence, correntropy effectively estimates a radial marginal density obtained by integrating along the bisector from samples with linear complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' This is unsuspected, because we are quantifying similarity in the structure of a time series beyond what we can achieve with the mean value of the product of samples in the autocorrelation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Note that here the kernel size should be made small for fine temporal resolution, but there is a compromise, because if we use a very small kernel size, the number of samples N must be sufficiently large to get sufficient number of samples around the bisector of the joint space for accurate statistical estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' The Relation between 𝑃(𝑥𝑡1 − 𝑥𝑡2) and the Conditional Density in the Joint Space The Dirac calculus is a short cut and here we provide a more precise derivation of the value of the radial margin as a conditional distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' As is well known the definition of conditional distribution of the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' X given Y is 𝑓(𝑥|𝑦) = 𝑓(𝑥, 𝑦) 𝑓(𝑦) = 𝑓(𝑥|𝑌 = 𝑦0) = 𝑓(𝑥, 𝑌 = 𝑦0) 𝑓(𝑌 = 𝑦0) The meaning of this conditional is that we pick a value for y = y0 and compute the area under the joint pdf at y0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Here we are interested in a radial marginal, which is the bisector of the joint space given by the equality in probability i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=', Y=X, and would like to see how to compute it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Let us start with the distribution function and write the conditional probability as 𝐹(𝑥|(𝑥 − 𝛿) < 𝑌 ≤ 𝑥) = 𝑃(𝑋 ≤ 𝑥|(𝑥 − 𝛿) < 𝑌 ≤ 𝑥) = 𝑃(𝑋 ≤ 𝑥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='(𝑥 − 𝛿) < 𝑌 ≤ 𝑥) 𝑃((𝑥 − 𝛿) < 𝑌 ≤ 𝑥) = lim 𝛿→0 ∫ ∫ 𝑓𝑋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='𝑌(𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' 𝑣)𝑑𝑢𝑑𝑣 𝑥 −∞ 𝑥 𝑥−𝛿 ∫ 𝑓𝑌(𝑣)𝑑𝑣 𝑥 𝑥−𝛿 = 𝑓𝑋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='𝑌(𝑥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' 𝑥) 𝑓𝑌(𝑥) So,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' when the concept of the radial margin is employed as a conditional probability,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' we see that there is a normalizing factor that guarantees that the result adds to one as required for probabilities,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' but the numerator is exactly what the Dirac calculus quantifies in (A4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Approximating 𝑃(𝑥𝑡1 − 𝑥𝑡2) with Correntropy lim 𝜎→0 𝑣𝜎 (𝑡1, 𝑡2) = ∬ 𝛿(𝑥𝑡1 − 𝑥𝑡2)𝑝𝑝𝑡1𝑝𝑡2(𝑥𝑡1,𝑥𝑡2)𝑑𝑥𝑡1𝑑𝑥𝑡2 = ∫ 𝑝𝑝𝑡1𝑝𝑡2(𝑥𝑡1,𝑥𝑡1)𝑑𝑥𝑡1 (A9) In practice, the kernel size is always finite so correntropy does not estimate the probability density over a line in the joint space but the probability on a “Gaussian shaped tunnel” of width \uf073 along the radial direction 𝑥𝑡1 = 𝑥𝑡2, which will be approximated by a parallelepiped of width 2\uf065 with \uf065 ~ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content='25\uf073.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' We can write 𝑃(|𝑥𝑡1 − 𝑥𝑡2| < 𝜀) = ∫ ∫ 𝑝𝑝𝑡1𝑝𝑡2(𝑥𝑡1, 𝑥𝑡2)𝑑𝑥𝑡1𝑑𝑥𝑡2 𝑥𝑡1+𝜀 𝑥𝑡2=𝑥𝑡1−𝜀 ∞ 𝑥𝑡1=−∞ (A10) If \uf065 is small and 𝑝𝑝𝑡1𝑝𝑡2(𝑥𝑡1,𝑥𝑡2) is continuous at every point along the 𝑥𝑡1 = 𝑥𝑡2 line, the function value does not change a lot along 𝑥(𝑡2) within the interval [𝑥𝑡1 − 𝜀, 𝑥𝑡1 + 𝜀] for any fixed 𝑥(𝑡1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Thus 𝑃(|𝑥𝑡1 − 𝑥𝑡2| < 𝜀) ≈ 2𝜀 ∫ 𝑝𝑝𝑡1𝑝𝑡2(𝑥𝑡1,𝑥𝑡1)𝑑𝑥𝑡1 ∞ 𝑥(𝑡1)=−∞ = 2𝜀𝑣𝜎(𝑡1,𝑡2) (A11) And finally, we have 𝑣𝜎(𝑡1,𝑡2) = 𝑃(|𝑥𝑡1−𝑥𝑡2|<𝜀) 2𝜀 (A12) which shows that correntropy estimates indeed the probability density of the event 𝑃(𝑥𝑡1 = 𝑥𝑡2) in the joint sample space for small kernel sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Wiener, Norbert (1949).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Extrapolation, Interpolation, and Smoothing of Stationary Time Series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' New York: Wiley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Wiener, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Hopf, "Ueber eine Klasse singulärer Integralgleichungen" Sitzungber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Akad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Wiss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Berlin (1931) pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' 696–706 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dAyT4oBgHgl3EQfcvf_/content/2301.00291v1.pdf'} +page_content=' Wiener, Norbert (1930).' metadata={'source': 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0000000000000000000000000000000000000000..1b2595b83562b81ae52a3618dd7561b83bc4d3c8 --- /dev/null +++ b/8tE0T4oBgHgl3EQffgDC/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:3d7f7bc30b82f0ba3736237f5d5d1564c88aba0c20b07db47428985f7f799441 +size 138383 diff --git a/8tFST4oBgHgl3EQfaTh3/content/tmp_files/2301.13795v1.pdf.txt b/8tFST4oBgHgl3EQfaTh3/content/tmp_files/2301.13795v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..8510bb2d78ebfabea2a10d696c8e1d077c59e4af --- /dev/null +++ b/8tFST4oBgHgl3EQfaTh3/content/tmp_files/2301.13795v1.pdf.txt @@ -0,0 +1,1409 @@ +Tunable BCS-BEC crossover, reentrant, and hidden quantum phase transitions in +two-band superconductors with tunable valence and conduction bands +Giovanni Midei1 and Andrea Perali2 +1School of Science and Technology, Physics Division, University of Camerino, +Via Madonna delle Carceri, 9B, 62032 - Camerino (MC), Italy +2School of Pharmacy, Physics Unit, University of Camerino, +Via Madonna delle Carceri, 9B, 62032 - Camerino (MC), Italy +Two-band electronic structures with a valence and a conduction band separated by a tunable en- +ergy gap and with pairing of electrons in different channels can be relevant to investigate the proper- +ties of two-dimensional multiband superconductors and electron-hole superfluids, as monolayer FeSe, +recently discovered superconducting bilayer graphene, and double-bilayer graphene electron-hole sys- +tems. This electronic configuration allows also to study the coexistence of superconductivity and +charge density waves in connection with underdoped cuprates and transition metal dichalcogenides. +By using a mean-field approach to study the system above mentioned, we have obtained numerical +results for superconducting gaps, chemical potential, condensate fractions, coherence lengths, and +superconducting mean-field critical temperature, considering a tunable band gap and different filling +of the conduction band, for parametric choice of the pairing interactions. By tuning these quantities, +the electrons redistribute among valence and conduction band in a complex way, leading to a new +physics with respect to single-band superconductors, such as density induced and band-selective +BCS-BEC crossover, quantum phase transitions, and hidden criticalities. At finite temperature, +this phenomenon is also responsible for the non-monotonic behavior of the superconducting gaps +resulting in a superconducting-normal state reentrant transition, without the need of disorder or +magnetic effects. +I. +INTRODUCTION +Multi-band and multi-gap superconductivity is a com- +plex quantum coherent phenomenon with peculiar fea- +tures that cannot be found in single-band and single- +gap superconductors [1]. The increased number of de- +grees of freedom in the condensate state allows for novel +quantum effects which are unattainable otherwise, for in- +stance enriching the physics of the BCS-BEC crossover +[2–5]. Proximity to the crossover regime of the BCS-BEC +crossover in multi-band superconductors having deep and +shallow bands can determine a notable increase of su- +perconducting gaps and critical temperature (Tc) [6–9], +associated with an higher mean-field Tc, together with +optimal conditions for the screening of superconduct- +ing fluctuations [10–12]. Furthermore, the interplay of +low-dimensional two-band systems allows for screening +of fluctuations in systems composed by coupled quasi-2D +bands or even in the vicinity of a van Hove singularity +(e.g., in the case of quasi-1D), enabling shrinking of the +pseudo-gap phase and robust high-critical temperatures +[13–15]. +Motivated by high temperature superconductivity +and anomalous metallic state properties in underdoped +cuprates, interest has grown in the pseudogap physics, +in which a blurred gap persists in the normal state near +the Fermi level. There are different models and explana- +tions for this pseudogap, the simplest one being a smooth +crossover from the BCS regime towards a Bose-Einstein +condensation regime in which bound pairs form first at +higher temperatures, and then below a critical temper- +ature Tc they condense, with the pseudogap being the +excitation energy of the quasi-molecular pairs. Another +explanation relevant for underdoped cuprates is the pres- +ence of other mechanisms different from pair fluctuations, +such as charge density waves (CDWs) [16–19] and their +fluctuations that can modify the energy spectrum with +opening of (pseudo)gaps and at the same time mediate +Cooper pairing. Thus, systems in which CDWs and su- +perconductivity coexist are of primary interest to study +the BCS-BEC crossover when an energy gap separates +the electronic spectrum in two bands, determining a va- +lence and a conduction band. +In addition to underdoped cuprates, an interesting ex- +ample is given by the transition metal dichalcogenide +(TMD) family, MX2, where M = Ti, Nb, Mo, Ta and X += S, Se, which exhibits a rich interplay between super- +conductivity and CDW order [20]. In these materials, su- +perconductivity occurs in an environment of pre-existing +CDW order [21, 22], making them an ideal platform to +study many-body ground states and competing phases +in the 2D regime. The relationship between CDW and +superconductivity in such systems is still under investi- +gation [23, 24]. In general, their mutual interaction is +competitive, but evidence to the contrary, indicating a +cooperative interplay, has also been reported in angle- +resolved photoemission spectroscopy (ARPES) studies +[22]. Among them, bulk Niobium diselenide (2H-NbSe2) +undergoes a CDW distortion at T=30 K and becomes su- +perconducting at 7 K. References [25, 26] reported that +Tc lowers to 1.9 K in 2H-NbSe2 single-layers and that the +CDW measured in the bulk is preserved. Theoretical sup- +port is given by Chao-Sheng Lian et al. [27]: they demon- +strate enhanced superconductivity in the CDW state of +monolayer tantalium diselenide (TaSe2) with DFT cal- +culations. In contrast with 2H-NbSe2, they report that +arXiv:2301.13795v1 [cond-mat.supr-con] 31 Jan 2023 + +2 +as TaSe2 is thinned to the monolayer limit, its super- +conducting critical temperature rises from 0.14 K in the +bulk to 2 K in the monolayer. Another appealing super- +conducting material is the monolayer FeSe grown on a +SrTiO3 substrate, which exhibits a huge increase of Tc +up to 100 K [28] and it is characterized by a valence and +a conduction band structure near the Fermi level. Fur- +thermore, very recently 2D superconductivity has been +found in bilayer graphene systems, in which conduction +and valence bands are separated by a small energy band- +gap (0 ÷ 100 meV) that can be precisely tuned by an +external electric field [29] (for a review see [30]). Cou- +pling a monolayer of WSe2 with bilayer graphene has +been found to enhance superconductivity by an order of +magnitude in Tc and superconductivity emerges already +at zero magnetic field [31]. +Finally, it turns out that +the two-band superconducting system considered in this +work is in close correspondence with two-band electron- +hole superfluids in double bilayer graphene [32]. +Therefore, the growing experimental realization of 2D +superconductors with valence and conduction bands sep- +arated by a tunable energy gap and electron-hole super- +fluidity in multilayer systems motivated us to investigate +the BCS-BEC crossover in this kind of systems. The de- +tailed analysis of this configuration is lacking in the liter- +ature to the best of our knowledge. A pioneering work on +a related system with valence and conduction parabolic +bands has been done by Nozi`eres and Pistolesi [33] to +study the phase transition from a semiconducting to a +superconducting state and the consequent (pseudo)gap +opening, in the specific case of equal pairing strengths +for all interaction channels considered. In our work we +consider a superconductor with two tight-binding bands +with different intra-band and pair-exchange couplings, in +order to probe the possibility to have coexisting Cooper +pairs of different average sizes [34] in the valence and con- +duction band. However, for most of multi-band supercon- +ductors the tuning of intra-band and pair-exchange inter- +actions is rather challenging and their properties cannot +be studied easily in a continuous way across the BCS- +BEC crossover. As shown in this work, a different way +to explore the BCS-BEC crossover in such systems can be +achieved by tuning the energy gap between the valence +and the conduction band. In fact, since the number of +particles in the single bands is not conserved, when the +energy band gap is modified the number of holes and of +electrons forming Cooper pair respectively in the valence +and in the conduction bands changes, allowing for the +occurrence of a density induced multi-band BCS-BEC +crossover [35]. +This redistribution of charges between +the valence and the conduction band leads also to novel +and interesting quantum phase transitions (QPTs) from +a superconducting to an insulating state, or hidden crit- +icalities evidenced by the analysis of the order parame- +ter coherence lengths [36, 37]. At finite temperature, a +new type of reentrant superconducting to normal state +transition has been also found and characterized. The +results reported and discussed in this work demonstrate +the richness of the proposed valence and conduction band +configuration to generate and tune new types of crossover +phenomena and quantum phases. +The manuscript is organized as follow. In section II +we describe the model for the physical system considered +and the theoretical approach for the evaluation of the +superconducting state properties. In section III we report +our results. The conclusions of our work will be reported +in Section IV. +II. +MODEL SYSTEM AND THEORETICAL +APPROACH +We consider a two-dimensional (2D) two-band super- +conductor with a valence and a conduction electronic +band in a square lattice. The valence and the conduction +bands are modelled by a tight-binding dispersion given, +respectively, by Eqs. (1) and (2): +ε1(k) = 2t[cos(kxa) + cos(kya)] − 8t − Eg +(1) +ε2(k) = −2t[cos(kxa) + cos(kya)] +(2) +where t is the nearest neighbour hopping parameter as- +sumed to be the same for both bands, a is the lattice +parameter and the wave-vectors belong to the first Bril- +louin zone − π +a ≤ kx,y ≤ π +a; Eg is the energy band-gap +between the conduction and the valence band. The band +dispersions are reported in Fig. 1. In order to study the +superconducting state properties of our system, we as- +sume that Cooper pairs formation is due to an attractive +interaction between opposite spin electrons. +The two- +particle interaction has been approximated by a separa- +ble potential Vij(k, k′) with an energy cutoff ω0, which +is given by: +Vij(k, k′) = −V 0 +ijΘ +� +ω0 − |ξi(k)| +� +Θ +� +ω0 − |ξi(k′)| +� +(3) +FIG. 1. Electronic band structure of the two-band 2D system +considered in this work. Eg is the energy gap between the +valence (i = 1) and the conduction (i = 2) band. + +CONDUCTION BAND +82 = - 2t(cos(akx) + cos(ak,) +E +VALENCEBAND +81 = 2t(cos(akx) + cos(ak,) - 8t - Eg3 +where V 0 +ij > 0 is the strength of the potential in the +different pairing channels and i, j label the bands. V 0 +11 +and V 0 +22 are the strength of the intra-band pairing inter- +actions (Cooper pairs are created and destroyed in the +same band). V 0 +12 and V 0 +21 are the strength of the pair- +exchange interactions (Cooper pairs are created in one +band and destroyed in the other band, and vice versa), +so that superconductivity in one band can induce super- +conductivity in the other band. The same energy cutoff +ω0 of the interaction for intra-band and pair-exchange +terms is considered. Through out this work, ω0 is con- +sidered an energy scale larger than the total bandwidth +of our system to model an effective pairing of electronic +origin, or a contact attractive potential. This is a key as- +sumption to make possible for the system to explore the +entire BCS-BEC crossover [38]. The terms corresponding +to Cooper pairs forming from electrons associated with +different bands (inter-band or cross-band pairing) are not +considered in this work (see [39]). ξi(k) = εi(k) − µ in +Eq. (3) is the energy dispersion for the band i with re- +spect to the chemical potential µ. The superconducting +state of the system and its evolution with relevant sys- +tem parameters is studied at a mean-field level. +The +BCS equations for the superconducting gaps have to be +coupled with the density equation which fixes the chemi- +cal potential, since the self-consistent renormalization of +the chemical potential is a key feature to account for the +BCS-BEC crossover physics. +Zero and finite tempera- +ture cases have been considered in this work. The BCS +equations for the superconducting gaps in the two-band +system at a given temperature T are +∆1(k) = − 1 +2Ω +� +k′ +� +V11(k, k′)∆1(k′) +E1(k′) tanh E1(k′) +2T ++ V12(k, k′)∆2(k′) +E2(k′) tanh E2(k′) +2T +� +(4) +∆2(k) = − 1 +2Ω +� +k′ +� +V22(k, k′)∆2(k′) +E2(k′) tanh E2(k′) +2T ++ V21(k, k′)∆1(k′) +E1(k′) tanh E1(k′) +2T +� +(5) +where Ei(k) = +� +ξi(k)2 + ∆i(k)2 is the dispersion of +single-particle excitations in the superconducting state +and Ω is the area occupied by the 2D system. ℏ = 1 and +kB = 1 throughout the manuscript. The superconduct- +ing gaps have the same energy cutoff of the separable +interaction: +∆i(k) = ∆iΘ +� +ω0 − |ξi(k)| +� +(6) +The total electron density of the two-band system is fixed +and given by the sum of the single-band densities, ntot = +n1 + n2, that can vary instead. The electronic density ni +in the band (i) at temperature T is given by, +ni = 2 +Ω +� +k +� +vi(k)2f +� +− Ei(k) +� ++ ui(k)2f +� +Ei(k) +�� +(7) +where f(E) is the Fermi-Dirac distribution function. The +BCS coherence weights vi(k) and ui(k) are: +vi(k)2 = 1 +2 +� +1 − +ξi(k) +� +ξi(k)2 + ∆i(k)2 +� +(8) +ui(k)2 = 1 − vi(k)2 +(9) +For the valence band the definition of the condensate +fraction is the ratio of the number of Cooper pairs in the +valence band to the number of holes in the valence band, +αh +1 = +� +k +� +u1(k)v1(k) +�2 +� +k u1(k)2 +(10) +For the conduction band instead, the expression already +used in the one-band case is generalized to the number +of Cooper pairs divided by the total number of carriers +in the conduction band +αe +2 = +� +k +� +u2(k)v2(k) +�2 +� +k v2(k)2 +(11) +The intra-pair coherence length ξpairi has the same form +for both the valence and the conduction bands, that is +ξ2 +pairi = +� +k +��∇ +� +ui(k)vi(k) +���2 +� +k +� +ui(k)vi(k) +�2 +(12) +Regarding the superconducting order parameter coher- +ence length, two characteristic length scales in the spatial +behavior of superconducting fluctuations are expected, +since the system is made up by two partial condensates. +When the pair-exchange interaction is not present, these +two lengths are simply the order parameter coherence +lengths of the condensates of the valence ξc1 and of the +conduction ξc2 band. When the pair-exchange interac- +tions is different from zero, one has to deal with coupled +condensates, and these length scales cannot be attributed +to the single bands involved, describing instead the col- +lective features of the whole two-component condensate. +The pair-exchange interactions mix the superconducting +order parameters of the initially non-interacting bands, +that acquire mixed character. The soft, or critical, co- +herence length ξs diverges at the phase transition point, +while the rigid, or non-critical, coherence length ξr re- +mains finite. Following the approach in [37], these char- +acteristic length scales are given by +ξ2 +s,r = G(T) ± +� +G2(T) − 4K(T)γ(T) +2K(T) +(13) + +4 +where ξs corresponds to the solution with the plus and +ξr to the one with the minus sign and +G(T) = (V 0 +12)2� +˜g1(T)β2(T) + ˜g2(T)β1(T) +� ++ +� +1 − V 0 +11˜g1(T) +� +V 0 +22β2(T)+ +� +1 − V 0 +22˜g2(T) +� +V 0 +11β1(T) +(14) +K(T) = +� +1 − V 0 +11˜g1(T) +�� +1 − V 0 +22˜g2(T) +� +− +(V 0 +12)2˜g1(T)˜g2(T) +(15) +γ(T) = +� +V 0 +11V 0 +22 − (V 0 +12)2� +β1(T)β2(T) +(16) +˜gi(T) = gi(T) − 3νi(T) +� +∆i(T) +�2 +(17) +gi(T) = +1 +2V +� +k +1 +ξi(k) tanh ξi(k) +2T +(18) +νi(T) = +− 1 +2V +� +k +∂ +∂|∆i|2 +� +1 +Ei(k) tanh ξi(k) +2T +� +∆i=0 +(19) +βi(T) = − 1 +4V +� +k +∂2 +∂q2 +l +� +1 +ξi(k) + ξi(k − q) +× +� +tanh ξi(k) +2T ++ tanh ξi(k − q) +2T +�� +q=0 +(20) +where l refers to the Cartesian axis in Eq. (20). +In order to describe the physics of the quantum phase +transition, the values of the coherence lengths at zero +temperature have been approximated by choosing a low +enough temperature so that the superconducting gaps +and the chemical potential retain the same behavior of +the zero temperature case. The energies are normalized +in units of the hopping t and the dimensionless couplings +λii are defined as λii = NV 0 +ii, where N = 1/4πa2t is +the density of states at the top / bottom of the valence / +conduction band, that coincide since the density of states +is not modified by the concavity of the band. The intra- +pair coherence lengths ξpairi are normalized using the +average inter-particle distance in the normal state li = +1/√πni, where ni is the density in the band i. +This +quantities differ by a factor of +√ +2 by the inverse of the +respective Fermi wave-vector KF i. The soft ξs and the +rigid ξr coherence lengths are normalized with respect to +the lattice constant a, since in the two-band case they +cannot be attributed to any of the two bands. +III. +RESULTS +In this section we study the properties of the super- +conducting ground state and give a full characteriza- +tion of the BCS–BEC crossover in the two-band system +considered in this work. First, we study the zero tem- +perature superconducting gaps in the conduction (∆2) +and in the valence (∆1) band through the BCS-BEC +crossover, for the case of unbalanced intra-band couplings +(λ11 ̸= λ22). The results are shown in Fig. 2, in which +the superconducting gaps are reported as functions of +the energy band-gap Eg, for different values of the total +density a2ntot and for different pair-exchange couplings +λ12 = λ21. +In the case of an empty conduction band + 0 + 0.004 + 0.008 + 0.012 + 0.016 + 1.58 1.59 1.6 1.61 1.62 + 0 + 0.6 + 1.2 + 1.8 + 2.4 + 3 +(a) +Δ2 / t +(b) +a2 ntot=2.00 +a2 ntot=2.07 +a2 ntot=2.26 +a2 ntot=2.35 + 0 + 0.2 + 0.4 + 0.6 + 0.8 + 0 + 0.4 0.8 1.2 + 1.6 + 2 +(c) +Δ1 / t +Eg / t + 0 + 0.4 0.8 1.2 + 1.6 + 2 +(d) +Eg / t +QCP +QCP +QCP +QCP +FIG. 2. Superconducting gaps ∆2/t opening in the conduc- +tion band (a)-(b) and in the valence band ∆1/t (c)-(d) as +functions of the band-gap energy Eg/t for an energy cutoff +of the attractive interactions ω0/t = 20. The intra-band cou- +plings are λ11 = 0.23 and λ22 = 0.75. +The pair-exchange +couplings are (λ12 = λ21): (a),(c) (0.001), (b),(d) (0.1). The +superconducting gaps are reported for different values of the +total density a2ntot. +and a completely filled valence band, corresponding to +a2ntot = 2.00, a quantum phase transition (QPT) to the +normal state takes place at a specific quantum critical +point (QCP), that occurs when Eg = E∗ +g. When the car- +rier concentration in the conduction band is non-zero, the +phase transition becomes a crossover and superconduc- +tivity extends for all values of the band gap Eg. However, +the system presents different behaviors if the value of the +band gap is smaller or larger of E∗ +g. For finite doping, +the valence band contributes very weakly to the super- +conducting state of the system for Eg > E∗ +g. +In this +regime the bands are almost decoupled and the super- +conducting gaps does not depend on Eg. +However, in +the case of Fig. 2(c) since the pair-exchange couplings +are weak the conduction band cannot sustain the super- +conductivity in the valence band and ∆1 is suppressed. +Thus, continuously tuning Eg to higher values will result +in ∆1 << ∆2 so that there is only one significant super- + +5 +conducting gap and one significant condensate. In the +other case instead (Fig. +2(d)), the pair-exchange cou- +plings are stronger and ∆1 is not much suppressed with +respect to its initial value, since in these cases the su- +perconductivity in the valence band is sustained by the +condensate of the conduction band. +Another interesting feature of this system is that ∆1 is +enhanced for lower values of the total density as long as +Eg < E∗ +g. When Eg > E∗ +g instead, the opposite situ- +ation occurs. +The value of E∗ +g at which this behavior +takes place depends on the level of filling of the conduc- +tion band, shifting to the left when higher total densities +are considered, and on the pair-exchange couplings that +shifts E∗ +g to the right when larger interactions strength +are considered. The reason behind the behavior of the + 0 + 0.1 + 0.2 + 0.3 + 0.4 +(a) +a2 n2 +e +(b) + 0 + 0.05 + 0.1 + 0.15 + 0.2 + 0 + 0.4 0.8 1.2 + 1.6 + 2 +(c) +a2 n1 +h +Eg / t + 0 + 0.4 0.8 + 1.2 + 1.6 + 2 +(d) +Eg / t +a2 ntot=2.00 +a2 ntot=2.07 +a2 ntot=2.26 +a2 ntot=2.35 +FIG. 3. Electron density a2ne +2 (a)-(b) in the conduction band +and hole density a2nh +1 (c)-(d) in the valence band as functions +of the band-gap Eg/t for different values of the total density +a2ntot, normalized to the area of the unit cell. ω0/t = 20. +The intra-band couplings are λ11 = 0.23 and λ22 = 0.75. +The pair-exchange couplings are (λ12 = λ21): (a),(c) (0.001), +(b),(d) (0.1). +superconducting gaps can be found by looking at the den- +sities of particles forming Cooper pairs, which are elec- +trons in the conduction band and holes in the valence +band. +While the total density is fixed, the density in +each band can vary. In this way, the density of particles +in the conduction band n2 is no longer controlled only by +doping as for a single band system, there are instead ad- +ditional particles excited from the valence band. Never- +theless, for larger values of Eg the gain in the interaction +energy due to superconductivity is much smaller than +the kinetic energy cost for transferring electrons from the +valence band to the conduction band, so that very few +electrons (compared to the total density of electrons in +the valence band) are excited into the conduction band. +This behavior is shown in Fig. 3. As one can see for +a2ntot = 2.00 the hole density in the valence band and +the electron density in the conduction band coincide and +are monotonically decreasing, both of them vanishing at +the QCP Eg = E∗ +g. This is a sign that superconductivity +is due to holes in the valence band and to electrons in +the conduction band. In the other cases the hole density +in the valence band is almost zero for Eg > E∗ +g, while the +electron density in the conduction band is approaching +the asymptotic value given by the total density minus the +density of the filled valence band a2n2 = a2ntot − 2.00. +-6.2 +-5.7 +-5.2 +-4.7 +-4.2 +-3.7 + 0 + 0.4 0.8 + 1.2 + 1.6 + 2 +(a) +µ / t +Eg / t + 0 + 0.4 0.8 + 1.2 + 1.6 + 2 +(b) +Eg / t +a2 ntot=2.00 +a2 ntot=2.07 +a2 ntot=2.26 +a2 ntot=2.35 +FIG. 4. Chemical potential µ/t as a function of the band- +gap Eg/t for ω0/t = 20. +The pair-exchange couplings are +λ11 = 0.23 and λ22 = 0.75. The pair-exchange couplings are +(λ12 = λ21): (a) (0.001),(b) (0.1). The chemical potential µ +is reported for different total densities a2ntot. The black and +the magenta dashed lines correspond to the bottom of the +conduction band and the top of the valence band, respectively. +In Fig. 4 the chemical potential is reported as a function +of Eg, for different total densities a2ntot and for different +pair-exchange couplings. For higher values of the total +density and of the pair-exchange couplings the chemical +potential shift toward higher energies, due to the larger +number of electrons in the conduction band. In particu- +lar, when Eg is increased, in the low density regime the +chemical potential starts deep inside the valence band +and then enters the gap between the two bands, mean- +ing that the condensate in the valence band spans a wide +region of the BCS-BEC crossover, while the conduction +band is always located in the BEC side of the crossover +regime or in the BEC regime, depending on whether the +chemical potential lies inside the conduction band or not. +When Eg > E∗ +g the chemical potential acquires a flat de- +pendence and is not modified by Eg, in a similar way to +what happens to the superconducting gaps and the den- +sities. +In Fig. +5 the condensate fraction is shown as a func- +tion of Eg, for different a2ntot and for different pair- +exchange couplings. The usual choice of the boundaries +between the different pairing regimes has been adopted: +for α < 0.2 the superconducting state is in the weak- +coupling BCS regime; for 0.2 < α < 0.8 the system is +in the crossover regime; for α > 0.8 the system is in +the strong-coupling BEC regime. Consistently with the +information obtained from the chemical potential, in the +low density regime the condensate in the valence band ex- +plores the entire BCS-BEC crossover by varying Eg. For +the considered pair-exchange interactions in (Fig. 5(c)) + +6 + 0 + 0.2 + 0.4 + 0.6 + 0.8 + 1 +(a) +α2 +e +(b) +a2 ntot=2.00 +a2 ntot=2.07 +a2 ntot=2.26 +a2 ntot=2.35 + 0 + 0.2 + 0.4 + 0.6 + 0.8 + 0 + 0.4 0.8 + 1.2 + 1.6 + 2 +(c) +α1 +h +Eg / t + 0 + 0.4 0.8 + 1.2 + 1.6 + 2 +(d) +Eg / t +Crossover +BEC +BCS +BEC +Crossover +BCS +BEC +Crossover +BCS +BCS +Crossover +BEC +FIG. 5. Condensate fractions in the conduction band αe +2 (a)- +(b) and in the valence band αh +1 (c)-(d) as functions of the +band-gap Eg/t for ω0/t = 20. The intra-band couplings are +λ11 = 0.23 and λ22 = 0.75. The pair-exchange couplings are +(λ12 = λ21): (a),(c) (0.001), (b),(d) (0.1). The condensate +fractions are reported for different total densities a2ntot. Thin +grey dashed lines correspond to α = 0.2, 0.8 from bottom to +top. +the valence band condensate is in the BCS regime for +small Eg, while for larger pair-exchange interactions (Fig. +5(d)) is in the crossover regime. When the energy gap or +the total density increases, the valence band condensate +enters the BEC regime, with the hole condensate fraction +αh +1 approaching unity, indicating that the remaining few +holes are all in the condensate. The situation in the con- +duction band is different, since due to the strong intra- +band coupling the condensate is always located in the +BEC side of the crossover regime or in the BEC regime. +In the case a2ntot = 2.00 both the condensate fractions +suddenly drop to zero when Eg = E∗ +g due to the quantum +phase transition. +In Fig. 6 the intra-pair coherence length is reported as a +function of Eg, for different a2ntot and for different pair- +exchange couplings. +Since for low densities and small +pair-exchange couplings the valence band condensate is +in the BCS regime (6(a)) when Eg is small, ξpair1 assumes +initially larger values with respect to the average inter- +particle distance l1. For larger Eg the system enters the +BEC regime and ξpair1 becomes much smaller than the +average inter-particle distance. The valence band con- +densate goes from the crossover to the BEC regime in a +small range of band gap values. This behavior is observed +also for larger values of the total density. The conduction +band instead, due to the strong intra-band coupling re- +tains a small value of the intra-pair coherence length with +respect to the the average inter-particle distance l2 for all +the considered values of the system density. In this way +we found Cooper pairs of different size coexisting in the +system for low density and low pair-exchange couplings + 0 + 0.1 + 0.2 + 0.3 +(a) +ξpair2 / l2 +(b) + 0.5 + 1 + 1.5 + 2 + 0 + 0.4 0.8 1.2 + 1.6 + 2 +(c) +ξpair1 / l1 +Eg / t + 0 + 0.4 0.8 1.2 + 1.6 + 2 +(d) +Eg / t +a2 ntot=2.00 +a2 ntot=2.07 +a2 ntot=2.26 +a2 ntot=2.35 +FIG. 6. Intra-pair coherence length ξpair2/l2 for the Cooper +pairs of the conduction band (a)-(b) and intra-pair coherence +length ξpair1/l1 for the Cooper pairs of the valence band (c)- +(d) as functions of the band-gap Eg/t for ω0/t = 20. The +intra-band couplings are λ11 = 0.23 and λ22 = 0.75. The pair- +exchange couplings are (λ12 = λ21): (a),(c) (0.001), (b),(d) +(0.1). The intra-pair coherence lengths ξpairi/li are reported +for different a2ntot. +values, in the regime of small Eg. For the zero doping +case the intra-pair coherence length is defined only for +Eg < E∗ +g, since in this regime the system is not super- +conducting and a intra-pair coherence length cannot be +defined. The fact that the intra-pair coherence length is +approaching zero at the QCP in the BEC regime is dif- +ferent from Ref. [34], where giant Cooper pairs are found +in the vicinity of the QCP in the BCS side. In this case +instead, what we have found is equivalent to the finite- +density to zero-density QCP of tightly bound molecules. +Namely, near the present QCP in the BEC side the pair +size is so small that pairs behave as point-like bosons and +the system can be described by its bosonic counterpart +[40]. +In Fig. 7 the order parameter coherence coherence length +is reported as a function of Eg, for different a2ntot and for +different pair-exchange couplings. In the case a2ntot = +2.00 the soft or critical coherence length ξs diverges when +the band gap reaches the critical value Eg = E∗ +g, since +the system undergoes a quantum phase transition to the +insulating state. In the other cases a2ntot ̸= 2.00, the soft +coherence length ξs is not diverging, since no quantum +phase transition occurs in the system for any Eg. In par- +ticular, in the cases of a2ntot = 2.07 and a2ntot = 2.26 the +soft coherence length ξs shows a maximum in correspon- +dence of the respective Eg = E∗ +g, showing its memory +about the quantum phase transition of the valence band +condensate, which takes place when the pair-exchange +interactions are absent. The increase of λ12 = λ21 sup- +presses the maximum, as shown in Figs. 7(a) and (b), +since the band-condensates become more coupled. In the + +7 + 0 + 2 + 4 + 6 +(a) +ξs / a +(b) +a2 ntot=2.00 +a2 ntot=2.07 +a2 ntot=2.26 +a2 ntot=2.35 + 0 + 0.2 + 0.4 + 0.6 + 0.8 + 1 + 1.2 + 0 + 0.4 + 0.8 + 1.2 + 1.6 +(c) +ξr / a +Eg / t + 0 + 0.4 + 0.8 + 1.2 + 1.6 +(d) +Eg / t +HC +QPT +QPT +HC +HC +HC +HC +HC +FIG. 7. +Soft ξs (a)-(b) and rigid ξr (c)-(d) order parame- +ter coherence length, normalized to the lattice constant a, as +functions of the band-gap Eg/t between the two bands at tem- +perature T/t = 0.00065 and for ω0/t = 20. The intra-band +couplings are λ11 = 0.23 and λ22 = 0.75. The pair-exchange +couplings are (λ12 = λ21): (a),(c) (0.001), (b),(d) (0.03). The +coherence lengths ξs,r are reported for different values of the +total density a2ntot. In the case a2ntot = 2.00 (orange dashed +line) ξr has been rescaled by a factor of 7 (c) and 4.5 (d) to +make the plot more visible. +case of a2ntot = 2.35 instead, since the valence band +is never superconducting for any Eg when the band- +condensates are decoupled, there is no quantum phase +transition and no peak. The rigid coherence length ξr in- +stead remains finite for all Eg and for all a2ntot. Anyway, +we find the memory of the quantum phase transition that +takes place when the conduction band is empty and the +valence band is filled (anntot = 2.00). In this case in fact, +also the conduction band returns to the normal state at +Eg = E∗ +g. Indeed, for zero pair-exchange couplings, the +rigid coherence length ξr reduces to the coherence length +of the conduction band ξ2. Even though for finite pair- +exchange coupling the coherence length is non-diverging, +it encodes the memory of the quantum phase transition +of the conduction band. Also the maximum value of the +rigid coherence length ξr is suppressed by the increase of +λ12 = λ21 in this case, as shown in Figs. 7(c) and (d). +We consider now finite temperature effects on the critical +energy band gap for the case of no doping. The super- +conducting gaps as functions of temperature for different +band gaps are reported in Fig. 8. The superconducting +gaps present a non-monotonic behavior, that is very dif- +ferent from the temperature dependence of the gaps in +conventional superconductors. The strong enhancement +of ∆2 at finite temperature is due to the thermal excita- +tion of the electrons from the valence band to the con- +duction band. This behavior becomes more pronounced +for larger Eg, especially in the case of Fig. 8(c) in which +the system is initially in the normal state for tempera- +tures close to zero, and then becomes superconducting for + 0 + 0.5 + 1 + 1.5 + 2 + 2.5 +(a) +(b) + 0 + 0.2 + 0.4 + 0.6 + 0.8 + 1 +(c) +Δ / t +(d) + 0 + 0.1 + 0.2 + 0.3 + 0.4 + 0 + 0.2 + 0.4 + 0.6 + 0.8 + 1 +(e) +T / Tc + 0 + 0.2 + 0.4 + 0.6 + 0.8 + 1 +(f) +T / Tc +Eg/t = 0 +Eg/t = 2 +Eg/t = 3 +NS +SC +Δ2 +Δ2 +Δ2 +Δ1 +Δ1 +Δ1 +Δ1 +Δ1 +Δ1 +Δ2 +Δ2 +Δ2 +NS +NS +NS +SC +FIG. 8. Superconducting gaps ∆2/t opening in the conduc- +tion band and in the valence band ∆1/t as functions of tem- +perature T, normalized with respect to the critical tempera- +ture Tc, for a2ntot = 2.00. The pair-exchange couplings are +(λ12 = λ21): (a), (c), (e) (0.03), (b), (d), (f) (0.1). +larger temperatures. This superconducting-normal state +reentrant transition that we have found in our two-band +system is based on a different mechanism with respect +to the reentrant transitions observed in superconductors +containing magnetic elements [41] or in granular super- +conducting systems [42–45]: in the former it is attributed +to the competition of magnetic ordering and supercon- +ductivity, while in the latter is attributed to tunneling +barriers effect, while in our valence-conduction bands sys- +tem the thermal excitation of electrons from the valence +into the conduction band play a crucial role. In Fig. 9 +we report the phase diagram T vs Eg for our system. In +Fig. 9 the branch of the phase transition from the su- +perconducting to the normal state corresponding to the +reentrant behavior results from the second solution at +lower temperatures of the linearized self-consistent equa- +tions for the superconducting gaps. From the left panel of +Fig. 9 it is clear how the reentrant transition is more pro- +nounced when the intra-band couplings are unbalanced +(λ22 ≃ 3λ11 in the figure), while the reentrance is reduced +when the intra-band couplings have similar values. This +effect occurs in a less evident manner also when the pair- +exchange couplings are increased. Therefore, the most +relevant parameter to control the reentrance phenomenon +is the intra-band coupling. +IV. +CONCLUSIONS +We have studied the superconducting properties of a +two-band system of electrons, interacting through a sep- + +8 +λ11 → λ22 +λ22 → λ11 +λ22 = 0.75 +λ11 = 0.23 + 0.01 + 0.1 + 1 + 0 + 1 + 2 + 3 + 4 +T / t +Eg / t + 0 + 1 + 2 + 3 + 4 +Eg / t +SC +NS +SC +NS +λ12 ↑→ +FIG. 9. +Phase diagrams in the temperature versus energy +band gap plane, for the zero doping case. In the left panel the +red dashed line is for λ11 = 0.23, λ22 = 0.4, the green dashed +line is for λ11 = 0.23, λ22 = 0.75 and the blue dashed line is +for λ11 = 0.65, λ22 = 0.75. The pair-exchange couplings are +the same for all curves, λ12 = λ21 = 0.1. In the right panel the +pair-exchange couplings from left to right are: λ12 = λ21 = +0.03, 0.1, 0.2, while the intra-band couplings are λ11 = 0.23 +and λ11 = 0.75. +arable attractive potential with a large energy cutoff and +multiple pairing channels, at a mean-field level. The su- +perconducting state properties are studied by varying the +energy gap between the bands. We have considered dif- +ferent levels of filling for the conduction band, while the +valence band is always completely filled. When the band- +gap is modified, the density of electrons in the two bands +changes, allowing for the occurrence of a density-induced +BCS-BEC crossover. When the pair-exchange couplings +are small, the condensate in the valence band remains su- +perconducting but with a strongly suppressed supercon- +ducting gap ∆1 for Eg > E∗ +g. Therefore, in the regime +of small pair-exchange coupling, after E∗ +g, there is only +one significant superconducting gap and one significant +condensate. Interestingly, in this case the soft coherence +length present a peak as a memory of the quantum phase +transition that the valence band condensate undergoes in +absence of pair exchanges. This peak is more pronounced +if the pair-exchange couplings are sufficiently weak and +disappears for higher values of the pair-exchange cou- +plings. +For higher values of λij, superconductivity in +the valence band is sustained by the condensate in the +conduction band. Furthermore, in this regime we have +found that superconductivity is enhanced in the valence +band for increasing doping as long as Eg < E∗ +g, while for +Eg > E∗ +g superconductivity is enhanced for lower doping. +We have also found that superconductivity may occur +even when no free carriers exist in the conduction band +in the normal state at T = 0, as soon as the gain in super- +conducting energy exceeds the cost in producing carriers +across the band gap Eg. If the binding energy is larger +than the energy band-gap, the system becomes unstable +under the formation of Cooper pairs and superconduc- +tivity emerges. However, there exists a critical value of +the energy band gap E∗ +g in correspondence of which the +process of creating Cooper pairs is not energetically fa- +vorable anymore, at this point a quantum phase transi- +tion occurs. This quantum phase transition is confirmed +by the soft coherence length, which is diverging in corre- +spondence of the critical band gap Eg = E∗ +g. Thus, the +ground state is superconducting if Eg < E∗ +g, insulating +if Eg > E∗ +g. At finite temperature, the value of E∗ +g is +larger than its zero temperature value, because the elec- +trons are thermally excited from the valence band. This +situation is responsible for the non-monotonic behavior +of the superconducting gap opening in the conduction +band, which is enhanced at low temperatures because of +the electrons that jump from the valence band into the +conduction band due to thermal excitation. When there +is a finite doping in the system, the sharp phase transi- +tion becomes a smooth crossover and superconductivity +extends for all Eg. In this case, for Eg > E∗ +g the va- +lence band contributes very weakly to the superconduct- +ing state, since the hole density becomes almost zero in +this regime. +To conclude, we have found that the system explores dif- +ferent regimes of the BCS-BEC crossover by tuning the +energy band-gap and the total density. The valence-band +condensate spans the entire BCS-BEC crossover for low +enough density by varying the band-gap Eg. For larger +values of the total density, the condensate of the valence +band is very dilute and results in the BEC regime for any +Eg. The condensate of the conduction band instead re- +sides in the BEC side of the crossover or completely inside +the BEC regime, due to the strength of the intra-band +coupling of electrons in the conduction band. This pic- +ture of the BCS-BEC crossover for the system has been +found by analyzing the consistent behavior of the chemi- +cal potential, the condensate fractions and the coherence +lengths. Finally, in the case of zero doping and at finite +temperature, an interesting new type of reentrant super- +conducting to normal state transition has been numer- +ically discovered for unbalanced intra-band couplings, +showing that in this configuration superconductivity is +assisted instead of being suppressed by increasing tem- +perature. This happens because the electrons in the va- +lence band are able to jump into the conduction band +even for larger values of the zero temperature critical +band gap, due to thermal excitation, making the super- +conducting state available for a wider range of Eg when +the temperature is higher. +V. +ACKNOWLEDGMENTS +We are grateful to Tiago Saraiva (HSE-Moscow) and +Hiroyuki Tajima (University of Tokyo) for interesting dis- +cussions and a critical reading of the manuscript. G. M. +acknowledges INFN for financial support of his Ph.D. +grant. This work has been partially supported by PNRR +MUR project PE0000023-NQSTI. + +9 +[1] Milorad V. 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Matter 14, +193 (2002). + diff --git a/8tFST4oBgHgl3EQfaTh3/content/tmp_files/load_file.txt b/8tFST4oBgHgl3EQfaTh3/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8931bc0fb27b5c0bda3882c05baf8f4b0b67e20a --- /dev/null +++ b/8tFST4oBgHgl3EQfaTh3/content/tmp_files/load_file.txt @@ -0,0 +1,954 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf,len=953 +page_content='Tunable BCS-BEC crossover,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' reentrant,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' and hidden quantum phase transitions in two-band superconductors with tunable valence and conduction bands Giovanni Midei1 and Andrea Perali2 1School of Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Physics Division,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' University of Camerino,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Via Madonna delle Carceri,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' 9B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' 62032 - Camerino (MC),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Italy 2School of Pharmacy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Physics Unit,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' University of Camerino,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Via Madonna delle Carceri,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' 9B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' 62032 - Camerino (MC),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Italy Two-band electronic structures with a valence and a conduction band separated by a tunable en- ergy gap and with pairing of electrons in different channels can be relevant to investigate the proper- ties of two-dimensional multiband superconductors and electron-hole superfluids,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' as monolayer FeSe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' recently discovered superconducting bilayer graphene,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' and double-bilayer graphene electron-hole sys- tems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' This electronic configuration allows also to study the coexistence of superconductivity and charge density waves in connection with underdoped cuprates and transition metal dichalcogenides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' By using a mean-field approach to study the system above mentioned, we have obtained numerical results for superconducting gaps, chemical potential, condensate fractions, coherence lengths, and superconducting mean-field critical temperature, considering a tunable band gap and different filling of the conduction band, for parametric choice of the pairing interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' By tuning these quantities, the electrons redistribute among valence and conduction band in a complex way, leading to a new physics with respect to single-band superconductors, such as density induced and band-selective BCS-BEC crossover, quantum phase transitions, and hidden criticalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' At finite temperature, this phenomenon is also responsible for the non-monotonic behavior of the superconducting gaps resulting in a superconducting-normal state reentrant transition, without the need of disorder or magnetic effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' INTRODUCTION Multi-band and multi-gap superconductivity is a com- plex quantum coherent phenomenon with peculiar fea- tures that cannot be found in single-band and single- gap superconductors [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The increased number of de- grees of freedom in the condensate state allows for novel quantum effects which are unattainable otherwise, for in- stance enriching the physics of the BCS-BEC crossover [2–5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Proximity to the crossover regime of the BCS-BEC crossover in multi-band superconductors having deep and shallow bands can determine a notable increase of su- perconducting gaps and critical temperature (Tc) [6–9], associated with an higher mean-field Tc, together with optimal conditions for the screening of superconduct- ing fluctuations [10–12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Furthermore, the interplay of low-dimensional two-band systems allows for screening of fluctuations in systems composed by coupled quasi-2D bands or even in the vicinity of a van Hove singularity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=', in the case of quasi-1D), enabling shrinking of the pseudo-gap phase and robust high-critical temperatures [13–15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Motivated by high temperature superconductivity and anomalous metallic state properties in underdoped cuprates, interest has grown in the pseudogap physics, in which a blurred gap persists in the normal state near the Fermi level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' There are different models and explana- tions for this pseudogap, the simplest one being a smooth crossover from the BCS regime towards a Bose-Einstein condensation regime in which bound pairs form first at higher temperatures, and then below a critical temper- ature Tc they condense, with the pseudogap being the excitation energy of the quasi-molecular pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Another explanation relevant for underdoped cuprates is the pres- ence of other mechanisms different from pair fluctuations, such as charge density waves (CDWs) [16–19] and their fluctuations that can modify the energy spectrum with opening of (pseudo)gaps and at the same time mediate Cooper pairing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Thus, systems in which CDWs and su- perconductivity coexist are of primary interest to study the BCS-BEC crossover when an energy gap separates the electronic spectrum in two bands, determining a va- lence and a conduction band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' In addition to underdoped cuprates, an interesting ex- ample is given by the transition metal dichalcogenide (TMD) family, MX2, where M = Ti, Nb, Mo, Ta and X = S, Se, which exhibits a rich interplay between super- conductivity and CDW order [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' In these materials, su- perconductivity occurs in an environment of pre-existing CDW order [21, 22], making them an ideal platform to study many-body ground states and competing phases in the 2D regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The relationship between CDW and superconductivity in such systems is still under investi- gation [23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' In general, their mutual interaction is competitive, but evidence to the contrary, indicating a cooperative interplay, has also been reported in angle- resolved photoemission spectroscopy (ARPES) studies [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Among them, bulk Niobium diselenide (2H-NbSe2) undergoes a CDW distortion at T=30 K and becomes su- perconducting at 7 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' References [25, 26] reported that Tc lowers to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='9 K in 2H-NbSe2 single-layers and that the CDW measured in the bulk is preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Theoretical sup- port is given by Chao-Sheng Lian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' [27]: they demon- strate enhanced superconductivity in the CDW state of monolayer tantalium diselenide (TaSe2) with DFT cal- culations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' In contrast with 2H-NbSe2, they report that arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='13795v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='supr-con] 31 Jan 2023 2 as TaSe2 is thinned to the monolayer limit, its super- conducting critical temperature rises from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='14 K in the bulk to 2 K in the monolayer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Another appealing super- conducting material is the monolayer FeSe grown on a SrTiO3 substrate, which exhibits a huge increase of Tc up to 100 K [28] and it is characterized by a valence and a conduction band structure near the Fermi level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Fur- thermore, very recently 2D superconductivity has been found in bilayer graphene systems, in which conduction and valence bands are separated by a small energy band- gap (0 ÷ 100 meV) that can be precisely tuned by an external electric field [29] (for a review see [30]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Cou- pling a monolayer of WSe2 with bilayer graphene has been found to enhance superconductivity by an order of magnitude in Tc and superconductivity emerges already at zero magnetic field [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Finally, it turns out that the two-band superconducting system considered in this work is in close correspondence with two-band electron- hole superfluids in double bilayer graphene [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Therefore, the growing experimental realization of 2D superconductors with valence and conduction bands sep- arated by a tunable energy gap and electron-hole super- fluidity in multilayer systems motivated us to investigate the BCS-BEC crossover in this kind of systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The de- tailed analysis of this configuration is lacking in the liter- ature to the best of our knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' A pioneering work on a related system with valence and conduction parabolic bands has been done by Nozi`eres and Pistolesi [33] to study the phase transition from a semiconducting to a superconducting state and the consequent (pseudo)gap opening, in the specific case of equal pairing strengths for all interaction channels considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' In our work we consider a superconductor with two tight-binding bands with different intra-band and pair-exchange couplings, in order to probe the possibility to have coexisting Cooper pairs of different average sizes [34] in the valence and con- duction band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' However, for most of multi-band supercon- ductors the tuning of intra-band and pair-exchange inter- actions is rather challenging and their properties cannot be studied easily in a continuous way across the BCS- BEC crossover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' As shown in this work, a different way to explore the BCS-BEC crossover in such systems can be achieved by tuning the energy gap between the valence and the conduction band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' In fact, since the number of particles in the single bands is not conserved, when the energy band gap is modified the number of holes and of electrons forming Cooper pair respectively in the valence and in the conduction bands changes, allowing for the occurrence of a density induced multi-band BCS-BEC crossover [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' This redistribution of charges between the valence and the conduction band leads also to novel and interesting quantum phase transitions (QPTs) from a superconducting to an insulating state, or hidden crit- icalities evidenced by the analysis of the order parame- ter coherence lengths [36, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' At finite temperature, a new type of reentrant superconducting to normal state transition has been also found and characterized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The results reported and discussed in this work demonstrate the richness of the proposed valence and conduction band configuration to generate and tune new types of crossover phenomena and quantum phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The manuscript is organized as follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' In section II we describe the model for the physical system considered and the theoretical approach for the evaluation of the superconducting state properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' In section III we report our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The conclusions of our work will be reported in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' MODEL SYSTEM AND THEORETICAL APPROACH We consider a two-dimensional (2D) two-band super- conductor with a valence and a conduction electronic band in a square lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The valence and the conduction bands are modelled by a tight-binding dispersion given, respectively, by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' (1) and (2): ε1(k) = 2t[cos(kxa) + cos(kya)] − 8t − Eg (1) ε2(k) = −2t[cos(kxa) + cos(kya)] (2) where t is the nearest neighbour hopping parameter as- sumed to be the same for both bands, a is the lattice parameter and the wave-vectors belong to the first Bril- louin zone − π a ≤ kx,y ≤ π a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Eg is the energy band-gap between the conduction and the valence band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The band dispersions are reported in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' In order to study the superconducting state properties of our system, we as- sume that Cooper pairs formation is due to an attractive interaction between opposite spin electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The two- particle interaction has been approximated by a separa- ble potential Vij(k, k′) with an energy cutoff ω0, which is given by: Vij(k, k′) = −V 0 ijΘ � ω0 − |ξi(k)| � Θ � ω0 − |ξi(k′)| � (3) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Electronic band structure of the two-band 2D system considered in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Eg is the energy gap between the valence (i = 1) and the conduction (i = 2) band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' CONDUCTION BAND 82 = - 2t(cos(akx) + cos(ak,) E VALENCEBAND 81 = 2t(cos(akx) + cos(ak,) - 8t - Eg3 where V 0 ij > 0 is the strength of the potential in the different pairing channels and i, j label the bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' V 0 11 and V 0 22 are the strength of the intra-band pairing inter- actions (Cooper pairs are created and destroyed in the same band).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' V 0 12 and V 0 21 are the strength of the pair- exchange interactions (Cooper pairs are created in one band and destroyed in the other band, and vice versa), so that superconductivity in one band can induce super- conductivity in the other band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The same energy cutoff ω0 of the interaction for intra-band and pair-exchange terms is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Through out this work, ω0 is con- sidered an energy scale larger than the total bandwidth of our system to model an effective pairing of electronic origin, or a contact attractive potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' This is a key as- sumption to make possible for the system to explore the entire BCS-BEC crossover [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The terms corresponding to Cooper pairs forming from electrons associated with different bands (inter-band or cross-band pairing) are not considered in this work (see [39]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' ξi(k) = εi(k) − µ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' (3) is the energy dispersion for the band i with re- spect to the chemical potential µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The superconducting state of the system and its evolution with relevant sys- tem parameters is studied at a mean-field level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The BCS equations for the superconducting gaps have to be coupled with the density equation which fixes the chemi- cal potential, since the self-consistent renormalization of the chemical potential is a key feature to account for the BCS-BEC crossover physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Zero and finite tempera- ture cases have been considered in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The BCS equations for the superconducting gaps in the two-band system at a given temperature T are ∆1(k) = − 1 2Ω � k′ � V11(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' k′)∆1(k′) E1(k′) tanh E1(k′) 2T + V12(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' k′)∆2(k′) E2(k′) tanh E2(k′) 2T � (4) ∆2(k) = − 1 2Ω � k′ � V22(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' k′)∆2(k′) E2(k′) tanh E2(k′) 2T + V21(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' k′)∆1(k′) E1(k′) tanh E1(k′) 2T � (5) where Ei(k) = � ξi(k)2 + ∆i(k)2 is the dispersion of single-particle excitations in the superconducting state and Ω is the area occupied by the 2D system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' ℏ = 1 and kB = 1 throughout the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The superconduct- ing gaps have the same energy cutoff of the separable interaction: ∆i(k) = ∆iΘ � ω0 − |ξi(k)| � (6) The total electron density of the two-band system is fixed and given by the sum of the single-band densities, ntot = n1 + n2, that can vary instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The electronic density ni in the band (i) at temperature T is given by, ni = 2 Ω � k � vi(k)2f � − Ei(k) � + ui(k)2f � Ei(k) �� (7) where f(E) is the Fermi-Dirac distribution function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The BCS coherence weights vi(k) and ui(k) are: vi(k)2 = 1 2 � 1 − ξi(k) � ξi(k)2 + ∆i(k)2 � (8) ui(k)2 = 1 − vi(k)2 (9) For the valence band the definition of the condensate fraction is the ratio of the number of Cooper pairs in the valence band to the number of holes in the valence band,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' αh 1 = � k � u1(k)v1(k) �2 � k u1(k)2 (10) For the conduction band instead,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' the expression already used in the one-band case is generalized to the number of Cooper pairs divided by the total number of carriers in the conduction band αe 2 = � k � u2(k)v2(k) �2 � k v2(k)2 (11) The intra-pair coherence length ξpairi has the same form for both the valence and the conduction bands,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' that is ξ2 pairi = � k ��∇ � ui(k)vi(k) ���2 � k � ui(k)vi(k) �2 (12) Regarding the superconducting order parameter coher- ence length,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' two characteristic length scales in the spatial behavior of superconducting fluctuations are expected,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' since the system is made up by two partial condensates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' When the pair-exchange interaction is not present, these two lengths are simply the order parameter coherence lengths of the condensates of the valence ξc1 and of the conduction ξc2 band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' When the pair-exchange interac- tions is different from zero, one has to deal with coupled condensates, and these length scales cannot be attributed to the single bands involved, describing instead the col- lective features of the whole two-component condensate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The pair-exchange interactions mix the superconducting order parameters of the initially non-interacting bands, that acquire mixed character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The soft, or critical, co- herence length ξs diverges at the phase transition point, while the rigid, or non-critical, coherence length ξr re- mains finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Following the approach in [37],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' these char- acteristic length scales are given by ξ2 s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='r = G(T) ± ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='G2(T) − 4K(T)γ(T) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='2K(T) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='(13) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='where ξs corresponds to the solution with the plus and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='ξr to the one with the minus sign and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='G(T) = (V 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='12)2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='˜g1(T)β2(T) + ˜g2(T)β1(T) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='1 − V 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='11˜g1(T) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='V 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='22β2(T)+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='1 − V 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='22˜g2(T) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='V 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='11β1(T) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='(14) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='K(T) = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='1 − V 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='11˜g1(T) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='1 − V 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='22˜g2(T) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='(V 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='12)2˜g1(T)˜g2(T) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='(15) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='γ(T) = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='V 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='11V 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='22 − (V 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='12)2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='β1(T)β2(T) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='(16) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='˜gi(T) = gi(T) − 3νi(T) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='∆i(T) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='�2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='(17) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='gi(T) = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='2V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='ξi(k) tanh ξi(k) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='2T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='(18) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='νi(T) = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='2V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='∂ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='∂|∆i|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='Ei(k) tanh ξi(k) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='2T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='∆i=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='(19) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='βi(T) = − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='4V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='∂2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='∂q2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='l ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='ξi(k) + ξi(k − q) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='tanh ξi(k) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='2T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='+ tanh ξi(k − q) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='2T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='q=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='(20) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='where l refers to the Cartesian axis in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' In order to describe the physics of the quantum phase transition, the values of the coherence lengths at zero temperature have been approximated by choosing a low enough temperature so that the superconducting gaps and the chemical potential retain the same behavior of the zero temperature case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The energies are normalized in units of the hopping t and the dimensionless couplings λii are defined as λii = NV 0 ii, where N = 1/4πa2t is the density of states at the top / bottom of the valence / conduction band, that coincide since the density of states is not modified by the concavity of the band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The intra- pair coherence lengths ξpairi are normalized using the average inter-particle distance in the normal state li = 1/√πni, where ni is the density in the band i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' This quantities differ by a factor of √ 2 by the inverse of the respective Fermi wave-vector KF i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The soft ξs and the rigid ξr coherence lengths are normalized with respect to the lattice constant a, since in the two-band case they cannot be attributed to any of the two bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' RESULTS In this section we study the properties of the super- conducting ground state and give a full characteriza- tion of the BCS–BEC crossover in the two-band system considered in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' First, we study the zero tem- perature superconducting gaps in the conduction (∆2) and in the valence (∆1) band through the BCS-BEC crossover, for the case of unbalanced intra-band couplings (λ11 ̸= λ22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' 2, in which the superconducting gaps are reported as functions of the energy band-gap Eg, for different values of the total density a2ntot and for different pair-exchange couplings λ12 = λ21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' In the case of an empty conduction band 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='016 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='58 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='59 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='61 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='62 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='4 3 (a) Δ2 / t (b) a2 ntot=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='00 a2 ntot=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='07 a2 ntot=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='26 a2 ntot=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='35 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='8 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='6 2 (c) Δ1 / t Eg / t 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='6 2 (d) Eg / t QCP QCP QCP QCP FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Superconducting gaps ∆2/t opening in the conduc- tion band (a)-(b) and in the valence band ∆1/t (c)-(d) as functions of the band-gap energy Eg/t for an energy cutoff of the attractive interactions ω0/t = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The intra-band cou- plings are λ11 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='23 and λ22 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The pair-exchange couplings are (λ12 = λ21): (a),(c) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='001), (b),(d) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The superconducting gaps are reported for different values of the total density a2ntot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' and a completely filled valence band, corresponding to a2ntot = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='00, a quantum phase transition (QPT) to the normal state takes place at a specific quantum critical point (QCP), that occurs when Eg = E∗ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' When the car- rier concentration in the conduction band is non-zero, the phase transition becomes a crossover and superconduc- tivity extends for all values of the band gap Eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' However, the system presents different behaviors if the value of the band gap is smaller or larger of E∗ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' For finite doping, the valence band contributes very weakly to the super- conducting state of the system for Eg > E∗ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' In this regime the bands are almost decoupled and the super- conducting gaps does not depend on Eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' However, in the case of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' 2(c) since the pair-exchange couplings are weak the conduction band cannot sustain the super- conductivity in the valence band and ∆1 is suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Thus, continuously tuning Eg to higher values will result in ∆1 << ∆2 so that there is only one significant super- 5 conducting gap and one significant condensate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' In the other case instead (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' 2(d)), the pair-exchange cou- plings are stronger and ∆1 is not much suppressed with respect to its initial value, since in these cases the su- perconductivity in the valence band is sustained by the condensate of the conduction band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Another interesting feature of this system is that ∆1 is enhanced for lower values of the total density as long as Eg < E∗ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' When Eg > E∗ g instead, the opposite situ- ation occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The value of E∗ g at which this behavior takes place depends on the level of filling of the conduc- tion band, shifting to the left when higher total densities are considered, and on the pair-exchange couplings that shifts E∗ g to the right when larger interactions strength are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The reason behind the behavior of the 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='4 (a) a2 n2 e (b) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='6 2 (c) a2 n1 h Eg / t 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='6 2 (d) Eg / t a2 ntot=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='00 a2 ntot=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='07 a2 ntot=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='26 a2 ntot=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='35 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Electron density a2ne 2 (a)-(b) in the conduction band and hole density a2nh 1 (c)-(d) in the valence band as functions of the band-gap Eg/t for different values of the total density a2ntot, normalized to the area of the unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' ω0/t = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The intra-band couplings are λ11 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='23 and λ22 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The pair-exchange couplings are (λ12 = λ21): (a),(c) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='001), (b),(d) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' superconducting gaps can be found by looking at the den- sities of particles forming Cooper pairs, which are elec- trons in the conduction band and holes in the valence band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' While the total density is fixed, the density in each band can vary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' In this way, the density of particles in the conduction band n2 is no longer controlled only by doping as for a single band system, there are instead ad- ditional particles excited from the valence band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Never- theless, for larger values of Eg the gain in the interaction energy due to superconductivity is much smaller than the kinetic energy cost for transferring electrons from the valence band to the conduction band, so that very few electrons (compared to the total density of electrons in the valence band) are excited into the conduction band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' This behavior is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' As one can see for a2ntot = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='00 the hole density in the valence band and the electron density in the conduction band coincide and are monotonically decreasing, both of them vanishing at the QCP Eg = E∗ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' This is a sign that superconductivity is due to holes in the valence band and to electrons in the conduction band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' In the other cases the hole density in the valence band is almost zero for Eg > E∗ g, while the electron density in the conduction band is approaching the asymptotic value given by the total density minus the density of the filled valence band a2n2 = a2ntot − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='7 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='6 2 (a) µ / t Eg / t 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='6 2 (b) Eg / t a2 ntot=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='00 a2 ntot=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='07 a2 ntot=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='26 a2 ntot=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='35 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Chemical potential µ/t as a function of the band- gap Eg/t for ω0/t = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The pair-exchange couplings are λ11 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='23 and λ22 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The pair-exchange couplings are (λ12 = λ21): (a) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='001),(b) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The chemical potential µ is reported for different total densities a2ntot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The black and the magenta dashed lines correspond to the bottom of the conduction band and the top of the valence band, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' 4 the chemical potential is reported as a function of Eg, for different total densities a2ntot and for different pair-exchange couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' For higher values of the total density and of the pair-exchange couplings the chemical potential shift toward higher energies, due to the larger number of electrons in the conduction band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' In particu- lar, when Eg is increased, in the low density regime the chemical potential starts deep inside the valence band and then enters the gap between the two bands, mean- ing that the condensate in the valence band spans a wide region of the BCS-BEC crossover, while the conduction band is always located in the BEC side of the crossover regime or in the BEC regime, depending on whether the chemical potential lies inside the conduction band or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' When Eg > E∗ g the chemical potential acquires a flat de- pendence and is not modified by Eg, in a similar way to what happens to the superconducting gaps and the den- sities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' 5 the condensate fraction is shown as a func- tion of Eg, for different a2ntot and for different pair- exchange couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The usual choice of the boundaries between the different pairing regimes has been adopted: for α < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='2 the superconducting state is in the weak- coupling BCS regime;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='2 < α < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='8 the system is in the crossover regime;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' for α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='8 the system is in the strong-coupling BEC regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Consistently with the information obtained from the chemical potential, in the low density regime the condensate in the valence band ex- plores the entire BCS-BEC crossover by varying Eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' For the considered pair-exchange interactions in (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' 5(c)) 6 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='8 1 (a) α2 e (b) a2 ntot=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='00 a2 ntot=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='07 a2 ntot=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='26 a2 ntot=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='35 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='8 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='6 2 (c) α1 h Eg / t 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='6 2 (d) Eg / t Crossover BEC BCS BEC Crossover BCS BEC Crossover BCS BCS Crossover BEC FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Condensate fractions in the conduction band αe 2 (a)- (b) and in the valence band αh 1 (c)-(d) as functions of the band-gap Eg/t for ω0/t = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The intra-band couplings are λ11 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='23 and λ22 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The pair-exchange couplings are (λ12 = λ21): (a),(c) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='001), (b),(d) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The condensate fractions are reported for different total densities a2ntot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Thin grey dashed lines correspond to α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='8 from bottom to top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' the valence band condensate is in the BCS regime for small Eg, while for larger pair-exchange interactions (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' 5(d)) is in the crossover regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' When the energy gap or the total density increases, the valence band condensate enters the BEC regime, with the hole condensate fraction αh 1 approaching unity, indicating that the remaining few holes are all in the condensate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The situation in the con- duction band is different, since due to the strong intra- band coupling the condensate is always located in the BEC side of the crossover regime or in the BEC regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' In the case a2ntot = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='00 both the condensate fractions suddenly drop to zero when Eg = E∗ g due to the quantum phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' 6 the intra-pair coherence length is reported as a function of Eg, for different a2ntot and for different pair- exchange couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Since for low densities and small pair-exchange couplings the valence band condensate is in the BCS regime (6(a)) when Eg is small, ξpair1 assumes initially larger values with respect to the average inter- particle distance l1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' For larger Eg the system enters the BEC regime and ξpair1 becomes much smaller than the average inter-particle distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The valence band con- densate goes from the crossover to the BEC regime in a small range of band gap values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' This behavior is observed also for larger values of the total density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The conduction band instead, due to the strong intra-band coupling re- tains a small value of the intra-pair coherence length with respect to the the average inter-particle distance l2 for all the considered values of the system density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' In this way we found Cooper pairs of different size coexisting in the system for low density and low pair-exchange couplings 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='3 (a) ξpair2 / l2 (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='5 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='6 2 (c) ξpair1 / l1 Eg / t 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='6 2 (d) Eg / t a2 ntot=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='00 a2 ntot=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='07 a2 ntot=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='26 a2 ntot=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='35 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Intra-pair coherence length ξpair2/l2 for the Cooper pairs of the conduction band (a)-(b) and intra-pair coherence length ξpair1/l1 for the Cooper pairs of the valence band (c)- (d) as functions of the band-gap Eg/t for ω0/t = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The intra-band couplings are λ11 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='23 and λ22 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The pair- exchange couplings are (λ12 = λ21): (a),(c) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='001), (b),(d) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The intra-pair coherence lengths ξpairi/li are reported for different a2ntot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' values, in the regime of small Eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' For the zero doping case the intra-pair coherence length is defined only for Eg < E∗ g, since in this regime the system is not super- conducting and a intra-pair coherence length cannot be defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The fact that the intra-pair coherence length is approaching zero at the QCP in the BEC regime is dif- ferent from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' [34], where giant Cooper pairs are found in the vicinity of the QCP in the BCS side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' In this case instead, what we have found is equivalent to the finite- density to zero-density QCP of tightly bound molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Namely, near the present QCP in the BEC side the pair size is so small that pairs behave as point-like bosons and the system can be described by its bosonic counterpart [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' 7 the order parameter coherence coherence length is reported as a function of Eg, for different a2ntot and for different pair-exchange couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' In the case a2ntot = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='00 the soft or critical coherence length ξs diverges when the band gap reaches the critical value Eg = E∗ g, since the system undergoes a quantum phase transition to the insulating state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' In the other cases a2ntot ̸= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='00, the soft coherence length ξs is not diverging, since no quantum phase transition occurs in the system for any Eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' In par- ticular, in the cases of a2ntot = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='07 and a2ntot = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='26 the soft coherence length ξs shows a maximum in correspon- dence of the respective Eg = E∗ g, showing its memory about the quantum phase transition of the valence band condensate, which takes place when the pair-exchange interactions are absent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The increase of λ12 = λ21 sup- presses the maximum, as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' 7(a) and (b), since the band-condensates become more coupled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' In the 7 0 2 4 6 (a) ξs / a (b) a2 ntot=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='00 a2 ntot=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='07 a2 ntot=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='26 a2 ntot=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='35 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='6 (c) ξr / a Eg / t 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='6 (d) Eg / t HC QPT QPT HC HC HC HC HC FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Soft ξs (a)-(b) and rigid ξr (c)-(d) order parame- ter coherence length, normalized to the lattice constant a, as functions of the band-gap Eg/t between the two bands at tem- perature T/t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='00065 and for ω0/t = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The intra-band couplings are λ11 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='23 and λ22 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The pair-exchange couplings are (λ12 = λ21): (a),(c) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='001), (b),(d) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='03).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The coherence lengths ξs,r are reported for different values of the total density a2ntot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' In the case a2ntot = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='00 (orange dashed line) ξr has been rescaled by a factor of 7 (c) and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='5 (d) to make the plot more visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' case of a2ntot = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='35 instead, since the valence band is never superconducting for any Eg when the band- condensates are decoupled, there is no quantum phase transition and no peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The rigid coherence length ξr in- stead remains finite for all Eg and for all a2ntot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Anyway, we find the memory of the quantum phase transition that takes place when the conduction band is empty and the valence band is filled (anntot = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='00).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' In this case in fact, also the conduction band returns to the normal state at Eg = E∗ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Indeed, for zero pair-exchange couplings, the rigid coherence length ξr reduces to the coherence length of the conduction band ξ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Even though for finite pair- exchange coupling the coherence length is non-diverging, it encodes the memory of the quantum phase transition of the conduction band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Also the maximum value of the rigid coherence length ξr is suppressed by the increase of λ12 = λ21 in this case, as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' 7(c) and (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' We consider now finite temperature effects on the critical energy band gap for the case of no doping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The super- conducting gaps as functions of temperature for different band gaps are reported in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The superconducting gaps present a non-monotonic behavior, that is very dif- ferent from the temperature dependence of the gaps in conventional superconductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The strong enhancement of ∆2 at finite temperature is due to the thermal excita- tion of the electrons from the valence band to the con- duction band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' This behavior becomes more pronounced for larger Eg, especially in the case of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' 8(c) in which the system is initially in the normal state for tempera- tures close to zero, and then becomes superconducting for 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='5 (a) (b) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='8 1 (c) Δ / t (d) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='8 1 (e) T / Tc 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='8 1 (f) T / Tc Eg/t = 0 Eg/t = 2 Eg/t = 3 NS SC Δ2 Δ2 Δ2 Δ1 Δ1 Δ1 Δ1 Δ1 Δ1 Δ2 Δ2 Δ2 NS NS NS SC FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Superconducting gaps ∆2/t opening in the conduc- tion band and in the valence band ∆1/t as functions of tem- perature T, normalized with respect to the critical tempera- ture Tc, for a2ntot = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The pair-exchange couplings are (λ12 = λ21): (a), (c), (e) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='03), (b), (d), (f) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' larger temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' This superconducting-normal state reentrant transition that we have found in our two-band system is based on a different mechanism with respect to the reentrant transitions observed in superconductors containing magnetic elements [41] or in granular super- conducting systems [42–45]: in the former it is attributed to the competition of magnetic ordering and supercon- ductivity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' while in the latter is attributed to tunneling barriers effect,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' while in our valence-conduction bands sys- tem the thermal excitation of electrons from the valence into the conduction band play a crucial role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' 9 we report the phase diagram T vs Eg for our system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' 9 the branch of the phase transition from the su- perconducting to the normal state corresponding to the reentrant behavior results from the second solution at lower temperatures of the linearized self-consistent equa- tions for the superconducting gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' From the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' 9 it is clear how the reentrant transition is more pro- nounced when the intra-band couplings are unbalanced (λ22 ≃ 3λ11 in the figure), while the reentrance is reduced when the intra-band couplings have similar values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' This effect occurs in a less evident manner also when the pair- exchange couplings are increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Therefore, the most relevant parameter to control the reentrance phenomenon is the intra-band coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' CONCLUSIONS We have studied the superconducting properties of a two-band system of electrons, interacting through a sep- 8 λ11 → λ22 λ22 → λ11 λ22 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='75 λ11 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='1 1 0 1 2 3 4 T / t Eg / t 0 1 2 3 4 Eg / t SC NS SC NS λ12 ↑→ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Phase diagrams in the temperature versus energy band gap plane, for the zero doping case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' In the left panel the red dashed line is for λ11 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='23, λ22 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='4, the green dashed line is for λ11 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='23, λ22 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='75 and the blue dashed line is for λ11 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='65, λ22 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The pair-exchange couplings are the same for all curves, λ12 = λ21 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' In the right panel the pair-exchange couplings from left to right are: λ12 = λ21 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='03, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='2, while the intra-band couplings are λ11 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='23 and λ11 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' arable attractive potential with a large energy cutoff and multiple pairing channels, at a mean-field level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The su- perconducting state properties are studied by varying the energy gap between the bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' We have considered dif- ferent levels of filling for the conduction band, while the valence band is always completely filled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' When the band- gap is modified, the density of electrons in the two bands changes, allowing for the occurrence of a density-induced BCS-BEC crossover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' When the pair-exchange couplings are small, the condensate in the valence band remains su- perconducting but with a strongly suppressed supercon- ducting gap ∆1 for Eg > E∗ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Therefore, in the regime of small pair-exchange coupling, after E∗ g, there is only one significant superconducting gap and one significant condensate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Interestingly, in this case the soft coherence length present a peak as a memory of the quantum phase transition that the valence band condensate undergoes in absence of pair exchanges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' This peak is more pronounced if the pair-exchange couplings are sufficiently weak and disappears for higher values of the pair-exchange cou- plings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' For higher values of λij, superconductivity in the valence band is sustained by the condensate in the conduction band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Furthermore, in this regime we have found that superconductivity is enhanced in the valence band for increasing doping as long as Eg < E∗ g, while for Eg > E∗ g superconductivity is enhanced for lower doping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' We have also found that superconductivity may occur even when no free carriers exist in the conduction band in the normal state at T = 0, as soon as the gain in super- conducting energy exceeds the cost in producing carriers across the band gap Eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' If the binding energy is larger than the energy band-gap, the system becomes unstable under the formation of Cooper pairs and superconduc- tivity emerges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' However, there exists a critical value of the energy band gap E∗ g in correspondence of which the process of creating Cooper pairs is not energetically fa- vorable anymore, at this point a quantum phase transi- tion occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' This quantum phase transition is confirmed by the soft coherence length, which is diverging in corre- spondence of the critical band gap Eg = E∗ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Thus, the ground state is superconducting if Eg < E∗ g, insulating if Eg > E∗ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' At finite temperature, the value of E∗ g is larger than its zero temperature value, because the elec- trons are thermally excited from the valence band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' This situation is responsible for the non-monotonic behavior of the superconducting gap opening in the conduction band, which is enhanced at low temperatures because of the electrons that jump from the valence band into the conduction band due to thermal excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' When there is a finite doping in the system, the sharp phase transi- tion becomes a smooth crossover and superconductivity extends for all Eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' In this case, for Eg > E∗ g the va- lence band contributes very weakly to the superconduct- ing state, since the hole density becomes almost zero in this regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' To conclude, we have found that the system explores dif- ferent regimes of the BCS-BEC crossover by tuning the energy band-gap and the total density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The valence-band condensate spans the entire BCS-BEC crossover for low enough density by varying the band-gap Eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' For larger values of the total density, the condensate of the valence band is very dilute and results in the BEC regime for any Eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The condensate of the conduction band instead re- sides in the BEC side of the crossover or completely inside the BEC regime, due to the strength of the intra-band coupling of electrons in the conduction band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' This pic- ture of the BCS-BEC crossover for the system has been found by analyzing the consistent behavior of the chemi- cal potential, the condensate fractions and the coherence lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Finally, in the case of zero doping and at finite temperature, an interesting new type of reentrant super- conducting to normal state transition has been numer- ically discovered for unbalanced intra-band couplings, showing that in this configuration superconductivity is assisted instead of being suppressed by increasing tem- perature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' This happens because the electrons in the va- lence band are able to jump into the conduction band even for larger values of the zero temperature critical band gap, due to thermal excitation, making the super- conducting state available for a wider range of Eg when the temperature is higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' ACKNOWLEDGMENTS We are grateful to Tiago Saraiva (HSE-Moscow) and Hiroyuki Tajima (University of Tokyo) for interesting dis- cussions and a critical reading of the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' acknowledges INFN for financial support of his Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' grant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' This work has been partially supported by PNRR MUR project PE0000023-NQSTI.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Matter 14, 193 (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} diff --git a/99AyT4oBgHgl3EQfqfgS/content/tmp_files/2301.00542v1.pdf.txt b/99AyT4oBgHgl3EQfqfgS/content/tmp_files/2301.00542v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..8944f0ceaf37b234b11cdd58ee50e1ebb87c4ceb --- /dev/null +++ b/99AyT4oBgHgl3EQfqfgS/content/tmp_files/2301.00542v1.pdf.txt @@ -0,0 +1,1774 @@ +Simple reactor model of relativistic runaway electron avalanche development +Egor Stadnichuk∗ +Moscow Institute of Physics and Technology - 1 “A” Kerchenskaya st., Moscow, 117303, Russian Federation +HSE University - 20 Myasnitskaya ulitsa, Moscow 101000 Russia +Daria Zemlianskaya,† Ekaterina Svechnikova,‡ Eduard Kim,§ Alexander Sedelnikov,¶ and Oraz Anuaruly∗∗ +(Dated: January 3, 2023) +High-energy gamma radiation in the Earth’s atmosphere is associated with the bremsstrahlung +of Relativistic Runaway Electron Avalanches (RREA) developing in thunderstorm electric fields. In +this paper, RREA development is studied in the system of two strong electric-field regions within +thunderstorms, which accelerate runaway electrons toward each other. Such a system is called the +simple reactor. It is discovered that the propagation of gamma rays and runaway electrons from one +region to another leads to positive feedback. This feedback called the reactor feedback can make +RREA self-sustaining, thus effectively multiplying high-energy particles inside thunderstorms con- +taining the simple reactor. The spectrum and characteristic time scale of the simple reactor gamma +radiation are in agreement with Terrestrial Gamma-ray Flashes (TGFs) data. The applicability of +the simple reactor model to TGF is discussed, and the distinguishing observable properties of the +simple reactor radiation during TGF and Thunderstorm Ground Enhancement are considered. +I. +KEYPOINTS +• RREA development in thunderstorms containing +the simple reactor is studied +• Two feedback mechanisms that can make RREA +self-sustaining in the simple reactor are discovered: +electron and gamma-ray reactor feedback +• Characteristics of gamma radiation of the simple +reactor are in agreement with TGF and TGE ex- +perimental data +∗ 1Moscow Institute of Physics and Technology - 1 “A” Kerchen- +skaya st., Moscow, 117303, Russian Federation; 2HSE Uni- +versity +- +20 +Myasnitskaya +ulitsa, +Moscow +101000 +Russia; +yegor.stadnichuk@phystech.edu +† zemlianskay.d@phystech.edu; Moscow Institute of Physics and +Technology - 1 “A” Kerchenskaya st., Moscow, 117303, Russian +Federation +Institute for Nuclear Research of RAS - prospekt 60-letiya Ok- +tyabrya 7a, Moscow 117312 +‡ svechnikova@ipfran.ru; Institute of Applied Physics of RAS - 46 +Ul’yanov str., 603950, Nizhny Novgorod, Russia +§ Moscow Institute of Physics and Technology - 1 “A” Kerchen- +skaya st., Moscow, 117303, Russian Federation +Institute for Nuclear Research of RAS - prospekt 60-letiya Ok- +tyabrya 7a, Moscow 117312; kim.e@phystech.edu +¶ Moscow Institute of Physics and Technology - 1 “A” Kerchen- +skaya st., Moscow, 117303, Russian Federation +Lebedev Physical Institute RAS; sedelnikov.as@phystech.edu +∗∗ Moscow Institute of Physics and Technology - 1 “A” Kerchen- +skaya st., Moscow, 117303, Russian Federation +Kurchatov Institute, Russian Research Centre - sq. +Academi- +cian Kurchatov, 1, Moscow, 123098, Russian Federation; orazan- +uaruly@gmail.com +II. +INTRODUCTION +Atmospheric physics is rich in mysterious natural phe- +nomena. +One of the new directions in atmospheric +research is high-energy atmospheric physics. +It sud- +denly appeared in 1992, when the Burst and Tran- +sient Source Experiment (BATSE) detector aboard the +Compton Gamma-Ray Observatory experiment discov- +ered short and intensive bursts of gamma rays originating +in the atmosphere of Earth [1]. These energetic bursts +are called Terrestrial Gamma-ray Flashes (TGFs). It is +established that the source of TGFs are thunderstorms +[2]. The characteristic duration of a TGF is 100 µs [3], +energy of detected TGF gamma-rays is up to 40 MeV +[4, 5]. Thunderstorm gamma radiation is also detected +on the Earth’s surface. It is called Thunderstorm Ground +Enhancement (TGE) [6] or gamma-ray glows [7], and +its characteristic duration is up to tens of minutes. It +is important to note that high-energy processes within +thunderstorms are closely related to lightning. TGE pre- +cede lightning and are always terminated by lightning +discharges [8]. TGFs are established to occur at the early +stage of the lightning leader propagation [9–11]. More- +over, many other interesting bright phenomena were reg- +istered in connection with the high-energy radiation from +thunderstorms [12–14]. +The underlying physics of high-energy atmospheric ra- +diation is the acceleration of electrons in thunderstorm +electric fields [2, 15–19]. In strong thunderstorm electric +fields, relativistic electrons obtain more energy from the +acceleration in the electric field than they on average lose +on interactions with atmosphere air molecules. Such elec- +trons are called runaway electrons [20]. When the electric +field strength exceeds the critical value, Ec = 276 +kV +m·atm, +runaway electrons are Møller scattered by air molecules, +which leads to the appearance of additional runaway elec- +trons [21]. +In this way, runaway electrons multiply in +the process of their propagation along the thunderstorm +arXiv:2301.00542v1 [physics.ao-ph] 2 Jan 2023 + +2 +electric field, forming the Relativistic Runaway Electron +Avalanche (RREA) [16, 17]. To start a RREA an initial +seed energetic particle is needed to appear within the +thunderstorm electric field [16, 22]. For example, it can +be a secondary cosmic ray particle [23] or a seed parti- +cle generated inside the thunderstorm [24–26]. Charac- +teristic RREA particle energies range from tens of keV +to tens of MeV [16]. +Thus, runaway electrons natu- +rally produce bremsstrahlung gamma rays in collisions +with air molecules, which is detected as TGF and TGE +[4, 7, 8, 10, 17, 22]. +The mystery of TGFs is that a large number of high- +energy particles appear almost instantly inside a thunder- +cloud [1, 5, 25]. There are two possible ways to explain +such phenomena. The first possible scenario is the gener- +ation of a large number of seed electrons within the thun- +derstorm super strong electric fields, possibly created by +the lightning leader propagation [18, 24, 25, 27–29]. Also +it is considered that lightning leader itself can radiate +synchrotron gamma-rays [30]. These ideas are supported +by the fact that x-rays are observed in association with +lightning leader propagation [31, 32]. The second pos- +sible scenario is the multiplication of RREAs by posi- +tive feedback mechanisms [21, 26, 33–37]. The relativis- +tic feedback works in the following way. Bremsstrahlung +gamma ray radiated by runaway electrons can produce +electron-positron pairs within thunderstorm supercriti- +cal electric field region. Positrons are accelerated by the +electric field in the direction opposite to the runaway +electrons acceleration direction. In this way, positrons +reach the beginning of the supercritical region, where +they produce seed runaway electrons by the Bhabha scat- +tering [33]. Thus, relativistic positron feedback multiplies +RREAs and, moreover, can make RREA self-sustaining +[21]. Similarly bremsstrahlung gamma rays can Compton +backscatter and thus produce seed runaway electrons at +the beginning of the supercritical region, which is the rel- +ativistic gamma ray feedback [38]. RREA models based +on the positive feedback are supported by the fact that +their characteristic time and spectrum coincide with the +characteristic time and spectrum of TGF [2–4, 25]. Nev- +ertheless, the relativistic feedback requires strong large- +scale electric fields, which have never been directly ex- +perimentally observed in thunderstorms [33, 34, 36, 39]. +It has been discovered that non-uniform thunderstorm +electric field geometry leads to another feedback mech- +anism called the reactor feedback [26, 40]. Let a thun- +derstorm consist of several separate electric-field regions +with electric-field strength sufficient for RREA produc- +tion. +Such regions, for simplicity, are called cells [26]. +If a seed electron starts a RREA within one of the +cells, the following processes occur. A RREA radiates +bremsstrahlung gamma-rays. Gamma-rays have a large +attenuation length at thunderstorm altitudes. +There- +fore, gamma-ray photons propagate through the thun- +derstorm and can reach other cells. +There is a prob- +ability that a gamma-ray photon will interact with air +molecules by compton scattering, photoelectric effect of +electron-positron pair production within a cell, which +can result in runaway electron generation. A runaway +electron can produce a RREA. In this way the reac- +tor gamma-ray feedback works: separate thunderstorm +cells irradiate each other with gamma radiation, which +results in RREA multiplication. Another reactor feed- +back mechanism - runaway electron transport between +cells. If the cells are close to each other, runaway elec- +trons are able to penetrate the air layer between them. +In this way, runaway electrons propagate from one cell +to another and, thus, multiply RREA. In general, reac- +tor feedback is defined as the multiplication of RREA by +high-energy particle exchange between separate thunder- +storm RREA-accelerating regions. Distant cells amplify +each other mostly by gamma-ray photon exchange be- +cause of their high penetrating power in the air. For cells +located close to each other, the reactor feedback works +mainly by runaway electron exchange, since RREA con- +sists mainly of runaway electrons. It has been established +that reactor feedback can lead to self-sustaining devel- +opment of RREA, and, moreover, requires lower electric +field strength in comparison with the relativistic feedback +[26]. +In this paper, the reactor feedback is studied in the +simplest case of non-uniform thunderstorm electric field, +when thunderstorm consists of two cells, oriented in the +way that they accelerate runaway electrons towards each +other. The system is called the simple reactor. The re- +search is motivated by observations of thunderstorm elec- +tric structures, part of which can be described as the sim- +ple reactor electric structure [41–47]. In this paper, it has +been discovered that both gamma-ray reactor feedback +and runaway electron transport feedback amplify RREA +in the simple reactor, and these feedback mechanisms +require a smaller electric field strength to provide self- +sustaining RREA development in comparison with the +relativistic feedback. In Section III the feedback mecha- +nisms of the simple reactor are described. In Section IV +the reactor feedback is theoretically described. Section V +provides Monte Carlo simulations of the simple reactor +using GEANT4. In Section VI the simple reactor is dis- +cussed as a possible mechanism for TGF and TGE, the +distinguishing properties of this model that are experi- +mentally observable are considered. +III. +SIMPLE REACTOR MODEL +The reactor feedback is the intensification of RREA +development in a thunderstorm supercritical electric field +region (cell) by the radiation of other cells [26]. The sim- +plest system capable of demonstrating the reactor feed- +back is the system of two cells with oppositely directed +electric fields accelerating electrons toward each other. +This system is called “the simple reactor” and we con- +sider it as the next step from uniform electric field mod- +els to the description of RREA in the electric field of a +real thundercloud. In the simple reactor, the distribu- + +3 +tion of the electric field corresponds to the system of two +flat capacitors placed one on top of the other. +It can +be considered as the approximation of the electric field +distribution in the region of the cloud with three charge +layers: a positive middle layer and two negative layers. +The described electric field distribution can be a part of +a natural thunderstorm [41–47]. +There are two reactor feedback mechanisms in the sim- +ple reactor. +The first mechanism, reactor gamma-ray +feedback, works in the following way. Let a seed elec- +tron form a RREA within one of the cells. RREA grows +towards the opposite cell and radiates bremsstrahlung +gamma rays [16]. Gamma rays have a significant pene- +tration power through the atmosphere at thunderstorm +altitudes. Thus, gamma rays reach the opposite cell and +propagate through it. Interactions of gamma rays with +air molecules of the opposite cell generate seed runaway +electrons. These electrons start a new RREA in the op- +posite cell. Further, the new RREA propagates towards +the initial cell, generating gamma rays, which similarly +produce RREA in the initial cell. In this way, the pro- +cess loops, and thus gamma-ray reactor feedback makes +RREA self-sustaining in the simple reactor. The second +mechanism, runaway electron transport feedback, works +in the case when cells are close to each other. In this +case, runaway electrons can penetrate the gap between +cells, reaching the opposite cell. When a runaway elec- +tron reaches the opposite cell, it penetrates inside the cell +until the electric field reverses it. After reversal runaway +electrons are accelerated toward the initial cell. Thus, +in the simple reactor, runaway electrons oscillate near +the border of the cells. During the oscillation, runaway +electrons are multiplied by the Møller scattering, which +also leads to a self-sustaining process. It is established +in this paper, that both feedback mechanisms not only +can make RREA self-sustaining but also can significantly +multiply the number of relativistic particles in the thun- +derstorm containing the simple reactor (Figure 1). +It +should be noted that the relativistic feedback naturally +impacts RREA development in the simple reactor, how- +ever, further it is shown that the influence of the rela- +tivistic feedback is negligible compared to the influence +of the reactor feedback. +IV. +ANALYTICAL SIMPLE REACTOR MODEL +A. +Gamma-ray reactor feedback +To describe the simple reactor gamma-ray feedback +theoretically, it is necessary to study the response of a cell +to gamma radiation falling into it [26]. Let N gamma-ray +photons enter the i-th cell (i =1,2) from above, along the +cell’s electric field vector. In order to find the feedback +coefficient, it should be calculated how much gamma will +fly back from the cell towards the other cell. Let λi +RREA +be the growth length of an avalanche of runaway elec- +trons, λi +− be the decay length of gamma radiation, λi +γ be +FIG. 1. The physics of the simple reactor in the GEANT4 +simulation [48]. Green lines - gamma-ray photon tracks, red +lines - runaway electron tracks. Blue arrows - electric field +lines, yellow dots - particle interaction points. The simula- +tion is started with a single seed electron. The simple reactor +consists of two supercritical electric field regions accelerat- +ing runaway electrons towards each other. High-energy par- +ticles exchange between these regions makes the process self- +sustaining. +The picture resembles the Eye of Sauron from +the Lord of the Rings trilogy: runaway electrons oscillating +near the cell boundary form a pupil, the halo of gamma-ray +photons and RREAs formed by gamma-ray reactor feedback +resembles the cornea of the eye. +the path of a runaway electron before the emission of a +gamma-ray photon with supercritical energy, λi +e− be the +path length of a gamma before the birth of a runaway +electron, and P i is the probability of a turn of an elec- +tron with further development of the runaway avalanche. +These parameters depend on the magnitude of the elec- +tric field and the density of the air. In the first approxi- +mation, the values for these parameters can be retrieved +from the article [34]. In general, the cells of the simple +reactor are assumed to have different field strengths, air +density, and cell lengths. +The gamma entering the cell along the electric field +will generate electrons with supercritical energy, while +losing its energy, which leads to an exponential decay of +the primary gamma flux. On the segment [z, z + dz], the +flown gammas will give birth to the following number of +avalanches of runaway electrons: +df i +e−(z) +dz +dz = NP ie +− +z +λi +− dz +λi +e− +(1) +The dynamics of the number of bremsstrahlung +gamma-ray photons during RREA propagation along the +z axis is described by the following equation [34]: +dNγ = e +z−z0 +λRREA dz +λγ +− Nγ +dz +λ− +(2) +The first term in 2 describes the production of +bremsstrahlung gamma-ray photons by runaway elec- +trons, and the second term describes a decrease in the + +4 +number of gamma-rays due to its interaction with air +molecules. The solution for 2 is 3: +Nγ(z, z0) = +λRREAλ− +λγ(λ− + λRREA) · +� +e +z−z0 +λRREA − 1 +� +(3) +Each RREA grows according to the well-known ex- +ponential law [20], spreading toward the initial gamma +rays entering the plane. In this case, depending on the +point of birth of the avalanche, the amount of secondary +gamma rays that will reach the end of the cell will be +as follows (since the avalanche born at the point z will +travel a distance equal to z): +dF i +γ(z) = +df i +e−(z) +dz +dz +λRREAλ− +λγ(λ−+λRREA) · +� +e +z +λRREA − 1 +� +(4) +Thus, the gamma-ray local multiplication factor [26] +can be calculated (Formula 5): +νi = +� Li +0 +dF i +γ(z) +dz +dz +N +(5) +Integration leads to the following formula for the +gamma-ray multiplication factor: +νi = +P iλi +RREAλi +− +λi +e−λiγ(λi +RREA + λi +−) +� +λi +RREAλi +− +λi +− − λi +RREA +� +e +Li λi +−−λi +RREA +λi +RREAλi +− − 1 +� +− λi +− +� +1 − e +− Li +λi +− +�� +(6) +The system with positive feedback can be character- +ized by the feedback coefficient [26, 33, 34]. +For the +simple reactor, the feedback coefficient shows how many +times the number of high-energy particles will increase in +one full reactor feedback cycle, and it is found with the +following formula: +Γ = ν1 · ν2 +(7) +The number of particles in the simple reactor grows ex- +ponentially with each feedback generation: N(n) = Γn, +where n is the number of feedback generation [26]. There- +fore, the criterion for self-sustaining RREA development +in a simple reactor is as follows: +Γ = ν1 · ν2 ≥ 1 +(8) +With the obtained criterion, the thunderstorm condi- +tions necessary for self-sustaining RREA development by +the reactor gamma-ray feedback can be calculated. These +conditions are presented in Figure 2. Conditions are pre- +sented for 3 types of feedback: relativistic positron feed- +back [21], simple reactor feedback, and multicell reactor +FIG. 2. The comparison of self-sustaining positron feedback +necessary conditions [34] and simple reactor self-sustaining +feedback necessary conditions (Formula 6, 8). RREA accel- +erating region length is normalized to λRREA, electric field +strength is normalized to critical electric field strength (the +electric field required for RREA development [16]). The sim- +ple reactor is also compared with necessary conditions for +self-sustaining gamma-ray feedback in multicell reactor model +[26]. It can be seen that reactor models require significantly +lower thunderstorm electric field strengths for self-sustaining +RREA development than the relativistic feedback. This im- +portant property of the model comes at expense of the com- +plexity of the electric field geometry. The more complex elec- +tric field geometry is, the lower electric field strength is re- +quired for the feedback to be effective. It should be noted that +the conditions for the reactor models are presented without +taking into account runaway electron transport between cells. +Moreover, it has been discovered that thundercloud hydrome- +teors can amplify RREA [49]. Thus, the exact self-sustaining +RREA conditions can be lower than the conditions presented +on this picture. +feedback [26]. The coordinates in the figure are chosen so +that the conditions are invariant with respect to altitude +[33, 34]. It can be seen from the figure that both the mul- +ticell and the simple reactor feedback mechanisms require +significantly lower electric field strength in comparison +to the relativistic feedback discharge model. This impor- +tant property of the reactor feedback comes with a price +of the thunderstorm electric field geometry complexity. +The most complex electric field geometry, the multicell +reactor, requires the lowest electric field strength for the +self-sustaining RREA development, while the simplest re- +actor structure, the simple reactor, requires the electric +field strength lying in between the multicell reactor and +the uniform electric field. It should be noted that if thun- +dercloud electric field parameters lie above the curve in +Figure 2 then the number of energetic particles within +the thundercloud grows exponentially [26, 33, 34]. Oth- +erwise, provided that there is no external source of seed +particles, the number of energetic particles decays, and +the decay rate depends on the feedback coefficient [26]. +Thus, even if the feedback does not make the RREA de- +velopment self-sustaining, it still increases its duration. + +Positron feedback +Multicell reactor +103 +Simple reactor +102 +入RREA +101 +100 +10-1 +2 ×100 +3 ×100 +4×100 +100 +E5 +B. +Runaway electron transport between cells in +the simple reactor +GEANT4 simulations of the simple reactor showed the +importance of runaway electron transport between cells +for the reactor feedback (Figure 1). In this section, it is +shown that the oscillations of runaway electrons between +cells in the simple reactor can become self-sustaining and +even lead to runaway electron multiplication. +At first +glance, this effect may seem paradoxical and contrary to +the law of energy conservation: While a single runaway +electron move from one cell to another, the total energy +it receive from the electric field in the full circle of its +oscillation is zero, and, thus, this runaway electron, on +average, loses energy in interaction with air. Therefore, a +single runaway electron will inevitably lose its energy and +stop. However, it is seen in the simulations that runaway +electrons oscillate and multiply in the strong electric field +of the simple reactor. Therefore, the following question +arises: Where do runaway electrons take energy when the +feedback becomes self-sustaining? +Moreover, the total +length of runaway electron motion between cells back and +forth cannot be longer than its energy divided by eEc, +where Ec - critical electric field, e — elementary electric +charge. This is not more than several tens of meters. +It turns out that the effect of runaway electron trans- +port between cells can be physically explained and that +the energy conservation paradox is resolved by runaway +electron multiplication. If a runaway electron multiplies +by Møller scattering [21], the result is that the initial and +generated runaway electrons receive twice as much energy +from the electric field compared to the single initial run- +away electron. When the initial runaway electron stops, +the secondary electron continues to oscillate and multi- +ply. The reactor feedback in the simple reactor caused by +runaway electrons can even become self-sustaining. If the +thunderstorm electric field is much stronger than the crit- +ical electric field, the runaway electron interaction with +the air becomes negligible. Moreover, by the interaction +with air, runaway electrons will multiply, which leads to +an enormous growth of the number of relativistic parti- +cles. Thus, when the electric field strength decreases to +values comparable to Ec, there is a point where the multi- +plication of runaway electrons compensates for the energy +losses in the air interaction. At this point, the runaway +electron transport feedback becomes self-sustaining. +An interesting property of the runaway electron trans- +port feedback is its spatial scale. A runaway electron af- +ter hitting an adjacent cell cannot propagate within the +cell deeper than its kinetic energy divided by e(E + Ec), +where E is the electric field strength of the cell. There- +fore, runaway electron transport feedback coefficient de- +pends only on the electric field strength for cell lengths +longer than runaway electron maximum energy divided +by e(E + Ec), which is about 100 meters for 10 km alti- +tude and 40 MeV maximum energy [15, 50]. Thus, run- +away electron transport occurs near the cell interface, +which softens the conditions required for self-sustaining +RREA development in the simple reactor, because long +cells are not needed as in other types of feedback 2. Nev- +ertheless, it should be noted that runaway electron feed- +back works effectively only for cells located close to each +other since electrons are quickly absorbed by air unless +they are accelerated by the electric field. +To theoretically analyze the runaway electron trans- +port feedback, it should first be understood how run- +away electrons are decelerated in the electric field of the +adjustment cell. Decelerated runaway electron attenua- +tion length can be found by substituting negative electric +field strength into the empirical formula for the RREA +e-folding length: +λdecay = 7300[kV ] +−E − Ec +(9) +In this way, number of runaway electrons in the beam +will decrease exponentially: +Nbeam(z) = N0e +z +λdecay +(10) +This analytic continuation of the RREA growth law +[21] can be justified in the following way. +Normalized +runaway electron spectrum can be described with the +function: +dfRREA +dε += 1 +ε0 +e− ε +ε0 +(11) +ε0 = 7.3 MeV - runaway electron mean energy [16]. +An electron with energy ε, on average, stops at the coor- +dinate: +z(ε) = +ε +e(E + Ec) +(12) +Number of runaway electrons leaving the beam in the +interval (z, z + dz) per one primary electron is: +Nbeam +dz +dz = −N0 +dfRREA +dε +dε +dz dz = N0 +E + Ec +7300[kV ]e− +E+Ec +7300[kV ] z +(13) +Thus, the formula 9 is obtained. +The runaway electron transport feedback coefficient +can be defined as the number of runway electrons leav- +ing the cell per one runaway electron entering the cell +(analogically to the gamma-ray reactor feedback). The +number of runaway electrons, which entered the cell, de- +creases according to the exponential law derived above +as these runaway electrons propagate into the cell (for- +mula 9). When a runaway electron leaves this beam it +can stop or it can reverse and form a RREA, which then +propagates to the entry plane of the cell. +If a RREA +starts at the point z, the number of runaway electrons +within this RREA reaches e +z +λRREA when RREA leaves + +6 +the cell [34]. Therefore, if the reversal probability of run- +away electrons from the primary beam is equal to P, the +runaway electron transport feedback coefficient can be +obtained as follows: +�νe− = +� L +0 +dzPe +z +λRREA dNbeam +dz += P +�λ +� L +0 +dze +1 +λRREA − 1 +�λ +(14) +Here �λ = −λdecay > 0. Thus, the following formula is +obtained: +�νe− = +PλRREA +λRREA − �λ +� +1 − exp +� +L +� +1 +λRREA +− 1 +�λ +��� +(15) +This formula can be simplified using the empirical for- +mula for λRREA [21]: +λRREA +λRREA − �λ += E + Ec +2Ec +(16) +Therefore: +�νe− = P E + Ec +2Ec +� +1 − exp +� +−L +2Ec +7300[kV ] +�� +(17) +This formula can be further simplified for cells with +cell length L ≫ +7300[kV ] +2Ec +, which works for cells larger +than 100 m: +�νe− = P E + Ec +2Ec +(18) +Generally, there is some space between cells within a +thunderstorm. Runaway electrons lose energy by inter- +acting with air molecules while propagating through the +gap between cells. A fraction of runaway electrons be- +come undercritical and leave the beam. +This fraction +can be estimated with the decay length from formula 9 +for E = 0 as exp +� +−l +Ec +7300[kV ] +� +, where l is the gap between +cells in the simple reactor. Since, in the first approxima- +tion, all runaway electrons lose the same amount of en- +ergy in the gap, the shape of their spectrum remains the +same. Thus, the formula for runaway electron transport +feedback coefficient, taking into account the gap between +cells, simply modifies in the following way: +�νe− = P E + Ec +2Ec +� +1 − exp +� +− +2EcL +7300[kV ] +�� +· +exp +� +− +Ecl +7300[kV ] +� +(19) +V. +GEANT4 SIMULATION +The Monte Carlo simulation of the simple reactor was +carried out using Geant4, version 4.10.06.p01. Geant4 is +recognized as a good tool to model RREA [37, 51]. The +physics list G4EmStandardPhysics option4 was chosen +as the reliable physics list for RREA simulations [51]. +This list includes all interactions of electrons, gamma- +rays and positrons for energies characteristic for RREA +processes [26, 48]. The energy cut for the particles was +chosen 50 keV based on the fact that low-energy particles +will quickly decay, as they do not run away [16], and will +not contribute to the feedback. The simulated geometry +is a large world volume filled with air, within which a +child volume is specified, also filled with air. A simple +reactor by definition consists of two child volumes: both +volumes are filled with air with a density of 0.414 kg/m3, +corresponding to altitude 10 km, and contain electric field +in the way that both volumes accelerate runaway elec- +trons towards each other (Figure 1). +The purpose of the GEANT4 simulation is to find the +parameters of the system necessary for the self-sustaining +RREA regime (when the generation of high-energy par- +ticles within the thunderstorm does not stop until the +electric field is discharged). The simulation was carried +out by varying the cell size and the strength of the elec- +tric field inside of it. At a certain electric field strength, +the number of gamma-ray photons and runaway electrons +crossing in both directions the boundary between the +simple reactor cells will not decrease over time. Thus, +RREA within the simple reactor will not die out over +time. +In this case, self-sustaining feedback is reached. +Thus, by increasing the electric field with a constant cell +length, one can find the critical point at which the reactor +will become self-sustaining. In this way, the achievement +of critical values is checked, and the conditions are cal- +culated. +The most important stage in modeling is the division +of the high-energy particles into generations. If directly +two cells are created with an oppositely directed field, +then it will be quite difficult to divide the process into +feedback generations, since, under certain conditions, a +self-sustaining feedback is formed and the simulation will +not stop. Thus, analogically to the theoretical model, it +was decided to simultaneously simulate only a half of +a simple reactor. This approach is possible due to the +symmetry of the simple reactor. The modeling scheme +is shown in Figure 3. +The model consists of a single +cell filled with air and electric field. At the beginning of +the cell an air detector is placed — the volume within +which particles are stopped and registered. In the first +simulation step, seed particles with an energy of 5 MeV +are launched from the beginning of the cell along the di- +rection of the electric field. Seed runaway electrons are +decelerated by the electric field. Some of them penetrate +into the cell, reverse, and form RREA towards the de- +tector. Seed gamma-ray photons propagate through the +cell and interact with air molecules. This interaction re- + +7 +detector +e- +FIG. 3. The design of a simple reactor has been simplified +in the GEANT4 simulation to consider only one cell as in +the figure. Runaway electrons and gamma-ray photons are +launched from the right side of the cell along the direction +of the electric field. +The interactions of launched particles +lead to RREA formation, which is accelerated by the electric +field to the right side of the cell. In the result, generated par- +ticles reach the detector and registered. In the next stages +of the simulation, registered particles are launched and new +generated particles are similarly registered. In this way, each +reactor feedback generation is studied separately, thus, allow- +ing the analysis of the model. +sults in runaway electrons generation, which reverse and +form RREA, also moving and growing toward the begin- +ning of the cell. All particles that reach the detector are +stopped and registered and the simulation stops. In this +way, the first feedback generation is modeled. In subse- +quent simulations, the particles registered in the previous +iteration are launched into the cell accordingly (thus im- +itating propagation of the high-energy particles from one +cell into another in the simple reactor). These particles +interact with the cell, which results in new particles gen- +erated that reach the detector. For each feedback gen- +eration this process repeats. Figure 4 shows the number +of gamma-rays reaching the detector in each simulated +feedback generation. The graph shows that depending on +the thunderstorm conditions the number of high-energy +particles can decay from generation to generation or vice +versa. The thunderstorm conditions when the number of +particles does not change are the necessary conditions for +the self-sustaining development of RREA in the simple +reactor. +To calculate the feedback coefficient, the simulation +was launched with seed runaway electrons. Number of +generated gamma-ray photons and runaway electrons for +each feedback generation was registered, thus forming +plots similar to Figure 4. Each plot was fitted with an +exponential function. The feedback coefficient 6, 19 is +obtained from the coefficient in the exponent by adding +0 +2 +4 +6 +8 +10 +Generation number +4 +5 +6 +7 +8 +9 +10 +log(N) +Dependence of the number of gamma in a generation on its number +100kV/m +150kV/m +170kV/m +200kV/m +220kV/m +240kV/m +250kV/m +260kV/m +300kV/m +FIG. 4. The dependence of the logarithm of the number of +gamma-ray photons that propagates from one cell to another +in the simple reactor depending on the number of feedback +generation. The number of generation is the number of an +iteration of a simple reactor simulation (Figure 3). It can be +seen that the number of gamma-rays produced by the simple +reactor exponentially grows or exponentially decays depend- +ing on the electric field strength. +1 to it [26]. The obtained dependence of the exponent +parameter on the electric field strength is shown in Fig- +ure 5. When the exponent parameter is positive, number +of high energy particles in the simple reactor thunder- +storm self-sustainably grows until the electric field is dis- +charged. +It is also interesting to calculate the spectrum of the +gamma-rays produced within the simple reactor and +compare it with the spectrum of an ordinary RREA +bremsstrahlung. To obtain the spectrum, a full simple +reactor with two cells oriented towards each other was +simulated. This simulation captures the particles with +their energies in a tracking action. The critical parame- +ters of the simple reactor were chosen — the field is 300 +kV/m and the length of one cell is 400 m for 10 km alti- +tude air density. +Similarly to the previous simulation +technique, the G4EmStandardPhysics option4 physics +list was used, and the energy cut for particles is 50 keV. +The simulation was stopped when the number of high- +energy particles reached 106, and the spectrum of regis- +tered gamma rays is obtained. In addition, a simulation +for a single cell with the same parameters was carried out +to obtain the spectrum of an ordinary RREA. The result- +ing spectra are shown in Figure 6. The graph shows that +the spectra are the same. It should be noted that sin- +gle cell gamma-ray spectrum contains more pronounced + +8 +100 +125 +150 +175 +200 +225 +250 +275 +300 +Field, kV/m +0.05 +0.00 +0.05 +0.10 +0.15 +Exponent parameter +FIG. 5. The dependence of the feedback generations expo- +nent parameter on the electric field in the simple reactor for +the cell length 400 m. Negative exponent parameters means +the decay of RREA in the simple reactor, while positive ex- +ponent parameter means self-sustaining RREA development +with high energy particles generation. The exponent parame- +ter includes both simple reactor feedback processes: gamma- +ray reactor feedback and runaway electron oscillations (Fig- +ure 1). The conditions necessary for self-sustainable regime +(when exponential parameter equals 0) are in agreement with +theoretical predictions (Figure 2). +positron peak. +However, when gamma rays propagate +from thunderstorm to the detector registering TGF or +TGE, they interact with the atmospheric layer and nat- +urally produce the positron peak. Thus, this peak will +also be present when the TGF or TGE produced by the +simple reactor is measured. Nowadays it has been reli- +ably established that the TGF and TGE source spectrum +is the RREA spectrum [50, 51]. Thus, the simple reactor +can be the mechanism for the TGF or TGE. +VI. +DISCUSSION +The discovered mechanism called the simple reactor +can be applied for a thundercloud containing two regions +with electric field exceeding the critical value, i.e. +al- +lowing the RREA development (for simplicity, such re- +gions are called cells [26]), electric field is oriented in the +way that cells accelerate runaway electrons towards each +other. It was established that there is a positive feed- +back in this system caused by two mechanisms (besides +the relativistic feedback [21], which impact is relatively +low (Figure 2)). The first mechanism is the transport +of runaway electrons from one strong field region to an- +other. +This leads to the effective high-energy electron +multiplication and runaway electron oscillation near the +edge between the strong electric-field regions. The elec- +tron transport feedback coefficient is very high for a small +gap between cells, and is a dominant RREA multiplica- +tion mechanism in the case of the small gap. On the other +FIG. 6. Comparison of the spectra obtained from the sim- +ulation of the simple reactor and ordinary RREA spectrum, +obtained from a single cell simulation with a uniform electric +field. +The simulation of the simple reactor was turned off +when enough statistics were collected. It can be seen that the +spectrum of the simple reactor gamma-radiation is the same +as the RREA bremsstrahlung spectrum. It is established that +the thunderstorm gamma-radiation spectrum agrees with the +RREA spectrum [4, 8]. Thus, the simple reactor can be one +of the mechanisms of TGF and TGE. +hand, in the case of a significant gap, when the distance +between strong field regions exceeds the characteristic +length of runaway electrons, too few runaway electrons +propagate through the gap between regions, thus an- +other feedback mechanism dominates. The second feed- +back mechanism is the gamma-ray reactor feedback [26]. +RREA bremsstrahlung gamma-rays have high penetra- +tion rate in the air. Thus, in the simple reactor, gamma- +rays effectively propagate from one cell to another. When +a gamma-ray photon propagates through the opposite +cell, it interacts with air, producing secondary RREAs, +which is the gamma-ray reactor feedback. +Both feed- +back mechanisms can lead to self-sustaining RREA de- +velopment and, moreover, to rapid multiplication of high- +energy particles within a thunderstorm. +The formulas derived in this paper allow one to pre- +dict the feedback coefficient for both feedback mecha- +nisms without complicated modeling; the theoretical pre- +dictions of this paper are verified by GEANT4. The dis- +covered feedback coefficients completely describe the be- +havior of the simple reactor, e.g. allow to calculate the +conditions required for the self-sustaining RREA devel- +opment (Figure 2). The limitations of the proposed an- +alytical model are as follows. Firstly, the model is one- +dimensional and, therefore, does not consider the trans- +verse dynamics of the avalanche, which affects the feed- +back coefficients in the case of narrow electric field regions +[34]. Second, though the description of runaway electron +transport feedback qualitatively matches Geant4 simu- +lations, it lacks quantitative accuracy. More theoretical + +10-1 +simple reactor +onebox +10-2 +10-3 +102 +Energy,kev9 +and modeling research is needed to establish the exact in- +fluence of the electron transport feedback on the RREA +development. +The simple reactor geometry corresponds to the charge +distribution with two negative charge layers on both sides +of the positive layer. +This structure can be a part of +a more complicated charge structure of a thunderstorm +[41–47]. In the simple reactor, the maximum density of +runaway electrons will be on the border between two op- +positely directed cells — in the center of the simple re- +actor, in the region of the positive charge (it should be +noted that the large value of a single positive charge in a +region of a cloud can be sufficient for RREA development +below and above this region, forming the simple reactor). +This feature distinguishes the simple reactor model from +models assuming the development of RREA in a single +cell with maximum particle density in the cloud top or +cloud base. Since in the simple reactor the maximum run- +away electron density is located in the center of the reac- +tor, it is harder for bremsstrahlung gamma-rays to reach +detectors registering TGF or TGE due to the greater +thickness of the atmosphere that they must penetrate. +However, this does not contradict the observed gamma- +ray fluxes, since the reactor feedback increases the num- +ber of generated bremsstrahlung gamma-rays within a +thunderstorm containing the simple reactor. +This in- +crease compensates for the decrease in gamma-ray flux +by extra atmosphere in has to penetrate. +Another distinguishing and important property of the +simple reactor is that it generates simultaneous gamma- +ray radiation directed upward and downward from a +thundercloud (or in other opposite directions if the simple +reactor is not oriented vertically). This means that theo- +retically it is possible to simultaneously detect a TGF or +a TGE from two opposite sides of a thunderstorm, e.g., +from the top and from the bottom. Such observation can +be performed, for example, with an airplane containing +particle detectors flying over an observatory with particle +detectors. Also a TGF generated by the simple reactor +can be registered simultaneously from space and ground +observatories, but the probability for the space station to +be located above the ground observatory at the moment +of TGF is very low due to the TGF short duration. It +should be noted that the time profile of the gamma-ray +flux in measurements from both sides of the thunder- +storm must match in order to conclude that upward and +downward gamma-ray radiation are connected by the re- +actor feedback. This requires a good temporal resolution +of the detectors. +The simple reactor with a large feedback coefficient can +be a source of TGF. Characteristic timescale of the sim- +ple reactor is its size divided by the speed of light, which +is in order of microsecond. Therefore, the timescale and +radiated gamma-ray spectrum satisfy the experimentally +observed TGF data [1, 3, 5]. Runaway electron accelera- +tion and its bremsstrahlung gamma-ray radiation in the +simple reactor precede the lightning leader and should co- +incide with the early stage of the lightning initiation. It +should be noted that a TGF generated by positive feed- +back has a characteristic exponential gamma-ray flux rise +time profile. Number of high-energy particles grows ex- +ponentially on TGF timescales as thunderstorm electric +field remains almost constant on these timescales. +At +the TGF peak, thunderstorm electric field lowers, thus +feedback coefficient drops and the feedback becomes fi- +nite: the flux of high-energy particles starts to decay or +even abruptly terminates, if the electric field required for +RREA development abruptly disappear. The disappear- +ance of the electric field can be connected either with lo- +cal discharges or with the initiation of a lightning leader. +From the rise profile of measured TGF flux the feedback +coefficient can be restored. The feedback coefficient is a +good source of information on the thunderstorm electric +field during the TGF (Formula 6, 19) [34]. +Another TGF model based on RREA, the relativis- +tic feedback discharge model, supposes significant posi- +tive feedback (the relativistic feedback) in the most sim- +ple thunderstorm geometry - uniform electric field [38]. +The disadvantage of this model is that it requires very +high values of electric field strength extended over a +large thunderstorm space [16, 34, 39]. +The significant +feature of the simple reactor is that it requires smaller +electric field strength for the self-sustaining RREA de- +velopment than it is in the uniform electric field (Fig- +ure 2) [26, 34, 38]. Moreover, provided that two strong +field regions are formed by the same positive charge layer, +the conditions for self-sustaining feedback in the simple +reactor are significantly more achievable than for self- +sustaining relativistic feedback. +For the simple reactor (as for any other RREA model +with positive feedback [21, 26]) the following time de- +pendence of the gamma radiation flux measured on the +ground is possible. Usually during a TGE measurement, +the gamma flux slowly increases exponentially [7, 8]. This +can be explained by the fact that when the cloud ap- +proaches the detector at a constant speed, so the distance +from the cloud to the TGE source decreases linearly in +time. The measured particle flux decays exponentially +with distance, thus, if the distance is decreased linearly, +the measured flux grows exponentially [7]. +If RREAs +are self-sustaining within the thunderstorm due to the +positive feedback, then their bremsstrahlung gamma-ray +flux grows exponentially within the thunderstorm itself +(it can grow slowly if the multiplication rate is slightly +higher than unity). +Moreover, even if the feedback is +present but the RREA is not self-sustaining due to the +low feedback coefficient, the RREA time profile is mod- +ified and its radiation time increases [26]. +Thus, with +the positive feedback, the time profile of the measured +gamma-ray flux is exponent superimposed on exponent. +The time profile can be more complicated if the electric +field within thunderstorm is changing. Such time pro- +file was measured during winter thunderstorms gamma- +ray glows [7], which supports the hypothesis about the +importance of the positive feedback in thunderstorm +physics. + +10 +Lightning initiation by RREA is a widely discussed +problem in the atmospheric electricity science commu- +nity [8, 11, 20, 23, 26, 38, 52, 53]. +Within the simple +reactor, RREAs are directed to the center of the system, +thus creating the maximal density of RREA electrons +and their products in the center. This also leads to max- +imum ionization in the middle part of the simple reactor +[54, 55]. The described case can be more favorable for +streamer initiation when compared to a single strong field +region with RREAs directed to the top or to the bottom +base of a cloud because the ionization has its maximum +at the end of a RREA, on the edge of the strong field +region. Moreover, the simple reactor naturally contains +more high-energy particles than the uniform electric-field +region because of the reactor feedback. Thus, the simple +reactor model can be a useful mechanism for lightning +initiation research. It should be noted, that if stream- +ers are generated with the reactor feedback, it can lead +to an exponential growth of radio signal preceding the +lightning leader. +VII. +CONCLUSION +This paper studies RREA physics in thunderstorms +containing two supercritical electric field regions accel- +erating runaway electrons toward each other. +Such a +system, named the simple reactor, can be a part of a +natural thunderstorm. +It is discovered that RREA in +the simple reactor has positive reactor feedback. The re- +actor feedback enhances RREA duration and can lead +to self-sustaining RREA development. +There are two +mechanisms of the reactor feedback in the simple reac- +tor. RREA is effectively multiplied by the gamma-ray ex- +change between regions even if they are far enough apart. +If regions are close to each other, high-energy particles +are generated by the runaway electron oscillations near +the border between regions. In this case, the small-scale +strong electric field is sufficient for self-sustaining RREA +development. It is shown that the reactor feedback in the +simple reactor requires significantly lower electric field +strength for RREA multiplication compared to relativis- +tic feedback. +The simple reactor in the self-sustaining regime rapidly +increases the number of high-energy particles within a +thunderstorm and can hypothetically precede or cause +lightning initiation. It is established that the time scale +and the spectrum of the simple reactor gamma radiation +agree with TGF data. +The distinguishing property of +the simple reactor is that it radiates gamma rays in two +opposite directions. 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Chilingarian, G. Hovsepyan, S. Soghomonyan, M. Za- +zyan, and M. Zelenyy, Structures of the intracloud elec- +tric field supporting origin of long-lasting thunderstorm +ground enhancements, Physical Review D 98 (2018). +[52] A. Kostinskiy, T. Marshall, and M. Stolzenburg, The +mechanism of the origin and development of light- +ning from initiating event to initial breakdown pulses +(v.2), Journal of Geophysical Research Atmospheres 125 + +13 +(2020). +[53] A. Kostinskiy, T. Marshall, and M. Stolzenburg, The +mechanism of the origin and development of lightning +from initiating event to initial breakdown pulses, (2019). +[54] T. Khamitov, A. Nozik, E. Stadnichuk, E. Svechnikova, +and M. Zelenyi, Estimation of number of runaway elec- +trons per avalanche in earth's atmosphere, EPL (Euro- +physics Letters) 132, 35001 (2020). +[55] J. Dwyer and L. Babich, Low-energy electron production +by relativistic runaway electron avalanches in air, Journal +of Geophysical Research 116 (2011). + diff --git a/99AyT4oBgHgl3EQfqfgS/content/tmp_files/load_file.txt b/99AyT4oBgHgl3EQfqfgS/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..fd7b447d4174aeb35408005f22f16be41fda622b --- /dev/null +++ b/99AyT4oBgHgl3EQfqfgS/content/tmp_files/load_file.txt @@ -0,0 +1,1123 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf,len=1122 +page_content='Simple reactor model of relativistic runaway electron avalanche development Egor Stadnichuk∗ Moscow Institute of Physics and Technology - 1 “A” Kerchenskaya st.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=', Moscow, 117303, Russian Federation HSE University - 20 Myasnitskaya ulitsa, Moscow 101000 Russia Daria Zemlianskaya,† Ekaterina Svechnikova,‡ Eduard Kim,§ Alexander Sedelnikov,¶ and Oraz Anuaruly∗∗ (Dated: January 3, 2023) High-energy gamma radiation in the Earth’s atmosphere is associated with the bremsstrahlung of Relativistic Runaway Electron Avalanches (RREA) developing in thunderstorm electric fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' In this paper, RREA development is studied in the system of two strong electric-field regions within thunderstorms, which accelerate runaway electrons toward each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Such a system is called the simple reactor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' It is discovered that the propagation of gamma rays and runaway electrons from one region to another leads to positive feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' This feedback called the reactor feedback can make RREA self-sustaining, thus effectively multiplying high-energy particles inside thunderstorms con- taining the simple reactor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' The spectrum and characteristic time scale of the simple reactor gamma radiation are in agreement with Terrestrial Gamma-ray Flashes (TGFs) data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' The applicability of the simple reactor model to TGF is discussed, and the distinguishing observable properties of the simple reactor radiation during TGF and Thunderstorm Ground Enhancement are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' KEYPOINTS RREA development in thunderstorms containing the simple reactor is studied Two feedback mechanisms that can make RREA self-sustaining in the simple reactor are discovered: electron and gamma-ray reactor feedback Characteristics of gamma radiation of the simple reactor are in agreement with TGF and TGE ex- perimental data ∗ 1Moscow Institute of Physics and Technology - 1 “A” Kerchen- skaya st.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=', Moscow, 117303, Russian Federation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' 2HSE Uni- versity 20 Myasnitskaya ulitsa, Moscow 101000 Russia;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' yegor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content='stadnichuk@phystech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content='edu † zemlianskay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content='d@phystech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content='edu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Moscow Institute of Physics and Technology - 1 “A” Kerchenskaya st.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=', Moscow, 117303, Russian Federation Institute for Nuclear Research of RAS - prospekt 60-letiya Ok- tyabrya 7a, Moscow 117312 ‡ svechnikova@ipfran.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content='ru;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Institute of Applied Physics of RAS - 46 Ul’yanov str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=', 603950, Nizhny Novgorod, Russia § Moscow Institute of Physics and Technology - 1 “A” Kerchen- skaya st.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=', Moscow, 117303, Russian Federation Institute for Nuclear Research of RAS - prospekt 60-letiya Ok- tyabrya 7a, Moscow 117312;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' kim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content='e@phystech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content='edu ¶ Moscow Institute of Physics and Technology - 1 “A” Kerchen- skaya st.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=', Moscow, 117303, Russian Federation Lebedev Physical Institute RAS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' sedelnikov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content='as@phystech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content='edu ∗∗ Moscow Institute of Physics and Technology - 1 “A” Kerchen- skaya st.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=', Moscow, 117303, Russian Federation Kurchatov Institute, Russian Research Centre - sq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Academi- cian Kurchatov, 1, Moscow, 123098, Russian Federation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' orazan- uaruly@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content='com II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' INTRODUCTION Atmospheric physics is rich in mysterious natural phe- nomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' One of the new directions in atmospheric research is high-energy atmospheric physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' It sud- denly appeared in 1992, when the Burst and Tran- sient Source Experiment (BATSE) detector aboard the Compton Gamma-Ray Observatory experiment discov- ered short and intensive bursts of gamma rays originating in the atmosphere of Earth [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' These energetic bursts are called Terrestrial Gamma-ray Flashes (TGFs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' It is established that the source of TGFs are thunderstorms [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' The characteristic duration of a TGF is 100 µs [3], energy of detected TGF gamma-rays is up to 40 MeV [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Thunderstorm gamma radiation is also detected on the Earth’s surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' It is called Thunderstorm Ground Enhancement (TGE) [6] or gamma-ray glows [7], and its characteristic duration is up to tens of minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' It is important to note that high-energy processes within thunderstorms are closely related to lightning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' TGE pre- cede lightning and are always terminated by lightning discharges [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' TGFs are established to occur at the early stage of the lightning leader propagation [9–11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' More- over, many other interesting bright phenomena were reg- istered in connection with the high-energy radiation from thunderstorms [12–14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' The underlying physics of high-energy atmospheric ra- diation is the acceleration of electrons in thunderstorm electric fields [2, 15–19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' In strong thunderstorm electric fields, relativistic electrons obtain more energy from the acceleration in the electric field than they on average lose on interactions with atmosphere air molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Such elec- trons are called runaway electrons [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' When the electric field strength exceeds the critical value, Ec = 276 kV m·atm, runaway electrons are Møller scattered by air molecules, which leads to the appearance of additional runaway elec- trons [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' In this way, runaway electrons multiply in the process of their propagation along the thunderstorm arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content='00542v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content='ao-ph] 2 Jan 2023 2 electric field, forming the Relativistic Runaway Electron Avalanche (RREA) [16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' To start a RREA an initial seed energetic particle is needed to appear within the thunderstorm electric field [16, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' For example, it can be a secondary cosmic ray particle [23] or a seed parti- cle generated inside the thunderstorm [24–26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Charac- teristic RREA particle energies range from tens of keV to tens of MeV [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Thus, runaway electrons natu- rally produce bremsstrahlung gamma rays in collisions with air molecules, which is detected as TGF and TGE [4, 7, 8, 10, 17, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' The mystery of TGFs is that a large number of high- energy particles appear almost instantly inside a thunder- cloud [1, 5, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' There are two possible ways to explain such phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' The first possible scenario is the gener- ation of a large number of seed electrons within the thun- derstorm super strong electric fields, possibly created by the lightning leader propagation [18, 24, 25, 27–29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Also it is considered that lightning leader itself can radiate synchrotron gamma-rays [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' These ideas are supported by the fact that x-rays are observed in association with lightning leader propagation [31, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' The second pos- sible scenario is the multiplication of RREAs by posi- tive feedback mechanisms [21, 26, 33–37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' The relativis- tic feedback works in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Bremsstrahlung gamma ray radiated by runaway electrons can produce electron-positron pairs within thunderstorm supercriti- cal electric field region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Positrons are accelerated by the electric field in the direction opposite to the runaway electrons acceleration direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' In this way, positrons reach the beginning of the supercritical region, where they produce seed runaway electrons by the Bhabha scat- tering [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Thus, relativistic positron feedback multiplies RREAs and, moreover, can make RREA self-sustaining [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Similarly bremsstrahlung gamma rays can Compton backscatter and thus produce seed runaway electrons at the beginning of the supercritical region, which is the rel- ativistic gamma ray feedback [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' RREA models based on the positive feedback are supported by the fact that their characteristic time and spectrum coincide with the characteristic time and spectrum of TGF [2–4, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Nev- ertheless, the relativistic feedback requires strong large- scale electric fields, which have never been directly ex- perimentally observed in thunderstorms [33, 34, 36, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' It has been discovered that non-uniform thunderstorm electric field geometry leads to another feedback mech- anism called the reactor feedback [26, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Let a thun- derstorm consist of several separate electric-field regions with electric-field strength sufficient for RREA produc- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Such regions, for simplicity, are called cells [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' If a seed electron starts a RREA within one of the cells, the following processes occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' A RREA radiates bremsstrahlung gamma-rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Gamma-rays have a large attenuation length at thunderstorm altitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' There- fore, gamma-ray photons propagate through the thun- derstorm and can reach other cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' There is a prob- ability that a gamma-ray photon will interact with air molecules by compton scattering, photoelectric effect of electron-positron pair production within a cell, which can result in runaway electron generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' A runaway electron can produce a RREA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' In this way the reac- tor gamma-ray feedback works: separate thunderstorm cells irradiate each other with gamma radiation, which results in RREA multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Another reactor feed- back mechanism - runaway electron transport between cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' If the cells are close to each other, runaway elec- trons are able to penetrate the air layer between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' In this way, runaway electrons propagate from one cell to another and, thus, multiply RREA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' In general, reac- tor feedback is defined as the multiplication of RREA by high-energy particle exchange between separate thunder- storm RREA-accelerating regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Distant cells amplify each other mostly by gamma-ray photon exchange be- cause of their high penetrating power in the air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' For cells located close to each other, the reactor feedback works mainly by runaway electron exchange, since RREA con- sists mainly of runaway electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' It has been established that reactor feedback can lead to self-sustaining devel- opment of RREA, and, moreover, requires lower electric field strength in comparison with the relativistic feedback [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' In this paper, the reactor feedback is studied in the simplest case of non-uniform thunderstorm electric field, when thunderstorm consists of two cells, oriented in the way that they accelerate runaway electrons towards each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' The system is called the simple reactor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' The re- search is motivated by observations of thunderstorm elec- tric structures, part of which can be described as the sim- ple reactor electric structure [41–47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' In this paper, it has been discovered that both gamma-ray reactor feedback and runaway electron transport feedback amplify RREA in the simple reactor, and these feedback mechanisms require a smaller electric field strength to provide self- sustaining RREA development in comparison with the relativistic feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' In Section III the feedback mecha- nisms of the simple reactor are described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' In Section IV the reactor feedback is theoretically described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Section V provides Monte Carlo simulations of the simple reactor using GEANT4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' In Section VI the simple reactor is dis- cussed as a possible mechanism for TGF and TGE, the distinguishing properties of this model that are experi- mentally observable are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' SIMPLE REACTOR MODEL The reactor feedback is the intensification of RREA development in a thunderstorm supercritical electric field region (cell) by the radiation of other cells [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' The sim- plest system capable of demonstrating the reactor feed- back is the system of two cells with oppositely directed electric fields accelerating electrons toward each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' This system is called “the simple reactor” and we con- sider it as the next step from uniform electric field mod- els to the description of RREA in the electric field of a real thundercloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' In the simple reactor, the distribu- 3 tion of the electric field corresponds to the system of two flat capacitors placed one on top of the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' It can be considered as the approximation of the electric field distribution in the region of the cloud with three charge layers: a positive middle layer and two negative layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' The described electric field distribution can be a part of a natural thunderstorm [41–47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' There are two reactor feedback mechanisms in the sim- ple reactor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' The first mechanism, reactor gamma-ray feedback, works in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Let a seed elec- tron form a RREA within one of the cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' RREA grows towards the opposite cell and radiates bremsstrahlung gamma rays [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Gamma rays have a significant pene- tration power through the atmosphere at thunderstorm altitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Thus, gamma rays reach the opposite cell and propagate through it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Interactions of gamma rays with air molecules of the opposite cell generate seed runaway electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' These electrons start a new RREA in the op- posite cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Further, the new RREA propagates towards the initial cell, generating gamma rays, which similarly produce RREA in the initial cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' In this way, the pro- cess loops, and thus gamma-ray reactor feedback makes RREA self-sustaining in the simple reactor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' The second mechanism, runaway electron transport feedback, works in the case when cells are close to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' In this case, runaway electrons can penetrate the gap between cells, reaching the opposite cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' When a runaway elec- tron reaches the opposite cell, it penetrates inside the cell until the electric field reverses it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' After reversal runaway electrons are accelerated toward the initial cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Thus, in the simple reactor, runaway electrons oscillate near the border of the cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' During the oscillation, runaway electrons are multiplied by the Møller scattering, which also leads to a self-sustaining process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' It is established in this paper, that both feedback mechanisms not only can make RREA self-sustaining but also can significantly multiply the number of relativistic particles in the thun- derstorm containing the simple reactor (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' It should be noted that the relativistic feedback naturally impacts RREA development in the simple reactor, how- ever, further it is shown that the influence of the rela- tivistic feedback is negligible compared to the influence of the reactor feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' ANALYTICAL SIMPLE REACTOR MODEL A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Gamma-ray reactor feedback To describe the simple reactor gamma-ray feedback theoretically, it is necessary to study the response of a cell to gamma radiation falling into it [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Let N gamma-ray photons enter the i-th cell (i =1,2) from above, along the cell’s electric field vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' In order to find the feedback coefficient, it should be calculated how much gamma will fly back from the cell towards the other cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Let λi RREA be the growth length of an avalanche of runaway elec- trons, λi − be the decay length of gamma radiation, λi γ be FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' The physics of the simple reactor in the GEANT4 simulation [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Green lines - gamma-ray photon tracks, red lines - runaway electron tracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Blue arrows - electric field lines, yellow dots - particle interaction points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' The simula- tion is started with a single seed electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' The simple reactor consists of two supercritical electric field regions accelerat- ing runaway electrons towards each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' High-energy par- ticles exchange between these regions makes the process self- sustaining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' The picture resembles the Eye of Sauron from the Lord of the Rings trilogy: runaway electrons oscillating near the cell boundary form a pupil, the halo of gamma-ray photons and RREAs formed by gamma-ray reactor feedback resembles the cornea of the eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' the path of a runaway electron before the emission of a gamma-ray photon with supercritical energy, λi e− be the path length of a gamma before the birth of a runaway electron, and P i is the probability of a turn of an elec- tron with further development of the runaway avalanche.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' These parameters depend on the magnitude of the elec- tric field and the density of the air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' In the first approxi- mation, the values for these parameters can be retrieved from the article [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' In general, the cells of the simple reactor are assumed to have different field strengths, air density, and cell lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' The gamma entering the cell along the electric field will generate electrons with supercritical energy, while losing its energy, which leads to an exponential decay of the primary gamma flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' On the segment [z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' z + dz],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' the flown gammas will give birth to the following number of avalanches of runaway electrons: df i e−(z) dz dz = NP ie − z λi − dz λi e− (1) The dynamics of the number of bremsstrahlung gamma-ray photons during RREA propagation along the z axis is described by the following equation [34]: dNγ = e z−z0 λRREA dz λγ − Nγ dz λ− (2) The first term in 2 describes the production of bremsstrahlung gamma-ray photons by runaway elec- trons,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' and the second term describes a decrease in the 4 number of gamma-rays due to its interaction with air molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' The solution for 2 is 3: Nγ(z, z0) = λRREAλ− λγ(λ− + λRREA) · � e z−z0 λRREA − 1 � (3) Each RREA grows according to the well-known ex- ponential law [20], spreading toward the initial gamma rays entering the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' In this case,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' depending on the point of birth of the avalanche,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' the amount of secondary gamma rays that will reach the end of the cell will be as follows (since the avalanche born at the point z will travel a distance equal to z): dF i γ(z) = df i e−(z) dz dz λRREAλ− λγ(λ−+λRREA) · � e z λRREA − 1 � (4) Thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' the gamma-ray local multiplication factor [26] can be calculated (Formula 5): νi = � Li 0 dF i γ(z) dz dz N (5) Integration leads to the following formula for the gamma-ray multiplication factor: νi = P iλi RREAλi − λi e−λiγ(λi RREA + λi −) � λi RREAλi − λi − − λi RREA � e Li λi −−λi RREA λi RREAλi − − 1 � − λi − � 1 − e − Li λi − �� (6) The system with positive feedback can be character- ized by the feedback coefficient [26,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' 33,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' For the simple reactor, the feedback coefficient shows how many times the number of high-energy particles will increase in one full reactor feedback cycle, and it is found with the following formula: Γ = ν1 · ν2 (7) The number of particles in the simple reactor grows ex- ponentially with each feedback generation: N(n) = Γn, where n is the number of feedback generation [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' There- fore, the criterion for self-sustaining RREA development in a simple reactor is as follows: Γ = ν1 · ν2 ≥ 1 (8) With the obtained criterion, the thunderstorm condi- tions necessary for self-sustaining RREA development by the reactor gamma-ray feedback can be calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' These conditions are presented in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Conditions are pre- sented for 3 types of feedback: relativistic positron feed- back [21], simple reactor feedback, and multicell reactor FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' The comparison of self-sustaining positron feedback necessary conditions [34] and simple reactor self-sustaining feedback necessary conditions (Formula 6, 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' RREA accel- erating region length is normalized to λRREA, electric field strength is normalized to critical electric field strength (the electric field required for RREA development [16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' The sim- ple reactor is also compared with necessary conditions for self-sustaining gamma-ray feedback in multicell reactor model [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' It can be seen that reactor models require significantly lower thunderstorm electric field strengths for self-sustaining RREA development than the relativistic feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' This im- portant property of the model comes at expense of the com- plexity of the electric field geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' The more complex elec- tric field geometry is, the lower electric field strength is re- quired for the feedback to be effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' It should be noted that the conditions for the reactor models are presented without taking into account runaway electron transport between cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Moreover, it has been discovered that thundercloud hydrome- teors can amplify RREA [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Thus, the exact self-sustaining RREA conditions can be lower than the conditions presented on this picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' feedback [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' The coordinates in the figure are chosen so that the conditions are invariant with respect to altitude [33, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' It can be seen from the figure that both the mul- ticell and the simple reactor feedback mechanisms require significantly lower electric field strength in comparison to the relativistic feedback discharge model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' This impor- tant property of the reactor feedback comes with a price of the thunderstorm electric field geometry complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' The most complex electric field geometry, the multicell reactor, requires the lowest electric field strength for the self-sustaining RREA development, while the simplest re- actor structure, the simple reactor, requires the electric field strength lying in between the multicell reactor and the uniform electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' It should be noted that if thun- dercloud electric field parameters lie above the curve in Figure 2 then the number of energetic particles within the thundercloud grows exponentially [26, 33, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Oth- erwise, provided that there is no external source of seed particles, the number of energetic particles decays, and the decay rate depends on the feedback coefficient [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Thus, even if the feedback does not make the RREA de- velopment self-sustaining, it still increases its duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Positron feedback Multicell reactor 103 Simple reactor 102 入RREA 101 100 10-1 2 ×100 3 ×100 4×100 100 E5 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Runaway electron transport between cells in the simple reactor GEANT4 simulations of the simple reactor showed the importance of runaway electron transport between cells for the reactor feedback (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' In this section, it is shown that the oscillations of runaway electrons between cells in the simple reactor can become self-sustaining and even lead to runaway electron multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' At first glance, this effect may seem paradoxical and contrary to the law of energy conservation: While a single runaway electron move from one cell to another, the total energy it receive from the electric field in the full circle of its oscillation is zero, and, thus, this runaway electron, on average, loses energy in interaction with air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Therefore, a single runaway electron will inevitably lose its energy and stop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' However, it is seen in the simulations that runaway electrons oscillate and multiply in the strong electric field of the simple reactor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Therefore, the following question arises: Where do runaway electrons take energy when the feedback becomes self-sustaining?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Moreover, the total length of runaway electron motion between cells back and forth cannot be longer than its energy divided by eEc, where Ec - critical electric field, e — elementary electric charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' This is not more than several tens of meters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' It turns out that the effect of runaway electron trans- port between cells can be physically explained and that the energy conservation paradox is resolved by runaway electron multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' If a runaway electron multiplies by Møller scattering [21], the result is that the initial and generated runaway electrons receive twice as much energy from the electric field compared to the single initial run- away electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' When the initial runaway electron stops, the secondary electron continues to oscillate and multi- ply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' The reactor feedback in the simple reactor caused by runaway electrons can even become self-sustaining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' If the thunderstorm electric field is much stronger than the crit- ical electric field, the runaway electron interaction with the air becomes negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Moreover, by the interaction with air, runaway electrons will multiply, which leads to an enormous growth of the number of relativistic parti- cles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Thus, when the electric field strength decreases to values comparable to Ec, there is a point where the multi- plication of runaway electrons compensates for the energy losses in the air interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' At this point, the runaway electron transport feedback becomes self-sustaining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' An interesting property of the runaway electron trans- port feedback is its spatial scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' A runaway electron af- ter hitting an adjacent cell cannot propagate within the cell deeper than its kinetic energy divided by e(E + Ec), where E is the electric field strength of the cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' There- fore, runaway electron transport feedback coefficient de- pends only on the electric field strength for cell lengths longer than runaway electron maximum energy divided by e(E + Ec), which is about 100 meters for 10 km alti- tude and 40 MeV maximum energy [15, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Thus, run- away electron transport occurs near the cell interface, which softens the conditions required for self-sustaining RREA development in the simple reactor, because long cells are not needed as in other types of feedback 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Nev- ertheless, it should be noted that runaway electron feed- back works effectively only for cells located close to each other since electrons are quickly absorbed by air unless they are accelerated by the electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' To theoretically analyze the runaway electron trans- port feedback, it should first be understood how run- away electrons are decelerated in the electric field of the adjustment cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Decelerated runaway electron attenua- tion length can be found by substituting negative electric field strength into the empirical formula for the RREA e-folding length: λdecay = 7300[kV ] −E − Ec (9) In this way, number of runaway electrons in the beam will decrease exponentially: Nbeam(z) = N0e z λdecay (10) This analytic continuation of the RREA growth law [21] can be justified in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Normalized runaway electron spectrum can be described with the function: dfRREA dε = 1 ε0 e− ε ε0 (11) ε0 = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content='3 MeV - runaway electron mean energy [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' An electron with energy ε, on average, stops at the coor- dinate: z(ε) = ε e(E + Ec) (12) Number of runaway electrons leaving the beam in the interval (z, z + dz) per one primary electron is: Nbeam dz dz = −N0 dfRREA dε dε dz dz = N0 E + Ec 7300[kV ]e− E+Ec 7300[kV ] z (13) Thus, the formula 9 is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' The runaway electron transport feedback coefficient can be defined as the number of runway electrons leav- ing the cell per one runaway electron entering the cell (analogically to the gamma-ray reactor feedback).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' The number of runaway electrons, which entered the cell, de- creases according to the exponential law derived above as these runaway electrons propagate into the cell (for- mula 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' When a runaway electron leaves this beam it can stop or it can reverse and form a RREA, which then propagates to the entry plane of the cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' If a RREA starts at the point z, the number of runaway electrons within this RREA reaches e z λRREA when RREA leaves 6 the cell [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Therefore, if the reversal probability of run- away electrons from the primary beam is equal to P, the runaway electron transport feedback coefficient can be obtained as follows: �νe− = � L 0 dzPe z λRREA dNbeam dz = P �λ � L 0 dze 1 λRREA − 1 �λ (14) Here �λ = −λdecay > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' the following formula is obtained: �νe− = PλRREA λRREA − �λ � 1 − exp � L � 1 λRREA − 1 �λ ��� (15) This formula can be simplified using the empirical for- mula for λRREA [21]: λRREA λRREA − �λ = E + Ec 2Ec (16) Therefore: �νe− = P E + Ec 2Ec � 1 − exp � −L 2Ec 7300[kV ] �� (17) This formula can be further simplified for cells with cell length L ≫ 7300[kV ] 2Ec ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' which works for cells larger than 100 m: �νe− = P E + Ec 2Ec (18) Generally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' there is some space between cells within a thunderstorm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Runaway electrons lose energy by inter- acting with air molecules while propagating through the gap between cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' A fraction of runaway electrons be- come undercritical and leave the beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' This fraction can be estimated with the decay length from formula 9 for E = 0 as exp � −l Ec 7300[kV ] � , where l is the gap between cells in the simple reactor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Since, in the first approxima- tion, all runaway electrons lose the same amount of en- ergy in the gap, the shape of their spectrum remains the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Thus, the formula for runaway electron transport feedback coefficient, taking into account the gap between cells, simply modifies in the following way: �νe− = P E + Ec 2Ec � 1 − exp � − 2EcL 7300[kV ] �� exp � − Ecl 7300[kV ] � (19) V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' GEANT4 SIMULATION The Monte Carlo simulation of the simple reactor was carried out using Geant4, version 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content='06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content='p01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Geant4 is recognized as a good tool to model RREA [37, 51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' The physics list G4EmStandardPhysics option4 was chosen as the reliable physics list for RREA simulations [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' This list includes all interactions of electrons, gamma- rays and positrons for energies characteristic for RREA processes [26, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' The energy cut for the particles was chosen 50 keV based on the fact that low-energy particles will quickly decay, as they do not run away [16], and will not contribute to the feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' The simulated geometry is a large world volume filled with air, within which a child volume is specified, also filled with air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' A simple reactor by definition consists of two child volumes: both volumes are filled with air with a density of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content='414 kg/m3, corresponding to altitude 10 km, and contain electric field in the way that both volumes accelerate runaway elec- trons towards each other (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' The purpose of the GEANT4 simulation is to find the parameters of the system necessary for the self-sustaining RREA regime (when the generation of high-energy par- ticles within the thunderstorm does not stop until the electric field is discharged).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' The simulation was carried out by varying the cell size and the strength of the elec- tric field inside of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' At a certain electric field strength, the number of gamma-ray photons and runaway electrons crossing in both directions the boundary between the simple reactor cells will not decrease over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Thus, RREA within the simple reactor will not die out over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' In this case, self-sustaining feedback is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Thus, by increasing the electric field with a constant cell length, one can find the critical point at which the reactor will become self-sustaining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' In this way, the achievement of critical values is checked, and the conditions are cal- culated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' The most important stage in modeling is the division of the high-energy particles into generations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' If directly two cells are created with an oppositely directed field, then it will be quite difficult to divide the process into feedback generations, since, under certain conditions, a self-sustaining feedback is formed and the simulation will not stop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Thus, analogically to the theoretical model, it was decided to simultaneously simulate only a half of a simple reactor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' This approach is possible due to the symmetry of the simple reactor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' The modeling scheme is shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' The model consists of a single cell filled with air and electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' At the beginning of the cell an air detector is placed — the volume within which particles are stopped and registered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' In the first simulation step, seed particles with an energy of 5 MeV are launched from the beginning of the cell along the di- rection of the electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Seed runaway electrons are decelerated by the electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Some of them penetrate into the cell, reverse, and form RREA towards the de- tector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Seed gamma-ray photons propagate through the cell and interact with air molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' This interaction re- 7 detector e- FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' The design of a simple reactor has been simplified in the GEANT4 simulation to consider only one cell as in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Runaway electrons and gamma-ray photons are launched from the right side of the cell along the direction of the electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' The interactions of launched particles lead to RREA formation, which is accelerated by the electric field to the right side of the cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' In the result, generated par- ticles reach the detector and registered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' In the next stages of the simulation, registered particles are launched and new generated particles are similarly registered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' In this way, each reactor feedback generation is studied separately, thus, allow- ing the analysis of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' sults in runaway electrons generation, which reverse and form RREA, also moving and growing toward the begin- ning of the cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' All particles that reach the detector are stopped and registered and the simulation stops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' In this way, the first feedback generation is modeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' In subse- quent simulations, the particles registered in the previous iteration are launched into the cell accordingly (thus im- itating propagation of the high-energy particles from one cell into another in the simple reactor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' These particles interact with the cell, which results in new particles gen- erated that reach the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' For each feedback gen- eration this process repeats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Figure 4 shows the number of gamma-rays reaching the detector in each simulated feedback generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' The graph shows that depending on the thunderstorm conditions the number of high-energy particles can decay from generation to generation or vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' The thunderstorm conditions when the number of particles does not change are the necessary conditions for the self-sustaining development of RREA in the simple reactor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' To calculate the feedback coefficient, the simulation was launched with seed runaway electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Number of generated gamma-ray photons and runaway electrons for each feedback generation was registered, thus forming plots similar to Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Each plot was fitted with an exponential function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' The feedback coefficient 6, 19 is obtained from the coefficient in the exponent by adding 0 2 4 6 8 10 Generation number 4 5 6 7 8 9 10 log(N) Dependence of the number of gamma in a generation on its number 100kV/m 150kV/m 170kV/m 200kV/m 220kV/m 240kV/m 250kV/m 260kV/m 300kV/m FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' The dependence of the logarithm of the number of gamma-ray photons that propagates from one cell to another in the simple reactor depending on the number of feedback generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' The number of generation is the number of an iteration of a simple reactor simulation (Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' It can be seen that the number of gamma-rays produced by the simple reactor exponentially grows or exponentially decays depend- ing on the electric field strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' 1 to it [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' The obtained dependence of the exponent parameter on the electric field strength is shown in Fig- ure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' When the exponent parameter is positive, number of high energy particles in the simple reactor thunder- storm self-sustainably grows until the electric field is dis- charged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' It is also interesting to calculate the spectrum of the gamma-rays produced within the simple reactor and compare it with the spectrum of an ordinary RREA bremsstrahlung.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' To obtain the spectrum, a full simple reactor with two cells oriented towards each other was simulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' This simulation captures the particles with their energies in a tracking action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' The critical parame- ters of the simple reactor were chosen — the field is 300 kV/m and the length of one cell is 400 m for 10 km alti- tude air density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Similarly to the previous simulation technique, the G4EmStandardPhysics option4 physics list was used, and the energy cut for particles is 50 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' The simulation was stopped when the number of high- energy particles reached 106, and the spectrum of regis- tered gamma rays is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' In addition, a simulation for a single cell with the same parameters was carried out to obtain the spectrum of an ordinary RREA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' The result- ing spectra are shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' The graph shows that the spectra are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' It should be noted that sin- gle cell gamma-ray spectrum contains more pronounced 8 100 125 150 175 200 225 250 275 300 Field, kV/m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content='15 Exponent parameter FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' The dependence of the feedback generations expo- nent parameter on the electric field in the simple reactor for the cell length 400 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Negative exponent parameters means the decay of RREA in the simple reactor, while positive ex- ponent parameter means self-sustaining RREA development with high energy particles generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' The exponent parame- ter includes both simple reactor feedback processes: gamma- ray reactor feedback and runaway electron oscillations (Fig- ure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' The conditions necessary for self-sustainable regime (when exponential parameter equals 0) are in agreement with theoretical predictions (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' positron peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' However, when gamma rays propagate from thunderstorm to the detector registering TGF or TGE, they interact with the atmospheric layer and nat- urally produce the positron peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Thus, this peak will also be present when the TGF or TGE produced by the simple reactor is measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Nowadays it has been reli- ably established that the TGF and TGE source spectrum is the RREA spectrum [50, 51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Thus, the simple reactor can be the mechanism for the TGF or TGE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' DISCUSSION The discovered mechanism called the simple reactor can be applied for a thundercloud containing two regions with electric field exceeding the critical value, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' al- lowing the RREA development (for simplicity, such re- gions are called cells [26]), electric field is oriented in the way that cells accelerate runaway electrons towards each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' It was established that there is a positive feed- back in this system caused by two mechanisms (besides the relativistic feedback [21], which impact is relatively low (Figure 2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' The first mechanism is the transport of runaway electrons from one strong field region to an- other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' This leads to the effective high-energy electron multiplication and runaway electron oscillation near the edge between the strong electric-field regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' The elec- tron transport feedback coefficient is very high for a small gap between cells, and is a dominant RREA multiplica- tion mechanism in the case of the small gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' On the other FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Comparison of the spectra obtained from the sim- ulation of the simple reactor and ordinary RREA spectrum, obtained from a single cell simulation with a uniform electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' The simulation of the simple reactor was turned off when enough statistics were collected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' It can be seen that the spectrum of the simple reactor gamma-radiation is the same as the RREA bremsstrahlung spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' It is established that the thunderstorm gamma-radiation spectrum agrees with the RREA spectrum [4, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Thus, the simple reactor can be one of the mechanisms of TGF and TGE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' hand, in the case of a significant gap, when the distance between strong field regions exceeds the characteristic length of runaway electrons, too few runaway electrons propagate through the gap between regions, thus an- other feedback mechanism dominates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' The second feed- back mechanism is the gamma-ray reactor feedback [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' RREA bremsstrahlung gamma-rays have high penetra- tion rate in the air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Thus, in the simple reactor, gamma- rays effectively propagate from one cell to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' When a gamma-ray photon propagates through the opposite cell, it interacts with air, producing secondary RREAs, which is the gamma-ray reactor feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Both feed- back mechanisms can lead to self-sustaining RREA de- velopment and, moreover, to rapid multiplication of high- energy particles within a thunderstorm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' The formulas derived in this paper allow one to pre- dict the feedback coefficient for both feedback mecha- nisms without complicated modeling;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' the theoretical pre- dictions of this paper are verified by GEANT4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' The dis- covered feedback coefficients completely describe the be- havior of the simple reactor, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' allow to calculate the conditions required for the self-sustaining RREA devel- opment (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' The limitations of the proposed an- alytical model are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Firstly, the model is one- dimensional and, therefore, does not consider the trans- verse dynamics of the avalanche, which affects the feed- back coefficients in the case of narrow electric field regions [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Second, though the description of runaway electron transport feedback qualitatively matches Geant4 simu- lations, it lacks quantitative accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' More theoretical 10-1 simple reactor onebox 10-2 10-3 102 Energy,kev9 and modeling research is needed to establish the exact in- fluence of the electron transport feedback on the RREA development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' The simple reactor geometry corresponds to the charge distribution with two negative charge layers on both sides of the positive layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' This structure can be a part of a more complicated charge structure of a thunderstorm [41–47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' In the simple reactor, the maximum density of runaway electrons will be on the border between two op- positely directed cells — in the center of the simple re- actor, in the region of the positive charge (it should be noted that the large value of a single positive charge in a region of a cloud can be sufficient for RREA development below and above this region, forming the simple reactor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' This feature distinguishes the simple reactor model from models assuming the development of RREA in a single cell with maximum particle density in the cloud top or cloud base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Since in the simple reactor the maximum run- away electron density is located in the center of the reac- tor, it is harder for bremsstrahlung gamma-rays to reach detectors registering TGF or TGE due to the greater thickness of the atmosphere that they must penetrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' However, this does not contradict the observed gamma- ray fluxes, since the reactor feedback increases the num- ber of generated bremsstrahlung gamma-rays within a thunderstorm containing the simple reactor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' This in- crease compensates for the decrease in gamma-ray flux by extra atmosphere in has to penetrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Another distinguishing and important property of the simple reactor is that it generates simultaneous gamma- ray radiation directed upward and downward from a thundercloud (or in other opposite directions if the simple reactor is not oriented vertically).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' This means that theo- retically it is possible to simultaneously detect a TGF or a TGE from two opposite sides of a thunderstorm, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=', from the top and from the bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Such observation can be performed, for example, with an airplane containing particle detectors flying over an observatory with particle detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Also a TGF generated by the simple reactor can be registered simultaneously from space and ground observatories, but the probability for the space station to be located above the ground observatory at the moment of TGF is very low due to the TGF short duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' It should be noted that the time profile of the gamma-ray flux in measurements from both sides of the thunder- storm must match in order to conclude that upward and downward gamma-ray radiation are connected by the re- actor feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' This requires a good temporal resolution of the detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' The simple reactor with a large feedback coefficient can be a source of TGF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Characteristic timescale of the sim- ple reactor is its size divided by the speed of light, which is in order of microsecond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Therefore, the timescale and radiated gamma-ray spectrum satisfy the experimentally observed TGF data [1, 3, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Runaway electron accelera- tion and its bremsstrahlung gamma-ray radiation in the simple reactor precede the lightning leader and should co- incide with the early stage of the lightning initiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' It should be noted that a TGF generated by positive feed- back has a characteristic exponential gamma-ray flux rise time profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Number of high-energy particles grows ex- ponentially on TGF timescales as thunderstorm electric field remains almost constant on these timescales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' At the TGF peak, thunderstorm electric field lowers, thus feedback coefficient drops and the feedback becomes fi- nite: the flux of high-energy particles starts to decay or even abruptly terminates, if the electric field required for RREA development abruptly disappear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' The disappear- ance of the electric field can be connected either with lo- cal discharges or with the initiation of a lightning leader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' From the rise profile of measured TGF flux the feedback coefficient can be restored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' The feedback coefficient is a good source of information on the thunderstorm electric field during the TGF (Formula 6, 19) [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Another TGF model based on RREA, the relativis- tic feedback discharge model, supposes significant posi- tive feedback (the relativistic feedback) in the most sim- ple thunderstorm geometry - uniform electric field [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' The disadvantage of this model is that it requires very high values of electric field strength extended over a large thunderstorm space [16, 34, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' The significant feature of the simple reactor is that it requires smaller electric field strength for the self-sustaining RREA de- velopment than it is in the uniform electric field (Fig- ure 2) [26, 34, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Moreover, provided that two strong field regions are formed by the same positive charge layer, the conditions for self-sustaining feedback in the simple reactor are significantly more achievable than for self- sustaining relativistic feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' For the simple reactor (as for any other RREA model with positive feedback [21, 26]) the following time de- pendence of the gamma radiation flux measured on the ground is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Usually during a TGE measurement, the gamma flux slowly increases exponentially [7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' This can be explained by the fact that when the cloud ap- proaches the detector at a constant speed, so the distance from the cloud to the TGE source decreases linearly in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' The measured particle flux decays exponentially with distance, thus, if the distance is decreased linearly, the measured flux grows exponentially [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' If RREAs are self-sustaining within the thunderstorm due to the positive feedback, then their bremsstrahlung gamma-ray flux grows exponentially within the thunderstorm itself (it can grow slowly if the multiplication rate is slightly higher than unity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Moreover, even if the feedback is present but the RREA is not self-sustaining due to the low feedback coefficient, the RREA time profile is mod- ified and its radiation time increases [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Thus, with the positive feedback, the time profile of the measured gamma-ray flux is exponent superimposed on exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' The time profile can be more complicated if the electric field within thunderstorm is changing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Such time pro- file was measured during winter thunderstorms gamma- ray glows [7], which supports the hypothesis about the importance of the positive feedback in thunderstorm physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' 10 Lightning initiation by RREA is a widely discussed problem in the atmospheric electricity science commu- nity [8, 11, 20, 23, 26, 38, 52, 53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Within the simple reactor, RREAs are directed to the center of the system, thus creating the maximal density of RREA electrons and their products in the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' This also leads to max- imum ionization in the middle part of the simple reactor [54, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' The described case can be more favorable for streamer initiation when compared to a single strong field region with RREAs directed to the top or to the bottom base of a cloud because the ionization has its maximum at the end of a RREA, on the edge of the strong field region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Moreover, the simple reactor naturally contains more high-energy particles than the uniform electric-field region because of the reactor feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Thus, the simple reactor model can be a useful mechanism for lightning initiation research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' It should be noted, that if stream- ers are generated with the reactor feedback, it can lead to an exponential growth of radio signal preceding the lightning leader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' CONCLUSION This paper studies RREA physics in thunderstorms containing two supercritical electric field regions accel- erating runaway electrons toward each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Such a system, named the simple reactor, can be a part of a natural thunderstorm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' It is discovered that RREA in the simple reactor has positive reactor feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' The re- actor feedback enhances RREA duration and can lead to self-sustaining RREA development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' There are two mechanisms of the reactor feedback in the simple reac- tor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' RREA is effectively multiplied by the gamma-ray ex- change between regions even if they are far enough apart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' If regions are close to each other, high-energy particles are generated by the runaway electron oscillations near the border between regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' In this case, the small-scale strong electric field is sufficient for self-sustaining RREA development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' It is shown that the reactor feedback in the simple reactor requires significantly lower electric field strength for RREA multiplication compared to relativis- tic feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' The simple reactor in the self-sustaining regime rapidly increases the number of high-energy particles within a thunderstorm and can hypothetically precede or cause lightning initiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' It is established that the time scale and the spectrum of the simple reactor gamma radiation agree with TGF data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' The distinguishing property of the simple reactor is that it radiates gamma rays in two opposite directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' This allows simultaneous and cor- related observation of TGF or TGE gamma rays from the top and from the bottom of a thundercloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' More- over, the feedback coefficient can be retrieved from TGF and TGE data, which can be a good source of infor- mation about gamma radiating thunderstorm parame- ters, including electric field strength, supercritical region length, and the electric field geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' ACKNOWLEDGEMENTS The work of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Stadnichuk was supported by the Foun- dation for the Advancement of Theoretical Physics and Mathematics “BASIS”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} +page_content=' Babich, Low-energy electron production by relativistic runaway electron avalanches in air, Journal of Geophysical Research 116 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99AyT4oBgHgl3EQfqfgS/content/2301.00542v1.pdf'} diff --git a/9dFQT4oBgHgl3EQf5zZD/content/tmp_files/2301.13436v1.pdf.txt b/9dFQT4oBgHgl3EQf5zZD/content/tmp_files/2301.13436v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..abd1b649c00fdfdface8a09ba0bc52070ab4aa48 --- /dev/null +++ b/9dFQT4oBgHgl3EQf5zZD/content/tmp_files/2301.13436v1.pdf.txt @@ -0,0 +1,2042 @@ +arXiv:2301.13436v1 [math-ph] 31 Jan 2023 +Closed Form Expressions for Certain Improper Integrals of +Mathematical Physics +B. Ananthanarayan * Tanay Pathak† Kartik Sharma‡ +Centre for High Energy Physics, Indian Institute of Science, +Bangalore-560012, Karnataka, India +Abstract +We present new closed-form expressions for certain improper integrals of Mathematical Physics such as Ising, Box, +and Associated integrals. The techniques we employ here include (a) the Method of Brackets and its modifications and +suitable extensions and (b) the evaluation of the resulting Mellin-Barnes representations via the recently discovered +Conic Hull method. Analytic continuations of these series solutions are then produced using the automated method +of Olsson. Thus, combining all the recent advances allows for closed-form solutions for the hitherto unknown B3(s) +and related integrals in terms of multivariable hypergeometric functions. Along the way, we also discuss certain com- +plications while using the Original Method of Brackets for these evaluations and how to rectify them. The interesting +cases of C5,k is also studied. It is not yet fully resolved for the reasons we discuss in this paper. +1 +Introduction +In studies of theoretical physics and mathematics, various integrals appear whose symbolic evaluation is sought +after. Gradshyteyn and Ryzik [1] compiled a long list of such integrals. Recently there have been attempts to provide +a derivation of a large number of these integrals, specifically the improper integral with limits from 0 to ∞ using the +Original Method of Brackets (OMOB) [2–7]. Apart from this, some of the present authors have also evaluated the +integral of quadratic and quartic types and their generalization using the OMOB, which has been reported in [8]. +In the present investigation, we turn to other interesting improper integrals that appear in Mathematical +Physics, such as the Ising integrals and the Box integrals. Our work is motivated by the need to express them in +terms of elegant closed-form expression or in terms of known functions of mathematical physics, especially the hyper- +geometric functions [9,10]. In the recent past, several tools have also been developed to facilitate tasks of symbolic +evaluation of these integrals. Our results here have been facilitated by the recent development of tools and ad- +vances in various theoretical treatments. Note for instance, the recently proposed solution to the problem of finding +the series solution of the N-dimensional Mellin-Barnes (MB) representation [11–13], using what has been termed +as the Conic Hull Mellin Barnes (CHMB) method. This has also been automated as the MATHEMATICA package +MBConichulls.wl [14, 15]. The series representation hence obtained, in general, can be written as hypergeometric +functions or their derivatives. Independently, the issue of finding the analytic continuations (ACs) of the multivari- +able hypergeometric function using the method of Olsson [16,17], which has also been automated as a MATHEMAT- +ICA package Olsson.wl [18] have been addressed recently. In this work, we show how these tools together, which +were primarily directed at solving Feynman integrals, are of sufficient generality to find their use in the evaluation +of the integrals considered here. +We will consider the Ising integrals which have been studied in the Ising model [19–22] and also have been in +the context of OMOB [3]. Apart from the evaluation with these newly developed tools, we will also consider certain +complications while doing similar evaluations with the OMOB [23]. One of them is the use of regulators for the +evaluation of the Ising integrals. This arises in the case of Ising integrals C3,1 and C4,1. For the case of C4,1, it is +further complicated due to the use of two regulators, which, when the proper limiting procedure is applied, will give +the final result. However, we point out that such a procedure is complicated and thus use the Modified Method of +Brackets (MMOB) [24] to get the MB-integral. This MB integral can then be evaluated without any introduction +of such regulators and thus provides an efficient way to deal with these integrals. Using a similar procedure, we +*anant@iisc.ac.in +†tanaypathak@iisc.ac.in +‡kartiksharma@iisc.ac.in +1 + +attempt to evaluate the elusive C5,k integral. However, we hit a roadblock for the same, as the resulting series does +not converge and would require a proper analytic continuation procedure. At present, we find this task beyond the +reach of the tools at hand, though we provide a possible way to achieve the same. Yet such results still shed some light +on the form that these integrals can be evaluated to. All the results are provided in the ancillary MATHEMATICA file +Ising.nb . +Box integrals [25–28] are another interesting integrals where such techniques can be applied to get new results. +They do carry a physical meaning in the sense that they provide the expected distance between two randomly chosen +points over the unit n-cube. We consider the two special cases of them, namely the Bn(s) and the ∆n(s). We use +the same techniques and derive the closed form results for already known B1(s) and B2(s) and new evaluation for +B3(s) and B4(s) for general values of s. The results are in terms of multi-variable hypergeometric function. These +evaluations further require the use of an analytic continuation procedure which has been done using Olsson.wl . +All the results are provided in the ancillary MATHEMATICA file Box.nb . These results for box integrals can then +be further used to evaluate the Jellium potential Jn, which can be related to box integral Bn(s) [26, 29]. Finally, +we give a general MB integral for Bn(s), which can be used to find the closed form result for all values of n and +s using MbConicHull.wl . With all this, we find new connections between the Box integrals and the multivariable +hypergeometric functions. All our calculations rely heavily on MATHEMATICA as we try to achieve the symbolic +results for all the problems. +The paper is structured as follows: In section (2) using an example given in [4], we point out the problem in the +OMOB and discuss the alternative to surpass this problem. We then, in section (3), proceed to the evaluation of Ising +integrals up to n = 4 while contrasting our method with the method used before to achieve the same in [3]. In section +(4) we attempt to solve the C5,k integral and point out a general integral C5,k(α,β) which gives C5,k as a special case. +Though we point out that it is not the final result, a proper analytic continuation procedure is required to get C5,k +from it. We then evaluate box integral Bn(s) for n = 3,4 in section (5). The new results for ∆n(s) and Jn with the above +new results are also provided. Finally, we conclude the paper with some conclusions and possible future directions +in section (6). In appendix C, we provide the table for all the MATHEMATICA files that we give and the packages +required. +2 +Method of Brackets revisited +We will first illustrate the OMOB using a simple example of integral evaluation as given in [4]. We will first evaluate +the integral by directly using the OMOB, then briefly propose a possible resolution while doing such evaluations, and +then illustrate the alternative method to do the same. +We consider the following integral +H1(a,b) = +�∞ +0 +K0(ax)K0(bx) +(1) +The integral is introduced to facilitate the evaluation of another integral, which is given by putting a = b +H(a) = +�∞ +0 +K2 +0(ax)dx +(2) +We can express K0(x) using the following series expansion: +K0(ax) = +� +n1 +φn1 +a2n1Γ(−n1) +22n1+1 +x2n1 +(3) +where φn = (−1)n +Γ(n+1). +This expansion uses a divergent series, and we can express the result in the form of an integral representation +as +K0(bx) = 1 +2 +�∞ +0 +exp +� +−t− b2x2 +4t +� dt +t +(4) +Using the OMOB, we get: +K0(bx) = +� +n2,n3 +φn2,n3 +b2n3 x2n3 +22n3+1 〈n2 − n3〉 +(5) +Substituting the bracket series in Eq.(1), we get +H1(a,b) = +� +n1,n2,n3 +φn1,n2,n3 +a2n1b2n3Γ(−n1) +22n1+2n3+2 +〈n2 − n3〉〈2n1 +2n3 +1〉 +(6) +2 + +Now, we need to solve the bracket equations, which involve 2 equations but 3 variables. Evaluating this we get +following 3 series, Ti where ni is the free variable: +T1 = 1 +4a +� +n +φnΓ(−n)Γ2 +� +n+ 1 +2 +�� b +a +�2n +T2 = 1 +4a +� +n +φnΓ(−n)Γ2 +� +n+ 1 +2 +�� b +a +�2n +T3 = 1 +4a +� +n +φnΓ(−n)Γ2 +� +n+ 1 +2 +�� b +a +�2n +(7) +Using the rules of the OMOB, all the 3 series of Eq.(7) have to be discarded as they are divergent. +A solution to such a problem, as implemented in [4], is to regularize the singularity. This amounts to modifying +the bracket 〈n2−n3〉 → 〈n2−n3+ǫ〉. With this modification, when n1 is a free variable, one gets the series that contains +Γ(−n), which is diverging and is thus discarded. While for the other cases, one gets two series with ǫ parameter (in +the form of Γ(−n + ǫ) and Γ(−n − ǫ)). In these series, when the proper limiting procedure is done, along with the +condition a = b to ease the calculation, they give the result for the integral of Eq.(2). Thus, the original integral of +Eq.(1) we started with still remains elusive, as the calculation is much more involved (the limiting procedure) within +this present framework. +An alternative to the above evaluation, free from choosing the regulator and doing the tedious limiting procedure, +is to use the MB representation derived using the MMOB [24]. Using it, we get the following MB representation for +the integral given by Eq.(1) +H1(a,b) = 1 +4 +c+i∞ +� +c−i∞ +dz +2πi a−2z−1b2zΓ(−z)2Γ +�1 +2(2z +1) +�2 +(8) +The above MB integral can be readily evaluated in MATHEMATICA to give the following result +H1(a,b) = +π +� +a2 +b2 K +� +1− a2 +b2 +� +2a +(9) +where K(x) is the complete elliptic integral of the first kind. Thus we get the value of the original integrals, Eq.(1) we +started with. +For the special case of a = b, using K(0) = π +2 we get +H1(a,a) = H(a) = π2 +4a +(10) +So we see that for the simple cases, too, using the MB representation to evaluate these integrals provides an efficient +way to evaluate these integrals. +3 +Ising integrals +In this section, we will analyze the integrals of the “Ising class". Ising models are extensively used to study the +statistical nature of ferromagnets [30–32]. The model accounts for the magnetic dipole moments of the spins. The n - +dimensional integrals are denoted by Cn,Dn,En, where Dn is found in the magnetic susceptibility integrals essential +to the Ising calculations. +Dn = 4 +n! +�∞ +0 +··· +�∞ +0 +� +i 2 +(62) +The n-th Jellium potential is defined as +Jn := 〈Vn(r)〉⃗r∈[−1/2,1/2]n +(63) +All the Jn can be written as a box integral up to an offset. The final result is +Jn = 2n−2(1−Bn(2− n)), +n > 2 +(64) +Using the result for Bn, J3 can be readily evaluated to: +J3 = π +2 +2−6tanh−1 +� 1 +� +3 +� +(65) +6 +Conclusion and Discussion +We show that using the MMOB [24] for the evaluation of improper integral with limits from 0 to ∞ combined with +tools to evaluate such MB integrals such as MbConicHull.wl results in more efficient evaluation of these integrals. +This method is particularly helpful to evaluate the integrals when using OMOB; one requires the use of ’regulators’ +and further a proper limiting procedure to evaluate the integrals. The choice of these regulators is somewhat arbi- +trary, and at times more than one regulator has to be used, which further complicates the process. With these tools +at hand, we then re-evaluate the Ising integral, which had been already evaluated in [3] but with regulators. We +further make an attempt to evaluate the sought-after integral C5,k with all these techniques. We are, though, able to +evaluate a more general integral C5,k(α,β) which, when properly analytically continued, will give the result for C5,k. +At present we are unable to do so with the techniques at hand. Though we believe that the result can be written as +a derivative of some multivariable hypergeometric function. Continuing further we evaluate the B3(s) and B4(s) and +give a general MB representation for Bn(s). For the case of B3(s), we use Olsson.wl to find the ACs of the hypergeo- +metric functions that appear in the solution. For B4(s), similar techniques would work. It is important to note that +though the OMOB and the evaluation of MB representation will give essentially the same number of series, grouping +10 + +them in the same ROC is not an easy task. For the case of 3 or more variables, the problem of finding the ROC is +still a problem yet to be solved in an efficient manner. This problem is essentially removed in the case of applying +the CHMB method, where such grouping is automatically done without prior knowledge of the ROC. As a byproduct +of these evaluations, we get the result for associated box integrals ∆n(s) and Jellium potential Jn. We through these +evaluation also discover the relations between these integrals and multivariable hypergeometric functions. +As a future direction, it would be interesting to modify the rules of the OMOB so that the final evaluation of +the bracket series doesn’t require regulators. For the case of C5,k(α,β) evaluated in the present work, one can try +to find a way to evaluate the ACs. One way towards this direction is to write the final result as a derivative of a +hypergeometric function and then find the ACs of it using Olsson.wl . After finding the ACs, the derivative can be +taken to get the final result which converges in the appropriate region. We also note that a similar process can be +used to evaluate C6,k, which also gives a 2-fold MB integral. Finally, it would be interesting to derive the result for +the various Box integrals Bn(s),∆n(s) and Jellium-potential Jn from the results given here. The result in the present +work matches numerically with those results; it would still be interesting to see how they can be obtained from the +present work by using various reduction formulas of multivariable hypergeometric functions. +7 +Acknowledgements +TP would like to thank Souvik Bera for his help and his useful comments. +A +Ruby’s formula +Ruby’s formula is another interesting physical problem where the OMOB can still be used. We provide an evaluation +of a general integral of which Ruby’s formula is a special case in this Appendix to highlight the application of the +OMOB when regulators are not required. Ruby’s formula gives the solid angle subtended at a disk source by a coaxial +parallel-disk detector [36]. It is given as follows +D = Rd +Rs +�∞ +0 +J1(kRd)J1(kRs) e−kd +k +dk +(66) +where Rd and Rs are the radii of the detector and the source, respectively, d is the distance between the source and +the detector, and J1(x) is the order one Bessel’s function of the first kind. We now consider the generalization of +integral 66, as discussed in [37]. We will use the MOB to evaluate the integral and show that it reproduces the result, +along with two ACs. +S = +�∞ +0 +kl e−kd +N +� +j=1 +Ja j(kR j)dk +(67) +we can again apply the method of brackets by using the series expansion of the functions +Ja j(kR j) = 1 +2a j +∞ +� +n j=0 +φn j +(kR j)2n j+a j +22n jΓ(a j + n j +1) +e−kd = +∞ +� +np=0 +φnpknpdnp +putting the series expansion in the above integral, we get +S = +�∞ +0 +∞ +� +np=0 +φnpknp+ldnp +N +� +j=1 +1 +2a j +∞ +� +n j=0 +φn j +(kR j)2n j+1 +22n jΓ(a j + n j +1) +dk +(68) +we can simplify the above by noting that +11 + +N +� +j=1 +1 +2a j +∞ +� +n j=1 +φn j +(kR j)2n j+a j +22n jΓ(a j + n j +1) += +∞ +� +n1=0 +··· +∞ +� +nN=0 +φ1,2,···,Nk +�N +j=1(2n j+a j) +2 +�N +j=1(2n j+a j) +× +�N +j=1(R j)(2n j+a j) +�N +j=1 Γ(a j + n j +1) +putting above value in Eq.(68) gives +S = +�∞ +0 +∞ +� +np=0 +φnpk(np+l+�N +j=1(2n j+a j))dnp +∞ +� +n1=0 +··· +∞ +� +nN=0 +φ1,2,···,N +2 +�N +j=1(2n j+a j) +× +�N +j=1(R j)(2n j+a j) +�N +j=1 Γ(a j + n j +1) +dk +(69) +Using the method of brackets, Eq.(69) can be written as +S = +∞ +� +n1=0 +··· +∞ +� +nN=0 +∞ +� +np=0 +φ1,2,···,N,p〈(np + l +1+ +N +� +j=1 +(2n j + a j))〉 +dnp +2 +�N +j=1(2n j+a j) +× +�N +j=1(R j)(2n j+a j) +�N +j=1Γ(a j + n j +1) +(70) +where φ1,2,···,N,p = φn1φn2 ···φnN φnp +The solutions to Eq.(70) are determined using the solution to the linear equation. +np + l +1+ +N +� +j=1 +(2n j + a j) = 0 +(71) +above equation has (N +1) variables. There are (N +1) different ways to write solutions to the above equation, taking +N free variables each time. +Out of (N+1) solutions, the solution with np as the dependent variable gives the Lauricella function of N variables, +as we will show. The rest of other solutions give the series representation that is the analytical continuation of the +earlier. +Denoting the solution to Eq.(71) by n∗ +i with ni being the dependent variable. +The solutions to equation Eq.(71) can be written as +n∗ +p = −(l +1)− +N +� +j=1 +(2n j + a j);a = 1 +n∗ +i = − +(np + l +1) +2 +− +N +� +j=1,i̸=j +(n j)− +N +� +j=1 +�a j +2 +� +;a = 1 +2 +a is the coefficient of the dependent variable if the set of linear equations obtained from brackets are written in the +form an+ b = 0 +where n is the dependent variable, and b includes all the free variables and the constants. +Denoting the solution of Eq.(70) by Si obtained by using n∗ +i (i = 1,2,··· ,N, p). +I) With np as the dependent variable +12 + +We write the solution to Eq.(70) as +Sp = 1 +a +∞ +� +n1=0 +··· +∞ +� +nN=0 +φ1,2,···,NF(n1,n2,··· ,nN,n∗ +p)Γ(−n∗ +p) +(72) +where F(n1,n2,··· ,nN,np) = +dnp �N +j=1(R j)(2nj +a j) +2 +�N +j=1(2nj +a j) �N +j=1 Γ(a j+n j+1) +. +Putting the values, we get +Sp = +∞ +� +n1=0 +··· +∞ +� +nN=0 +φ1,2,···,N +d−(l+1)−�N +j=1(2n j+a j) �N +j=1(R j)(2n j+a j) +2 +�N +j=1(2n j+a j) �N +j=1Γ(a j + n j +1) +Γ +� +(l +1)+ +N +� +j=1 +(2n j + a j) +� +(73) +Using Legendre’s duplication formula +Γ +� +2 +�l +1 +2 ++ +N +� +j=1 +� +n j + a j +2 +��� += +2 +� +l+�N +j=1(2n j+a j) +� +Γ +� +l+1 +2 +�N +j=1 +� +n j + +a j +2 +�� +Γ +� +l +2 +1+�N +j=1 +� +n j + +a j +2 +�� +�π +(74) +putting above value in equation Eq.(73) and simplifying gives +Sp = +∞ +� +n1=0 +··· +∞ +� +nN=0 +φ1,2,···,N +d−(l+1)−�N +j=1(2n j+a j) �N +j=1(R j)(2n j+a j) +2 +�N +j=1(2n j+a j) �N +j=1 Γ(a j + n j +1) +× +Γ +� +l+1 +2 +�N +j=1 +� +n j + +a j +2 +�� +Γ +� +l +2 +1+�N +j=1 +� +n j + +a j +2 +�� +�π +(75) +this equation can be written in compact form as follow +Sp = 1 +�π +� 2 +d +�l� 1 +d +� +Γ +� N +� +j=1 +a j +2 + l +1 +2 +� +Γ +� N +� +j=1 +a j +2 + l +2 +1 +� N +� +j=1 +�R j +d +�a j +× +∞ +� +n1=0 +··· +∞ +� +nN=0 +(−1) +�N +j=1 n j �N +j=1 +� R j +d +�2n j +�N +j=1 +�� +a j +1 +� +n jΓ(n j +1) +� +× +��N +j=1 +a j +2 + l+1 +2 +� +(�N +j=1 n j) +��N +j=1 +a j +2 + l +2 +1 +� +(�N +j=1 n j) +�N +j=1Γ(a j +1) +(76) +(a)m is the Pochhammer symbol +which exactly matches the series representation obtained in [37] with ROC +N +� +i=1 +|R j| < d +13 + +The above series corresponds to the Lauricella function of N variables. +Sp = 1 +�π +� 2 +d +�l� 1 +d +�� +1 +�N +j=1Γ(a j +1) +� +Γ +� N +� +j=1 +a j +2 + l +1 +2 +� +Γ +� N +� +j=1 +a j +2 + l +2 +1 +� N +� +j=1 +�R j +d +�a j +×Fc +�� N +� +j=1 +a j +2 + l +1 +2 +� +, +� N +� +j=1 +a j +2 + l +2 +1 +� +;(1+ a1),··· ,(1+ aN);− +�R1 +d +�2 +,··· ,− +� RN +d +�2� +(77) +where Fc in the above equation is the Lauricella function for N variables. +II) With ni as the dependent variable +We write the solution to Eq.(70) as +Si = 1 +a +∞ +� +n1=0 +··· +∞ +� +nN=0 +φ1,2,···,(i−1),(i+1),···,N,pF(n1,n2,··· ,n∗ +i ,··· ,nN,np)Γ(−n∗ +i ) +(78) +putting the values, we get +Si = 1 +2 +∞ +� +n1=0 +··· +∞ +� +ni−1=0 +∞ +� +ni+1=0 +··· +∞ +� +nN=0 +∞ +� +np=0 +φ1,2,···,(i−1),(i+1),···,N,p +dnp +��N +j=1,j̸=i(R j)(2n j+a j)� +� +2 +�N +j=1,j̸=i(2n j+a j)���N +j=1,j̸=i Γ(a j + n j +1) +� +× +� +1 +� +Γ(ai − +(np+l+1) +2 +−�N +n j=1,i̸=j(n j)−�N +j=1 +� a j +2 +� ++1 +� +�� +(Ri)−(np+l+1)−�N +j=1,i̸=j(2n j)−�N +j=1 a j+ai� +× +� +1 +2−(np+l+1)−�N +j=1,i̸=j(2n j)−�N +j=1 a j+ai +�� +Γ +� np + l +1 +2 ++ +N +� +j=1,j̸=i +(n j)+ +N +� +j=1 +�a j +2 +��� +(79) +Eq.(79) gives series representation for all values of i = 1,2,··· ,N and is the most general form of all the analytically +continued series. +B +∆n Relations +∆n can be expressed in terms of Bn as has already been shown in the subsection (5.2). Here, the relations for ∆4 and +∆5 are provided: +∆4(s) = 64 +� +3·2 +s +2 +3 +2s+6 −3 +s +2 +4� +(s+7)+1 +(s+2)(s+4)(s+6)(s+7)(s+8) ++ +96 +(s+2)(s+4) B2(s+4)− +96(s+8) +(s+2)(s+4)(s+6) B2(s+6) +(80) ++ 64 +s+2 B3(s+2)− +96(s+7) +(s+2)(s+4) B3(s+4)+ +32(s+8)(s+9) +(s+2)(s+4)(s+6) B3(s+6)+16B4(s) +− 88(s+6) +3(s+2) B4(s+2)+ 8(s+8)(6s+43) +3(s+2)(s+4) B4(s+4)− 8(s+8)(s+9)(s+10) +3(s+2)(s+4)(s+6) B4(s+6) +∆5(s) = 1601+(9+ s) +� +26+s/2 +210+s −54+s/2 −2·35+s/2� +(2+ s)(4+ s)(6+ s)(8+ s)(9+ s)(10+ s) ++ +320 +(2+ s)(4+ s)(6+ s) B2(6+ s)+ +320 +(2+ s)(4+ s) B3(4+ s) +(81) +− +320(10+ s) +(2+ s)(4+ s)(6+ s)(8+ s) B2(8+ s)− +480(9+ s) +(2+ s)(4+ s)(6+ s) B3(6+ s)+ 160 +2+ s B4(2+ s) +− 880 +3 +(8+ s) +(2+ s)(4+ s) B4(4+ s)+ 80 +3 +(10+ s)(55+6s) +(2+ s)(4+ s)(6+ s) B4(6+ s)− 80 +3 +(10+ s)(11+ s)(12+ s) +(2+ s)(4+ s)(6+ s)(8+ s) B4(8+ s) ++32B5(s)−200(7+ s) +6+3s B5(2+ s)+ 4 +3 +(9+ s)(291+35s) +(2+ s)(4+ s) +B5(4+ s)− 8 +3 +(10+ s)(11+ s)(47+5s) +(2+ s)(4+ s)(6+ s) +B5(6+ s) ++ 4 +3 +(10+ s)(11+ s)(12+ s)(13+ s) +(2+ s)(4+ s)(6+ s)(8+ s) +B5(8+ s) +14 + +C +MATHEMATICA files +Here, we give a list of the MATHEMATICA files and packages that we provide, which contains the derivation of the +various results of the paper. +Files Provided +Description +Ising.nb +Contains the evaluation of the Ising integrals C3,k, C4,k, C5,k(α,β) and C6,k(α,β) +Box.nb +Contains the evaluation of the Box integrals B3(s) and B4(s) +MbConicHull.wl +Package required to evaluate multidimensional MB integrals. Used in the +evaluation of C5,k(α,β), C6,k(α,β), B3(s) and B4(s) +MultivariateResidues.m +Used by the package MbConicHull.wl internally +Olsson.wl +Package required for finding the ACs. Used for the case B3(s) +ROC2.wl +Package required for finding the region of convergence of the 2-variable hypergeometric series. +Table 2: +References +[1] Izrail Solomonovich Gradshteyn and Iosif Moiseevich Ryzhik. Table of integrals, series, and products. Academic +press, 2014. +[2] Ivan Gonzalez and Victor H. Moll. Definite integrals by the method of brackets. Advances in Applied Mathemat- +ics, 45(1):50–73, 2010. +[3] Ivan Gonzalez, Victor H. Moll, and Armin Straub. The Method of brackets. Part 2. Examples and applications. +4 2010. +[4] Ivan Gonzalez, Karen Kohl, Lin Jiu, and Victor H. Moll. An extension of the method of brackets. part 1. Open +Mathematics, 15(1):1181–1211, 2017. +[5] Ivan Gonzalez, Lin Jiu, and Victor H. Moll. An extension of the method of brackets. part 2. Open Mathematics, +18(1):983–995, 2020. +[6] Ivan Gonzalez, Igor Kondrashuk, Victor H Moll, and Luis M Recabarren. +Mellin–barnes integrals and the +method of brackets. The European Physical Journal C, 82(1):28, 2022. +[7] Ivan Gonzalez, Igor Kondrashuk, Victor H Moll, and Alfredo Vega. Analytic expressions for debye functions and +the heat capacity of a solid. Mathematics, 10(10):1745, 2022. +[8] B. Ananthanarayan, Sumit Banik, Sudeepan Datta, and Tanay Pathak. Quadratic and quartic integrals using +the method of brackets. Scientia, 29:45–59, 2019. +[9] Hari M Srivastava and Per Wennerberg Karlsson. Multiple Gaussian hypergeometric series. E. Horwood, 1985. +[10] Harold Exton. Multiple hypergeometric functions and applications. Ellis Horwood, 1976. +[11] B. Ananthanarayan, Sumit Banik, Samuel Friot, and Shayan Ghosh. Double box and hexagon conformal Feyn- +man integrals. Phys. Rev. D, 102(9):091901, 2020. +[12] B. Ananthanarayan, Sumit Banik, Samuel Friot, and Shayan Ghosh. Massive One-loop Conformal Feynman +Integrals and Quadratic Transformations of Multiple Hypergeometric Series. Phys. Rev. D, 103(9):096008, 2021. +[13] Sumit Banik. On Hypergeometric solutions of Feynman integrals using Mellin-Barnes Integrals with Applica- +tions. PhD thesis, Bangalore, Indian Inst. Sci., 9 2022. +[14] B. Ananthanarayan, Sumit Banik, Samuel Friot, and Shayan Ghosh. Multiple Series Representations of N-fold +Mellin-Barnes Integrals. Phys. Rev. Lett., 127(15):151601, 2021. +[15] Sumit Banik and Samuel Friot. +Multiple Mellin-Barnes integrals with straight contours. +Phys. Rev. D, +107(1):016007, 2023. +15 + +[16] Olsson, Per O. M. . Integration of the Partial Differential Equations for the Hypergeometric Functions F1 and +FD of Two and More Variables. Journal of Mathematical Physics, 5(3):420–430, 1964. +[17] B. Ananthanarayan, Souvik Bera, S. Friot, O. Marichev, and Tanay Pathak. On the evaluation of the Appell F2 +double hypergeometric function. Comput. Phys. Commun., 284:108589, 2023. +[18] B. Ananthanarayan, Souvik Bera, S. Friot, and Tanay Pathak. Olsson.wl : a Mathematica package for the +computation of linear transformations of multivariable hypergeometric functions. 12 2021. +[19] D.H. Bailey and J.M. Borwein. +High-precision numerical integration: Progress and challenges. +Journal of +Symbolic Computation, 46(7):741–754, 2011. Special Issue in Honour of Keith Geddes on his 60th Birthday. +[20] David H Bailey, Jonathan M Borwein, and Richard E Crandall. Integrals of the ising class. Journal of Physics +A: Mathematical and General, 39(40):12271, 2006. +[21] Flavia Stan. On recurrences for ising integrals. Advances in Applied Mathematics, 45(3):334–345, 2010. +[22] David H Bailey, David Borwein, Jonathan M Borwein, and Richard E Crandall. Hypergeometric forms for ising- +class integrals. Experimental Mathematics, 16(3):257–276, 2007. +[23] B. Ananthanarayan, Sumit Banik, Samuel Friot, and Tanay Pathak. On the Method of Brackets. 12 2021. +[24] Mario Prausa. Mellin–barnes meets method of brackets: a novel approach to mellin–barnes representations of +feynman integrals. The European Physical Journal C, 77(9):1–10, 2017. +[25] David H Bailey, Jonathan M Borwein, and Richard E Crandall. Box integrals. Journal of Computational and +Applied Mathematics, 206(1):196–208, 2007. +[26] D Bailey, J Borwein, and R Crandall. Advances in the theory of box integrals. Mathematics of Computation, +79(271):1839–1866, 2010. +[27] R. S. Anderssen, R. P. Brent, D. J. Daley, and P. A. P. Moran. Concerning +�1 +0 ··· +�1 +0 (x2 +1 +···+ x2 +k)1/2dx1 ··· ,dxk and +a taylor series method. SIAM Journal on Applied Mathematics, 30(1):22–30, 1976. +[28] Johan Philip. The distance between two random points in a 4-and 5-cube. KTH mathematics, 2008. +[29] D.H. Bailey, J.M. Borwein, and R.E. Crandall. Box integrals. Journal of Computational and Applied Mathemat- +ics, 206(1):196–208, 2007. +[30] WP Orrick, Bernie Nickel, AJ Guttmann, and Jacques HH Perk. The susceptibility of the square lattice ising +model: new developments. Journal of Statistical Physics, 102:795–841, 2001. +[31] Tai Tsun Wu, Barry M. McCoy, Craig A. Tracy, and Eytan Barouch. Spin-spin correlation functions for the +two-dimensional ising model: Exact theory in the scaling region. Phys. Rev. B, 13:316–374, Jan 1976. +[32] N Zenine, S Boukraa, S Hassani, and JM Maillard. Square lattice ising model susceptibility: series expansion +method and differential equation for χ (3). Journal of Physics A: Mathematical and General, 38(9):1875, 2005. +[33] David H Bailey, Jonathan M Borwein, David Broadhurst, and Wadim Zudilin. Experimental mathematics and +mathematical physics. Contemp. Math, 517:41–58, 2010. +[34] David H Bailey and Jonathan M Borwein. +High-precision numerical integration: Progress and challenges. +Journal of Symbolic Computation, 46(7):741–754, 2011. +[35] Per OM Olsson. Integration of the partial differential equations for the hypergeometric functions f 1 and fd of +two and more variables. Journal of Mathematical Physics, 5(3):420–430, 1964. +[36] Lawrence Ruby. Further comments on the geometrical efficiency of a parallel-disk source and detector system. +Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and +Associated Equipment, 337(2):531–533, 1994. +[37] Samuel Friot. On Ruby’s solid angle formula and some of its generalizations. Nucl. Instrum. Meth. A, 773:150– +153, 2015. +16 + diff --git a/9dFQT4oBgHgl3EQf5zZD/content/tmp_files/load_file.txt b/9dFQT4oBgHgl3EQf5zZD/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..11e283686d397230fc982ad3d426a48184600bee --- /dev/null +++ b/9dFQT4oBgHgl3EQf5zZD/content/tmp_files/load_file.txt @@ -0,0 +1,785 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf,len=784 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content='13436v1 [math-ph] 31 Jan 2023 Closed Form Expressions for Certain Improper Integrals of Mathematical Physics B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content=' Ananthanarayan * Tanay Pathak† Kartik Sharma‡ Centre for High Energy Physics, Indian Institute of Science, Bangalore-560012, Karnataka, India Abstract We present new closed-form expressions for certain improper integrals of Mathematical Physics such as Ising, Box, and Associated integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content=' The techniques we employ here include (a) the Method of Brackets and its modifications and suitable extensions and (b) the evaluation of the resulting Mellin-Barnes representations via the recently discovered Conic Hull method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content=' Analytic continuations of these series solutions are then produced using the automated method of Olsson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content=' Thus, combining all the recent advances allows for closed-form solutions for the hitherto unknown B3(s) and related integrals in terms of multivariable hypergeometric functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content=' Along the way, we also discuss certain com- plications while using the Original Method of Brackets for these evaluations and how to rectify them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content=' The interesting cases of C5,k is also studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content=' It is not yet fully resolved for the reasons we discuss in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content=' 1 Introduction In studies of theoretical physics and mathematics, various integrals appear whose symbolic evaluation is sought after.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content=' Gradshyteyn and Ryzik [1] compiled a long list of such integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content=' Recently there have been attempts to provide a derivation of a large number of these integrals, specifically the improper integral with limits from 0 to ∞ using the Original Method of Brackets (OMOB) [2–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content=' Apart from this, some of the present authors have also evaluated the integral of quadratic and quartic types and their generalization using the OMOB, which has been reported in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content=' In the present investigation, we turn to other interesting improper integrals that appear in Mathematical Physics, such as the Ising integrals and the Box integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content=' Our work is motivated by the need to express them in terms of elegant closed-form expression or in terms of known functions of mathematical physics, especially the hyper- geometric functions [9,10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content=' In the recent past, several tools have also been developed to facilitate tasks of symbolic evaluation of these integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content=' Our results here have been facilitated by the recent development of tools and ad- vances in various theoretical treatments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content=' Note for instance, the recently proposed solution to the problem of finding the series solution of the N-dimensional Mellin-Barnes (MB) representation [11–13], using what has been termed as the Conic Hull Mellin Barnes (CHMB) method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content=' This has also been automated as the MATHEMATICA package MBConichulls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content='wl [14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content=' The series representation hence obtained, in general, can be written as hypergeometric functions or their derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content=' Independently, the issue of finding the analytic continuations (ACs) of the multivari- able hypergeometric function using the method of Olsson [16,17], which has also been automated as a MATHEMAT- ICA package Olsson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content='wl [18] have been addressed recently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content=' In this work, we show how these tools together, which were primarily directed at solving Feynman integrals, are of sufficient generality to find their use in the evaluation of the integrals considered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content=' We will consider the Ising integrals which have been studied in the Ising model [19–22] and also have been in the context of OMOB [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content=' Apart from the evaluation with these newly developed tools, we will also consider certain complications while doing similar evaluations with the OMOB [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content=' One of them is the use of regulators for the evaluation of the Ising integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content=' This arises in the case of Ising integrals C3,1 and C4,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content=' For the case of C4,1, it is further complicated due to the use of two regulators, which, when the proper limiting procedure is applied, will give the final result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content=' However, we point out that such a procedure is complicated and thus use the Modified Method of Brackets (MMOB) [24] to get the MB-integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content=' This MB integral can then be evaluated without any introduction of such regulators and thus provides an efficient way to deal with these integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content=' Using a similar procedure, we anant@iisc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content='in †tanaypathak@iisc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content='in ‡kartiksharma@iisc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content='in 1 attempt to evaluate the elusive C5,k integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content=' However, we hit a roadblock for the same, as the resulting series does not converge and would require a proper analytic continuation procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content=' At present, we find this task beyond the reach of the tools at hand, though we provide a possible way to achieve the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content=' Yet such results still shed some light on the form that these integrals can be evaluated to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content=' All the results are provided in the ancillary MATHEMATICA file Ising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content='nb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content=' Box integrals [25–28] are another interesting integrals where such techniques can be applied to get new results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content=' They do carry a physical meaning in the sense that they provide the expected distance between two randomly chosen points over the unit n-cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content=' We consider the two special cases of them, namely the Bn(s) and the ∆n(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content=' We use the same techniques and derive the closed form results for already known B1(s) and B2(s) and new evaluation for B3(s) and B4(s) for general values of s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content=' The results are in terms of multi-variable hypergeometric function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content=' These evaluations further require the use of an analytic continuation procedure which has been done using Olsson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content='wl .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content=' All the results are provided in the ancillary MATHEMATICA file Box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content='nb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content=' These results for box integrals can then be further used to evaluate the Jellium potential Jn, which can be related to box integral Bn(s) [26, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content=' Finally, we give a general MB integral for Bn(s), which can be used to find the closed form result for all values of n and s using MbConicHull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content='wl .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content=' With all this, we find new connections between the Box integrals and the multivariable hypergeometric functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content=' All our calculations rely heavily on MATHEMATICA as we try to achieve the symbolic results for all the problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content=' The paper is structured as follows: In section (2) using an example given in [4], we point out the problem in the OMOB and discuss the alternative to surpass this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content=' We then, in section (3), proceed to the evaluation of Ising integrals up to n = 4 while contrasting our method with the method used before to achieve the same in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content=' In section (4) we attempt to solve the C5,k integral and point out a general integral C5,k(α,β) which gives C5,k as a special case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content=' Though we point out that it is not the final result, a proper analytic continuation procedure is required to get C5,k from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content=' We then evaluate box integral Bn(s) for n = 3,4 in section (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content=' The new results for ∆n(s) and Jn with the above new results are also provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content=' Finally, we conclude the paper with some conclusions and possible future directions in section (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content=' In appendix C, we provide the table for all the MATHEMATICA files that we give and the packages required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content=' 2 Method of Brackets revisited We will first illustrate the OMOB using a simple example of integral evaluation as given in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content=' We will first evaluate the integral by directly using the OMOB, then briefly propose a possible resolution while doing such evaluations, and then illustrate the alternative method to do the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content=' We consider the following integral H1(a,b) = �∞ 0 K0(ax)K0(bx) (1) The integral is introduced to facilitate the evaluation of another integral, which is given by putting a = b H(a) = �∞ 0 K2 0(ax)dx (2) We can express K0(x) using the following series expansion: K0(ax) = � n1 φn1 a2n1Γ(−n1) 22n1+1 x2n1 (3) where φn = (−1)n Γ(n+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content=' This expansion uses a divergent series, and we can express the result in the form of an integral representation as K0(bx) = 1 2 �∞ 0 exp � −t− b2x2 4t � dt t (4) Using the OMOB, we get: K0(bx) = � n2,n3 φn2,n3 b2n3 x2n3 22n3+1 〈n2 − n3〉 (5) Substituting the bracket series in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content=' (1), we get H1(a,b) = � n1,n2,n3 φn1,n2,n3 a2n1b2n3Γ(−n1) 22n1+2n3+2 〈n2 − n3〉〈2n1 +2n3 +1〉 (6) 2 Now, we need to solve the bracket equations, which involve 2 equations but 3 variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content=' Evaluating this we get following 3 series, Ti where ni is the free variable: T1 = 1 4a � n φnΓ(−n)Γ2 � n+ 1 2 �� b a �2n T2 = 1 4a � n φnΓ(−n)Γ2 � n+ 1 2 �� b a �2n T3 = 1 4a � n φnΓ(−n)Γ2 � n+ 1 2 �� b a �2n (7) Using the rules of the OMOB, all the 3 series of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content=' (7) have to be discarded as they are divergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content=' A solution to such a problem, as implemented in [4], is to regularize the singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content=' This amounts to modifying the bracket 〈n2−n3〉 → 〈n2−n3+ǫ〉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content=' With this modification, when n1 is a free variable, one gets the series that contains Γ(−n), which is diverging and is thus discarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content=' While for the other cases, one gets two series with ǫ parameter (in the form of Γ(−n + ǫ) and Γ(−n − ǫ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content=' In these series, when the proper limiting procedure is done, along with the condition a = b to ease the calculation, they give the result for the integral of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content='(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content=' Thus, the original integral of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content=' (1) we started with still remains elusive, as the calculation is much more involved (the limiting procedure) within this present framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content=' An alternative to the above evaluation, free from choosing the regulator and doing the tedious limiting procedure, is to use the MB representation derived using the MMOB [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content=' Using it, we get the following MB representation for the integral given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content=' (1) H1(a,b) = 1 4 c+i∞ � c−i∞ dz 2πi a−2z−1b2zΓ(−z)2Γ �1 2(2z +1) �2 (8) The above MB integral can be readily evaluated in MATHEMATICA to give the following result H1(a,b) = π � a2 b2 K � 1− a2 b2 � 2a (9) where K(x) is the complete elliptic integral of the first kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content=' Thus we get the value of the original integrals, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content=' (1) we started with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content=' For the special case of a = b, using K(0) = π 2 we get H1(a,a) = H(a) = π2 4a (10) So we see that for the simple cases, too, using the MB representation to evaluate these integrals provides an efficient way to evaluate these integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content=' 3 Ising integrals In this section, we will analyze the integrals of the “Ising class".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content=' Ising models are extensively used to study the statistical nature of ferromagnets [30–32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content=' The model accounts for the magnetic dipole moments of the spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content=' The n - dimensional integrals are denoted by Cn,Dn,En, where Dn is found in the magnetic susceptibility integrals essential to the Ising calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content=' Dn = 4 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQf5zZD/content/2301.13436v1.pdf'} +page_content=' �∞ 0 ··· �∞ 0 � i 0. + +ON THE LIMIT SPECTRUM OF A DEGENERATE OPERATOR +11 +Multiplying the first equation of (1.10) by uk +0 and using 1 +µ1 +as the constant (it is in fact the best one) in the +Poincaré’s inequality, we get +(2.13) +� +D +uk +0(y) dy = +� +D +|∇′ +yuk +0(y)|2 dy − λk +� +D +|uk +0(y)|2 dy ≥ +� +1 − λk +µ1 +� � +D +|∇′ +yuk +0(y)|2 dy. +On the other hand, the first eigenvalue µ1 is characterized by +(2.14) +µ1 = +inf +u∈H1 +0(D) +∥ ∇′ +yu ∥2 +L2(D) +∥ u ∥2 +L2(D) +. +Hence the following estimate holds true +(2.15) +� +D +|∇′ +yuk +0(y)|2 dy ≥ µ1 +� +D +|uk +0(y)|2 dy. +From (2.13), we derive with the help of (2.15) +(2.16) +(µ1 − λk) +� +D +|uk +0(y)|2 dy ≤ +� +D +uk +0(y) dy ≤ +� +|D| +�� +D +|uk +0(y)|2 dy +� 1 +2 , +and then from (2.16) we deduce +(2.17) +0 < +� +D +uk +0(y) dy ≤ +|D| +µ1 − λk +. +By virtue of the last equation in (1.10), ˆλk := λk +� +1 + +|D| +|C \ D| + +λk +|C \ D| +� +D +uk +0 dy +� +is an eigenvalue of − d2 +dx2 +3 +so +that ˆλk ≥ λ0 where λ0 denotes the first eigenvalue of − d2 +dx2 +3 +. Using the second inequality of (2.17) we get +(2.18) +λk +� +1 + +|D| +|C \ D| + λk +|D| +|C \ D|(µ1 − λk) +� +≥ ˆλk ≥ λ0. +Hence, λk ≥ µ0 := φ−1(λ0) where φ is the continuous increasing function defined on (0, µ1) by +φ(t) = t +� +1 + +|D| +|C \ D| + t +|D| +|C \ D|(µ1 − t) +� +. +□ +So far, we have not yet proved that (uk, vk) is indeed an eigenvector of the limit operator; this is the purpose +of the next subsection. +2.3. The strong convergence of the eigenvectors. We prove the following compactness result +Proposition 2.4. For each k, there exists a subsequence of ε such that the sequence of solutions uk +ε of (1.9) converges +strongly in L2(Ω) to the eigenvector uk of (2.10). +Proof. One can extend uk +ε from M to the whole Ω in such a way the extension U k +ε fulfills U k +ε ∈ Vs, U k +ε = uk +ε in M +and +(2.19) +∥ ∇′U k +ε ∥L2(Ω)≤ K ∥ ∇′uk +ε ∥L2(M), +���� +∂U k +ε +∂x3 +���� +L2(Ω) +≤ K +���� +∂uk +ε +∂x3 +���� +L2(M) +. +Note that the extension only affects the horizontal variable y so that the Dirichlet boundary condition on the upper +and lower faces of Ω (x3 = 0 or x3 = L ) is preserved, see for instance [6], [10], [29]. +In addition, one can assume that such extension satisfies the following equation +(2.20) +� +−∆′ +yU k +ε − ε2 ∂2U k +ε +∂x2 +3 += 0 +in F. + +12 +KAÏS AMMARI AND ALI SILI +Indeed, if (2.20) is not true for U k +ε , then one can introduce the function W k +ε as the unique solution of +(2.21) +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +W k +ε ∈ V, +� +F +� 1 +ε2 ∇′ +yW k +ε ∇′ +yφ + ∂W k +ε +∂x3 +∂φ +∂x3 +� +dy dx3 = +� +F +� 1 +ε2 ∇′ +yU k +ε ∇′ +yφ + ∂U k +ε +∂x3 +∂φ +∂x3 +� +dy dx3 +∀ φ ∈ V, +where V := +� +u ∈ Vs, u = 0 on ∂D × (0, L) +� +(recall that V := V ε +s with ε = 1 where V ε +s is defined by (1.5)). +Hence, V is the subspace of Vs of functions vanishing in M. By the Lax-Milgram Theorem we get existence and +uniqueness for W k +ε . Choosing φ ∈ C∞ +0 (F), the last equation leads to +(2.22) +− 1 +ε2 ∆′ +yW k +ε − ∂2W k +ε +∂x2 +3 += − 1 +ε2 ∆′ +yU k +ε − ∂2U k +ε +∂x2 +3 +in F. +On the other hand, using equation (2.21) with φ = W k +ε , we get the following estimate with the help of (2.19) and +(2.7) +(2.23) +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +���� +1 +ε∇′W k +ε +���� +L2(F ) ++ +���� +∂W k +ε +∂x3 +���� +L2(F ) +≤ K +����� +1 +ε∇′U k +ε +���� +L2(F ) ++ +���� +∂U k +ε +∂x3 +���� +L2(F ) +� +≤ +≤ K +����� +1 +ε∇′uk +ε +���� +L2(M) ++ +���� +∂uk +ε +∂x3 +���� +L2(M) +� +≤ K. +Multiplying equation (2.22) by ε2, we see that ˜uk +ε defined by ˜uk +ε = U k +ε − W k +ε is indeed an extension which fulfills +equation (2.20) and preserves the apriori estimate (2.19). Note that functions of V may be extended by zero inside +M so that ˜uk +ε is still an extension of uk +ε from M to the whole Ω. +In the sequel, we will still denote the extension of uk +ε satisfying (2.19) and (2.20) by U k +ε . +Consider now the sequence defined in Ω by zk +ε = uk +ε − U k +ε . If we prove that zk +ε admits a strongly converging +subsequence in L2(Ω) then we can deduce the existence of such subsequence for uk +ε since U k +ε is bounded in H1(Ω) +by virtue of (2.19) and (2.7) and therefore admits a strongly converging subsequence in L2(Ω) according to the +Rellich imbedding Theorem. +We first derive the following equation on zk +ε by the use of (1.7) together with (2.20) +(2.24) +� +� +� +� +� +� +� +zk +ε ∈ Vs, +−∆′ +yzk +ε − ε2 ∂2zk +ε +∂x2 +3 += λk +εzk +ε + λk +εU k +ε +in F, +zk +ε = 0 +on ∂D × (0, L). +Since uk +ε and U k +ε are bounded respectively in L2(0, L; H1(C)) and H1(Ω), the sequence zk +ε is bounded in +L2(0, L; H1(C)). Hence, there exist a subsequence and zk ∈ L2(0, L; H1(C)) such that +zk +ε ⇀ zk weakly in L2(0, L; H1(C)). +Therefore, denoting by Uk the weak limit in H1(Ω) of the corresponding subsequence U k +ε , one can pass easily to +the limit in (2.24) to get the equation +(2.25) +� +� +� +zk ∈ L2(0, L; H1(C)), +−∆′ +yzk = λkzk + λkUk +in F, +zk = 0 +on ∂D × (0, L). +Note that by construction, zk +ε = 0 in M = (C \ D) × (0, L) so that the convergence +zk +ε χM(y) ⇀ zkχM(y) weakly in L2(Ω) +shows that zk = 0 in M which is equivalently expressed by the boundary condition of (2.25). + +ON THE LIMIT SPECTRUM OF A DEGENERATE OPERATOR +13 +More generally, given a bounded sequence (fε) in L2(Ω) and f ∈ L2(Ω), we now consider equations of the +form +(2.26) +� +� +� +� +� +� +� +wε ∈ Vs, +−∆′ +ywε − ε2 ∂2wε +∂x2 +3 += λk +εwε + fε +in F, +wε = 0 +on ∂D × (0, L), +and +(2.27) +� +� +� +w ∈ L2(0, L; H1(C)), +−∆′ +yw = λkw + f +in F, +w = 0 +on ∂D × (0, L). +Regarding the sequence of solutions of (2.26), the following lemma holds true. +Lemma 2.5. Assume that λk +ε → λk with λk < µ1 and that fε ⇀ f weakly in L2(Ω). Then the sequence wε is +bounded in L2(0, L; H1(C)) and for the whole sequence ε, wε ⇀ w weakly in L2(0, L; H1(C)) where w is the +unique solution of (2.27). +Proof. We only have to prove that wε is bounded in L2(0, L; H1(C)), the limit problem (2.27) satisfied by w can +be established exactly by the same process already used in the proof of (2.25). +The main ingredient to get that apriori estimate relies on the Poincaré inequality +(2.28) +� +D +|u|2 dy ≤ 1 +µ1 +� +D +|∇′ +yu|2 dy +∀ u ∈ H1 +0(D), +combined with the assumption λk < µ1. +Multiplying equation (2.26) by wε and integrating, we get +(2.29) +� L +0 +� +D +|∇′wε|2 dydx3 ≤ λk +ε +� L +0 +� +D +|wε|2 dydx3+ ∥ fε ∥L2(Ω)∥ wε ∥L2(F ) . +Choosing u = wε(., x3) with x3 ∈ (0, L) and integrating (2.28) over (0, L), we infer +(2.30) +� L +0 +� +D +|wε|2 dydx3 ≤ 1 +µ1 +� L +0 +� +D +|∇′ +ywε|2 dydx3. +Let δ > 0 be such that 0 < λk < δ < µ1. Turning back to (2.29) and using (2.30), we get for ε sufficiently small, +(2.31) +� +1 − δ +µ1 +� � L +0 +� +D +|∇′wε|2 dydx3 ≤∥ fε ∥L2(Ω)∥ wε ∥L2(F ) . +Since fε is bounded in L2(Ω), applying once again inequality (2.30), we derive from (2.31) the estimate +(2.32) +� L +0 +� +D +|∇′wε|2 dydx3 ≤ K. +The estimates (2.30) and (2.32) show that wε is bounded in L2(0, L; H1(D)) and thus in L2(0, L; H1(C)) since wε +is equal to zero in C \ D. +□ +We continue the proof of the Proposition 2.4 in the following way. +Multiplying equations (2.24) and (2.26) respectively by wε and by zk +ε and integrating we get +(2.33) +� +� +� +� +� +� +� +� +� +� +� +� +F +� +∇′zk +ε ∇′wε + ε2 ∂zk +ε +∂x3 +∂wε +∂x3 +� +dydx3 = λk +ε +� +F +zk +ε wε dydx3 + λk +ε +� +F +U k +ε wε dydx3 = +λk +ε +� +F +wεzk +ε dydx3 + +� +F +fεzk +ε dydx3. + +14 +KAÏS AMMARI AND ALI SILI +Since U k +ε is bounded in H1(Ω), there exist a subsequence of ε and Uk ∈ H1(Ω) such that U k +ε ⇀ Uk weakly in +H1(Ω) and strongly in L2(Ω) by virtue of the Rellich imbedding Theorem. Therefore for that a subsequence, we +get from (2.33) with the help of Lemma 2.6 +(2.34) +lim +ε→0 +� +F +fεzk +ε dydx3 = lim +ε→0 λk +ε +� +F +U k +ε wε dydx3 = λk +� +F +Ukw dydx3. +On the other hand, one can multiply (2.25) and (2.27) respectively by w and by zk and integrate to obtain +(2.35) +� +� +� +� +� +� +� +� +� +� +F +∇′zk∇′w dydx3 = +� +F +∇′w∇′zk dydx3 = λk +� +F +zkw dydx3 + λk +� +F +Ukw dydx3 += λk +� +F +wzk dydx3 + +� +F +fzk dydx3. +Combining (2.34) and (2.35), we get +(2.36) +lim +ε→0 +� +F +fεzk +ε dydx3 = λk +� +F +Ukw dydx3 = +� +F +fzk dydx3. +Choosing in particular fε = zk +ε which converges weakly in L2(Ω) to f = zk, we obtain +(2.37) +lim +ε→0 +� +F +(zk +ε )2 dydx3 = +� +F +(zk)2 dydx3, +which implies the strong convergence of the subsequence zk +ε and therefore the strong convergence of the corre- +sponding subsequence of uk +ε. Hence Proposition 2.4 is proved. +□ +We now proceed to complete the proof of Theorem 1.3. +2.4. Proof of Theorem 1.3. The strong convergence in L2(Ω) of the eigenvectors when λk < µ1 is proved in +Proposition 2.4. We use it to prove the convergence of the sequence of energies from which we obtain immediately +(1.16) and (1.17). +Consider the sequence +(2.38) +Jε = +� +Ω +�� +��∇′uk +ε − ∇′uk +��2 + ε2 +���� +∂uk +ε +∂x3 +���� +2� +χF + +� +1 +ε2 +��∇′uk +ε +��2 + +���� +∂uk +ε +∂x3 +− dvk +dx3 +���� +2� +χM +� +dydx3. +Choosing uk +ε and (uk, vk) as test functions respectively in (1.9) and in (2.9), we get with the help of the weak +convergences proved in Proposition 2.1 and of the strong convergence proved in Proposition 2.4, +(2.39) +� +� +� +� +� +� +� +� +� +� +� +Jε = λk +ε +� +Ω +(|uk +ε|2dydx3 + λk +� +Ω +|uk|2dydx3 − 2 +� +Ω +� +∇′uk +ε∇′ukχF + ∂uk +ε +∂x3 +dvk +dx3 +χM +� +dydx3 +−→ 2λk +� +Ω +|uk|2dydx3 − 2λk +� +Ω +|uk|2dydx3 = 0. +Hence the weak convergences stated in Proposition 2.1 are in fact strong convergences; in particular, keeping in +mind Proposition 2.4, we get the strong convergences stated in Theorem 1.3. +We have proved above that λk is an eigenvalue of the limit problem (in the sense of (1.13)) if and only if λk +satisfies (1.10). In the sequel, a number λ satisfying (1.10) will be called an eigenvalue of the limit problem (1.10). +We now prove that there exist non trivial solutions for the system (1.10) and that any λ ∈ (µ0, µ1) which +satisfies (1.10) may be attained as a limit of a sequence (λk +ε)ε; by this we can conclude that (1.13) has no other +eigenvalues on the left of µ1 than those obtained from the limits of the eigenvalues λk +ε and thus we can list all its +eigenvalues in increasing order. It is then clear that for a fixed k, we cannot have two subsequences ε and ε′ with +two different limits for λk +ε and λk +ε′ since this would lead to add a new element to the set of eigenvalues of (1.10); +hence for each k, (1.15) holds for the whole sequence ε. + +ON THE LIMIT SPECTRUM OF A DEGENERATE OPERATOR +15 +To prove the existence of non trivial solutions +� +uk +0 +vk +� +for the system (1.10) with λk < µ1 leading to non +trivial solutions +�uk +vk +� +for (1.13)) where uk := (λkuk +0 + 1)vk, it is sufficient to show that one can find solutions +� +uk +0 +vk +� +of (1.10) with vk ̸= 0. +uk +0 is uniquely determined by the first equation of (1.10) since λk < µ1 and if (fn)n is the orthonormal basis in +L2(D) made up of eigenfunctions associated to the increasing sequence (µn)n of eigenvalues of −∆′ +y, one can get +from the first equation of (1.10) +(2.40) +uk +0 = +∞ +� +n=1 +cnfn +µn − λk +; where cn = +� +D +fn dy. +Replacing the mean value of uk +0 in the second equation of (1.10), we derive +(2.41) +− d2vk +dx2 +3 += δ(λk)vk; with δ(λ) := Cλ + C′ +∞ +� +n=1 +c2 +nλ2 +µn − λ, +where C, C′ denote positive constants. +Let (γj, vj) be an eigenelement of − d2 +dx2 +3 +in H1 +0(0, L). Since δ is a strictly positive increasing function over +(0, µ1), there exists λkj ∈ (0, µ1) such that γj = δ(λkj), so that the second equation of (1.10) may be written +as −d2vj +dx2 +3 += δ(λkj)vj, taking vkj := vj. Hence for λk < µ1, the pair (uk +0, vkj) is a non trivial solution for any +j = 1, 2, ... +We now argue by contradiction to prove that any λ ∈ (µ0, µ1[ which is an eigenvalue of (1.10) may be +attained as a limit of a sequence (λk +ε)ε for some k. +If for any k and for any sequence ε, λk +ε does not converge to λ, then there exists a neighborhood of λ which +does not contain any λk +ε for all k. In other words, λ belongs to the resolvent of the operator Aε defined by (1.7). +Hence, for any f ∈ L2(0, L) ⊂ L2(Ω), there exists uε ∈ D(Aε) such that +(2.42) +Aεuε = λuε + f +in Ω. +Multiplying (2.42) by φ ∈ Vs and integrating we get +(2.43) +� +� +� +� +� +� +� +� +� +� +Ω +�� +∇′uε∇′φ + ε2 ∂uε +∂x3 +∂φ +∂x3 +� +χF + +� 1 +ε2 ∇′uε∇′φ + ∂uε +∂x3 +∂φ +∂x3 +� +χM +� +dy dx3 = +λ +� +Ω +uεφ dy dx3 + +� +Ω +fφ dy dx3, +∀ φ ∈ Vs. +To get apriori estimates on the sequence uε, we will use the following Poincaré type inequality. +Lemma 2.6. There exists a positive constant K such that +(2.44) +� +� +� +� +� +� +� +� +� +∥u∥L2(Ω) ≤ K +� +∥∇′u∥L2(Ω) + +���� +∂u +∂x3 +χM +���� +L2(Ω) +� +, +∀ u ∈ L2(0, L; H1(C)) ∩ L2(C \ D; H1 +0(0, L)). +Proof. We argue by contradiction. Assuming inequality (2.44) false, one can find a sequence +un ∈ L2(0, L; H1(C)) ∩ L2(C \ D; H1 +0(0, L) + +16 +KAÏS AMMARI AND ALI SILI +such that +(2.45) +∥un∥L2(Ω) = 1 +∀ n, +and +� +∥∇′un∥L2(Ω) + +���� +∂un +∂x3 +χM +���� +L2(Ω) +� +−→ 0. +Thanks to the classical Poincaré inequality +�����u − +1 +|C \ D| +� +C\D +u dy +����� +L2(C) +≤ K ∥∇′u∥L2(C) applied to u = +un(., x3) ∈ H1(C), x3 ∈ (0, L), we get after integrating with respect to x3, (remember that Ω = C × (0, L)) +(2.46) +�����un − +1 +|C \ D| +� +C\D +un dy +����� +L2(Ω) +≤ K ∥∇′un∥L2(Ω) . +On the other hand, the one-dimensional Poincaré inequality for functions of H1 +0(0, L) applied with u(x3) = +� +C\D +un(y, x3) dy ∈ H1 +0(0, L) leads to the estimate +(2.47) +����� +� +C\D +un dy +����� +L2(Ω) +≤ K +���� +∂un +∂x3 +���� +L2(M) +. +Combining (2.46) and (2.47) with (2.45), we come to a contradiction. +□ +Taking φ = uε in (2.43) and applying (2.44) with u = uε (note that Vs ⊂ L2(0, L; H1(C)) ∩ L2(C \ +D; H1 +0(0, L)), we get the same apriori estimates as those obtained for the sequence uk +ε in (2.7). Indeed all the apriori +estimates on the sequence uk +ε are based on its L2(Ω)- apriori estimate which still holds true for the sequence uε. +Hence by the same arguments that led to (2.10) one can pass to the limit ε → 0 in (2.43) to get at the limit +(2.48) +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +u(y, x3) ∈ L2((0, L); H1(C)), +−∆′ +yu(y, x3) = λu + f in D × (0, L), +u = v +on ∂D × (0, L), +v ∈ H1 +0(0, L), +−d2v +dx2 +3 += λv + +λ +|C \ D| +� +D +u dy + +1 +|C \ D| +� +C +f dy +in (0, L). +Choosing f(y, x3) = g(x3)χC\D(y) (which implies f = 0 in D) with an arbitrary g ∈ L2(0, L), the second +equation in (2.48) reduces to +(2.49) +v ∈ H1 +0(0, L), +−d2v +dx2 +3 += λv + +λ +|C \ D| +� +D +u dy + g +in (0, L). +Note that v ̸= 0 for g ̸= 0. Indeed if v = 0, the first equation in (2.48) would imply u = 0 since we have chosen f +such that f = 0 in D and λ < µ1 is not an eigenvalue of −∆′ +y. Therefore equation (2.49) would give g = 0 which +is a contradiction. +Therefore, one can express u as u = (λu0 + 1)v where the pair (λ, u0) solves the first equation of (1.10). +Therefore (2.49) takes the form +(2.50) +v ∈ H1 +0(0, L)), +−d2v +dx2 +3 += λ +� +1 + +|D| +|C \ D| + +λ +|C \ D| +� +D +u0 dy +� +v + g +in (0, L). +On the other hand, by hypothesis, λ is an eigenvalue of (1.10) so that the last equation of (1.10) with the same u0 as +in (2.50) shows that λ +� +1 + +|D| +|C \ D| + +λ +|C \ D| +� +D +u0 dy +� +is an eigenvalue of − d2 +dx2 +3 +. This is a contradiction since + +ON THE LIMIT SPECTRUM OF A DEGENERATE OPERATOR +17 +equation (2.50) valid for all g ∈ L2(0, L) means that the number λ +� +1 + +|D| +|C \ D| + +λ +|C \ D| +� +D +u0 dy +� +belongs +to the resolvent of − d2 +dx2 +3 +. +We prove now that +lim +k→+∞ λk = µ1. +Since λk ∈ (µ0, µ1) for any k, the sequence (λk)k admits at least an accumulation point and each accumu- +lation point λ is such that µ0 ≤ λ ≤ µ1. Assume that there exists an accumulation point λ such that λ < µ1. There +exists a subsequence (λkn, ukn +0 , vkn) of solutions of (1.10) such that +lim +n→+∞ λkn = λ. Hence the following equation +takes place for all n +(2.51) +− ∆′ukn +0 += λknukn +0 + 1 +in D. +Let δ be a positive number such that λ < δ < µ1. For n large enough we have λkn ≤ δ so that applying the Poincaré +inequality +(2.52) +� +D +|u|2 dy ≤ 1 +µ1 +� +D +|∇′ +yu|2 dy +∀ u ∈ H1 +0(D), +after multiplying (2.51) by ukn +0 , we get for n large enough +(2.53) +� +D +|∇′ +yukn +0 |2 dy ≤ δ +µ1 +� +D +|∇′ +yukn +0 |2 dy + +� +D +|ukn +0 | dy. +Applying successively the Cauchy-Schwarz inequality and (2.52) in the last integral of (2.53), we infer +(2.54) +� +1 − δ +µ1 +� � +D +|∇′ +yukn +0 |2 dy ≤ +� +|D| +� 1 +µ1 +�� +D +|∇′yukn +0 |2 dy. +Therefore, (ukn +0 )n is bounded in H1 +0(D) and one can assume (possibly for another subsequence) that (ukn +0 )n con- +verges weakly to u0 in H1 +0(D). In particular we have that +lim +n→+∞ +� +D +ukn +0 +dy = +� +D +u0 dy. On the other hand +(λkn, ukn +0 , vkn) being a solution of (1.10), the following equation (recall that vkn ̸= 0 ) +(2.55) +− d2vkn +dx2 +3 += λkn +� +1 + +|D| +|C \ D| + +λkn +|C \ D| +� +D +ukn +0 +dy +� +vkn +∀ n, +shows that the number µ defined by µ := λ +� +1 + +|D| +|C \ D| + +λ +|C \ D| +� +D +u0 dy +� +is a finite accumulation point of +the spectrum of − d2 +dx2 +3 +since µ = +lim +n→+∞ µn where µn := λkn +� +1 + +|D| +|C \ D| + +λkn +|C \ D| +� +D +ukn +0 +dy +� +. This is a +contradiction since it is well known that such spectrum is in fact an increasing sequence which tends to +∞. +The last point which remains to prove is that all the limiting eigenvalues are simple and that uk +ε converges +to uk for the whole sequence ε. Assuming that λk is a simple eigenvalue, the proof of the convergence of the +eigenvectors for the whole sequence ε is known since the work of [26] (see also [10]). We sketch it in the vectorial +setting for the convenience of the reader. +Assume that +�uk +vk +� +is an eigenvector associated to the simple eigenvalue λk. Using the fact that the eigenval- +ues converge for the whole sequence ε, it is easy to check that the multiplicity of λk is equal or greater than that of +λk +ε; hence λk +ε is simple and there are only two eigenvectors satisfying +� +Ω +|uk +ε|2 dx = 1, namely uk +ε and −uk +ε. Among +these two eigenvectors, we choose the one satisfying the inequality +(2.56) +� +Ω +� +uk +εχF uk + uk +εχMvk +� +dydx3 > 0. + +18 +KAÏS AMMARI AND ALI SILI +Therefore if ε′ is a subsequence such that +�uk +ε′χF +uk +ε′χM +� +strongly converges in (L2(Ω))2 to the eigenvector +�ˆuχF +ˆvχM +� +associated to λk , we get by passing to the limit in (2.56), +(2.57) +� +Ω +(ˆuχF uk + ˆvχMvk) dydx3 > 0. +On the other hand, +�ukχF +vkχM +� += +�ˆukχF +ˆvkχM +� +or +�ukχF +vkχM +� += − +� +ˆukχF +ˆvkχM +� +since λk is a simple eigenvalue. The last +equality is excluded thanks to (2.57) so that any subsequence is such that +�uk +ε′χF +uk +ε′χM +� +strongly converges in (L2(Ω))2 +to +�ukχF +vkχM +� +. +Let us now prove that all the limit eigenvalues are simple eigenvalues. +Assume that for some k, (1.13) holds true for two orthogonal eigenvectors +�uk +vk +� +and +�¯uk +¯vk +� +in L2(D) × L2(0, L). +By assumption, we have +(2.58) +� L +0 +� +D +uk¯ukdydx3 + |C \ D| +� L +0 +vk¯vkdx3 = 0. +We know that uk and ¯uk are given respectively by uk(y, x3) = (λkuk +0(y) + 1)vk(x3) and ¯uk(y, x3) = (λkuk +0(y) + +1)¯vk(x3) where uk +0(y) given by the first equation of (1.10) depends only on the eigenvalue λk. +Turning back to (2.58), we infer +(2.59) +� L +0 +��� +D +� +λkuk +0(y) + 1 +� +dy +�2 ++ |C \ D| +� +vk(x3)¯vk(x3)dx3 = 0. +As remarked above vk and ¯vk are always eigenvectors of the operator − d2 +dx2 +3 +with Dirichlet condition so that (2.59) +and the second equation of (1.10) would mean that vk and ¯vk eigenvectors associated to the eigenvalue λk +� +1 + +|D| +|C \ D| + +λk +|C \ D| +� +D +uk +0 dy +� +are othogonal in L2(0, L). This is a contradiction since all the eigenvalues of − d2 +dx2 +3 +with Dirichlet condition are simple eigenvalues. +The proof of Theorem 1.3 is now complete. +Finally, let us indicate briefly in the following short section how to derive the analogous theorem in the +homogenization setting using the same approach as in the reduction of dimension. +3. PROOF OF THEOREM 1.5 +In the spirit of the above section, the natural idea is to choose a test function vanishing outside the set Fε +of fibers to get the apriori estimate on the sequence of eigenvalues. To that aim, we consider an eigenvector φ(y) +corresponding to the first eigenvalue of −∆′ +y in H1 +0(D). We extend φ by zero over C \ D and then by periodicity to +the whole R2. The k-th eigenvalue λk +ε of (1.8) is given by the same min-max formula, namely +(3.1) +λk +ε = min +V k⊂Vh +max +u∈V k +� +Ω +� +ε2|∇u|2χFε + |∇u|2χMε +� +dx′ dx3 +� +Ω +|u|2 dx′ dx3 +. + +ON THE LIMIT SPECTRUM OF A DEGENERATE OPERATOR +19 +For each ε, we choose V k +ε +⊂ Vh as the subspace spanned by +� +φ( x′ +ε )v1, φ( x′ +ε )v2, ..., φ( x′ +ε )vk� +with the same +v1, v2, ..., vk as those defined in the previous section, i.e., k normalized orthogonal eigenvectors associated to the +first k eigenvalues of − d2 +dx2 +3 +in H1 +0(0, L). +Hence, by construction, the functions of V k +ε vanish in Mε so that making the change of variable x′ := +εy + εi, y ∈ D in each cell, we can perform the same calculations as those of (2.6) to get for u ∈ V k +ε , +(3.2) +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +Ω +u2dx′ dx3 = +� +i∈Iε +� +εD+εi +φ2 +�x′ +ε +� +dx′ +� L +0 +� +α2 +1(v1)2 + ... + α2 +k(vk)2� +dx3 += +� +α2 +1 + ... + α2 +k +� +ε2 � +i∈Iε +� +D +φ2(y) dy, +� +Ω +ε2|∇′ +x′u|2dx′ dx3 = +� +α2 +1 + ... + α2 +k +� � +i∈Iε +ε2 +� +εD+εi +����∇′ +x′φ +�x′ +ε +����� +2 +dx′ = +� +α2 +1 + ... + α2 +k +� +ε4 � +i∈Iε +� +D +1 +ε2 |∇′ +yφ(y)|2dy = +� +α2 +1 + ... + α2 +k +� +ε2µ1 +� +i∈Iε +� +D +|φ(y)|2dy, +� +Ω +ε2 +���� +∂u +∂x3 +���� +2 +dx′ dx3 = ε2 +� L +0 +� +α2 +1 +�dv1 +dx3 +�2 ++ ... + α2 +k +�dvk +dx3 +�2� +dx3 ε2 � +i∈Iε +� +D +|φ(y)|2dy += ε4� +α2 +1λ0 +1 + ... + α2 +kλ0 +k +� � +i∈Iε +� +D +|φ|2 dy ≤ ε4λ0 +k +� +α2 +1 + ... + α2 +k +� � +i∈Iε +� +D +|φ|2dy, +in such a way the following estimate holds true +(3.3) +λk +ε ≤ +� +µ1 + ε2λ0 +k +�� +α2 +1 + ... + α2 +k +� +ε2 � +i∈Iε +� +D +|φ|2 dy +ε2� +α2 +1 + ... + α2 +k +� � +i∈Iε +� +D +φ2(y) dy += µ1 + ε2λ0 +k, +which is exactly the same estimate as that obtained in (2.4). +Remark 3.1. It is interesting to note in the proof of (3.3), we have chosen a test function verifying the same prop- +erties as those of the 3d − 1d case, namely: null in the matrix and with the regularity H1 +0(0, L) for almost all +x′. +Remark 3.1 is of general relevance since the other proofs in the homogenization setting are similar in all +points to the corresponding ones in the 3d − 1d problem, the main reason being that the vertical variable is not +concerned by the homogenization process which occurs only with respect to the horizontal variable x′ in such a +way basically, the local 3d − 1d effect is repeated periodically in the horizontal plane. Hence all the proofs take +up exactly the 3d-1d case while sticking to two principles: Dirichlet condition on x3 = 0 or x3 = L both for the +3d − 1d problem and the homogenization problem and when x3 plays the role of parameter as it is the case for +example in equation (2.25), it is x that will play the role of parameter in the homogenization problem. Indeed for +instance, the natural formulation of equation (2.24) in the homogenization setting is the following one +(3.4) +� +� +� +� +� +� +� +zk +ε ∈ Vh, +−∆′ +x′zk +ε − ε2 ∂2zk +ε +∂x2 +3 += λk +εzk +ε + λk +εU k +ε +in Fε, +zk +ε = 0 +on ∂Di +ε × (0, L), + +20 +KAÏS AMMARI AND ALI SILI +in such a way passing to the two-scale limit in (3.4), we get the equivalent of (2.25) +(3.5) +� +� +� +zk ∈ L2(Ω; H1 +#(C)), +−∆′ +yzk = λkzk + λkUk +in Ω × D, +zk = 0 +on Ω × ∂D. +The same approach may be applied to the other proofs following exactly the same steps and replacing the weak +(resp. strong) convergence in L2(Ω) by the two-scale (resp. strong two-scale) convergence. +REFERENCES +[1] G. ALLAIRE, Homogenization and Two-Scale Convergence, SIAM J. Math Anal. 23 (1992), 6, 1482-1518, +[2] G. ALLAIRE & Y. CAPDEBOSC, Homogenization of a spectral problem in neutronic multigroup diffusion, Comput. Methods Appl. Mech. +Engrg. 187 (2000), 1-2, 91-117, +[3] T. ARBOGAST, J. DOUGLAS & U. 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YOSIFIAN, Springer-Verlag. +LR ANALYSIS AND CONTROL OF PDES, LR 22ES03, DEPARTMENT OF MATHEMATICS, FACULTY OF SCIENCES OF MONASTIR, +UNIVERSITY OF MONASTIR, TUNISIA +Email address: kais.ammari@fsm.rnu.tn +INSTITUT DE MATHÉMATIQUES DE MARSEILLE (I2M), UMR 7373, AIX-MARSEILLE UNIVERSITÉ, CNRS, CMI, 39 RUE F. +JOLIOT-CURIE, 13453 MARSEILLE CEDEX 13, FRANCE +Email address: ali.sili@univ-amu.fr + diff --git a/C9E2T4oBgHgl3EQf9QlY/content/tmp_files/load_file.txt b/C9E2T4oBgHgl3EQf9QlY/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..468df8fce3af759f78538112f19ffea66f4cdb50 --- /dev/null +++ b/C9E2T4oBgHgl3EQf9QlY/content/tmp_files/load_file.txt @@ -0,0 +1,904 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf,len=903 +page_content='ON THE LIMIT SPECTRUM OF A DEGENERATE OPERATOR IN THE FRAMEWORK OF PERIODIC HOMOGENIZATION OR SINGULAR PERTURBATION PROBLEMS KAÏS AMMARI AND ALI SILI ABSTRACT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' In this paper we perform the analysis of the spectrum of a degenerate operator Aε corresponding to the stationary heat equation in a ε-periodic composite medium having two components with high contrast diffusivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' We prove that although Aε is a bounded self-adjoint operator with compact resolvent, the limits of its eigenvalues when the size ε of the medium tends to zero, make up a part of the spectrum of a unbounded operator A0, namely the eigenvalues of A0 located on the left of the first eigenvalue of the bi-dimensional Laplacian with homogeneous Dirichlet condition on the boundary of the representative cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' We also show that the homogenized problem does not differ in any way from the one-dimensional problem obtained in the study of the local reduction of dimension induced by the homogenization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' CONTENTS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Introduction, setting of the problem and statement of the results 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Proof of the results in the case of a single thin structure: the reduction of dimension 3d − 1d 8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Apriori estimate on the sequence of eigenvalues and eigenvectors 8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' The limit problem associated to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='9) 9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' The strong convergence of the eigenvectors 11 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='3 14 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='5 18 References 20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' INTRODUCTION, SETTING OF THE PROBLEM AND STATEMENT OF THE RESULTS The purpose of the present work is the asymptotic analysis of the eigenelements of a spectral problem in the framework of the homogenization of a periodic composite medium made up of a ε-periodic set of parallel vertical fibers Fε surrounded by a matrix Mε having better properties;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' more precisely, we consider the following problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='1) � � � � � � � � � � � � � � � Aεuε = λεuε in Ω, where Aεu = −ε2∆uχFε − ∆uχMε ∀ u ∈ D(Aε), with D(Aε) = � u ∈ Vh, Aεu ∈ L2(Ω), ∂u ∂nχ∂Fε = − 1 ε2 ∂u ∂nχ∂Mε � , with the following notations: ∆ denotes the classical Laplacian operator, Ω denotes a bounded rectangular open set of R3 of the form Ω := ω × (0, L), ω being a domain of R2 and L is a positive number, ∂u ∂nχ∂Mε (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' ∂u ∂nχ∂Fε) denotes the outer normal to the lateral boundary of Mε (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Fε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' The space Vh (h stands for homogenization) is defined by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='2) Vh := � u ∈ H1(Ω), u(x′, 0) = u(x′, L) = 0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' x′ = (x1, x2) ∈ ω � , 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' 35B25;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' 35B27;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' 35B40;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' 35B45;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' 35J25;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' 35J57;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' 35J70;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' 35P20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Spectrum, Degenerate, High contrast, Homogenization, Singular perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='04226v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='AP] 10 Jan 2023 2 KAÏS AMMARI AND ALI SILI hence, Vh is the subspace of functions in H1(Ω) which vanish on the lower and the upper faces of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' In the sequel, the two horizontal variables x′ := (x1, x2) or y := (y1, y2) will play a different role from that of the vertical variable x3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' The gradient and the Laplacian with respect to the horizontal variables will be denoted respectively by ∇′ and ∆′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' We assume that Ω is the reference configuration of a composite medium whose two components are a set Fε of vertical cylindrical fibers and its complement, the matrix Mε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Hence, the projection on the horizontal x′-plane of the set Fε is made up of a ε-periodic set of disks while the complement of such set represents the projection of Mε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' The characteristic functions of Fε (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Mε) are denoted by χFε (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' χMε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' The fibers are distributed in Ω with a period of size ε and the ratio between the conductivity coefficients of the two components is 1 ε2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Throughout the paper, for a measurable set B we denote by |B| its Lebesgue measure and by χB its characteristic function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' A generic positive constant the value of which may change from a line to another will be denoted by K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Let C be a square of R2 and let D be a disk strictly contained in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' The complement of D in C will be denoted by M ′ in such a way that C = M ′ ∪ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' The geometry of the domain is described as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='3) � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � Ci ε = (εC + εi) × (0, L);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' ω = � i∈Iε (εC + εi);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Ω = � i∈Iε Y i ε = ω × (0, L), Fε = � i∈Iε F i ε, F i ε = Di ε × (0, L) = (εD + εi) × (0, L), Mε = � i∈Iε M i ε, M i ε = M ′i ε × (0, L) = (Ci ε \\ D i ε) × (0, L), Iε = {i ∈ Z2, Ci ε ⊂ Ω}, Ω = Fε � Mε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' FIGURE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' The composite structure after dilation which is also the reference cell in the homoge- nization setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' In Figure 1 we have represented the representative cell C = D ∪ M ′ which represents also the composite structure after dilation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' When dealing with the homogenization of a problem posed on a domain Ω with a geometry given by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='3), a reduction of dimension 3d − 1d appears locally in each cell;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' it is then natural to study separately such a reduction of dimension problem which can be seen as a special case of the homogenization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' The geometry of the ON THE LIMIT SPECTRUM OF A DEGENERATE OPERATOR 3 reduction of dimension 3d − 1d problem is the following: the composite medium consists of a single fiber Fε := (εD)×(0, L) surrounded by the matrix Mε = εM ′ ×(0, L) = � ε(C \\D) � ×(0, L) in such a way the global domain depends now on the small parameter ε;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' it is defined by Ωε := (εC) × (0, L) = Fε ∪ Mε and it may be viewed as the configuration of a thin structure with the characteristic parameter ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' In this setting, the spectral problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='1) takes the following form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='4) � � � � � � � � � � � � � � � Aεvε = λεvε in Ωε, where Aεv = −ε2∆vχFε − ∆vχMε ∀ v ∈ D(Aε), with D(Aε) = � v ∈ V ε s , Aεv ∈ L2(Ωε), ∂v ∂nχ∂(εD) = − 1 ε2 ∂v ∂nχ∂(εM ′) � , where the space V ε s (the subscript "s" stands for singular perturbation) is now defined by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='5) V ε s := � v ∈ H1(Ωε), v(x′, 0) = v(x′, L) = 0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' x′ = (x1, x2) ∈ εC � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' In order to deal with a problem posed on the fixed domain Ω := C ×(0, L), we introduce the classical scaling uε(y′, x3) = vε(εy′, x3), y′ ∈ C which implies (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='6) ∇′ yuε(y, x3) = ε∇′vε(εy, x3) = ε∇′vε(x′, x3), ∀ (x′, x3) ∈ (εC) × (0, L), (this approach is of course not applicable in the homogenization setting in which we have to deal with 1 ε2 such thin structures).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' This change of variables transforms the problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='4) into the following singular perturbation problem, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='7) � � � � � � � � � � � � � � � � � � � � � � � � � � � Aεu = λεu in Ω, where Aεu = � −∆′u − ε2 ∂2u ∂x2 3 � χD + � − 1 ε2 ∆′u − ∂2u ∂x2 3 � χM ′, ∀ u ∈ D(Aε), with D(Aε) = � u ∈ Vs, Aεu ∈ L2(Ω), ∂u ∂nχ∂D = − 1 ε2 ∂u ∂nχ∂M ′ � , Vs being the space V ε s corresponding to ε = 1 and defined in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Note that the study of the asymptotic behavior of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='4) is the so-called reduction of dimension problem 3d − 1d since when ε goes to zero the three dimensional domain Ωε = (εC) × (0, L) looks like the segment (0, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Remarkably, it appears that the homogenized problem is very similar to the limit problem describing the one- dimensional model in the local 3d−1d reduction of dimension as explained in [23] (see also [19, 22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' This similarity is essentially due to the absence of oscillations in the vertical direction, whereas oscillations in the horizontal plane induce a local reduction of dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' We take advantage of that remark to limit ourselves to the complete study of the 3d − 1d problem which is technically simpler than the homogenization problem and we will only state the results within the framework of homogenization by referring to [23] for an adaptation of the proofs to the homogenization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Homogenization of a medium with high contrast between its components leads in general to a limit model described by an equation with significant differences compared with the equation of the media at the scale ε, see [3], [4], [7], [12], [9], [19], [22], [24], [25], [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Other settings have been studied in [2], [13], [14], [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Of course, this rule also fits for spectral problems, see for instance [28], [17], [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' To describe the behavior of the eigenvalues of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='7) ( 3d − 1d problem) or (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='1) (homogenization) we use the variational formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Note that for a fixed ε, Aε defined either by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='1) or by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='7) is a bounded selfadjoint operator with compact resolvent so that one can state the following well known result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' 4 KAÏS AMMARI AND ALI SILI Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='1) (or problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='7)) admits a sequence of eigenvalues (λk ε)k, 0 < λ1 ε ≤ λ2 ε ≤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' ≤ λn ε ≤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=', with lim k→∞ λk ε = +∞ while the associate eigenvectors (uk ε)k may be chosen as an orthonormal basis of L2(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Taking into account this result, the variational formulation of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='1) and of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='7) are respectively the following ones (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='8) � � � � � � � � � � � � � uk ε ∈ Vh, � Ω (ε2∇uk ε∇φχFε + ∇uk ε∇φχMε � dx = λk ε � Ω uk εφ dx, ∀ φ ∈ Vh, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='9) � � � � � � � � � � � � � � � � � � � uk ε ∈ Vs, � Ω �� ∇′uk ε∇′φ + ε2 ∂uk ε ∂x3 ∂φ ∂x3 � χF + � 1 ε2 ∇′uk ε∇′φ + ∂uk ε ∂x3 ∂φ ∂x3 � χM � dy dx3 = λk ε � Ω uk εφ dy dx3, ∀ φ ∈ Vs, where F := D × (0, L) and M := (C \\ D) × (0, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' We prove in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='3 below that for each k, the limit λk of the sequence of eigenvalues (λk ε)ε of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='7) ( 3d − 1d problem) is either equal to the first eigenvalue µ1 of the bidimensional Laplacian in the disk D with homogeneous Dirichlet boundary condition or is on the left of µ1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' furthermore, if λk fulfills 0 < λk < µ1, then s(λk) := λk � 1 + |D| |C \\ D| + λk |C \\ D| � D uk 0 dy � is an eigenvalue of − d2 dx2 3 in (0, L) with homogeneous Dirichlet boundary condition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' more precisely, λk is a solution of the following system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='10) � � � � � � � � � � � � � � � uk 0(y) ∈ H1(C)), −∆′ yuk 0 = λkuk 0 + 1 in D, uk 0 = 0 on ∂D, vk ∈ H1 0(0, L)), −dvk dx2 3 = λk � 1 + |D| |C \\ D| + λk |C \\ D| � D uk 0 dy � vk in (0, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Similar results (Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='5) are obtained for the homogenisation problem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' the limit λk of the sequence of eigenvalues (λk ε)ε of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='1) is either equal to the first eigenvalue µ1 of the bidimensional Laplacian in the disk D with homogeneous Dirichlet boundary condition or is on the left of µ1 and satisfies the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='11) � � � � � � � � � � � � � � � � � � � � � � � uk 0 ∈ H1 #(C), −∆′ yuk 0 = λkuk 0 + 1 in D, uk 0 = 0 on ∂D, vk ∈ L2(ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' H1 0(0, L)), −∂2vk ∂x2 3 = λk � 1 + |D| |C \\ D| + λk |C \\ D| � D uk 0 dy � vk in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Let us notice the very close analogy between the two limit problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' The first equation of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='11) is exactly the first one in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='10) (the boundary condition uk 0 = 0 on ∂D allows to consider uk 0 as an element of H1 #(C) in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='11)) so that the only difference between (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='11) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='10) lies in vk arising in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='11) is a function depending also ON THE LIMIT SPECTRUM OF A DEGENERATE OPERATOR 5 on the variable x′ ∈ ω while in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='10) it depends only on the vertical variable x3 ∈ (0, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' The dependence of vk with respect to x′ in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='11) is natural and it simply means that the homogenized problem is a duplication through ω of the phenomenon occurring in each cell of the horizontal plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' On the other hand, the limit system is nonlocal:the main vibrations at the limit are that of the matrix (the stiff part of the medium) in which the reduction of dimension occurs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' however the vibrations in the fibers must also be taken into account at the limit through the term � D uk 0 dy given by the first equation of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' The last term can be seen as a memory term intended to highlight the contribution of the soft part of the medium (here the fiber) to the limit vibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' This situation is in contrast with the one usually occurring with uniformly bounded operators with respect to the small parameter leading to limit problems of the same nature as the original ones, see for instance [13], [14], [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Note also that the existence and the uniqueness of uk 0 in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='10) is ensured by the fact that λk belongs to the resolvent ρ(−∆′ y) of −∆′ y since λk < µ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' The fact that � D uk 0 dy ̸= 0 will be proved in section 2, see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='17), using the constant 1 µ1 in the Poincaré inequality (in fact such value is the best constant for the Poincaré inequality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' It is natural to ask what is the relationship between the problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='10) (or the problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='11)) and the classical formulation of eigenvalue problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' In fact, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='10) is derived from the system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='10) which in turn is derived from the equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='9) satisfied by the pair (uk, vk), see the details of the proof in section 2 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' If one integrates the first equation of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='10) over D, we get an equivalent formulation of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='10) as follows (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='12) � � � � � � � � � � � � � � � uk(y, x3) ∈ L2((0, L);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' H1(C)), −∆′ yuk(y, x3) = λkuk in D × (0, L), uk = vk on ∂D × (0, L), vk ∈ H1 0(0, L), −d2vk dx2 3 + 1 |C \\ D| � ∂D ∂uk ∂n dσ = λkvk in (0, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Another equivalent formulation of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='12) is the following (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='13) A0 �uk vk � = λk �uk vk � where the operator A0 is defined by A0 : D(A0) → H := L2(Ω) × L2(0, L) with (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='14) � � � � � � � � � � � � � � � D(A0) = ��u v � ∈ L2(0, L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' H1(D) × H1 0(0, L);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' A0 �u v � ∈ H, u = v on ∂D � , A0 �u v � = � � −∆′ yu −d2v dx2 3 + 1 |C \\ D| � ∂D ∂u ∂n dσ � � , ∀ �u v � ∈ D(A0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' We see from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='13) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='14) the sharp difference between the bounded selfadjoint operator Aε and the limit operator A0 which is no more a bounded selfadjoint operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Of course, the same remark may be made about the homogenized problem given by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' From the technical point of view the main difficulty in the asymptotic analysis comes from the lack of com- pactness since we have to consider sequences of eigenvectors not bounded in H1(Ω) so that the strong convergence in L2(Ω) (or strong two-scale convergence in the case of homogenization) which allows to conclude that the limit of an eigenvector uk ε is still an eigenvector (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' ̸= 0) is not straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' To overcome this difficulty, we will use an extension technique (see [10], [29]) combined with another slightly more intricate argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' From now on and based on the previous comments, we will focus on the asymptotic analysis of the singular perturbation problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='9) (the study of the reduction of dimension occurring in each cell).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' This kind of problems is usually encountered in the study of thin structures, see for instance [16] and [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' 6 KAÏS AMMARI AND ALI SILI Our main results may be stated as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' For each k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=', the sequence of eigenvalues (λk ε)ε of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='9) is bounded above by the first eigen- value µ1 of −∆′ in H1 0(D) and the associated sequence of eigenvectors (uk ε)ε is bounded in L2(0, L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' H1(C));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' if for a subsequence of ε, λk ε → λk with λk ̸= µ1, then there exists a solution (λk, uk 0, vk) ∈ (0, µ1[×L2(0, L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' H1(C)) × H1 0(0, L) of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='10) with vk ̸= 0 such that for the whole sequence ε, one has (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='15) λk ε → λk, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='16) uk ε −→ uk(y, x3) := (λkuk 0 + 1)vk strongly in L2(0, L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' H1(C)), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='17) uk εχM −→ vkχM strongly in L2(C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' H1 0(0, L)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Any λk such that 0 < λk < µ1 is a simple eigenvalue of the limit operator A0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Conversely, problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='10) admits non trivial solutions such that 0 < λk < µ1 and any λ ∈ (0, µ1[ which is an eigenvalue of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='10) is a limit of a sequence (λk ε)ε of eigenvalues of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' The unique accumulation point of the sequence (λk)k is the first eigenvalue µ1 of −∆′ y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' hence lim k→+∞ λk = µ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' The property vk ̸= 0 may be deduced from the strong convergence (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='16) of the eigenvectors but we prefer to write it explicitly to highlight the fact that vk is always an eigenvector of − d2 dx2 3 with Dirichlet condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Regarding the homogenization problem, the result is in all respects similar to that of 3d − 1d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' We state it through the following theorem which is the homogenized version of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' To state the results, we need the use of the two-scale convergence, see [1], [20], [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' We use the notation 2−sc ⇀ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' 2−sc −→) for the two-scale convergence (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' the strong two-scale convergence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' For each k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=', the sequence of eigenvalues (λk ε)ε of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='8) is bounded above by the first eigen- value µ1 of −∆′ in H1 0(D) and the associated sequence of eigenvectors (uk ε)ε is bounded in L2(0, L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' H1(ω));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' if for a subsequence of ε, λk ε → λk with λk ̸= µ1, then there exists a solution (λk, uk 0, vk) ∈ (µ0, µ1[×L2(0, L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' H1 #(C))× L2(ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' H1 0(0, L)) of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='11) with vk ̸= 0 such that for the whole sequence ε, one has (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='18) λk ε → λk, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='19) uk ε 2−sc −→ uk(x, y) := (λkuk 0 + 1)vk, with the following corrector result (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='20) � Ω ������ε∇′uk ε − ∇′ yuk � x, x′ ε ����� 2 + ε2 ���� ∂uk ε ∂x3 ���� 2� χFε(x′) + � ��∇′uk ε ��2 + ���� ∂uk ε ∂x3 − ∂vk ∂x3 ���� 2� χMε(x′) � dx → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Any λk such that 0 < λk < µ1 is a simple eigenvalue of the limit operator A0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Conversely, any eigenvalue λ ∈ (0, µ1[ of problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='11) is a limit of a sequence (λk ε)ε of eigenvalues of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' The sequence (λk)k converges to µ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Note that the structure of the limit spectrum is quite complicated because not only the mean value � D uk 0 dy arising in the second equation of the limit system must be calculated by the use of the first equation of the system but the function uk 0 itself depends on the corresponding eigenvalue as shown by the first equation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' hence, λk � 1 + |D| |C \\ D| + λk |C \\ D| � D uk 0 dy � which is an eigenvalue of − d2 dx2 3 is not completely known in terms of λk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' ON THE LIMIT SPECTRUM OF A DEGENERATE OPERATOR 7 However, we will prove (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='41)) that for 0 < λk < µ1, the second equation describing the vibrations of the string (0, L) may be written as (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='21) − d2vk dx2 3 = δ(λk)vk with δ(λ) := Cλ + C′ ∞ � n=1 c2 nλ2 µn − λ, where C, C′ denote positive constants and cn := � D fndy where (fn)n denotes the orthonormal basis in L2(D) made up of the eigenfunctions associated to the increasing sequence (µn)n of eigenvalues of −∆′ y with Dirichlet boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Of course, the spectrum σ0 of the limit operator A0 contains eigenvalues on the right of µ1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' in particular, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='21) shows that any eigenvalue µn of −∆′ such that cn = � D fndy ̸= 0 is an accumulation point of σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Our result states that the limits λk make up a part of the spectrum σ0 of A0, namely the values of σ0 located on the left of µ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Remark also that in the homogenization setting, the analogous result of the convergence (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='17) is the conver- gence � Ω ���� ∂uk ε ∂x3 − ∂vk ∂x3 ���� 2 χMε(x′) dx → 0 obtained from the corrector result (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' However, the latter does not mean that the sequence uk εχMε converges strongly in L2(ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' H1 0(0, L)) to |C\\D| |C| vk = |C \\ D|vk (we have assumed |C| = 1) in which case this convergence would be the exact analogue of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Unfortunately, because of the oscillations induced by the homogenization process, such exact analogue of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='17) is false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' This is one of the few differences between the 3d − 1d problem and the homogenization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Finally we point out other possible scalings of the form εγχFε + εδχMε as addressed in [11], [12], [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' For instance in the static case, one can refer to [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' The critical case giving rise to a coupled system at the limit is the one corresponding to lim εδ−2 = l ∈]0, +∞[ which we consider here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' In order to highlight the close analogy between the 3d − 1d limit problem and the homogenized problem, the macroscopic variable x will be denoted by x = (y, x3), y ∈ C in the study of the 3d − 1d problem for which Ω := C × (0, L) while in the homogenization problem x will be denoted by x = (x′, x3), x′ ∈ ω := � i∈Iε (εC + εi) since Ω := � i∈Iε (εC + εi) × (0, L) so that each x′ ∈ ω may be written as x′ = εy + εi, i ∈ Iε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' In the case of a single thin structure Ωε = (εC) × (0, L), Ω := C × (0, L) is obtained from Ωε by the scaling x′ = εy, y ∈ C, thus making our notations homogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Before proceeding to prove the results in the next sections, it should be pointed out that the study can be extended to the case of operators in divergence form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' In that case, we have to take into account at the limit the contribution of the anisotropy of the heavy part of the material (here the matrix) as shown in [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' On the other hand, one can consider other scalings of the form εγχFε + εδχMε as addressed in [11], [12], [25] in the static case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' For instance in the static case and under convenient assumptions on the source term, one can consider coefficients of order εδ in the fiber Fε and 1 in Mε, then loosely speaking the structure of the limit problem depends on the limit of the ratio εδ−2, the critical case giving rise to a coupled system at the limit is the one corresponding to lim εδ−2 = l ∈]0, +∞[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Here we address the critical case in the framework of the Laplacian operator for the sake of simplicity and brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' In order to highlight the close analogy between the 3d − 1d limit problem and the homogenized problem, the macroscopic variable x will be denoted by x = (y, x3), y ∈ C in the study of the 3d − 1d problem for which Ω := C × (0, L) while in the homogenization problem x will be denoted by x = (x′, x3), x′ ∈ ω := � i∈Iε (εC + εi) since Ω := � i∈Iε (εC + εi) × (0, L) so that each x′ ∈ ω may be written as x′ = εy + εi, i ∈ Iε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' In the case of a 8 KAÏS AMMARI AND ALI SILI single thin structure Ωε = (εC) × (0, L), Ω := C × (0, L) is obtained from Ωε by the scaling x′ = εy, y ∈ C, thus making our notations homogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' In the following we study in detail the dimension reduction problem and then indicate briefly the few technical changes needed in the proofs of the result in the framework of homogenization, see also [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' PROOF OF THE RESULTS IN THE CASE OF A SINGLE THIN STRUCTURE: THE REDUCTION OF DIMENSION 3d − 1d 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Apriori estimate on the sequence of eigenvalues and eigenvectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' For each k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=', the sequence (λk ε, uk ε) of eigenpairs of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='9) is bounded in R×L2(0, L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' H1(C)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' There exist (λk, uk, vk) ∈ (0, µ1) × L2(0, L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' H1(C)) × H1 0(0, L) and a subsequence of ε still denoted by ε such that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='1) uk ε ⇀ uk weakly in L2(0, L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' H1(C)) and uk(y, x3) = vk(x3) in M = (C \\ D) × (0, L), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='2) ∂uk ε ∂x3 χM ⇀ dvk dx3 χM weakly in L2(Ω), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='3) λk ε → λk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' We first prove an apriori estimate on the sequence of eigenvalues which will play a key role in the sequel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Let λ0 k be the k-th eigenvalue of − d2 dx2 3 in (0, L) with homogeneous Dirichlet boundary conditions and let µ1 be the first eigenvalue of −∆′ y in D with homogeneous Dirichlet boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' We claim that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='4) ∀ ε, ∀ k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=', λk ε ≤ µ1 + ε2λ0 k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Indeed, we use the well known min-max formula giving the k-th eigenvalue λk ε of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='9), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='5) λk ε = min V k⊂Vs max u∈V k � Ω �� ��∇′ yu ��2 + ε2 ���� ∂u ∂x3 ���� 2� χF + � 1 ε2 ��∇′ yu ��2 + ���� ∂u ∂x3 ���� 2� χM � dy dx3 � Ω |u|2 dy dx3 , where the space Vs is defined by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='5) (with ε = 1) and the min runs over all subspaces V k of Vs with finite dimension k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Let φ(y) be an eigenvector associated to µ1 extended by zero in C \\ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Then φ(y)ψ(x3) belongs to Vs for any ψ ∈ H1 0(0, L) and φψ = 0 in M := (C \\ D) × (0, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Let V k be the subspace of Vs spanned by � φv1, φv2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=', φvk� where v1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=', vk denote the associated eigenvectors to the first k eigenvalues λ0 1, λ0 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=', λ0 k of − d2 dx2 3 with homogeneous Dirichlet boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' ON THE LIMIT SPECTRUM OF A DEGENERATE OPERATOR 9 For any u = α1φv1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' + αkφvk ∈ V k, we have u = 0 in M and since v1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=', vk, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=', is an orthonormal basis in H1 0(0, L) we also have (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='6) � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � Ω u2dy dx3 = � D φ2 dy � L 0 � α2 1(v1)2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' + α2 k(vk)2� dx3 = � α2 1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' + α2 k � � D φ2 dy, � Ω |∇′ yu|2dy dx3 = � α2 1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' + α2 k � � D |∇′ yφ|2 dy = � α2 1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' + α2 k � µ1 � D |φ|2 dy, � Ω ε2 ���� ∂u ∂x3 ���� 2 dy dx3 = ε2 � L 0 � α2 1 ���� dv1 dx3 ���� 2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' + α2 k ���� dvk dx3 ���� 2� dx3 � D |φ|2 dy, = ε2� α2 1λ0 1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' + α2 kλ0 k � � D |φ|2 dy ≤ ε2λ0 k � α2 1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' + α2 k � � D |φ|2 dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Note that the equality occurring in the fifth line of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='6) is a consequence of the equation −∆′ yφ = µ1φ in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Hence, using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='6) in the min-max formula above, we get estimate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' We obtain that λk ∈ (0, µ1) by passing to the limit (for a subsequence of ε) in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' We will prove later that the value µ1 cannot be attained by λk for all k and that the whole sequence (λk)k converges to µ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Turning back to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='9) and taking uk ε (with ∥ uk ε ∥L2(Ω)= 1) as a test function, we get (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='7) � Ω �� ��∇′uk ε ��2 + ε2 ���� ∂uk ε ∂x3 ���� 2� χF + � 1 ε2 ��∇′uk ε ��2 + ���� ∂uk ε ∂x3 ���� 2� χM � dy dx3 = λk ε ≤ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' The last estimate implies that ∇′uk ε is bounded in L2(Ω) and thus uk ε is bounded in L2(0, L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' H1(C)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Hence, there exist a sequence of ε and uk ∈ L2(C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' H1 0(0, L)) such that the convergence (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='1) holds true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' One has ∇′uk εχM(y) ⇀ ∇′ukχM weakly in L2(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' But ∇′uk εχM which is bounded in L2(Ω) by Cε strongly converges to zero in L2(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Hence, ∇′ukχM = 0 which means that uk = vk(x3) for some vk ∈ L2(0, L) a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' The sequence uk εχM(y) (note that the characteristic functions χF and χM depend only on the horizontal variable y) is bounded in L2(C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' H1 0(0, L)) since ∂uk ε ∂x3 χM is bounded in L2(Ω) so that for a subsequence ∂uk ε ∂x3 χM ⇀ ∂uk ∂x3 χM = dvk dx3 χM weakly in L2(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Hence vk ∈ H1 0(0, L) and the convergence (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='2) holds true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' The proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='1 is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' The limit problem associated to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Choosing a test function in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='9) in the form φ = ¯u with ¯u = ¯v(x3) in M and (¯u, ¯v) ∈ Vs × H1 0(0, L), we get from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='9) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='8) � Ω �� ∇′uk ε∇′¯u + ε2 ∂uk ε ∂x3 ∂¯u ∂x3 � χF + ∂uk ε ∂x3 d¯v dx3 χM � dy dx3 = λk ε � Ω uk ε ¯u dy dx3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' 10 KAÏS AMMARI AND ALI SILI Passing to the limit in this equation, we get with the help of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='1) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='9) � � � � � � � � � � � � � (uk, vk) ∈ L2(0, L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' H1(C)) × H1 0(0, L), uk = vk in M, � Ω � ∇′uk∇′¯uχF + dvk dx3 d¯v dx3 χM � dy dx3 = λk � Ω uk¯u dy dx3, ∀ (¯u, ¯v) ∈ Vs × H1 0(0, L), ¯u = ¯v in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Finally a density argument allows to extend (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='9) to all test functions ¯u ∈ L2(0, L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' H1(C)) such that ¯u = ¯v in M and ¯v ∈ H1 0(0, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Choosing successively in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='9) ¯u ∈ L2(0, L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' H1(C)) such that ¯u = 0 in M and then ¯u ∈ L2(0, L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' H1(C)) such that ¯u = ¯v ∈ H1 0(0, L) almost everywhere in Ω and bearing in mind the geometry of Ω := C × (0, L) = � (C \\ D) ∪ D � × (0, L), we get that the limit problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='8) may be split into two equations leading to the following equivalent system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='10) � � � � � � � � � � � � � � � uk(y, x3) ∈ L2((0, L);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' H1(C)), −∆′ yuk(y, x3) = λkuk in D × (0, L), uk = vk on ∂D × (0, L), vk ∈ H1 0(0, L)), −d2vk dx2 3 = λkvk + λk |C \\ D| � D uk dy in (0, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Eigenvectors of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='10) corresponding to eigenvalues λk < µ1 are pairs (uk, vk) made up of two inseparable elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' In particular, if vk = 0 then uk = 0 as shown by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Indeed, otherwise uk should be an eigenvector of −∆′ y associated to the eigenvalue λk < µ1 which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Conversely if uk = 0 then vk = 0 since almost everywhere in (0, L), we have vk = uk on the boundary of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Hence, the eigenvectors (uk, vk) of the limit operator are such that uk ̸= 0 and vk ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' We now prove that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='10) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='10) are equivalent if one defines uk by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='11) and then we will improve the lower bound of the limit eigenvalues using (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' If (λk, uk, vk) solves the system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='10) with 0 < λk < µ1, then vk ̸= 0 and uk writes as (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='11) uk(y, x3) = (λkuk 0(y) + 1)vk(x3) where (λk, uk 0, vk) solves (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Furthermore, there exists a positive constant µ0 depending both on µ1 and on the first eigenvalue of − d2 dx2 3 in H1 0(0, L) such that λk ≥ µ0 for all k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Assume that (uk, vk) is a non trivial solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='10), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='e, (uk, vk) is an eigenvector of the limit operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Then according to the Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='2 above, vk ̸= 0 and uk ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Dividing by vk in the first system of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='10), one can check easily that wk := uk vk − 1 is the unique solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='12) � � � −∆′ ywk = λkwk + λk in D wk = 0 on ∂D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Note that the uniqueness of wk is ensured since λk < µ1 belongs to the resolvent of −∆′ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' On the other hand, the function λkuk 0 where uk 0 is defined in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='10) is also a solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='12) so that the equality wk := uk vk − 1 = λkuk 0 holds true and therefore (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='11) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='11) in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='10) we get (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' We now make more precise the lower bound of the sequence of eigenvalues and we prove at the meanwhile that � D uk 0(y) dy > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' ON THE LIMIT SPECTRUM OF A DEGENERATE OPERATOR 11 Multiplying the first equation of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='10) by uk 0 and using 1 µ1 as the constant (it is in fact the best one) in the Poincaré’s inequality, we get (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='13) � D uk 0(y) dy = � D |∇′ yuk 0(y)|2 dy − λk � D |uk 0(y)|2 dy ≥ � 1 − λk µ1 � � D |∇′ yuk 0(y)|2 dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' On the other hand, the first eigenvalue µ1 is characterized by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='14) µ1 = inf u∈H1 0(D) ∥ ∇′ yu ∥2 L2(D) ∥ u ∥2 L2(D) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Hence the following estimate holds true (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='15) � D |∇′ yuk 0(y)|2 dy ≥ µ1 � D |uk 0(y)|2 dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' From (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='13), we derive with the help of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='15) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='16) (µ1 − λk) � D |uk 0(y)|2 dy ≤ � D uk 0(y) dy ≤ � |D| �� D |uk 0(y)|2 dy � 1 2 , and then from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='16) we deduce (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='17) 0 < � D uk 0(y) dy ≤ |D| µ1 − λk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' By virtue of the last equation in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='10), ˆλk := λk � 1 + |D| |C \\ D| + λk |C \\ D| � D uk 0 dy � is an eigenvalue of − d2 dx2 3 so that ˆλk ≥ λ0 where λ0 denotes the first eigenvalue of − d2 dx2 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Using the second inequality of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='17) we get (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='18) λk � 1 + |D| |C \\ D| + λk |D| |C \\ D|(µ1 − λk) � ≥ ˆλk ≥ λ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Hence, λk ≥ µ0 := φ−1(λ0) where φ is the continuous increasing function defined on (0, µ1) by φ(t) = t � 1 + |D| |C \\ D| + t |D| |C \\ D|(µ1 − t) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' □ So far, we have not yet proved that (uk, vk) is indeed an eigenvector of the limit operator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' this is the purpose of the next subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' The strong convergence of the eigenvectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' We prove the following compactness result Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' For each k, there exists a subsequence of ε such that the sequence of solutions uk ε of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='9) converges strongly in L2(Ω) to the eigenvector uk of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' One can extend uk ε from M to the whole Ω in such a way the extension U k ε fulfills U k ε ∈ Vs, U k ε = uk ε in M and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='19) ∥ ∇′U k ε ∥L2(Ω)≤ K ∥ ∇′uk ε ∥L2(M), ���� ∂U k ε ∂x3 ���� L2(Ω) ≤ K ���� ∂uk ε ∂x3 ���� L2(M) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Note that the extension only affects the horizontal variable y so that the Dirichlet boundary condition on the upper and lower faces of Ω (x3 = 0 or x3 = L ) is preserved, see for instance [6], [10], [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' In addition, one can assume that such extension satisfies the following equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='20) � −∆′ yU k ε − ε2 ∂2U k ε ∂x2 3 = 0 in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' 12 KAÏS AMMARI AND ALI SILI Indeed, if (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='20) is not true for U k ε , then one can introduce the function W k ε as the unique solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='21) � � � � � � � � � � � � � � � W k ε ∈ V, � F � 1 ε2 ∇′ yW k ε ∇′ yφ + ∂W k ε ∂x3 ∂φ ∂x3 � dy dx3 = � F � 1 ε2 ∇′ yU k ε ∇′ yφ + ∂U k ε ∂x3 ∂φ ∂x3 � dy dx3 ∀ φ ∈ V, where V := � u ∈ Vs, u = 0 on ∂D × (0, L) � (recall that V := V ε s with ε = 1 where V ε s is defined by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='5)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Hence, V is the subspace of Vs of functions vanishing in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' By the Lax-Milgram Theorem we get existence and uniqueness for W k ε .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Choosing φ ∈ C∞ 0 (F), the last equation leads to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='22) − 1 ε2 ∆′ yW k ε − ∂2W k ε ∂x2 3 = − 1 ε2 ∆′ yU k ε − ∂2U k ε ∂x2 3 in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' On the other hand, using equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='21) with φ = W k ε , we get the following estimate with the help of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='19) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='7) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='23) � � � � � � � � � � � � � � � ���� 1 ε∇′W k ε ���� L2(F ) + ���� ∂W k ε ∂x3 ���� L2(F ) ≤ K ����� 1 ε∇′U k ε ���� L2(F ) + ���� ∂U k ε ∂x3 ���� L2(F ) � ≤ ≤ K ����� 1 ε∇′uk ε ���� L2(M) + ���� ∂uk ε ∂x3 ���� L2(M) � ≤ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Multiplying equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='22) by ε2, we see that ˜uk ε defined by ˜uk ε = U k ε − W k ε is indeed an extension which fulfills equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='20) and preserves the apriori estimate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Note that functions of V may be extended by zero inside M so that ˜uk ε is still an extension of uk ε from M to the whole Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' In the sequel, we will still denote the extension of uk ε satisfying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='19) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='20) by U k ε .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Consider now the sequence defined in Ω by zk ε = uk ε − U k ε .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' If we prove that zk ε admits a strongly converging subsequence in L2(Ω) then we can deduce the existence of such subsequence for uk ε since U k ε is bounded in H1(Ω) by virtue of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='19) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='7) and therefore admits a strongly converging subsequence in L2(Ω) according to the Rellich imbedding Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' We first derive the following equation on zk ε by the use of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='7) together with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='20) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='24) � � � � � � � zk ε ∈ Vs, −∆′ yzk ε − ε2 ∂2zk ε ∂x2 3 = λk εzk ε + λk εU k ε in F, zk ε = 0 on ∂D × (0, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Since uk ε and U k ε are bounded respectively in L2(0, L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' H1(C)) and H1(Ω), the sequence zk ε is bounded in L2(0, L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' H1(C)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Hence, there exist a subsequence and zk ∈ L2(0, L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' H1(C)) such that zk ε ⇀ zk weakly in L2(0, L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' H1(C)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Therefore, denoting by Uk the weak limit in H1(Ω) of the corresponding subsequence U k ε , one can pass easily to the limit in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='24) to get the equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='25) � � � zk ∈ L2(0, L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' H1(C)), −∆′ yzk = λkzk + λkUk in F, zk = 0 on ∂D × (0, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Note that by construction, zk ε = 0 in M = (C \\ D) × (0, L) so that the convergence zk ε χM(y) ⇀ zkχM(y) weakly in L2(Ω) shows that zk = 0 in M which is equivalently expressed by the boundary condition of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' ON THE LIMIT SPECTRUM OF A DEGENERATE OPERATOR 13 More generally, given a bounded sequence (fε) in L2(Ω) and f ∈ L2(Ω), we now consider equations of the form (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='26) � � � � � � � wε ∈ Vs, −∆′ ywε − ε2 ∂2wε ∂x2 3 = λk εwε + fε in F, wε = 0 on ∂D × (0, L), and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='27) � � � w ∈ L2(0, L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' H1(C)), −∆′ yw = λkw + f in F, w = 0 on ∂D × (0, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Regarding the sequence of solutions of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='26), the following lemma holds true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Assume that λk ε → λk with λk < µ1 and that fε ⇀ f weakly in L2(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Then the sequence wε is bounded in L2(0, L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' H1(C)) and for the whole sequence ε, wε ⇀ w weakly in L2(0, L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' H1(C)) where w is the unique solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' We only have to prove that wε is bounded in L2(0, L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' H1(C)), the limit problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='27) satisfied by w can be established exactly by the same process already used in the proof of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' The main ingredient to get that apriori estimate relies on the Poincaré inequality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='28) � D |u|2 dy ≤ 1 µ1 � D |∇′ yu|2 dy ∀ u ∈ H1 0(D), combined with the assumption λk < µ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Multiplying equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='26) by wε and integrating, we get (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='29) � L 0 � D |∇′wε|2 dydx3 ≤ λk ε � L 0 � D |wε|2 dydx3+ ∥ fε ∥L2(Ω)∥ wε ∥L2(F ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Choosing u = wε(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=', x3) with x3 ∈ (0, L) and integrating (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='28) over (0, L), we infer (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='30) � L 0 � D |wε|2 dydx3 ≤ 1 µ1 � L 0 � D |∇′ ywε|2 dydx3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Let δ > 0 be such that 0 < λk < δ < µ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Turning back to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='29) and using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='30), we get for ε sufficiently small, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='31) � 1 − δ µ1 � � L 0 � D |∇′wε|2 dydx3 ≤∥ fε ∥L2(Ω)∥ wε ∥L2(F ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Since fε is bounded in L2(Ω), applying once again inequality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='30), we derive from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='31) the estimate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='32) � L 0 � D |∇′wε|2 dydx3 ≤ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' The estimates (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='30) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='32) show that wε is bounded in L2(0, L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' H1(D)) and thus in L2(0, L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' H1(C)) since wε is equal to zero in C \\ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' □ We continue the proof of the Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='4 in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Multiplying equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='24) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='26) respectively by wε and by zk ε and integrating we get (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='33) � � � � � � � � � � � � F � ∇′zk ε ∇′wε + ε2 ∂zk ε ∂x3 ∂wε ∂x3 � dydx3 = λk ε � F zk ε wε dydx3 + λk ε � F U k ε wε dydx3 = λk ε � F wεzk ε dydx3 + � F fεzk ε dydx3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' 14 KAÏS AMMARI AND ALI SILI Since U k ε is bounded in H1(Ω), there exist a subsequence of ε and Uk ∈ H1(Ω) such that U k ε ⇀ Uk weakly in H1(Ω) and strongly in L2(Ω) by virtue of the Rellich imbedding Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Therefore for that a subsequence, we get from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='33) with the help of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='6 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='34) lim ε→0 � F fεzk ε dydx3 = lim ε→0 λk ε � F U k ε wε dydx3 = λk � F Ukw dydx3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' On the other hand, one can multiply (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='25) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='27) respectively by w and by zk and integrate to obtain (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='35) � � � � � � � � � � F ∇′zk∇′w dydx3 = � F ∇′w∇′zk dydx3 = λk � F zkw dydx3 + λk � F Ukw dydx3 = λk � F wzk dydx3 + � F fzk dydx3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Combining (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='34) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='35), we get (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='36) lim ε→0 � F fεzk ε dydx3 = λk � F Ukw dydx3 = � F fzk dydx3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Choosing in particular fε = zk ε which converges weakly in L2(Ω) to f = zk, we obtain (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='37) lim ε→0 � F (zk ε )2 dydx3 = � F (zk)2 dydx3, which implies the strong convergence of the subsequence zk ε and therefore the strong convergence of the corre- sponding subsequence of uk ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Hence Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='4 is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' □ We now proceed to complete the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' The strong convergence in L2(Ω) of the eigenvectors when λk < µ1 is proved in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' We use it to prove the convergence of the sequence of energies from which we obtain immediately (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='16) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Consider the sequence (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='38) Jε = � Ω �� ��∇′uk ε − ∇′uk ��2 + ε2 ���� ∂uk ε ∂x3 ���� 2� χF + � 1 ε2 ��∇′uk ε ��2 + ���� ∂uk ε ∂x3 − dvk dx3 ���� 2� χM � dydx3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Choosing uk ε and (uk, vk) as test functions respectively in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='9) and in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='9), we get with the help of the weak convergences proved in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='1 and of the strong convergence proved in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='4, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='39) � � � � � � � � � � � Jε = λk ε � Ω (|uk ε|2dydx3 + λk � Ω |uk|2dydx3 − 2 � Ω � ∇′uk ε∇′ukχF + ∂uk ε ∂x3 dvk dx3 χM � dydx3 −→ 2λk � Ω |uk|2dydx3 − 2λk � Ω |uk|2dydx3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Hence the weak convergences stated in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='1 are in fact strong convergences;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' in particular, keeping in mind Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='4, we get the strong convergences stated in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' We have proved above that λk is an eigenvalue of the limit problem (in the sense of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='13)) if and only if λk satisfies (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' In the sequel, a number λ satisfying (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='10) will be called an eigenvalue of the limit problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' We now prove that there exist non trivial solutions for the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='10) and that any λ ∈ (µ0, µ1) which satisfies (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='10) may be attained as a limit of a sequence (λk ε)ε;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' by this we can conclude that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='13) has no other eigenvalues on the left of µ1 than those obtained from the limits of the eigenvalues λk ε and thus we can list all its eigenvalues in increasing order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' It is then clear that for a fixed k, we cannot have two subsequences ε and ε′ with two different limits for λk ε and λk ε′ since this would lead to add a new element to the set of eigenvalues of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='10);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' hence for each k, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='15) holds for the whole sequence ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' ON THE LIMIT SPECTRUM OF A DEGENERATE OPERATOR 15 To prove the existence of non trivial solutions � uk 0 vk � for the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='10) with λk < µ1 leading to non trivial solutions �uk vk � for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='13)) where uk := (λkuk 0 + 1)vk, it is sufficient to show that one can find solutions � uk 0 vk � of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='10) with vk ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' uk 0 is uniquely determined by the first equation of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='10) since λk < µ1 and if (fn)n is the orthonormal basis in L2(D) made up of eigenfunctions associated to the increasing sequence (µn)n of eigenvalues of −∆′ y, one can get from the first equation of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='10) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='40) uk 0 = ∞ � n=1 cnfn µn − λk ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' where cn = � D fn dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Replacing the mean value of uk 0 in the second equation of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='10), we derive (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='41) − d2vk dx2 3 = δ(λk)vk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' with δ(λ) := Cλ + C′ ∞ � n=1 c2 nλ2 µn − λ, where C, C′ denote positive constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Let (γj, vj) be an eigenelement of − d2 dx2 3 in H1 0(0, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Since δ is a strictly positive increasing function over (0, µ1), there exists λkj ∈ (0, µ1) such that γj = δ(λkj), so that the second equation of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='10) may be written as −d2vj dx2 3 = δ(λkj)vj, taking vkj := vj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Hence for λk < µ1, the pair (uk 0, vkj) is a non trivial solution for any j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' We now argue by contradiction to prove that any λ ∈ (µ0, µ1[ which is an eigenvalue of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='10) may be attained as a limit of a sequence (λk ε)ε for some k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' If for any k and for any sequence ε, λk ε does not converge to λ, then there exists a neighborhood of λ which does not contain any λk ε for all k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' In other words, λ belongs to the resolvent of the operator Aε defined by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Hence, for any f ∈ L2(0, L) ⊂ L2(Ω), there exists uε ∈ D(Aε) such that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='42) Aεuε = λuε + f in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Multiplying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='42) by φ ∈ Vs and integrating we get (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='43) � � � � � � � � � � Ω �� ∇′uε∇′φ + ε2 ∂uε ∂x3 ∂φ ∂x3 � χF + � 1 ε2 ∇′uε∇′φ + ∂uε ∂x3 ∂φ ∂x3 � χM � dy dx3 = λ � Ω uεφ dy dx3 + � Ω fφ dy dx3, ∀ φ ∈ Vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' To get apriori estimates on the sequence uε, we will use the following Poincaré type inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' There exists a positive constant K such that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='44) � � � � � � � � � ∥u∥L2(Ω) ≤ K � ∥∇′u∥L2(Ω) + ���� ∂u ∂x3 χM ���� L2(Ω) � , ∀ u ∈ L2(0, L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' H1(C)) ∩ L2(C \\ D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' H1 0(0, L)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' We argue by contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Assuming inequality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='44) false, one can find a sequence un ∈ L2(0, L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' H1(C)) ∩ L2(C \\ D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' H1 0(0, L) 16 KAÏS AMMARI AND ALI SILI such that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='45) ∥un∥L2(Ω) = 1 ∀ n, and � ∥∇′un∥L2(Ω) + ���� ∂un ∂x3 χM ���� L2(Ω) � −→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Thanks to the classical Poincaré inequality �����u − 1 |C \\ D| � C\\D u dy ����� L2(C) ≤ K ∥∇′u∥L2(C) applied to u = un(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=', x3) ∈ H1(C), x3 ∈ (0, L), we get after integrating with respect to x3, (remember that Ω = C × (0, L)) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='46) �����un − 1 |C \\ D| � C\\D un dy ����� L2(Ω) ≤ K ∥∇′un∥L2(Ω) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' On the other hand, the one-dimensional Poincaré inequality for functions of H1 0(0, L) applied with u(x3) = � C\\D un(y, x3) dy ∈ H1 0(0, L) leads to the estimate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='47) ����� � C\\D un dy ����� L2(Ω) ≤ K ���� ∂un ∂x3 ���� L2(M) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Combining (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='46) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='47) with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='45), we come to a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' □ Taking φ = uε in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='43) and applying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='44) with u = uε (note that Vs ⊂ L2(0, L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' H1(C)) ∩ L2(C \\ D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' H1 0(0, L)), we get the same apriori estimates as those obtained for the sequence uk ε in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Indeed all the apriori estimates on the sequence uk ε are based on its L2(Ω)- apriori estimate which still holds true for the sequence uε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Hence by the same arguments that led to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='10) one can pass to the limit ε → 0 in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='43) to get at the limit (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='48) � � � � � � � � � � � � � � � u(y, x3) ∈ L2((0, L);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' H1(C)), −∆′ yu(y, x3) = λu + f in D × (0, L), u = v on ∂D × (0, L), v ∈ H1 0(0, L), −d2v dx2 3 = λv + λ |C \\ D| � D u dy + 1 |C \\ D| � C f dy in (0, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Choosing f(y, x3) = g(x3)χC\\D(y) (which implies f = 0 in D) with an arbitrary g ∈ L2(0, L), the second equation in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='48) reduces to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='49) v ∈ H1 0(0, L), −d2v dx2 3 = λv + λ |C \\ D| � D u dy + g in (0, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Note that v ̸= 0 for g ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Indeed if v = 0, the first equation in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='48) would imply u = 0 since we have chosen f such that f = 0 in D and λ < µ1 is not an eigenvalue of −∆′ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Therefore equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='49) would give g = 0 which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Therefore, one can express u as u = (λu0 + 1)v where the pair (λ, u0) solves the first equation of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Therefore (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='49) takes the form (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='50) v ∈ H1 0(0, L)), −d2v dx2 3 = λ � 1 + |D| |C \\ D| + λ |C \\ D| � D u0 dy � v + g in (0, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' On the other hand, by hypothesis, λ is an eigenvalue of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='10) so that the last equation of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='10) with the same u0 as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='50) shows that λ � 1 + |D| |C \\ D| + λ |C \\ D| � D u0 dy � is an eigenvalue of − d2 dx2 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' This is a contradiction since ON THE LIMIT SPECTRUM OF A DEGENERATE OPERATOR 17 equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='50) valid for all g ∈ L2(0, L) means that the number λ � 1 + |D| |C \\ D| + λ |C \\ D| � D u0 dy � belongs to the resolvent of − d2 dx2 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' We prove now that lim k→+∞ λk = µ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Since λk ∈ (µ0, µ1) for any k, the sequence (λk)k admits at least an accumulation point and each accumu- lation point λ is such that µ0 ≤ λ ≤ µ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Assume that there exists an accumulation point λ such that λ < µ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' There exists a subsequence (λkn, ukn 0 , vkn) of solutions of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='10) such that lim n→+∞ λkn = λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Hence the following equation takes place for all n (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='51) − ∆′ukn 0 = λknukn 0 + 1 in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Let δ be a positive number such that λ < δ < µ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' For n large enough we have λkn ≤ δ so that applying the Poincaré inequality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='52) � D |u|2 dy ≤ 1 µ1 � D |∇′ yu|2 dy ∀ u ∈ H1 0(D), after multiplying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='51) by ukn 0 , we get for n large enough (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='53) � D |∇′ yukn 0 |2 dy ≤ δ µ1 � D |∇′ yukn 0 |2 dy + � D |ukn 0 | dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Applying successively the Cauchy-Schwarz inequality and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='52) in the last integral of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='53), we infer (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='54) � 1 − δ µ1 � � D |∇′ yukn 0 |2 dy ≤ � |D| � 1 µ1 �� D |∇′yukn 0 |2 dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Therefore, (ukn 0 )n is bounded in H1 0(D) and one can assume (possibly for another subsequence) that (ukn 0 )n con- verges weakly to u0 in H1 0(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' In particular we have that lim n→+∞ � D ukn 0 dy = � D u0 dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' On the other hand (λkn, ukn 0 , vkn) being a solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='10), the following equation (recall that vkn ̸= 0 ) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='55) − d2vkn dx2 3 = λkn � 1 + |D| |C \\ D| + λkn |C \\ D| � D ukn 0 dy � vkn ∀ n, shows that the number µ defined by µ := λ � 1 + |D| |C \\ D| + λ |C \\ D| � D u0 dy � is a finite accumulation point of the spectrum of − d2 dx2 3 since µ = lim n→+∞ µn where µn := λkn � 1 + |D| |C \\ D| + λkn |C \\ D| � D ukn 0 dy � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' This is a contradiction since it is well known that such spectrum is in fact an increasing sequence which tends to +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' The last point which remains to prove is that all the limiting eigenvalues are simple and that uk ε converges to uk for the whole sequence ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Assuming that λk is a simple eigenvalue, the proof of the convergence of the eigenvectors for the whole sequence ε is known since the work of [26] (see also [10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' We sketch it in the vectorial setting for the convenience of the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Assume that �uk vk � is an eigenvector associated to the simple eigenvalue λk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Using the fact that the eigenval- ues converge for the whole sequence ε, it is easy to check that the multiplicity of λk is equal or greater than that of λk ε;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' hence λk ε is simple and there are only two eigenvectors satisfying � Ω |uk ε|2 dx = 1, namely uk ε and −uk ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Among these two eigenvectors, we choose the one satisfying the inequality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='56) � Ω � uk εχF uk + uk εχMvk � dydx3 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' 18 KAÏS AMMARI AND ALI SILI Therefore if ε′ is a subsequence such that �uk ε′χF uk ε′χM � strongly converges in (L2(Ω))2 to the eigenvector �ˆuχF ˆvχM � associated to λk , we get by passing to the limit in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='56), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='57) � Ω (ˆuχF uk + ˆvχMvk) dydx3 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' On the other hand, �ukχF vkχM � = �ˆukχF ˆvkχM � or �ukχF vkχM � = − � ˆukχF ˆvkχM � since λk is a simple eigenvalue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' The last equality is excluded thanks to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='57) so that any subsequence is such that �uk ε′χF uk ε′χM � strongly converges in (L2(Ω))2 to �ukχF vkχM � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Let us now prove that all the limit eigenvalues are simple eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Assume that for some k, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='13) holds true for two orthogonal eigenvectors �uk vk � and �¯uk ¯vk � in L2(D) × L2(0, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' By assumption, we have (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='58) � L 0 � D uk¯ukdydx3 + |C \\ D| � L 0 vk¯vkdx3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' We know that uk and ¯uk are given respectively by uk(y, x3) = (λkuk 0(y) + 1)vk(x3) and ¯uk(y, x3) = (λkuk 0(y) + 1)¯vk(x3) where uk 0(y) given by the first equation of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='10) depends only on the eigenvalue λk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Turning back to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='58), we infer (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='59) � L 0 ��� D � λkuk 0(y) + 1 � dy �2 + |C \\ D| � vk(x3)¯vk(x3)dx3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' As remarked above vk and ¯vk are always eigenvectors of the operator − d2 dx2 3 with Dirichlet condition so that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='59) and the second equation of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='10) would mean that vk and ¯vk eigenvectors associated to the eigenvalue λk � 1 + |D| |C \\ D| + λk |C \\ D| � D uk 0 dy � are othogonal in L2(0, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' This is a contradiction since all the eigenvalues of − d2 dx2 3 with Dirichlet condition are simple eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' The proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='3 is now complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Finally, let us indicate briefly in the following short section how to derive the analogous theorem in the homogenization setting using the same approach as in the reduction of dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' PROOF OF THEOREM 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='5 In the spirit of the above section, the natural idea is to choose a test function vanishing outside the set Fε of fibers to get the apriori estimate on the sequence of eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' To that aim, we consider an eigenvector φ(y) corresponding to the first eigenvalue of −∆′ y in H1 0(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' We extend φ by zero over C \\ D and then by periodicity to the whole R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' The k-th eigenvalue λk ε of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='8) is given by the same min-max formula, namely (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='1) λk ε = min V k⊂Vh max u∈V k � Ω � ε2|∇u|2χFε + |∇u|2χMε � dx′ dx3 � Ω |u|2 dx′ dx3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' ON THE LIMIT SPECTRUM OF A DEGENERATE OPERATOR 19 For each ε, we choose V k ε ⊂ Vh as the subspace spanned by � φ( x′ ε )v1, φ( x′ ε )v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=', φ( x′ ε )vk� with the same v1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=', vk as those defined in the previous section, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=', k normalized orthogonal eigenvectors associated to the first k eigenvalues of − d2 dx2 3 in H1 0(0, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Hence, by construction, the functions of V k ε vanish in Mε so that making the change of variable x′ := εy + εi, y ∈ D in each cell, we can perform the same calculations as those of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='6) to get for u ∈ V k ε , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='2) � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � Ω u2dx′ dx3 = � i∈Iε � εD+εi φ2 �x′ ε � dx′ � L 0 � α2 1(v1)2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' + α2 k(vk)2� dx3 = � α2 1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' + α2 k � ε2 � i∈Iε � D φ2(y) dy, � Ω ε2|∇′ x′u|2dx′ dx3 = � α2 1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' + α2 k � � i∈Iε ε2 � εD+εi ����∇′ x′φ �x′ ε ����� 2 dx′ = � α2 1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' + α2 k � ε4 � i∈Iε � D 1 ε2 |∇′ yφ(y)|2dy = � α2 1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' + α2 k � ε2µ1 � i∈Iε � D |φ(y)|2dy, � Ω ε2 ���� ∂u ∂x3 ���� 2 dx′ dx3 = ε2 � L 0 � α2 1 �dv1 dx3 �2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' + α2 k �dvk dx3 �2� dx3 ε2 � i∈Iε � D |φ(y)|2dy = ε4� α2 1λ0 1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' + α2 kλ0 k � � i∈Iε � D |φ|2 dy ≤ ε4λ0 k � α2 1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' + α2 k � � i∈Iε � D |φ|2dy, in such a way the following estimate holds true (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='3) λk ε ≤ � µ1 + ε2λ0 k �� α2 1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' + α2 k � ε2 � i∈Iε � D |φ|2 dy ε2� α2 1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' + α2 k � � i∈Iε � D φ2(y) dy = µ1 + ε2λ0 k, which is exactly the same estimate as that obtained in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' It is interesting to note in the proof of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='3), we have chosen a test function verifying the same prop- erties as those of the 3d − 1d case, namely: null in the matrix and with the regularity H1 0(0, L) for almost all x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='1 is of general relevance since the other proofs in the homogenization setting are similar in all points to the corresponding ones in the 3d − 1d problem, the main reason being that the vertical variable is not concerned by the homogenization process which occurs only with respect to the horizontal variable x′ in such a way basically, the local 3d − 1d effect is repeated periodically in the horizontal plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Hence all the proofs take up exactly the 3d-1d case while sticking to two principles: Dirichlet condition on x3 = 0 or x3 = L both for the 3d − 1d problem and the homogenization problem and when x3 plays the role of parameter as it is the case for example in equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='25), it is x that will play the role of parameter in the homogenization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Indeed for instance, the natural formulation of equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='24) in the homogenization setting is the following one (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='4) � � � � � � � zk ε ∈ Vh, −∆′ x′zk ε − ε2 ∂2zk ε ∂x2 3 = λk εzk ε + λk εU k ε in Fε, zk ε = 0 on ∂Di ε × (0, L), 20 KAÏS AMMARI AND ALI SILI in such a way passing to the two-scale limit in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='4), we get the equivalent of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='25) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='5) � � � zk ∈ L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' H1 #(C)), −∆′ yzk = λkzk + λkUk in Ω × D, zk = 0 on Ω × ∂D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' The same approach may be applied to the other proofs following exactly the same steps and replacing the weak (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' strong) convergence in L2(Ω) by the two-scale (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' strong two-scale) convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' REFERENCES [1] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' ALLAIRE, Homogenization and Two-Scale Convergence, SIAM J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' Math Anal.' metadata={'source': 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MATHEMATICS, FACULTY OF SCIENCES OF MONASTIR, UNIVERSITY OF MONASTIR, TUNISIA Email address: kais.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='ammari@fsm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='rnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='tn INSTITUT DE MATHÉMATIQUES DE MARSEILLE (I2M), UMR 7373, AIX-MARSEILLE UNIVERSITÉ, CNRS, CMI, 39 RUE F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content=' JOLIOT-CURIE, 13453 MARSEILLE CEDEX 13, FRANCE Email address: ali.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E2T4oBgHgl3EQf9QlY/content/2301.04226v1.pdf'} +page_content='sili@univ-amu.' metadata={'source': 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and Strongly Convex Objective Functions +Unifying Nesterov’s Accelerated Gradient Methods for +Convex and Strongly Convex Objective Functions: From +Continuous-Time Dynamics to Discrete-Time Algorithms +Jungbin Kim +kjb2952@snu.ac.kr +Department of Electrical and Computer Engineering +Seoul National University +Seoul 08826, Korea +Insoon Yang +insoonyang@snu.ac.kr +Department of Electrical and Computer Engineering +Seoul National University +Seoul 08826, Korea +Abstract +Although Nesterov’s accelerated gradient (NAG) methods have been studied from various +perspectives, it remains unclear why the most popular forms of NAG must handle convex +and strongly convex objective functions separately. Motivated by this inconsistency, we +propose an NAG method that unifies the existing ones for the convex and strongly convex +cases. We first design a Lagrangian function that continuously extends the first Bregman +Lagrangian to the strongly convex setting. As a specific case of the Euler–Lagrange equation +for this Lagrangian, we derive an ordinary differential equation (ODE) model, which we +call the unified NAG ODE, that bridges the gap between the ODEs that model NAG for +convex and strongly convex objective functions. We then design the unified NAG, a novel +momentum method whereby the continuous-time limit corresponds to the unified ODE. The +coefficients and the convergence rates of the unified NAG and unified ODE are continuous +in the strong convexity parameter µ on [0, +∞). Unlike the existing popular algorithm +and ODE for strongly convex objective functions, the unified NAG and the unified NAG +ODE always have superior convergence guarantees compared to the known algorithms and +ODEs for non-strongly convex objective functions. This property is beneficial in practical +perspective when considering strongly convex objective functions with small µ. Furthermore, +we extend our unified dynamics and algorithms to the higher-order setting. Last but not +least, we propose the unified NAG-G ODE, a novel ODE model for minimizing the gradient +norm of strongly convex objective functions. Our unified Lagrangian framework is crucial +in the process of constructing this ODE. Fascinatingly, using our novel tool, called the +differential kernel, we observe that the unified NAG ODE and the unified NAG-G ODE +have an anti-transpose relationship. +Keywords: +Convex optimization, first-order methods, Nesterov acceleration +1. Introduction +We consider the optimization problem +min +x∈Rn f(x), +(1) +where f : Rn → R is a continuously differentiable function whose gradient is L-Lipschitz +continuous. We assume that the objective function f has a minimizer x∗. One of the most +1 +arXiv:2301.03576v1 [math.OC] 9 Jan 2023 + +Kim and Yang +popular first-order method for solving this problem is gradient descent (GD): +xk+1 = xk − s∇f (xk) +(2) +with the algorithmic stepsize s > 0. When f is convex, GD with s ≤ 1/L achieves an +O(∥x0 − x∗∥2/k) convergence rate (see d’Aspremont et al., 2021, Section 4.2). When f is +µ-strongly convex, GD with s ≤ 1/L achieves an O((1 − µs)k∥x0 − x∗∥2) convergence rate +(see d’Aspremont et al., 2021, Section 4.5). +Nesterov acceleration. +A natural and important question is whether there are other first- +order methods that outperform gradient descent. Nesterov (1983) proposed an accelerated +gradient method that achieves a faster convergence rate compared to gradient descent. Given +the initial point x0 = z0, a general three-sequence scheme for Nesterov’s accelerated gradient +(NAG) methods can be written as +yk = xk + τk (zk − xk) +(3a) +xk+1 = yk − s∇f (yk) +(3b) +zk+1 = zk + δk (µyk − µzk − ∇f (yk)) +(3c) +with s > 0, where the parameters τk and δk usually satisfy the collinearity condition1 +1 − µδk − (1/s − µ)τkδk = 0. +(4) +In particular, for µ-strongly (possibly with µ = 0) convex objective functions, Nesterov +considered the following algorithm: Given an initial point x0 = z0 ∈ Rn and γ0 > 0, the +constant step scheme I (Nesterov, 2018, Equation 2.2.19) (we will refer to this algorithm as +the original NAG) updates the iterates as +γk+1 = (1 − αk) γk + µαk +yk = +1 +γk + µαk +(αkγkzk + γk+1xk) +xk+1 = yk − s∇f (yk) +zk+1 = +1 +γk+1 +((1 − αk) γkzk + µαkyk − αk∇f (yk)) , +(5) +where the sequence (αk)∞ +k=0 in (0, 1) is inductively defined by the equation +1 +sα2 +k = (1 − αk) γk + µαk. +(6) +Using the estimate sequence technique, Nesterov (2018, Theorem 2.2.1) showed that the +iterates of the original NAG (5) satisfy the inequality +f (xk) − f (x∗) ≤ +�k−1 +� +i=0 +(1 − αi) +� � +f (x0) − f (x∗) + γ0 +2 ∥x0 − x∗∥2� +(7) +1. This condition ensures that the points xk, xk+1, zk+1 are collinear (see Section 2.4.1). Thus, one can +write the updating rule for yk as yk+1 = xk+1 + βk(xk+1 − xk) for some βk ∈ R. This property provides +a clear momentum effect: The point yk+1 is defined by adding a momentum term βk (xk+1 − xk) to the +previous point xk+1. This property is useful when generalizing NAG methods to handle non-smooth +terms (see d’Aspremont et al., 2021, Algorithm 20). +2 + +Unifying NAG for Convex and Strongly Convex Objective Functions +when s ≤ 1/L. Although the original NAG achieves a faster convergence rate than gradient +descent, it is difficult to analyze this algorithm because it involves auxiliary sequences αk +and γk which are defined inductively. However, when γ0 = µ (here we need µ > 0 because +γ0 > 0 is assumed), we simply have αk = √µs and γk = µ for all k ≥ 0. In this case, the +original NAG (5) can be expressed as the three-sequence scheme (3) with τk = +√µs +1+√µs and +δk = +� +s +µ: +yk = xk + +√µs +1 + √µs (zk − xk) +xk+1 = yk − s∇f (yk) +zk+1 = zk + +� s +µ (µyk − µzk − ∇f (yk)) . +(8) +We refer to this algorithm as NAG-SC. Letting αi = √µs in (7), we can see that this +algorithm achieves an O((1 − √µs)k(f(x0) − f(x∗) + µ +2∥x0 − x∗∥2)) convergence rate when +s ≤ 1/L. A major drawback of NAG-SC is that we cannot apply it to non-strongly convex +objective functions (µ = 0). For non-strongly convex objective functions, Tseng (2008) +proposed a simple alternative algorithm to the original NAG (5). They set the algorithmic +parameters as τk = +2 +k+1 and δk = s(k+1) +2 +to obtain the following simple algorithm, which we +call NAG-C: +yk = xk + +2 +k + 1 (zk − xk) +xk+1 = yk − s∇f (yk) +zk+1 = zk − s(k + 1) +2 +∇f (yk) . +(9) +When s ≤ 1/L, this algorithm achieves an O +� +∥x0 − x∗∥2/k2� +convergence rate (see Sec- +tion 2.2). +Although there are many variants of NAG, most recent studies on acceleration (Di- +akonikolas and Orecchia, 2019; Shi et al., 2019; Siegel, 2019; Alimisis et al., 2020; Shi et al., +2021; Wilson et al., 2021; Kim and Yang, 2022) focus on these two particular algorithms +because of their simplicity. Unfortunately, these two algorithms should be handled separately +because NAG-SC (8) does not recover NAG-C (9) as µ → 0. +Inconsistency I. NAG-SC does not recover NAG-C as µ → 0. +Moreover, NAG-SC has the following drawbacks: +• It cannot be applied to non-strongly convex objective functions. +• When µ is very small, the convergence guarantee for NAG-SC is worse than that for +NAG-C in early stages because (1 − √µs)k converges to 0 very slowly. +• The convergence rate of NAG-SC depends on both the initial squared distance ∥x0−x∗∥2 +and the initial function value accuracy f(x0) − f(x∗), while the convergence rate of +NAG-C depends only on the squared initial distance ∥x0 − x∗∥2. +As most of recent works on Nesterov acceleration are based on these two specific algorithms, +similar inconsistencies can be found in the literature. We discuss more inconsistencies below. +3 + +Kim and Yang +1.1 Inconsistencies between convex and strongly convex cases +1.1.1 Continuous-time models. +In this subsection, we first informally derive the limiting ODE of the three-sequence scheme +(3). To identify a discrete-time sequence (xk)∞ +k=0 with a continuous-time curve X : [0, ∞) → +Rn, given the algorithmic stepsize s, we introduce a strictly increasing sequence (tk)∞ +k=0 +(depending on s) in [0, ∞) and make the identification X(tk) = xk. We denote the inverse +of the sequence t : {0, 1, 2, . . .} → R as k, that is, k(tk) = k for all k ≥ 0. For convenience, +we extend the function k to a piecewise linear function defined on [0, ∞). +We assume that +lim +s→0 t0 = 0 +(10) +and that the timesteps are asymptotically equivalent to √s as s → 0 in the sense that +lim +s→0 +tk(t)+1 − t +√s += 1 for all t ∈ (0, ∞) . +(11) +Note that the popular choice tk = tk := k√s (we will use the notation tk for this specific +sequence throughout the paper) used in (Su et al., 2014; Wibisono et al., 2016; Shi et al., +2021) satisfies these conditions. +For the iterates of three-sequence scheme (3), we have +xk+1 − xk +√s += τk +√s (zk − xk) − √s∇f (yk) +zk+1 − zk +√s += δk +√s (µyk − µzk − ∇f (yk)) . +We introduce two sufficiently smooth curves X, Z : [0, ∞) → Rn (possibly depending on +s now) such that X(t) = xk(t) and Z(t) = zk(t). Since ∥xk+1 − yk∥ = o(√s) and ∇f is +Lipschitz continuous, we have +˙X(t) = lim +s→0 +xk(t)+1 − xk(t) +tk(t)+1 − t += lim +s→0 +xk(t)+1 − xk(t) +√s += lim +s→0 +�τk(t) +√s +� +(Z(t) − X(t)) +˙Z(t) = lim +s→0 +zk(t)+1 − zk(t) +tk(t)+1 − t += lim +s→0 +zk(t)+1 − zk(t) +√s += lim +s→0 +�δk(t) +√s +� +(µX(t) − µZ(t) − ∇f(X(t))) +for all t > 0. Thus, if the limits +τ(t) = lim +s→0 +τk(t) +√s +δ(t) = lim +s→0 +δk(t) +√s +(12) +exist for all t ∈ (0, ∞), then as s → 0, the iterates generated by the three-sequence scheme +(3) converge to a solution to the following system of ODEs: +˙X(t) = τ(t)(Z(t) − X(t)) +˙Z(t) = δ(t)(µX(t) − µZ(t) − ∇f(X(t))) +(13) +4 + +Unifying NAG for Convex and Strongly Convex Objective Functions +with the initial conditions X(0) = Z(0) = x0. We can equivalently write this as the following +second-order ODE: +¨X + +� +τ(t) − ˙τ(t) +τ(t) + µδ(t) +� +˙X + τ(t)δ(t)∇f(X) = 0. +(14) +Furthermore, when the collinearity condition (4) holds, we have +δ(t) = lim +s→0 +δk +√s = lim +s→0 +1 +√s (µ + (1/s − µ)τk) = lim +s→0 +√s +µs + (1 − µs)τk += +1 +τ(t). +(15) +Limiting ODE of NAG-C. +Recall that NAG-C (9) is the three-sequence scheme (3) with +τk = +2 +k+1 and δk = s(k+1) +2 +. With the sequence tk = k√s, we have +τ(t) = lim +s→0 +τk(t) +√s = lim +s→0 +2 +√s (t/√s + 1) = 2 +t +δ(t) = lim +s→0 +δk(t) +√s = lim +s→0 +√s (t/√s + 1) +2 += t +2. +Thus, as s → 0, NAG-C converges to the following ODE system, which we call NAG-C +system: +˙X = 2 +t (Z − X) +˙Z = − t +2∇f(X) +(16) +with X(0) = Z(0) = x0. This system can be written in the following second-order ODE, +which we call NAG-C ODE: +¨X + 3 +t +˙X + ∇f(X) = 0 +(17) +with X(0) = x0 and ˙X(0) = 0. Su et al. (2014) first derived this ODE and showed that the +solution to (17) satisfies an O(∥x0 − x∗∥2/t2) convergence rate. +Limiting ODE of NAG-SC. +Recall that NAG-SC (8) is the three-sequence scheme (3) +with τk = +√µs +1+√µs and δk = +� +s +µ. With the sequence tk = −k log(1−√µs) +√µ +,2 we have +τ(t) = lim +s→0 +τk(t) +√s = lim +s→0 +√µ +1 + √µs = √µ +δ(t) = lim +s→0 +δk(t) +√s = lim +s→0 +1 +√µ = +1 +√µ. +Thus, as s → 0, NAG-SC converges to the following ODE system, which we call NAG-SC +system: +˙X = √µ(Z − X) +˙Z = +1 +√µ (µX − µZ − ∇f(X)) +(18) +2. Although the sequence tk = k√s leads to the same limiting dynamics, this particular sequence makes a +clear connection between the convergence analysis of NAG-SC and that of NAG-SC ODE (see Section 2.2). +5 + +Kim and Yang +with X(0) = Z(0) = x0, or equivalently, the following NAG-SC ODE: +¨X + 2√µ ˙X + ∇f(X) = 0 +(19) +with X(0) = x0 and +˙X(0) = 0. Wilson et al. (2021) showed that the solution to this +ODE satisfies an O(e−√µt(f(x0) − f(x∗) + µ +2∥x0 − x∗∥2)) convergence rate. Just like in the +discrete-time case, NAG-C ODE (17) and NAG-SC ODE (19) should be handled as separate +cases because NAG-SC ODE does not recover NAG-C ODE as µ → 0. +Inconsistency II. NAG-SC ODE does not recover NAG-C ODE as µ → 0. +Moreover, NAG-SC ODE has the following drawbacks: +• The solution to NAG-SC ODE with µ = 0 may not converge to the minimizer of f: +For the objective function f(x) = 1 +2x2 on R, the solution to NAG-SC ODE with x0 = 1 +is X(t) = cos(t), which does not converge to the minimizer x∗ = 0. +• When µ is very small, the convergence guarantee for NAG-SC ODE is worse than that +for NAG-C ODE in early stages because e−√µt converges to 0 very slowly. +• The convergence rate of NAG-SC ODE depends on both the initial squared distance +∥x0 −x∗∥2 and the initial function value accuracy f(x0)−f(x∗), while the convergence +rate of NAG-C ODE depends only on the squared initial distance ∥x0 − x∗∥2. +1.1.2 Bregman Lagrangians +To systematically study the acceleration phenomenon of momentum methods, Wibisono +et al. (2016) introduced the following first Bregman Lagrangian: +L1st +� +X, ˙X, t +� += eα+γ � +Dh +� +X + e−α ˙X, X +� +− eβf(X) +� +, +(20) +where α, β, γ : [0, ∞) → R are continuously differentiable functions, h is a continuously +differentiable strictly convex function, and Dh is the Bregman divergence (see Section 2.1 for +its definition). In order to obtain accelerated convergence rates, the following ideal scaling +conditions are introduced: +˙γ = eα +(21a) +˙β ≤ eα. +(21b) +Under the ideal scaling condition (21a), the Euler–Lagrange equation +d +dt +� ∂L +∂ ˙X +� +X, ˙X, t +�� += ∂L +∂X +� +X, ˙X, t +� +(22) +for the first Bregman Lagrangian (20) reduces to the following system of first-order equations: +˙X = eα(Z − X) +(23a) +d +dt∇h(Z) = −eα+β∇f(X). +(23b) +6 + +Unifying NAG for Convex and Strongly Convex Objective Functions +When f is convex, any solution to the system of ODEs (23) reduces the objective function +value accuracy at an O(e−β(t)) convergence rate (see Section 2.2). In particular, setting +α(t) = log 2 +t and β(t) = log t2 +4 , we recover NAG-C system (16) and its convergence rate. +Although the first Bregman Lagrangian (20) generates a large family of momentum +dynamics, it does not include NAG-SC system (18). To handle strongly convex cases, Wilson +et al. (2021) introduced the second Bregman Lagrangian, defined as +L2nd +� +X, ˙X, t +� += eα+β+γ � +µDh +� +X + e−α ˙X, X +� +− f(X) +� +. +(24) +Under the ideal scaling condition (21a), the Euler–Lagrange equation (22) for the second +Bregman Lagrangian (24) reduces to the following system of first-order equations: +˙X = eα(Z − X) +(25a) +d +dt∇h(Z) = ˙β (∇h(X) − ∇h(Z)) − eα +µ ∇f(X). +(25b) +When f is µ-uniformly convex with respect to h (see Section 2.1), any solution to the system +of ODEs (25) satisfies an O(e−β(t)) convergence rate (see Section 2.2). In particular, letting +α(t) = log √µ and β(t) = √µt, we recover NAG-SC system (18) and its convergence rate. +Here, we observe an inconsistency between the two Bregman Lagrangians. +Inconsistency III. The second Bregman Lagrangian does not recover the first Bregman +Lagrangian as µ → 0. +1.2 Contributions +In this paper, we propose a novel unified framework for Lagrangians, ODE models and +algorithms to address the inconsistencies between the convex case and the strongly convex +case mentioned above. The proposed framework seamlessly bridges the gap between the two +cases as illustrated in Figure 1. The main contributions of this work can be summarized as +follows: +• We propose the unified Bregman Lagrangian (Section 3). Unlike the second Bregman +Lagrangian, the unified Bregman Lagrangian recovers the first Bregman Lagrangian +when µ = 0. As the Euler–Lagrange equation for the unified Bregman Lagrangian, +we obtain a family of continuous-time dynamics (Proposition 2). Using a Lyapunov +function, we analyze the convergence rate for these flows (Theorem 3). +• We derive the unified NAG ODE (59) as a special case of the unified Bregman La- +grangian flows (Section 4.1). Unlike NAG-SC ODE (19), for non-strongly convex +objective functions (µ = 0), the unified NAG ODE and its convergence rate (The- +orem 10) recover NAG-C ODE (17) and its convergence rate. Furthermore, for any +µ > 0, the unified NAG ODE and its convergence rate (Corollary 8) recover NAG-SC +ODE (19) and its convergence rate as t → ∞. +• We devise the unified NAG family (63), a family of momentum algorithms that +converge to the unified NAG ODE as s → 0 (Section 4.2). As a special case, we have +7 + +Kim and Yang +NAG-C ODE +(Su et al., 2014) +First Bregman La- +grangian (Wibisono +et al., 2016) +Unified Bregman +Lagrangian +(Section 3) +Second Bregman +Lagrangian (Wilson +et al., 2021) +Unified NAG +ODE (Section 4.1) +NAG-SC ODE +(Wilson et al., 2021) +NAG-C +(Tseng, 2008) +Unified NAG +(Section 4.2) +NAG-SC (Nes- +terov, 2018) +Higher-order +optimization +(Section 5) +Gradient norm +minimization +(Section 6) +special case +special case +discretize +limit +discritize +limit +unify +unify +recover +µ = 0 +recover +µ = 0 +recover +t → ∞ +recover +k → ∞ +Figure 1: An illustration of our framework and contributions. +the unified NAG (70), a simple algorithm which unifies NAG-C (9) and NAG-SC (8). +Moreover, using an adaptive timestep in the unified NAG family, we constructively +recover the original NAG (5) with γ0 > µ and its convergence rate (7). +• We extend the unified NAG ODE and the unified NAG family to the higher-order +non-Euclidean setting (mirror descent setup) (Section 5). Our novel dynamics and +algorithms can be viewed as continuous extensions of the accelerated tensor method +(convex case) and its limiting ODE in (Wibisono et al., 2016) to the strongly convex +setting. +We also made the following contributions that are not closely related to our major goal but +may deserve independent attention: +• We compute the general limiting ODEs of the three-sequence scheme (3), the two- +sequence scheme (42), and the fixed-step first-order scheme (46). In particular, we +introduce a novel tool, called the differential kernel H(t, τ), to derive the limiting +ODE of the fixed-step first-order scheme. We show that an anti-transpose relationship +(95) between OGM and OGM-G can be naturally shifted to a continuous-time setting +by this tool. +• We propose the unified NAG-G ODE, an ODE model for minimizing the gradient +norm of strongly convex objective functions (Section 6). Surprisingly, the differential +kernels corresponding to the unified NAG ODE and the unified NAG-G ODE have an +anti-transpose relationship, just like it does between OGM ODE and OGM-G ODE. +8 + +Unifying NAG for Convex and Strongly Convex Objective Functions +Dynamics +Convergence rate +Unified NAG ODE +f +� +X +� +t +�� +− f +� +x∗� +≤ O +� +min +� +1/t2, e−√µt���x0 − x∗��2� +Unified accelerated tensor flow +f +� +X +� +t +�� +− f +� +x∗� +≤ O +� +min +� +1/tp, e−pC1/pµ1/pt� +Dh +� +x∗, x0 +�� +Unified NAG-G ODE +��∇f(X(T)) +��2 ≤ O +� +min +� +1/T 2, e−√µT �� +f +� +x0 +� +− f +� +x∗��� +Algorithm +Convergence rate +Unified NAG +f +� +xk +� +− f +� +x∗� +≤ O +� +min +� +1/k2, +� +1 − √µs +�k���x0 − x∗��2� +Unified accelerated tensor method +f +� +xk +� +− f +� +x∗� +≤ O +� +min +� +1/kp, +� +1 + C1/ppµ1/ps1/p�−k� +Dh +� +x∗, x0 +��� +Table 1: Convergence rates of the momentum dynamics and algorithms proposed in this +paper. +We summarize the convergence rates for our continuous-time dynamics and discrete-time +algorithms in Table 1. In addition to theoretical and algorithmic perspectives, we discuss +the need for unified acceleration methods from a practical perspective. +Practical perspective. +Many optimization problems in machine learning can be formu- +lated as +min +x∈Rn f(x) = 1 +m +� m +� +i=1 +fi(x) + λR(x) +� +, +(26) +where fi is the loss function corresponding to the i-th sample, λ > 0 is the regularization +parameter, and R(x) is the regularization term (Bubeck et al., 2015, Equation 1.1). Consider +the problem (26) where the functions fi are convex and L-smooth, and R(x) = ∥x∥2. Then, +f is µ-strongly convex and L-smooth, where µ = 2λ/m. As the sample size m grows or the +regularization parameter λ decreases, the strong convexity parameter µ decreases. Thus, +improving the convergence rate for ill-conditioned strongly convex objective functions (where +µ is small) is quite significant, as emphasized in (Bubeck et al., 2015, Section 3.6). +As mentioned above, the convergence guarantee of NAG-SC (8) is no better than that of +NAG-C (9) when µ is small. In our numerical experiments (see Section 7), it is observed that +the performance of NAG-SC is worse than that of NAG-C when µ is very small. Thus, it is +desirable to design a strongly convex optimization algorithm whose convergence guarantee +is not worse than that of NAG-C even when µ is very small. In the experiments, we observe +that for a logistic regression problem, when µ is small, our algorithm is comparable to +NAG-C, while NAG-SC underperforms NAG-C. +Existing unified methods and dynamics. +To clarify what is our novel contribution and +what is not, we review existing algorithms and dynamics that can handle the non-strongly +convex case and the strongly convex case in a unified way. The original NAG (5) is an +accelerated algorithm that can handle both convex objective functions and strongly convex +objective functions. In Section 4.2.2, we show that the original NAG can be constructively +recovered by our unified Lagrangian formulation. Luo and Chen (2021) designed the following +9 + +Kim and Yang +ODE model for the original NAG, which we call the original NAG system: +˙γ = µ − γ +˙X = Z − X +˙Z = 1 +γ (µX − µZ − ∇f(X)) +(27) +with X(0) = Z(0) = x0 and γ(0) = γ0 > 0. Luo and Chen (2021, Section 6.2) showed that +the original NAG can be viewed as a discretization scheme with the timestep αi, which is +inductively defined in (6). +Using time rescaling technique, Luo and Chen (2021) also proposed the following system +of ODEs (although most of their results directly deal with Equation 27): +˙X(t) = a(t)(Z(t) − X(t)) +b(t) ˙Z(t) = a(t)(µX(t) − µZ(t) − ∇f(X(t))), +(28) +where a : [0, ∞) → [0, ∞) is an arbitrary function and +b(t) = γ +�� t +0 +a(s) ds +� +. +This ODE system is closely related to the unified Bregman Lagrangian flow (56) and the +unified NAG system (58) proposed in this paper. In Appendix A.1, we show that the rescaled +original NAG flow (28) can be expressed as the unified Bregman Lagrangian flow (56). +Conversely, the unified Bregman Lagrangian flow can be expressed as the rescaled original +NAG flow if the ideal scaling condition (21b) holds with equality and the distance-generating +function h is Euclidean (h(x) = +1 +2∥x∥2). +Therefore, our unified Bregman Lagrangian +generates a strictly larger family compared to (28). To emphasize, only our family can deal +with the non-Euclidean setup (mirror descent setup). In addition, the derivation of our +unified family (56) is more constructive because it comes from a Lagrangian formulation, +whereas Luo and Chen (2021) designed the family (28) through heuristic speculation. +1.3 Related work +Nesterov (1983) first proposed the original NAG (5) with µ = 0. The original NAG with +µ > 0 was first analyzed using the estimate sequence technique (Nesterov, 2018). Tseng +(2008) proposed NAG-C (9) and its generalization to composite optimization problems. Su +et al. (2014) derived NAG-C ODE (17) by taking the limit s → 0 in NAG-C. This ODE has +further been generalized and investigated in (Krichene et al., 2015; Attouch et al., 2018). +Wibisono et al. (2016) proposed the first Bregman Lagrangian (20) that systematically +generates a family of ODEs (23) including NAG-C ODE and its higher-order extensions. +Wilson et al. (2021) extended this framework to the strongly convex case. They proposed +the second Bregman Lagrangian (24), which generates a family of continuous-time flows (25) +including NAG-SC ODE (18), and strengthened the connection between continuous-time +dynamics and discrete-time algorithms via Lyapunov function arguments. However, as +mentioned in Section 1.1, their work is not consistent with (Wibisono et al., 2016) because +the second Bregman Lagrangian does not recover the first Bregman Lagrangian as µ → 0. +10 + +Unifying NAG for Convex and Strongly Convex Objective Functions +Based on Lagrangian formulations, Betancourt et al. (2018) studied a symplectic integrator +to obtain discrete-time algorithms from continuous-time dynamics. Shi et al. (2019, 2021) +derived high-resolution ODEs for NAG-C and NAG-SC, and then obtained algorithms with +accelerated convergence rates by applying the symplectic Euler method to the high-resolution +ODEs. Luo and Chen (2021) understood acceleration using the A-stability theory and +designed an ODE model for the original NAG method. Zhang et al. (2021) obtained an +accelerated algorithm by applying the explicit Euler method to a variant of high-resolution +ODEs. Diakonikolas and Orecchia (2019) proposed the approximate duality gap technique to +construct and analyze accelerated algorithms. Using conservation laws in dilated coordinate +systems, Suh et al. (2022) recovered NAG-C ODE and NAG-SC ODE and showed that a +semi-second-order symplectic Euler discretization in the dilated coordinate yields accelerated +methods. +2. Preliminaries +In this section, we review the basic notions that we will use throughout the paper. While +Sections 2.1 and 2.2 review the standard concepts in the literature, Sections 2.3 and 2.4 +contain novel ideas and results. +2.1 Convex analysis +Convexity and smoothness. +Let f : Rn → R be a C∞ function. Then for µ ≥ 0, the +function f is called µ-strongly convex if the inequality +f(y) ≥ f(x) + ⟨∇f(x), y − x⟩ + µ +2 ∥y − x∥2 +holds for all x, y ∈ Rn. In particular, the function f is called convex if it is strongly convex +with the strong convexity parameter µ = 0. For L > 0, the function f is called L-smooth if +its gradient is L-Lipschitz continuous, that is, the inequality +∥∇f(x) − ∇f(y)∥ ≤ L ∥x − y∥ +holds for all x, y ∈ Rn. It is known that when f is L-smooth, the inequality +f(y) ≤ f(x) + ⟨∇f(x), y − x⟩ + L +2 ∥y − x∥2 +holds for all x, y ∈ Rn. For most of the remaining sections of this paper (Sections 4 and +6), we make the following assumptions, which we call the standard smooth strongly convex +setting: +• The objective function f is (1/s)-smooth, where s > 0 is the algorithmic stepsize. +• The objective function f is µ-strongly (possibly with µ = 0) convex. +Higher-order convexity and smoothness. +The notions of convexity and smoothness +can be generalized to the higher-order setting. The function f is called µ-uniformly convex +of order p ≥ 2 if the inequality +f(y) ≥ f(x) + ⟨∇f(x), y − x⟩ + µ +p ∥y − x∥p +(29) +11 + +Kim and Yang +holds for all x, y ∈ Rn. The function f is called L-smooth of order p − 1 if the inequality +��∇p−1f(y) − ∇p−1f(x) +�� ≤ L ∥y − x∥ +(30) +holds for all x, y ∈ Rn. Note that these definitions recover the standard notions of convexity +and smoothness when p = 2. +Bregman divergences. +In the optimization literature, a common way to consider a +non-Euclidean setting is by using the Bregman divergence, instead of the Euclidean distance. +For a continuously differentiable function h : Rn → R which is convex and essentially smooth +(∥∇h(x)∥ → ∞ as ∥x∥ → ∞), the Bregman divergence Dh : Rn ×Rn → [0, ∞) of h is defined +as +Dh(y, x) = h(y) − h(x) − ⟨∇h(x), y − x⟩ . +(31) +Note that when h(x) = 1 +2∥x∥2, the Bregman divergence of h is the squared Euclidean +distance 1 +2∥y − x∥2. For all x, y, z ∈ Rn, the three-point identity (see Wilson et al., 2021, +Proposition 5) +Dh(x, y) − Dh(x, z) = − ⟨∇h(y) − ∇h(z), x − y⟩ − Dh(y, z) +(32) +holds. For µ ≥ 0, the function f is called µ-uniformly convex with respect to h if the +inequality +Df(x, y) ≥ µDh(x, y) +(33) +holds for all x, y ∈ Rn. Note that this condition is equivalent to the µ-storng convexity of f +when h(x) = 1 +2 ∥x∥2. +2.2 Lyapunov arguments for convergence analyses +A popular method for proving the convergence rates of momentum dynamics and algorithms +is constructing an energy function decreasing over time, called the Lyapunov function +(Lyapunov, 1992). The particular analyses presented in this section handle discrete-time +algorithms and the corresponding continuous-time dynamics using a single Lyapunov function, +as in (Krichene et al., 2015). To prove the convergence rates of the given algorithm and +associated dynamics, we take the following steps: +1. Define a time-dependent Lyapunov function V : Rn × Rn × [0, ∞) → [0, ∞). +2. Show that the continuous-time energy functional E(t) = V (X(t), Z(t), t) is monotoni- +cally decreasing along the solution trajectory (X, Z) : [0, ∞) → Rn × Rn of the ODE +system. +3. Show that the discrete-time energy functional Ek = V (xk, zk, tk) is monotonically +decreasing along the iterates (xk, zk) : {0, 1, 2, . . .} → Rn × Rn of the algorithm. +The remainder of this subsection shows how we can apply this strategy to known algorithms. +We assume the standard smooth (strongly) convex setting (see Section 2.1). +12 + +Unifying NAG for Convex and Strongly Convex Objective Functions +NAG-C and NAG-C ODE. +We define a time-dependent Lyapunov function as +V (X, Z, t) := 1 +2 ∥Z − x∗∥2 + t2 +4 (f(X) − f (x∗)) . +(34) +Then, the continuous-time energy functional +E(t) = V (X(t), Z(t), t) = 1 +2 ∥Z(t) − x∗∥2 + t2 +4 (f(X(t)) − f (x∗)) +is monotonically decreasing along the solution trajectory of NAG-C ODE (16) (see Su et al., +2016). Writing E(t) ≤ E(0) explicitly, we obtain an O(1/t2) convergence rate as +f(X(t)) − f(x∗) ≤ 4 +t2 E(t) ≤ 4 +t2 E(0) = 2 +t2 ∥x0 − x∗∥2 . +For the iterates of NAG-C (9), the discrete-time energy function +Ek = V (xk, zk, tk) = 1 +2 ∥zk − x∗∥2 + sk2 +4 (f(xk) − f (x∗)) , +(35) +where tk = k√s, is monotonically decreasing (see Ryu and Yin, 2022, Chapter 12). Hence, +we obtain an O(1/k2) convergence rate. +NAG-SC and NAG-SC ODE. +We define a time-dependent Lyapunov function as +V (X, Z, t) := e +√µt �µ +2 ∥Z − x∗∥2 + f(X) − f (x∗) +� +. +(36) +Then we can show that NAG-SC ODE (18) achieves an O(e−√µt) convergence rate by showing +that the energy functional +E(t) = V (X(t), Z(t), t) = e +√µt �µ +2 ∥Z(t) − x∗∥2 + f(X(t)) − f (x∗) +� +is monotonically decreasing along the solution trajectory of NAG-SC ODE (see Wilson et al., +2021). Similarly, we can show that NAG-SC (8) achieves an O((1 − √µs)k) convergence rate +by showing that the energy functional +Ek = V (xk, zk, tk) = (1 − √µs)−k �µ +2 ∥zk − x∗∥2 + f(xk) − f (x∗) +� +, +where tk = −k log(1−√µs) +√µ +, is monotonically decreasing along the iterates of NAG-SC (see +d’Aspremont et al., 2021, Section 4.5). +Bregman Lagrangians. +We can show that the first Bregman Lagrangian flow (23) and +the second Bregman Lagrangian flow (25) achieve an O(e−β(t)) convergence rate by showing +that the energy functional E(t) = V (X(t), Z(t), t) is monotonically decreasing, where the +Lyapunov function V is defined as +V1st(X, Z, t) := Dh (x∗, Z) + eβ(t) (f(X) − f (x∗)) +(37) +for the first Bregman Lagrangian flow and +V2nd(X, Z, t) := eβ(t) (µDh (x∗, Z) + f(X) − f (x∗)) +(38) +for the second Bregman Lagrangian flow. See (Wibisono et al., 2016; Wilson et al., 2021) +for the proofs. +13 + +Kim and Yang +2.3 Hyperbolic functions and their higher-order generalization +Hyperbolic functions. +We first review the definitions and properties of hyperbolic +functions. The sinh, cosh, tanh, coth, sech, and csch functions are defined as +sinh x = ex − e−x +2 +, +sinh x ∼ x as x → 0, +sinh x ∼ ex +2 as x → ∞ +cosh x = ex + e−x +2 +, +cosh x ∼ 1 as x → 0, +cosh x ∼ ex +2 as x → ∞ +tanh x = sinh x +cosh x, +tanh x ∼ x as x → 0, +tanh x ∼ 1 as x → ∞ +coth x = cosh x +sinh x , +coth x ∼ 1 +x as x → 0, +coth x ∼ 1 as x → ∞ +sech x = +1 +cosh x, +sech x ∼ 1 as x → 0, +sech x ∼ 2e−x as x → ∞ +csch x = +1 +sinh x, +csch x ∼ 1 +x as x → 0, +csch x ∼ 2e−x as x → ∞. +(39) +Furthermore, the sinhc, tanhc, cothc, and cschc functions are defined as follows (see ten +Thije Boonkkamp et al., 2012): +sinhc x := +� +sinh x +x +, +if x ̸= 0 +1, +if x = 0 +sinhc x ∼ 1 as x → 0, +sinhc x ∼ ex +2x as x → ∞ +tanhc x := sinhc x +cosh x +tanhc x ∼ 1 as x → 0, +tanhc x ∼ 1 +x as x → ∞ +cothc x := +1 +tanhc x, +cothc x ∼ 1 as x → 0, +cothc x ∼ x as x → ∞ +cschc x := +1 +sinhc x, +cschc x ∼ 1 as x → 0, +cschc x ∼ 2xe−x as x → ∞. +(40) +The graphs of these functions are shown in Figure 2. +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +x +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +f(x) +f(x) = sinh(x) +f(x) = cosh(x) +f(x) = tanh(x) +(a) sinh, cosh, tanh +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +x +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +f(x) +f(x) = csch(x) +f(x) = sech(x) +f(x) = coth(x) +(b) coth, sech, csch +0 +1 +2 +3 +4 +5 +6 +x +0 +1 +2 +3 +4 +5 +6 +f(x) +f(x) = sinhc(x) +f(x) = tanhc(x) +f(x) = cothc(x) +f(x) = cschc(x) +(c) sinhc, tanhc, cothc, cschc +Figure 2: Hyperbolic functions and their variants. +14 + +Unifying NAG for Convex and Strongly Convex Objective Functions +Higher-order hyperbolic functions. +We now define the higher-order hyperbolic func- +tions that will be used to design higher-order accelerated optimization algorithms. We define +the p-th order hyperbolic sine function sinhp : [0, ∞) → R as the solution of the initial value +problem +sinh′ +p(t) = coshp(t) := +� +1 + sinhp +p(t) +�1/p , +sinhp(0) = 0. +(41) +Furthermore, we define the tanhp, cothp, sechp, and cschp functions as +tanhp(t) = sinhp(t) +coshp(t), +cothp(t) = +1 +tanhp(t), +sechp(t) = +1 +sinhp(t), +cschp(t) = +1 +coshp(t). +We define the sinhcp, tanhcp, cothcp, and cschcp functions as +sinhcp x := +� sinhp x +x +, +if x ̸= 0 +1, +if x = 0 +tanhcp x := sinhcp x +coshp x , +cothcp x := +1 +tanhcp x, +cschcp x := +1 +sinhcp x. +Note that the higher-order hyperbolic functions recover the usual hyperbolic functions when +p = 2. The following proposition says that the sinhp function grows exponentially. +Proposition 1 There is a constant Cp > 0 such that sinhp(t) ∼ Cpet as t → ∞. +In +particular, we have Cp = 1/2 for p = 2. +The proof of Proposition 1 can be found in Appendix B.1. Using (41) and Proposition 1, it +is straightforward to check the following asymptotic properties: +sinhp x ∼ x as x → 0, +sinhp x ∼ Cpex as x → ∞ +coshp x ∼ 1 as x → 0, +coshp x ∼ Cpex as x → ∞ +tanhp x ∼ x as x → 0, +tanhp x ∼ 1 as x → ∞. +2.4 Limiting arguments and examples +We investigate two additional ways to derive the limiting ODEs of first-order algorithms. +The first approach is to write the algorithm as a two-sequence scheme and then derive +the limiting ODE via the second-order Taylor series expansion. This argument frequently +appears in the literature (see Su et al., 2016; Shi et al., 2021). The second approach, which +is novel, is to express the algorithm using the difference matrix H = (hij) and then derive +the differential kernel H(t, τ) corresponding to the matrix (hij). We only present the results +here and defer the detailed computations to Appendices C.1 and C.2. +2.4.1 Limiting ODEs of two-sequence algorithms +We consider the following two-sequence scheme: +xk+1 = yk − s∇f (yk) +yk+1 = xk+1 + βk (xk+1 − xk) + γk (xk+1 − yk) . +(42) +15 + +Kim and Yang +If we have +lim +s→0 +1 − βt/√s +√s += b(t) and lim +s→0 γt/√s = c(t) for all t > 0 +(43) +for some smooth functions b, c : (0, ∞) → R, then under the identification X(tk) = xk with +tk = k√s, the two-sequence scheme (42) converges to the ODE +¨X(t) + b(t) ˙X(t) + (1 + c(t))∇f(X(t)) = 0 +(44) +as s → 0. +Recovering the limiting ODE of three-sequence scheme. +We can write the three- +sequence scheeme (3) as the two-sequence scheme (42) with the following parameters (see +Lee et al., 2021, Appendix B): +βk = (1 − τk) τk+1 (1 − µδk) +τk +γk = τk+1 ((1/s − µ)δkτk − 1 + µδk) +τk +. +(45) +If the limits (12) with tk = k√s exist, then we have +lim +s→0 +1 − βt/√s +√s += τ(t) − ˙τ(t) +τ(t) + µδ(t) +lim +s→0 γt/√s = τ(t)δ(t) − 1 +for all t > 0. Therefore, we recover the limiting ODE (14) of the three-sequence scheme. In +particular, if the algorithmic parameters (τk) and (δk) satisfy the collinearity condition (4), +then we have γk = 0 for all k ≥ 0, and thus c(t) = 0. +Two-sequence form of NAG-C. +Because NAG-C is the three-sequence scheme (3) with +τk = +2 +k+1, δk = s(k+1) +2 +, and µ = 0, we can rewrite it as the two-sequence scheme (42) with +βk = +� +1 − +2 +k+1 +� +2 +k+2 +2 +k+1 += k − 1 +k + 2 +γk = +2 +k+2 · s(k+1) +2 +s +− +2 +k+2 +2 +k+1 += 0. +Thus, NAG-C converges to the ODE (44) with +b(t) = lim +s→0 +1 − t/√s−1 +t/√s+2 +√s += 3 +t +c(t) = 0, +which recovers NAG-C ODE (17). +16 + +Unifying NAG for Convex and Strongly Convex Objective Functions +Two-sequence form of NAG-SC. +Because NAG-SC is the three-sequence scheme (3) +with τk = +√µs +1+√µs and δk = +� +s +µ, it can be written as the two-sequence scheme (42) with +βk = +� +1 − +√µs +1+√µs +� +√µs +1+√µs +� +1 − µ +� +s +µ +� +√µs +1+√µs += 1 − √µs +1 + √µs +γk = +√µs +1+√µs +� +s +µ +s +− +� +1 − µ +� s +µ + µ +√µs +1 + √µs +� s +µ +� += 0. +Thus NAG-SC converges to the ODE (44) with +b(t) = lim +s→0 +1 − 1−√µs +1+√µs +√s += 2√µ +c(t) = 0, +which recovers NAG-SC ODE (19). +2.4.2 Difference matrices and differential kernels +We can formulate most of the practical first-order momentum methods as the following +fixed-step first-order scheme (see Drori and Teboulle, 2014): +yi+1 = yi − s +i +� +j=0 +hij∇f (yj) for i = 0, . . . , N − 1, +(46) +where N is the number of iterations. We can write this scheme equivalently as +� +���� +y1 − y0 +y2 − y1 +... +yN − yN−1 +� +���� = −s +� +���� +h0,0 +0 +· · · +0 +h1,0 +h1,1 +· · · +0 +... +... +... +... +hN−1,0 +hN−1,2 +· · · +hN−1,N−1 +� +���� +� +���� +∇f (y0) +∇f (y1) +... +∇f (yN−1) +� +���� +Here, we call the lower triangular matrix H = (hij) the difference matrix for the algo- +rithm (46). +To derive the limiting ODE of the algorithm (46), we introduce a smooth curve X : +[0, T] → Rn with the identifications X(k√s) = yk and T = N√s. As a continuous-time +analog of the difference matrix (hij), we intoduce a continuously differentiable function +H (possibly depending on s now) defined on {(t, τ) ∈ R2 : 0 < τ ≤ t < T} with the +identification H(ti, τj) = hij, where ti = i√s and τj = j√s. Substituting X(ti) = yi in (46) +yields +X (ti+1) − X (ti) +√s += − (τj+1 − τj) +i +� +j=0 +H (ti, τj) ∇f (X (τj)) . +(47) +Then, we can observe that the right-hand side of (47) is a Riemann sum of the function +τ �→ −H(ti, τ)∇f(X(τ)) over [0, ti+1]. Thus, taking the limit s → 0 yields +˙X(t) = − +� t +0 +H(t, τ)∇f(X(τ)) dτ, where H(t, τ) = lim +s→0 h t +√s , τ +√s +(48) +17 + +Kim and Yang +as the limiting ODE of the fixed-step first-order scheme (46). Note that the form of this +equation clearly reflects the momentum effect because the gradient ∇f(X(τ)) at time τ +affects the velocity ˙X(t) at all times t after τ. Inspired by the observation that the function +H(t, τ) plays a role similar to the kernel function in the integral transform, we call it the +differential kernel (or the H-kernel) corresponding to the difference matrix (hij). +From differential kernels to second-order ODEs. +Differentiating both sides of (48) +and applying the Leibniz integral rule, we obtain +¨X(t) = −H(t, t)∇f(X(t)) − +� t +0 +∂H(t, τ) +∂t +∇f(X(τ)) dτ. +(49) +If there exists a function b(t) such that +∂H(t, τ) +∂t += −b(t)H(t, τ), +then it follows from (48) that the equation (49) is expressed as the following second-order +ODE: +¨X(t) + b(t) ˙X + H(t, t)∇f(X(t)) = 0. +(50) +Recovering the limiting ODE of two-sequence scheme. +We can write the two- +sequence scheme (42) as the fixed-step first-order scheme with +hij = (βj + γj) +i� +ν=j+1 +βν + δij, +where δij is the Kronecker delta funciton. For i > j,3 we have +hi+1,j − hi,j = (βi+1 − 1) hij. +Under the identification H(ti, τj) = hij, we have +hi+1,j − hi,j = H (ti+1, τj) − H (ti, τj) = ∂H (ti, τj) +∂t +√s + o +�√s +� +. +Thus, when the limits (43) exist, taking the limit s → 0 yields +∂H (t, τ) +∂t += −b(t)H (t, τ) . +(51) +Also, because hk+1,k = βk+1 +γk and lims→0 βt/√s = 1 by (43), we have H(t, t) = 1+c(t) for +all t ∈ (0, T). Therefore, the ODE (50) recovers the limiting ODE (44) of the two-sequence +scheme. Moreover, we can explicitly write the differential kernel H as +H(t, τ) = (1 + c(τ)) e− +� t +τ b(s) ds. +(52) +3. We exclude the case i = j because the difference matrix hij has singularities at these points due to the +Kronecker delta function. +18 + +Unifying NAG for Convex and Strongly Convex Objective Functions +Difference matrix for NAG-C. +Because we can write NAG-C as the two-sequence scheme +(42) with βk = k−1 +k+2 and γk = 0, we can rewrite it as the fixed-step first-order scheme (46) +with +hij = +i� +ν=j +ν − 1 +ν + 2 + δij = (j − 1)j(j + 1) +i(i + 1)(i + 2) + δij. +By definition, the differential kernel corresponding to this matrix (hij) is +H(t, τ) = lim +s→0 +� +τ +√s − 1 +� +τ +√s +� +τ +√s + 1 +� +t +√s +� +t +√s + 1 +� � +t +√s + 2 +� = τ 3 +t3 . +(53) +This can be also obtained by substituting b(t) = 3/t and c(t) = 0 into (52): +H(t, τ) = e− +� t +τ +3 +s ds = e−3(log(t)−log(τ)) = τ 3 +t3 . +Because +∂H(t, τ) +∂t += −3τ 3 +t4 = −3 +t H(t, τ), +the ODE (49) with (53) recovers NAG-C ODE (17). +Difference matrix for NAG-SC. +Because we can write NAG-SC as the two-sequence +scheme (42) with βk = 1−√µs +1+√µs and γk = 0, we can rewrite it as the fixed-step first-order +scheme (46) with +hij = +i� +ν=j +1 − √µs +1 + √µs + δij = +�1 − √µs +1 + √µs +�i−j+1 ++ δij. +By definition, the differential kernel corresponding to this matrix (hij) is +H(t, τ) = lim +s→0 +�1 − √µs +1 + √µs +� t +√s − τ +√s +1 += e2√µτ +e2√µt . +(54) +This can be also obtained by substituting b(t) = 2√µ and c(t) = 0 into (52): +H(t, τ) = e− +� t +τ 2√µ ds = e−2√µ(t−τ) = e2√µτ +e2√µt . +It follows from +∂H(t, τ) +∂t += −2√µe2√µ(τ−t) = −2√µH(t, τ) +that the ODE (49) with (54) recovers NAG-SC ODE (17). +19 + +Kim and Yang +3. Unified Bregman Lagrangian +In this section, we address the inconsistency between the first Bregman Lagrangian (20) +and the second Bregman Lagrangian (24). For a continuously differentiable strictly convex +function h, we define the unified Bregman Lagrangian as +L +� +X, ˙X, t +� += L1st +� +X, ˙X, t +� ++ L2nd +� +X, ˙X, t +� +− +� +L2nd +� +X, ˙X, t +�� +µ=0 += eα+γ �� +1 + µeβ� +Dh +� +X + e−αV, X +� +− eβf(X) +� +. +(55) +Then by construction, this Lagrangian recovers the first Bregman Lagrangian (20) when +µ = 0. Because the Lagrangian (55) is continuous in the strong convexity parameter µ, it is +a continuous extension of the first Bregman Lagrangian to the strongly convex case. +Proposition 2 Under the ideal scaling condition (21a), the Euler–Lagrange equation (22) +for the unified Bregman Lagrangian (55) reduces to the following system of ODEs: +˙X = eα(Z − X) +(56a) +d +dt∇h(Z) = +µ ˙βeβ +1 + µeβ (∇h(X) − ∇h(Z)) − +eα+β +1 + µeβ ∇f(X). +(56b) +The proof of Proposition 2 can be found in Appendix D.1. To analyze the convergence rate +of this dynamics, we define the time-dependent Lyapunov function V : Rn ×Rn ×[0, ∞) → R +as +V (X, Z, t) = +� +1 + µeβ(t)� +Dh (x∗, Z) + eβ(t) (f(X) − f (x∗)) . +(57) +Theorem 3 Suppose that the ideal scaling condition (21b) holds. Let f be a µ-uniformly +(possibly with µ = 0) convex function with respect to h. Then, for any solution (X, Z) to the +unified Bregman Lagrangian flow (56), the continuous-time energy function +E(t) = V (X(t), Z(t), t) = +� +1 + µeβ(t)� +Dh (x∗, Z(t)) + eβ(t) (f(X(t)) − f (x∗)) +is monotonically decreasing on [0, ∞). +The proof of Theorem 3 can be found in Appendix D.2. Writing E(t) ≤ E(0) explicitly, we +obtain an O(e−β(t)) convergence rate for the dynamics (56). +Corollary 4 Suppose that the ideal scaling condition (21b) holds. Let f be a µ-uniformly +(possibly with µ = 0) convex function with respect to h. Then, any solution (X, Z) to the +unified Bregman Lagrangian flow (56) satisfies the inequality +f(X(t)) − f (x∗) ≤ e−β(t) �� +1 + µeβ(0)� +Dh (x∗, Z(0)) + eβ(0) (f (X(0)) − f (x∗)) +� +for all t > 0. +Similarly to the first Bregman Lagrangian flow (23) and the second Bregman Lagrangian +flow (25) (see Wibisono et al., 2016; Wilson et al., 2021), the dynamical system (56) is closed +under time-dilation. +20 + +Unifying NAG for Convex and Strongly Convex Objective Functions +Theorem 5 Let T : I2 → I1 be an increasing continuously differentiable bijective function, +where I1 and I2 are intervals in [0, ∞). If (X1, Z1) is a solution to the unified Bregman +Lagrangian flow (56) on I1 with parameters α1, β1 : I1 → R, then the reparametrized curves +X2(t) = X1(T(t)) and Z2(t) = Z1(T(t)) is a solution to the unified Bregman Lagrangian +flow on I2 with the parameters α2, β2 : I2 → R defined by +α2(t) = α1(T(t)) + log ˙T(t) +β2(t) = β1(T(t)). +The proof of Theorem 5 can be found in Appendix D.3. +Recovering the first and second Bregman Lagrangians. +We now discuss how the +first Bregman Lagrangian flow (23), the second Bregman Lagrangian flow (25), and the +corresponding Lyapunov analyses can be recovered from the proposed unified Bregman +Lagrangian flow (56) and the corresponding Lyapunov analysis (Theorem 3). When µ = 0, it +is easy to check that the unified Bregman Lagrangian flow and the corresponding Lyapunov +function (57) recover the first Bregman Lagrangian flow and the corresponding Lyapunov +function (37). When the limits α(∞) := limt→∞ α(t) and ˙β(∞) := limt→∞ ˙β(t) > 0 exist, +the second Bregman Lagrangian flow with α2nd(t) :≡ α(∞) and β2nd(t) := ˙β(∞)t is the +asymptotic version of the unified Bregman Lagrangian flow with α(t) and β(t) in the sense +that the coefficients of (56) converge to the ones of (25) as t → ∞. In Appendix D.4, we +show that the Lyapunov analysis for the second Bregman Lagrangian flow with ˜α and ˜β can +be recovered from Theorem 3 by taking the limit t → ∞ of some inequalities. +4. Unified Methods for Minimizing Convex and Strongly Convex +Functions +In Section 4.1, we address the inconsistency between NAG-C ODE (17) and NAG-SC ODE (19) +by proposing an ODE model that unifies NAG-C ODE and NAG-SC ODE. In Section 4.2, we +address the inconsistency between NAG-C (9) and NAG-SC (8) by proposing novel algorithms +that can be viewed as a discrete-time counterpart of the unified NAG ODE. Throughout +this section, we assume the standard smooth strongly convex setting in Section 2.1. +4.1 Proposed dynamics: Unified NAG ODE +We consider the unified Bregman Lagrangian flow (56) with α(t) = log( 2 +t cothc( +√µ +2 t)), +β(t) = log( t2 +4 sinhc2( +√µ +2 t)),4 h(x) = 1 +2 ∥x∥2, and the initial conditions X(0) = Z(0) = x0, +which we call the unified NAG system: +˙X = 2 +t cothc +�√µ +2 t +� +(Z − X) +˙Z = t +2 tanhc +�√µ +2 t +� +(µX − µZ − ∇f(X)) . +(58) +4. We can constructively choose these functions (see Appendix E.1). Note that when µ = 0, we have +α(t) = log 2 +t and β(t) = log t2 +4 , which recover NAG-C ODE (17) from the first Bregman Lagrangian flow +(23). Also, as t → ∞, we have α(t) ∼ log √µ and β(t) ∼ √µt − log (4µ), which recover NAG-SC ODE +(19) from the second Bregman Lagrangian flow (25). +21 + +Kim and Yang +Writing this system in a single equation, we obtain the unified NAG ODE (see Appendix E.2): +¨X + +�√µ +2 tanh +�√µ +2 t +� ++ 3 +t cothc +�√µ +2 t +�� +˙X + ∇f(X) = 0 +(59) +with X(0) = x0 and ˙X(0) = 0. +Existence and uniqueness of the solution. +To prove the existence and uniqueness of +solution to the unified NAG system (58), we cannot directly apply the classical existence +and uniqueness theorem because the coefficient 2 +t cothc +� √µ +2 t +� +has a singularity at t = 0. +Thus, we follow the arguments in (Krichene et al., 2015; Su et al., 2016). +Theorem 6 The unified NAG system (58) has a unique solution (X, Z) in C1([0, ∞), Rn × +Rn). +The proof of Theorem 6 can be found in Appendix H.1. +1 +2 +3 +4 +5 +t +1 +2 +3 +4 +5 +6 +Coefficient of ˙X +NAG-C ODE +NAG-SC ODE +Unified ODE +(a) µ = 0.3 +1 +2 +3 +4 +5 +t +1 +2 +3 +4 +5 +6 +Coefficient of ˙X +NAG-C ODE +NAG-SC ODE +Unified ODE +(b) µ = 1 +1 +2 +3 +4 +5 +t +1 +2 +3 +4 +5 +6 +7 +Coefficient of ˙X +NAG-C ODE +NAG-SC ODE +Unified ODE +(c) µ = 5 +Figure 3: Plots for the coefficient of ˙X, which can be interpreted as a measure of friction. +Damping system interpretation. +As mentioned in (Su et al., 2014), the second-order +ODE (59) can be viewed as a damping system, and the coefficient of ˙X can be viewed as +a measure of friction. Because the coefficient of ˙X in NAG-SC ODE (19) is 2√µ, NAG-SC +ODE behaves like an underdamped system when µ is small. Thus, the flow generated by +NAG-SC ODE may present excessive oscillatory behaviors (see Figure 5). In the unified +NAG ODE (59), the coefficient of ˙X is large when t is small and converges to 2√µ as t → ∞ +(see Figure 3). Thus, the unified NAG ODE behaves like an overdamped system (which +displays less severe oscillations) when t is small, regardless of the value of µ. +Convergence analysis. +For the unified NAG system, the Lyapunov function (57) can be +written as +V (X, Z, t) = 1 +2 cosh2 +�√µ +2 t +� +∥Z − x∗∥2 + t2 +4 sinhc2 +�√µ +2 t +� +(f(X) − f (x∗)) . +(60) +Furthermore, we can rewrite Theorem 3 and Corollary 4 for this ODE model as follows: +22 + +Unifying NAG for Convex and Strongly Convex Objective Functions +Theorem 7 For the solution (X, Z) to the unified NAG system (58), the continuous-time +energy functional +E(t) = V (X(t), Z(t), t) += 1 +2 cosh2 +�√µ +2 t +� +∥Z(t) − x∗∥2 + t2 +4 sinhc2 +�√µ +2 t +� +(f(X(t)) − f (x∗)) +is monotonically decreasing on [0, ∞). +Corollary 8 The solution (X, Z) to the unified NAG system (58) satisfies the inequality +f(X(t)) − f (x∗) ≤ 2 +t2 cschc2 +�√µ +2 t +� +∥x0 − x∗∥2 +(61) +for all t > 0. +Since cschc2 is decreasing on [0, ∞), Corollary 8 implies that the unified NAG ODE +(59) achieves an O(∥x0 − x∗∥2/t2) convergence rate regardless of the value of µ ≥ 0. When +µ > 0, since +1 +t2 cschc2 � √µ +2 t +� +∼ µe−√µt as t → ∞, the unified NAG ODE achieves an +O(e−√µt∥x0 −x∗∥2) convergence rate. Combining these bounds, we conclude that the unified +NAG ODE achieves an +O +� +min +� +1/t2, e−√µt� +∥x0 − x∗∥2� +convergence rate. +Advantages of the unified NAG ODE compared to NAG-SC ODE. +We now remark +that our novel ODE model resolves the three drawbacks of NAG-SC ODE (19) discussed in +Section 1.1. +• While the solution to NAG-SC ODE may not converge to the minimizer of f when +µ = 0, the solution to the unified NAG ODE always converges to the minimizer +regardless of the value of µ. +• While the convergence guarantee for NAG-SC ODE may be worse than that for NAG-C +ODE in early stages, the convergence guarantee (61) for the unified NAG ODE is +always better than that for NAG-C ODE because cschc2 is decreasing on [0, ∞) and +the rate (61) recovers the exact convergence guarantee of NAG-C ODE when µ = 0. +• While the convergence rate of NAG-SC ODE involves both the initial squared distance +∥x0 − x∗∥2 and the initial function value accuracy f(x0) − f(x∗), the convergence rate +of the unified NAG ODE involves only the initial squared distance ∥x0 − x∗∥2. +Recovering NAG-C ODE and NAG-SC ODE. +We now discuss how NAG-C ODE (17), +NAG-SC ODE (19), and their convergence analyses can be recovered from the proposed +unified NAG ODE (59). When µ = 0, it is easy to check that the unified ODE recovers +NAG-C ODE and that the Lyapunov function (60) recovers (34) for NAG-C ODE. In the +unified NAG ODE, because the coefficient of ˙X converges to 2√µ as t → ∞ (see Figure 3), +NAG-SC ODE is the asymptotic version of the unified NAG ODE. In Appendix D.4, we +show that the Lyapunov analysis for NAG-SC ODE can be recovered from Theorem 7 by +taking the limit t → ∞ of some inequalities. +23 + +Kim and Yang +4.2 Proposed family of algorithms: Unified NAG family +Given the algorithmic stepsize s and a strictly increasing sequence (tk)∞ +k=0 (depending on +s) in [0, ∞) satisfying lims→0 t0 = 0, we consider the three-sequence scheme (3) with the +algorithmic paramameters5 +τk = +2√s +tk+1 cothc +� √µ +2 tk+1 +� +− µs +1 − µs +δk = +√stk+1 +2 +tanhc +�√µ +2 tk+1 +� +, +(62) +that is, we consider the following unified NAG family: +yk = xk + +2√s +tk+1 cothc +� √µ +2 tk+1 +� +− µs +1 − µs +(zk − xk) +xk+1 = yk − s∇f (yk) +zk+1 = zk + +√stk+1 +2 +tanhc +�√µ +2 tk+1 +� +(µyk − µzk − ∇f (yk)) . +(63) +Then, it is straightforward to check that the sequences (τk) and (δk) satisfy the collinearity +condition (4). The following remark indicates that this algorithm can be regarded as a +discretized version of the unified NAG system (58). +Remark 9 When the sequence (tk)∞ +k=0 in [0, ∞) satisfies the conditions (10) and (11), we +have +lim +s→0 +τk(t) +√s = 2 +t cothc +�√µ +2 t +� +lim +s→0 +δk(t) +√s = lim +s→0 +t +2 tanhc +�√µ +2 t +� += t +2 tanhc +�√µ +2 t +� +for all t > 0, where k is the inverse function of the sequence t. Thus, the result in Section 1.1 +implies that the unified NAG family (63) converges to the unified NAG system (58) as s → 0. +When µ > 0 and limk→∞ tk = ∞, we have limk→∞ τk = +√q +1+√q and limk→∞ δk = +� +s +µ. Thus, +NAG-SC (8) is the asymptotic version of the unified NAG family (63) in the sense that the +coefficients of the unified NAG family converge to the coefficients of NAG-SC. To obtain the +convergence rate of the unified NAG family, we introduce the following assumptions on the +sequence (tk):6 +2√s +tk +cothc +�√µ +2 tk +� +≤ 1 for k ≥ 2 +(64) +5. We can constructively choose these sequences: First, we observe the relationship δk = √sδ(tk+1), where +tk = k√s, between the algorithmic parameter δk = s(k+1) +2 +of NAG-C and the coefficient δ(t) = t +2 of NAG-C +system. Inspired by this relationship, for our algorithm, we define the sequence δk as δk = √sδ(tk+1), +where δ(t) = t +2 tanhc +� √µ +2 t +� +, and then set the sequence τk so that the collinearity condition (4) holds. +6. These assumption is purely inspired from the proof of Theorem 10. Note that the assumptions (10) +and (11) are not required for the convergence analysis. Note that when µ = 0, under the identification +θk = 2√s +tk , the unified NAG family is equivalent to (Tseng, 2008, Algorithm 1) and the condition (65) is +equivalent to (Tseng, 2008, Equation 15). +24 + +Unifying NAG for Convex and Strongly Convex Objective Functions +and +� +1 − 2√s +tk+1 +cothc +�√µ +2 tk+1 +�� t2 +k+1 +4 +sinhc2 +�√µ +2 tk+1 +� +≤ t2 +k +4 sinhc2 +�√µ +2 tk +� +for k ≥ 0. +(65) +The following results are the discrete-time analogs of Theorem 7 and Corollary 8. +Theorem 10 For the iterates of the unified NAG family (63) with (tk)∞ +k=0 satisfying the +conditions (64) and (65), the following discrete-time energy function is monotonically +decreasing: +Ek = V (xk, zk, tk) = 1 +2 cosh2 +�√µ +2 tk +� +∥zk − x∗∥2 + t2 +k +4 sinhc2 +�√µ +2 tk +� +(f (xk) − f (x∗)) , +(66) +where the Lyapunov function V is defined in (60). +The proof of Theorem 10 can be found in Appendix F.1. Writing Ek ≤ E0 explicitly, we +obtain the following result. +Corollary 11 For the iterates of the unified NAG family (63) with (tk)∞ +k=0 satisfying the +conditions (64) and (65), the following inequality holds for all k ≥ 0: +f (xk) − f (x∗) ≤ 4 +t2 +k +cschc2 +�√µ +2 tk +� +× +�1 +2 cosh2 +�√µ +2 t0 +� +∥x0 − x∗∥2 + t2 +0 +4 sinhc2 +�√µ +2 t0 +� +(f (x0) − f (x∗)) +� +. +(67) +In the following subsections, we propose two concrete algorithms with specific choices of +the sequence (tk). In Section 4.2.1, we propose the unified NAG, a simple unified algorithm +which continuously extend NAG-C (9) to the strongly convex setting. In Section 4.2.2, we +constructively recover the original NAG (5) and its convergence rate from the unified NAG +family (63). +4.2.1 Constant timestep scheme: Unified NAG +First, we set the constant timestep δ as +δ = +� +� +� +− +log(1−√µs) +√µ +, +if µ > 0 +√s, +if µ = 0, +(68) +and then define the sequence tk = kδ, that is, +tk = +� +− log(1−√µs) +√µ +k, +if µ > 0 +√sk, +if µ = 0. +(69) +Note that this choice is same as the previous choices of tk for NAG-C and NAG-SC in +Section 2.2. For this specific sequence (tk), the unified NAG family (63) can be written +25 + +Kim and Yang +simply as +yk = xk + +2 +ι(k+1) cothc +� k+1 +2 ι√µs +� +− µs +1 − µs +(zk − xk) +xk+1 = yk − s∇f (yk) +zk+1 = zk + ιs(k + 1) +2 +tanhc +�k + 1 +2 +ι√µs +� +(µyk − µzk − ∇f (yk)) , +(70) +where ι = − log(1−√µs) +√µs +for µ > 0 and ι = 1 for µ = 0. We refer to this algorithm as the +unified NAG. +The sequence (tk) in (69) can be shown to satisfy the conditions (64) and (65) (see +Section F.2), and thus the convergence guarantee (67) holds for this specific algorithm. Also +it is straightforward to check that the conditions (10) and (11) hold, and thus the unified +NAG (70) converges to the unified NAG system (58) as s → 0. Because cschc2 is decreasing +on [0, ∞) and δ ≥ √s, we have +4 +t2 +k +cschc2 +�√µ +2 tk +� +≤ 4 +t2 +k += +4 +δ2k2 ≤ +4 +sk2 . +This implies that the convergence guarantee of the unified NAG is always better than that of +NAG-C and that the unified NAG achieves an O(∥x0 − x∗∥2/k2) convergence rate, regardless +of the value of µ. When µ > 0, since +4 +t2 +k +cschc2 +�√µ +2 tk +� +∼ 4µe−√µtk = 4µ (1 − √µs)k as k → ∞, +the unified NAG achieves an O((1 − √µs)k∥x0 − x∗∥2) convergence rate. Combining these +two guarantees, we conclude that the unified NAG achieves an +O +� +min +� +1/k2, (1 − √µs)k� +∥x0 − x∗∥2� +convergence rate. +Advantages of the unified NAG compared to NAG-SC. +We now highlight that the +unified NAG resolves the three drawbacks of NAG-SC (9) discussed in Section 1. +• While NAG-SC cannot handle the non-strongly convex case, the unified NAG can +handle the case µ = 0. Moreover, when µ = 0, the unified NAG and its convergence +rate (67) recover NAG-C and its convergence rate. +• While the convergence guarantee for NAG-SC may be worse than that for NAG-C in +early stages, the convergence guarantee for the unified NAG is always better than that +for NAG-C. +• While the convergence rate of NAG-SC involves both the initial squared distance +∥x0 − x∗∥2 and the initial function value accuracy f(x0) − f(x∗), the convergence rate +of the unified NAG involves only the initial squared distance ∥x0 − x∗∥2. +26 + +Unifying NAG for Convex and Strongly Convex Objective Functions +4.2.2 Adaptive timestep scheme: Recovering the original NAG +The constant timestep scheme (unified NAG) in the previous section can be improved in +terms of the convergence rate by defining the sequence (tk)∞ +k=0 more aggressively as +tk+1 = +� +Given constant t0 > 0 (possibly depending on s), +k + 1 = 0 +The largest real number satisfying (65), +k + 1 ≥ 1. +(71) +Then, it is easy to check that the sequence (tk)∞ +k=0 is well-defined and strictly increasing. +We refer to the unified NAG family (63) with this sequence as the adaptive timestep scheme. +Note that the conditions (64) and (65) hold by construction.7 Therefore, the convergence +guarantee (67) holds for the adaptive timestep scheme. In Section F.3, we show that if +t0 → 0 as s → 0, then the conditions (10) and (11) hold, and thus the adaptive timestep +scheme converges to the unified NAG system (58) as s → 0. By construction, we have +tk+1 − tk > δ, where δ is defined in (68), which implies that tk > t0 + kδ for all k ≥ 0. +Thus, the adaptive timestep scheme has a (slightly) better convergence rate than the unified +NAG. Surprisingly, our new algorithm, which is purely obtained from the unified Lagrangian +framework, is equivalent to the original Nesterov’s method (5). +Proposition 12 The adaptive timestep scheme is equivalent to the original NAG (5) with +γ0 = 4 +t2 +0 cothc2 � √µ +2 t0 +� +> µ. Moreover, the sequence γk and αk in the original NAG can be +written as γk = 4 +t2 +k cothc2 � √µ +2 tk +� +and αk = 2√s +tk+1 cothc +� √µ +2 tk+1 +� +. Conversely, the original +NAG (5) with γ0 > µ is equivalent to the adaptive timestep scheme, where t0 satisfies +γ0 = 4 +t2 +0 cothc2 � √µ +2 t0 +� +. +The proof of Proposition 12 can be found in Section F.4. The following remark shows +that under the identification in Theorem 12, the convergence rate (67) of the adaptive +timestep scheme is equivalent to the convergence rate (7) of the original NAG obtained by +Nesterov (2018). +7. The first condition follows from the facts that (64) holds for the sequence tk = kδ (see Section 4.2.1) and +we have tk > 2δ for k ≥ 2 for the sequence (71). +27 + +Kim and Yang +Remark 13 By Corollary 11, the iterates of the adaptive timestep scheme satisfy +f (xk) − f (x∗) +≤ 4 +t2 +k +cschc2 +�√µ +2 tk +� +× +�1 +2 cosh2 +�√µ +2 t0 +� +∥x0 − x∗∥2 + t2 +0 +4 sinhc2 +�√µ +2 t0 +� +(f (x0) − f (x∗)) +� += 4 +t2 +k +cschc2 +�√µ +2 tk +� t2 +0 +4 sinhc2 +�√µ +2 t0 +� +× +� 2 +t2 +0 +cothc2 +�√µ +2 t0 +� +∥x0 − x∗∥2 + (f (x0) − f (x∗)) +� += +k−1 +� +i=0 +� +1 − 2√s +ti+1 +cothc +�√µ +2 ti+1 +�� � 2 +t2 +0 +cothc2 +�√µ +2 t0 +� +∥x0 − x∗∥2 + (f (x0) − f (x∗)) +� +, +(72) +where the last equality follows from our updating rule (71) of the sequence (tk). Therefore, +we recover the convergence rate (7) of the original NAG with γk = +4 +t2 +k cothc2 � √µ +2 tk +� +and +αk = 2√s +tk+1 cothc +� √µ +2 tk+1 +� +. +5. Extension to Higher-order Non-Euclidean Setting +Based on the first Bregman Lagrangian (20) and the prior work (Baes, 2009), Wibisono +et al. (2016) proposed the accelerated tensor flow and accelerated tensor method for convex +objective functions to achieve a polynomial O(1/tp) or O(1/kp) convergence rate. They also +tried to design accelerated tensor methods for uniformly convex objective functions to achieve +an exponential convergence rate. They were able to obtain an exponential convergence rate +for continuous-time flows obtained from the first Bregman Lagrangian, but a rate-matching +discretization was not identified. Instead, they showed that the accelerated tensor method +(convex case) with a restart scheme achieves an exponential convergence rate for uniformly +convex objective functions. However, as they admitted, understanding the connection +between the discrete-time algorithm and the continuous-time flow is unclear and remains as +an open problem. +In this section, using the unified Bregman Lagrangian (55), we continuously extend to +the accelerated tensor flow and the accelerated tensor method in (Wibisono et al., 2016) to +the strongly convex case. Our novel dynamics and algorithm achieve exponential convergence +rates without using a restarting technique. +We make the following assumptions throughout this section: +• The distance-generating function h is 1-uniformly convex (29) of order p ≥ 2. +• The objective function f is µ-uniformly (possibly with µ = 0) convex (33) with respect +to the distance-generating function h. +• The objective function f is (p−1)! +s +-smooth of order p−1 (30), where s is the algorithmic +stepsize. +28 + +Unifying NAG for Convex and Strongly Convex Objective Functions +These assumptions are standard in the literature of higher-order optimization (see Nesterov, +2008; Baes, 2009; Wibisono et al., 2016; Gasnikov et al., 2019; Wilson et al., 2021). In +particular, when p = 2 and h(x) = ∥x∥2, these assumptions recover the standard smooth +strongly convex setting in Section 2.1. +Following (Wibisono et al., 2016), we define the tensor update operator Gp,s,N : Rn → Rn +as +Gp,s,N(y) = arg min +x +� +fp−1(x; y) + N +ps ∥x − y∥p +� +, +(73) +where the function x �→ fp−1(x, y) = �p−1 +i=0 +1 +i!∇if(y)(x − y)i is the (p − 1)-st order Taylor +approximation of the objective function f at y ∈ Rn. Wibisono et al. (2016, Lemma 2.2) +showed that one can choose N > 0 so that there exists a constant M > 0 for which the +inequality +⟨∇f (x) , y − x⟩ ≥ Ms +1 +p−1 ∥∇f (x)∥ +p +p−1 +(74) +holds for x = Gp,s,N(y). From now on, we denote the tensor update operator satisfying +the inequality (74) by Gp,M. As a special case, when p = 2, the operator (73) with N = 1 +satisfies the inequality (74) with M = 1/2.8 +5.1 Proposed dynamics: Unified accelerated tensor flow +We consider the unified Bregman Lagrangian flow (56) with the parameters +α(t) = log p − log t + log +� +cothcp +� +C1/pµ1/pt +�� +β(t) = p log t + log C + p log +� +sinhcp +� +C1/pµ1/pt +�� +(75) +and the initial conditions X(0) = Z(0) = x0, where C > 0 is a constant. It is straightforward +to check that the ideal scaling condition (21b) holds. This dynamical system can be written +as +˙X = p +t cothcp +� +C1/pµ1/pt +� +(Z − X) +d +dt∇h(Z) = Cptp−1 tanhcp−1 +p +� +C1/pµ1/pt +� +(µ∇h(X) − µ∇h(Z) − ∇f(X)) . +(76) +From now on, we refer to this system of ODEs as the unified accelerated tensor flow. Using +the existence and uniqueness of solution to the unified NAG system (Theorem 6) and the +time-dilation property (Theorem 5), we can prove the following theorem (see Appendix H.2). +Theorem 14 The unified accelerated tensor flow (58) has a unique solution (X, Z) in +C1([0, ∞), Rn × Rn). +For this dynamical system, the Lyapunov function (57) can be expressed as +V (X, Z, t) = coshp +p +� +C1/pµ1/pt +� +Dh (x∗, Z) + Ctp sinhcp +p +� +C1/pµ1/pt +� +(f(X) − f (x∗)) . (77) +We can rewrite Theorem 3 and Corollary 4 for the unified accelerated tensor flow (76) as +follows: +8. See, for example, the proof of Lemma 6 in the arXiv version of (Wilson et al., 2021): arXiv:1611.02635v4. +29 + +Kim and Yang +Theorem 15 For the solution (X, Z) to the unified accelerated tensor flow (76), the +continuous-time energy function +E(t) = V (X(t), Z(t), t) += coshp +p +� +C1/pµ1/pt +� +Dh (x∗, Z(t)) + Ctp sinhcp +p +� +C1/pµ1/pt +� +(f(X(t)) − f (x∗)) +is monotonically decreasing on [0, ∞). +Corollary 16 The solution (X, Z) to the unified accelerated tensor flow (76) satisfies the +inequality +f(X(t)) − f (x∗) ≤ +1 +Ctp sinhcp +p +� +C1/pµ1/pt +�Dh (x∗, x0) +(78) +for all t > 0. +Since sinhcp(0) = 1 and sinhcp is increasing on [0, ∞) (see Appendix B.2), Corollary 16 +implies that the unified accelerated tensor flow (76) achieves an O (Dh (x∗, x0) /tp) conver- +gence rate regardless of the value of µ ≥ 0. On the other hand, when µ > 0, it follows from +Proposition 1 that +1 +Ctp sinhcp +p +� +C1/pµ1/pt +� = O +� +e−pC1/pµ1/pt� +as t → ∞. +Therefore, the unified accelerated tensor flow achieves an O(e−pC1/pµ1/ptDh (x∗, x0)) conver- +gence rate. Combining these bounds, we conclude that the unified accelerated tensor flow +achieves an +O +� +min +� +1/tp, e−pC1/pµ1/pt� +Dh (x∗, x0) +� +convergence rate. +5.2 Proposed algorithm: Unified accelerated tensor method +As a discretization scheme for the unified accelerated tensor flow (76), we propose the +following unified accelerated tensor method family: +Ak = Ctp +k sinhcp +p +� +C1/pµ1/ptk +� +(79a) +yk = xk + Ak+1 − Ak +Ak+1 +(zk − xk) +(79b) +xk+1 = Gp,M (yk) +(79c) +zk+1 = arg min +z +�Ak+1 − Ak +1 + µAk +(⟨∇f (xk+1) , z⟩ + µDh (z, xk+1)) + Dh (z, zk) +� +, +(79d) +where (tk) is a strictly increasing sequence (depending on the algorithmic stepsize s) in +[0, ∞) and Gp,M is the tensor update operator satisfying (74). Because the algorithm (79) is +continuous in the strong convexity parameter µ, it handles the convex case and the strongly +30 + +Unifying NAG for Convex and Strongly Convex Objective Functions +convex case in a unified way. By the first-order optimality condition, the step (79d) is +equivalent to +∇h (zk+1) − ∇h (zk) = Ak+1 − Ak +1 + µAk +(µ∇h (xk+1) − µ∇h (zk+1) − ∇f (xk+1)) . +(80) +Although the scheme (79) cannot be written in the three-sequence form (3), we observe that +the step (79b) plays a role of (3a) (updating yk as a convex combination of xk and zk), the +step (79d) plays a role similar to (3c) (updating zk by gradient/mirror step), and that the +tensor update step (79c) corresponds to the gradient update step (3b). +Limiting ODE. +In Appendix G.1, we show that if +lim +s→0 t0 = 0 +(81) +and the timesteps are asymptotically equivalent to s1/p as s → 0 in the sense that +lim +s→0 +tk(t)+1 − t +s1/p += 1 for all t ∈ (0, ∞) , +(82) +where k is the inverse of t, then the unified accelerated tensor method family (79) converges +to the unified accelerated tensor flow (76) when letting xk = X(tk) and zk = Z(tk). +Convergence analysis. +To prove the convergence rate, we introduce the following as- +sumption on the sequence (tk) (note that Ak is uniquely determined by tk and vice versa): +(Ak+1 − Ak)p − CppsAp−1 +k+1 (1 + µAk) ≤ 0 with C = 1 +p +� M +p − 1 +�p−1 +, +(83) +where M is the constant involved in (74). The following results are the discrete-time analogs +of Theorem 15 and Corollary 16. +Theorem 17 For the iterates of the unified accelerated tensor method family (79) with (tk) +satisfying the condition (83), the discrete-time energy function +Ek = (1 + µAk) Dh (x∗, zk) + Ak (f (xk) − f (x∗)) +(84) +is monotonically decreasing. +The proof of Theorem 17 can be found in Appendix G.2. Writing Ek ≤ E0 explicitly, we +obtain the following result. +Corollary 18 For the iterates of the unified accelerated tensor method family (79) with (tk) +satisfying the condition (83), the following inequality holds for all k ≥ 0: +f (xk) − f (x∗) ≤ 1 +Ak +((1 + µA0) Dh (x∗, x0) + A0 (f (x0) − f (x∗))) . +(85) +31 + +Kim and Yang +Specific algorithm: Unified accelerated tensor method. +We now consider the fol- +lowing specific choice of sequence (tk): +tk+1 = +� +0, +k + 1 = 0 +The largest real number satisfying (83), +k + 1 ≥ 1. +(86) +Then, the condition (83) clearly holds, and thus the convergence results hold. In addition, we +can show that this sequence satisfies the conditions (81) and (82) (see Appendix G.1). Hence, +the algorithm converges to the unified accelerated tensor flow (76) as s → 0. Furthermore, +we can show that the inequalities +Ak ≥ O (kp) , +Ak ≥ O +�� +1 + C1/ppµ1/ps1/p�k� +hold (see Appendix G.3). Therefore, Corollary 18 implies the following convergence rate: +f (xk) − f (x∗) ≤ O +� +min +� +1/kp, +� +1 + C1/ppµ1/ps1/p�−k�� +. +5.3 Recovering the non-strongly convex case +When µ = 0, the system of ODEs (76) recovers the following accelerated tensor flow (convex +case) given in (Wibisono et al., 2016):9 +˙X = p +t (Z − X) +d +dt∇h(Z) = −Cptp−1∇f(X). +(87) +Moreover, the unified accelerated tensor method family (79) becomes the following family: +Ak = Ctp +k +yk = xk + Ak+1 − Ak +Ak+1 +(zk − xk) +xk+1 = Gp,M (yk) +zk+1 = arg min +z +{(Ak+1 − Ak) ⟨∇f (xk+1) , z⟩ + Dh (z, zk)} . +(88) +This recovers the accelerated tensor method (convex case) in (Wibisono et al., 2016) if the +sequence (tk) is chosen as +tk = s1/pk1/p(k + 1)1/p · · · (k + p − 1)1/p, +(89) +for which the inequality (83) holds with µ = 0. +9. This flow can be obtained by putting α(t) = log p − log t and β(t) = p log t + log C (Equation 75 with +µ = 0) in the first Bregman Lagrangian flow (23). +32 + +Unifying NAG for Convex and Strongly Convex Objective Functions +6. Further Exploration: ODE Model for Minimizing Gradient Norms of +Strongly Convex Functions +So far, we have focused on ODEs and algorithms that achieve a fast convergence rate +for the accuracy of objective function values f(X(t)) − f(x∗) or f(xk) − f(x∗). Typically, +the goal of numerically solving a convex optimization problem is to reduce the deviation +from the minimum value. Alternatively, the gradient norm ∥∇f(xk)∥2 can be used as a +performance measure. This criterion is often reasonable for both theoretical and practical +purposes (see Nesterov, 2012; Diakonikolas and Wang, 2022). Recently, Kim and Fessler +(2021) proposed OGM-G, which is a method that achieves the optimal convergence rate (up +to a constant factor) for minimizing the gradient norm ∥∇f(xN)∥2 of non-strongly convex +functions. Recently, this method has attracted some attention: Lee et al. (2021) provided a +Lyapunov argument for its convergence analysis. Suh et al. (2022) derived and analyzed +the limiting ODE of OGM-G. However, most studies on OGM-G have focused only on the +non-strongly convex case. +In this section, we propose a novel continuous-time dynamical system that reduces the +squared gradient norm ∥∇f(X(T))∥2 of strongly convex objective functions f with an +O +� +min +� +1/T 2, e−√µT � � +f(x0) − f (x∗) + µ +2 ∥x0 − X(T)∥2�� +convergence rate. Interestingly, the ODE model presented in this section and the unified +NAG ODE (59) have an anti-transpose relationship between the corresponding differential +kernels. +6.1 Motivation: Symmetric relationship between OGM ODE and OGM-G +ODE +For non-strongly convex objective functions, Suh et al. (2022) proposed OGM-G ODE, an +ODE model whose solution X : [0, T] → Rn reduces the squared gradient norm ∥∇f(X(T))∥2 +with an O((f(x0) − f(x∗))/T 2) convergence rate. In this section, we investigate a symmetric +relationship between OGM ODE (which we will discuss later) and OGM-G ODE. This +relationship will give us a hint for designing our novel ODE model. +Anti-transpose relationship between OGM and OGM-G. +We first review a sym- +metric relationship between OGM (Kim and Fessler, 2016), an algorithm for reducing the +function value accuracy f(xN) − f(x∗), and OGM-G (Kim and Fessler, 2021), an algorithm +for reducing the squared gradient norm ∥∇f(xN)∥2. Given the number N of total iterations, +define a sequence (θk)N +k=0 as +θk = +� +� +� +� +� +� +� +� +� +1 +if k = 0 +1+ +� +4θ2 +k−1+1 +2 +if 1 ≤ k ≤ N − 1 +1+ +� +8θ2 +k−1+1 +2 +if k = N. +(90) +Then, OGM is equivalent to the fixed-step first-order scheme (46) with the difference matrix +HF, and OGM-G is equivalent to the fixed-step first-order scheme (46) with the difference +33 + +Kim and Yang +matrix HG, where the entries of HF and HG are defined as +hF +ij = +� +� +� +� +� +� +� +θi−1 +θi+1 hi−1,j +if k = 0, . . . , i − 2, +θi−1 +θi+1 (hi−1,i−1 − 1) +if k = i − 1, +1 + 2θi−1 +θi+1 +if k = i, +hG +ij = +� +� +� +� +� +� +� +θN−i−1−1 +θN−i +hi,j+1 +if k = 0, . . . , i − 2, +θN−i−1−1 +θN−i +(hi,i − 1) +if k = i − 1, +1 + 2θN−i−1−1 +θN−i +if k = i. +(91) +Kim and Fessler (2021) observed the following relationship between the difference kernels +for OGM and OGM-G: +hF +ij = hG +N−1−j,N−1−i for all i and j. +(92) +When the condition (92) holds, we say there is an anti-transpose relationship between HF +and HG because the matrix HF can be obtained by reflecting HG about its anti-diagonal +and vice versa. +A (naive) symmetric relationship between OGM ODE and OGM-G ODE. +Next, +we look at the relationship between the limiting ODEs of OGM and OGM-G. When letting +T = N√s and xk = X(tk), OGM converges to the ODE +¨X + 3 +t +˙X + 2∇f(X) = 0 +(93) +with X(0) = x0 and ˙X(0) = 0 (see Appendix I.1). Because this ODE is equivalent to the +first Bregman Lagrangian flow (23) with α(t) = log 2 +t and β(t) = log t2 +2 , its solution reduces +the function value accuracy f(X(T)) − f(x∗) with an O(∥x0 − x∗∥2/T 2) convergence rate. +Under the same setting, Suh et al. (2022) showed that OGM-G converges to the ODE +¨X + +3 +T − t +˙X + 2∇f(X) = 0 +(94) +with X(0) = x0 and +˙X(0) = 0, and showed that the solution to this ODE reduces the +squared gradient norm ∥∇f(X(T))∥2 with an O(f(x0) − f(x∗)/T 2) convergence rate. We +can observe that the coefficients in (94) can be obtained by substituting t with T − t into +the coefficient in (93) and vice versa. +Based on the symmetric relationship between OGM ODE and OGM-G ODE, one +might intuitively think that “OGM-G ODE is a time-reversed version of OGM ODE.” +This interpretation, however, might be misleading because the solution to OGM ODE and +the solution to OGM-G ODE do not have a time-reversed relationship. In the following +paragraph, using the differential kernel (48), we present a different, conceivably more accurate, +symmetrical relationship between the two ODEs. +34 + +Unifying NAG for Convex and Strongly Convex Objective Functions +Anti-transpose relationship between OGM ODE and OGM-G ODE. +Substitut- +ing bF(t) = 3/t, bG(t) = 3/(T − t), and cF(t) = cG(t) = 1 in (52), the differential kernels +HF(t, τ) corresponding to OGM ODE and HG(t, τ) corresponding to OGM-G ODE can be +computed as +HF(t, τ) = 2τ 3 +t3 +HG(t, τ) = 2(T − t)3 +(T − τ)3 . +Here, we can observe the following anti-transpose relationship between two differential +kernels: +HF(t, τ) = HG(T − τ, T − t). +(95) +Note that this can also be obtained by using the definition of the differential kernel and +the anti-transpose relationship (92) between two matrices HF and HG defined in (91). To +summarize, the relationships between OGM, OGM-G, and their limiting ODEs are illustrated +in Figure 4. +˙X(t) = − +� t +0 HF(t, τ)∇f(X(τ)) dτ +(OGM ODE) +y = −sHF∇f(y) +(OGM) +y = −sHG∇f(y) +(OGM-G) +˙X(t) = − +� t +0 HG(t, τ)∇f(X(τ)) dτ +(OGM-G ODE) +limiting +limiting +HF(t, τ) = HG(T − τ, T − t) +hF +i,j = hG +N−1−j,N−1−i +Figure 4: Anti-transpose relationships between OGM (reducing the function value accuracy), +OGM-G (reducing the gradient norm), and their limiting ODEs. +A failed attempt to design an ODE that minimizes the gradient norm of strongly +convex functions. +A downside of OGM-G ODE (94) is that it exploits only the non- +strong convexity of the objective function f. Thus, one might want to design an ODE model +that minimizes the gradient norm of strongly convex objective functions. Inspired by the +symmetric relationship between OGM ODE and OGM-G ODE, one might substitute t with +T − t into the coefficients in NAG-SC ODE (19) to yield the following ODE: +¨X + 2√µ ˙X + ∇f(X) = 0, +(96) +and one might guess that the solution to this ODE reduces the squared gradient norm +∥∇f(X(T))∥2 with an O(e−√µT ) convergence rate. However, one cannot easily modify the +argument in (Suh et al., 2022) to prove the convergence rate of the gradient norm for (96) +because their argument depends on the property ˙X(T) = 0, which is not true for the solution +to (96). +35 + +Kim and Yang +6.2 Proposed dynamics: Unified NAG-G ODE +In this subsection, we claim that the symmetric counterpart of the unified NAG ODE (59) +works well for our purpose, unlike the aforementioned failed attempt. The property that the +unified NAG ODE is a continuous extension of NAG-C ODE allows us to use the argument +in (Suh et al., 2022, Section 4.1). Substituting t with T − t into the coefficients in the unified +NAG ODE (59), we obtain the following ODE: +¨X + +�√µ +2 tanh +�√µ +2 (T − t) +� ++ +3 +T − t cothc +�√µ +2 (T − t) +�� +˙X + ∇f(X) = 0. +(97) +We refer to this ODE with the initial conditions X(0) = x0 and ˙X(0) = 0 as the unified NAG - +G ODE. Clearly, this ODE has a unique solution in C1([0, T), Rn).10 We can continuously +extend this solution to t = T with ˙X(T) = 0 and ¨X(T) = limt→T − +˙X(t) +t−T = 1 +2∇f(X(T)) (see +Appendix I.3). To analyze the convergence rate, we use the Lyapunov analysis again. +Theorem 19 For the solution X : [0, T] → Rn to the unified NAG-G ODE (97), the +continuous-time energy function +E(t) = +4 +(T − t)2 cschc2 +�√µ +2 (T − t) +� +(f(X(t)) − f(X(T))) +− +8 +(T − t)4 cschc4 +�√µ +2 (T − t) +� +∥X(t) − X(T)∥2 ++ +8 +(T − t)4 cschc2 +�√µ +2 (T − t) +� +cothc2 +�√µ +2 (T − t) +� +× +����X(t) + T − t +2 +tanhc +�√µ +2 (T − t) +� +˙X(t) − X(T) +���� +2 +. +(98) +is monotonically decreasing on [0, T). +The proof of Theorem 19 can be found in Appendix I.2. By L’Hˆopital’s rule, we have +lim +t→T − +f(X(t)) − f(X(T)) +(T − t)2 += lim +t→T − +1 +2 +� ˙X(t) +t − T , ∇f(X) +� += 1 +4 ∥∇f(X(T))∥2 +lim +t→T − +X(t) − X(T) +(T − t)2 += lim +t→T − +˙X(t) +2(t − T) = 1 +4∇f(X(T)). +It follows from cschc(0) = cothc(0) = 1 that +lim +t→T − E(t) += lim +t→T − +� +�4 · f(X(t)) − f(X(T)) +(T − t)2 +− 8 +���� +X(t) − X(T) +(T − t)2 +���� +2 ++ 8 +����� +X(t) − X(T) +(T − t)2 +− +˙X(t) +2(t − T) +����� +2� +� +10. Sketch of the proof: For any ϵ ∈ (0, T/2), the existence and uniqueness of solution on [0, T −ϵ] follows from +Cauchy-Lipschitz theorem (Teschl, 2012, Theorem 25). Paste these solutions on [0, T) = ∪ϵ∈(0,T/2)[0, T−ϵ). +36 + +Unifying NAG for Convex and Strongly Convex Objective Functions += ∥∇f(X(T))∥2 − 1 +2 ∥∇f(X(T))∥2 + 0 += 1 +2 ∥∇f(X(T))∥2 . +Writing limt→T − E(t) ≤ E(0) explicitly, we obtain the following result. +Corollary 20 The solution X to the unified NAG-G ODE (97) satisfies the inequality +∥∇f(X(T))∥2 ≤ 8 +T 2 cschc2 +�√µ +2 T +� � +f(x0) − f (X(T)) + µ +2 ∥x0 − X(T)∥2� +≤ 8 +T 2 cschc2 +�√µ +2 T +� � +f(x0) − f (x∗) + µ +2 ∥x0 − X(T)∥2� +. +(99) +Since cschc2 is decreasing on [0, ∞), Corollary 20 implies that the unified NAG-G ODE (58) +reduces the squared gradient norm with an O +� +1/T 2� +convergence rate regardless of the value +of µ ≥ 0. When µ > 0, since +1 +T 2 cschc2 � √µ +2 T +� +∼ µe−√µT as T → ∞, the unified NAG-G +ODE reduces the squared gradient norm with an O +� +e−√µT � +convergence rate. Combining +these bounds, we conclude that the unified NAG-G ODE reduces the squared gradient norm +with the following convergence rate: +∥∇f(X(T))∥2 ≤ O +� +min +� +1/T 2, e−√µT � � +f(x0) − f (x∗) + µ +2 ∥x0 − X(T)∥2�� +. +Anti-transpose relationship between the unified NAG ODE and the unified +NAG-G ODE. +The differential kernels HF(t, τ) corresponding to the unified NAG ODE +and HG(t, τ) corresponding to the unified NAG-G ODE can be computed as (see Ap- +pendix E.2) +HF(t, τ) = +τ 3 sinhc3 � √µ +2 τ +� +cosh +� √µ +2 τ +� +t3 sinhc3 � √µ +2 t +� +cosh +� √µ +2 t +� +HG(t, τ) = +(T − t)3 sinhc3 � √µ +2 (T − t) +� +cosh +� √µ +2 (T − t) +� +(T − τ)3 sinhc3 � √µ +2 (T − τ) +� +cosh +� √µ +2 (T − τ) +�. +Remarkably, there is an anti-transpose relationship (95) between these differential kernels, +like the one between the differential kernels corresponding to OGM ODE (which minimizes +the function value accuracy, similarly to what the unified NAG ODE does) and OGM-G +ODE (which minimizes the gradient norm, similarly to what the unified NAG-G ODE does). +7. Numerical Experiments +In this section, we validate the performance of the unified NAG (70) for a toy problem +and the logistic regression problem, and we also compare our method with NAG-C (9) and +NAG-SC (8). For each problem, we empirically observed that the unified NAG attains the +advantages of both NAG-C and NAG-SC. +37 + +Kim and Yang +Toy problem. +We consider the problem +min +(x,y)∈R2 f(x, y) = µ +2 x2 + 0.005y2. +(100) +This problem is strongly convex with parameter min {µ, 0.01}. We set the initial point +and the algorithmic stepsize as (x0, y0) = (1, 1) and s = 1. When µ is large (µ = 10−3), +Figure 5(a) shows that NAG-SC outperforms NAG-C and that the unified NAG behaves like +NAG-SC. When µ = 10−4, Figure 5(b) shows that the unified NAG behaves like NAG-C in +the early stages and behaves like NAG-SC in the late stages. When µ is small (µ = 10−7), +Figure 5(c) shows that NAG-C outperforms NAG-SC at least in the early stages and that +the unified NAG behaves like NAG-C. In each case, the performance of the unified NAG +is comparable to the better choice between NAG-C and NAG-SC. The trajectories of the +algorithms are shown in Figures 5(d), 5(e), and 5(f). We can see that NAG-SC converges +with more severe oscillation compared to NAG-C and the unified NAG, particularly when the +strong convexity parameter µ is small. This result matches the damping system interpretation +in Section 4.1: NAG-SC behaves like an underdamped system when µ is small, while our +unified NAG always behaves like an overdamped system in the early stages. +100 +101 +102 +103 +Iterations +10−26 +10−22 +10−18 +10−14 +10−10 +10−6 +10−2 +f (xk) − f (x∗) +GD +NAG-C +NAG-SC +Unified NAG +(a) Errors f − f ∗, µ = 10−3 +100 +101 +102 +103 +Iterations +10−10 +10−8 +10−6 +10−4 +10−2 +f (xk) − f (x∗) +GD +NAG-C +NAG-SC +Unified NAG +(b) Errors f − f ∗, µ = 10−4 +100 +101 +102 +103 +Iterations +10−7 +10−6 +10−5 +10−4 +10−3 +f (xk) − f (x∗) +GD +NAG-C +NAG-SC +Unified NAG +(c) Errors f − f ∗, µ = 10−7 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +x +−0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +y +GD +NAG-C +NAG-SC +Unified NAG +(d) Trajectories, µ = 10−3 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +x +−0.6 +−0.4 +−0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +y +GD +NAG-C +NAG-SC +Unified NAG +(e) Trajectories, µ = 10−4 +0.960 +0.965 +0.970 +0.975 +0.980 +0.985 +0.990 +0.995 +1.000 +x +−0.75 +−0.50 +−0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +y +GD +NAG-C +NAG-SC +Unified NAG +(f) Trajectories, µ = 10−7 +Figure 5: Results for the problem with the objective function f(x, y) = µ +2x2 + 0.005y2 and +the initial state x0 = (1, 1). +ℓ2-regularized logistic regression. +We now consider the ℓ2-regularized logistic regression +problem +min +x∈Rn f(x) = 1 +m +� m +� +i=1 +� +−yiaT +i x + log +� +1 + eaT +i x�� ++ λ ∥x∥2 +� +, +(101) +38 + +Unifying NAG for Convex and Strongly Convex Objective Functions +where ai ∈ Rn and yi ∈ {0, 1} for i = 1, 2, . . . , m. Then, (101) is the problem (26) with the +convex functions fi(x) = −yiaT +i x+log(1+eaT +i x) and the ℓ2-regularization term R(x) = ∥x∥2. +As mentioned in Section 1.2, the function f is µ-strongly convex with µ = 2λ +m . We set +s = 0.01 and choose the sample size and the dimension as m = 100 and n = 20, respectively. +Following (Su et al., 2014), we use a synthetically generated data set: the entries of ai are +generated by the Gaussian distribution N(0, 1), and the labels yi ∈ {0, 1} are generated +by the logistic model P (yi) = 1 = +1 +1+e−aT +i x0 , where the entries of x0 are generated by the +Gaussian distribution N(0, 1/100). The results are shown in Figure 6. Again, we can observe +that NAG-SC outperforms NAG-C when µ is large and underperforms NAG-C when µ is +small. In each case, the performance of unified NAG is on par with the better one among +NAG-C and NAG-SC. +100 +101 +102 +103 +104 +Iterations +10−12 +10−10 +10−8 +10−6 +10−4 +10−2 +100 +f (xk) − f (x∗) +GD +NAG-C +NAG-SC +Unified NAG +(a) Errors f − f ∗, λ = 5 +100 +101 +102 +103 +104 +Iterations +10−13 +10−11 +10−9 +10−7 +10−5 +10−3 +10−1 +f (xk) − f (x∗) +GD +NAG-C +NAG-SC +Unified NAG +(b) Errors f − f ∗, λ = 5 · 10−2 +100 +101 +102 +103 +104 +Iterations +10−12 +10−10 +10−8 +10−6 +10−4 +10−2 +100 +f (xk) − f (x∗) +GD +NAG-C +NAG-SC +Unified NAG +(c) Errors f − f ∗, λ = 5 · 10−4 +Figure 6: Results for the ℓ2-regularized logistic regression problem. +8. Conclusions +In this paper, we examined and resolved inconsistencies between the momentum algorithms +and ODE models for convex and strongly convex cases. To bridge the gap between the two +cases, we proposed the unified Bregman Lagrangian (55), the unified NAG ODE (59), and +the unified NAG (70). Because our algorithm, ODE model and Lagrangian are continuous +in µ and recover the corresponding counterparts for non-strongly convex cases (see Figure 1), +they can be viewed as continuous extensions of the NAG-C, NAG-C ODE, and the first +Bregman Lagrangian. We theoretically and empirically showed that unlike NAG-SC, the +unified NAG has a better convergence rate compared to NAG-C regardless of the values +of µ, which is quite significant in practice, as mentioned in Section 1.2. Based on the +Lagrangian formalism, we proposed the unified accelerated tensor flow (76) and scheme +(79), achieving exponential convergence rates in the higher-order setting. Lastly, hinted +from the unified NAG ODE, we designed the unified NAG-G ODE (97), a novel dynamical +system that minimizes the gradient norm of strongly convex functions. Using our novel +tool, the differential kernel (48), we discovered an anti-transpose relationship (95) between +OGM ODE and OGM-G ODE. Surprisingly, such relationship can also be found between +the unified NAG ODE and the unified NAG-G ODE. +39 + +Kim and Yang +Acknowledgments and Disclosure of Funding +We thank Prof. Ernest K. Ryu at Seoul National University for providing feedback on this +work. This work was supported in part by Samsung Electronics, the National Research +Foundation of Korea funded by MSIT(2020R1C1C1009766), and the Information and +Communications Technology Planning and Evaluation (IITP) grant funded by MSIT(2022- +0-00124, 2022-0-00480). +40 + +Unifying NAG for Convex and Strongly Convex Objective Functions +A. Existing Unified Dynamics +A.1 Relationship between the rescaled original NAG flow and the unified +Bregman Lagranfian flow +First, we show that the rescaled original NAG flow (28) can be expressed as the unified +Bregman Lagrangian flow (56). Given the parameter function a(t) and the constant γ0 of +the rescaled original NAG flow, we can write the functions γ(t) and b(t) involved in (27) +and (28) as +γ(t) = µ + (γ0 − µ) e−t +b(t) = µ + (γ0 − µ) e− +� t +0 a(s) ds. +We define the functions α(t) and β(t) as +α(t) = log a(t) +β(t) = log +� +1 +γ0 − µ +� ++ +� t +0 +a(s) ds. +(102) +Then, we have +˙βeβ +1 + µeβ = +eα+β +1 + µeβ = +eα +µ + e−β = +a(t) +µ + (γ0 − µ) e− +� t +0 a(s) ds = a(t) +b(t) . +Thus, the rescaled original NAG flow is equivalent to the unified Bregman Lagrangian +flow with the parameter functions (102) and the Euclidean distance-generating function +h(x) = 1 +2∥x∥2. +Conversely, we show that if the ideal scaling conditon (21b) holds with equality and the +distance-generating function h is Euclidean, then the unified Bregman Lagrangian flow can +be written as the rescaled original NAG flow. Given the parameter functions α(t) and β(t) +of the unified Bregman Lagrangian flow, we define the function a(t) and the constant γ0 as +a(t) = eα(t) +γ0 = µ + e−β(0). +Then, because +b(t) = µ + (γ0 − µ) e− +� t +0 a(s) ds = µ + e−β(t), +we can write the rescaled original NAG flow as +˙X(t) = eα(t)(Z(t) − X(t)) +� +µ + e−β(t)� +˙Z(t) = eα(t)(µX(t) − µZ(t) − ∇f(X(t))), +which is equivalent to the unified Bregman Lagrangian flow if the ideal scaling conditon +(21b) holds with equality and h(x) = 1 +2∥x∥2. +41 + +Kim and Yang +A.2 Relationship between the rescaled original NAG flow with specific +parameters and the unified NAG system +In particular, given γ > 0, one can choose the function a(t) in the rescaled original NAG +flow as (see Luo and Chen, 2021, Equation 70) +a(t) = +� +� +� +� +� +2√γ0 +√γ0t+2, +if µ = 0, +√µ · +e +√µt− +√µ−√γ0 +√µ+√γ0 +e +√µt+ +√µ−√γ0 +√µ+√γ0 +, +if µ > 0. +(103) +In this case, we have b(t) = (a(t))2. Thus, the rescaled original flow with these functions +can be written as +˙X(t) = +2√γ0 +√γ0t + 2(Z(t) − X(t)) +˙Z(t) = − +√γ0t + 2 +2√γ0 +− ∇f(X(t)) +when µ = 0, and +˙X(t) = √µ · +e +√µt − +√µ−√γ0 +√µ+√γ0 +e +√µt + +√µ−√γ0 +√µ+√γ0 +(Z(t) − X(t)) +˙Z(t) = +1 +√µ · +e +√µt + +√µ−√γ0 +√µ+√γ0 +e +√µt − +√µ−√γ0 +√µ+√γ0 +(µX(t) − µZ(t) − ∇f(X(t))) +when µ > 0. In the non-strongly convex case, it is easy to observe that this ODE system +converges to NAG-C system (16) as γ0 → ∞. In the strongly convex case, because +√µ−√γ0 +√µ+√γ0 → +−1 as γ0 → ∞ and e +√µt+1 +e +√µt−1 = coth( +√µ +2 t), the ODE system converges to the unified NAG +system (58) as γ0 → ∞. +B. Higher-Order Hyperbolic Functions +B.1 Proof of Proposition 1 +Fix T > 0. We will show that +log (sinhp(T + t)) − t +(104) +converges to some constant as t → ∞. We can bound the derivative of (104) as +d +dt {log (sinhp(T + t)) − t} = sinh′ +p(T + t) +sinhp(T + t) − 1 += coshp(T + t) +sinhp(T + t) − 1 += +� +1 + +1 +sinhp +p(T + t) +�1/p +− 1 +42 + +Unifying NAG for Convex and Strongly Convex Objective Functions +∈ +� +0, +1 +sinhp(T + t) +� +, +where the last line follows from the fact that 1 ≤ (1 + x)1/p ≤ 1 + x1/p holds for x ≥ 0.11 +Thus, if the integral +� ∞ +0 +1 +sinhp(T + t) dt +(105) +is finite, then (104) converges to some constant because it is monotonically increasing and +bounded above, and thus this completes the proof. To show that the integral (105) is finite, +it is enough to show that the inequality +sinhp(T + t) ≥ sinhp(T)et +holds for all t ≥ 0. This can be shown by the following calculation: +log (sinhp(T + t)) = log (sinhp(T)) + +� t +0 +d +ds {log (sinhp(T + s))} ds += log (sinhp(T)) + +� t +0 +sinh′ +p(T + s) +sinhp(T + s) ds += log (sinhp(T)) + +� t +0 +� +1 + sinhp +p(T + s) +�1/p +sinhp(T + s) +ds +≥ log (sinhp(T)) + +� t +0 +1 ds += log (sinhp(T)) + t += log +� +sinhp(T)et� +. +B.2 The function sinhcp is non-decreasing +It is easy to see that sinhp and coshp are increasing. Since +tanh′ +p(t) = d +dt +� sinhp(t) +coshp(t) +� += sinh′ +p(t) coshp(t) − cosh′ +p(t) sinhp(t) +cosh2 +p(t) +≤ sinh′ +p(t) coshp(t) +cosh2 +p(t) += 1, +we have tanhp(t) ≤ t for all t ≥ 0. Now, we deduce that +sinhc′ +p(t) = d +dt +�sinhp(t) +t +� +11. To check this basic inequality, one can consider the p-th power of each side. +43 + +Kim and Yang += t sinh′ +p(t) − sinhp(t) +t2 += t coshp(t) − sinhp(t) +t2 += coshp(t) +t2 +(t − tanhp(t)) +≥ 0, +and thus sinhc is non-decreasing. +C. Limiting Arguments +C.1 Limiting argument for two-sequence scheme +Limiting ODE of two-sequence scheme. +For the iterates of the two-sequence scheme +(42), we have +xk+1 − xk +√s += 1 +√s (yk − s∇f (yk) − xk) += 1 +√s (βk−1 (xk − xk−1) + γk (xk − yk−1) − s∇f (yk)) += 1 +√s (βk−1 (xk − xk−1) − sγk∇f (yk−1) − s∇f (yk)) += βk−1 +xk − xk−1 +√s +− √sγk∇f (yk−1) − √s∇f (yk) . +Using the Taylor expansions +xk+1 − xk +√s += ˙X(tk) + 1 +2 +¨X(tk)√s + o +�√s +� +xk − xk−1 +√s += ˙X(tk) − 1 +2 +¨X(tk)√s + o +�√s +� +, +we obtain +˙X(tk)+1 +2 +¨X(tk)√s+o +�√s +� += βk−1 +� +˙X(tk) − 1 +2 +¨X(tk)√s + o +�√s +�� +−√sγk∇f (yk−1)−√s∇f (yk) . +It follows from ∥xk − yk−1∥ = o(√s) and the Lipschitz continuity of ∇f that +√s∇f (yk−1) = √s∇f(X(tk)) + o +�√s +� +√s∇f (yk) = √s∇f (yk−1) + o +�√s +� += √s∇f(X(tk)) + o +�√s +� +. +Substituting these into the ODE yields +1 + βk−1 +2 +¨X(tk)√s + (1 − βk−1) ˙X(tk) + (1 + γk) ∇f(X(tk))√s + o +�√s +� += 0. +Dividing both sides by √s, substituting k = t/√s and the limits (43), and then letting +s → 0, we obtain (note that βt/√s−1 → 1 by Equation (43)) +¨X(t) + b(t) ˙X(t) + (1 + c(t))∇f(X(t)) = 0. +44 + +Unifying NAG for Convex and Strongly Convex Objective Functions +Recovering the limiting ODE of three-sequence scheme. +It follows from the Taylor +expansion that +τk = τ (tk) √s +τk+1 = τ (tk) √s + ˙τ (tk) s + √so +�√s +� +δk = δ (tk) √s. +Thus, for the sequences (βk) and (γk) in (45), we have +1 − βk +√s += 1 +√s +� +1 − (1 − τk) (1 − µδk) τk+1 +τk +� += 1 +√s +� +1 − +� +1 − √sτ (tk) +� � +1 − µ√sδ (tk) +� � +1 + ˙τ (tk) s + √so (√s) +τ (tk) √s +�� += 1 +√s +�√sτ (tk) + µ√sδ (tk) − √s ˙τ (tk) +τ (tk) + o +�√s +�� += τ (tk) + µδ (tk) − ˙τ (tk) +τ (tk) + o (√s) +√s +and +γk = τk+1 +τk +((1/s − µ)δkτk − 1 + µδk) += +� +1 + ˙τ (tk) √s + o (√s) +τ (tk) +� � +(1 − µs)δ (tk) τ (tk) − 1 + µ√sδ (tk) +� += δ (tk) τ (tk) − 1 + o (1) . +Therefore, we have +lim +s→0 +1 − βt/√s +√s += τ(t) + µδ(t) − ˙τ(t) +τ(t) +lim +s→0 γt/√s = τ(t)δ(t) − 1, +which recovers the limiting ODE (14) of the three-sequence scheme. +C.2 Difference matrix and differential kernel +From the two-sequence scheme to the difference matrix. +The iterates of the two- +sequence scheme (42) satisfy +yk+1 − yk = xk+1 − yk + βk (xk+1 − xk) − sγk∇f (yk) += βk (yk − yk−1) + sβk∇f (yk−1) − s (1 + βk + γk) ∇f (yk) . +Substituting +yk+1 − yk = −s +k +� +i=0 +hk,i∇f (yi) +45 + +Kim and Yang +yk − yk−1 = −s +k−1 +� +i=0 +hk−1,i∇f (yi) +into the equality and comparing the coefficients of each ∇f(yi), we obtain +hk,j = +� +� +� +� +� +1 + βk + γk, +if j = k +βk (hk−1,k−1 − 1) , +if j = k − 1 +βkhk−1,i, +if j ≤ k − 2. +Using mathematical induction, it is straightforward to show that +hij = (βj + γj) +i� +ν=j+1 +βν + δij. +Differential kernel for the two-sequence scheme. +By (51), we have +∂ +∂s log(H(s, t)) = ∂H(s, τ) +∂s +1 +H(s, τ) = −b(s). +Integrating over s, we obtain +log (H(t, τ)) − log (H(τ, τ)) = − +� t +τ +b(s) ds. +Thus, we have +H(t, τ) = H(τ, τ)e− +� t +τ b(s) ds = (1 + c(τ)) e− +� t +τ b(s) ds. +D. Unified Bregman Lagrangian +D.1 Proof of Proposition 2 +For the unified Bregman Lagrangian (55), the partial derivatives ∂L +∂ ˙X +� +X, ˙X, t +� +and ∂L +∂X +� +X, ˙X, t +� +are given by +∂L +∂ ˙X +� +X, ˙X, t +� += eγ � +1 + µeβ� � +∇h +� +X + e−α ˙X +� +− ∇h(X) +� +∂L +∂X +� +X, ˙X, t +� += eα+γ � +1 + µeβ� � +∇h +� +X + e−α ˙X +� +− ∇h(X) +� +− eγ � +1 + µeβ� d +dt∇h(X) − eα+β+γ∇f(X). +The time derivative of ∂L +∂ ˙X can be computed as +d +dt +� ∂L +∂ ˙X +� +X, ˙X, t +�� += +� +˙γeγ + µ +� +˙β + ˙γ +� +eβ+γ� � +∇h +� +X + e−α ˙X +� +− ∇h(X) +� ++ eγ � +1 + µeβ� � d +dt∇h +� +X + e−α ˙X +� +− d +dt∇h(X) +� +. +46 + +Unifying NAG for Convex and Strongly Convex Objective Functions +Thus, the Euler–Lagrange equation (22) can be written as +eγ � +1 + µeβ� d +dt∇h +� +X + e−α ˙X +� += +� +eα+γ � +1 + µeβ� +− ˙γeγ − µ +� +˙β + ˙γ +� +eβ+γ� +× +� +∇h +� +X + e−α ˙X +� +− ∇h(X) +� +−eα+β+γ∇f(X). +Substituting ˙γ = eα (21a) into the equation and dividing both sides by eγ � +1 + µeβ� +> 0, we +obtain +d +dt∇h +� +X + e−α ˙X +� += − µ ˙βeβ +1 + µeβ +� +∇h +� +X + e−α ˙X +� +− ∇h(X) +� +− +eα+β +1 + µeβ ∇f(X). +Letting Z = X + e−α ˙X yields the system of ODEs (56). +D.2 Proof of Theorem 3 +Note that +d +dtDh (x∗, Z) = d +dt {h (x∗) − h(Z) − ⟨∇h(Z), x∗ − Z⟩} += − +� +∇h(Z), ˙Z +� +− +� d +dt∇h(Z), x∗ − Z +� ++ +� +∇h(Z), ˙Z +� += − +� d +dt∇h(Z), x∗ − Z +� +. +Using this equation, we have +d +dt {φ(X(t), Z(t), t)} = − +� +1 + µeβ� � d +dt∇h(Z), x∗ − Z +� ++ µ ˙βeβDh (x∗, Z) ++ ˙βeβ (f(X) − f (x∗)) + eβ � +∇f(X), ˙X +� += +� +µ ˙βeβ (∇h(Z) − ∇h(X)) + eα+β∇f(X), x∗ − Z +� ++ µ ˙βeβDh (x∗, Z) ++ ˙βeβ (f(X) − f (x∗)) + eβ � +∇f(X), ˙X +� +, +where the second equality follows from (56b). It follows from the Bregman three-point +identity (32), the non-negativity of Bregman divergence, and the µ-uniform convexity of f +with respect to h (33) that +⟨∇h(Z) − ∇h(X), x∗ − Z⟩ + Dh (x∗, Z) = Dh (x∗, X) − Dh(Z, X) +≤ Dh (x∗, X) +≤ 1 +µDf (x∗, X) . +Thus, we have +d +dt {φ(X(t), Z(t), t)} ≤ ˙βeβDf (x∗, X) + eα+β ⟨∇f(X), x∗ − Z⟩ ++ ˙βeβ (f(X) − f (x∗)) + eβ � +∇f(X), ˙X +� +47 + +Kim and Yang += ˙βeβDf (x∗, X) + eα+β ⟨∇f(X), x∗ − X⟩ + ˙βeβ (f(X) − f (x∗)) += +� +eα − ˙β +� +eβ ⟨∇f(X), x∗ − X⟩ +≤ +� +eα − ˙β +� +eβ (f (x∗) − f(X)) +≤ 0, +where the last two inequalities follows from the ideal scaling condition (21b), the convexity +of f, and the fact that x∗ is a minimizer of f. +D.3 Proof of Theorem 5 +The derivatives of X2 and ∇h(Z2) can be computed as +˙X2(t) = ˙T(t) ˙X1(T(t)) += ˙T(t)eα1(T(t))(Z1(T(t)) − X1(T(t)) += ˙T(t)eα1(T(t))(Z2(t) − X2(t)) += eα2(t)(Z2(t) − X2(t)) +and +d +dt∇h(Z2(t)) = ˙T(t)d(∇h ◦ Z1) +dt +(T(t)) += ˙T(t) +� +µ ˙β1(T(t))eβ1(T(t)) +1 + µeβ1(T(t)) +(∇h(X1(T(t)) − ∇h(Z1(T(t)))) +− eα1(T(t))+β1(T(t)) +1 + µeβ1(T(t)) ∇f(X1(T(t))) +� += µ ˙β2(t)eβ2(t) +1 + µeβ2(t) (∇h(X2(t)) − ∇h(Z2(t))) − eα2(t)+β2(t) +1 + µeβ2(t) ∇f(X2(t)). +Thus, we obtain the desired system of ODEs. +D.4 Recovering Lyapunov analysis for the second Bregman Lagrangian flow +In this section, we recover the second Bregman Lagrangian flow (25) with constant coefficients +and its Lyapunov analysis from the unified Bregman Lagrangian flow (56) and its Lyapunov +analysis (Theorem 3). In particular, we recover NAG-SC ODE (19) and its Lyapunov analysis +from the unified NAG ODE (59) and its Lyapunov analysis (Theorem 7). +For the parameter functions α, β : [0, ∞) → R of the unified Bregman Lagrangian flow +(56), assume that the limits α(∞) := limt→∞ α(t) and ˙β(∞) := limt→∞ ˙β(t) > 0 exist. We +consider the following second Bregman Lagrangian flow (25) with α2nd(t) :≡ α(∞) and +β2nd(t) := ˙β(∞)t: +˙X = eα(∞)(Z − X) +d +dt∇h(Z) = ˙β(∞) (∇h(X) − ∇h(Z)) − eα(∞) +µ +∇f(X). +(106) +48 + +Unifying NAG for Convex and Strongly Convex Objective Functions +Then, it follows from limt→∞ eα(t) = eα(∞), limt→∞ +µ ˙βeβ +1+µeβ = ˙β(∞), and limt→∞ eα+β +1+µeβ = +eα(∞) +µ +that the coefficients in the unified Bregman Lagrangian flow (56) converge to those in +the dynamics (106) as t → ∞. Thus, roughly speaking, the dynamics (106) is the asymptotic +version of the unified Bregman Lagrangian flow in the sense that [the flow corresponding to +(56), starting at time t0] converges to [the flow corresponding to (106), starting at time 0] as +t0 → ∞. +Note that the time derivative of the Lyapunov function (57) for the unified Bregman +Lagrangian flow can be written as +d +dt {V (X(t), Z(t), t)} = d +dt +� +1 + µeβ� +Dh (x∗, Z) + +� +1 + µeβ� d +dt {Dh (x∗, Z)} ++ d +dt +� +eβ� +(f(X) − f (x∗)) + eβ d +dt {f(X) − f (x∗)} . +Thus, we have +0 ≥ e−β(t0+t) d +dt {V (X(t0 + t), Z(t0 + t), t0 + t)} += µ ˙β(t0 + t)Dh (x∗, Z(t0 + t)) + 1 + µeβ(t0+t) +eβ(t0+t) +d +dt {Dh (x∗, Z(t0 + t))} ++ ˙β(t0 + t) (f(X(t0 + t)) − f (x∗)) + d +dt {f(X(t0 + t)) − f (x∗)} +for all t > 0, where t0 > 0 is the initial time of the flow. Fix x0 = X(t0) and z0 = Z(t0) +in Rn. Note that as t0 → ∞, the flow t �→ (X(t0 + t), Z(t0 + t)) converges to the flow +t �→ (X2nd(t), Z2nd(t)) corresponding to (106) with X2nd(0) = x0 and Z2nd(0) = z0. Now, +taking the limit t0 → ∞ in the inequality above yields yields +0 ≥ µ ˙β(∞)Dh (x∗, Z(t)) + µ d +dt {Dh (x∗, Z(t))} ++ ˙β(∞) (f(X(t)) − f (x∗)) + d +dt {f(X(t)) − f (x∗)} += e−β2nd(t) d +dt {V2nd(X(t), Z(t), t)} , +where V2nd is the Lyapunov function (38) for the second Bregman Lagrangian flow with the +parameters α2nd and β2nd. Because e−β2nd(t) > 0, we recover the Lyapunov analysis for the +second Bregman Lagrangian flow. +Recovering NAG-SC ODE from the unified ODE. +Note that the unified Bregman +Lagrangian flow (56) and its Lyapunov analysis (Theorem 3) with h(x) = 1 +2 ∥x∥2, α(t) = +log +� +2 +t cothc +� √µ +2 t +�� +, and β(t) = log +� +t2 +4 sinhc2 � √µ +2 t +�� +recover the unified NAG system (58) +and its Lyapunov analysis (Theorem 7). Also, note that the second Bregman Lagrangian +flow (25) and the corresponding Lyapunov function (38) with α2nd(t) = log +�√µ +� +and +β2nd(t) = √µt recover NAG-SC system (18) and the corresponding Lyapunov function (36). +When µ > 0, because α(∞) = log +�√µ +� +and ˙β(∞) = √µ, the results above shows that +NAG-SC ODE is the asymptotic version of the unified NAG ODE and that the Lyapunov +analysis of NAG-SC ODE can be obtained by taking the limit t → ∞ into the coeffiicients of +49 + +Kim and Yang +the inequality (rigorously, taking the limit t0 → ∞ of the initial time as in the preceding +paragraph) +4 +t2 cschc2 +�√µ +2 t +� d +dt {V (X(t), Z(t), t)} ≤ 0, +where V is the Lyapunov function (60) for the unified NAG ODE. +E. Unified NAG ODE +E.1 Choosing α and β +We first note some properties of the functions α and β that recover NAG-C ODE (or NAG-SC +ODE) from the first Bregman Lagrangian flow (or the second Bregman Lagrangian flow, +respectively). +The first Bregman Lagrangian flow (23) with h(x) = 1 +2 ∥x∥2 can be written as the +following ODE: +¨X + (− ˙α + eα) ˙X + e2α+β∇f(X) = 0. +The choices α(t) = log 2 +t and β(t) = log t2 +4 , which recover NAG-C ODE, satisfy the ideal +scaling condition (21b) with equality and make the coefficient of ∇f(X) equal to the +coefficient of ¨X. +The second Bregman Lagrangian flow (25) with h(x) = 1 +2 ∥x∥2 can be written as +¨X + +� +− ˙α + eα + ˙β +� +˙X + e2α +µ ∇f(X) = 0. +The choices α(t) = log √µ and β(t) = log +�√µt +� +, which recover NAG-SC ODE, satisfy the +ideal scaling condition (21b) with equality and make the coefficient of ∇f(X) equal to the +coefficient of ¨X. +Inspired by these facts, for the unified Bregman Lagrangian, we construct functions α(t) +and β(t) so that the ideal scaling condition (21b) holds with equality and that the coefficient +of ∇f(X) is equal to the coefficient of ¨X. The unified Bregman Lagrangian flow (56) with +h(x) = 1 +2 ∥x∥2 can be written as +¨X + +� +− ˙α + eα + +µ ˙βeβ +1 + µeβ +� +˙X + e2α+β +1 + µeβ ∇f(X) = 0. +Now, we solve the following system of ODEs: +˙β = eα +e2α+β = 1 + µeβ. +Let A(t) = eβ(t) > 0. Then, we have ˙A = ˙βeβ = eα+β > 0. Because ( ˙A)2 = e2α+βeβ = +A(1 + µA), we have ˙A = +� +A(1 + µA). Solving this differential equation with the initial +condition A(0) = 0 yields A = t2 +4 sinhc2( +√µ +2 t). Thus, we have β(t) = log( t2 +4 sinhc2( +√µ +2 t)) and +α(t) = log( ˙β(t)) = log( 2 +t cothc( +√µ +2 t)). +50 + +Unifying NAG for Convex and Strongly Convex Objective Functions +E.2 Equivalent forms of the unified NAG system and the unified NAG-G +system +When µ = 0, the unified NAG system is equivalent to NAG-C system. Thus, we assume +µ > 0 for the sake of simplicity. +Second-order ODE form of the unified NAG system. +When µ > 0, we can write +the unified NAG system (58) as +˙X = √µ coth +�√µ +2 t +� +(Z − X) +˙Z = +1 +√µ tanh +�√µ +2 t +� +(µX − µZ − ∇f(X)) . +Substituting Z = X + +1 +√µ tanh( +√µ +2 t) ˙X into ˙Z = +1 +√µ tanh( +√µ +2 t)(µX − µZ − ∇f(X)), we have +1 +√µ tanh +�√µ +2 t +� +¨X + +� +1 + 1 +2 sech2 +�√µ +2 t +�� += +1 +√µ tanh +�√µ +2 t +� +(µX − µZ − ∇f(X)) += −√µ tanh +�√µ +2 t +� +(Z − X) − +1 +√µ tanh +�√µ +2 t +� +∇f(X) += − tanh2 +�√µ +2 t +� +˙X − +1 +√µ tanh +�√µ +2 t +� +∇f(X). +Multiplying by √µ coth( +√µ +2 t) and rearranging the terms, we have +¨X + +�√µ tanh +�√µ +2 t +� ++ √µ coth +�√µ +2 t +� ++ +√µ +2 sech +�√µ +2 t +� +csch +�√µ +2 t +� � +˙X + ∇f(X) = 0. +Using the identity tanh(x) − coth(x) + sech(x) csch(x) = 0, we can equivalently write this +ODE as +¨X + +�√µ +2 tanh +�√µ +2 t +� ++ 3√µ +2 +coth +�√µ +2 t +�� +˙X + ∇f(X) = 0. +Differential kernel for the unified NAG ODE. +Substituting b(t) = +√µ +2 tanh( +√µ +2 t) + +3√µ +2 +coth( +√µ +2 t) and c(t) = 0 into (52), we yield the following differential kernel corresponding +to the unified NAG ODE: +H(t, τ) = e− +� t +τ +� √µ +2 tanh +� √µ +2 s +� ++ 3√µ +2 +coth +� √µ +2 s +�� +ds += e +− +� +3 log +� +sinh +� √µ +2 s +�� ++log +� +cosh +� √µ +2 s +���t +τ += +sinh3 � √µ +2 τ +� +cosh +� √µ +2 τ +� +sinh3 � √µ +2 t +� +cosh +� √µ +2 t +� . +51 + +Kim and Yang +Differential kernel for the unified NAG-G ODE. +Substituting b(t) = +√µ +2 tanh( +√µ +2 (T− +t)) + 3√µ +2 +coth( +√µ +2 (T − t)) and c(t) = 0 into (52), we yield the following differential kernel +corresponding to the unified NAG-G ODE: +H(t, τ) = e− +� t +τ +� √µ +2 tanh +� √µ +2 (T−s) +� ++ 3√µ +2 +coth +� √µ +2 (T−s) +�� +ds += e +� +3 log +� +sinh +� √µ +2 (T−s) +�� ++log +� +cosh +� √µ +2 (T−s) +���t +τ += +sinh3 � √µ +2 (T − t) +� +cosh +� √µ +2 (T − t) +� +sinh3 � √µ +2 (T − τ) +� +cosh +� √µ +2 (T − τ) +�. +F. Unified NAG Family +F.1 Proof of Theorem 10 +Note that when µ > 0, the inequality (65) can be written as +0 ≥ +� +1 − √µs coth +�√µ +2 tk+1 +�� 1 +µ sinh2 +�√µ +2 tk+1 +� +− 1 +µ sinh2 +�√µ +2 tk +� += 1 +µ sinh2 +�√µ +2 tk+1 +� +− +� s +µ sinh +�√µ +2 tk+1 +� +cosh +�√µ +2 tk+1 +� +− 1 +µ sinh2 +�√µ +2 tk +� += 1 +µ cosh2 +�√µ +2 tk+1 +� +− +� s +µ sinh +�√µ +2 tk+1 +� +cosh +�√µ +2 tk+1 +� +− 1 +µ cosh2 +�√µ +2 tk +� += +� +1 − √µs tanh +�√µ +2 tk+1 +�� 1 +µ cosh2 +�√µ +2 tk+1 +� +− 1 +µ cosh2 +�√µ +2 tk +� +. +Thus, the following inequality holds for all µ ≥ 0 (it clearly holds for µ = 0): +� +1 − µ√stk+1 +2 +tanhc +�√µ +2 tk+1 +�� +cosh2 +�√µ +2 tk+1 +� +≤ cosh2 +�√µ +2 tk +� +. +(107) +Using (65) and (107), we have +Ek+1 − Ek += 1 +2 cosh2 +�√µ +2 tk+1 +� +∥zk+1 − x∗∥2 − 1 +2 cosh2 +�√µ +2 tk +� +∥zk − x∗∥2 ++ t2 +k+1 +4 +sinhc2 +�√µ +2 tk+1 +� +(f (xk+1) − f (x∗)) − t2 +k +4 sinhc2 +�√µ +2 tk +� +(f (xk) − f (x∗)) +≤ 1 +2 cosh2 +�√µ +2 tk+1 +� +∥zk+1 − x∗∥2 − 1 +2 +� +1 − µ√stk+1 +2 +tanhc +�√µ +2 tk+1 +�� +cosh2 +�√µ +2 tk+1 +� +∥zk − x∗∥2 ++ t2 +k+1 +4 +sinhc2 +�√µ +2 tk+1 +� +(f (xk+1) − f (x∗)) +− +� +1 − 2√s +tk+1 +cothc +�√µ +2 tk+1 +�� t2 +k+1 +4 +sinhc2 +�√µ +2 tk+1 +� +(f (xk) − f (x∗)) . +52 + +Unifying NAG for Convex and Strongly Convex Objective Functions +Substituting +zk+1 = yk+ +� +1 − µ√stk+1 +2 +tanhc +�√µ +2 tk+1 +�� +(zk − yk)− +√stk+1 +2 +tanhc +�√µ +2 tk+1 +� +∇f (yk) +into the inequality above, we have +Ek+1 − Ek +≤ 1 +2 cosh2 +�√µ +2 tk+1 +� +× +����� +� +1 − µ√stk+1 +2 +tanhc +�√µ +2 tk+1 +�� +(zk − yk) +− +√stk+1 +2 +tanhc +�√µ +2 tk+1 +� +∇f (yk) − (x∗ − yk) +����� +2 +− 1 +2 +� +1 − µ√stk+1 +2 +tanhc +�√µ +2 tk+1 +�� +cosh2 +�√µ +2 tk+1 +� +∥(zk − yk) − (x∗ − yk)∥2 ++ t2 +k+1 +4 +sinhc2 +�√µ +2 tk+1 +� +(f (xk+1) − f (x∗)) +− +� +1 − 2√s +tk+1 +cothc +�√µ +2 tk+1 +�� t2 +k+1 +4 +sinhc2 +�√µ +2 tk+1 +� +(f (xk) − f (x∗)) += 1 +2 cosh2 +�√µ +2 tk+1 +� � � +1 − µ√stk+1 +2 +tanhc +�√µ +2 tk+1 +��2 +− +� +1 − µ√stk+1 +2 +tanhc +�√µ +2 tk+1 +�� � +∥zk − yk∥2 ++ +√stk+1 +2 +sinhc +�√µ +2 tk+1 +� +cosh +�√µ +2 tk+1 +� +⟨∇f (yk) , x∗ − yk⟩ ++ µ√stk+1 +4 +sinhc +�√µ +2 tk+1 +� +cosh +�√µ +2 tk+1 +� +∥x∗ − yk∥2 +− +√stk+1 +2 +sinhc +�√µ +2 tk+1 +� +cosh +�√µ +2 tk+1 +� +× +� +1 − µ√stk+1 +2 +tanhc +�√µ +2 tk+1 +�� +⟨∇f(yk), zk − yk⟩ ++ st2 +k+1 +8 +sinhc2 +�√µ +2 tk+1 +� +∥∇f (yk)∥2 + t2 +k+1 +4 +sinhc2 +�√µ +2 tk+1 +� +(f (xk+1) − f (x∗)) +− +� +1 − 2√s +tk+1 +cothc +�√µ +2 tk+1 +�� t2 +k+1 +4 +sinhc2 +�√µ +2 tk+1 +� +(f (xk) − f (x∗)) . +Since +0 ≤ 1 − √µs ≤ 1 − µ√stk+1 +2 +tanhc +�√µ +2 tk+1 +� +≤ 1, +we have +53 + +Kim and Yang +1 +2 cosh2 +�√µ +2 tk+1 +� � � +1 − µ√stk+1 +2 +tanhc +�√µ +2 tk+1 +��2 +− +� +1 − µ√stk+1 +2 +tanhc +�√µ +2 tk+1 +�� � +∥zk − yk∥2 ≤ 0. +Therefore, we deduce that +Ek+1 − Ek +≤ +√stk+1 +2 +sinhc +�√µ +2 tk+1 +� +cosh +�√µ +2 tk+1 +� +⟨∇f (yk) , x∗ − yk⟩ ++ µ√stk+1 +4 +sinhc +�√µ +2 tk+1 +� +cosh +�√µ +2 tk+1 +� +∥x∗ − yk∥2 +− +√stk+1 +2 +sinhc +�√µ +2 tk+1 +� +cosh +�√µ +2 tk+1 +� +× +� +1 − µ√stk+1 +2 +tanhc +�√µ +2 tk+1 +�� +⟨∇f(yk), zk − yk⟩ ++ st2 +k+1 +8 +sinhc2 +�√µ +2 tk+1 +� +∥∇f (yk)∥2 ++ t2 +k+1 +4 +sinhc2 +�√µ +2 tk+1 +� +(f (xk+1) − f (x∗)) +− +� +1 − 2√s +tk+1 +cothc +�√µ +2 tk+1 +�� t2 +k+1 +4 +sinhc2 +�√µ +2 tk+1 +� +(f (xk) − f (x∗)) . +Now, it suffices to show that the right-hand side (RHS) of the inequality above is non-positive. +By the µ-strong convexity of f, we have +0 ≥ f (yk) − f (x∗) + ⟨∇f (yk) , x∗ − yk⟩ + µ +2 ∥x∗ − yk∥2 . +Moreover, it follows from the convexity and the 1 +s-smoothness of f that +0 ≥ f (yk) − f (xk) + ⟨∇f(yk), xk − yk⟩ +and +0 ≥ f(xk+1) − f(yk) + s +2 ∥∇f(yk)∥2 , +respectively. Note that +xk − yk = − +τk +1 − τk +(zk − yk) = − +2√s +tk+1 cothc +� √µ +2 tk+1 +� +− µs +1 − 2√s +tk+1 cothc +� √µ +2 tk+1 +� (zk − yk) . +Taking a weighted sum of the inequalities above yields (the assumption (64) ensures that +these weights are non-negative for k ≥ 1, and the case k = 0 is trivial because y0 = x0) +0 ≥ 2√s +tk+1 +cothc +�√µ +2 tk+1 +� t2 +k+1 +4 +sinhc2 +�√µ +2 tk+1 +� +54 + +Unifying NAG for Convex and Strongly Convex Objective Functions +× +� +f (yk) − f (x∗) + ⟨∇f (yk) , x∗ − yk⟩ + µ +2 ∥x∗ − yk∥2� ++ +� +1 − 2√s +tk+1 +cothc +�√µ +2 tk+1 +�� t2 +k+1 +4 +sinhc2 +�√µ +2 tk+1 +� +× [f (yk) − f (xk) + ⟨∇f(yk), xk − yk⟩] ++ t2 +k+1 +4 +sinhc2 +�√µ +2 tk+1 +� � +f(xk+1) − f(yk) + s +2 ∥∇f(yk)∥2� += +√stk+1 +2 +sinhc +�√µ +2 tk+1 +� +cosh +�√µ +2 tk+1 +� +⟨∇f (yk) , x∗ − yk⟩ ++ µ√stk+1 +4 +sinhc +�√µ +2 tk+1 +� +cosh +�√µ +2 tk+1 +� +∥x∗ − yk∥2 +− +� 2√s +tk+1 +cothc +�√µ +2 tk+1 +� +− µs +� t2 +k+1 +4 +sinhc2 +�√µ +2 tk+1 +� +⟨∇f(yk), zk − yk⟩ ++ st2 +k+1 +8 +sinhc2 +�√µ +2 tk+1 +� +∥∇f (yk)∥2 ++ 2√s +tk+1 +cothc +�√µ +2 tk+1 +� t2 +k+1 +4 +sinhc2 +�√µ +2 tk+1 +� +(f (yk) − f (x∗)) ++ +� +1 − 2√s +tk+1 +cothc +�√µ +2 tk+1 +�� t2 +k+1 +4 +sinhc2 +�√µ +2 tk+1 +� +(f (yk) − f (xk)) ++ t2 +k+1 +4 +sinhc2 +�√µ +2 tk+1 +� +(f(xk+1) − f(yk)) += +√stk+1 +2 +sinhc +�√µ +2 tk+1 +� +cosh +�√µ +2 tk+1 +� +⟨∇f (yk) , x∗ − yk⟩ ++ µ√stk+1 +4 +sinhc +�√µ +2 tk+1 +� +cosh +�√µ +2 tk+1 +� +∥x∗ − yk∥2 +− +√stk+1 +2 +sinhc +�√µ +2 tk+1 +� +cosh +�√µ +2 tk+1 +� +× +� +1 − µ√stk+1 +2 +tanhc +�√µ +2 tk+1 +�� +⟨∇f(yk), zk − yk⟩ ++ st2 +k+1 +8 +sinhc2 +�√µ +2 tk+1 +� +∥∇f (yk)∥2 ++ t2 +k+1 +4 +sinhc2 +�√µ +2 tk+1 +� +(f (xk+1) − f (x∗)) +− +� +1 − 2√s +tk+1 +cothc +�√µ +2 tk+1 +�� t2 +k+1 +4 +sinhc2 +�√µ +2 tk+1 +� +(f (xk) − f (x∗)) . +This completes the proof. +55 + +Kim and Yang +F.2 Constant timestep scheme +In this section, we show that the sequence (tk) defined in (69) satisfies the conditions (64) +and (65). For convenience, we assume µ > 0 (the case µ = 0 can be handled easily). The +condition (64) follows from +2√s +tk +cothc +�√µ +2 tk +� += √µs coth +�√µ +2 tk +� +≤ √µs coth +�√µ +2 t2 +� += √µs coth (− log (1 − √µs)) += √µs1 + e2 log(1−√µs) +1 − e2 log(1−√µs) += √µs1 + +� +1 − √µs +�2 +1 − +� +1 − √µs +�2 +≤ 1, +where the last inequality holds because √µs ∈ (0, 1). To prove (65), it suffices to show that +the inequality +sinh2 +�√µ +2 t +� +− √µs sinh +�√µ +2 t +� +cosh +�√µ +2 t +� +− sinh2 +�√µ +2 t + 1 +2 log (1 − √µs) +� +≤ 0 +holds for all t ∈ R. Letting r = e +√µ +2 t, this inequality can be expressed as +r2 + r−2 − 2 +4 +− √µsr2 − r−2 +4 +− +� +1 − √µs +� +r2 + +� +1 − √µs +�−1 r−2 − 2 +4 +≤ 0. +Letting q = r2 and multiplying both sides by 4q, the inequality can be rewritten as +0 ≥ q2 + 1 − 2q − √µs +� +q2 − 1 +� +− (1 − √µs) q2 − (1 − √µs)−1 + 2q += 1 + √µs − +1 +1 − √µs += +−µs +1 − √µs, +which clearly holds. +F.3 Adaptive timestep scheme +In this section, we show that for the sequence (tk) defined by (71), +• the sequence (tk) is well-defined, and +• the conditions (10) and (11) hold when lims→0 t0 = 0. +56 + +Unifying NAG for Convex and Strongly Convex Objective Functions +The sequence (tk) is well-defined. +Because +4 +t2 +k+1 +cschc2 +�√µ +2 tk+1 +� ++ µ = +4 +t2 +k+1 +cothc2 +�√µ +2 tk+1 +� +, +the updating rule (71) is equivalent to +4 +t2 +k+1 +cothc2 +�√µ +2 tk+1 +� += +� +1 − 2√s +tk+1 +cothc +�√µ +2 tk+1 +�� 4 +t2 +k +cothc2 +�√µ +2 tk +� ++ 2µ√s +tk+1 +cothc +�√µ +2 tk+1 +� +, tk+1 > 0. +(108) +Introduce a sequence (αk)∞ +k=−1 such that αk = 2√s +tk+1 cothc +� √µ +2 tk+1 +� +. As t �→ 2√s +t +cothc +� √µ +2 t +� +is a bijective map from (0, ∞) to +�√µs, ∞ +� +, the sequences (tk) and (αk) have a one-to-one +relationship. Thus, the updating rule (108) is equivalent to +α2 +k = (1 − αk) α2 +k−1 + µsαk, +αk > √µs, +(109) +which admits a unique solution in (√µs, ∞) when αk−1 > √µs. Thus, the sequence (tk) is +well-defined. +The sequence (tk) satisfies the conditions (10) and (11). +Define a function A(t) as +A(t) := t2 +4 sinhc2 +�√µ +2 t +� +. +(110) +For t ∈ (0, ∞), it follows from (71) that +˙A +� +tk(t)+1 +� += A +� +tk(t)+1 +� +− A (t) +√s += A +� +tk(t)+1 +� +− A (t) +tk(t)+1 − t +tk(t)+1 − t +√s +. +Because tk(t)+1 → t as s → 0, taking the limit s → 0 in the equation above yields +1 = lim +s→0 +tk(t)+1 − t +√s +. +Thus, the condition (65) holds. +F.4 Equivalence between the adaptive timestep scheme and the original NAG +In this section, we show that the adaptive timestep scheme (Section 4.2.2) with t0 > 0 is +equivalent to the original NAG (5) with γ0 = 4 +t2 +0 cothc2 � √µ +2 t0 +� +> µ. +We first show that the sequences (αk)∞ +k=0 and (γk)∞ +k=0 generated in the original NAG +(5) with γ0 = +4 +t2 +0 cothc2 � √µ +2 t0 +� +> µ can be written as αk = +2√s +tk+1 cothc +� √µ +2 tk+1 +� +and +γk = 4 +t2 +k cothc2 � √µ +2 tk +� +, where the sequence (tk)∞ +k=0 is defined as (71). Note that the equality +(6) implies +γk+1 = (1 − αk) γk + µαk = α2 +k +s . +57 + +Kim and Yang +Thus, the updating rule for αk (6) can be written as +1 +sα2 +k = (1 − αk) α2 +k−1 +s ++ µαk, +where we define α−1 := √sγ0 = 2√s +t0 cothc +� √µ +2 t0 +� +> √µs. This implies that the sequence +(αk)∞ +k=−1 in the original NAG and the sequence (αk)∞ +k=−1 defined in Section F.3 are identical. +Thus, we have αk = 2√s +tk+1 cothc +� √µ +2 tk+1 +� +and γk = +α2 +k−1 +s += 4 +t2 +k cothc2 � √µ +2 tk +� +. +Now, we show that the parameters τk and δk for the original NAG are equal to those for +our adaptive timestep scheme. In the original NAG, we have +(αk − µs) (γk + µαk) = αkγk + µα2 +k − µsγk − µ2sαk += µsγk+1 + αkγk − µsγk − µ2sαk += µs ((1 − αk) γk + µαk) + αkγk − µsγk − µ2sαk += (1 − µs)αkγk. +Therefore, we have +τk = +αkγk +γk + µαk += αk − µs +1 − µs = +2√s +tk+1 cothc +� √µ +2 tk+1 +� +− µs +1 − µs +and +δk = +αk +γk+1 += s +αk += +√stk+1 +2 +tanhc +�√µ +2 tk+1 +� +. +Thus, the ogirinal Nesterov’s method with γ0 = 4 +t2 +0 cothc2 � √µ +2 t0 +� +> µ is equivalent to the +adaptive timestep scheme. +G. Higher-Order Extension +G.1 Limiting ODE +Limiting ODE of the unified accelerated tensor method family. +We show that +if the sequence (tk) satisfies the conditions (81) and (82), then the unified accelerated +tensor method family (79) converges to the unified accelerated tensor flow (76) under the +identifications xk = X(tk) and zk = Z(tk). +For convenience, we assume that µ > 0 (the case µ = 0 can be handled easily). Define a +function A : [0, ∞) → R as +A(t) = Ctp sinhcp +p +� +C1/pµ1/pt +� += 1 +µ sinhp +p +� +C1/pµ1/pt +� +(111) +so that Ak = A(tk). It follows from the step (79b) that +˙X(t) = lim +s→0 +xk(t)+1 − xk(t) +tk(t)+1 − t +58 + +Unifying NAG for Convex and Strongly Convex Objective Functions += lim +s→0 +xk(t)+1 − xk(t) +s1/p += lim +s→0 +yk(t) − xk(t) +s1/p += lim +s→0 +Ak(t)+1 − Ak(t) +s1/pAk(t)+1 +� +zk(t) − xk(t) +� += lim +s→0 +A +� +tk(t)+1 +� +− A(t) +s1/pA +� +tk(t)+1 +� +(Z(t) − X(t)) += +˙A(t) +A(t) (Z(t) − X(t)) += pC1/pµ1/p cothp +� +C1/pµ1/pt +� +(Z(t) − X(t)) , +where we used ∥xk+1 − yk∥ = o(s1/p) (see Wibisono et al., 2016, Lemma 2.2) for the third +equality. Using the step (80), we have +d +dt∇h(Z(t)) = lim +s→0 +∇h +� +zk(t)+1 +� +− ∇h +� +zk(t) +� +tk(t)+1 − t += lim +s→0 +∇h +� +zk(t)+1 +� +− ∇h +� +zk(t) +� +s1/p += lim +s→0 +Ak(t)+1 − Ak(t) +s1/p � +1 + µAk(t) +� � +µ∇h +� +xk(t)+1 +� +− µ∇h +� +zk(t)+1 +� +− ∇f +� +xk(t)+1 +�� += lim +s→0 +A +� +tk(t)+1 +� +− A(t) +s1/p (1 + µA(t)) (µ∇h (X(t)) − µ∇h (X(t)) − ∇f (X(t))) += +˙A(t) +1 + µA(t) (µ∇h (X(t)) − µ∇h (X(t)) − ∇f (X(t))) += +C1/pp +µ(p−1)/p tanhp−1 +p +� +C1/pµ1/pt +� +(µ∇h (X(t)) − µ∇h (X(t)) − ∇f (X(t))) . +Thus, we obtain the system of ODEs (76). +Limiting ODE of the unified accelerated tensor method. +We check that the se- +quence (tk) defined in (86) satisfies the condition (82). It is easy to check that the function +A(t) defined in (111) satisfies +˙A(t) = C1/ppµ +1−p +p sinhp−1 +p +� +C1/pµ1/pt +� +coshp +� +C1/pµ1/pt +� += C1/ppA(t) +p−1 +p (1 + µA(t)) +1 +p +and that the sequence (tk) defined in (86) satisfies +A (tk+1) − A (tk) +s1/p +− C1/ppA (tk+1) +p−1 +p (1 + µA (tk)) +1 +p = 0. +Now, substituting k = k(t) into the above equality and taking the limit s → 0, we have +lims→0 +tk(t)+1−t +s1/p += 1. +59 + +Kim and Yang +G.2 Proof of Theorem 17 +By the Bregman three-point identity (32) with x = x∗, y = zk+1, z = xk+1 and the +non-negativity of Bregman divergence, we have +Dh (x∗, zk+1) = Dh (x∗, xk+1) − ⟨∇h (zk+1) − ∇h (xk+1) , x∗ − zk+1⟩ − Dh (zk+1, xk+1) +≤ Dh (x∗, xk+1) − ⟨∇h (zk+1) − ∇h (xk+1) , x∗ − zk+1⟩ . +Thus, we can bound the difference of the discrete-time energy function (84) as follows: +Ek+1 − Ek += (1 + µAk+1) Dh (x∗, zk+1) − (1 + µAk) Dh (x∗, zk) ++ Ak+1 (f (xk+1) − f (x∗)) − Ak (f (xk) − f (x∗)) += µ (Ak+1 − Ak) Dh (x∗, zk+1) ++ (Ak+1 − Ak) (f (xk+1) − f (x∗)) + Ak (f (xk+1) − f (xk)) ++ (1 + µAk) (−h (zk+1) − ⟨∇h (zk+1) , x∗ − zk+1⟩ + h (zk) + ⟨∇h (zk) , x∗ − zk⟩) +≤ µ (Ak+1 − Ak) Dh (x∗, xk+1) − µ (Ak+1 − Ak) ⟨∇h (zk+1) − ∇h (xk+1) , x∗ − zk+1⟩ ++ (Ak+1 − Ak) (f (xk+1) − f (x∗)) + Ak (f (xk+1) − f (xk)) ++ (1 + µAk) (−h (zk+1) − ⟨∇h (zk+1) , x∗ − zk+1⟩ + h (zk) + ⟨∇h (zk) , x∗ − zk⟩) . +By the (µ-uniform) convexity of f with respect to h, the p-th order 1-uniform convexity +of h, and the property (74) of the higher-order gradient update operator Gp,M, the following +inequalities hold: +0 ≥ f (xk+1) − f (x∗) + ⟨∇f (xk+1) , x∗ − xk+1⟩ + µDh (x∗, xk+1) +0 ≥ f (xk+1) − f (xk) + ⟨∇f (xk+1) , xk − xk+1⟩ +0 ≥ Ms +1 +p−1 ∥∇f (xk+1)∥ +p +p−1 − ⟨∇f (xk+1) , yk − xk+1⟩ +0 ≥ h (zk) − h (zk+1) + ⟨∇h (zk) , zk+1 − zk⟩ + 1 +p ∥zk+1 − zk∥p . +Taking a weighted sum of these inequalities yields +0 ≥ (Ak+1 − Ak) [f (xk+1) − f (x∗) + ⟨∇f (xk+1) , x∗ − xk+1⟩ + µDh (x∗, xk+1)] ++ Ak [f (xk+1) − f (xk) + ⟨∇f (xk+1) , xk − xk+1⟩] ++ Ak+1 +� +Ms +1 +p−1 ∥∇f (xk+1)∥ +p +p−1 − ⟨∇f (xk+1) , yk − xk+1⟩ +� ++ (1 + µAk) +� +h (zk) − h (zk+1) + ⟨∇h (zk) , zk+1 − zk⟩ + 1 +p ∥zk+1 − zk∥p +� +≥ Ek+1 − Ek +− µ (Ak+1 − Ak) Dh (x∗, xk+1) + µ (Ak+1 − Ak) ⟨∇h (zk+1) − ∇h (xk+1) , x∗ − zk+1⟩ +− (Ak+1 − Ak) (f (xk+1) − f (x∗)) − Ak (f (xk+1) − f (xk)) +− (1 + µAk+1) (−h (zk+1) − ⟨∇h (zk+1) , x∗ − zk+1⟩ + h (zk) + ⟨∇h (zk) , x∗ − zk⟩) ++ (Ak+1 − Ak) [f (xk+1) − f (x∗) + ⟨∇f (xk+1) , x∗ − xk+1⟩ + µDh (x∗, xk+1)] +60 + +Unifying NAG for Convex and Strongly Convex Objective Functions ++ Ak [f (xk+1) − f (xk) + ⟨∇f (xk+1) , xk − xk+1⟩] ++ Ak+1 +� +Ms +1 +p−1 ∥∇f (xk+1)∥ +p +p−1 − ⟨∇f (xk+1) , yk − xk+1⟩ +� ++ (1 + µAk) +� +h (zk) − h (zk+1) + ⟨∇h (zk) , zk+1 − zk⟩ + 1 +p ∥zk+1 − zk∥p +� += Ek+1 − Ek ++ ⟨∇f (xk+1) , (Ak+1 − Ak) (x∗ − xk+1) + Ak (xk − xk+1) + Ak+1 (xk+1 − yk)⟩ ++ (1 + µAk) ⟨∇h (zk+1) − ∇h (zk) , x∗ − zk+1⟩ + 1 + µAk +p +∥zk+1 − zk∥p ++ µ (Ak+1 − Ak) ⟨∇h (zk+1) − ∇h (xk+1) , x∗ − zk+1⟩ + MAk+1s +1 +p−1 ∥∇f (xk+1)∥ +p +p−1 . +Substituting (80) with the term (1 + µAk) ⟨∇h (zk+1) − ∇h (zk) , x∗ − zk+1⟩, we have +0 ≥ Ek+1 − Ek ++ ⟨∇f (xk+1) , (Ak+1 − Ak) (x∗ − xk+1) + Ak (xk − xk+1) + Ak+1 (xk+1 − yk)⟩ ++ (Ak+1 − Ak) ⟨µ∇h (xk+1) − µ∇h (zk+1) − ∇f (xk+1) , x∗ − zk+1⟩ ++ 1 + µAk +p +∥zk+1 − zk∥p ++ µ (Ak+1 − Ak) ⟨∇h (zk+1) − ∇h (xk+1) , x∗ − zk+1⟩ + MAk+1s +1 +p−1 ∥∇f (xk+1)∥ +p +p−1 += Ek+1 − Ek ++ ⟨∇f (xk+1) , (Ak+1 − Ak) (zk+1 − xk+1) + Ak (xk − xk+1) + Ak+1 (xk+1 − yk)⟩ ++ 1 + µAk +p +∥zk+1 − zk∥p + MAk+1s +1 +p−1 ∥∇f (xk+1)∥ +p +p−1 . +We also notice that +(Ak+1 − Ak) (zk+1 − xk+1) + Ak (xk − xk+1) + Ak+1 (xk+1 − yk) += (Ak+1 − Ak) zk+1 + Akxk − Ak+1yk += (Ak+1 − Ak) (zk+1 − zk) + (Ak+1 − Ak) zk + Akxk − Ak+1yk += (Ak+1 − Ak) (zk+1 − zk) , +where the last equality follows from yk = xk + Ak+1−Ak +Ak+1 +(zk − xk). Therefore, +0 ≥ Ek+1 − Ek ++ (Ak+1 − Ak) ⟨∇f (xk+1) , zk+1 − zk⟩ ++ 1 + µAk +p +∥zk+1 − zk∥p + MAk+1s +1 +p−1 ∥∇f (xk+1)∥ +p +p−1 . +Now, we use the Fenchel-Young inequality ⟨s, u⟩ + 1 +p ∥u∥p ≥ − p−1 +p ∥s∥ +p +p−1 with u = +(1 + µAk) +1 +p (zk+1 − zk) and s = (Ak+1 − Ak) (1 + µAk)− 1 +p ∇f (xk+1) to obtain that +(Ak+1 − Ak) ⟨∇f (xk+1) , zk+1 − zk⟩ + 1 + µAk +p +∥zk+1 − zk∥p +61 + +Kim and Yang +≥ −p − 1 +p +(Ak+1 − Ak) +p +p−1 (1 + µAk)− +1 +p−1 ∥∇f (xk+1)∥ +p +p−1 . +Hence, we have +0 ≥ Ek+1 − Ek ++ +� +MAk+1s +1 +p−1 − p − 1 +p +(Ak+1 − Ak) +p +p−1 (1 + µAk)− +1 +p−1 +� +∥∇f (xk+1)∥ +p +p−1 += Ek+1 − Ek ++ +� +(p − 1)p +1 +p−1 C +1 +p−1 Ak+1s +1 +p−1 − p − 1 +p +(Ak+1 − Ak) +p +p−1 (1 + µAk)− +1 +p−1 +� +∥∇f (xk+1)∥ +p +p−1 , +where C = 1 +p( M +p−1)p−1. It is easy to see that the condition (83) implies that the term +� +(p − 1)p +1 +p−1 C +1 +p−1 Ak+1s +1 +p−1 − p − 1 +p +(Ak+1 − Ak) +p +p−1 (1 + µAk)− +1 +p−1 +� +∥∇f (xk+1)∥ +p +p−1 +is non-negative. Thus, we conclude that +0 ≥ Ek+1 − Ek +as desired. +G.3 Lower bounds for the sequence (Ak) +Let (Abest +k +) denote the sequence (Ak) determined by (86). In this section, we prove that +that the following inequality holds: +Abest +k +≥ max +� +O (kp) , O +�� +1 + C1/ppµ1/ps1/p�k�� +. +We use the following lemma. +Lemma 21 For any sequence (Ak) satisfying A0 = 0 and the condition (83), we have +Ak ≤ Abest +k +∀k ≥ 0. +(112) +Its proof can be found in the following subsection. Now, we claim that the following two +sequences satisfy the condition (83): +Ak = Csk(k + 1) · · · (k + p − 1) +and +Ak = +� +0, +k = 0 +Cpps +� +1 + C1/ppµ1/ps1/p�k−1 +k = 1. +For the first sequence, we have +(Ak+1 − Ak)p − CppsAp−1 +k+1 (1 + µAk) +62 + +Unifying NAG for Convex and Strongly Convex Objective Functions +≤ (Ak+1 − Ak)p − CppsAp−1 +k+1 += (Cps(k + 1) · · · (k + p − 1))p − Cpps (Cs(k + 1) · · · (k + p))p−1 += Cpppsp � +((k + 1) · · · (k + p − 1))p − ((k + 1) · · · (k + p))p−1� +≤ 0, +which implies that (83) holds. +For the second sequence, (83) holds because +(Ak+1 − Ak)p − CppsAp−1 +k+1 (1 + µAk) +≤ (Ak+1 − Ak)p − CµppsAp−1 +k+1Ak +≤ (Ak+1 − Ak)p − CµppsAp +k += +��Ak+1 +Ak +− 1 +�p +− Cµpps +� +Ap +k += +�� +C1/ppµ1/ps1/p�p +− Cµpps +� +Ap +k += 0 +for all k ≥ 1 (the case k = 0 is trivial). Thus, it follows from Lemma 21 that +Abest +k +≥ max +� +Csk(k + 1) · · · (k + p − 1), Cpps +� +1 + C1/ppµ1/ps1/p�k−1� += max +� +O (kp) , O +�� +1 + C1/ppµ1/ps1/p�k�� +, +as desired. +G.3.1 Proof of Lemma 21 +For r ≥ 0, we define +S(r) := +� +x : (x − r)p − Cppsxp−1(1 + µr) ≤ 0 +� +U(r) := max Sr. +Then, it is straightforward to see the following: +• The set S(r) is nonempty. In particular, r ∈ S(r) (which implies U(r) ≥ r). +• For any sequence (Ak) satisfying the condition (83), we have Ak+1 ∈ S(Ak) for all +k ≥ 0. +• For the sequence (Ak) defined in (86), we have Ak+1 = U(Ak) for all k ≥ 0. +If we have +U (r1) ≤ U (r2) whenever r1 ≤ r2, +(113) +then we can prove (112) using mathematical induction on k. It clearly holds when k = 0. If +(112) holds for k, then it holds for k + 1 because +Ak+1 ≤ U (Ak) ≤ U(Abest +k +) = Abest +k+1. +63 + +Kim and Yang +It remains to prove (113). Let r1 and r2 be positive real numbers with r1 ≤ r2. Then, it +is easy to check that r2 + U(r1) − r1 ∈ S(r2). Thus, we have +U (r1) ≤ U (r1) + (r2 − r1) ≤ U (r2) . +This completes the proof. +H. Existence and Uniqueness Theorems +H.1 Proof of Theorem 6 +We prove a stronger result, that the unified Bregman Lagrangian flow (56) with α(t) = +log( 2 +t cothc( +√µ +2 t)), β(t) = log( t2 +4 sinhc2( +√µ +2 t)): +˙X = 2 +t cothc +�√µ +2 t +� +(Z − X) +d +dt∇h(Z) = t +2 tanhc +�√µ +2 t +� +(µ∇h(X) − µ∇h(Z) − ∇f(X)) +(114) +with the initial conditions X(0) = Z(0) = x0 has a unique global solution (X, Z) in +C1([0, ∞), Rn × Rn). Following (Krichene et al., 2015), we assume that ∇f is Lf-Lipschitz +continuous and ∇h is Lh-Lipschitz continuous. The strong convexity of h implies a Lh∗- +Lipschitz continuity of ∇h∗ for some Lh∗ > 0 (see Rockafellar and Wets, 2009, Proposi- +tion 12.60). +H.1.1 Proof of existence +Fix t1 > 0. We show the existence of solution to the system (114) on [0, t1]. To remove the +singularity of the system (114) at t = 0, fix δ > 0, and consider the following system of +ODEs: +˙X = +2 +max{δ, t} cothc +�√µ +2 max{δ, t} +� +(Z − X) +d +dt∇h(Z) = t +2 tanhc +�√µ +2 t +� +(µ∇h(X) − µ∇h(Z) − ∇f(X)) +(115) +with X(0) = Z(0) = x0, which does not have singularities. Denote the image of Z under the +mirror map as W(t) = ∇h(Z(t)). Denote the convex conjugate of h by h∗ : Rn → R. Then, +∇h and ∇h∗ are inverses of each other (see Rockafellar and Wets, 2009, Section 11). Now, +we can equivalently write the system (115) as +˙X = +2 +max{δ, t} cothc +�√µ +2 max{δ, t} +� +(∇h∗(W) − X) +(116a) +˙W = t +2 tanhc +�√µ +2 t +� +(µ∇h(X) − µW − ∇f(X)) +(116b) +with X(0) = x0 and W(0) = w0 := ∇h (x0). By the Cauchy-Lipschitz theorem, the system of +ODEs (116) has a unique solution (Xδ, Wδ) in C1([0, t1], Rn × Rn). If we prove the following +lemma, then one can prove the existence of solution to the ODE system (115) following the +argument in (Krichene et al., 2015, Section 3.2). +64 + +Unifying NAG for Convex and Strongly Convex Objective Functions +Lemma 22 Define a constant T as +T = min +�� 2 +µ, 1 +2 +� +1 +K2K3 +� +, +where K2 and K3 are constants defined in (118). Then, the family of solutions ((Xδ, Zδ)|[0,T])δ∈(0,T] +is equi-Lipschitz-continuous and uniformly bounded. +We now prove this lemma. We follow the argument of Krichene et al. (2015) and omit the +detailed calculations that can be found in (Krichene et al., 2015, Appendix 2). Fix δ. For +t > 0, define +Aδ(t) := sup +u∈[0,t] +��� ˙Wδ(u) +��� +u +Bδ(t) := sup +u∈[0,t] +∥Xδ(u) − x0∥ +u +Cδ(t) := sup +u∈[0,t] +��� ˙Xδ(u) +��� . +Then, these quantities are finite. We first prove the following inequalities, which correspond +to (Krichene et al., 2015, Lemma 3). +Aδ(t) ≤ µ ∥w0∥ + µ ∥∇h (x0)∥ + ∥∇f (x0)∥ + (µLh + Lf) tBδ(t) +(117a) +Bδ(t) ≤ Lh∗t +3 +cothc +�√µ +2 T +� +Aδ(t) +(117b) +Cδ(t) ≤ cothc +�√µ +2 T +� +(Lh∗TAδ(t) + 2Bδ(t)) . +(117c) +Proof of (117a). +Using Aδ and Bδ, we can bound ∥Wδ(t) − w0∥ and ∥Xδ(t) − x0∥ as +∥Wδ(t) − w0∥ ≤ t2 +2 Aδ(t) +∥Xδ(t) − x0∥ ≤ tBδ(t). +From (116b), we have +2 +��� ˙Wδ(t) +��� +t += tanhc +�√µ +2 t +� +∥µ∇h(Xδ) − µWδ − ∇f(Xδ)∥ +≤ ∥µ∇h(Xδ) − µWδ − ∇f(Xδ)∥ +≤ µ ∥Wδ∥ + µ ∥∇h(Xδ)∥ + ∥∇f(Xδ)∥ +≤ µ ∥w0∥ + µt2 +2 Aδ(t) + µ ∥∇h (x0)∥ + µLhtBδ(t) + ∥∇f (x0)∥ + LftBδ(t). +Thus, +2Aδ(t) ≤ µ ∥w0∥ + µ ∥∇h (x0)∥ + ∥∇f (x0)∥ ++ µt2 +2 Aδ(t) + (µLh + Lf) tBδ(t). +Because T ≤ +� +2/µ, we obtain the inequality (117a). +65 + +Kim and Yang +Proof of (117b). +To bound the function Bδ(t) = supu∈[0,t] +∥Xδ(u)−x0∥ +u +, we first compute +an upper bound of ∥Xδ(t) − x0∥ in the case 0 ≤ t ≤ δ and the case t ≥ δ separately. First, +consider the case t ∈ [0, δ]. By (116a), we have +˙Xδ + 2 +δ cothc +�√µ +2 δ +� +(Xδ − x0) = 2 +δ cothc +�√µ +2 δ +� +(∇h∗(Wδ − ∇h∗ (w0)) . +Multiplying e +2 +δ cothc +� √µ +2 δ +� +t, we obtain +e +2 +δ cothc +� √µ +2 δ +� +t +� +˙Xδ + 2 +δ cothc +�√µ +2 δ +� +(Xδ − x0) +� += 2 +δ cothc +�√µ +2 δ +� +e +2 +δ cothc +� √µ +2 δ +� +t (∇h∗(Wδ) − ∇h∗ (w0)) . +This equality can be written as +d +dt +� +(Xδ(t) − x0) e +2 +δ cothc +� √µ +2 δ +� +t +� += 2 +δ cothc +�√µ +2 δ +� +e +2 +δ cothc +� √µ +2 δ +� +t (∇h∗ (Wδ(t)) − ∇h∗ (w0)) . +Integrating both sides yields +(Xδ(t) − x0) e +2 +δ cothc +� √µ +2 δ +� +t += 2 +δ cothc +�√µ +2 δ +� � t +0 +� +e +2 +δ cothc +� √µ +2 δ +� +s (∇h∗ (Wδ(s)) − ∇h∗ (w0)) +� +ds. +Taking norms, we have +∥Xδ(t) − x0∥ ≤ 2 +δ cothc +�√µ +2 δ +� � t +0 +∥∇h∗ (Wδ(s)) − ∇h∗ (w0)∥ ds +≤ 2Lh∗ +δ +cothc +�√µ +2 δ +� � t +0 +∥Wδ(s) − w0∥ ds +≤ 2Lh∗ +δ +cothc +�√µ +2 δ +� � t +0 +s2 +2 Aδ(t) ds += 2Lh∗ +δ +cothc +�√µ +2 δ +� +Aδ(t)t3 +6 +≤ 2Lh∗ +t +cothc +�√µ +2 δ +� +Aδ(t)t3 +6 += Lh∗t2 +3 +cothc +�√µ +2 δ +� +Aδ(t). +So far, we provide an upper bound of ∥Xδ(t) − x0∥ in the case 0 ≤ t ≤ δ. We now consider +the case t ≥ δ. By (116a), we have +˙Xδ + 2 +t cothc +�√µ +2 t +� +(Xδ − x0) = 2 +t cothc +�√µ +2 t +� +(∇h∗(Wδ) − ∇h∗ (w0)) . +66 + +Unifying NAG for Convex and Strongly Convex Objective Functions +Multiplying t2 +4 sinhc2( +√µ +2 t) to both sides, we obtain +t2 +4 sinhc2 +�√µ +2 t +� +˙Xδ + t +2 sinhc +�√µ +2 t +� +cosh +�√µ +2 t +� +(Xδ − x0) += t +2 sinhc +�√µ +2 t +� +cosh +�√µ +2 t +� +(∇h∗(Wδ) − ∇h∗ (w0)) . +This equality can be written as +d +dt +�t2 +4 sinhc2 +�√µ +2 t +� +(Xδ(t) − x0) +� += t +2 sinhc +�√µ +2 t +� +cosh +�√µ +2 t +� +(∇h∗ (Wδ(t)) − ∇h∗ (w0)) . +Integrating both sides, we obtain +t2 +4 sinhc2 +�√µ +2 t +� +(Xδ(t) − x0) += +� t +0 +�s +2 sinhc +�√µ +2 s +� +cosh +�√µ +2 s +� +(∇h∗ (Wδ(s)) − ∇h∗ (w0)) +� +ds. +Taking norms, we have the following upper bound on ∥Xδ(t) − x0∥: +∥Xδ(t) − x0∥ ≤ 2 +t cothc +�√µ +2 t +� � t +0 +∥∇h∗ (Wδ(s)) − ∇h∗ (w0)∥ ds. +≤ 2Lh∗ +t +cothc +�√µ +2 t +� � t +0 +∥Wδ(s) − w0∥ ds. +≤ 2Lh∗ +t +cothc +�√µ +2 t +� � t +0 +s2 +2 Aδ(t) ds += 2Lh∗ +t +cothc +�√µ +2 t +� +Aδ(t)t3 +6 += Lh∗t2 +3 +cothc +�√µ +2 t +� +Aδ(t). +Combining both cases 0 ≤ t ≤ δ and t ≥ δ, we have +∥Xδ(t) − x0∥ ≤ Lh∗t2 +3 +cothc +�√µ +2 T +� +Aδ(t) +for all t ≥ 0. Dividing by t and taking the supremum, we obtain +Bδ(t) ≤ Lh∗t +3 +cothc +�√µ +2 T +� +Aδ(t). +67 + +Kim and Yang +Proof of (117c). +By (116a), we have +��� ˙X +��� = +2 +max{δ, t} cothc +�√µ +2 max{δ, t} +� +∥∇h∗ (Wδ(t)) − Xδ(t)∥ +≤ +2 +max{δ, t} cothc +�√µ +2 max{δ, t} +� +(∥∇h∗ (Wδ(t)) − ∇h∗ (z0)∥ + ∥Xδ(t) − x0∥) +≤ +2 +max{δ, t} cothc +�√µ +2 max{δ, t} +� �t2 +2 Lh∗Aδ(t) + tBδ(t) +� +≤ cothc +�√µ +2 T +� 2 +t +�t2 +2 Lh∗Aδ(t) + tBδ(t) +� +≤ cothc +�√µ +2 T +� +(Lh∗TAδ(t) + 2Bδ(t)) . +Complete the proof of Lemma 22. +Define five positive constants K1, . . ., K5 as +K1 := µ ∥w0∥ + µ ∥∇h (x0)∥ + ∥∇f (x0)∥ +K2 := µLh + Lf +K3 := 2Lh∗ +3 +K4 := 2Lh∗ +K5 := 4. +(118) +Because T ≤ +2 +√µ, we have cothc( +√µ +2 T) ≤ cothc(1) ≤ 2. Thus, the inequalities (117) imply +Aδ(t) ≤ K1 + K2TBδ(t) +(119a) +Bδ(t) ≤ K3TAδ(t) +(119b) +Cδ(t) ≤ K4TAδ(t) + K5Bδ(t). +(119c) +Combining (119a) and (119b), we have +� +1 +K3T − K2T +� +Bδ(t) ≤ K1. +Because T �→ +1 +K3T − K2T is a positive decreasing funtion on [0, 1 +2 +� +1 +K2K3 ] and T ≤ 1 +2 +� +1 +K2K3 , +we have +Bδ(T) ≤ +� +� +1 +K3 · 1 +2 +� +1 +K2K3 +− K2 · 1 +2 +� +1 +K2K3 +� +� +−1 +K1 = 2 +3K1 +� +K3 +K2 +. +(120) +The inequalities (119a), (120), and T ≤ 1 +2 +� +1 +K2K3 imply +Aδ(T) ≤ K1 + K2TBδ(T) ≤ K1 + K2 +�1 +2 +� +1 +K2K3 +� � +2 +3K1 +� +K3 +K2 +� +. +(121) +68 + +Unifying NAG for Convex and Strongly Convex Objective Functions +The inequalities (119a), (120), (121), and T ≤ 1 +2 +� +1 +K2K3 imply +Cδ(T) ≤ K4TAδ(T) + K5Bδ(T) +≤ K4 +�1 +2 +� +1 +K2K3 +� � +K1 + K2 +�1 +2 +� +1 +K2K3 +� � +2 +3K1 +� +K3 +K2 +�� ++ K5 +� +2 +3K1 +� +K3 +K2 +� +. +(122) +Therefore, ∥ ˙W∥ and ∥ ˙X∥ are bounded uniformly in δ because +��� ˙Wδ(t) +��� ≤ TAδ(T) +��� ˙Xδ(t) +��� ≤ Cδ(T) +for all t ∈ [0, T]. This implies that the family of solutions ((Xδ, Zδ)|[0,T])δ∈(0,T] is equi- +Lipschitz-continuous and uniformly bounded. +H.1.2 Proof of uniqueness +We follow the argument in (Krichene et al., 2015, Appendix 3) and omit the detailed +calculations that can be found in (Krichene et al., 2015). Because we only need to prove +the uniqueness of solution near t = 0, we assume t < T for some T > 0. Let (X, W) and +� ¯X, ¯W +� +be solutions to the following system of ODEs, which is equivalent to (114): +˙X = 2 +t cothc +�√µ +2 t +� +(∇h∗(W) − X) +˙W = t +2 tanhc +�√µ +2 t +� +(µ∇h(X) − µW − ∇f(X)) . +Let ∆W = W − ¯W and ∆X = X − ¯X. Then, we have +˙∆W = t +2 tanhc +�√µ +2 t +� � +µ∇h(X) − µW − ∇f(X) − µ∇h +� ¯X +� ++ µ ¯W + ∇f +� ¯X +�� +˙∆X = 2 +t cothc +�√µ +2 t +� � +∇h∗(W) − ∇h∗ � ¯W +� +− ∆X +� +with ∆X(0) = ∆W (0) = 0. Define +A(t) := sup +[0,t] +��� ˙∆W (u) +��� +u +B(t) := sup +[0,t] +∥∆X∥ . +69 + +Kim and Yang +Then, B(t) and C(t) are finite because ∆X and ∆W are continuous. First, we compute an +upper bound of A(t). We have +��� ˙∆W (t) +��� = t +2 tanhc +�√µ +2 t +� ��µ∇h(X) − µW − ∇f(X) − µ∇h +� ¯X +� ++ µ ¯W + ∇f +� ¯X +��� +≤ t +2 tanhc +�√µ +2 t +� � +µ +��∇h(X) − ∇h +� ¯X +��� + µ +��W − ¯W +�� + +��∇f(X) − ∇f +� ¯X +���� +≤ t +2 tanhc +�√µ +2 t +� +((µLh + Lf) ∥∆X∥ + µ ∥∆W ∥) +≤ t +2 tanhc +�√µ +2 t +� � +(µLh + Lf) B(t) + µt2 +2 A(t) +� +, +(123) +where we used ∥∆W (t)∥ ≤ ∥ +� t +0 ˙∆W (s) ds∥ ≤ +� t +0 sA(s) ds ≤ +� t +0 sA(t) ds = t2 +2 A(t) for the last +inequality. Dividing both sides of (123) by t and then taking the supremum, we obtain +A(t) ≤ 1 +2 tanhc +�√µ +2 t +� � +(µLh + Lf) B(t) + µt2 +2 A(t) +� +. +(124) +Nest, we compute an upper boudn of B(t). We have +˙∆X + 2 +t cothc +�√µ +2 t +� +∆X = 2 +t cothc +�√µ +2 t +� � +∇h∗(W) − ∇h∗ � ¯W +�� +. +Multiplying both sides by t2 +4 sinhc2 � √µ +2 t +� +, we have +t2 +4 sinhc2 +�√µ +2 t +� +˙∆X + t +2 sinhc +�√µ +2 t +� +cosh +�√µ +2 t +� +∆X += t +2 sinhc +�√µ +2 t +� +cosh +�√µ +2 t +� � +∇h∗(W) − ∇h∗ � ¯W +�� +. +This equality can be written as +d +dt +�t2 +4 sinhc2 +�√µ +2 t +� +∆X +� += t +2 sinhc +�√µ +2 t +� +cosh +�√µ +2 t +� � +∇h∗(W) − ∇h∗ � ¯W +�� +. +Integrating both sides, we obtain +t2 +4 sinhc2 +�√µ +2 t +� +∆X = +� t +0 +�s +2 sinhc +�√µ +2 s +� +cosh +�√µ +2 s +� � +∇h∗(W(s)) − ∇h∗ � ¯W(s) +��� +ds. +Taking norms, we have +∥∆X(t)∥ ≤ 2 +t cothc +�√µ +2 t +� � t +0 +��∇h∗(W(s)) − ∇h∗ � ¯W(s) +��� ds. +≤ 2Lh∗ +t +cothc +�√µ +2 t +� � t +0 +∥∆W (s)∥ ds. +70 + +Unifying NAG for Convex and Strongly Convex Objective Functions +≤ 2Lh∗ +t +cothc +�√µ +2 t +� � t +0 +s2 +2 A(t) ds += Lh∗2 +t +cothc +�√µ +2 t +� +A(t)t3 +6 += Lh∗t2 +3 +cothc +�√µ +2 t +� +A(t). +Taking the supremum yields +B(t) ≤ Lh∗t2 +3 +cothc +�√µ +2 t +� +A(t). +(125) +Now, combining the inequalities (124) and (125), we have +A(t) ≤ 1 +2 tanhc +�√µ +2 t +� � +(µLh + Lf) B(t) + µt2 +2 A(t) +� +≤ 1 +2 tanhc +�√µ +2 t +� � +(µLh + Lf) Lh∗t2 +3 +cothc +�√µ +2 t +� ++ µt2 +2 +� +A(t). +Using continuity, it is easy to see that there is Tsmall > 0 such that the following inequality +holds whenever t ∈ (0, Tsmall): +1 +2 tanhc +�√µ +2 t +� � +(µLh + Lf) Lh∗t2 +3 +cothc +�√µ +2 t +� ++ µt2 +2 +� +< 1. +Thus, for t ∈ (0, Tsmall), we have A(t) ≤ 1 · A(t), which implies A(t) = 0 because A(t) is +nonnegative by its definition. Finally, B(t) = 0 follows from (125). This completes the proof. +H.2 Existence and uniqueness of solution to the unified accelerated tensor flow +We first note that +• The system of ODEs (114) is the unified Bregman Lagrangian flow (56) with β1 = +log +� +t2 +4 sinhc2 � √µ +2 t +�� +and α1 = log ˙β1. +• The unified accelerated tensor flow (76) is the unified Bregman Lagrangian flow (56) +with β2 = p log t + log C + p log +� +sinhcp +� +C1/pµ1/pt +�� +and α2 = log ˙β2. +Define a function T : [0, ∞) → [0, ∞) as T = β−1 +1 +◦ β2. Then, we have +α2(t) = α1(T(t)) + log ˙T(t) +β2(t) = β1(T(t)). +Thus, by Theorem 5, if (X1, Z1) is a solution to the unified NAG system, then X2(t) = +X1(T(t)) and Z2(t) = Z1(T(t)) is a solution to the unified accelerated tensor system. Thus, +the existence of solution to the unified NAG system implies the existence of solution to the +unified accelerated tensor system. +A similar argument shows that if (X2, Z2) is a solution to the unified accelerated tensor +system, then X1(t) = X2(T−1(t)) and Z1(t) = Z2(T−1(t)) is a solution to the unified NAG +system. It is easy to show that this correspondence is one-to-one. Thus, the uniqueness +of solution to the unified NAG system implies the uniqueness of solution to the unified +accelerated tensor system. +71 + +Kim and Yang +I. Further Exploration: ODE Model for Minimizing Gradient Norms of +Strongly Convex Functions +I.1 Limiting ODE of OGM +For the sequence θk defined in (90), Su et al. (2016) showed that the algorithm +yk = +� +1 − 1 +θk +� +xk + 1 +θk +zk +xk+1 = yk − s∇f (yk) +zk+1 = zk − sθk∇f (yk) +(126) +converges to NAG-C ODE as s → 0 (see Su et al., 2016, Section 2) (in fact, this algorithm is +equivalent to the original NAG (5) with µ = 0 and γ0 = ∞). Because ∥xk+1 − yk∥ = o(√s), +we can ignore the gradient descent step xk+1 = yk −s∇f (yk) in both (126) and OGM. Then, +applying OGM to the objective function f is equivalent to applying the algorithm (126) to +the objective function 2f. Thus, the limiting ODE of OGM is given by +¨X + 3 +t +˙X + 2∇f(X) = 0. +I.2 Proof of Theorem 19 +For convenience, we assume µ > 0 (the case µ = 0 can be handled easily). We denote +X := X(t) and xT := X(T). We also omit the input +√µ +2 (T − t) of each hyperbolic function. +For example, we write the unified NAG-G ODE (97) as +¨X + +�√µ +2 tanh +3√µ +2 +coth +� +˙X + ∇f(X) = 0 +and the continuous-time energy function (98) as +E(t) = µ2 csch4 +� +sinh2 +µ +� +f(X) − f +� +xT �� +− 1 +2 +��X − xT ��2 + cosh2 +2 +����X + tanh +√µ +˙X − xT +���� +2� +. +Then, we have +sinh4 +µ2 +˙E(t) += sinh4 d +dt +� +csch4� +� +sinh2 +µ +� +f(X) − f +� +xT �� +− 1 +2 +��X − xT ��2 + cosh2 +2 +����X + tanh +√µ +˙X − xT +���� +2� ++ d +dt +� +sinh2 +µ +� +f(X) − f +� +xT �� +− 1 +2 +��X − xT ��2 + cosh2 +2 +����X + tanh +√µ +˙X − xT +���� +2� += 2√µ coth +� +sinh2 +µ +� +f(X) − f +� +xT �� +− 1 +2 +��X − xT ��2 + cosh2 +2 +����X + tanh +√µ +˙X − xT +���� +2� +− sinh cosh +√µ +� +f(X) − f +� +xT �� ++ sinh2 +µ +� +∇f(X), ˙X +� +− +� +X − xT , ˙X +� +72 + +Unifying NAG for Convex and Strongly Convex Objective Functions +− +√µ sinh cosh +2 +����X + tanh +√µ +˙X − xT +���� +2 ++ cosh2 +� +X + tanh +√µ +˙X − xT , − ˙X − tanh +√µ ∇f(X) +� +, +where we used +d +dt +� +X + tanh +√µ +˙X − xT +� += tanh +√µ +¨X + +� +1 − 1 +2 sech2 +� +˙X += +� +−1 +2 tanh2 −1 +2 − 1 +2 sech2 +� +˙X − tanh +√µ ∇f(X) += − ˙X − tanh +√µ ∇f(X) +for the last equality. We further simplify as +sinh4 +µ2 +˙E(t) = 2 sinh cosh +√µ +� +f(X) − f +� +xT �� +− √µ coth +��X − xT ��2 ++ √µ coth cosh2 +���X − xT ��2 + tanh2 +µ +��� ˙X +��� +2 ++ 2 tanh +√µ +� +X − xT , ˙X +�� +− sinh cosh +√µ +� +f(X) − f +� +xT �� ++ sinh2 +µ +� +∇f(X), ˙X +� +− +� +X − xT , ˙X +� +− +√µ sinh cosh +2 +���X − xT ��2 + tanh2 +µ +��� ˙X +��� +2 ++ 2 tanh +√µ +� +X − xT , ˙X +�� +− cosh2 +� � +X − xT , ˙X +� ++ tanh +√µ +��� ˙X +��� +2 ++ tanh +√µ +� +X − xT , ∇f(X) +� ++ tanh2 +µ +� +˙X, ∇f(X) +� � += +�2 sinh cosh +√µ +− sinh cosh +√µ +� � +f(X) − f +� +xT �� ++ +� +−√µ coth +√µ coth cosh2 − +√µ sinh cosh +2 +� ��X − xT ��2 ++ +�sinh cosh +√µ +− sinh2 tanh +2√µ +− sinh cosh +√µ +� ��� ˙X +��� +2 ++ +� +2 cosh2 −1 − sinh2 − cosh2� � +X − xT , ˙X +� ++ +�sinh2 +µ +− sinh2 +µ +� � +∇f(X), ˙X +� +− sinh cosh +√µ +� +X − xT , ∇f(X) +� += sinh cosh +√µ +� +f(X) − f +� +xT �� ++ +√µ sinh cosh +2 +��X − xT ��2 − sinh2 tanh +2√µ +��� ˙X +��� +2 +− sinh cosh +√µ +� +X − xT , ∇f(X) +� +. +73 + +Kim and Yang +It follows from the µ-strong convexity of f that f(X) − f +� +xT � +≤ +� +X − xT , ∇f(X) +� +− +µ +2 +��X − xT ��2. Thus, we have +sinh4 +µ2 +˙E(t) ≤ sinh cosh +√µ +�� +X − xT , ∇f(X) +� +− µ +2 +��X − xT ��2� ++ +√µ sinh cosh +2 +��X − xT ��2 − sinh2 tanh +2√µ +��� ˙X +��� +2 +− sinh cosh +√µ +� +X − xT , ∇f(X) +� += −sinh2 tanh +2√µ +��� ˙X +��� +2 +≤ 0. +I.3 Computing ˙X(T) and ¨X(T) +For simplicity, we assume that the limits limt→T − ˙X(T) and limt→T − ¨X(T) exist.12 Consider +the energy function +E(t) = 1 +2 +��� ˙X(t) +��� +2 ++ (f(X(t)) − f (x∗)) ++ +� t +0 +�√µ +2 tanh +�√µ +2 (T − s) +� ++ +3 +T − s cothc +�√µ +2 (T − s) +�� ��� ˙X(s) +��� +2 +ds. +(127) +Then, it is easy to show that E(t) = E(0) for all t ∈ [0, T). Because the terms 1 +2 +��� ˙X(t) +��� +2 +and f(X(t)) − f (x∗) are non-negtive, we have +� T +0 +�√µ +2 tanh +�√µ +2 (T − s) +� ++ +3 +T − s cothc +�√µ +2 (T − s) +�� ��� ˙X(s) +��� +2 +ds < ∞. +This implies limt→T − ˙X(t) = 0. 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On accelerated proximal gradient methods for convex-concave optimization. +submitted to SIAM Journal on Optimization, 2(3), 2008. +76 + +Unifying NAG for Convex and Strongly Convex Objective Functions +A. Wibisono, A. C. Wilson, and M. I. Jordan. A variational perspective on accelerated +methods in optimization. arXiv preprint arXiv:1603.04245, 2016. +A. C. Wilson, B. Recht, and M. I. Jordan. A lyapunov analysis of accelerated methods in +optimization. Journal of Machine Learning Research, 22(113):1–34, 2021. +P. Zhang, A. Orvieto, H. Daneshmand, T. Hofmann, and R. S. Smith. Revisiting the role +of euler numerical integration on acceleration and stability in convex optimization. In +International Conference on Artificial Intelligence and Statistics, pages 3979–3987. PMLR, +2021. +77 + diff --git a/GdE1T4oBgHgl3EQf_Aah/content/tmp_files/load_file.txt b/GdE1T4oBgHgl3EQf_Aah/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..4fdd68a524637ce3ab33a4b5bbe88b4711f31cef --- /dev/null +++ b/GdE1T4oBgHgl3EQf_Aah/content/tmp_files/load_file.txt @@ -0,0 +1,1742 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf,len=1741 +page_content='Unifying NAG for Convex and Strongly Convex Objective Functions Unifying Nesterov’s Accelerated Gradient Methods for Convex and Strongly Convex Objective Functions: From Continuous-Time Dynamics to Discrete-Time Algorithms Jungbin Kim kjb2952@snu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='kr Department of Electrical and Computer Engineering Seoul National University Seoul 08826, Korea Insoon Yang insoonyang@snu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='kr Department of Electrical and Computer Engineering Seoul National University Seoul 08826, Korea Abstract Although Nesterov’s accelerated gradient (NAG) methods have been studied from various perspectives, it remains unclear why the most popular forms of NAG must handle convex and strongly convex objective functions separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Motivated by this inconsistency, we propose an NAG method that unifies the existing ones for the convex and strongly convex cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' We first design a Lagrangian function that continuously extends the first Bregman Lagrangian to the strongly convex setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' As a specific case of the Euler–Lagrange equation for this Lagrangian, we derive an ordinary differential equation (ODE) model, which we call the unified NAG ODE, that bridges the gap between the ODEs that model NAG for convex and strongly convex objective functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' We then design the unified NAG, a novel momentum method whereby the continuous-time limit corresponds to the unified ODE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' The coefficients and the convergence rates of the unified NAG and unified ODE are continuous in the strong convexity parameter µ on [0, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Unlike the existing popular algorithm and ODE for strongly convex objective functions, the unified NAG and the unified NAG ODE always have superior convergence guarantees compared to the known algorithms and ODEs for non-strongly convex objective functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' This property is beneficial in practical perspective when considering strongly convex objective functions with small µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Furthermore, we extend our unified dynamics and algorithms to the higher-order setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Last but not least, we propose the unified NAG-G ODE, a novel ODE model for minimizing the gradient norm of strongly convex objective functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Our unified Lagrangian framework is crucial in the process of constructing this ODE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Fascinatingly, using our novel tool, called the differential kernel, we observe that the unified NAG ODE and the unified NAG-G ODE have an anti-transpose relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Keywords: Convex optimization, first-order methods, Nesterov acceleration 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Introduction We consider the optimization problem min x∈Rn f(x), (1) where f : Rn → R is a continuously differentiable function whose gradient is L-Lipschitz continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' We assume that the objective function f has a minimizer x∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' One of the most 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='03576v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='OC] 9 Jan 2023 Kim and Yang popular first-order method for solving this problem is gradient descent (GD): xk+1 = xk − s∇f (xk) (2) with the algorithmic stepsize s > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' When f is convex, GD with s ≤ 1/L achieves an O(∥x0 − x∗∥2/k) convergence rate (see d’Aspremont et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=', 2021, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' When f is µ-strongly convex, GD with s ≤ 1/L achieves an O((1 − µs)k∥x0 − x∗∥2) convergence rate (see d’Aspremont et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=', 2021, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Nesterov acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' A natural and important question is whether there are other first- order methods that outperform gradient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Nesterov (1983) proposed an accelerated gradient method that achieves a faster convergence rate compared to gradient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Given the initial point x0 = z0, a general three-sequence scheme for Nesterov’s accelerated gradient (NAG) methods can be written as yk = xk + τk (zk − xk) (3a) xk+1 = yk − s∇f (yk) (3b) zk+1 = zk + δk (µyk − µzk − ∇f (yk)) (3c) with s > 0, where the parameters τk and δk usually satisfy the collinearity condition1 1 − µδk − (1/s − µ)τkδk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (4) In particular, for µ-strongly (possibly with µ = 0) convex objective functions, Nesterov considered the following algorithm: Given an initial point x0 = z0 ∈ Rn and γ0 > 0, the constant step scheme I (Nesterov, 2018, Equation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='19) (we will refer to this algorithm as the original NAG) updates the iterates as γk+1 = (1 − αk) γk + µαk yk = 1 γk + µαk (αkγkzk + γk+1xk) xk+1 = yk − s∇f (yk) zk+1 = 1 γk+1 ((1 − αk) γkzk + µαkyk − αk∇f (yk)) , (5) where the sequence (αk)∞ k=0 in (0, 1) is inductively defined by the equation 1 sα2 k = (1 − αk) γk + µαk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (6) Using the estimate sequence technique, Nesterov (2018, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='1) showed that the iterates of the original NAG (5) satisfy the inequality f (xk) − f (x∗) ≤ �k−1 � i=0 (1 − αi) � � f (x0) − f (x∗) + γ0 2 ∥x0 − x∗∥2� (7) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' This condition ensures that the points xk, xk+1, zk+1 are collinear (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Thus, one can write the updating rule for yk as yk+1 = xk+1 + βk(xk+1 − xk) for some βk ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' This property provides a clear momentum effect: The point yk+1 is defined by adding a momentum term βk (xk+1 − xk) to the previous point xk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' This property is useful when generalizing NAG methods to handle non-smooth terms (see d’Aspremont et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=', 2021, Algorithm 20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' 2 Unifying NAG for Convex and Strongly Convex Objective Functions when s ≤ 1/L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Although the original NAG achieves a faster convergence rate than gradient descent, it is difficult to analyze this algorithm because it involves auxiliary sequences αk and γk which are defined inductively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' However, when γ0 = µ (here we need µ > 0 because γ0 > 0 is assumed), we simply have αk = √µs and γk = µ for all k ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' In this case, the original NAG (5) can be expressed as the three-sequence scheme (3) with τk = √µs 1+√µs and δk = � s µ: yk = xk + √µs 1 + √µs (zk − xk) xk+1 = yk − s∇f (yk) zk+1 = zk + � s µ (µyk − µzk − ∇f (yk)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (8) We refer to this algorithm as NAG-SC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Letting αi = √µs in (7), we can see that this algorithm achieves an O((1 − √µs)k(f(x0) − f(x∗) + µ 2∥x0 − x∗∥2)) convergence rate when s ≤ 1/L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' A major drawback of NAG-SC is that we cannot apply it to non-strongly convex objective functions (µ = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' For non-strongly convex objective functions, Tseng (2008) proposed a simple alternative algorithm to the original NAG (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' They set the algorithmic parameters as τk = 2 k+1 and δk = s(k+1) 2 to obtain the following simple algorithm, which we call NAG-C: yk = xk + 2 k + 1 (zk − xk) xk+1 = yk − s∇f (yk) zk+1 = zk − s(k + 1) 2 ∇f (yk) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (9) When s ≤ 1/L, this algorithm achieves an O � ∥x0 − x∗∥2/k2� convergence rate (see Sec- tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Although there are many variants of NAG, most recent studies on acceleration (Di- akonikolas and Orecchia, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Siegel, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Alimisis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Wilson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Kim and Yang, 2022) focus on these two particular algorithms because of their simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Unfortunately, these two algorithms should be handled separately because NAG-SC (8) does not recover NAG-C (9) as µ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Inconsistency I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' NAG-SC does not recover NAG-C as µ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Moreover, NAG-SC has the following drawbacks: It cannot be applied to non-strongly convex objective functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' When µ is very small, the convergence guarantee for NAG-SC is worse than that for NAG-C in early stages because (1 − √µs)k converges to 0 very slowly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' The convergence rate of NAG-SC depends on both the initial squared distance ∥x0−x∗∥2 and the initial function value accuracy f(x0) − f(x∗), while the convergence rate of NAG-C depends only on the squared initial distance ∥x0 − x∗∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' As most of recent works on Nesterov acceleration are based on these two specific algorithms, similar inconsistencies can be found in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' We discuss more inconsistencies below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' 3 Kim and Yang 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='1 Inconsistencies between convex and strongly convex cases 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='1 Continuous-time models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' In this subsection, we first informally derive the limiting ODE of the three-sequence scheme (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' To identify a discrete-time sequence (xk)∞ k=0 with a continuous-time curve X : [0, ∞) → Rn, given the algorithmic stepsize s, we introduce a strictly increasing sequence (tk)∞ k=0 (depending on s) in [0, ∞) and make the identification X(tk) = xk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' We denote the inverse of the sequence t : {0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='} → R as k, that is, k(tk) = k for all k ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' For convenience, we extend the function k to a piecewise linear function defined on [0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' We assume that lim s→0 t0 = 0 (10) and that the timesteps are asymptotically equivalent to √s as s → 0 in the sense that lim s→0 tk(t)+1 − t √s = 1 for all t ∈ (0, ∞) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (11) Note that the popular choice tk = tk := k√s (we will use the notation tk for this specific sequence throughout the paper) used in (Su et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Wibisono et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=', 2021) satisfies these conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' For the iterates of three-sequence scheme (3), we have xk+1 − xk √s = τk √s (zk − xk) − √s∇f (yk) zk+1 − zk √s = δk √s (µyk − µzk − ∇f (yk)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' We introduce two sufficiently smooth curves X, Z : [0, ∞) → Rn (possibly depending on s now) such that X(t) = xk(t) and Z(t) = zk(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Since ∥xk+1 − yk∥ = o(√s) and ∇f is Lipschitz continuous, we have ˙X(t) = lim s→0 xk(t)+1 − xk(t) tk(t)+1 − t = lim s→0 xk(t)+1 − xk(t) √s = lim s→0 �τk(t) √s � (Z(t) − X(t)) ˙Z(t) = lim s→0 zk(t)+1 − zk(t) tk(t)+1 − t = lim s→0 zk(t)+1 − zk(t) √s = lim s→0 �δk(t) √s � (µX(t) − µZ(t) − ∇f(X(t))) for all t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Thus, if the limits τ(t) = lim s→0 τk(t) √s δ(t) = lim s→0 δk(t) √s (12) exist for all t ∈ (0, ∞), then as s → 0, the iterates generated by the three-sequence scheme (3) converge to a solution to the following system of ODEs: ˙X(t) = τ(t)(Z(t) − X(t)) ˙Z(t) = δ(t)(µX(t) − µZ(t) − ∇f(X(t))) (13) 4 Unifying NAG for Convex and Strongly Convex Objective Functions with the initial conditions X(0) = Z(0) = x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' We can equivalently write this as the following second-order ODE: ¨X + � τ(t) − ˙τ(t) τ(t) + µδ(t) � ˙X + τ(t)δ(t)∇f(X) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (14) Furthermore, when the collinearity condition (4) holds, we have δ(t) = lim s→0 δk √s = lim s→0 1 √s (µ + (1/s − µ)τk) = lim s→0 √s µs + (1 − µs)τk = 1 τ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (15) Limiting ODE of NAG-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Recall that NAG-C (9) is the three-sequence scheme (3) with τk = 2 k+1 and δk = s(k+1) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' With the sequence tk = k√s, we have τ(t) = lim s→0 τk(t) √s = lim s→0 2 √s (t/√s + 1) = 2 t δ(t) = lim s→0 δk(t) √s = lim s→0 √s (t/√s + 1) 2 = t 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Thus, as s → 0, NAG-C converges to the following ODE system, which we call NAG-C system: ˙X = 2 t (Z − X) ˙Z = − t 2∇f(X) (16) with X(0) = Z(0) = x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' This system can be written in the following second-order ODE, which we call NAG-C ODE: ¨X + 3 t ˙X + ∇f(X) = 0 (17) with X(0) = x0 and ˙X(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Su et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (2014) first derived this ODE and showed that the solution to (17) satisfies an O(∥x0 − x∗∥2/t2) convergence rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Limiting ODE of NAG-SC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Recall that NAG-SC (8) is the three-sequence scheme (3) with τk = √µs 1+√µs and δk = � s µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' With the sequence tk = −k log(1−√µs) √µ ,2 we have τ(t) = lim s→0 τk(t) √s = lim s→0 √µ 1 + √µs = √µ δ(t) = lim s→0 δk(t) √s = lim s→0 1 √µ = 1 √µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Thus, as s → 0, NAG-SC converges to the following ODE system, which we call NAG-SC system: ˙X = √µ(Z − X) ˙Z = 1 √µ (µX − µZ − ∇f(X)) (18) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Although the sequence tk = k√s leads to the same limiting dynamics, this particular sequence makes a clear connection between the convergence analysis of NAG-SC and that of NAG-SC ODE (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' 5 Kim and Yang with X(0) = Z(0) = x0, or equivalently, the following NAG-SC ODE: ¨X + 2√µ ˙X + ∇f(X) = 0 (19) with X(0) = x0 and ˙X(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Wilson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (2021) showed that the solution to this ODE satisfies an O(e−√µt(f(x0) − f(x∗) + µ 2∥x0 − x∗∥2)) convergence rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Just like in the discrete-time case, NAG-C ODE (17) and NAG-SC ODE (19) should be handled as separate cases because NAG-SC ODE does not recover NAG-C ODE as µ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Inconsistency II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' NAG-SC ODE does not recover NAG-C ODE as µ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Moreover, NAG-SC ODE has the following drawbacks: The solution to NAG-SC ODE with µ = 0 may not converge to the minimizer of f: For the objective function f(x) = 1 2x2 on R, the solution to NAG-SC ODE with x0 = 1 is X(t) = cos(t), which does not converge to the minimizer x∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' When µ is very small, the convergence guarantee for NAG-SC ODE is worse than that for NAG-C ODE in early stages because e−√µt converges to 0 very slowly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' The convergence rate of NAG-SC ODE depends on both the initial squared distance ∥x0 −x∗∥2 and the initial function value accuracy f(x0)−f(x∗), while the convergence rate of NAG-C ODE depends only on the squared initial distance ∥x0 − x∗∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2 Bregman Lagrangians To systematically study the acceleration phenomenon of momentum methods, Wibisono et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (2016) introduced the following first Bregman Lagrangian: L1st � X, ˙X, t � = eα+γ � Dh � X + e−α ˙X, X � − eβf(X) � , (20) where α, β, γ : [0, ∞) → R are continuously differentiable functions, h is a continuously differentiable strictly convex function, and Dh is the Bregman divergence (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='1 for its definition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' In order to obtain accelerated convergence rates, the following ideal scaling conditions are introduced: ˙γ = eα (21a) ˙β ≤ eα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (21b) Under the ideal scaling condition (21a), the Euler–Lagrange equation d dt � ∂L ∂ ˙X � X, ˙X, t �� = ∂L ∂X � X, ˙X, t � (22) for the first Bregman Lagrangian (20) reduces to the following system of first-order equations: ˙X = eα(Z − X) (23a) d dt∇h(Z) = −eα+β∇f(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (23b) 6 Unifying NAG for Convex and Strongly Convex Objective Functions When f is convex, any solution to the system of ODEs (23) reduces the objective function value accuracy at an O(e−β(t)) convergence rate (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' In particular, setting α(t) = log 2 t and β(t) = log t2 4 , we recover NAG-C system (16) and its convergence rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Although the first Bregman Lagrangian (20) generates a large family of momentum dynamics, it does not include NAG-SC system (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' To handle strongly convex cases, Wilson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (2021) introduced the second Bregman Lagrangian, defined as L2nd � X, ˙X, t � = eα+β+γ � µDh � X + e−α ˙X, X � − f(X) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (24) Under the ideal scaling condition (21a), the Euler–Lagrange equation (22) for the second Bregman Lagrangian (24) reduces to the following system of first-order equations: ˙X = eα(Z − X) (25a) d dt∇h(Z) = ˙β (∇h(X) − ∇h(Z)) − eα µ ∇f(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (25b) When f is µ-uniformly convex with respect to h (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='1), any solution to the system of ODEs (25) satisfies an O(e−β(t)) convergence rate (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' In particular, letting α(t) = log √µ and β(t) = √µt, we recover NAG-SC system (18) and its convergence rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Here, we observe an inconsistency between the two Bregman Lagrangians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Inconsistency III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' The second Bregman Lagrangian does not recover the first Bregman Lagrangian as µ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2 Contributions In this paper, we propose a novel unified framework for Lagrangians, ODE models and algorithms to address the inconsistencies between the convex case and the strongly convex case mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' The proposed framework seamlessly bridges the gap between the two cases as illustrated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' The main contributions of this work can be summarized as follows: We propose the unified Bregman Lagrangian (Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Unlike the second Bregman Lagrangian, the unified Bregman Lagrangian recovers the first Bregman Lagrangian when µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' As the Euler–Lagrange equation for the unified Bregman Lagrangian, we obtain a family of continuous-time dynamics (Proposition 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Using a Lyapunov function, we analyze the convergence rate for these flows (Theorem 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' We derive the unified NAG ODE (59) as a special case of the unified Bregman La- grangian flows (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Unlike NAG-SC ODE (19), for non-strongly convex objective functions (µ = 0), the unified NAG ODE and its convergence rate (The- orem 10) recover NAG-C ODE (17) and its convergence rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Furthermore, for any µ > 0, the unified NAG ODE and its convergence rate (Corollary 8) recover NAG-SC ODE (19) and its convergence rate as t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' We devise the unified NAG family (63), a family of momentum algorithms that converge to the unified NAG ODE as s → 0 (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' As a special case, we have 7 Kim and Yang NAG-C ODE (Su et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=', 2014) First Bregman La- grangian (Wibisono et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=', 2016) Unified Bregman Lagrangian (Section 3) Second Bregman Lagrangian (Wilson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=', 2021) Unified NAG ODE (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='1) NAG-SC ODE (Wilson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=', 2021) NAG-C (Tseng, 2008) Unified NAG (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2) NAG-SC (Nes- terov, 2018) Higher-order optimization (Section 5) Gradient norm minimization (Section 6) special case special case discretize limit discritize limit unify unify recover µ = 0 recover µ = 0 recover t → ∞ recover k → ∞ Figure 1: An illustration of our framework and contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' the unified NAG (70), a simple algorithm which unifies NAG-C (9) and NAG-SC (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Moreover, using an adaptive timestep in the unified NAG family, we constructively recover the original NAG (5) with γ0 > µ and its convergence rate (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' We extend the unified NAG ODE and the unified NAG family to the higher-order non-Euclidean setting (mirror descent setup) (Section 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Our novel dynamics and algorithms can be viewed as continuous extensions of the accelerated tensor method (convex case) and its limiting ODE in (Wibisono et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=', 2016) to the strongly convex setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' We also made the following contributions that are not closely related to our major goal but may deserve independent attention: We compute the general limiting ODEs of the three-sequence scheme (3), the two- sequence scheme (42), and the fixed-step first-order scheme (46).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' In particular, we introduce a novel tool, called the differential kernel H(t, τ), to derive the limiting ODE of the fixed-step first-order scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' We show that an anti-transpose relationship (95) between OGM and OGM-G can be naturally shifted to a continuous-time setting by this tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' We propose the unified NAG-G ODE, an ODE model for minimizing the gradient norm of strongly convex objective functions (Section 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Surprisingly, the differential kernels corresponding to the unified NAG ODE and the unified NAG-G ODE have an anti-transpose relationship, just like it does between OGM ODE and OGM-G ODE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' 8 Unifying NAG for Convex and Strongly Convex Objective Functions Dynamics Convergence rate Unified NAG ODE f � X � t �� − f � x∗� ≤ O � min � 1/t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' e−√µt���x0 − x∗��2� Unified accelerated tensor flow f � X � t �� − f � x∗� ≤ O � min � 1/tp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' e−pC1/pµ1/pt� Dh � x∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' x0 �� Unified NAG-G ODE ��∇f(X(T)) ��2 ≤ O � min � 1/T 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' e−√µT �� f � x0 � − f � x∗��� Algorithm Convergence rate Unified NAG f � xk � − f � x∗� ≤ O � min � 1/k2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' � 1 − √µs �k���x0 − x∗��2� Unified accelerated tensor method f � xk � − f � x∗� ≤ O � min � 1/kp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' � 1 + C1/ppµ1/ps1/p�−k� Dh � x∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' x0 ��� Table 1: Convergence rates of the momentum dynamics and algorithms proposed in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' We summarize the convergence rates for our continuous-time dynamics and discrete-time algorithms in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' In addition to theoretical and algorithmic perspectives, we discuss the need for unified acceleration methods from a practical perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Practical perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Many optimization problems in machine learning can be formu- lated as min x∈Rn f(x) = 1 m � m � i=1 fi(x) + λR(x) � , (26) where fi is the loss function corresponding to the i-th sample, λ > 0 is the regularization parameter, and R(x) is the regularization term (Bubeck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=', 2015, Equation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Consider the problem (26) where the functions fi are convex and L-smooth, and R(x) = ∥x∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Then, f is µ-strongly convex and L-smooth, where µ = 2λ/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' As the sample size m grows or the regularization parameter λ decreases, the strong convexity parameter µ decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Thus, improving the convergence rate for ill-conditioned strongly convex objective functions (where µ is small) is quite significant, as emphasized in (Bubeck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=', 2015, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' As mentioned above, the convergence guarantee of NAG-SC (8) is no better than that of NAG-C (9) when µ is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' In our numerical experiments (see Section 7), it is observed that the performance of NAG-SC is worse than that of NAG-C when µ is very small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Thus, it is desirable to design a strongly convex optimization algorithm whose convergence guarantee is not worse than that of NAG-C even when µ is very small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' In the experiments, we observe that for a logistic regression problem, when µ is small, our algorithm is comparable to NAG-C, while NAG-SC underperforms NAG-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Existing unified methods and dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' To clarify what is our novel contribution and what is not, we review existing algorithms and dynamics that can handle the non-strongly convex case and the strongly convex case in a unified way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' The original NAG (5) is an accelerated algorithm that can handle both convex objective functions and strongly convex objective functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2, we show that the original NAG can be constructively recovered by our unified Lagrangian formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Luo and Chen (2021) designed the following 9 Kim and Yang ODE model for the original NAG, which we call the original NAG system: ˙γ = µ − γ ˙X = Z − X ˙Z = 1 γ (µX − µZ − ∇f(X)) (27) with X(0) = Z(0) = x0 and γ(0) = γ0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Luo and Chen (2021, Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2) showed that the original NAG can be viewed as a discretization scheme with the timestep αi, which is inductively defined in (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Using time rescaling technique, Luo and Chen (2021) also proposed the following system of ODEs (although most of their results directly deal with Equation 27): ˙X(t) = a(t)(Z(t) − X(t)) b(t) ˙Z(t) = a(t)(µX(t) − µZ(t) − ∇f(X(t))), (28) where a : [0, ∞) → [0, ∞) is an arbitrary function and b(t) = γ �� t 0 a(s) ds � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' This ODE system is closely related to the unified Bregman Lagrangian flow (56) and the unified NAG system (58) proposed in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' In Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='1, we show that the rescaled original NAG flow (28) can be expressed as the unified Bregman Lagrangian flow (56).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Conversely, the unified Bregman Lagrangian flow can be expressed as the rescaled original NAG flow if the ideal scaling condition (21b) holds with equality and the distance-generating function h is Euclidean (h(x) = 1 2∥x∥2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Therefore, our unified Bregman Lagrangian generates a strictly larger family compared to (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' To emphasize, only our family can deal with the non-Euclidean setup (mirror descent setup).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' In addition, the derivation of our unified family (56) is more constructive because it comes from a Lagrangian formulation, whereas Luo and Chen (2021) designed the family (28) through heuristic speculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='3 Related work Nesterov (1983) first proposed the original NAG (5) with µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' The original NAG with µ > 0 was first analyzed using the estimate sequence technique (Nesterov, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Tseng (2008) proposed NAG-C (9) and its generalization to composite optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Su et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (2014) derived NAG-C ODE (17) by taking the limit s → 0 in NAG-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' This ODE has further been generalized and investigated in (Krichene et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Attouch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Wibisono et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (2016) proposed the first Bregman Lagrangian (20) that systematically generates a family of ODEs (23) including NAG-C ODE and its higher-order extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Wilson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (2021) extended this framework to the strongly convex case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' They proposed the second Bregman Lagrangian (24), which generates a family of continuous-time flows (25) including NAG-SC ODE (18), and strengthened the connection between continuous-time dynamics and discrete-time algorithms via Lyapunov function arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' However, as mentioned in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='1, their work is not consistent with (Wibisono et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=', 2016) because the second Bregman Lagrangian does not recover the first Bregman Lagrangian as µ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' 10 Unifying NAG for Convex and Strongly Convex Objective Functions Based on Lagrangian formulations, Betancourt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (2018) studied a symplectic integrator to obtain discrete-time algorithms from continuous-time dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (2019, 2021) derived high-resolution ODEs for NAG-C and NAG-SC, and then obtained algorithms with accelerated convergence rates by applying the symplectic Euler method to the high-resolution ODEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Luo and Chen (2021) understood acceleration using the A-stability theory and designed an ODE model for the original NAG method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (2021) obtained an accelerated algorithm by applying the explicit Euler method to a variant of high-resolution ODEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Diakonikolas and Orecchia (2019) proposed the approximate duality gap technique to construct and analyze accelerated algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Using conservation laws in dilated coordinate systems, Suh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (2022) recovered NAG-C ODE and NAG-SC ODE and showed that a semi-second-order symplectic Euler discretization in the dilated coordinate yields accelerated methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Preliminaries In this section, we review the basic notions that we will use throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' While Sections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2 review the standard concepts in the literature, Sections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='3 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='4 contain novel ideas and results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='1 Convex analysis Convexity and smoothness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Let f : Rn → R be a C∞ function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Then for µ ≥ 0, the function f is called µ-strongly convex if the inequality f(y) ≥ f(x) + ⟨∇f(x), y − x⟩ + µ 2 ∥y − x∥2 holds for all x, y ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' In particular, the function f is called convex if it is strongly convex with the strong convexity parameter µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' For L > 0, the function f is called L-smooth if its gradient is L-Lipschitz continuous, that is, the inequality ∥∇f(x) − ∇f(y)∥ ≤ L ∥x − y∥ holds for all x, y ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' It is known that when f is L-smooth, the inequality f(y) ≤ f(x) + ⟨∇f(x), y − x⟩ + L 2 ∥y − x∥2 holds for all x, y ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' For most of the remaining sections of this paper (Sections 4 and 6), we make the following assumptions, which we call the standard smooth strongly convex setting: The objective function f is (1/s)-smooth, where s > 0 is the algorithmic stepsize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' The objective function f is µ-strongly (possibly with µ = 0) convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Higher-order convexity and smoothness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' The notions of convexity and smoothness can be generalized to the higher-order setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' The function f is called µ-uniformly convex of order p ≥ 2 if the inequality f(y) ≥ f(x) + ⟨∇f(x), y − x⟩ + µ p ∥y − x∥p (29) 11 Kim and Yang holds for all x, y ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' The function f is called L-smooth of order p − 1 if the inequality ��∇p−1f(y) − ∇p−1f(x) �� ≤ L ∥y − x∥ (30) holds for all x, y ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Note that these definitions recover the standard notions of convexity and smoothness when p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Bregman divergences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' In the optimization literature, a common way to consider a non-Euclidean setting is by using the Bregman divergence, instead of the Euclidean distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' For a continuously differentiable function h : Rn → R which is convex and essentially smooth (∥∇h(x)∥ → ∞ as ∥x∥ → ∞), the Bregman divergence Dh : Rn ×Rn → [0, ∞) of h is defined as Dh(y, x) = h(y) − h(x) − ⟨∇h(x), y − x⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (31) Note that when h(x) = 1 2∥x∥2, the Bregman divergence of h is the squared Euclidean distance 1 2∥y − x∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' For all x, y, z ∈ Rn, the three-point identity (see Wilson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=', 2021, Proposition 5) Dh(x, y) − Dh(x, z) = − ⟨∇h(y) − ∇h(z), x − y⟩ − Dh(y, z) (32) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' For µ ≥ 0, the function f is called µ-uniformly convex with respect to h if the inequality Df(x, y) ≥ µDh(x, y) (33) holds for all x, y ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Note that this condition is equivalent to the µ-storng convexity of f when h(x) = 1 2 ∥x∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2 Lyapunov arguments for convergence analyses A popular method for proving the convergence rates of momentum dynamics and algorithms is constructing an energy function decreasing over time, called the Lyapunov function (Lyapunov, 1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' The particular analyses presented in this section handle discrete-time algorithms and the corresponding continuous-time dynamics using a single Lyapunov function, as in (Krichene et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' To prove the convergence rates of the given algorithm and associated dynamics, we take the following steps: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Define a time-dependent Lyapunov function V : Rn × Rn × [0, ∞) → [0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Show that the continuous-time energy functional E(t) = V (X(t), Z(t), t) is monotoni- cally decreasing along the solution trajectory (X, Z) : [0, ∞) → Rn × Rn of the ODE system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Show that the discrete-time energy functional Ek = V (xk, zk, tk) is monotonically decreasing along the iterates (xk, zk) : {0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='} → Rn × Rn of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' The remainder of this subsection shows how we can apply this strategy to known algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' We assume the standard smooth (strongly) convex setting (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' 12 Unifying NAG for Convex and Strongly Convex Objective Functions NAG-C and NAG-C ODE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' We define a time-dependent Lyapunov function as V (X, Z, t) := 1 2 ∥Z − x∗∥2 + t2 4 (f(X) − f (x∗)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (34) Then, the continuous-time energy functional E(t) = V (X(t), Z(t), t) = 1 2 ∥Z(t) − x∗∥2 + t2 4 (f(X(t)) − f (x∗)) is monotonically decreasing along the solution trajectory of NAG-C ODE (16) (see Su et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Writing E(t) ≤ E(0) explicitly, we obtain an O(1/t2) convergence rate as f(X(t)) − f(x∗) ≤ 4 t2 E(t) ≤ 4 t2 E(0) = 2 t2 ∥x0 − x∗∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' For the iterates of NAG-C (9), the discrete-time energy function Ek = V (xk, zk, tk) = 1 2 ∥zk − x∗∥2 + sk2 4 (f(xk) − f (x∗)) , (35) where tk = k√s, is monotonically decreasing (see Ryu and Yin, 2022, Chapter 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Hence, we obtain an O(1/k2) convergence rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' NAG-SC and NAG-SC ODE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' We define a time-dependent Lyapunov function as V (X, Z, t) := e √µt �µ 2 ∥Z − x∗∥2 + f(X) − f (x∗) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (36) Then we can show that NAG-SC ODE (18) achieves an O(e−√µt) convergence rate by showing that the energy functional E(t) = V (X(t), Z(t), t) = e √µt �µ 2 ∥Z(t) − x∗∥2 + f(X(t)) − f (x∗) � is monotonically decreasing along the solution trajectory of NAG-SC ODE (see Wilson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Similarly, we can show that NAG-SC (8) achieves an O((1 − √µs)k) convergence rate by showing that the energy functional Ek = V (xk, zk, tk) = (1 − √µs)−k �µ 2 ∥zk − x∗∥2 + f(xk) − f (x∗) � , where tk = −k log(1−√µs) √µ , is monotonically decreasing along the iterates of NAG-SC (see d’Aspremont et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=', 2021, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Bregman Lagrangians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' We can show that the first Bregman Lagrangian flow (23) and the second Bregman Lagrangian flow (25) achieve an O(e−β(t)) convergence rate by showing that the energy functional E(t) = V (X(t), Z(t), t) is monotonically decreasing, where the Lyapunov function V is defined as V1st(X, Z, t) := Dh (x∗, Z) + eβ(t) (f(X) − f (x∗)) (37) for the first Bregman Lagrangian flow and V2nd(X, Z, t) := eβ(t) (µDh (x∗, Z) + f(X) − f (x∗)) (38) for the second Bregman Lagrangian flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' See (Wibisono et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Wilson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=', 2021) for the proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' 13 Kim and Yang 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='3 Hyperbolic functions and their higher-order generalization Hyperbolic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' We first review the definitions and properties of hyperbolic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' The sinh, cosh, tanh, coth, sech, and csch functions are defined as sinh x = ex − e−x 2 , sinh x ∼ x as x → 0, sinh x ∼ ex 2 as x → ∞ cosh x = ex + e−x 2 , cosh x ∼ 1 as x → 0, cosh x ∼ ex 2 as x → ∞ tanh x = sinh x cosh x, tanh x ∼ x as x → 0, tanh x ∼ 1 as x → ∞ coth x = cosh x sinh x , coth x ∼ 1 x as x → 0, coth x ∼ 1 as x → ∞ sech x = 1 cosh x, sech x ∼ 1 as x → 0, sech x ∼ 2e−x as x → ∞ csch x = 1 sinh x, csch x ∼ 1 x as x → 0, csch x ∼ 2e−x as x → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (39) Furthermore, the sinhc, tanhc, cothc, and cschc functions are defined as follows (see ten Thije Boonkkamp et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=', 2012): sinhc x := � sinh x x , if x ̸= 0 1, if x = 0 sinhc x ∼ 1 as x → 0, sinhc x ∼ ex 2x as x → ∞ tanhc x := sinhc x cosh x tanhc x ∼ 1 as x → 0, tanhc x ∼ 1 x as x → ∞ cothc x := 1 tanhc x, cothc x ∼ 1 as x → 0, cothc x ∼ x as x → ∞ cschc x := 1 sinhc x, cschc x ∼ 1 as x → 0, cschc x ∼ 2xe−x as x → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (40) The graphs of these functions are shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='0 x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='0 f(x) f(x) = sinh(x) f(x) = cosh(x) f(x) = tanh(x) (a) sinh, cosh, tanh 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='0 x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='0 f(x) f(x) = csch(x) f(x) = sech(x) f(x) = coth(x) (b) coth, sech, csch 0 1 2 3 4 5 6 x 0 1 2 3 4 5 6 f(x) f(x) = sinhc(x) f(x) = tanhc(x) f(x) = cothc(x) f(x) = cschc(x) (c) sinhc, tanhc, cothc, cschc Figure 2: Hyperbolic functions and their variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' 14 Unifying NAG for Convex and Strongly Convex Objective Functions Higher-order hyperbolic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' We now define the higher-order hyperbolic func- tions that will be used to design higher-order accelerated optimization algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' We define the p-th order hyperbolic sine function sinhp : [0, ∞) → R as the solution of the initial value problem sinh′ p(t) = coshp(t) := � 1 + sinhp p(t) �1/p , sinhp(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (41) Furthermore, we define the tanhp, cothp, sechp, and cschp functions as tanhp(t) = sinhp(t) coshp(t), cothp(t) = 1 tanhp(t), sechp(t) = 1 sinhp(t), cschp(t) = 1 coshp(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' We define the sinhcp, tanhcp, cothcp, and cschcp functions as sinhcp x := � sinhp x x , if x ̸= 0 1, if x = 0 tanhcp x := sinhcp x coshp x , cothcp x := 1 tanhcp x, cschcp x := 1 sinhcp x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Note that the higher-order hyperbolic functions recover the usual hyperbolic functions when p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' The following proposition says that the sinhp function grows exponentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Proposition 1 There is a constant Cp > 0 such that sinhp(t) ∼ Cpet as t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' In particular, we have Cp = 1/2 for p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' The proof of Proposition 1 can be found in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Using (41) and Proposition 1, it is straightforward to check the following asymptotic properties: sinhp x ∼ x as x → 0, sinhp x ∼ Cpex as x → ∞ coshp x ∼ 1 as x → 0, coshp x ∼ Cpex as x → ∞ tanhp x ∼ x as x → 0, tanhp x ∼ 1 as x → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='4 Limiting arguments and examples We investigate two additional ways to derive the limiting ODEs of first-order algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' The first approach is to write the algorithm as a two-sequence scheme and then derive the limiting ODE via the second-order Taylor series expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' This argument frequently appears in the literature (see Su et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' The second approach, which is novel, is to express the algorithm using the difference matrix H = (hij) and then derive the differential kernel H(t, τ) corresponding to the matrix (hij).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' We only present the results here and defer the detailed computations to Appendices C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='1 and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='1 Limiting ODEs of two-sequence algorithms We consider the following two-sequence scheme: xk+1 = yk − s∇f (yk) yk+1 = xk+1 + βk (xk+1 − xk) + γk (xk+1 − yk) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (42) 15 Kim and Yang If we have lim s→0 1 − βt/√s √s = b(t) and lim s→0 γt/√s = c(t) for all t > 0 (43) for some smooth functions b, c : (0, ∞) → R, then under the identification X(tk) = xk with tk = k√s, the two-sequence scheme (42) converges to the ODE ¨X(t) + b(t) ˙X(t) + (1 + c(t))∇f(X(t)) = 0 (44) as s → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Recovering the limiting ODE of three-sequence scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' We can write the three- sequence scheeme (3) as the two-sequence scheme (42) with the following parameters (see Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=', 2021, Appendix B): βk = (1 − τk) τk+1 (1 − µδk) τk γk = τk+1 ((1/s − µ)δkτk − 1 + µδk) τk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (45) If the limits (12) with tk = k√s exist, then we have lim s→0 1 − βt/√s √s = τ(t) − ˙τ(t) τ(t) + µδ(t) lim s→0 γt/√s = τ(t)δ(t) − 1 for all t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Therefore, we recover the limiting ODE (14) of the three-sequence scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' In particular, if the algorithmic parameters (τk) and (δk) satisfy the collinearity condition (4), then we have γk = 0 for all k ≥ 0, and thus c(t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Two-sequence form of NAG-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Because NAG-C is the three-sequence scheme (3) with τk = 2 k+1, δk = s(k+1) 2 , and µ = 0, we can rewrite it as the two-sequence scheme (42) with βk = � 1 − 2 k+1 � 2 k+2 2 k+1 = k − 1 k + 2 γk = 2 k+2 · s(k+1) 2 s − 2 k+2 2 k+1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Thus, NAG-C converges to the ODE (44) with b(t) = lim s→0 1 − t/√s−1 t/√s+2 √s = 3 t c(t) = 0, which recovers NAG-C ODE (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' 16 Unifying NAG for Convex and Strongly Convex Objective Functions Two-sequence form of NAG-SC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Because NAG-SC is the three-sequence scheme (3) with τk = √µs 1+√µs and δk = � s µ, it can be written as the two-sequence scheme (42) with βk = � 1 − √µs 1+√µs � √µs 1+√µs � 1 − µ � s µ � √µs 1+√µs = 1 − √µs 1 + √µs γk = √µs 1+√µs � s µ s − � 1 − µ � s µ + µ √µs 1 + √µs � s µ � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Thus NAG-SC converges to the ODE (44) with b(t) = lim s→0 1 − 1−√µs 1+√µs √s = 2√µ c(t) = 0, which recovers NAG-SC ODE (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2 Difference matrices and differential kernels We can formulate most of the practical first-order momentum methods as the following fixed-step first-order scheme (see Drori and Teboulle, 2014): yi+1 = yi − s i � j=0 hij∇f (yj) for i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' , N − 1, (46) where N is the number of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' We can write this scheme equivalently as � ���� y1 − y0 y2 − y1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' yN − yN−1 � ���� = −s � ���� h0,0 0 · · 0 h1,0 h1,1 · · 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' hN−1,0 hN−1,2 · · hN−1,N−1 � ���� � ���� ∇f (y0) ∇f (y1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' ∇f (yN−1) � ���� Here, we call the lower triangular matrix H = (hij) the difference matrix for the algo- rithm (46).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' To derive the limiting ODE of the algorithm (46), we introduce a smooth curve X : [0, T] → Rn with the identifications X(k√s) = yk and T = N√s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' As a continuous-time analog of the difference matrix (hij), we intoduce a continuously differentiable function H (possibly depending on s now) defined on {(t, τ) ∈ R2 : 0 < τ ≤ t < T} with the identification H(ti, τj) = hij, where ti = i√s and τj = j√s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Substituting X(ti) = yi in (46) yields X (ti+1) − X (ti) √s = − (τj+1 − τj) i � j=0 H (ti, τj) ∇f (X (τj)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (47) Then, we can observe that the right-hand side of (47) is a Riemann sum of the function τ �→ −H(ti, τ)∇f(X(τ)) over [0, ti+1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Thus, taking the limit s → 0 yields ˙X(t) = − � t 0 H(t, τ)∇f(X(τ)) dτ, where H(t, τ) = lim s→0 h t √s , τ √s (48) 17 Kim and Yang as the limiting ODE of the fixed-step first-order scheme (46).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Note that the form of this equation clearly reflects the momentum effect because the gradient ∇f(X(τ)) at time τ affects the velocity ˙X(t) at all times t after τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Inspired by the observation that the function H(t, τ) plays a role similar to the kernel function in the integral transform, we call it the differential kernel (or the H-kernel) corresponding to the difference matrix (hij).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' From differential kernels to second-order ODEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Differentiating both sides of (48) and applying the Leibniz integral rule, we obtain ¨X(t) = −H(t, t)∇f(X(t)) − � t 0 ∂H(t, τ) ∂t ∇f(X(τ)) dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (49) If there exists a function b(t) such that ∂H(t, τ) ∂t = −b(t)H(t, τ), then it follows from (48) that the equation (49) is expressed as the following second-order ODE: ¨X(t) + b(t) ˙X + H(t, t)∇f(X(t)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (50) Recovering the limiting ODE of two-sequence scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' We can write the two- sequence scheme (42) as the fixed-step first-order scheme with hij = (βj + γj) i� ν=j+1 βν + δij, where δij is the Kronecker delta funciton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' For i > j,3 we have hi+1,j − hi,j = (βi+1 − 1) hij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Under the identification H(ti, τj) = hij, we have hi+1,j − hi,j = H (ti+1, τj) − H (ti, τj) = ∂H (ti, τj) ∂t √s + o �√s � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Thus, when the limits (43) exist, taking the limit s → 0 yields ∂H (t, τ) ∂t = −b(t)H (t, τ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (51) Also, because hk+1,k = βk+1 +γk and lims→0 βt/√s = 1 by (43), we have H(t, t) = 1+c(t) for all t ∈ (0, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Therefore, the ODE (50) recovers the limiting ODE (44) of the two-sequence scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Moreover, we can explicitly write the differential kernel H as H(t, τ) = (1 + c(τ)) e− � t τ b(s) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (52) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' We exclude the case i = j because the difference matrix hij has singularities at these points due to the Kronecker delta function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' 18 Unifying NAG for Convex and Strongly Convex Objective Functions Difference matrix for NAG-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Because we can write NAG-C as the two-sequence scheme (42) with βk = k−1 k+2 and γk = 0, we can rewrite it as the fixed-step first-order scheme (46) with hij = i� ν=j ν − 1 ν + 2 + δij = (j − 1)j(j + 1) i(i + 1)(i + 2) + δij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' By definition, the differential kernel corresponding to this matrix (hij) is H(t, τ) = lim s→0 � τ √s − 1 � τ √s � τ √s + 1 � t √s � t √s + 1 � � t √s + 2 � = τ 3 t3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (53) This can be also obtained by substituting b(t) = 3/t and c(t) = 0 into (52): H(t, τ) = e− � t τ 3 s ds = e−3(log(t)−log(τ)) = τ 3 t3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Because ∂H(t, τ) ∂t = −3τ 3 t4 = −3 t H(t, τ), the ODE (49) with (53) recovers NAG-C ODE (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Difference matrix for NAG-SC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Because we can write NAG-SC as the two-sequence scheme (42) with βk = 1−√µs 1+√µs and γk = 0, we can rewrite it as the fixed-step first-order scheme (46) with hij = i� ν=j 1 − √µs 1 + √µs + δij = �1 − √µs 1 + √µs �i−j+1 + δij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' By definition, the differential kernel corresponding to this matrix (hij) is H(t, τ) = lim s→0 �1 − √µs 1 + √µs � t √s − τ √s +1 = e2√µτ e2√µt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (54) This can be also obtained by substituting b(t) = 2√µ and c(t) = 0 into (52): H(t, τ) = e− � t τ 2√µ ds = e−2√µ(t−τ) = e2√µτ e2√µt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' It follows from ∂H(t, τ) ∂t = −2√µe2√µ(τ−t) = −2√µH(t, τ) that the ODE (49) with (54) recovers NAG-SC ODE (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' 19 Kim and Yang 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Unified Bregman Lagrangian In this section, we address the inconsistency between the first Bregman Lagrangian (20) and the second Bregman Lagrangian (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' For a continuously differentiable strictly convex function h, we define the unified Bregman Lagrangian as L � X, ˙X, t � = L1st � X, ˙X, t � + L2nd � X, ˙X, t � − � L2nd � X, ˙X, t �� µ=0 = eα+γ �� 1 + µeβ� Dh � X + e−αV, X � − eβf(X) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (55) Then by construction, this Lagrangian recovers the first Bregman Lagrangian (20) when µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Because the Lagrangian (55) is continuous in the strong convexity parameter µ, it is a continuous extension of the first Bregman Lagrangian to the strongly convex case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Proposition 2 Under the ideal scaling condition (21a), the Euler–Lagrange equation (22) for the unified Bregman Lagrangian (55) reduces to the following system of ODEs: ˙X = eα(Z − X) (56a) d dt∇h(Z) = µ ˙βeβ 1 + µeβ (∇h(X) − ∇h(Z)) − eα+β 1 + µeβ ∇f(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (56b) The proof of Proposition 2 can be found in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' To analyze the convergence rate of this dynamics, we define the time-dependent Lyapunov function V : Rn ×Rn ×[0, ∞) → R as V (X, Z, t) = � 1 + µeβ(t)� Dh (x∗, Z) + eβ(t) (f(X) − f (x∗)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (57) Theorem 3 Suppose that the ideal scaling condition (21b) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Let f be a µ-uniformly (possibly with µ = 0) convex function with respect to h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Then, for any solution (X, Z) to the unified Bregman Lagrangian flow (56), the continuous-time energy function E(t) = V (X(t), Z(t), t) = � 1 + µeβ(t)� Dh (x∗, Z(t)) + eβ(t) (f(X(t)) − f (x∗)) is monotonically decreasing on [0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' The proof of Theorem 3 can be found in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Writing E(t) ≤ E(0) explicitly, we obtain an O(e−β(t)) convergence rate for the dynamics (56).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Corollary 4 Suppose that the ideal scaling condition (21b) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Let f be a µ-uniformly (possibly with µ = 0) convex function with respect to h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Then, any solution (X, Z) to the unified Bregman Lagrangian flow (56) satisfies the inequality f(X(t)) − f (x∗) ≤ e−β(t) �� 1 + µeβ(0)� Dh (x∗, Z(0)) + eβ(0) (f (X(0)) − f (x∗)) � for all t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Similarly to the first Bregman Lagrangian flow (23) and the second Bregman Lagrangian flow (25) (see Wibisono et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Wilson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=', 2021), the dynamical system (56) is closed under time-dilation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' 20 Unifying NAG for Convex and Strongly Convex Objective Functions Theorem 5 Let T : I2 → I1 be an increasing continuously differentiable bijective function, where I1 and I2 are intervals in [0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' If (X1, Z1) is a solution to the unified Bregman Lagrangian flow (56) on I1 with parameters α1, β1 : I1 → R, then the reparametrized curves X2(t) = X1(T(t)) and Z2(t) = Z1(T(t)) is a solution to the unified Bregman Lagrangian flow on I2 with the parameters α2, β2 : I2 → R defined by α2(t) = α1(T(t)) + log ˙T(t) β2(t) = β1(T(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' The proof of Theorem 5 can be found in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Recovering the first and second Bregman Lagrangians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' We now discuss how the first Bregman Lagrangian flow (23), the second Bregman Lagrangian flow (25), and the corresponding Lyapunov analyses can be recovered from the proposed unified Bregman Lagrangian flow (56) and the corresponding Lyapunov analysis (Theorem 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' When µ = 0, it is easy to check that the unified Bregman Lagrangian flow and the corresponding Lyapunov function (57) recover the first Bregman Lagrangian flow and the corresponding Lyapunov function (37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' When the limits α(∞) := limt→∞ α(t) and ˙β(∞) := limt→∞ ˙β(t) > 0 exist, the second Bregman Lagrangian flow with α2nd(t) :≡ α(∞) and β2nd(t) := ˙β(∞)t is the asymptotic version of the unified Bregman Lagrangian flow with α(t) and β(t) in the sense that the coefficients of (56) converge to the ones of (25) as t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' In Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='4, we show that the Lyapunov analysis for the second Bregman Lagrangian flow with ˜α and ˜β can be recovered from Theorem 3 by taking the limit t → ∞ of some inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Unified Methods for Minimizing Convex and Strongly Convex Functions In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='1, we address the inconsistency between NAG-C ODE (17) and NAG-SC ODE (19) by proposing an ODE model that unifies NAG-C ODE and NAG-SC ODE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2, we address the inconsistency between NAG-C (9) and NAG-SC (8) by proposing novel algorithms that can be viewed as a discrete-time counterpart of the unified NAG ODE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Throughout this section, we assume the standard smooth strongly convex setting in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='1 Proposed dynamics: Unified NAG ODE We consider the unified Bregman Lagrangian flow (56) with α(t) = log( 2 t cothc( √µ 2 t)), β(t) = log( t2 4 sinhc2( √µ 2 t)),4 h(x) = 1 2 ∥x∥2, and the initial conditions X(0) = Z(0) = x0, which we call the unified NAG system: ˙X = 2 t cothc �√µ 2 t � (Z − X) ˙Z = t 2 tanhc �√µ 2 t � (µX − µZ − ∇f(X)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (58) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' We can constructively choose these functions (see Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Note that when µ = 0, we have α(t) = log 2 t and β(t) = log t2 4 , which recover NAG-C ODE (17) from the first Bregman Lagrangian flow (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Also, as t → ∞, we have α(t) ∼ log √µ and β(t) ∼ √µt − log (4µ), which recover NAG-SC ODE (19) from the second Bregman Lagrangian flow (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' 21 Kim and Yang Writing this system in a single equation, we obtain the unified NAG ODE (see Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2): ¨X + �√µ 2 tanh �√µ 2 t � + 3 t cothc �√µ 2 t �� ˙X + ∇f(X) = 0 (59) with X(0) = x0 and ˙X(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Existence and uniqueness of the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' To prove the existence and uniqueness of solution to the unified NAG system (58), we cannot directly apply the classical existence and uniqueness theorem because the coefficient 2 t cothc � √µ 2 t � has a singularity at t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Thus, we follow the arguments in (Krichene et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Su et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Theorem 6 The unified NAG system (58) has a unique solution (X, Z) in C1([0, ∞), Rn × Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' The proof of Theorem 6 can be found in Appendix H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' 1 2 3 4 5 t 1 2 3 4 5 6 Coefficient of ˙X NAG-C ODE NAG-SC ODE Unified ODE (a) µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='3 1 2 3 4 5 t 1 2 3 4 5 6 Coefficient of ˙X NAG-C ODE NAG-SC ODE Unified ODE (b) µ = 1 1 2 3 4 5 t 1 2 3 4 5 6 7 Coefficient of ˙X NAG-C ODE NAG-SC ODE Unified ODE (c) µ = 5 Figure 3: Plots for the coefficient of ˙X, which can be interpreted as a measure of friction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Damping system interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' As mentioned in (Su et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=', 2014), the second-order ODE (59) can be viewed as a damping system, and the coefficient of ˙X can be viewed as a measure of friction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Because the coefficient of ˙X in NAG-SC ODE (19) is 2√µ, NAG-SC ODE behaves like an underdamped system when µ is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Thus, the flow generated by NAG-SC ODE may present excessive oscillatory behaviors (see Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' In the unified NAG ODE (59), the coefficient of ˙X is large when t is small and converges to 2√µ as t → ∞ (see Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Thus, the unified NAG ODE behaves like an overdamped system (which displays less severe oscillations) when t is small, regardless of the value of µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Convergence analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' For the unified NAG system, the Lyapunov function (57) can be written as V (X, Z, t) = 1 2 cosh2 �√µ 2 t � ∥Z − x∗∥2 + t2 4 sinhc2 �√µ 2 t � (f(X) − f (x∗)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (60) Furthermore, we can rewrite Theorem 3 and Corollary 4 for this ODE model as follows: 22 Unifying NAG for Convex and Strongly Convex Objective Functions Theorem 7 For the solution (X, Z) to the unified NAG system (58), the continuous-time energy functional E(t) = V (X(t), Z(t), t) = 1 2 cosh2 �√µ 2 t � ∥Z(t) − x∗∥2 + t2 4 sinhc2 �√µ 2 t � (f(X(t)) − f (x∗)) is monotonically decreasing on [0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Corollary 8 The solution (X, Z) to the unified NAG system (58) satisfies the inequality f(X(t)) − f (x∗) ≤ 2 t2 cschc2 �√µ 2 t � ∥x0 − x∗∥2 (61) for all t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Since cschc2 is decreasing on [0, ∞), Corollary 8 implies that the unified NAG ODE (59) achieves an O(∥x0 − x∗∥2/t2) convergence rate regardless of the value of µ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' When µ > 0, since 1 t2 cschc2 � √µ 2 t � ∼ µe−√µt as t → ∞, the unified NAG ODE achieves an O(e−√µt∥x0 −x∗∥2) convergence rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Combining these bounds, we conclude that the unified NAG ODE achieves an O � min � 1/t2, e−√µt� ∥x0 − x∗∥2� convergence rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Advantages of the unified NAG ODE compared to NAG-SC ODE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' We now remark that our novel ODE model resolves the three drawbacks of NAG-SC ODE (19) discussed in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' While the solution to NAG-SC ODE may not converge to the minimizer of f when µ = 0, the solution to the unified NAG ODE always converges to the minimizer regardless of the value of µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' While the convergence guarantee for NAG-SC ODE may be worse than that for NAG-C ODE in early stages, the convergence guarantee (61) for the unified NAG ODE is always better than that for NAG-C ODE because cschc2 is decreasing on [0, ∞) and the rate (61) recovers the exact convergence guarantee of NAG-C ODE when µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' While the convergence rate of NAG-SC ODE involves both the initial squared distance ∥x0 − x∗∥2 and the initial function value accuracy f(x0) − f(x∗), the convergence rate of the unified NAG ODE involves only the initial squared distance ∥x0 − x∗∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Recovering NAG-C ODE and NAG-SC ODE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' We now discuss how NAG-C ODE (17), NAG-SC ODE (19), and their convergence analyses can be recovered from the proposed unified NAG ODE (59).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' When µ = 0, it is easy to check that the unified ODE recovers NAG-C ODE and that the Lyapunov function (60) recovers (34) for NAG-C ODE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' In the unified NAG ODE, because the coefficient of ˙X converges to 2√µ as t → ∞ (see Figure 3), NAG-SC ODE is the asymptotic version of the unified NAG ODE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' In Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='4, we show that the Lyapunov analysis for NAG-SC ODE can be recovered from Theorem 7 by taking the limit t → ∞ of some inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' 23 Kim and Yang 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2 Proposed family of algorithms: Unified NAG family Given the algorithmic stepsize s and a strictly increasing sequence (tk)∞ k=0 (depending on s) in [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' ∞) satisfying lims→0 t0 = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' we consider the three-sequence scheme (3) with the algorithmic paramameters5 τk = 2√s tk+1 cothc � √µ 2 tk+1 � − µs 1 − µs δk = √stk+1 2 tanhc �√µ 2 tk+1 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (62) that is,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' we consider the following unified NAG family: yk = xk + 2√s tk+1 cothc � √µ 2 tk+1 � − µs 1 − µs (zk − xk) xk+1 = yk − s∇f (yk) zk+1 = zk + √stk+1 2 tanhc �√µ 2 tk+1 � (µyk − µzk − ∇f (yk)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (63) Then, it is straightforward to check that the sequences (τk) and (δk) satisfy the collinearity condition (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' The following remark indicates that this algorithm can be regarded as a discretized version of the unified NAG system (58).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Remark 9 When the sequence (tk)∞ k=0 in [0, ∞) satisfies the conditions (10) and (11), we have lim s→0 τk(t) √s = 2 t cothc �√µ 2 t � lim s→0 δk(t) √s = lim s→0 t 2 tanhc �√µ 2 t � = t 2 tanhc �√µ 2 t � for all t > 0, where k is the inverse function of the sequence t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Thus, the result in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='1 implies that the unified NAG family (63) converges to the unified NAG system (58) as s → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' When µ > 0 and limk→∞ tk = ∞, we have limk→∞ τk = √q 1+√q and limk→∞ δk = � s µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Thus, NAG-SC (8) is the asymptotic version of the unified NAG family (63) in the sense that the coefficients of the unified NAG family converge to the coefficients of NAG-SC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' To obtain the convergence rate of the unified NAG family, we introduce the following assumptions on the sequence (tk):6 2√s tk cothc �√µ 2 tk � ≤ 1 for k ≥ 2 (64) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' We can constructively choose these sequences: First, we observe the relationship δk = √sδ(tk+1), where tk = k√s, between the algorithmic parameter δk = s(k+1) 2 of NAG-C and the coefficient δ(t) = t 2 of NAG-C system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Inspired by this relationship, for our algorithm, we define the sequence δk as δk = √sδ(tk+1), where δ(t) = t 2 tanhc � √µ 2 t � , and then set the sequence τk so that the collinearity condition (4) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' These assumption is purely inspired from the proof of Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Note that the assumptions (10) and (11) are not required for the convergence analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Note that when µ = 0, under the identification θk = 2√s tk , the unified NAG family is equivalent to (Tseng, 2008, Algorithm 1) and the condition (65) is equivalent to (Tseng, 2008, Equation 15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' 24 Unifying NAG for Convex and Strongly Convex Objective Functions and � 1 − 2√s tk+1 cothc �√µ 2 tk+1 �� t2 k+1 4 sinhc2 �√µ 2 tk+1 � ≤ t2 k 4 sinhc2 �√µ 2 tk � for k ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (65) The following results are the discrete-time analogs of Theorem 7 and Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Theorem 10 For the iterates of the unified NAG family (63) with (tk)∞ k=0 satisfying the conditions (64) and (65), the following discrete-time energy function is monotonically decreasing: Ek = V (xk, zk, tk) = 1 2 cosh2 �√µ 2 tk � ∥zk − x∗∥2 + t2 k 4 sinhc2 �√µ 2 tk � (f (xk) − f (x∗)) , (66) where the Lyapunov function V is defined in (60).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' The proof of Theorem 10 can be found in Appendix F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Writing Ek ≤ E0 explicitly, we obtain the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Corollary 11 For the iterates of the unified NAG family (63) with (tk)∞ k=0 satisfying the conditions (64) and (65), the following inequality holds for all k ≥ 0: f (xk) − f (x∗) ≤ 4 t2 k cschc2 �√µ 2 tk � × �1 2 cosh2 �√µ 2 t0 � ∥x0 − x∗∥2 + t2 0 4 sinhc2 �√µ 2 t0 � (f (x0) − f (x∗)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (67) In the following subsections, we propose two concrete algorithms with specific choices of the sequence (tk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='1, we propose the unified NAG, a simple unified algorithm which continuously extend NAG-C (9) to the strongly convex setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2, we constructively recover the original NAG (5) and its convergence rate from the unified NAG family (63).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='1 Constant timestep scheme: Unified NAG First, we set the constant timestep δ as δ = � � � − log(1−√µs) √µ , if µ > 0 √s, if µ = 0, (68) and then define the sequence tk = kδ, that is, tk = � − log(1−√µs) √µ k, if µ > 0 √sk, if µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (69) Note that this choice is same as the previous choices of tk for NAG-C and NAG-SC in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' For this specific sequence (tk), the unified NAG family (63) can be written 25 Kim and Yang simply as yk = xk + 2 ι(k+1) cothc � k+1 2 ι√µs � − µs 1 − µs (zk − xk) xk+1 = yk − s∇f (yk) zk+1 = zk + ιs(k + 1) 2 tanhc �k + 1 2 ι√µs � (µyk − µzk − ∇f (yk)) , (70) where ι = − log(1−√µs) √µs for µ > 0 and ι = 1 for µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' We refer to this algorithm as the unified NAG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' The sequence (tk) in (69) can be shown to satisfy the conditions (64) and (65) (see Section F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2), and thus the convergence guarantee (67) holds for this specific algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Also it is straightforward to check that the conditions (10) and (11) hold, and thus the unified NAG (70) converges to the unified NAG system (58) as s → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Because cschc2 is decreasing on [0, ∞) and δ ≥ √s, we have 4 t2 k cschc2 �√µ 2 tk � ≤ 4 t2 k = 4 δ2k2 ≤ 4 sk2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' This implies that the convergence guarantee of the unified NAG is always better than that of NAG-C and that the unified NAG achieves an O(∥x0 − x∗∥2/k2) convergence rate, regardless of the value of µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' When µ > 0, since 4 t2 k cschc2 �√µ 2 tk � ∼ 4µe−√µtk = 4µ (1 − √µs)k as k → ∞, the unified NAG achieves an O((1 − √µs)k∥x0 − x∗∥2) convergence rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Combining these two guarantees, we conclude that the unified NAG achieves an O � min � 1/k2, (1 − √µs)k� ∥x0 − x∗∥2� convergence rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Advantages of the unified NAG compared to NAG-SC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' We now highlight that the unified NAG resolves the three drawbacks of NAG-SC (9) discussed in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' While NAG-SC cannot handle the non-strongly convex case, the unified NAG can handle the case µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Moreover, when µ = 0, the unified NAG and its convergence rate (67) recover NAG-C and its convergence rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' While the convergence guarantee for NAG-SC may be worse than that for NAG-C in early stages, the convergence guarantee for the unified NAG is always better than that for NAG-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' While the convergence rate of NAG-SC involves both the initial squared distance ∥x0 − x∗∥2 and the initial function value accuracy f(x0) − f(x∗), the convergence rate of the unified NAG involves only the initial squared distance ∥x0 − x∗∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' 26 Unifying NAG for Convex and Strongly Convex Objective Functions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2 Adaptive timestep scheme: Recovering the original NAG The constant timestep scheme (unified NAG) in the previous section can be improved in terms of the convergence rate by defining the sequence (tk)∞ k=0 more aggressively as tk+1 = � Given constant t0 > 0 (possibly depending on s), k + 1 = 0 The largest real number satisfying (65), k + 1 ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (71) Then, it is easy to check that the sequence (tk)∞ k=0 is well-defined and strictly increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' We refer to the unified NAG family (63) with this sequence as the adaptive timestep scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Note that the conditions (64) and (65) hold by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='7 Therefore, the convergence guarantee (67) holds for the adaptive timestep scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' In Section F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='3, we show that if t0 → 0 as s → 0, then the conditions (10) and (11) hold, and thus the adaptive timestep scheme converges to the unified NAG system (58) as s → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' By construction, we have tk+1 − tk > δ, where δ is defined in (68), which implies that tk > t0 + kδ for all k ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Thus, the adaptive timestep scheme has a (slightly) better convergence rate than the unified NAG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Surprisingly, our new algorithm, which is purely obtained from the unified Lagrangian framework, is equivalent to the original Nesterov’s method (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Proposition 12 The adaptive timestep scheme is equivalent to the original NAG (5) with γ0 = 4 t2 0 cothc2 � √µ 2 t0 � > µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Moreover, the sequence γk and αk in the original NAG can be written as γk = 4 t2 k cothc2 � √µ 2 tk � and αk = 2√s tk+1 cothc � √µ 2 tk+1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Conversely, the original NAG (5) with γ0 > µ is equivalent to the adaptive timestep scheme, where t0 satisfies γ0 = 4 t2 0 cothc2 � √µ 2 t0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' The proof of Proposition 12 can be found in Section F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' The following remark shows that under the identification in Theorem 12, the convergence rate (67) of the adaptive timestep scheme is equivalent to the convergence rate (7) of the original NAG obtained by Nesterov (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' The first condition follows from the facts that (64) holds for the sequence tk = kδ (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='1) and we have tk > 2δ for k ≥ 2 for the sequence (71).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' 27 Kim and Yang Remark 13 By Corollary 11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' the iterates of the adaptive timestep scheme satisfy f (xk) − f (x∗) ≤ 4 t2 k cschc2 �√µ 2 tk � × �1 2 cosh2 �√µ 2 t0 � ∥x0 − x∗∥2 + t2 0 4 sinhc2 �√µ 2 t0 � (f (x0) − f (x∗)) � = 4 t2 k cschc2 �√µ 2 tk � t2 0 4 sinhc2 �√µ 2 t0 � × � 2 t2 0 cothc2 �√µ 2 t0 � ∥x0 − x∗∥2 + (f (x0) − f (x∗)) � = k−1 � i=0 � 1 − 2√s ti+1 cothc �√µ 2 ti+1 �� � 2 t2 0 cothc2 �√µ 2 t0 � ∥x0 − x∗∥2 + (f (x0) − f (x∗)) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (72) where the last equality follows from our updating rule (71) of the sequence (tk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Therefore, we recover the convergence rate (7) of the original NAG with γk = 4 t2 k cothc2 � √µ 2 tk � and αk = 2√s tk+1 cothc � √µ 2 tk+1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Extension to Higher-order Non-Euclidean Setting Based on the first Bregman Lagrangian (20) and the prior work (Baes, 2009), Wibisono et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (2016) proposed the accelerated tensor flow and accelerated tensor method for convex objective functions to achieve a polynomial O(1/tp) or O(1/kp) convergence rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' They also tried to design accelerated tensor methods for uniformly convex objective functions to achieve an exponential convergence rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' They were able to obtain an exponential convergence rate for continuous-time flows obtained from the first Bregman Lagrangian, but a rate-matching discretization was not identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Instead, they showed that the accelerated tensor method (convex case) with a restart scheme achieves an exponential convergence rate for uniformly convex objective functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' However, as they admitted, understanding the connection between the discrete-time algorithm and the continuous-time flow is unclear and remains as an open problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' In this section, using the unified Bregman Lagrangian (55), we continuously extend to the accelerated tensor flow and the accelerated tensor method in (Wibisono et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=', 2016) to the strongly convex case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Our novel dynamics and algorithm achieve exponential convergence rates without using a restarting technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' We make the following assumptions throughout this section: The distance-generating function h is 1-uniformly convex (29) of order p ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' The objective function f is µ-uniformly (possibly with µ = 0) convex (33) with respect to the distance-generating function h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' The objective function f is (p−1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' s smooth of order p−1 (30), where s is the algorithmic stepsize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' 28 Unifying NAG for Convex and Strongly Convex Objective Functions These assumptions are standard in the literature of higher-order optimization (see Nesterov, 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Baes, 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Wibisono et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Gasnikov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Wilson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' In particular, when p = 2 and h(x) = ∥x∥2, these assumptions recover the standard smooth strongly convex setting in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Following (Wibisono et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=', 2016), we define the tensor update operator Gp,s,N : Rn → Rn as Gp,s,N(y) = arg min x � fp−1(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' y) + N ps ∥x − y∥p � , (73) where the function x �→ fp−1(x, y) = �p−1 i=0 1 i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='∇if(y)(x − y)i is the (p − 1)-st order Taylor approximation of the objective function f at y ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Wibisono et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (2016, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2) showed that one can choose N > 0 so that there exists a constant M > 0 for which the inequality ⟨∇f (x) , y − x⟩ ≥ Ms 1 p−1 ∥∇f (x)∥ p p−1 (74) holds for x = Gp,s,N(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' From now on, we denote the tensor update operator satisfying the inequality (74) by Gp,M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' As a special case, when p = 2, the operator (73) with N = 1 satisfies the inequality (74) with M = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='1 Proposed dynamics: Unified accelerated tensor flow We consider the unified Bregman Lagrangian flow (56) with the parameters α(t) = log p − log t + log � cothcp � C1/pµ1/pt �� β(t) = p log t + log C + p log � sinhcp � C1/pµ1/pt �� (75) and the initial conditions X(0) = Z(0) = x0, where C > 0 is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' It is straightforward to check that the ideal scaling condition (21b) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' This dynamical system can be written as ˙X = p t cothcp � C1/pµ1/pt � (Z − X) d dt∇h(Z) = Cptp−1 tanhcp−1 p � C1/pµ1/pt � (µ∇h(X) − µ∇h(Z) − ∇f(X)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (76) From now on, we refer to this system of ODEs as the unified accelerated tensor flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Using the existence and uniqueness of solution to the unified NAG system (Theorem 6) and the time-dilation property (Theorem 5), we can prove the following theorem (see Appendix H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Theorem 14 The unified accelerated tensor flow (58) has a unique solution (X, Z) in C1([0, ∞), Rn × Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' For this dynamical system, the Lyapunov function (57) can be expressed as V (X, Z, t) = coshp p � C1/pµ1/pt � Dh (x∗, Z) + Ctp sinhcp p � C1/pµ1/pt � (f(X) − f (x∗)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (77) We can rewrite Theorem 3 and Corollary 4 for the unified accelerated tensor flow (76) as follows: 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' See, for example, the proof of Lemma 6 in the arXiv version of (Wilson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=', 2021): arXiv:1611.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='02635v4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' 29 Kim and Yang Theorem 15 For the solution (X, Z) to the unified accelerated tensor flow (76), the continuous-time energy function E(t) = V (X(t), Z(t), t) = coshp p � C1/pµ1/pt � Dh (x∗, Z(t)) + Ctp sinhcp p � C1/pµ1/pt � (f(X(t)) − f (x∗)) is monotonically decreasing on [0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Corollary 16 The solution (X, Z) to the unified accelerated tensor flow (76) satisfies the inequality f(X(t)) − f (x∗) ≤ 1 Ctp sinhcp p � C1/pµ1/pt �Dh (x∗, x0) (78) for all t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Since sinhcp(0) = 1 and sinhcp is increasing on [0, ∞) (see Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2), Corollary 16 implies that the unified accelerated tensor flow (76) achieves an O (Dh (x∗, x0) /tp) conver- gence rate regardless of the value of µ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' On the other hand, when µ > 0, it follows from Proposition 1 that 1 Ctp sinhcp p � C1/pµ1/pt � = O � e−pC1/pµ1/pt� as t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Therefore, the unified accelerated tensor flow achieves an O(e−pC1/pµ1/ptDh (x∗, x0)) conver- gence rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Combining these bounds, we conclude that the unified accelerated tensor flow achieves an O � min � 1/tp, e−pC1/pµ1/pt� Dh (x∗, x0) � convergence rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2 Proposed algorithm: Unified accelerated tensor method As a discretization scheme for the unified accelerated tensor flow (76),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' we propose the following unified accelerated tensor method family: Ak = Ctp k sinhcp p � C1/pµ1/ptk � (79a) yk = xk + Ak+1 − Ak Ak+1 (zk − xk) (79b) xk+1 = Gp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='M (yk) (79c) zk+1 = arg min z �Ak+1 − Ak 1 + µAk (⟨∇f (xk+1) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' z⟩ + µDh (z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' xk+1)) + Dh (z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' zk) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (79d) where (tk) is a strictly increasing sequence (depending on the algorithmic stepsize s) in [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' ∞) and Gp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='M is the tensor update operator satisfying (74).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Because the algorithm (79) is continuous in the strong convexity parameter µ, it handles the convex case and the strongly 30 Unifying NAG for Convex and Strongly Convex Objective Functions convex case in a unified way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' By the first-order optimality condition, the step (79d) is equivalent to ∇h (zk+1) − ∇h (zk) = Ak+1 − Ak 1 + µAk (µ∇h (xk+1) − µ∇h (zk+1) − ∇f (xk+1)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (80) Although the scheme (79) cannot be written in the three-sequence form (3), we observe that the step (79b) plays a role of (3a) (updating yk as a convex combination of xk and zk), the step (79d) plays a role similar to (3c) (updating zk by gradient/mirror step), and that the tensor update step (79c) corresponds to the gradient update step (3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Limiting ODE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' In Appendix G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='1, we show that if lim s→0 t0 = 0 (81) and the timesteps are asymptotically equivalent to s1/p as s → 0 in the sense that lim s→0 tk(t)+1 − t s1/p = 1 for all t ∈ (0, ∞) , (82) where k is the inverse of t, then the unified accelerated tensor method family (79) converges to the unified accelerated tensor flow (76) when letting xk = X(tk) and zk = Z(tk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Convergence analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' To prove the convergence rate, we introduce the following as- sumption on the sequence (tk) (note that Ak is uniquely determined by tk and vice versa): (Ak+1 − Ak)p − CppsAp−1 k+1 (1 + µAk) ≤ 0 with C = 1 p � M p − 1 �p−1 , (83) where M is the constant involved in (74).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' The following results are the discrete-time analogs of Theorem 15 and Corollary 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Theorem 17 For the iterates of the unified accelerated tensor method family (79) with (tk) satisfying the condition (83), the discrete-time energy function Ek = (1 + µAk) Dh (x∗, zk) + Ak (f (xk) − f (x∗)) (84) is monotonically decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' The proof of Theorem 17 can be found in Appendix G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Writing Ek ≤ E0 explicitly, we obtain the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Corollary 18 For the iterates of the unified accelerated tensor method family (79) with (tk) satisfying the condition (83), the following inequality holds for all k ≥ 0: f (xk) − f (x∗) ≤ 1 Ak ((1 + µA0) Dh (x∗, x0) + A0 (f (x0) − f (x∗))) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (85) 31 Kim and Yang Specific algorithm: Unified accelerated tensor method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' We now consider the fol- lowing specific choice of sequence (tk): tk+1 = � 0, k + 1 = 0 The largest real number satisfying (83), k + 1 ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (86) Then, the condition (83) clearly holds, and thus the convergence results hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' In addition, we can show that this sequence satisfies the conditions (81) and (82) (see Appendix G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Hence, the algorithm converges to the unified accelerated tensor flow (76) as s → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Furthermore, we can show that the inequalities Ak ≥ O (kp) , Ak ≥ O �� 1 + C1/ppµ1/ps1/p�k� hold (see Appendix G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Therefore, Corollary 18 implies the following convergence rate: f (xk) − f (x∗) ≤ O � min � 1/kp, � 1 + C1/ppµ1/ps1/p�−k�� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='3 Recovering the non-strongly convex case When µ = 0, the system of ODEs (76) recovers the following accelerated tensor flow (convex case) given in (Wibisono et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=', 2016):9 ˙X = p t (Z − X) d dt∇h(Z) = −Cptp−1∇f(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (87) Moreover, the unified accelerated tensor method family (79) becomes the following family: Ak = Ctp k yk = xk + Ak+1 − Ak Ak+1 (zk − xk) xk+1 = Gp,M (yk) zk+1 = arg min z {(Ak+1 − Ak) ⟨∇f (xk+1) , z⟩ + Dh (z, zk)} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (88) This recovers the accelerated tensor method (convex case) in (Wibisono et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=', 2016) if the sequence (tk) is chosen as tk = s1/pk1/p(k + 1)1/p · · · (k + p − 1)1/p, (89) for which the inequality (83) holds with µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' This flow can be obtained by putting α(t) = log p − log t and β(t) = p log t + log C (Equation 75 with µ = 0) in the first Bregman Lagrangian flow (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' 32 Unifying NAG for Convex and Strongly Convex Objective Functions 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Further Exploration: ODE Model for Minimizing Gradient Norms of Strongly Convex Functions So far, we have focused on ODEs and algorithms that achieve a fast convergence rate for the accuracy of objective function values f(X(t)) − f(x∗) or f(xk) − f(x∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Typically, the goal of numerically solving a convex optimization problem is to reduce the deviation from the minimum value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Alternatively, the gradient norm ∥∇f(xk)∥2 can be used as a performance measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' This criterion is often reasonable for both theoretical and practical purposes (see Nesterov, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Diakonikolas and Wang, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Recently, Kim and Fessler (2021) proposed OGM-G, which is a method that achieves the optimal convergence rate (up to a constant factor) for minimizing the gradient norm ∥∇f(xN)∥2 of non-strongly convex functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Recently, this method has attracted some attention: Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (2021) provided a Lyapunov argument for its convergence analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Suh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (2022) derived and analyzed the limiting ODE of OGM-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' However, most studies on OGM-G have focused only on the non-strongly convex case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' In this section, we propose a novel continuous-time dynamical system that reduces the squared gradient norm ∥∇f(X(T))∥2 of strongly convex objective functions f with an O � min � 1/T 2, e−√µT � � f(x0) − f (x∗) + µ 2 ∥x0 − X(T)∥2�� convergence rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Interestingly, the ODE model presented in this section and the unified NAG ODE (59) have an anti-transpose relationship between the corresponding differential kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='1 Motivation: Symmetric relationship between OGM ODE and OGM-G ODE For non-strongly convex objective functions, Suh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (2022) proposed OGM-G ODE, an ODE model whose solution X : [0, T] → Rn reduces the squared gradient norm ∥∇f(X(T))∥2 with an O((f(x0) − f(x∗))/T 2) convergence rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' In this section, we investigate a symmetric relationship between OGM ODE (which we will discuss later) and OGM-G ODE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' This relationship will give us a hint for designing our novel ODE model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Anti-transpose relationship between OGM and OGM-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' We first review a sym- metric relationship between OGM (Kim and Fessler, 2016), an algorithm for reducing the function value accuracy f(xN) − f(x∗), and OGM-G (Kim and Fessler, 2021), an algorithm for reducing the squared gradient norm ∥∇f(xN)∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Given the number N of total iterations, define a sequence (θk)N k=0 as θk = � � � � � � � � � 1 if k = 0 1+ � 4θ2 k−1+1 2 if 1 ≤ k ≤ N − 1 1+ � 8θ2 k−1+1 2 if k = N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (90) Then, OGM is equivalent to the fixed-step first-order scheme (46) with the difference matrix HF, and OGM-G is equivalent to the fixed-step first-order scheme (46) with the difference 33 Kim and Yang matrix HG, where the entries of HF and HG are defined as hF ij = � � � � � � � θi−1 θi+1 hi−1,j if k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' , i − 2, θi−1 θi+1 (hi−1,i−1 − 1) if k = i − 1, 1 + 2θi−1 θi+1 if k = i, hG ij = � � � � � � � θN−i−1−1 θN−i hi,j+1 if k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' , i − 2, θN−i−1−1 θN−i (hi,i − 1) if k = i − 1, 1 + 2θN−i−1−1 θN−i if k = i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (91) Kim and Fessler (2021) observed the following relationship between the difference kernels for OGM and OGM-G: hF ij = hG N−1−j,N−1−i for all i and j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (92) When the condition (92) holds, we say there is an anti-transpose relationship between HF and HG because the matrix HF can be obtained by reflecting HG about its anti-diagonal and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' A (naive) symmetric relationship between OGM ODE and OGM-G ODE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Next, we look at the relationship between the limiting ODEs of OGM and OGM-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' When letting T = N√s and xk = X(tk), OGM converges to the ODE ¨X + 3 t ˙X + 2∇f(X) = 0 (93) with X(0) = x0 and ˙X(0) = 0 (see Appendix I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Because this ODE is equivalent to the first Bregman Lagrangian flow (23) with α(t) = log 2 t and β(t) = log t2 2 , its solution reduces the function value accuracy f(X(T)) − f(x∗) with an O(∥x0 − x∗∥2/T 2) convergence rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Under the same setting, Suh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (2022) showed that OGM-G converges to the ODE ¨X + 3 T − t ˙X + 2∇f(X) = 0 (94) with X(0) = x0 and ˙X(0) = 0, and showed that the solution to this ODE reduces the squared gradient norm ∥∇f(X(T))∥2 with an O(f(x0) − f(x∗)/T 2) convergence rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' We can observe that the coefficients in (94) can be obtained by substituting t with T − t into the coefficient in (93) and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Based on the symmetric relationship between OGM ODE and OGM-G ODE, one might intuitively think that “OGM-G ODE is a time-reversed version of OGM ODE.” This interpretation, however, might be misleading because the solution to OGM ODE and the solution to OGM-G ODE do not have a time-reversed relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' In the following paragraph, using the differential kernel (48), we present a different, conceivably more accurate, symmetrical relationship between the two ODEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' 34 Unifying NAG for Convex and Strongly Convex Objective Functions Anti-transpose relationship between OGM ODE and OGM-G ODE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Substitut- ing bF(t) = 3/t, bG(t) = 3/(T − t), and cF(t) = cG(t) = 1 in (52), the differential kernels HF(t, τ) corresponding to OGM ODE and HG(t, τ) corresponding to OGM-G ODE can be computed as HF(t, τ) = 2τ 3 t3 HG(t, τ) = 2(T − t)3 (T − τ)3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Here, we can observe the following anti-transpose relationship between two differential kernels: HF(t, τ) = HG(T − τ, T − t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (95) Note that this can also be obtained by using the definition of the differential kernel and the anti-transpose relationship (92) between two matrices HF and HG defined in (91).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' To summarize, the relationships between OGM, OGM-G, and their limiting ODEs are illustrated in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' ˙X(t) = − � t 0 HF(t, τ)∇f(X(τ)) dτ (OGM ODE) y = −sHF∇f(y) (OGM) y = −sHG∇f(y) (OGM-G) ˙X(t) = − � t 0 HG(t, τ)∇f(X(τ)) dτ (OGM-G ODE) limiting limiting HF(t, τ) = HG(T − τ, T − t) hF i,j = hG N−1−j,N−1−i Figure 4: Anti-transpose relationships between OGM (reducing the function value accuracy), OGM-G (reducing the gradient norm), and their limiting ODEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' A failed attempt to design an ODE that minimizes the gradient norm of strongly convex functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' A downside of OGM-G ODE (94) is that it exploits only the non- strong convexity of the objective function f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Thus, one might want to design an ODE model that minimizes the gradient norm of strongly convex objective functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Inspired by the symmetric relationship between OGM ODE and OGM-G ODE, one might substitute t with T − t into the coefficients in NAG-SC ODE (19) to yield the following ODE: ¨X + 2√µ ˙X + ∇f(X) = 0, (96) and one might guess that the solution to this ODE reduces the squared gradient norm ∥∇f(X(T))∥2 with an O(e−√µT ) convergence rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' However, one cannot easily modify the argument in (Suh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=', 2022) to prove the convergence rate of the gradient norm for (96) because their argument depends on the property ˙X(T) = 0, which is not true for the solution to (96).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' 35 Kim and Yang 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2 Proposed dynamics: Unified NAG-G ODE In this subsection, we claim that the symmetric counterpart of the unified NAG ODE (59) works well for our purpose, unlike the aforementioned failed attempt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' The property that the unified NAG ODE is a continuous extension of NAG-C ODE allows us to use the argument in (Suh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=', 2022, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Substituting t with T − t into the coefficients in the unified NAG ODE (59), we obtain the following ODE: ¨X + �√µ 2 tanh �√µ 2 (T − t) � + 3 T − t cothc �√µ 2 (T − t) �� ˙X + ∇f(X) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (97) We refer to this ODE with the initial conditions X(0) = x0 and ˙X(0) = 0 as the unified NAG - G ODE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Clearly, this ODE has a unique solution in C1([0, T), Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='10 We can continuously extend this solution to t = T with ˙X(T) = 0 and ¨X(T) = limt→T − ˙X(t) t−T = 1 2∇f(X(T)) (see Appendix I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' To analyze the convergence rate, we use the Lyapunov analysis again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Theorem 19 For the solution X : [0, T] → Rn to the unified NAG-G ODE (97), the continuous-time energy function E(t) = 4 (T − t)2 cschc2 �√µ 2 (T − t) � (f(X(t)) − f(X(T))) − 8 (T − t)4 cschc4 �√µ 2 (T − t) � ∥X(t) − X(T)∥2 + 8 (T − t)4 cschc2 �√µ 2 (T − t) � cothc2 �√µ 2 (T − t) � × ����X(t) + T − t 2 tanhc �√µ 2 (T − t) � ˙X(t) − X(T) ���� 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (98) is monotonically decreasing on [0, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' The proof of Theorem 19 can be found in Appendix I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' By L’Hˆopital’s rule, we have lim t→T − f(X(t)) − f(X(T)) (T − t)2 = lim t→T − 1 2 � ˙X(t) t − T , ∇f(X) � = 1 4 ∥∇f(X(T))∥2 lim t→T − X(t) − X(T) (T − t)2 = lim t→T − ˙X(t) 2(t − T) = 1 4∇f(X(T)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' It follows from cschc(0) = cothc(0) = 1 that lim t→T − E(t) = lim t→T − � �4 · f(X(t)) − f(X(T)) (T − t)2 − 8 ���� X(t) − X(T) (T − t)2 ���� 2 + 8 ����� X(t) − X(T) (T − t)2 − ˙X(t) 2(t − T) ����� 2� � 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Sketch of the proof: For any ϵ ∈ (0, T/2), the existence and uniqueness of solution on [0, T −ϵ] follows from Cauchy-Lipschitz theorem (Teschl, 2012, Theorem 25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Paste these solutions on [0, T) = ∪ϵ∈(0,T/2)[0, T−ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' 36 Unifying NAG for Convex and Strongly Convex Objective Functions = ∥∇f(X(T))∥2 − 1 2 ∥∇f(X(T))∥2 + 0 = 1 2 ∥∇f(X(T))∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Writing limt→T − E(t) ≤ E(0) explicitly, we obtain the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Corollary 20 The solution X to the unified NAG-G ODE (97) satisfies the inequality ∥∇f(X(T))∥2 ≤ 8 T 2 cschc2 �√µ 2 T � � f(x0) − f (X(T)) + µ 2 ∥x0 − X(T)∥2� ≤ 8 T 2 cschc2 �√µ 2 T � � f(x0) − f (x∗) + µ 2 ∥x0 − X(T)∥2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (99) Since cschc2 is decreasing on [0, ∞), Corollary 20 implies that the unified NAG-G ODE (58) reduces the squared gradient norm with an O � 1/T 2� convergence rate regardless of the value of µ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' When µ > 0, since 1 T 2 cschc2 � √µ 2 T � ∼ µe−√µT as T → ∞, the unified NAG-G ODE reduces the squared gradient norm with an O � e−√µT � convergence rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Combining these bounds, we conclude that the unified NAG-G ODE reduces the squared gradient norm with the following convergence rate: ∥∇f(X(T))∥2 ≤ O � min � 1/T 2, e−√µT � � f(x0) − f (x∗) + µ 2 ∥x0 − X(T)∥2�� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Anti-transpose relationship between the unified NAG ODE and the unified NAG-G ODE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' The differential kernels HF(t, τ) corresponding to the unified NAG ODE and HG(t, τ) corresponding to the unified NAG-G ODE can be computed as (see Ap- pendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2) HF(t, τ) = τ 3 sinhc3 � √µ 2 τ � cosh � √µ 2 τ � t3 sinhc3 � √µ 2 t � cosh � √µ 2 t � HG(t, τ) = (T − t)3 sinhc3 � √µ 2 (T − t) � cosh � √µ 2 (T − t) � (T − τ)3 sinhc3 � √µ 2 (T − τ) � cosh � √µ 2 (T − τ) �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Remarkably, there is an anti-transpose relationship (95) between these differential kernels, like the one between the differential kernels corresponding to OGM ODE (which minimizes the function value accuracy, similarly to what the unified NAG ODE does) and OGM-G ODE (which minimizes the gradient norm, similarly to what the unified NAG-G ODE does).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Numerical Experiments In this section, we validate the performance of the unified NAG (70) for a toy problem and the logistic regression problem, and we also compare our method with NAG-C (9) and NAG-SC (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' For each problem, we empirically observed that the unified NAG attains the advantages of both NAG-C and NAG-SC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' 37 Kim and Yang Toy problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' We consider the problem min (x,y)∈R2 f(x, y) = µ 2 x2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='005y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (100) This problem is strongly convex with parameter min {µ, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='01}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' We set the initial point and the algorithmic stepsize as (x0, y0) = (1, 1) and s = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' When µ is large (µ = 10−3), Figure 5(a) shows that NAG-SC outperforms NAG-C and that the unified NAG behaves like NAG-SC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' When µ = 10−4, Figure 5(b) shows that the unified NAG behaves like NAG-C in the early stages and behaves like NAG-SC in the late stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' When µ is small (µ = 10−7), Figure 5(c) shows that NAG-C outperforms NAG-SC at least in the early stages and that the unified NAG behaves like NAG-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' In each case, the performance of the unified NAG is comparable to the better choice between NAG-C and NAG-SC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' The trajectories of the algorithms are shown in Figures 5(d), 5(e), and 5(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' We can see that NAG-SC converges with more severe oscillation compared to NAG-C and the unified NAG, particularly when the strong convexity parameter µ is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' This result matches the damping system interpretation in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='1: NAG-SC behaves like an underdamped system when µ is small, while our unified NAG always behaves like an overdamped system in the early stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' 100 101 102 103 Iterations 10−26 10−22 10−18 10−14 10−10 10−6 10−2 f (xk) − f (x∗) GD NAG-C NAG-SC Unified NAG (a) Errors f − f ∗, µ = 10−3 100 101 102 103 Iterations 10−10 10−8 10−6 10−4 10−2 f (xk) − f (x∗) GD NAG-C NAG-SC Unified NAG (b) Errors f − f ∗, µ = 10−4 100 101 102 103 Iterations 10−7 10−6 10−5 10−4 10−3 f (xk) − f (x∗) GD NAG-C NAG-SC Unified NAG (c) Errors f − f ∗, µ = 10−7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='0 x −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='0 y GD NAG-C NAG-SC Unified NAG (d) Trajectories, µ = 10−3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='0 x −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='6 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='0 y GD NAG-C NAG-SC Unified NAG (e) Trajectories, µ = 10−4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='960 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='965 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='970 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='975 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='980 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='985 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='990 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='995 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='000 x −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='75 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='50 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='00 y GD NAG-C NAG-SC Unified NAG (f) Trajectories, µ = 10−7 Figure 5: Results for the problem with the objective function f(x, y) = µ 2x2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='005y2 and the initial state x0 = (1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' ℓ2-regularized logistic regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' We now consider the ℓ2-regularized logistic regression problem min x∈Rn f(x) = 1 m � m � i=1 � −yiaT i x + log � 1 + eaT i x�� + λ ∥x∥2 � , (101) 38 Unifying NAG for Convex and Strongly Convex Objective Functions where ai ∈ Rn and yi ∈ {0, 1} for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Then, (101) is the problem (26) with the convex functions fi(x) = −yiaT i x+log(1+eaT i x) and the ℓ2-regularization term R(x) = ∥x∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' As mentioned in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2, the function f is µ-strongly convex with µ = 2λ m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' We set s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='01 and choose the sample size and the dimension as m = 100 and n = 20, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Following (Su et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=', 2014), we use a synthetically generated data set: the entries of ai are generated by the Gaussian distribution N(0, 1), and the labels yi ∈ {0, 1} are generated by the logistic model P (yi) = 1 = 1 1+e−aT i x0 , where the entries of x0 are generated by the Gaussian distribution N(0, 1/100).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' The results are shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Again, we can observe that NAG-SC outperforms NAG-C when µ is large and underperforms NAG-C when µ is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' In each case, the performance of unified NAG is on par with the better one among NAG-C and NAG-SC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' 100 101 102 103 104 Iterations 10−12 10−10 10−8 10−6 10−4 10−2 100 f (xk) − f (x∗) GD NAG-C NAG-SC Unified NAG (a) Errors f − f ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' λ = 5 100 101 102 103 104 Iterations 10−13 10−11 10−9 10−7 10−5 10−3 10−1 f (xk) − f (x∗) GD NAG-C NAG-SC Unified NAG (b) Errors f − f ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' λ = 5 · 10−2 100 101 102 103 104 Iterations 10−12 10−10 10−8 10−6 10−4 10−2 100 f (xk) − f (x∗) GD NAG-C NAG-SC Unified NAG (c) Errors f − f ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' λ = 5 · 10−4 Figure 6: Results for the ℓ2-regularized logistic regression problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Conclusions In this paper, we examined and resolved inconsistencies between the momentum algorithms and ODE models for convex and strongly convex cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' To bridge the gap between the two cases, we proposed the unified Bregman Lagrangian (55), the unified NAG ODE (59), and the unified NAG (70).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Because our algorithm, ODE model and Lagrangian are continuous in µ and recover the corresponding counterparts for non-strongly convex cases (see Figure 1), they can be viewed as continuous extensions of the NAG-C, NAG-C ODE, and the first Bregman Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' We theoretically and empirically showed that unlike NAG-SC, the unified NAG has a better convergence rate compared to NAG-C regardless of the values of µ, which is quite significant in practice, as mentioned in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Based on the Lagrangian formalism, we proposed the unified accelerated tensor flow (76) and scheme (79), achieving exponential convergence rates in the higher-order setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Lastly, hinted from the unified NAG ODE, we designed the unified NAG-G ODE (97), a novel dynamical system that minimizes the gradient norm of strongly convex functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Using our novel tool, the differential kernel (48), we discovered an anti-transpose relationship (95) between OGM ODE and OGM-G ODE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Surprisingly, such relationship can also be found between the unified NAG ODE and the unified NAG-G ODE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' 39 Kim and Yang Acknowledgments and Disclosure of Funding We thank Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Ernest K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Ryu at Seoul National University for providing feedback on this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' This work was supported in part by Samsung Electronics, the National Research Foundation of Korea funded by MSIT(2020R1C1C1009766), and the Information and Communications Technology Planning and Evaluation (IITP) grant funded by MSIT(2022- 0-00124, 2022-0-00480).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' 40 Unifying NAG for Convex and Strongly Convex Objective Functions A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Existing Unified Dynamics A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='1 Relationship between the rescaled original NAG flow and the unified Bregman Lagranfian flow First, we show that the rescaled original NAG flow (28) can be expressed as the unified Bregman Lagrangian flow (56).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Given the parameter function a(t) and the constant γ0 of the rescaled original NAG flow, we can write the functions γ(t) and b(t) involved in (27) and (28) as γ(t) = µ + (γ0 − µ) e−t b(t) = µ + (γ0 − µ) e− � t 0 a(s) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' We define the functions α(t) and β(t) as α(t) = log a(t) β(t) = log � 1 γ0 − µ � + � t 0 a(s) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (102) Then, we have ˙βeβ 1 + µeβ = eα+β 1 + µeβ = eα µ + e−β = a(t) µ + (γ0 − µ) e− � t 0 a(s) ds = a(t) b(t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Thus, the rescaled original NAG flow is equivalent to the unified Bregman Lagrangian flow with the parameter functions (102) and the Euclidean distance-generating function h(x) = 1 2∥x∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Conversely, we show that if the ideal scaling conditon (21b) holds with equality and the distance-generating function h is Euclidean, then the unified Bregman Lagrangian flow can be written as the rescaled original NAG flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Given the parameter functions α(t) and β(t) of the unified Bregman Lagrangian flow, we define the function a(t) and the constant γ0 as a(t) = eα(t) γ0 = µ + e−β(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Then, because b(t) = µ + (γ0 − µ) e− � t 0 a(s) ds = µ + e−β(t), we can write the rescaled original NAG flow as ˙X(t) = eα(t)(Z(t) − X(t)) � µ + e−β(t)� ˙Z(t) = eα(t)(µX(t) − µZ(t) − ∇f(X(t))), which is equivalent to the unified Bregman Lagrangian flow if the ideal scaling conditon (21b) holds with equality and h(x) = 1 2∥x∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' 41 Kim and Yang A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2 Relationship between the rescaled original NAG flow with specific parameters and the unified NAG system In particular, given γ > 0, one can choose the function a(t) in the rescaled original NAG flow as (see Luo and Chen, 2021, Equation 70) a(t) = � � � � � 2√γ0 √γ0t+2, if µ = 0, √µ · e √µt− √µ−√γ0 √µ+√γ0 e √µt+ √µ−√γ0 √µ+√γ0 , if µ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (103) In this case, we have b(t) = (a(t))2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Thus, the rescaled original flow with these functions can be written as ˙X(t) = 2√γ0 √γ0t + 2(Z(t) − X(t)) ˙Z(t) = − √γ0t + 2 2√γ0 − ∇f(X(t)) when µ = 0, and ˙X(t) = √µ · e √µt − √µ−√γ0 √µ+√γ0 e √µt + √µ−√γ0 √µ+√γ0 (Z(t) − X(t)) ˙Z(t) = 1 √µ · e √µt + √µ−√γ0 √µ+√γ0 e √µt − √µ−√γ0 √µ+√γ0 (µX(t) − µZ(t) − ∇f(X(t))) when µ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' In the non-strongly convex case, it is easy to observe that this ODE system converges to NAG-C system (16) as γ0 → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' In the strongly convex case, because √µ−√γ0 √µ+√γ0 → −1 as γ0 → ∞ and e √µt+1 e √µt−1 = coth( √µ 2 t), the ODE system converges to the unified NAG system (58) as γ0 → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Higher-Order Hyperbolic Functions B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='1 Proof of Proposition 1 Fix T > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' We will show that log (sinhp(T + t)) − t (104) converges to some constant as t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' We can bound the derivative of (104) as d dt {log (sinhp(T + t)) − t} = sinh′ p(T + t) sinhp(T + t) − 1 = coshp(T + t) sinhp(T + t) − 1 = � 1 + 1 sinhp p(T + t) �1/p − 1 42 Unifying NAG for Convex and Strongly Convex Objective Functions ∈ � 0, 1 sinhp(T + t) � , where the last line follows from the fact that 1 ≤ (1 + x)1/p ≤ 1 + x1/p holds for x ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='11 Thus, if the integral � ∞ 0 1 sinhp(T + t) dt (105) is finite, then (104) converges to some constant because it is monotonically increasing and bounded above, and thus this completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' To show that the integral (105) is finite, it is enough to show that the inequality sinhp(T + t) ≥ sinhp(T)et holds for all t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' This can be shown by the following calculation: log (sinhp(T + t)) = log (sinhp(T)) + � t 0 d ds {log (sinhp(T + s))} ds = log (sinhp(T)) + � t 0 sinh′ p(T + s) sinhp(T + s) ds = log (sinhp(T)) + � t 0 � 1 + sinhp p(T + s) �1/p sinhp(T + s) ds ≥ log (sinhp(T)) + � t 0 1 ds = log (sinhp(T)) + t = log � sinhp(T)et� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2 The function sinhcp is non-decreasing It is easy to see that sinhp and coshp are increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Since tanh′ p(t) = d dt � sinhp(t) coshp(t) � = sinh′ p(t) coshp(t) − cosh′ p(t) sinhp(t) cosh2 p(t) ≤ sinh′ p(t) coshp(t) cosh2 p(t) = 1, we have tanhp(t) ≤ t for all t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Now, we deduce that sinhc′ p(t) = d dt �sinhp(t) t � 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' To check this basic inequality, one can consider the p-th power of each side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' 43 Kim and Yang = t sinh′ p(t) − sinhp(t) t2 = t coshp(t) − sinhp(t) t2 = coshp(t) t2 (t − tanhp(t)) ≥ 0, and thus sinhc is non-decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Limiting Arguments C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='1 Limiting argument for two-sequence scheme Limiting ODE of two-sequence scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' For the iterates of the two-sequence scheme (42), we have xk+1 − xk √s = 1 √s (yk − s∇f (yk) − xk) = 1 √s (βk−1 (xk − xk−1) + γk (xk − yk−1) − s∇f (yk)) = 1 √s (βk−1 (xk − xk−1) − sγk∇f (yk−1) − s∇f (yk)) = βk−1 xk − xk−1 √s − √sγk∇f (yk−1) − √s∇f (yk) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Using the Taylor expansions xk+1 − xk √s = ˙X(tk) + 1 2 ¨X(tk)√s + o �√s � xk − xk−1 √s = ˙X(tk) − 1 2 ¨X(tk)√s + o �√s � , we obtain ˙X(tk)+1 2 ¨X(tk)√s+o �√s � = βk−1 � ˙X(tk) − 1 2 ¨X(tk)√s + o �√s �� −√sγk∇f (yk−1)−√s∇f (yk) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' It follows from ∥xk − yk−1∥ = o(√s) and the Lipschitz continuity of ∇f that √s∇f (yk−1) = √s∇f(X(tk)) + o �√s � √s∇f (yk) = √s∇f (yk−1) + o �√s � = √s∇f(X(tk)) + o �√s � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Substituting these into the ODE yields 1 + βk−1 2 ¨X(tk)√s + (1 − βk−1) ˙X(tk) + (1 + γk) ∇f(X(tk))√s + o �√s � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Dividing both sides by √s, substituting k = t/√s and the limits (43), and then letting s → 0, we obtain (note that βt/√s−1 → 1 by Equation (43)) ¨X(t) + b(t) ˙X(t) + (1 + c(t))∇f(X(t)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' 44 Unifying NAG for Convex and Strongly Convex Objective Functions Recovering the limiting ODE of three-sequence scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' It follows from the Taylor expansion that τk = τ (tk) √s τk+1 = τ (tk) √s + ˙τ (tk) s + √so �√s � δk = δ (tk) √s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Thus, for the sequences (βk) and (γk) in (45), we have 1 − βk √s = 1 √s � 1 − (1 − τk) (1 − µδk) τk+1 τk � = 1 √s � 1 − � 1 − √sτ (tk) � � 1 − µ√sδ (tk) � � 1 + ˙τ (tk) s + √so (√s) τ (tk) √s �� = 1 √s �√sτ (tk) + µ√sδ (tk) − √s ˙τ (tk) τ (tk) + o �√s �� = τ (tk) + µδ (tk) − ˙τ (tk) τ (tk) + o (√s) √s and γk = τk+1 τk ((1/s − µ)δkτk − 1 + µδk) = � 1 + ˙τ (tk) √s + o (√s) τ (tk) � � (1 − µs)δ (tk) τ (tk) − 1 + µ√sδ (tk) � = δ (tk) τ (tk) − 1 + o (1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Therefore, we have lim s→0 1 − βt/√s √s = τ(t) + µδ(t) − ˙τ(t) τ(t) lim s→0 γt/√s = τ(t)δ(t) − 1, which recovers the limiting ODE (14) of the three-sequence scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2 Difference matrix and differential kernel From the two-sequence scheme to the difference matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' The iterates of the two- sequence scheme (42) satisfy yk+1 − yk = xk+1 − yk + βk (xk+1 − xk) − sγk∇f (yk) = βk (yk − yk−1) + sβk∇f (yk−1) − s (1 + βk + γk) ∇f (yk) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Substituting yk+1 − yk = −s k � i=0 hk,i∇f (yi) 45 Kim and Yang yk − yk−1 = −s k−1 � i=0 hk−1,i∇f (yi) into the equality and comparing the coefficients of each ∇f(yi), we obtain hk,j = � � � � � 1 + βk + γk, if j = k βk (hk−1,k−1 − 1) , if j = k − 1 βkhk−1,i, if j ≤ k − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Using mathematical induction, it is straightforward to show that hij = (βj + γj) i� ν=j+1 βν + δij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Differential kernel for the two-sequence scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' By (51), we have ∂ ∂s log(H(s, t)) = ∂H(s, τ) ∂s 1 H(s, τ) = −b(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Integrating over s, we obtain log (H(t, τ)) − log (H(τ, τ)) = − � t τ b(s) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Thus, we have H(t, τ) = H(τ, τ)e− � t τ b(s) ds = (1 + c(τ)) e− � t τ b(s) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Unified Bregman Lagrangian D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='1 Proof of Proposition 2 For the unified Bregman Lagrangian (55), the partial derivatives ∂L ∂ ˙X � X, ˙X, t � and ∂L ∂X � X, ˙X, t � are given by ∂L ∂ ˙X � X, ˙X, t � = eγ � 1 + µeβ� � ∇h � X + e−α ˙X � − ∇h(X) � ∂L ∂X � X, ˙X, t � = eα+γ � 1 + µeβ� � ∇h � X + e−α ˙X � − ∇h(X) � − eγ � 1 + µeβ� d dt∇h(X) − eα+β+γ∇f(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' The time derivative of ∂L ∂ ˙X can be computed as d dt � ∂L ∂ ˙X � X, ˙X, t �� = � ˙γeγ + µ � ˙β + ˙γ � eβ+γ� � ∇h � X + e−α ˙X � − ∇h(X) � + eγ � 1 + µeβ� � d dt∇h � X + e−α ˙X � − d dt∇h(X) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' 46 Unifying NAG for Convex and Strongly Convex Objective Functions Thus, the Euler–Lagrange equation (22) can be written as eγ � 1 + µeβ� d dt∇h � X + e−α ˙X � = � eα+γ � 1 + µeβ� − ˙γeγ − µ � ˙β + ˙γ � eβ+γ� × � ∇h � X + e−α ˙X � − ∇h(X) � −eα+β+γ∇f(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Substituting ˙γ = eα (21a) into the equation and dividing both sides by eγ � 1 + µeβ� > 0, we obtain d dt∇h � X + e−α ˙X � = − µ ˙βeβ 1 + µeβ � ∇h � X + e−α ˙X � − ∇h(X) � − eα+β 1 + µeβ ∇f(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Letting Z = X + e−α ˙X yields the system of ODEs (56).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2 Proof of Theorem 3 Note that d dtDh (x∗, Z) = d dt {h (x∗) − h(Z) − ⟨∇h(Z), x∗ − Z⟩} = − � ∇h(Z), ˙Z � − � d dt∇h(Z), x∗ − Z � + � ∇h(Z), ˙Z � = − � d dt∇h(Z), x∗ − Z � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Using this equation, we have d dt {φ(X(t), Z(t), t)} = − � 1 + µeβ� � d dt∇h(Z), x∗ − Z � + µ ˙βeβDh (x∗, Z) + ˙βeβ (f(X) − f (x∗)) + eβ � ∇f(X), ˙X � = � µ ˙βeβ (∇h(Z) − ∇h(X)) + eα+β∇f(X), x∗ − Z � + µ ˙βeβDh (x∗, Z) + ˙βeβ (f(X) − f (x∗)) + eβ � ∇f(X), ˙X � , where the second equality follows from (56b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' It follows from the Bregman three-point identity (32), the non-negativity of Bregman divergence, and the µ-uniform convexity of f with respect to h (33) that ⟨∇h(Z) − ∇h(X), x∗ − Z⟩ + Dh (x∗, Z) = Dh (x∗, X) − Dh(Z, X) ≤ Dh (x∗, X) ≤ 1 µDf (x∗, X) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Thus, we have d dt {φ(X(t), Z(t), t)} ≤ ˙βeβDf (x∗, X) + eα+β ⟨∇f(X), x∗ − Z⟩ + ˙βeβ (f(X) − f (x∗)) + eβ � ∇f(X), ˙X � 47 Kim and Yang = ˙βeβDf (x∗, X) + eα+β ⟨∇f(X), x∗ − X⟩ + ˙βeβ (f(X) − f (x∗)) = � eα − ˙β � eβ ⟨∇f(X), x∗ − X⟩ ≤ � eα − ˙β � eβ (f (x∗) − f(X)) ≤ 0, where the last two inequalities follows from the ideal scaling condition (21b), the convexity of f, and the fact that x∗ is a minimizer of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='3 Proof of Theorem 5 The derivatives of X2 and ∇h(Z2) can be computed as ˙X2(t) = ˙T(t) ˙X1(T(t)) = ˙T(t)eα1(T(t))(Z1(T(t)) − X1(T(t)) = ˙T(t)eα1(T(t))(Z2(t) − X2(t)) = eα2(t)(Z2(t) − X2(t)) and d dt∇h(Z2(t)) = ˙T(t)d(∇h ◦ Z1) dt (T(t)) = ˙T(t) � µ ˙β1(T(t))eβ1(T(t)) 1 + µeβ1(T(t)) (∇h(X1(T(t)) − ∇h(Z1(T(t)))) − eα1(T(t))+β1(T(t)) 1 + µeβ1(T(t)) ∇f(X1(T(t))) � = µ ˙β2(t)eβ2(t) 1 + µeβ2(t) (∇h(X2(t)) − ∇h(Z2(t))) − eα2(t)+β2(t) 1 + µeβ2(t) ∇f(X2(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Thus, we obtain the desired system of ODEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='4 Recovering Lyapunov analysis for the second Bregman Lagrangian flow In this section, we recover the second Bregman Lagrangian flow (25) with constant coefficients and its Lyapunov analysis from the unified Bregman Lagrangian flow (56) and its Lyapunov analysis (Theorem 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' In particular, we recover NAG-SC ODE (19) and its Lyapunov analysis from the unified NAG ODE (59) and its Lyapunov analysis (Theorem 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' For the parameter functions α, β : [0, ∞) → R of the unified Bregman Lagrangian flow (56), assume that the limits α(∞) := limt→∞ α(t) and ˙β(∞) := limt→∞ ˙β(t) > 0 exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' We consider the following second Bregman Lagrangian flow (25) with α2nd(t) :≡ α(∞) and β2nd(t) := ˙β(∞)t: ˙X = eα(∞)(Z − X) d dt∇h(Z) = ˙β(∞) (∇h(X) − ∇h(Z)) − eα(∞) µ ∇f(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (106) 48 Unifying NAG for Convex and Strongly Convex Objective Functions Then, it follows from limt→∞ eα(t) = eα(∞), limt→∞ µ ˙βeβ 1+µeβ = ˙β(∞), and limt→∞ eα+β 1+µeβ = eα(∞) µ that the coefficients in the unified Bregman Lagrangian flow (56) converge to those in the dynamics (106) as t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Thus, roughly speaking, the dynamics (106) is the asymptotic version of the unified Bregman Lagrangian flow in the sense that [the flow corresponding to (56), starting at time t0] converges to [the flow corresponding to (106), starting at time 0] as t0 → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Note that the time derivative of the Lyapunov function (57) for the unified Bregman Lagrangian flow can be written as d dt {V (X(t), Z(t), t)} = d dt � 1 + µeβ� Dh (x∗, Z) + � 1 + µeβ� d dt {Dh (x∗, Z)} + d dt � eβ� (f(X) − f (x∗)) + eβ d dt {f(X) − f (x∗)} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Thus, we have 0 ≥ e−β(t0+t) d dt {V (X(t0 + t), Z(t0 + t), t0 + t)} = µ ˙β(t0 + t)Dh (x∗, Z(t0 + t)) + 1 + µeβ(t0+t) eβ(t0+t) d dt {Dh (x∗, Z(t0 + t))} + ˙β(t0 + t) (f(X(t0 + t)) − f (x∗)) + d dt {f(X(t0 + t)) − f (x∗)} for all t > 0, where t0 > 0 is the initial time of the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Fix x0 = X(t0) and z0 = Z(t0) in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Note that as t0 → ∞, the flow t �→ (X(t0 + t), Z(t0 + t)) converges to the flow t �→ (X2nd(t), Z2nd(t)) corresponding to (106) with X2nd(0) = x0 and Z2nd(0) = z0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Now, taking the limit t0 → ∞ in the inequality above yields yields 0 ≥ µ ˙β(∞)Dh (x∗, Z(t)) + µ d dt {Dh (x∗, Z(t))} + ˙β(∞) (f(X(t)) − f (x∗)) + d dt {f(X(t)) − f (x∗)} = e−β2nd(t) d dt {V2nd(X(t), Z(t), t)} , where V2nd is the Lyapunov function (38) for the second Bregman Lagrangian flow with the parameters α2nd and β2nd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Because e−β2nd(t) > 0, we recover the Lyapunov analysis for the second Bregman Lagrangian flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Recovering NAG-SC ODE from the unified ODE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Note that the unified Bregman Lagrangian flow (56) and its Lyapunov analysis (Theorem 3) with h(x) = 1 2 ∥x∥2, α(t) = log � 2 t cothc � √µ 2 t �� , and β(t) = log � t2 4 sinhc2 � √µ 2 t �� recover the unified NAG system (58) and its Lyapunov analysis (Theorem 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Also, note that the second Bregman Lagrangian flow (25) and the corresponding Lyapunov function (38) with α2nd(t) = log �√µ � and β2nd(t) = √µt recover NAG-SC system (18) and the corresponding Lyapunov function (36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' When µ > 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' because α(∞) = log �√µ � and ˙β(∞) = √µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' the results above shows that NAG-SC ODE is the asymptotic version of the unified NAG ODE and that the Lyapunov analysis of NAG-SC ODE can be obtained by taking the limit t → ∞ into the coeffiicients of 49 Kim and Yang the inequality (rigorously,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' taking the limit t0 → ∞ of the initial time as in the preceding paragraph) 4 t2 cschc2 �√µ 2 t � d dt {V (X(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Z(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' t)} ≤ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' where V is the Lyapunov function (60) for the unified NAG ODE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Unified NAG ODE E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='1 Choosing α and β We first note some properties of the functions α and β that recover NAG-C ODE (or NAG-SC ODE) from the first Bregman Lagrangian flow (or the second Bregman Lagrangian flow, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' The first Bregman Lagrangian flow (23) with h(x) = 1 2 ∥x∥2 can be written as the following ODE: ¨X + (− ˙α + eα) ˙X + e2α+β∇f(X) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' The choices α(t) = log 2 t and β(t) = log t2 4 , which recover NAG-C ODE, satisfy the ideal scaling condition (21b) with equality and make the coefficient of ∇f(X) equal to the coefficient of ¨X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' The second Bregman Lagrangian flow (25) with h(x) = 1 2 ∥x∥2 can be written as ¨X + � − ˙α + eα + ˙β � ˙X + e2α µ ∇f(X) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' The choices α(t) = log √µ and β(t) = log �√µt � , which recover NAG-SC ODE, satisfy the ideal scaling condition (21b) with equality and make the coefficient of ∇f(X) equal to the coefficient of ¨X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Inspired by these facts, for the unified Bregman Lagrangian, we construct functions α(t) and β(t) so that the ideal scaling condition (21b) holds with equality and that the coefficient of ∇f(X) is equal to the coefficient of ¨X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' The unified Bregman Lagrangian flow (56) with h(x) = 1 2 ∥x∥2 can be written as ¨X + � − ˙α + eα + µ ˙βeβ 1 + µeβ � ˙X + e2α+β 1 + µeβ ∇f(X) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Now, we solve the following system of ODEs: ˙β = eα e2α+β = 1 + µeβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Let A(t) = eβ(t) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Then, we have ˙A = ˙βeβ = eα+β > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Because ( ˙A)2 = e2α+βeβ = A(1 + µA), we have ˙A = � A(1 + µA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Solving this differential equation with the initial condition A(0) = 0 yields A = t2 4 sinhc2( √µ 2 t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Thus, we have β(t) = log( t2 4 sinhc2( √µ 2 t)) and α(t) = log( ˙β(t)) = log( 2 t cothc( √µ 2 t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' 50 Unifying NAG for Convex and Strongly Convex Objective Functions E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2 Equivalent forms of the unified NAG system and the unified NAG-G system When µ = 0, the unified NAG system is equivalent to NAG-C system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Thus, we assume µ > 0 for the sake of simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Second-order ODE form of the unified NAG system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' When µ > 0, we can write the unified NAG system (58) as ˙X = √µ coth �√µ 2 t � (Z − X) ˙Z = 1 √µ tanh �√µ 2 t � (µX − µZ − ∇f(X)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Substituting Z = X + 1 √µ tanh( √µ 2 t) ˙X into ˙Z = 1 √µ tanh( √µ 2 t)(µX − µZ − ∇f(X)), we have 1 √µ tanh �√µ 2 t � ¨X + � 1 + 1 2 sech2 �√µ 2 t �� = 1 √µ tanh �√µ 2 t � (µX − µZ − ∇f(X)) = −√µ tanh �√µ 2 t � (Z − X) − 1 √µ tanh �√µ 2 t � ∇f(X) = − tanh2 �√µ 2 t � ˙X − 1 √µ tanh �√µ 2 t � ∇f(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Multiplying by √µ coth( √µ 2 t) and rearranging the terms, we have ¨X + �√µ tanh �√µ 2 t � + √µ coth �√µ 2 t � + √µ 2 sech �√µ 2 t � csch �√µ 2 t � � ˙X + ∇f(X) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Using the identity tanh(x) − coth(x) + sech(x) csch(x) = 0, we can equivalently write this ODE as ¨X + �√µ 2 tanh �√µ 2 t � + 3√µ 2 coth �√µ 2 t �� ˙X + ∇f(X) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Differential kernel for the unified NAG ODE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Substituting b(t) = √µ 2 tanh( √µ 2 t) + 3√µ 2 coth( √µ 2 t) and c(t) = 0 into (52), we yield the following differential kernel corresponding to the unified NAG ODE: H(t, τ) = e− � t τ � √µ 2 tanh � √µ 2 s � + 3√µ 2 coth � √µ 2 s �� ds = e − � 3 log � sinh � √µ 2 s �� +log � cosh � √µ 2 s ���t τ = sinh3 � √µ 2 τ � cosh � √µ 2 τ � sinh3 � √µ 2 t � cosh � √µ 2 t � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' 51 Kim and Yang Differential kernel for the unified NAG-G ODE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Substituting b(t) = √µ 2 tanh( √µ 2 (T− t)) + 3√µ 2 coth( √µ 2 (T − t)) and c(t) = 0 into (52), we yield the following differential kernel corresponding to the unified NAG-G ODE: H(t, τ) = e− � t τ � √µ 2 tanh � √µ 2 (T−s) � + 3√µ 2 coth � √µ 2 (T−s) �� ds = e � 3 log � sinh � √µ 2 (T−s) �� +log � cosh � √µ 2 (T−s) ���t τ = sinh3 � √µ 2 (T − t) � cosh � √µ 2 (T − t) � sinh3 � √µ 2 (T − τ) � cosh � √µ 2 (T − τ) �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Unified NAG Family F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='1 Proof of Theorem 10 Note that when µ > 0, the inequality (65) can be written as 0 ≥ � 1 − √µs coth �√µ 2 tk+1 �� 1 µ sinh2 �√µ 2 tk+1 � − 1 µ sinh2 �√µ 2 tk � = 1 µ sinh2 �√µ 2 tk+1 � − � s µ sinh �√µ 2 tk+1 � cosh �√µ 2 tk+1 � − 1 µ sinh2 �√µ 2 tk � = 1 µ cosh2 �√µ 2 tk+1 � − � s µ sinh �√µ 2 tk+1 � cosh �√µ 2 tk+1 � − 1 µ cosh2 �√µ 2 tk � = � 1 − √µs tanh �√µ 2 tk+1 �� 1 µ cosh2 �√µ 2 tk+1 � − 1 µ cosh2 �√µ 2 tk � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Thus, the following inequality holds for all µ ≥ 0 (it clearly holds for µ = 0): � 1 − µ√stk+1 2 tanhc �√µ 2 tk+1 �� cosh2 �√µ 2 tk+1 � ≤ cosh2 �√µ 2 tk � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (107) Using (65) and (107),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' we have ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='Ek+1 − Ek ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='= 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2 cosh2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='�√µ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2 tk+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='∥zk+1 − x∗∥2 − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2 cosh2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='�√µ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2 tk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='∥zk − x∗∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='+ t2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='k+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='sinhc2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='�√µ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2 tk+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='(f (xk+1) − f (x∗)) − t2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='4 sinhc2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='�√µ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2 tk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='(f (xk) − f (x∗)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='≤ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2 cosh2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='�√µ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2 tk+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='∥zk+1 − x∗∥2 − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='1 − µ√stk+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='tanhc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='�√µ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2 tk+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='cosh2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='�√µ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2 tk+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='∥zk − x∗∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='+ t2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='k+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='sinhc2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='�√µ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2 tk+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='(f (xk+1) − f (x∗)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='1 − 2√s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='tk+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='cothc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='�√µ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2 tk+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='�� t2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='k+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='sinhc2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='�√µ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2 tk+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='(f (xk) − f (x∗)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' 52 Unifying NAG for Convex and Strongly Convex Objective Functions Substituting zk+1 = yk+ � 1 − µ√stk+1 2 tanhc �√µ 2 tk+1 �� (zk − yk)− √stk+1 2 tanhc �√µ 2 tk+1 � ∇f (yk) into the inequality above,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' we have ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='Ek+1 − Ek ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='≤ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2 cosh2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='�√µ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2 tk+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='����� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='1 − µ√stk+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='tanhc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='�√µ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2 tk+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='(zk − yk) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='√stk+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='tanhc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='�√µ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2 tk+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='∇f (yk) − (x∗ − yk) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='����� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='1 − µ√stk+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='tanhc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='�√µ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2 tk+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='cosh2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='�√µ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2 tk+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='∥(zk − yk) − (x∗ − yk)∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='+ t2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='k+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='sinhc2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='�√µ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2 tk+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='(f (xk+1) − f (x∗)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='1 − 2√s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='tk+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='cothc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='�√µ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2 tk+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='�� t2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='k+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='sinhc2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='�√µ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2 tk+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='(f (xk) − f (x∗)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='= 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2 cosh2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='�√µ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2 tk+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='� � � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='1 − µ√stk+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='tanhc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='�√µ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2 tk+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='��2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='1 − µ√stk+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='tanhc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='�√µ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2 tk+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='�� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='∥zk − yk∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='√stk+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='sinhc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='�√µ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2 tk+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='cosh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='�√µ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2 tk+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='⟨∇f (yk) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' x∗ − yk⟩ + µ√stk+1 4 sinhc �√µ 2 tk+1 � cosh �√µ 2 tk+1 � ∥x∗ − yk∥2 − √stk+1 2 sinhc �√µ 2 tk+1 � cosh �√µ 2 tk+1 � × � 1 − µ√stk+1 2 tanhc �√µ 2 tk+1 �� ⟨∇f(yk),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' zk − yk⟩ + st2 k+1 8 sinhc2 �√µ 2 tk+1 � ∥∇f (yk)∥2 + t2 k+1 4 sinhc2 �√µ 2 tk+1 � (f (xk+1) − f (x∗)) − � 1 − 2√s tk+1 cothc �√µ 2 tk+1 �� t2 k+1 4 sinhc2 �√µ 2 tk+1 � (f (xk) − f (x∗)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Since 0 ≤ 1 − √µs ≤ 1 − µ√stk+1 2 tanhc �√µ 2 tk+1 � ≤ 1, we have 53 Kim and Yang 1 2 cosh2 �√µ 2 tk+1 � � � 1 − µ√stk+1 2 tanhc �√µ 2 tk+1 ��2 − � 1 − µ√stk+1 2 tanhc �√µ 2 tk+1 �� � ∥zk − yk∥2 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' we deduce that Ek+1 − Ek ≤ √stk+1 2 sinhc �√µ 2 tk+1 � cosh �√µ 2 tk+1 � ⟨∇f (yk) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' x∗ − yk⟩ + µ√stk+1 4 sinhc �√µ 2 tk+1 � cosh �√µ 2 tk+1 � ∥x∗ − yk∥2 − √stk+1 2 sinhc �√µ 2 tk+1 � cosh �√µ 2 tk+1 � × � 1 − µ√stk+1 2 tanhc �√µ 2 tk+1 �� ⟨∇f(yk),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' zk − yk⟩ + st2 k+1 8 sinhc2 �√µ 2 tk+1 � ∥∇f (yk)∥2 + t2 k+1 4 sinhc2 �√µ 2 tk+1 � (f (xk+1) − f (x∗)) − � 1 − 2√s tk+1 cothc �√µ 2 tk+1 �� t2 k+1 4 sinhc2 �√µ 2 tk+1 � (f (xk) − f (x∗)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Now, it suffices to show that the right-hand side (RHS) of the inequality above is non-positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' By the µ-strong convexity of f, we have 0 ≥ f (yk) − f (x∗) + ⟨∇f (yk) , x∗ − yk⟩ + µ 2 ∥x∗ − yk∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Moreover, it follows from the convexity and the 1 s-smoothness of f that 0 ≥ f (yk) − f (xk) + ⟨∇f(yk), xk − yk⟩ and 0 ≥ f(xk+1) − f(yk) + s 2 ∥∇f(yk)∥2 , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Note that xk − yk = − τk 1 − τk (zk − yk) = − 2√s tk+1 cothc � √µ 2 tk+1 � − µs 1 − 2√s tk+1 cothc � √µ 2 tk+1 � (zk − yk) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Taking a weighted sum of the inequalities above yields (the assumption (64) ensures that these weights are non-negative for k ≥ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' and the case k = 0 is trivial because y0 = x0) 0 ≥ 2√s tk+1 cothc �√µ 2 tk+1 � t2 k+1 4 sinhc2 �√µ 2 tk+1 � 54 Unifying NAG for Convex and Strongly Convex Objective Functions × � f (yk) − f (x∗) + ⟨∇f (yk) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' x∗ − yk⟩ + µ 2 ∥x∗ − yk∥2� + � 1 − 2√s tk+1 cothc �√µ 2 tk+1 �� t2 k+1 4 sinhc2 �√µ 2 tk+1 � × [f (yk) − f (xk) + ⟨∇f(yk),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' xk − yk⟩] + t2 k+1 4 sinhc2 �√µ 2 tk+1 � � f(xk+1) − f(yk) + s 2 ∥∇f(yk)∥2� = √stk+1 2 sinhc �√µ 2 tk+1 � cosh �√µ 2 tk+1 � ⟨∇f (yk) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' x∗ − yk⟩ + µ√stk+1 4 sinhc �√µ 2 tk+1 � cosh �√µ 2 tk+1 � ∥x∗ − yk∥2 − � 2√s tk+1 cothc �√µ 2 tk+1 � − µs � t2 k+1 4 sinhc2 �√µ 2 tk+1 � ⟨∇f(yk),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' zk − yk⟩ + st2 k+1 8 sinhc2 �√µ 2 tk+1 � ∥∇f (yk)∥2 + 2√s tk+1 cothc �√µ 2 tk+1 � t2 k+1 4 sinhc2 �√µ 2 tk+1 � (f (yk) − f (x∗)) + � 1 − 2√s tk+1 cothc �√µ 2 tk+1 �� t2 k+1 4 sinhc2 �√µ 2 tk+1 � (f (yk) − f (xk)) + t2 k+1 4 sinhc2 �√µ 2 tk+1 � (f(xk+1) − f(yk)) = √stk+1 2 sinhc �√µ 2 tk+1 � cosh �√µ 2 tk+1 � ⟨∇f (yk) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' x∗ − yk⟩ + µ√stk+1 4 sinhc �√µ 2 tk+1 � cosh �√µ 2 tk+1 � ∥x∗ − yk∥2 − √stk+1 2 sinhc �√µ 2 tk+1 � cosh �√µ 2 tk+1 � × � 1 − µ√stk+1 2 tanhc �√µ 2 tk+1 �� ⟨∇f(yk),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' zk − yk⟩ + st2 k+1 8 sinhc2 �√µ 2 tk+1 � ∥∇f (yk)∥2 + t2 k+1 4 sinhc2 �√µ 2 tk+1 � (f (xk+1) − f (x∗)) − � 1 − 2√s tk+1 cothc �√µ 2 tk+1 �� t2 k+1 4 sinhc2 �√µ 2 tk+1 � (f (xk) − f (x∗)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' 55 Kim and Yang F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2 Constant timestep scheme In this section, we show that the sequence (tk) defined in (69) satisfies the conditions (64) and (65).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' For convenience, we assume µ > 0 (the case µ = 0 can be handled easily).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' The condition (64) follows from 2√s tk cothc �√µ 2 tk � = √µs coth �√µ 2 tk � ≤ √µs coth �√µ 2 t2 � = √µs coth (− log (1 − √µs)) = √µs1 + e2 log(1−√µs) 1 − e2 log(1−√µs) = √µs1 + � 1 − √µs �2 1 − � 1 − √µs �2 ≤ 1, where the last inequality holds because √µs ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' To prove (65), it suffices to show that the inequality sinh2 �√µ 2 t � − √µs sinh �√µ 2 t � cosh �√µ 2 t � − sinh2 �√µ 2 t + 1 2 log (1 − √µs) � ≤ 0 holds for all t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Letting r = e √µ 2 t, this inequality can be expressed as r2 + r−2 − 2 4 − √µsr2 − r−2 4 − � 1 − √µs � r2 + � 1 − √µs �−1 r−2 − 2 4 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Letting q = r2 and multiplying both sides by 4q, the inequality can be rewritten as 0 ≥ q2 + 1 − 2q − √µs � q2 − 1 � − (1 − √µs) q2 − (1 − √µs)−1 + 2q = 1 + √µs − 1 1 − √µs = −µs 1 − √µs, which clearly holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='3 Adaptive timestep scheme In this section, we show that for the sequence (tk) defined by (71), the sequence (tk) is well-defined, and the conditions (10) and (11) hold when lims→0 t0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' 56 Unifying NAG for Convex and Strongly Convex Objective Functions The sequence (tk) is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Because 4 t2 k+1 cschc2 �√µ 2 tk+1 � + µ = 4 t2 k+1 cothc2 �√µ 2 tk+1 � , the updating rule (71) is equivalent to 4 t2 k+1 cothc2 �√µ 2 tk+1 � = � 1 − 2√s tk+1 cothc �√µ 2 tk+1 �� 4 t2 k cothc2 �√µ 2 tk � + 2µ√s tk+1 cothc �√µ 2 tk+1 � , tk+1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (108) Introduce a sequence (αk)∞ k=−1 such that αk = 2√s tk+1 cothc � √µ 2 tk+1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' As t �→ 2√s t cothc � √µ 2 t � is a bijective map from (0, ∞) to �√µs, ∞ � , the sequences (tk) and (αk) have a one-to-one relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Thus, the updating rule (108) is equivalent to α2 k = (1 − αk) α2 k−1 + µsαk, αk > √µs, (109) which admits a unique solution in (√µs, ∞) when αk−1 > √µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Thus, the sequence (tk) is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' The sequence (tk) satisfies the conditions (10) and (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Define a function A(t) as A(t) := t2 4 sinhc2 �√µ 2 t � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (110) For t ∈ (0, ∞), it follows from (71) that ˙A � tk(t)+1 � = A � tk(t)+1 � − A (t) √s = A � tk(t)+1 � − A (t) tk(t)+1 − t tk(t)+1 − t √s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Because tk(t)+1 → t as s → 0, taking the limit s → 0 in the equation above yields 1 = lim s→0 tk(t)+1 − t √s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Thus, the condition (65) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='4 Equivalence between the adaptive timestep scheme and the original NAG In this section, we show that the adaptive timestep scheme (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2) with t0 > 0 is equivalent to the original NAG (5) with γ0 = 4 t2 0 cothc2 � √µ 2 t0 � > µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' We first show that the sequences (αk)∞ k=0 and (γk)∞ k=0 generated in the original NAG (5) with γ0 = 4 t2 0 cothc2 � √µ 2 t0 � > µ can be written as αk = 2√s tk+1 cothc � √µ 2 tk+1 � and γk = 4 t2 k cothc2 � √µ 2 tk � , where the sequence (tk)∞ k=0 is defined as (71).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Note that the equality (6) implies γk+1 = (1 − αk) γk + µαk = α2 k s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' 57 Kim and Yang Thus, the updating rule for αk (6) can be written as 1 sα2 k = (1 − αk) α2 k−1 s + µαk, where we define α−1 := √sγ0 = 2√s t0 cothc � √µ 2 t0 � > √µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' This implies that the sequence (αk)∞ k=−1 in the original NAG and the sequence (αk)∞ k=−1 defined in Section F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='3 are identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Thus, we have αk = 2√s tk+1 cothc � √µ 2 tk+1 � and γk = α2 k−1 s = 4 t2 k cothc2 � √µ 2 tk � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Now, we show that the parameters τk and δk for the original NAG are equal to those for our adaptive timestep scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' In the original NAG, we have (αk − µs) (γk + µαk) = αkγk + µα2 k − µsγk − µ2sαk = µsγk+1 + αkγk − µsγk − µ2sαk = µs ((1 − αk) γk + µαk) + αkγk − µsγk − µ2sαk = (1 − µs)αkγk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Therefore, we have τk = αkγk γk + µαk = αk − µs 1 − µs = 2√s tk+1 cothc � √µ 2 tk+1 � − µs 1 − µs and δk = αk γk+1 = s αk = √stk+1 2 tanhc �√µ 2 tk+1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Thus, the ogirinal Nesterov’s method with γ0 = 4 t2 0 cothc2 � √µ 2 t0 � > µ is equivalent to the adaptive timestep scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Higher-Order Extension G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='1 Limiting ODE Limiting ODE of the unified accelerated tensor method family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' We show that if the sequence (tk) satisfies the conditions (81) and (82), then the unified accelerated tensor method family (79) converges to the unified accelerated tensor flow (76) under the identifications xk = X(tk) and zk = Z(tk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' For convenience, we assume that µ > 0 (the case µ = 0 can be handled easily).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Define a function A : [0, ∞) → R as A(t) = Ctp sinhcp p � C1/pµ1/pt � = 1 µ sinhp p � C1/pµ1/pt � (111) so that Ak = A(tk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' It follows from the step (79b) that ˙X(t) = lim s→0 xk(t)+1 − xk(t) tk(t)+1 − t 58 Unifying NAG for Convex and Strongly Convex Objective Functions = lim s→0 xk(t)+1 − xk(t) s1/p = lim s→0 yk(t) − xk(t) s1/p = lim s→0 Ak(t)+1 − Ak(t) s1/pAk(t)+1 � zk(t) − xk(t) � = lim s→0 A � tk(t)+1 � − A(t) s1/pA � tk(t)+1 � (Z(t) − X(t)) = ˙A(t) A(t) (Z(t) − X(t)) = pC1/pµ1/p cothp � C1/pµ1/pt � (Z(t) − X(t)) , where we used ∥xk+1 − yk∥ = o(s1/p) (see Wibisono et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=', 2016, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2) for the third equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Using the step (80),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' we have ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='dt∇h(Z(t)) = lim ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='s→0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='∇h ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='zk(t)+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='− ∇h ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='zk(t) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='tk(t)+1 − t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='= lim ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='s→0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='∇h ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='zk(t)+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='− ∇h ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='zk(t) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='s1/p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='= lim ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='s→0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='Ak(t)+1 − Ak(t) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='s1/p � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='1 + µAk(t) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='µ∇h ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='xk(t)+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='− µ∇h ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='zk(t)+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='− ∇f ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='xk(t)+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='= lim ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='s→0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='tk(t)+1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='− A(t) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='s1/p (1 + µA(t)) (µ∇h (X(t)) − µ∇h (X(t)) − ∇f (X(t))) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='˙A(t) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='1 + µA(t) (µ∇h (X(t)) − µ∇h (X(t)) − ∇f (X(t))) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='C1/pp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='µ(p−1)/p tanhp−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='C1/pµ1/pt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='(µ∇h (X(t)) − µ∇h (X(t)) − ∇f (X(t))) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Thus, we obtain the system of ODEs (76).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Limiting ODE of the unified accelerated tensor method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' We check that the se- quence (tk) defined in (86) satisfies the condition (82).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' It is easy to check that the function A(t) defined in (111) satisfies ˙A(t) = C1/ppµ 1−p p sinhp−1 p � C1/pµ1/pt � coshp � C1/pµ1/pt � = C1/ppA(t) p−1 p (1 + µA(t)) 1 p and that the sequence (tk) defined in (86) satisfies A (tk+1) − A (tk) s1/p − C1/ppA (tk+1) p−1 p (1 + µA (tk)) 1 p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Now, substituting k = k(t) into the above equality and taking the limit s → 0, we have lims→0 tk(t)+1−t s1/p = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' 59 Kim and Yang G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2 Proof of Theorem 17 By the Bregman three-point identity (32) with x = x∗, y = zk+1, z = xk+1 and the non-negativity of Bregman divergence, we have Dh (x∗, zk+1) = Dh (x∗, xk+1) − ⟨∇h (zk+1) − ∇h (xk+1) , x∗ − zk+1⟩ − Dh (zk+1, xk+1) ≤ Dh (x∗, xk+1) − ⟨∇h (zk+1) − ∇h (xk+1) , x∗ − zk+1⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' we can bound the difference of the discrete-time energy function (84) as follows: Ek+1 − Ek = (1 + µAk+1) Dh (x∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' zk+1) − (1 + µAk) Dh (x∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' zk) + Ak+1 (f (xk+1) − f (x∗)) − Ak (f (xk) − f (x∗)) = µ (Ak+1 − Ak) Dh (x∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' zk+1) + (Ak+1 − Ak) (f (xk+1) − f (x∗)) + Ak (f (xk+1) − f (xk)) + (1 + µAk) (−h (zk+1) − ⟨∇h (zk+1) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' x∗ − zk+1⟩ + h (zk) + ⟨∇h (zk) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' x∗ − zk⟩) ≤ µ (Ak+1 − Ak) Dh (x∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' xk+1) − µ (Ak+1 − Ak) ⟨∇h (zk+1) − ∇h (xk+1) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' x∗ − zk+1⟩ + (Ak+1 − Ak) (f (xk+1) − f (x∗)) + Ak (f (xk+1) − f (xk)) + (1 + µAk) (−h (zk+1) − ⟨∇h (zk+1) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' x∗ − zk+1⟩ + h (zk) + ⟨∇h (zk) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' x∗ − zk⟩) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' By the (µ-uniform) convexity of f with respect to h, the p-th order 1-uniform convexity of h, and the property (74) of the higher-order gradient update operator Gp,M, the following inequalities hold: 0 ≥ f (xk+1) − f (x∗) + ⟨∇f (xk+1) , x∗ − xk+1⟩ + µDh (x∗, xk+1) 0 ≥ f (xk+1) − f (xk) + ⟨∇f (xk+1) , xk − xk+1⟩ 0 ≥ Ms 1 p−1 ∥∇f (xk+1)∥ p p−1 − ⟨∇f (xk+1) , yk − xk+1⟩ 0 ≥ h (zk) − h (zk+1) + ⟨∇h (zk) , zk+1 − zk⟩ + 1 p ∥zk+1 − zk∥p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Taking a weighted sum of these inequalities yields 0 ≥ (Ak+1 − Ak) [f (xk+1) − f (x∗) + ⟨∇f (xk+1) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' x∗ − xk+1⟩ + µDh (x∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' xk+1)] + Ak [f (xk+1) − f (xk) + ⟨∇f (xk+1) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' xk − xk+1⟩] + Ak+1 � Ms 1 p−1 ∥∇f (xk+1)∥ p p−1 − ⟨∇f (xk+1) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' yk − xk+1⟩ � + (1 + µAk) � h (zk) − h (zk+1) + ⟨∇h (zk) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' zk+1 − zk⟩ + 1 p ∥zk+1 − zk∥p � ≥ Ek+1 − Ek − µ (Ak+1 − Ak) Dh (x∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' xk+1) + µ (Ak+1 − Ak) ⟨∇h (zk+1) − ∇h (xk+1) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' x∗ − zk+1⟩ − (Ak+1 − Ak) (f (xk+1) − f (x∗)) − Ak (f (xk+1) − f (xk)) − (1 + µAk+1) (−h (zk+1) − ⟨∇h (zk+1) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' x∗ − zk+1⟩ + h (zk) + ⟨∇h (zk) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' x∗ − zk⟩) + (Ak+1 − Ak) [f (xk+1) − f (x∗) + ⟨∇f (xk+1) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' x∗ − xk+1⟩ + µDh (x∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' xk+1)] 60 Unifying NAG for Convex and Strongly Convex Objective Functions + Ak [f (xk+1) − f (xk) + ⟨∇f (xk+1) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' xk − xk+1⟩] + Ak+1 � Ms 1 p−1 ∥∇f (xk+1)∥ p p−1 − ⟨∇f (xk+1) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' yk − xk+1⟩ � + (1 + µAk) � h (zk) − h (zk+1) + ⟨∇h (zk) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' zk+1 − zk⟩ + 1 p ∥zk+1 − zk∥p � = Ek+1 − Ek + ⟨∇f (xk+1) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (Ak+1 − Ak) (x∗ − xk+1) + Ak (xk − xk+1) + Ak+1 (xk+1 − yk)⟩ + (1 + µAk) ⟨∇h (zk+1) − ∇h (zk) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' x∗ − zk+1⟩ + 1 + µAk p ∥zk+1 − zk∥p + µ (Ak+1 − Ak) ⟨∇h (zk+1) − ∇h (xk+1) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' x∗ − zk+1⟩ + MAk+1s 1 p−1 ∥∇f (xk+1)∥ p p−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Substituting (80) with the term (1 + µAk) ⟨∇h (zk+1) − ∇h (zk) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' x∗ − zk+1⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' we have 0 ≥ Ek+1 − Ek + ⟨∇f (xk+1) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (Ak+1 − Ak) (x∗ − xk+1) + Ak (xk − xk+1) + Ak+1 (xk+1 − yk)⟩ + (Ak+1 − Ak) ⟨µ∇h (xk+1) − µ∇h (zk+1) − ∇f (xk+1) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' x∗ − zk+1⟩ + 1 + µAk p ∥zk+1 − zk∥p + µ (Ak+1 − Ak) ⟨∇h (zk+1) − ∇h (xk+1) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' x∗ − zk+1⟩ + MAk+1s 1 p−1 ∥∇f (xk+1)∥ p p−1 = Ek+1 − Ek + ⟨∇f (xk+1) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (Ak+1 − Ak) (zk+1 − xk+1) + Ak (xk − xk+1) + Ak+1 (xk+1 − yk)⟩ + 1 + µAk p ∥zk+1 − zk∥p + MAk+1s 1 p−1 ∥∇f (xk+1)∥ p p−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' We also notice that (Ak+1 − Ak) (zk+1 − xk+1) + Ak (xk − xk+1) + Ak+1 (xk+1 − yk) = (Ak+1 − Ak) zk+1 + Akxk − Ak+1yk = (Ak+1 − Ak) (zk+1 − zk) + (Ak+1 − Ak) zk + Akxk − Ak+1yk = (Ak+1 − Ak) (zk+1 − zk) , where the last equality follows from yk = xk + Ak+1−Ak Ak+1 (zk − xk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Therefore, 0 ≥ Ek+1 − Ek + (Ak+1 − Ak) ⟨∇f (xk+1) , zk+1 − zk⟩ + 1 + µAk p ∥zk+1 − zk∥p + MAk+1s 1 p−1 ∥∇f (xk+1)∥ p p−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Now, we use the Fenchel-Young inequality ⟨s, u⟩ + 1 p ∥u∥p ≥ − p−1 p ∥s∥ p p−1 with u = (1 + µAk) 1 p (zk+1 − zk) and s = (Ak+1 − Ak) (1 + µAk)− 1 p ∇f (xk+1) to obtain that (Ak+1 − Ak) ⟨∇f (xk+1) , zk+1 − zk⟩ + 1 + µAk p ∥zk+1 − zk∥p 61 Kim and Yang ≥ −p − 1 p (Ak+1 − Ak) p p−1 (1 + µAk)− 1 p−1 ∥∇f (xk+1)∥ p p−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Hence, we have 0 ≥ Ek+1 − Ek + � MAk+1s 1 p−1 − p − 1 p (Ak+1 − Ak) p p−1 (1 + µAk)− 1 p−1 � ∥∇f (xk+1)∥ p p−1 = Ek+1 − Ek + � (p − 1)p 1 p−1 C 1 p−1 Ak+1s 1 p−1 − p − 1 p (Ak+1 − Ak) p p−1 (1 + µAk)− 1 p−1 � ∥∇f (xk+1)∥ p p−1 , where C = 1 p( M p−1)p−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' It is easy to see that the condition (83) implies that the term � (p − 1)p 1 p−1 C 1 p−1 Ak+1s 1 p−1 − p − 1 p (Ak+1 − Ak) p p−1 (1 + µAk)− 1 p−1 � ∥∇f (xk+1)∥ p p−1 is non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Thus, we conclude that 0 ≥ Ek+1 − Ek as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='3 Lower bounds for the sequence (Ak) Let (Abest k ) denote the sequence (Ak) determined by (86).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' In this section, we prove that that the following inequality holds: Abest k ≥ max � O (kp) , O �� 1 + C1/ppµ1/ps1/p�k�� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' We use the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Lemma 21 For any sequence (Ak) satisfying A0 = 0 and the condition (83), we have Ak ≤ Abest k ∀k ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (112) Its proof can be found in the following subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Now, we claim that the following two sequences satisfy the condition (83): Ak = Csk(k + 1) · · · (k + p − 1) and Ak = � 0, k = 0 Cpps � 1 + C1/ppµ1/ps1/p�k−1 k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' For the first sequence, we have (Ak+1 − Ak)p − CppsAp−1 k+1 (1 + µAk) 62 Unifying NAG for Convex and Strongly Convex Objective Functions ≤ (Ak+1 − Ak)p − CppsAp−1 k+1 = (Cps(k + 1) · · · (k + p − 1))p − Cpps (Cs(k + 1) · · · (k + p))p−1 = Cpppsp � ((k + 1) · · · (k + p − 1))p − ((k + 1) · · · (k + p))p−1� ≤ 0, which implies that (83) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' For the second sequence, (83) holds because (Ak+1 − Ak)p − CppsAp−1 k+1 (1 + µAk) ≤ (Ak+1 − Ak)p − CµppsAp−1 k+1Ak ≤ (Ak+1 − Ak)p − CµppsAp k = ��Ak+1 Ak − 1 �p − Cµpps � Ap k = �� C1/ppµ1/ps1/p�p − Cµpps � Ap k = 0 for all k ≥ 1 (the case k = 0 is trivial).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Thus, it follows from Lemma 21 that Abest k ≥ max � Csk(k + 1) · · · (k + p − 1), Cpps � 1 + C1/ppµ1/ps1/p�k−1� = max � O (kp) , O �� 1 + C1/ppµ1/ps1/p�k�� , as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='1 Proof of Lemma 21 For r ≥ 0, we define S(r) := � x : (x − r)p − Cppsxp−1(1 + µr) ≤ 0 � U(r) := max Sr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Then, it is straightforward to see the following: The set S(r) is nonempty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' In particular, r ∈ S(r) (which implies U(r) ≥ r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' For any sequence (Ak) satisfying the condition (83), we have Ak+1 ∈ S(Ak) for all k ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' For the sequence (Ak) defined in (86), we have Ak+1 = U(Ak) for all k ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' If we have U (r1) ≤ U (r2) whenever r1 ≤ r2, (113) then we can prove (112) using mathematical induction on k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' It clearly holds when k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' If (112) holds for k, then it holds for k + 1 because Ak+1 ≤ U (Ak) ≤ U(Abest k ) = Abest k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' 63 Kim and Yang It remains to prove (113).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Let r1 and r2 be positive real numbers with r1 ≤ r2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Then, it is easy to check that r2 + U(r1) − r1 ∈ S(r2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Thus, we have U (r1) ≤ U (r1) + (r2 − r1) ≤ U (r2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Existence and Uniqueness Theorems H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='1 Proof of Theorem 6 We prove a stronger result, that the unified Bregman Lagrangian flow (56) with α(t) = log( 2 t cothc( √µ 2 t)), β(t) = log( t2 4 sinhc2( √µ 2 t)): ˙X = 2 t cothc �√µ 2 t � (Z − X) d dt∇h(Z) = t 2 tanhc �√µ 2 t � (µ∇h(X) − µ∇h(Z) − ∇f(X)) (114) with the initial conditions X(0) = Z(0) = x0 has a unique global solution (X, Z) in C1([0, ∞), Rn × Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Following (Krichene et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=', 2015), we assume that ∇f is Lf-Lipschitz continuous and ∇h is Lh-Lipschitz continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' The strong convexity of h implies a Lh∗- Lipschitz continuity of ∇h∗ for some Lh∗ > 0 (see Rockafellar and Wets, 2009, Proposi- tion 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='60).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='1 Proof of existence Fix t1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' We show the existence of solution to the system (114) on [0, t1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' To remove the singularity of the system (114) at t = 0, fix δ > 0, and consider the following system of ODEs: ˙X = 2 max{δ, t} cothc �√µ 2 max{δ, t} � (Z − X) d dt∇h(Z) = t 2 tanhc �√µ 2 t � (µ∇h(X) − µ∇h(Z) − ∇f(X)) (115) with X(0) = Z(0) = x0, which does not have singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Denote the image of Z under the mirror map as W(t) = ∇h(Z(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Denote the convex conjugate of h by h∗ : Rn → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Then, ∇h and ∇h∗ are inverses of each other (see Rockafellar and Wets, 2009, Section 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Now, we can equivalently write the system (115) as ˙X = 2 max{δ, t} cothc �√µ 2 max{δ, t} � (∇h∗(W) − X) (116a) ˙W = t 2 tanhc �√µ 2 t � (µ∇h(X) − µW − ∇f(X)) (116b) with X(0) = x0 and W(0) = w0 := ∇h (x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' By the Cauchy-Lipschitz theorem, the system of ODEs (116) has a unique solution (Xδ, Wδ) in C1([0, t1], Rn × Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' If we prove the following lemma, then one can prove the existence of solution to the ODE system (115) following the argument in (Krichene et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=', 2015, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' 64 Unifying NAG for Convex and Strongly Convex Objective Functions Lemma 22 Define a constant T as T = min �� 2 µ, 1 2 � 1 K2K3 � , where K2 and K3 are constants defined in (118).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Then, the family of solutions ((Xδ, Zδ)|[0,T])δ∈(0,T] is equi-Lipschitz-continuous and uniformly bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' We now prove this lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' We follow the argument of Krichene et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (2015) and omit the detailed calculations that can be found in (Krichene et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=', 2015, Appendix 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Fix δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' For t > 0, define Aδ(t) := sup u∈[0,t] ��� ˙Wδ(u) ��� u Bδ(t) := sup u∈[0,t] ∥Xδ(u) − x0∥ u Cδ(t) := sup u∈[0,t] ��� ˙Xδ(u) ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Then, these quantities are finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' We first prove the following inequalities, which correspond to (Krichene et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=', 2015, Lemma 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Aδ(t) ≤ µ ∥w0∥ + µ ∥∇h (x0)∥ + ∥∇f (x0)∥ + (µLh + Lf) tBδ(t) (117a) Bδ(t) ≤ Lh∗t 3 cothc �√µ 2 T � Aδ(t) (117b) Cδ(t) ≤ cothc �√µ 2 T � (Lh∗TAδ(t) + 2Bδ(t)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (117c) Proof of (117a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Using Aδ and Bδ, we can bound ∥Wδ(t) − w0∥ and ∥Xδ(t) − x0∥ as ∥Wδ(t) − w0∥ ≤ t2 2 Aδ(t) ∥Xδ(t) − x0∥ ≤ tBδ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' From (116b), we have 2 ��� ˙Wδ(t) ��� t = tanhc �√µ 2 t � ∥µ∇h(Xδ) − µWδ − ∇f(Xδ)∥ ≤ ∥µ∇h(Xδ) − µWδ − ∇f(Xδ)∥ ≤ µ ∥Wδ∥ + µ ∥∇h(Xδ)∥ + ∥∇f(Xδ)∥ ≤ µ ∥w0∥ + µt2 2 Aδ(t) + µ ∥∇h (x0)∥ + µLhtBδ(t) + ∥∇f (x0)∥ + LftBδ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Thus, 2Aδ(t) ≤ µ ∥w0∥ + µ ∥∇h (x0)∥ + ∥∇f (x0)∥ + µt2 2 Aδ(t) + (µLh + Lf) tBδ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Because T ≤ � 2/µ, we obtain the inequality (117a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' 65 Kim and Yang Proof of (117b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' To bound the function Bδ(t) = supu∈[0,t] ∥Xδ(u)−x0∥ u , we first compute an upper bound of ∥Xδ(t) − x0∥ in the case 0 ≤ t ≤ δ and the case t ≥ δ separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' First, consider the case t ∈ [0, δ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' By (116a), we have ˙Xδ + 2 δ cothc �√µ 2 δ � (Xδ − x0) = 2 δ cothc �√µ 2 δ � (∇h∗(Wδ − ∇h∗ (w0)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Multiplying e 2 δ cothc � √µ 2 δ � t, we obtain e 2 δ cothc � √µ 2 δ � t � ˙Xδ + 2 δ cothc �√µ 2 δ � (Xδ − x0) � = 2 δ cothc �√µ 2 δ � e 2 δ cothc � √µ 2 δ � t (∇h∗(Wδ) − ∇h∗ (w0)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' This equality can be written as d dt � (Xδ(t) − x0) e 2 δ cothc � √µ 2 δ � t � = 2 δ cothc �√µ 2 δ � e 2 δ cothc � √µ 2 δ � t (∇h∗ (Wδ(t)) − ∇h∗ (w0)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Integrating both sides yields (Xδ(t) − x0) e 2 δ cothc � √µ 2 δ � t = 2 δ cothc �√µ 2 δ � � t 0 � e 2 δ cothc � √µ 2 δ � s (∇h∗ (Wδ(s)) − ∇h∗ (w0)) � ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Taking norms, we have ∥Xδ(t) − x0∥ ≤ 2 δ cothc �√µ 2 δ � � t 0 ∥∇h∗ (Wδ(s)) − ∇h∗ (w0)∥ ds ≤ 2Lh∗ δ cothc �√µ 2 δ � � t 0 ∥Wδ(s) − w0∥ ds ≤ 2Lh∗ δ cothc �√µ 2 δ � � t 0 s2 2 Aδ(t) ds = 2Lh∗ δ cothc �√µ 2 δ � Aδ(t)t3 6 ≤ 2Lh∗ t cothc �√µ 2 δ � Aδ(t)t3 6 = Lh∗t2 3 cothc �√µ 2 δ � Aδ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' So far, we provide an upper bound of ∥Xδ(t) − x0∥ in the case 0 ≤ t ≤ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' We now consider the case t ≥ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' By (116a), we have ˙Xδ + 2 t cothc �√µ 2 t � (Xδ − x0) = 2 t cothc �√µ 2 t � (∇h∗(Wδ) − ∇h∗ (w0)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' 66 Unifying NAG for Convex and Strongly Convex Objective Functions Multiplying t2 4 sinhc2( √µ 2 t) to both sides, we obtain t2 4 sinhc2 �√µ 2 t � ˙Xδ + t 2 sinhc �√µ 2 t � cosh �√µ 2 t � (Xδ − x0) = t 2 sinhc �√µ 2 t � cosh �√µ 2 t � (∇h∗(Wδ) − ∇h∗ (w0)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' This equality can be written as d dt �t2 4 sinhc2 �√µ 2 t � (Xδ(t) − x0) � = t 2 sinhc �√µ 2 t � cosh �√µ 2 t � (∇h∗ (Wδ(t)) − ∇h∗ (w0)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Integrating both sides, we obtain t2 4 sinhc2 �√µ 2 t � (Xδ(t) − x0) = � t 0 �s 2 sinhc �√µ 2 s � cosh �√µ 2 s � (∇h∗ (Wδ(s)) − ∇h∗ (w0)) � ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Taking norms, we have the following upper bound on ∥Xδ(t) − x0∥: ∥Xδ(t) − x0∥ ≤ 2 t cothc �√µ 2 t � � t 0 ∥∇h∗ (Wδ(s)) − ∇h∗ (w0)∥ ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' ≤ 2Lh∗ t cothc �√µ 2 t � � t 0 ∥Wδ(s) − w0∥ ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' ≤ 2Lh∗ t cothc �√µ 2 t � � t 0 s2 2 Aδ(t) ds = 2Lh∗ t cothc �√µ 2 t � Aδ(t)t3 6 = Lh∗t2 3 cothc �√µ 2 t � Aδ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Combining both cases 0 ≤ t ≤ δ and t ≥ δ, we have ∥Xδ(t) − x0∥ ≤ Lh∗t2 3 cothc �√µ 2 T � Aδ(t) for all t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Dividing by t and taking the supremum, we obtain Bδ(t) ≤ Lh∗t 3 cothc �√µ 2 T � Aδ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' 67 Kim and Yang Proof of (117c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' By (116a), we have ��� ˙X ��� = 2 max{δ, t} cothc �√µ 2 max{δ, t} � ∥∇h∗ (Wδ(t)) − Xδ(t)∥ ≤ 2 max{δ, t} cothc �√µ 2 max{δ, t} � (∥∇h∗ (Wδ(t)) − ∇h∗ (z0)∥ + ∥Xδ(t) − x0∥) ≤ 2 max{δ, t} cothc �√µ 2 max{δ, t} � �t2 2 Lh∗Aδ(t) + tBδ(t) � ≤ cothc �√µ 2 T � 2 t �t2 2 Lh∗Aδ(t) + tBδ(t) � ≤ cothc �√µ 2 T � (Lh∗TAδ(t) + 2Bδ(t)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Complete the proof of Lemma 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Define five positive constants K1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=', K5 as K1 := µ ∥w0∥ + µ ∥∇h (x0)∥ + ∥∇f (x0)∥ K2 := µLh + Lf K3 := 2Lh∗ 3 K4 := 2Lh∗ K5 := 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (118) Because T ≤ 2 √µ, we have cothc( √µ 2 T) ≤ cothc(1) ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Thus, the inequalities (117) imply Aδ(t) ≤ K1 + K2TBδ(t) (119a) Bδ(t) ≤ K3TAδ(t) (119b) Cδ(t) ≤ K4TAδ(t) + K5Bδ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (119c) Combining (119a) and (119b), we have � 1 K3T − K2T � Bδ(t) ≤ K1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Because T �→ 1 K3T − K2T is a positive decreasing funtion on [0, 1 2 � 1 K2K3 ] and T ≤ 1 2 � 1 K2K3 , we have Bδ(T) ≤ � � 1 K3 · 1 2 � 1 K2K3 − K2 · 1 2 � 1 K2K3 � � −1 K1 = 2 3K1 � K3 K2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (120) The inequalities (119a), (120), and T ≤ 1 2 � 1 K2K3 imply Aδ(T) ≤ K1 + K2TBδ(T) ≤ K1 + K2 �1 2 � 1 K2K3 � � 2 3K1 � K3 K2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (121) 68 Unifying NAG for Convex and Strongly Convex Objective Functions The inequalities (119a), (120), (121), and T ≤ 1 2 � 1 K2K3 imply Cδ(T) ≤ K4TAδ(T) + K5Bδ(T) ≤ K4 �1 2 � 1 K2K3 � � K1 + K2 �1 2 � 1 K2K3 � � 2 3K1 � K3 K2 �� + K5 � 2 3K1 � K3 K2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (122) Therefore, ∥ ˙W∥ and ∥ ˙X∥ are bounded uniformly in δ because ��� ˙Wδ(t) ��� ≤ TAδ(T) ��� ˙Xδ(t) ��� ≤ Cδ(T) for all t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' This implies that the family of solutions ((Xδ, Zδ)|[0,T])δ∈(0,T] is equi- Lipschitz-continuous and uniformly bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2 Proof of uniqueness We follow the argument in (Krichene et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=', 2015, Appendix 3) and omit the detailed calculations that can be found in (Krichene et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Because we only need to prove the uniqueness of solution near t = 0, we assume t < T for some T > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Let (X, W) and � ¯X, ¯W � be solutions to the following system of ODEs, which is equivalent to (114): ˙X = 2 t cothc �√µ 2 t � (∇h∗(W) − X) ˙W = t 2 tanhc �√µ 2 t � (µ∇h(X) − µW − ∇f(X)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Let ∆W = W − ¯W and ∆X = X − ¯X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Then, we have ˙∆W = t 2 tanhc �√µ 2 t � � µ∇h(X) − µW − ∇f(X) − µ∇h � ¯X � + µ ¯W + ∇f � ¯X �� ˙∆X = 2 t cothc �√µ 2 t � � ∇h∗(W) − ∇h∗ � ¯W � − ∆X � with ∆X(0) = ∆W (0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Define A(t) := sup [0,t] ��� ˙∆W (u) ��� u B(t) := sup [0,t] ∥∆X∥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' 69 Kim and Yang Then, B(t) and C(t) are finite because ∆X and ∆W are continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' First, we compute an upper bound of A(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' We have ��� ˙∆W (t) ��� = t 2 tanhc �√µ 2 t � ��µ∇h(X) − µW − ∇f(X) − µ∇h � ¯X � + µ ¯W + ∇f � ¯X ��� ≤ t 2 tanhc �√µ 2 t � � µ ��∇h(X) − ∇h � ¯X ��� + µ ��W − ¯W �� + ��∇f(X) − ∇f � ¯X ���� ≤ t 2 tanhc �√µ 2 t � ((µLh + Lf) ∥∆X∥ + µ ∥∆W ∥) ≤ t 2 tanhc �√µ 2 t � � (µLh + Lf) B(t) + µt2 2 A(t) � , (123) where we used ∥∆W (t)∥ ≤ ∥ � t 0 ˙∆W (s) ds∥ ≤ � t 0 sA(s) ds ≤ � t 0 sA(t) ds = t2 2 A(t) for the last inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Dividing both sides of (123) by t and then taking the supremum, we obtain A(t) ≤ 1 2 tanhc �√µ 2 t � � (µLh + Lf) B(t) + µt2 2 A(t) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (124) Nest, we compute an upper boudn of B(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' We have ˙∆X + 2 t cothc �√µ 2 t � ∆X = 2 t cothc �√µ 2 t � � ∇h∗(W) − ∇h∗ � ¯W �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Multiplying both sides by t2 4 sinhc2 � √µ 2 t � , we have t2 4 sinhc2 �√µ 2 t � ˙∆X + t 2 sinhc �√µ 2 t � cosh �√µ 2 t � ∆X = t 2 sinhc �√µ 2 t � cosh �√µ 2 t � � ∇h∗(W) − ∇h∗ � ¯W �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' This equality can be written as d dt �t2 4 sinhc2 �√µ 2 t � ∆X � = t 2 sinhc �√µ 2 t � cosh �√µ 2 t � � ∇h∗(W) − ∇h∗ � ¯W �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Integrating both sides, we obtain t2 4 sinhc2 �√µ 2 t � ∆X = � t 0 �s 2 sinhc �√µ 2 s � cosh �√µ 2 s � � ∇h∗(W(s)) − ∇h∗ � ¯W(s) ��� ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Taking norms, we have ∥∆X(t)∥ ≤ 2 t cothc �√µ 2 t � � t 0 ��∇h∗(W(s)) − ∇h∗ � ¯W(s) ��� ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' ≤ 2Lh∗ t cothc �√µ 2 t � � t 0 ∥∆W (s)∥ ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' 70 Unifying NAG for Convex and Strongly Convex Objective Functions ≤ 2Lh∗ t cothc �√µ 2 t � � t 0 s2 2 A(t) ds = Lh∗2 t cothc �√µ 2 t � A(t)t3 6 = Lh∗t2 3 cothc �√µ 2 t � A(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Taking the supremum yields B(t) ≤ Lh∗t2 3 cothc �√µ 2 t � A(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (125) Now, combining the inequalities (124) and (125), we have A(t) ≤ 1 2 tanhc �√µ 2 t � � (µLh + Lf) B(t) + µt2 2 A(t) � ≤ 1 2 tanhc �√µ 2 t � � (µLh + Lf) Lh∗t2 3 cothc �√µ 2 t � + µt2 2 � A(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Using continuity, it is easy to see that there is Tsmall > 0 such that the following inequality holds whenever t ∈ (0, Tsmall): 1 2 tanhc �√µ 2 t � � (µLh + Lf) Lh∗t2 3 cothc �√µ 2 t � + µt2 2 � < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Thus, for t ∈ (0, Tsmall), we have A(t) ≤ 1 · A(t), which implies A(t) = 0 because A(t) is nonnegative by its definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Finally, B(t) = 0 follows from (125).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2 Existence and uniqueness of solution to the unified accelerated tensor flow We first note that The system of ODEs (114) is the unified Bregman Lagrangian flow (56) with β1 = log � t2 4 sinhc2 � √µ 2 t �� and α1 = log ˙β1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' The unified accelerated tensor flow (76) is the unified Bregman Lagrangian flow (56) with β2 = p log t + log C + p log � sinhcp � C1/pµ1/pt �� and α2 = log ˙β2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Define a function T : [0, ∞) → [0, ∞) as T = β−1 1 β2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Then, we have α2(t) = α1(T(t)) + log ˙T(t) β2(t) = β1(T(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Thus, by Theorem 5, if (X1, Z1) is a solution to the unified NAG system, then X2(t) = X1(T(t)) and Z2(t) = Z1(T(t)) is a solution to the unified accelerated tensor system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Thus, the existence of solution to the unified NAG system implies the existence of solution to the unified accelerated tensor system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' A similar argument shows that if (X2, Z2) is a solution to the unified accelerated tensor system, then X1(t) = X2(T−1(t)) and Z1(t) = Z2(T−1(t)) is a solution to the unified NAG system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' It is easy to show that this correspondence is one-to-one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Thus, the uniqueness of solution to the unified NAG system implies the uniqueness of solution to the unified accelerated tensor system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' 71 Kim and Yang I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Further Exploration: ODE Model for Minimizing Gradient Norms of Strongly Convex Functions I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='1 Limiting ODE of OGM For the sequence θk defined in (90), Su et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (2016) showed that the algorithm yk = � 1 − 1 θk � xk + 1 θk zk xk+1 = yk − s∇f (yk) zk+1 = zk − sθk∇f (yk) (126) converges to NAG-C ODE as s → 0 (see Su et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=', 2016, Section 2) (in fact, this algorithm is equivalent to the original NAG (5) with µ = 0 and γ0 = ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Because ∥xk+1 − yk∥ = o(√s), we can ignore the gradient descent step xk+1 = yk −s∇f (yk) in both (126) and OGM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Then, applying OGM to the objective function f is equivalent to applying the algorithm (126) to the objective function 2f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Thus, the limiting ODE of OGM is given by ¨X + 3 t ˙X + 2∇f(X) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='2 Proof of Theorem 19 For convenience, we assume µ > 0 (the case µ = 0 can be handled easily).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' We denote X := X(t) and xT := X(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' We also omit the input √µ 2 (T − t) of each hyperbolic function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' For example, we write the unified NAG-G ODE (97) as ¨X + �√µ 2 tanh +3√µ 2 coth � ˙X + ∇f(X) = 0 and the continuous-time energy function (98) as E(t) = µ2 csch4 � sinh2 µ � f(X) − f � xT �� − 1 2 ��X − xT ��2 + cosh2 2 ����X + tanh √µ ˙X − xT ���� 2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' we have sinh4 µ2 ˙E(t) = sinh4 d dt � csch4� � sinh2 µ � f(X) − f � xT �� − 1 2 ��X − xT ��2 + cosh2 2 ����X + tanh √µ ˙X − xT ���� 2� + d dt � sinh2 µ � f(X) − f � xT �� − 1 2 ��X − xT ��2 + cosh2 2 ����X + tanh √µ ˙X − xT ���� 2� = 2√µ coth � sinh2 µ � f(X) − f � xT �� − 1 2 ��X − xT ��2 + cosh2 2 ����X + tanh √µ ˙X − xT ���� 2� − sinh cosh √µ � f(X) − f � xT �� + sinh2 µ � ∇f(X),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' ˙X � − � X − xT ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' ˙X � 72 Unifying NAG for Convex and Strongly Convex Objective Functions − √µ sinh cosh 2 ����X + tanh √µ ˙X − xT ���� 2 + cosh2 � X + tanh √µ ˙X − xT ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' − ˙X − tanh √µ ∇f(X) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' where we used d dt � X + tanh √µ ˙X − xT � = tanh √µ ¨X + � 1 − 1 2 sech2 � ˙X = � −1 2 tanh2 −1 2 − 1 2 sech2 � ˙X − tanh √µ ∇f(X) = − ˙X − tanh √µ ∇f(X) for the last equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' We further simplify as sinh4 µ2 ˙E(t) = 2 sinh cosh √µ � f(X) − f � xT �� − √µ coth ��X − xT ��2 + √µ coth cosh2 ���X − xT ��2 + tanh2 µ ��� ˙X ��� 2 + 2 tanh √µ � X − xT ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' ˙X �� − sinh cosh √µ � f(X) − f � xT �� + sinh2 µ � ∇f(X),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' ˙X � − � X − xT ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' ˙X � − √µ sinh cosh 2 ���X − xT ��2 + tanh2 µ ��� ˙X ��� 2 + 2 tanh √µ � X − xT ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' ˙X �� − cosh2 � � X − xT ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' ˙X � + tanh √µ ��� ˙X ��� 2 + tanh √µ � X − xT ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' ∇f(X) � + tanh2 µ � ˙X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' ∇f(X) � � = �2 sinh cosh √µ − sinh cosh √µ � � f(X) − f � xT �� + � −√µ coth +√µ coth cosh2 − √µ sinh cosh 2 � ��X − xT ��2 + �sinh cosh √µ − sinh2 tanh 2√µ − sinh cosh √µ � ��� ˙X ��� 2 + � 2 cosh2 −1 − sinh2 − cosh2� � X − xT ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' ˙X � + �sinh2 µ − sinh2 µ � � ∇f(X),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' ˙X � − sinh cosh √µ � X − xT ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' ∇f(X) � = sinh cosh √µ � f(X) − f � xT �� + √µ sinh cosh 2 ��X − xT ��2 − sinh2 tanh 2√µ ��� ˙X ��� 2 − sinh cosh √µ � X − xT ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' ∇f(X) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' 73 Kim and Yang It follows from the µ-strong convexity of f that f(X) − f � xT � ≤ � X − xT , ∇f(X) � − µ 2 ��X − xT ��2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Thus, we have sinh4 µ2 ˙E(t) ≤ sinh cosh √µ �� X − xT , ∇f(X) � − µ 2 ��X − xT ��2� + √µ sinh cosh 2 ��X − xT ��2 − sinh2 tanh 2√µ ��� ˙X ��� 2 − sinh cosh √µ � X − xT , ∇f(X) � = −sinh2 tanh 2√µ ��� ˙X ��� 2 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='3 Computing ˙X(T) and ¨X(T) For simplicity, we assume that the limits limt→T − ˙X(T) and limt→T − ¨X(T) exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content='12 Consider the energy function E(t) = 1 2 ��� ˙X(t) ��� 2 + (f(X(t)) − f (x∗)) + � t 0 �√µ 2 tanh �√µ 2 (T − s) � + 3 T − s cothc �√µ 2 (T − s) �� ��� ˙X(s) ��� 2 ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' (127) Then, it is easy to show that E(t) = E(0) for all t ∈ [0, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Because the terms 1 2 ��� ˙X(t) ��� 2 and f(X(t)) − f (x∗) are non-negtive, we have � T 0 �√µ 2 tanh �√µ 2 (T − s) � + 3 T − s cothc �√µ 2 (T − s) �� ��� ˙X(s) ��� 2 ds < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' This implies limt→T − ˙X(t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' By L’Hˆopital’s rule, we obtain that lim t→T − �√µ 2 tanh �√µ 2 (T − t) � + 3 T − t cothc �√µ 2 (T − t) �� ˙X(t) = −3 ¨X(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Now, we have 0 = lim t→T − � ¨X(t) + �√µ 2 tanh �√µ 2 (T − t) � + 3 T − t cothc �√µ 2 (T − t) �� ˙X(t) + ∇f(X(t)) � = −2 ¨X(T) + ∇f(X(T)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Thus, ¨X(T) = 1 2∇f(X(T)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' References F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Alimisis, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Orvieto, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Becigneul, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' Lucchi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' A continuous-time perspective for modeling acceleration in riemannian optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' In Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics, pages 1297–1307, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} +page_content=' The proof to prove the existence of these limits is similar to that in (Suh et al.' 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77' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQf_Aah/content/2301.03576v1.pdf'} diff --git a/HNFJT4oBgHgl3EQfuC2G/content/tmp_files/2301.11620v1.pdf.txt b/HNFJT4oBgHgl3EQfuC2G/content/tmp_files/2301.11620v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..e68b8d2ca24df54de062815232e048597ddfe50b --- /dev/null +++ b/HNFJT4oBgHgl3EQfuC2G/content/tmp_files/2301.11620v1.pdf.txt @@ -0,0 +1,576 @@ +Email: Mandana Kariminejad + Mandana.kariminejad@mail.itsligo.ie +Optimization of a Commercial Injection-Moulded component by Using +DOE and Simulation + +Mandana Kariminejad, Centre for Precision Engineering, Material and Manufacturing, Institute of Technology Sligo +David Tormey, Centre for Precision Engineering, Material and Manufacturing, Institute of Technology Sligo +Saif Huq, School of Computing and Digital Media, London Metropolitan University +Jim Morrison, Department of Electronics and Mechanical Engineering, Letterkenny Institute of Technology +Jeff Redmond, Combination Products, Science and Technology, AbbVie Inc. +Carlos Souto, Engineering Moulding, AbbVie Ballytivnan +Marion McAfee, Centre for Precision Engineering, Material and Manufacturing, Institute of Technology Sligo +Abstract +Injection moulding is an important industry, providing a significant percentage of the demand for plastic +products throughout the world. The process consists of many variables which directly or indirectly +influence the part quality and cycle time. The first step in optimizing the process parameters is identifying +the most significant variables affecting the desired output. For this purpose, various Design of Experiments +methods (DOE) have been developed to investigate the effect of the experimental variables on the output +and predict the required settings to achieve the optimal value of the output. In this study we investigate the +application of DOE for a commercial injection moulded component which suffers from a long cycle time +and high shrinkage. The Taguchi method has been used to analyze the effect of four input variables on the +two output variables: cycle time and shrinkage. The component has been simulated in the Moldflow +software to validate the predicted output and optimized settings of the variables from the DOE. +Comparison of the simulation result and the predicted value from the DOE illustrated good accordance. +The calculated optimal setting with the Taguchi method reduced the cycle time from the 40s to about 23s +and met the shrinkage criteria for this commercial part. +Key Words: Injection Moulding, Design of Experiment, Taguchi Method, Moldflow Simulation, Cycle time +1. INTRODUCTION +One of the most developed processes for the production of plastic components is injection moulding. In general, +this process contains three main steps: the filling stage in which melted polymer pellets are injected into the +cavity, the packing stage which prevents excessive shrinkage by injection of extra polymer, and the cooling stage +where the polymer solidifies and gets ready for ejection (Kazmer, 2007). During these stages, many process +parameters such as mould temperature, melt temperature, and injection pressure should be controlled and +adjusted, directly affecting the part quality and efficiency of the process. Non-optimal process settings not only +lead to defects in injection moulded parts such as warpage, shrinkage and residual stresses, but also cause long +cycle time and low process efficiency (Kim et al., 2009; Xu et al., 2015; Zhang & Jiang, 2007). + +The first step for improving quality and enhancing efficiency is to identify the most significant process parameters +influencing the quality factors. For this purpose, various Design of Experiment (DOE) methods have been +developed. One of the developed DOE methods for prediction, optimization, and selection of the key variables +is the Taguchi method. The main advantage of this method is designing the experiments based on an orthogonal +array with a minimum number of experiments which saves time and cost (Van Nostrand, 2002). This method has +been used in injection moulding for optimization of the process in various studies. Ozcelic and Erzurumlu +(Ozcelik & Erzurumlu, 2006) investigated the effect of seven factors on the warpage of thin shell plastic +components using the Taguchi method and specified the key parameters influencing the warpage. Zhang et al. +(Zhang & Jiang, 2007) first used a fractional factorial design to identify the main factors on the part quality and +then used Taguchi method to optimize these process factors. Altan (Altan, 2010) investigated the impact of +different process parameters on the shrinkage of polypropylene (PP) and polystyrene (PS) injection moulded +parts using Taguchi method and ANOVA. They concluded that the most significant factor in the shrinkage is +packing pressure for PP and melt temperature for the PS. Then a neural network based method was applied to +predict shrinkage for these two parts based on the optimal process levels from the Taguchi result. Jan et al. (Jan +et al., 2016) applied Taguchi method and response surface method to predict sink marks in the injection moulding +process. Moayyedian et al. (Moayyedian et al., 2018) used a combination of Taguchi method and fuzzy logic to + +Kariminejad • Tormey • Huq • Morrison • Redmond • Souto • McAfee + +optimize three key defects: shrinkage, warpage and short shot, in injection moulding. Hentati et al. (Hentati et +al., 2019) studied the effect of four process parameters on the shear stress in PC/ABS blended part and validated +the result by simulation in SOLIDWORKS software. + +The optimization of the cycle time and shrinkage of a commercially moulded component from an industrial +partner, AbbVie, is studied and presented in this paper. For this purpose, the effect of four input factors, melt +temperature, mould temperature, injection pressure, and holding time, has been studied with respect to two critical +outputs: cycle time and shrinkage for this product. +2. METHODOLOGY +2.1 Part description +In this study, we investigate a component which we refer to as a ‘clip’. The isometric view of the clip is illustrated +in Figure 1. The initial process setting for optimization has been provided by AbbVie Ballytivnan, Sligo. The +material of the Clip component is Delrin 500P NC010 and the dimension is 32.36×26.33×11.9 mm. + +Figure 1. Isometric view of the Clip injection moulded component +2.2. Simulation +Autodesk Moldflow Insight 2019 software has been used to simulate the injection moulding process and validate +the data from DOE for the Clip. The simulated part with the designed cooling channels and two cavities and two +injection locations has been shown in Figure 2. (a). The conventional cooling channels (blue channels) with two +baffles (yellow channels) at the middle of cooling circuits have been indicated in Figure 2. (a). The baffle is a +type of cooling channel with a blade at the centre, placed at the hot spots, which causes an increase in the +turbulency and heat transfer, thus a reduction in the cooling time. Figure 2. (b) shows the simulated component +with immobile and mobile moulds and ejector pin spots. For the finite element analysis, the Dual-domain mesh +(fusion) has been selected because of the part geometry and the mesh tool has been applied to eliminate the mesh +defects. + + + +Figure 2. (a) Simulated Clip part with the designed cooling channels. (b) The Clip with mould and cavity. + +In this study, for the initial optimization of the process and saving cost and time, instead of running the designed +experiments from Taguchi in the real process, each experiment has been run in the simulation. For examining the +(a) +(b) + +Cooling Channels +BaffleImmobileMould +MobileMouldKariminejad • Tormey • Huq • Morrison • Redmond • Souto • McAfee + +trustworthiness of the simulation, the result of a specific injection moulding process setting has been compared +to the simulation in Moldflow. The result of this comparison has been summarized in Table 1. Figure 3 indicates +the result from the simulation for the filling time. The relatively small error percentage between the simulation +and the actual process demonstrates that the simulation can be used for initial optimization instead of the real +experiment. + +Table 1. Comparison of the real process and simulation +Parameters +Real Process +Moldflow Simulation +Error % +Cycle time (s) +40 - 46 +43.02 +6.9 +Filling time (s) +0.355 +0.37 +4.05 +Cooling time (s) +30 +28.03 +7.02 + + +Figure 3. The result of filling time from Moldflow simulation +2.3. Taguchi method +Taguchi method is a type of Design of Experiments method that can be used not only for the screening of +variables, but also for optimization. This method is a combination of fractional factorial design and orthogonal +array. The orthogonal experimental setting in this method refers to an equivalent number of all levels for each +variable in the designed experiments, ensuring the balance of the array (Butler, 1992; Kr Dwiwedi et al., 2015; +Van Nostrand, 2002). + +This method has been used in this study to investigate the effect of injection moulding process parameters on the +part shrinkage and cycle time. Each of the input factors has three levels based on the primary process setting from +the industrial partner. Minitab 19 software has been used to find the optimal process parameters via the Taguchi +method. The detailed description of the input parameters has been summarized in Table 2. + +Table 2. Input process Parameters details +Input Parameters +Level 1 +Level 2 +Level 3 +Mould temperature (°C) +75 +80 +85 +Melt temperature (°C) +215 +220 +230 +Injection pressure (bar) +470 +530 +580 +Holding time (s) +3.5 +4.5 +5.5 + +The L9 orthogonal array has been used based on the Taguchi method shown in Table 3. The optimal output (𝑅𝑜𝑝𝑡) +can be calculated from equation (1) for four input variables (A, B, C, and D). 𝑅̅ is the average of all outputs from +nine experiments and 𝐴̅𝑥, 𝐵̅𝑥, 𝐶̅𝑥 𝑎𝑛𝑑 𝐷̅𝑥 are the average of the desired output at the optimum level of x. As it is +clear from Table 3, the number of experiments for four input variables and three-levels is just nine with the +Taguchi method, while for the full factorial design, this number would increase to 34 = 81. + +𝑅𝑜𝑝𝑡 = 𝑅̅ + (𝐴̅𝑥 − 𝑅̅) + (𝐵̅𝑥 − 𝑅̅) + (𝐶̅𝑥 − 𝑅̅) + (𝐷̅𝑥 − 𝑅̅) +(1-a) + +Filltime += 0.3744[s] +[s] +0.3744 +0.2808 +0.1872 +0.0936 +0.0000 +AUTODESK +MOLDFLOWINSIGHT +27 +scale(1uumm)Kariminejad • Tormey • Huq • Morrison • Redmond • Souto • McAfee + +𝑅̅ = 𝑅1 + 𝑅2 + 𝑅3 + ⋯ + 𝑅9 +9 + +(1-b) + +Table 3. L9 orthogonal array Taguchi method +No. +Mould Temperature(°C) +Melt Temperature (°C) +Injection Pressure (bar) +Holding time (s) +1 +75 +215 +470 +3.5 +2 +75 +220 +530 +4.5 +3 +75 +230 +580 +5.5 +4 +80 +215 +530 +5.5 +5 +80 +220 +580 +3.5 +6 +80 +230 +470 +4.5 +7 +85 +215 +580 +4.5 +8 +85 +220 +470 +5.5 +9 +85 +230 +530 +3.5 + +The signal-to-noise ratio is a quality indicator to evaluate the variation of a specific variable on the final output +(Ross PJ., 1996). In the injection moulding process, the aim is to minimize the cycle time and shrinkage as much +as possible. Hence, in this study, the Taguchi signal-to-noise ratio 𝑆/𝑁 should be defined as ‘’the-smaller- the- +better’’ described in Equation 2. ‘n’ is the number of experiments (here 9), and ‘𝑦𝑖’ is the response value for the +ith experiment. + +𝑆/𝑁 = −10𝑙𝑜𝑔10( +∑ +𝑦𝑖2 +𝑛 +𝑖=1 +𝑛 +) +(2) +3. RESULTS AND DISSCUSSION +The designed experiments based on Table 3 have been simulated in the Moldflow software and the result for +cycle time and shrinkage and the related signal-to-noise ratio have been summarized in Table 4. + +The cycle time in this simulation is made up of the filling time, packing time, cooling time, and mould open time. +For the shrinkage simulation, first, the critical dimensions and the related tolerances provided by AbbVie are +defined. The shrinkage has been examined based on the average linear shrinkage, that is, the equally-weighted +mean of parallel and perpendicular shrinkage. The nominal parallel and perpendicular shrinkage is 1.934% and +2.082% for Delrin 500P NC010, respectively. The shrinkage result should be below these nominal values to +prevent excessive shrinkage in part. + +Table 4. Simulation result for L9 orthogonal array +No. +Mould +Temperature(°C) +Melt +Temperature +(°C) +Injection +Pressure +(MPa) +Holding +time (s) +Cycle +time (s) +Shrinkage +(%) +S/N +Cycle +time +S/N +shrinkage +1 +75 +215 +47 +3.5 +49.4161 +2.2 +-33.87 +-6.84 +2 +75 +220 +53 +4.5 +51.0519 +2.183 +-34.16 +-6.78 +3 +75 +230 +58 +5.5 +54.4495 +2.571 +-34.7 +-8.2 +4 +80 +215 +53 +5.5 +29.3798 +1.992 +-29.36 +-5.98 +5 +80 +220 +58 +3.5 +30.4038 +2.093 +-29.65 +-6.41 +6 +80 +230 +47 +4.5 +32.3585 +2.062 +-30.1 +-6.28 +7 +85 +215 +58 +4.5 +22.925 +1.972 +-27.2 +-5.89 +8 +85 +220 +47 +5.5 +23.4541 +1.961 +-27.4 +-5.84 +9 +85 +230 +53 +3.5 +24.4298 +2.144 +-27.75 +-6.62 + 3.1 Screening of input parameters +The Taguchi method is able to assess the most effective level and the importance rate of each input variable on +the desired output. The result of average values for cycle time and shrinkage has been summarized in Figure 4. + +Kariminejad • Tormey • Huq • Morrison • Redmond • Souto • McAfee + + +Regarding Figure 4. (a), the most significant factor on cycle time is mould temperature (Tmold). The minimum +value of cycle time will be obtained if the mould temperature is set to the highest level (85°C). Melt temperature +(Tmelt), holding time (tholding) and injection pressure (Pinj) also affect cycle time in that order of importance; +however, their influence is not considerable. + +Figure 4. (b) indicates mould temperature is also the leading variable affecting shrinkage, and to minimize the +shrinkage, the mould temperature should be set at the highest level of 85°C. The influence of melt temperature +is almost major and for the optimization of shrinkage, the minimum level of 215 °C should be adjusted. The +holding time and injection pressure have similar effects on the linear shrinkage. Injection pressure should be fixed +at the minimum level (47 MPa) and holding time should be set at the medium level, which is 4.5 s. The importance +of each input variable on the outputs has been presented in Table 5, where the input with the highest and lowest +impact has been defined by Rank ‘1’ and Rank ‘4’, respectively. + + + + +Figure 4. Average values plot for (a) cycle time, (b) Shrinkage at three levels + +Table 5. The effect of each input variables on the desired outputs +Desired Outputs +Mould +Temperature(°C) +Melt +Temperature(°C) +Injection Pressure (MPa) +Holding Time(s) +Cycle Time(s) +1 +2 +4 +3 +Shrinkage% +1 +2 +3 +4 + +3.2 Optimization of outputs with Taguchi method and simulation +The Taguchi method estimates the optimum output based on the optimal setting from screening in section 3.1 by +Equation 1. For validation of the predicted values from the Taguchi method, the predicted optimal settings were +simulated in Moldflow. As shown in Table 6, the difference between the prediction from the Taguchi method +and the Moldflow simulation is below 10% which validates that the Taguchi method can successfully predict +optimal settings. The shrinkage percentage is below the nominal value of the Delrin 500P NC010, which verifies +that under this process setting, excessive shrinkage will not occur in the part. Obviously the simulation should be +followed by optimisation of the settings in the actual process, however based on the Taguchi method (Table 6) +applied to the simulation environment, the initial mould temperature should be fixed at the highest level and the +initial melt temperature at the lowest level. Besides that with this optimal setting, the cyle time declined from +almost 40 s to 23s, improving the process efficiency + +Table 6. Comparison of the outputs from Taguchi method and Moldflow simulation +Output +Parameters +Mould +Temperature +(°C) +Melt +Temperature +(°C) +Injection +Pressure +(MPa) +Holding +Time (s) +Taguchi +Predicted +Value +Moldflow +Simulation +Value +Error +% +Cycle Time(s) +85 +215 +53 +3.5 +21.2575 +22.92 +7.27 +Shrinkage% +85 +215 +47 +4.5 +1.83 +1.98 +7.57 +(a) +(b) + +Main Effects Plot for Means +Data Means +Tmold (°C) +Tmelt (C) +Pini(MPa) +tholding (s) +55 +50 +Mean of Means +45 +40 +35 +30 +25 +20 - +75 +80 +85 +215 +220 +230 +47 +53 +58 +3.5 +4.5 +5.5Main Effects Plot for Means +Data Means +Tmold ('C) +Tmelt('C) +pini +(MPa) +tholding(s) +2.35 +2.30 +(%) +Mean of Means +2.25 +2.20 +2.15 +2.10 +2.05 +2.00 - +75 +80 +85 +215 +220 +230 +47 +53 +58 +3.5 +4.5 +5.5Kariminejad • Tormey • Huq • Morrison • Redmond • Souto • McAfee + +4. CONCLUSION +In this paper, Taguchi method and simulation are applied together to study the effect of melt temperature, mould +temperature, packing temperature and holding time on the shrinkage and cycle time of the commercial injection +moulded part. The experiments were initially simulated in the Moldflow software instead of the actual process to +save time and cost. + +The most significant factor on both shrinkage and cycle time is mould temperature. The result indicated that 85°C +of mould temperature, 215°C of melt temperature, 53 Mpa of injection pressure, and 3.5 s of holding time +minimize the cycle time to almost 23 s, much less than the current cycle time of the part in the process which is +about 40 s. The simulation obtained a minimum shrinkage of 1.98% with a mould temperature of 85°C, melt +temperature of 215°C, injection pressure of 47 Mpa, and 4.5 s of holding time (See Table 6). This value is lower +than the nominal shrinkage of the material (nominal parallel and perpendicular shrinkage are 1.934% and +2.082%). Based on this study, the mould temperature should be set at the highest level and melt temperature at +the lowest level to optimize shrinkage and cycle time. Changing the injection pressure and holding time is not +significant on the cycle time, so they should be fixed at the minimum and middle levels for minimum shrinkage, +respectively. + +Further research to improve the optimization results includes validation of the simulation data by running the L9 +in the real injection moulding process, increasing the number of experiments from L9 to L27 to investigate the +interactions between the factors and study other input variables such as ejection temperature, flow rate, coolant +temperature, gate type and cooling channels on the shrinkage and cycle time. +5. REFERENCES +Altan, M. (2010). Reducing shrinkage in injection moldings via the Taguchi, ANOVA and neural network methods. +Materials & Design, 31(1), 599–604. https://doi.org/10.1016/j.matdes.2009.06.049 +Butler, C. (1992). A primer on the Taguchi method. Computer Integrated Manufacturing Systems, 5(3), 246. +https://doi.org/10.1016/0951-5240(92)90037-D +Hentati, F., Hadriche, I., Masmoudi, N., & Bradai, C. (2019). Optimization of the injection molding process for the +PC/ABS parts by integrating Taguchi approach and CAE simulation. International Journal of Advanced +Manufacturing Technology, 104(9–12), 4353–4363. https://doi.org/10.1007/s00170-019-04283-z +Jan, M., Khalid, M. S., Awan, A. A., & Nisar, S. (2016). Optimization of injection molding process for sink marks +reduction by integrating response surface design methodology &taguchi approach. Journal of Quality and +Technology Management Volume XII, Issue I, XII(I), 45–79. Retrieved from +http://pu.edu.pk/images/journal/iqtm/PDF-FILES/02-Optimization_jun_16.pdf +Kazmer, D. O. (2007). Injection Mold Design Engineering. In Injection Mold Design Engineering (pp. I–XX). München: +Carl Hanser Verlag GmbH & Co. KG. https://doi.org/10.3139/9783446434196.fm +Kim, S. Y., Kim, C. H., Kim, S. H., Oh, H. J., & Youn, J. R. (2009). Measurement of residual stresses in film insert +molded parts with complex geometry. Polymer Testing, 28(5), 500–507. +https://doi.org/10.1016/j.polymertesting.2009.03.009 +Kr Dwiwedi, A., Kumar, S., Noor Rahbar, N., & Kumar, D. (2015). Practical Application of Taguchi Method for +Optimization of Process Parameters in Injection Molding Machine for PP Material. International Research Journal +of Engineering and Technology (IRJET), 2(4), 264–268. +Moayyedian, M., Abhary, K., & Marian, R. (2018). Optimization of injection molding process based on fuzzy quality +evaluation and Taguchi experimental design. CIRP Journal of Manufacturing Science and Technology, 21, 150–160. +https://doi.org/10.1016/j.cirpj.2017.12.001 +Ozcelik, B., & Erzurumlu, T. (2006). Comparison of the warpage optimization in the plastic injection molding using +ANOVA, neural network model and genetic algorithm. Journal of Materials Processing Technology, 171(3), 437– +445. https://doi.org/10.1016/j.jmatprotec.2005.04.120 +Ross PJ. (1996). Taguchi techniques for quality engineering. McGraw Hill. Retrieved from +https://books.google.ie/books/about/Taguchi_Techniques_for_Quality_Engineeri.html?id=CiunygZ90TsC&redir_es +c=y +Van Nostrand, R. C. (2002). Design of Experiments Using the Taguchi Approach: 16 Steps to Product and Process +Improvement. Technometrics, 44(3), 289–289. https://doi.org/10.1198/004017002320256440 +Xu, Y., Zhang, Q. W., Zhang, W., & Zhang, P. (2015). Optimization of injection molding process parameters to improve +the mechanical performance of polymer product against impact. International Journal of Advanced Manufacturing +Technology, 76(9–12), 2199–2208. https://doi.org/10.1007/s00170-014-6434-y + +Kariminejad • Tormey • Huq • Morrison • Redmond • Souto • McAfee + +Zhang, Z., & Jiang, B. (2007). Optimal process design of shrinkage and sink marks in injection molding. Journal of +Wuhan University of Technology-Mater. Sci. Ed., 22(3), 404–407. https://doi.org/10.1007/s11595-006-3404-8 + + diff --git a/HNFJT4oBgHgl3EQfuC2G/content/tmp_files/load_file.txt b/HNFJT4oBgHgl3EQfuC2G/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..67956a6c1617beee69d1496197269f4e07a3ad03 --- /dev/null +++ b/HNFJT4oBgHgl3EQfuC2G/content/tmp_files/load_file.txt @@ -0,0 +1,391 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf,len=390 +page_content='Email: Mandana Kariminejad Mandana.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='kariminejad@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='itsligo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='ie Optimization of a Commercial Injection-Moulded component by Using DOE and Simulation Mandana Kariminejad,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' Centre for Precision Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' Material and Manufacturing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' Institute of Technology Sligo David Tormey,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' Centre for Precision Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' Material and Manufacturing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' Institute of Technology Sligo Saif Huq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' School of Computing and Digital Media,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' London Metropolitan University Jim Morrison,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' Department of Electronics and Mechanical Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' Letterkenny Institute of Technology Jeff Redmond,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' Combination Products,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' AbbVie Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' Carlos Souto, Engineering Moulding, AbbVie Ballytivnan Marion McAfee, Centre for Precision Engineering, Material and Manufacturing, Institute of Technology Sligo Abstract Injection moulding is an important industry, providing a significant percentage of the demand for plastic products throughout the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' The process consists of many variables which directly or indirectly influence the part quality and cycle time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' The first step in optimizing the process parameters is identifying the most significant variables affecting the desired output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' For this purpose, various Design of Experiments methods (DOE) have been developed to investigate the effect of the experimental variables on the output and predict the required settings to achieve the optimal value of the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' In this study we investigate the application of DOE for a commercial injection moulded component which suffers from a long cycle time and high shrinkage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' The Taguchi method has been used to analyze the effect of four input variables on the two output variables: cycle time and shrinkage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' The component has been simulated in the Moldflow software to validate the predicted output and optimized settings of the variables from the DOE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' Comparison of the simulation result and the predicted value from the DOE illustrated good accordance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' The calculated optimal setting with the Taguchi method reduced the cycle time from the 40s to about 23s and met the shrinkage criteria for this commercial part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' Key Words: Injection Moulding, Design of Experiment, Taguchi Method, Moldflow Simulation, Cycle time 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' INTRODUCTION One of the most developed processes for the production of plastic components is injection moulding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' In general, this process contains three main steps: the filling stage in which melted polymer pellets are injected into the cavity, the packing stage which prevents excessive shrinkage by injection of extra polymer, and the cooling stage where the polymer solidifies and gets ready for ejection (Kazmer, 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' During these stages, many process parameters such as mould temperature, melt temperature, and injection pressure should be controlled and adjusted, directly affecting the part quality and efficiency of the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' Non-optimal process settings not only lead to defects in injection moulded parts such as warpage, shrinkage and residual stresses, but also cause long cycle time and low process efficiency (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=', 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' Zhang & Jiang, 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' The first step for improving quality and enhancing efficiency is to identify the most significant process parameters influencing the quality factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' For this purpose, various Design of Experiment (DOE) methods have been developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' One of the developed DOE methods for prediction, optimization, and selection of the key variables is the Taguchi method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' The main advantage of this method is designing the experiments based on an orthogonal array with a minimum number of experiments which saves time and cost (Van Nostrand, 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' This method has been used in injection moulding for optimization of the process in various studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' Ozcelic and Erzurumlu (Ozcelik & Erzurumlu, 2006) investigated the effect of seven factors on the warpage of thin shell plastic components using the Taguchi method and specified the key parameters influencing the warpage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' (Zhang & Jiang, 2007) first used a fractional factorial design to identify the main factors on the part quality and then used Taguchi method to optimize these process factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' Altan (Altan, 2010) investigated the impact of different process parameters on the shrinkage of polypropylene (PP) and polystyrene (PS) injection moulded parts using Taguchi method and ANOVA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' They concluded that the most significant factor in the shrinkage is packing pressure for PP and melt temperature for the PS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' Then a neural network based method was applied to predict shrinkage for these two parts based on the optimal process levels from the Taguchi result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' Jan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' (Jan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=', 2016) applied Taguchi method and response surface method to predict sink marks in the injection moulding process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' Moayyedian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' (Moayyedian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=', 2018) used a combination of Taguchi method and fuzzy logic to Kariminejad • Tormey • Huq • Morrison • Redmond • Souto • McAfee optimize three key defects: shrinkage, warpage and short shot, in injection moulding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' Hentati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' (Hentati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=', 2019) studied the effect of four process parameters on the shear stress in PC/ABS blended part and validated the result by simulation in SOLIDWORKS software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' The optimization of the cycle time and shrinkage of a commercially moulded component from an industrial partner, AbbVie, is studied and presented in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' For this purpose, the effect of four input factors, melt temperature, mould temperature, injection pressure, and holding time, has been studied with respect to two critical outputs: cycle time and shrinkage for this product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' METHODOLOGY 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='1 Part description In this study, we investigate a component which we refer to as a ‘clip’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' The isometric view of the clip is illustrated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' The initial process setting for optimization has been provided by AbbVie Ballytivnan, Sligo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' The material of the Clip component is Delrin 500P NC010 and the dimension is 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='36×26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='33×11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='9 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' Isometric view of the Clip injection moulded component 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' Simulation Autodesk Moldflow Insight 2019 software has been used to simulate the injection moulding process and validate the data from DOE for the Clip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' The simulated part with the designed cooling channels and two cavities and two injection locations has been shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' The conventional cooling channels (blue channels) with two baffles (yellow channels) at the middle of cooling circuits have been indicated in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' The baffle is a type of cooling channel with a blade at the centre, placed at the hot spots, which causes an increase in the turbulency and heat transfer, thus a reduction in the cooling time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' (b) shows the simulated component with immobile and mobile moulds and ejector pin spots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' For the finite element analysis, the Dual-domain mesh (fusion) has been selected because of the part geometry and the mesh tool has been applied to eliminate the mesh defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' (a) Simulated Clip part with the designed cooling channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' (b) The Clip with mould and cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' In this study, for the initial optimization of the process and saving cost and time, instead of running the designed experiments from Taguchi in the real process, each experiment has been run in the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' For examining the (a) (b) Cooling Channels BaffleImmobileMould MobileMouldKariminejad • Tormey • Huq • Morrison • Redmond • Souto • McAfee trustworthiness of the simulation, the result of a specific injection moulding process setting has been compared to the simulation in Moldflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' The result of this comparison has been summarized in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' Figure 3 indicates the result from the simulation for the filling time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' The relatively small error percentage between the simulation and the actual process demonstrates that the simulation can be used for initial optimization instead of the real experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' Comparison of the real process and simulation Parameters Real Process Moldflow Simulation Error % Cycle time (s) 40 - 46 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='02 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='9 Filling time (s) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='355 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='37 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='05 Cooling time (s) 30 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='03 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='02 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' The result of filling time from Moldflow simulation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' Taguchi method Taguchi method is a type of Design of Experiments method that can be used not only for the screening of variables, but also for optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' This method is a combination of fractional factorial design and orthogonal array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' The orthogonal experimental setting in this method refers to an equivalent number of all levels for each variable in the designed experiments, ensuring the balance of the array (Butler, 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' Kr Dwiwedi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' Van Nostrand, 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' This method has been used in this study to investigate the effect of injection moulding process parameters on the part shrinkage and cycle time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' Each of the input factors has three levels based on the primary process setting from the industrial partner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' Minitab 19 software has been used to find the optimal process parameters via the Taguchi method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' The detailed description of the input parameters has been summarized in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' Input process Parameters details Input Parameters Level 1 Level 2 Level 3 Mould temperature (°C) 75 80 85 Melt temperature (°C) 215 220 230 Injection pressure (bar) 470 530 580 Holding time (s) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='5 The L9 orthogonal array has been used based on the Taguchi method shown in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' The optimal output (𝑅𝑜𝑝𝑡) can be calculated from equation (1) for four input variables (A, B, C, and D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' 𝑅̅ is the average of all outputs from nine experiments and 𝐴̅𝑥, 𝐵̅𝑥, 𝐶̅𝑥 𝑎𝑛𝑑 𝐷̅𝑥 are the average of the desired output at the optimum level of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' As it is clear from Table 3, the number of experiments for four input variables and three-levels is just nine with the Taguchi method, while for the full factorial design, this number would increase to 34 = 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' 𝑅𝑜𝑝𝑡 = 𝑅̅ + (𝐴̅𝑥 − 𝑅̅) + (𝐵̅𝑥 − 𝑅̅) + (𝐶̅𝑥 − 𝑅̅) + (𝐷̅𝑥 − 𝑅̅) (1-a) Filltime = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='3744[s] [s] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='3744 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='2808 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='1872 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='0936 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='0000 AUTODESK MOLDFLOWINSIGHT 27 scale(1uumm)Kariminejad • Tormey • Huq • Morrison • Redmond • Souto • McAfee 𝑅̅ = 𝑅1 + 𝑅2 + 𝑅3 + ⋯ + 𝑅9 9 (1-b) Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' L9 orthogonal array Taguchi method No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' Mould Temperature(°C) Melt Temperature (°C) Injection Pressure (bar) Holding time (s) 1 75 215 470 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='5 2 75 220 530 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='5 3 75 230 580 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='5 4 80 215 530 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='5 5 80 220 580 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='5 6 80 230 470 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='5 7 85 215 580 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='5 8 85 220 470 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='5 9 85 230 530 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='5 The signal-to-noise ratio is a quality indicator to evaluate the variation of a specific variable on the final output (Ross PJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=', 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' In the injection moulding process, the aim is to minimize the cycle time and shrinkage as much as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' Hence, in this study, the Taguchi signal-to-noise ratio 𝑆/𝑁 should be defined as ‘’the-smaller- the- better’’ described in Equation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' ‘n’ is the number of experiments (here 9), and ‘𝑦𝑖’ is the response value for the ith experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' 𝑆/𝑁 = −10𝑙𝑜𝑔10( ∑ 𝑦𝑖2 𝑛 𝑖=1 𝑛 ) (2) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' RESULTS AND DISSCUSSION The designed experiments based on Table 3 have been simulated in the Moldflow software and the result for cycle time and shrinkage and the related signal-to-noise ratio have been summarized in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' The cycle time in this simulation is made up of the filling time, packing time, cooling time, and mould open time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' For the shrinkage simulation, first, the critical dimensions and the related tolerances provided by AbbVie are defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' The shrinkage has been examined based on the average linear shrinkage, that is, the equally-weighted mean of parallel and perpendicular shrinkage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' The nominal parallel and perpendicular shrinkage is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='934% and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='082% for Delrin 500P NC010, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' The shrinkage result should be below these nominal values to prevent excessive shrinkage in part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' Simulation result for L9 orthogonal array No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' Mould Temperature(°C) Melt Temperature (°C) Injection Pressure (MPa) Holding time (s) Cycle time (s) Shrinkage (%) S/N Cycle time S/N shrinkage 1 75 215 47 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='5 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='4161 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='2 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='87 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='84 2 75 220 53 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='5 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='0519 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='183 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='16 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='78 3 75 230 58 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='5 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='4495 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='571 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='7 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='2 4 80 215 53 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='5 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='3798 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='992 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='36 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='98 5 80 220 58 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='5 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='4038 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='093 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='65 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='41 6 80 230 47 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='5 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='3585 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='062 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='28 7 85 215 58 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='5 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='925 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='972 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='89 8 85 220 47 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='5 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='4541 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='961 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='84 9 85 230 53 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='5 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='4298 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='144 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='75 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='62 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='1 Screening of input parameters The Taguchi method is able to assess the most effective level and the importance rate of each input variable on the desired output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' The result of average values for cycle time and shrinkage has been summarized in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' Kariminejad • Tormey • Huq • Morrison • Redmond • Souto • McAfee Regarding Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' (a), the most significant factor on cycle time is mould temperature (Tmold).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' The minimum value of cycle time will be obtained if the mould temperature is set to the highest level (85°C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' Melt temperature (Tmelt), holding time (tholding) and injection pressure (Pinj) also affect cycle time in that order of importance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' however, their influence is not considerable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' (b) indicates mould temperature is also the leading variable affecting shrinkage, and to minimize the shrinkage, the mould temperature should be set at the highest level of 85°C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' The influence of melt temperature is almost major and for the optimization of shrinkage, the minimum level of 215 °C should be adjusted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' The holding time and injection pressure have similar effects on the linear shrinkage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' Injection pressure should be fixed at the minimum level (47 MPa) and holding time should be set at the medium level, which is 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='5 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' The importance of each input variable on the outputs has been presented in Table 5, where the input with the highest and lowest impact has been defined by Rank ‘1’ and Rank ‘4’, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' Average values plot for (a) cycle time, (b) Shrinkage at three levels Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' The effect of each input variables on the desired outputs Desired Outputs Mould Temperature(°C) Melt Temperature(°C) Injection Pressure (MPa) Holding Time(s) Cycle Time(s) 1 2 4 3 Shrinkage% 1 2 3 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='2 Optimization of outputs with Taguchi method and simulation The Taguchi method estimates the optimum output based on the optimal setting from screening in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='1 by Equation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' For validation of the predicted values from the Taguchi method, the predicted optimal settings were simulated in Moldflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' As shown in Table 6, the difference between the prediction from the Taguchi method and the Moldflow simulation is below 10% which validates that the Taguchi method can successfully predict optimal settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' The shrinkage percentage is below the nominal value of the Delrin 500P NC010, which verifies that under this process setting, excessive shrinkage will not occur in the part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' Obviously the simulation should be followed by optimisation of the settings in the actual process, however based on the Taguchi method (Table 6) applied to the simulation environment, the initial mould temperature should be fixed at the highest level and the initial melt temperature at the lowest level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' Besides that with this optimal setting, the cyle time declined from almost 40 s to 23s, improving the process efficiency Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' Comparison of the outputs from Taguchi method and Moldflow simulation Output Parameters Mould Temperature (°C) Melt Temperature (°C) Injection Pressure (MPa) Holding Time (s) Taguchi Predicted Value Moldflow Simulation Value Error % Cycle Time(s) 85 215 53 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='5 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='2575 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='92 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='27 Shrinkage% 85 215 47 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='83 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='98 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='57 (a) (b) Main Effects Plot for Means Data Means Tmold (°C) Tmelt (C) Pini(MPa) tholding (s) 55 50 Mean of Means 45 40 35 30 25 20 - 75 80 85 215 220 230 47 53 58 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content="5Main Effects Plot for Means Data Means Tmold ('C) Tmelt('C) pini (MPa) tholding(s) 2." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='35 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='30 (%) Mean of Means 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='20 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='15 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='00 - 75 80 85 215 220 230 47 53 58 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='5Kariminejad • Tormey • Huq • Morrison • Redmond • Souto • McAfee 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' CONCLUSION In this paper, Taguchi method and simulation are applied together to study the effect of melt temperature, mould temperature, packing temperature and holding time on the shrinkage and cycle time of the commercial injection moulded part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' The experiments were initially simulated in the Moldflow software instead of the actual process to save time and cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' The most significant factor on both shrinkage and cycle time is mould temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' The result indicated that 85°C of mould temperature, 215°C of melt temperature, 53 Mpa of injection pressure, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='5 s of holding time minimize the cycle time to almost 23 s, much less than the current cycle time of the part in the process which is about 40 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' The simulation obtained a minimum shrinkage of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='98% with a mould temperature of 85°C, melt temperature of 215°C, injection pressure of 47 Mpa, and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='5 s of holding time (See Table 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' This value is lower than the nominal shrinkage of the material (nominal parallel and perpendicular shrinkage are 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='934% and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='082%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' Based on this study, the mould temperature should be set at the highest level and melt temperature at the lowest level to optimize shrinkage and cycle time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' Changing the injection pressure and holding time is not significant on the cycle time, so they should be fixed at the minimum and middle levels for minimum shrinkage, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' Further research to improve the optimization results includes validation of the simulation data by running the L9 in the 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' Design of Experiments Using the Taguchi Approach: 16 Steps to Product and Process Improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' Technometrics, 44(3), 289–289.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='1198/004017002320256440 Xu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=', Zhang, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=', Zhang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=', & Zhang, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' Optimization of injection molding process parameters to improve the mechanical performance of polymer product against impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' International Journal of Advanced Manufacturing Technology, 76(9–12), 2199–2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='1007/s00170-014-6434-y Kariminejad • Tormey • Huq • Morrison • Redmond • Souto • McAfee Zhang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=', & Jiang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' Optimal process design of shrinkage and sink marks in injection molding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' Journal of Wuhan University of Technology-Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' Ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=', 22(3), 404–407.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} +page_content='1007/s11595-006-3404-8' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'} diff --git a/HdFIT4oBgHgl3EQfXith/vector_store/index.faiss b/HdFIT4oBgHgl3EQfXith/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..ac7ce068c1ea3e9ba4ff3ffa7d01f97d5aad27d5 --- /dev/null +++ b/HdFIT4oBgHgl3EQfXith/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:40580fc9f29a4a679c9cea6b78df65a7341980114297760364b4650c7d4dd12a +size 5177389 diff --git a/I9AyT4oBgHgl3EQfsPkA/content/tmp_files/load_file.txt b/I9AyT4oBgHgl3EQfsPkA/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..995b36eda3edbb6eb9a5a803b542299e4e2c73f4 --- /dev/null +++ b/I9AyT4oBgHgl3EQfsPkA/content/tmp_files/load_file.txt @@ -0,0 +1,723 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf,len=722 +page_content='January 3, 2023 1:32 Konika˙2 Modern Physics Letters A © World Scientific Publishing Company Modified Hawking temperature and entropy of Kerr-de Sitter black hole in Lorentz violation theory Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' Onika Laxmi Mathematics Department, Manipur University Canchipur, Manipur, India T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' Ibungochouba Singh Mathematics Department, Manipur University Canchipur, Manipur, India ibungochouba@rediffmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content='com Received (Day Month Year) Revised (Day Month Year) In this paper, we discuss the tunneling of scalar particles near the event horizon of stationary and nonstationary Kerr-de Sitter black hole using Lorentz violation theory in curved space time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' The modified form of Hamilton-Jacobi equation is derived from the Klein-Gordon equation by applying Lorentz violation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' The Hawking temperatures derived from stationary and nonstationary Kerr-de Sitter black holes are modified due to Lorentz violation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' It is noted that the change in Bekenstein-Hawking entropy and modified Hawking temperatures of stationary and nonstationary Kerr-de Sitter black hole not only depend on the black hole parameters but also on ether like vectors uα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' Keywords: Hawking radiation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' Bekenstein-Hawking entropy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' Specific Heat;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' Lorentz vio- lation theory PACS Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' : 04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content='70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content='Dy, 04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content='Gz, 03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content='65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content='-w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' Introduction Hawking1,2 showed theoretically that a black hole radiates like a black body in which the temperature of radiation is dependent on the surface gravity of black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' The discovery of Hawking radiation leads to black hole thermodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content='3−5 Since then, many scientists have proposed different techniques to study the quan- tum tunneling of black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' [6,7] proposed a new method to investigate the quantum thermal and nonthermal radiations of stationary and nonstationary black holes by applying the tortoise coordinate transformation, Maxwell’s electromagnetic field equation, Klein-Gordon equation and Dirac equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' Applying this method, many interesting results in different black holes have been derived in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content='8−11 Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' [12-14] introduced the semiclassical tunneling technique to investigate the Hawking radiation of black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' In this method, the Hawking radiation is taken as a tun- 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content='00571v1 [gr-qc] 2 Jan 2023 January 3, 2023 1:32 Konika˙2 2 Authors’ names neling process near the event horizon of the black hole and the outgoing particle produces the tunneling barrier of black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' They obtain well-behaved coordinate system which has no singularity near the event horizon to derive the emission rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' [15-17], as an extension of Parikh and Wilczek approach, studied the Hawking radiation as a tunneling of charged massive particle at the event horizon of black hole by developing the relation between phase and group velocity of the tunneling particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' Applying the Hamilton-Jacobi equation, Feynman prescription and WKB approximation, Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' [18] investigated the Hawking radiation at the event horizon of rotating and nonrotating black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' They showed that the isotropic coordinate or invariant coordinate gives the correct Hawking temperature whereas naive coor- dinate leads to half of correct Hawking temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' [19] studied the Hawking radiation at the event horizon of different black holes by using Dirac equation, Feynman prescription and Pauli Sigma matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' In this method, the appropriate gamma matrices obtained from black hole and suitable wave function are substi- tuted in Dirac equations, then the action related to Boltzman factor for emission in accordance with semiclassical approximation is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' [20-23] discussed the corrected Hawking temperature and entropy of black holes using first law of black hole thermodynamics and Hamilton-Jacobi equation beyond the semiclassical approximation in Schwarzschild like coordinate system and Painleve coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' Kruglov24,25 proposed the quantum tunneling of boson near the event horizon of black hole using Proca equation, WKB approximation and Feynman prescription.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' The emission temperature of Schwarzschild background geometry which is the same as the Hawking temperature corresponding to scalar particle is also obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' Following their work, many interesting results have been obtained in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content='26−29 The study of Lorentz symmetry violation theory has been discussed during the past decades and many researchers have proposed different gravity models induced by Lorentz symmetry violation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content='30−34 Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' [35-37] proposed the Lorentz symmetry violation in flat space time using Dirac equation and ether like vectors uα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' Since then, the Lorentz violation has been extended to curved space time by choosing ether like vectors uα so that it could hold uαuα = constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content='38−42 The paper is outlined as follows : In section 2, the modified Hawking tempera- ture, heat capacity and change in entropy near the event horizon of Kerr-de Sitter black hole are derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' In section 3, the modified surface gravity and the Hawking temperature of nonstationary KdS black hole are discussed using tortoise coordinate transformation in Lorentz violation theory in curved space time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' Some discussions and conclusions are given in section 4 and 5 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' Modified Hawking temperature of Kerr-de Sitter black hole The Kerr-de Sitter (KdS) solution indicates the space time geometry of a rotating black hole with non-zero cosmological constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' The vacuum solution of KdS black hole is well known Boyer-Lindquist coordinates (t, r, θ, φ) with geometrical unit January 3, 2023 1:32 Konika˙2 3 (c = G = 1) which is given by43 ds2 = −∆ − ∆θa2 sin2 θ R2Ξ2 dt2 + R2 ∆θ dθ2 − 2a[∆θ(r2 + a2) − ∆] sin2 θ R2Ξ2 dtdφ +R2 ∆ dr2 + ∆θ(r2 + a2)2 − ∆a2 sin2 θ R2Ξ2 sin2 θdφ2, (1) where R2 = r2 + a2 cos2 θ, Ξ = 1 + 1 3Λa2, ∆θ = 1 + 1 3Λa2 cos2 θ, ∆ = (r2 + a2) � 1 − 1 3Λr2� − 2Mr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (2) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (1) represents rotating KdS black hole for cosmological constant Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' The KdS black hole is well defined in the region −∞ ≤ t ≤ ∞, 0 ≤ θ ≤ π and 0 ≤ φ ≤ 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' M and a are the mass and rotational parameter of KdS black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' If 1 Λ ≥ M 2 > a2, then ∆ = 0 gives four distinct roots i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' r+, rh, r− and r−− (r+ > rh > r− > r−−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' The biggest root r+ denotes the location of cosmological horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' rh and r− represent the location of the event horizon and Cauchy horizon respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' If r = 0, θ = π 2 , then the other side of r = 0, r = r−− is taken as another cosmological horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content='44 According to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' [45], the event horizon of KdS black hole r = rh can be written as rh = 1 Ξ � 1 + 4ΛM 2 3β2Ξ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' � � M + � M 2 − a2Ξ � , (3) where β = � 1 − Λ 3 a2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' There is a frame dragging effect near the event horizon of KdS black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' Let φ = ϕ − Ωt and Ω = − g03 g33 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' Then Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (1) reduces to ds2 = − ∆∆θR2 Ξ2[∆θ(r2 + a2)2 − ∆a2 sin2 θ]dt2 + R2 ∆ dr2 + R2 ∆θ dθ2 +[∆θ(r2 + a2)2 − ∆a2 sin2 θ] sin2 θ R2Ξ2 dφ2, (4) where the angular velocity at the event horizon of KdS black hole is given as Ω = a r2 h + a2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (5) According to Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' [46, 47], the surface gravity of KdS black hole at the event horizon r = rh is derived as κ = lim g00→0 � − 1 2 � −g11 g00 dg00 dr � = (rh − M − 2 3Λr3 h − 1 3Λa2rh) Ξ(r2 h + a2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (6) The Hawking temperature of KdS black hole is connected with surface gravity via TH = κ 2π as TH = 1 2π � (rh − M − 2 3Λr3 h − 1 3Λa2rh) Ξ(r2 h + a2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (7) January 3, 2023 1:32 Konika˙2 4 Authors’ names To discuss heat capacity near the event horizon of black hole, the mass parameter KdS black hole might be obtained from ∆(rh) = 0 as M = rh 2 + a2 2rh − Λr3 h 6 − Λa2rh 6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (8) The heat capacity (Ch) of black hole is defined by Ch = ∂M ∂TH = � ∂M ∂rh �� ∂rh ∂TH � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (9) The heat capacity near the event horizon of KdS black hole is calculated as Ch = ∂M ∂TH = 2πΞ(r2 h + a2)2[3r2 h − 3a2 − Λr2 h(3r2 h + a2)] 3(a4 − r4 h) + 4a2r2 h(3 − 2Λr2 h) − Λr2 h(a4 + 3r4 h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (10) The modified form of Hamilton-Jacobi equation in Lorentz theory in curved space time is given by48 (gµν + λuµuν)∂µI∂νI + m2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (11) where λ and uα are the correction parameter and ether like vectors respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' As λ tends to zero in the above equation, the Lorentz violation theory is cancelled and the original Hamilton-Jacobi equation in curved space time is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' The ether like vectors uα are constant in the flat space time of the canonical coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' The ether like vectors uα are not constant in curved space time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' But we can take the vectors uα from curved space time that satisfies the condition uαuα = constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' To investigate the change in entropy of stationary KdS black hole, we can choose uα from (4) that satisfies uαuα = constant and uα are related to coordinate system acquired by the space time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' The expressions of ut, ur, uθ and uφ are defined by ut = ct √−gtt = ctΞ � ∆θ(r2 + a2)2 − ∆a2 sin2 θ √∆∆θR2 , ur = cr √grr = cr � ∆ R2 , uθ = cθ √gθθ = cθ � ∆θ R2 , uφ = cφ √gφφ = cφΞ √ R2 � [∆θ(r2 + a2)2 − ∆a2 sin2 θ] sin2 θ , (12) where ct, cr, cθ and cφ are arbitrary constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' uα satisfies the condition uαuα = −c2 t + c2 r + c2 θ + c2 φ = constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (13) Using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (12) and (4) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (11), the dynamical equation of scalar particle with mass m in stationary KdS black hole is obtained as g00(∂I ∂t )2 + g11(∂I ∂r )2 + g22(∂I ∂θ )2 + g33( ∂I ∂φ)2 + λuµuν∂µI∂νI + m2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (14) January 3, 2023 1:32 Konika˙2 5 It is known that the above equation involves the variables t, r, θ and φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' To separate the variables on the Hamilton principal functions I, we can choose the action I as follows I = −ωt + S(r, θ) + jφ + δ, (15) where S(r, θ), ω and j are the generalized momentum, particle energy and angular momentum along the φ-axis respectively and δ is a complex constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (15) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (14), a quadratic equation in ∂S ∂r is obtained as A � ∂S ∂r �2 + B �∂S ∂r � + C = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (16) Then the two roots of the above equation are given by S = � −B ± √ B2 − 4AC 2A dr, (17) where A = g11 + λurur, B = 2λuruθ �∂I ∂θ � − 2λutur(ω − jΩ) + 2λuruφj, C = (g00 + λutut)(ω − jΩ)2 + (g22 + λuθuθ) �∂I ∂θ �2 + (g33 + λuφuφ)j2 −2λut(ω − jΩ) � uθ �∂I ∂θ � + uφj � + 2λuθuφj �∂I ∂θ � + m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (18) Applying residue theorem of complex analysis and Feynman prescription near the event horizon of KdS black hole, the integration of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (17) can be written as S± = iπΞ(r2 h + a2) � λctcr ± � 1 − λc2 t + λc2r � (w − jΩ) 2(1 + λc2r){(1 − Λ 3 a2)rh − M − 2Λr3 h 3 } , (19) where S+ and S− are the outgoing particle and ingoing particle respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' The probabilities which cross the black hole near the event horizon are given by Γemission = exp(−2ImI) = exp[−2(ImS+ + Imδ)] (20) and Γabsorption = exp(−2ImI) = exp[−2(ImS− + Imδ)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (21) There is a 100% chance the ingoing particle to enter the KdS black hole according to WKB approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' This indicates that ImS+ = −ImS−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' We calculate the probability of outgoing particle as Γrate = Γemission Γabsorption = exp � − 2γΞπ(r2 h + a2)(ω − jΩ) {(1 − Λ 3 a2)rh − M − 2Λr3 h 3 } � , (22) January 3, 2023 1:32 Konika˙2 6 Authors’ names where γ = √ 1−λc2 t +λc2r 1+λc2r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (22) is similar to Boltzmann factor according to semi- classical approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' The Hawking temperature near the event horizon of KdS black hole in Lorentz violation theory is given by T = {(1 − Λ 3 a2)rh − M − 2Λr3 h 3 } 2γπΞ(r2 h + a2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (23) If λ = 0 in the above equation, the Lorentz violation has been cancelled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' In such case Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (23) is consistent with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' If γ > 1 or γ < 1, the Hawking temperature decreases or increases due to the presence of correction term λ and ether like vectors uα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' The modified heat capacity at the event horizon of KdS black hole is obtained as C ′ h = ∂M ∂T = 2πγΞ(r2 h + a2)2[3r2 h − 3a2 − Λr2 h(3r2 h + a2)] 3(a4 − r4 h) + 4a2r2 h(3 − 2Λr2 h) − Λr2 h(a4 + 3r4 h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (24) As γ tends to unity, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (24) is consistent with original heat capacity given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' If γ > 1 or γ < 1, the heat capacity increases or decreases near the event horizon of KdS black hole depending upon the choices of correction term λ and ether like vectors uα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' Using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (3) and (5) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (19) for the outgoing particle, we get ImS = γ ′ 2 � πk2 1k2 2 Ξ β2 Ξ k1k2 − M − δ ω + πΞa2 β2 Ξ k1k2 − M − δ ω − πΞa β2 Ξ k1k2 − M − δ j � , (25) where γ ′ = λctcr + � 1 − λc2 t + λc2r 1 + λc2r , k1 = � 1 + 4ΛM 2 3β2Ξ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' � , k2 = � M + � M 2 − a2Ξ � , δ = 2Λ 3Ξ3 � 1 + 4ΛM 2 3β2Ξ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' �3 � M + � M 2 − a2Ξ �3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (26) To obtain the biggest value of the integration, we ignore second order terms of KdS black hole mass parameter at the numerator and denominator in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' Then we find as ImS = γ ′ 2 � � πk2 2 β2 � k2 − MΞ β2 �ω + πΞ2a2 β2 � k2 − MΞ β2 �ω − πΞ2a β2 � k2 − MΞ β2 �j � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (27) Taking the self-gravitational interaction into account, the mass parameter KdS black hole is allowed to fluctuate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' If a black hole emits a particle ω and angular momentum j, the KdS black hole parameter will be M − ω and J − j respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' To find the January 3, 2023 1:32 Konika˙2 7 change in Bekenstein-Hawking entropy of KdS black hole, the term (1 − Ξ β2 )M is ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' Then we get ImS = γ ′ 2 �� ω 0 πk2 2 β2√ M 2 − a2Ξ dω ′ + � ω 0 πΞ2a2 β2√ M 2 − a2Ξ dω ′ − � j 0 πΞ2a β2√ M 2 − a2Ξ dj ′� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (28) Changing M by M − ω and j by J − j and putting the value of k2 in the above equation, we obtain ImS = γ ′ 2 � −π β2 � M−ω M k2 2 � (M ′ − ω′)2 − a2Ξ d(M ′ − ω ′) −πΞ2a2 β2 � M−ω M 1 � (M ′ − ω′)2 − a2Ξ d(M ′ − ω ′) +πΞ2a β2 � J−j j 1 � (M ′ − ω′)2 − a2Ξ d(J − j′) � , (29) where J − j = (M − ω)a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' The imaginary part of the action finally yields ImS = −γ ′π 2β2 � � M−ω M 2(M ′ − ω′)2 + 2(M ′ − ω′) � (M ′ − ω′)2 − a2Ξ � (M ′ − ω′)2 − a2Ξ d(M ′ − ω ′) − � M−ω M a2Ξ � (M ′ − ω′)2 − a2Ξ d(M ′ − ω ′) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (30) Calculating the ω ′ integral, that gives ImS = −γ ′π 2β2 � (M − ω) � (M − ω)2 − a2Ξ + (M − ω)2 − M � M 2 − a2Ξ − M 2 � , = −γ ′π 4β2 �� (M − ω) + � (M − ω)2 − a2Ξ �2 − � M + � M 2 − a2Ξ �2� , = −γ ′π 2 (r2 f − r2 i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (31) Using WKB approximation, the tunneling rate is obtained as Γ ∼ exp(−2ImS), = exp[γ ′π(r2 f − r2 i )], = exp[γ ′∆SBH], (32) where γ′∆SBH = γ′(SBH(M − ω) − SBH(M)) is the modified entropy of KdS black hole in Lorentz violation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' ri = 1 √ 2β � M + √ M 2 − a2Ξ � and rf = 1 √ 2β � (M − ω) + � (M − ω)2 − a2Ξ � are the locations of horizons before and after emission of particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' If λ = 0, the Lorentz violation theory has been cancelled, the original change in Bekenstein-Hawking entropy is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' When January 3, 2023 1:32 Konika˙2 8 Authors’ names γ ′ > 1, the change in Bekenstein-Hawking entropy increases and γ ′ < 1, the change in Bekenstein-Hawking entropy decreases near the event horizon of KdS black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' In above cases, the change in Bekenstein-Hawking entropy depends on correction parameter λ and ether like vectors uα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' Nonstationary rotating KdS black hole The metric of rotating nonstationary KdS black hole in retarded time coordinates10(u, r, θ, φ) is defined by ds2 = 1 R2Ξ2 [∆λ − ∆θa2 sin2 θ]du2 − R2 ∆θ dθ2 + 2a R2Ξ2 [∆θ(r2 + a2) − ∆λ] sin2 θdudφ + 2 Ξ[du − a sin2 θdφ]dr − 1 R2Ξ2 [∆θ(r2 + a2)2 − ∆λa2 sin2 θ] sin2 θdφ2, (33) where R2, Ξ, ∆θ are given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' The term ∆λ is given by ∆λ = r2 + a2 − 2M(u)r − 1 3Λr2(r2 + a2), (34) where M(u) is the mass of nonstationary KdS black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' The location of event horizon of stationary and nonstationary KdS black hole can be obtained from null surface equation F(u, r, θ, φ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' The expression of null surface equation is given by gµν ∂F ∂xµ ∂F ∂xν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (35) The location of event horizon of stationary and nonstationary black holes can be obtained from the null surface equation (35) using generalized tortoise coordinate transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' The space time geometry outside the event horizon of nonstationary black hole can be described by the tortoise coordinate and in such case, r∗ tends to positive infinity when tending to infinite point and r∗ tends to negative infinity at the event horizon of black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' To study the Hawking radiation of nonstationary black hole, the tortoise coordinate transformation is defined by49−54 r∗ = r + 1 2κ(u0, θ0, φ0)lnr − rh(u, θ, φ) rh(u, θ, φ) , u∗ = u − u0, θ∗ = θ − θ0, φ∗ = φ − φ0, (36) where u0, θ0 and φ0 are the arbitrary constants under the coordinate transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' From the above equation, we get ∂ ∂r = � 1 + 1 2κ(r − rh) � ∂ ∂r∗ , ∂ ∂u = ∂ ∂u∗ − rrh,u 2κrh(r − rh) ∂ ∂r∗ , ∂ ∂θ = ∂ ∂θ∗ − rrh,θ 2κrh(r − rh) ∂ ∂r∗ , January 3, 2023 1:32 Konika˙2 9 ∂ ∂φ = ∂ ∂φ∗ − rrh,φ 2κrj(r − rh) ∂ ∂r∗ , (37) where rh,u = ∂rh ∂u , rh,θ = ∂rh ∂θ and rh,φ = ∂rh ∂φ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' rh,u represents the evaporation rate at the event horizon of KdS space time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' If ∂rh ∂u > 0, the event horizon of KdS space time is expanded (absorbing black hole ) and when ∂rh ∂u < 0, the event horizon of KdS space time is contracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' κ ≡ κ(u0, θ0, φ0) is taken as surface gravity of KdS space time which depends on retarded time and angular coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' Using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (33) and (36) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (35) and taking r → rh, the horizon equation of nonstationary KdS black hole is derived as a2Ξ2 sin2 θr2 h,u ∆θ + 2(r2 h + a2)Ξrh,u + 2aΞ2rh,urh,φ ∆θ + 2aΞrh,u +∆λ(rh) + ∆θr2 h,θ + Ξ2r2 h,φ ∆θ sin2 θ = 0, (38) where ∆λ(rh) = r2 h + a2 − 2M(u)rh − 1 3Λr2 h(r2 h + a2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' It is noted that the location of event horizon of nonstationary black hole varies with retarded time u = t − r∗ and angular co-ordinates θ, φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' From Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (11) and (33), the dynamical equation of scalar particle with mass m in curved space time is obtained as g00�∂I ∂u �2 + 2g01�∂I ∂u ��∂I ∂r � + 2g03�∂S ∂u �� ∂I ∂φ � + g11�∂I ∂r �2 + 2g13�∂I ∂r �� ∂I ∂φ � +g22�∂I ∂θ �2 + g33� ∂I ∂φ �2 + λuµuν∂µI∂νI + m2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (39) Since ether like vectors are not constant in curved space time, we can choose uα from nonstationary space time Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (33) so that we can make uαuα = constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' The ether like vectors uα are related to the properties of black hole and system of coordinate adopted by the metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' The ether like vectors uα are choosen as uu = ku √guu = ku √ R2Ξ � ∆λ − ∆θa2 sin2 θ , ur = kr√guu gur = kr � ∆λ − ∆θa2 sin2 θ √ R2 , uθ = kθ √gθθ = kθ � −∆θ R2 , uφ = ∆λkφ √gφφ = ∆λkφ √ R2Ξ � −[∆θ(r2 + a2)2 − ∆λa2 sin2 θ] sin2 θ , (40) where ku, kr, kθ and kφ are arbitrary constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' Using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (37) and (40) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (39),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' we get D E � ∂I ∂r∗ �2 + 2 � ∂I ∂u∗ �� ∂I ∂r∗ � + 2F E � ∂I ∂r∗ � + 2κ(r − rh)G E = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (41) where D = 1 2κ(r − rh)r2 h � (g00 + λuuuu)r2r2 h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content='u − 2rh � (g01 + λuuur)rrh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content='u + (g13 January 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' 2023 1:32 Konika˙2 10 Authors’ names +λuφur)rrh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content='φ + λuruθrrh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content='θ � {2k(r − rh) + 1} + 2(g03 + λuuuφ)r2rh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content='urh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content='φ +r2 h(g11 + λurur){2κ(r − rh) + 1}2 + (g22 + λuθuθ)r2r2 h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content='θ +(g33 + λuφuφ)r2r2 h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content='φ + 2λuθr2rh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content='θ(uurh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content='u + uφrh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content='φ) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' E = −(g00 + λuuuu)rrh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content='u rh + (g01 + λuuur){2k(r − rh) + 1} −(g03 + λuuuφ)rrh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content='φ rh − λuuuθrrh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content='θ rh ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' F = −(g03 + λuuuφ)rrh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content='upφ rh + {g13pφ + λur(uφpφ + uθpθ)}{2k(r − rh) + 1} −(g22 + λuθuθ)rrh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content='θpθ rh − (g33 + λuφuφ)rrh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content='φpφ rh − λuuuθrrh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content='upθ rh −λuθuφrrh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content='φpθ rh − λuθuφrrh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content='θpφ rh ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' G = g00ω2 − 2g03pφω + g22p2 θ + g33p2 φ + λuuuuω2 − 2λuuuφωpφ + λuθuθp2 θ +λuφuφp2 φ − 2λuuuθωpθ + 2λuθuφpθpφ + m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (42) To study the Hawking temperature, the action I in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (15) can be writen as I = −ωu∗ + I0(r∗, θ∗, φ∗), then we get ∂I ∂u∗ = −ω, ∂I ∂θ∗ = pθ, ∂I ∂φ∗ = pφ, (43) where ω is the energy of emitted scalar particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' pθ and pφ are the components of generalized momenta of scalar particle along the angular coordinates θ and φ respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' To obtain surface gravity and Hawking temperature at the event horizon of nonstationary KdS black hole, we assume that the coefficient of � ∂I ∂r∗ �2 approaches to unity as r → rh, u → u0, θ → θ0 and φ → φ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (41), an infinite limit of the form 0 0 is obtained near the event horizon of KdS black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' Using L’Hopital’s rule, the surface gravity is obtained as κ = rh(1 + 2Ξrh,u) − M − 2 3Λr3 h − Λ 3 a2rh − r−1 h {∆λ + Ξ(r2 h + a2)rh,u + aΞrh,φ} {Ξ(r2 h + a2) + a2Ξ2 sin2 θrh,u ∆θ }(1 + 2rh,u) + Z + λY , (44) where Z and Y are given by Z = 2∆θr2 h,θ + rh,φ �4aΞ2rh,u ∆θ + aΞ2 ∆θ + 2Ξ2rh,φ ∆θ sin2 θ + 2aΞ � , Y = k2 uρ4rh,uΞ2 ∆θa2 sin2 θ + ρ2kukrΞ − ρ2kukθΞrh,θ a sin θ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (45) Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (43) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (41) and taking r → rh, then we obtain � ∂I ∂r∗ �2 + 2(ω − ω0) ∂I ∂r∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (46) January 3, 2023 1:32 Konika˙2 11 The value of chemical potential, ω0 near the event horizon of nonstationary KdS black hole is w0 = 1 g01 − g00rh,u − g03rh,φ − λuuuurh,u + λurur − λuuuφrh,φ − λuuuθrh,θ ×[g13pφ − g03rh,upφ − g22rh,θpθ − g33rh,φpφ − λuurh,u(uφpφ + uθpθ) −λ(uθrh,θ + uφrh,φ)(uθpθ + uφpφ) + λur(uθpθ + uφpφ)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (47) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (47) represents the chemical potential of nonstationary KdS space time due to tortoise coordinate transformation (36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' The chemical potential, w0 depends on the black hole mass, cosmological constant, generalized momenta of scalar particle, retarded time, angular coordinates, correction parameter λ and ether like vectors uα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' If λ and uα tend to zero, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (47) is consistent with earlier literatures [10, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' From the solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (37), we obtain ∂I ∂r∗ = [2κ(r − rh) + 1](ω − ω0) ± (ω − ω0) [2κ(r − rh)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (48) It is observed that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (48) has a singularity near the event horizon of KdS black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' Integrating Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (48) by applying residue theorem of complex analysis and Feynman prescription, the imaginary part of the radial action I is derived as ImI± = π 2κ[(ω − ω0) ± (ω − ω0)], (49) where I+ and I− represent the outgoing scalar particle and ingoing scalar particle at the event horizon of KdS black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' Taking outgoing and ingoing of scalar particle, the tunneling probability which crosses at the event horizon of KdS black hole is calculated as Γ = Γemission Γabsorption = exp[−(ω − ω0) T ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (50) The modified Hawking temperature near the event horizon of nonstationary KdS black hole due to Lorentz violation theory is given by T = Th(1 + λH)−1 = Th(1 − λH + λ2H2 − λ3H3 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content='), (51) where the value of Th and H are Th = 1 2π � rh(1 + 2Ξrh,u) − M − 2 3Λr3 h − Λ 3 a2rh {Ξ(r2 h + a2) + a2Ξ2 sin2 θrh,u ∆θ }(1 + 2rh,u) + Z − r−1 h {∆λ + Ξ(r2 h + a2)rh,u + aΞrh,φ} {Ξ(r2 h + a2) + a2Ξ2 sin2 θrh,u ∆θ }(1 + 2rh,u) + Z � (52) and H = k2 uρ4rh,uΞ2 ∆θa2 sin2 θ + ρ2kukrΞ − ρ2kukθΞrh,θ a sin θ {Ξ(r2 h + a2) + a2Ξ2 sin2 θrh,u ∆θ }(1 + 2rh,u) + Z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (53) January 3, 2023 1:32 Konika˙2 12 Authors’ names From Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (52) and (53), it is known that the Hawking temperature near the event horizon of KdS black hole is modified due to Lorentz violation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' The modified Hawking temperature (T) of nonstationary KdS black hole depends not only on the mass of the black hole but also on the properties of event horizon, cosmological constant Λ, retarded time u, correction term λ and on the ether like vectors uα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' If kuρ2rh,uΞ + ∆θa sin θ(a sin θkr − kθrh,θ) = 0, then H −→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' In such case, our result is consistent with the earlier literatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content='10,11,46 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' Discussion First the line element of stationary KdS black hole is transformed into static form using frame dragging effect given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' Using modified form of Hamilton-Jacobi Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' Plot of original and modified Hawking temperature with radius of event horizon, rh of KdS black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' Here a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content='1, Λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content='6, λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content='005, cr = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content='5, ct = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' equation, Feynman prescription and WKB approximation, the modified Hawking temperature, heat capacity and change in Bekenstein-Hawking entropy are derived in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (23), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (24) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (31) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' It is noted that both are dependent on correction term λ and ether like vectors uα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' If γ > 1, the modified Hawking temperature near the event horizon of KdS black hole decreases and if γ < 1, the modified Hawking temperature increases in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' If γ = 1, the modified heat capacity (24) approaches to original heat capacity (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content='4 ^ Modified-Hawking 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content='2 口1 Hawking 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content='8 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content='5 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content='5 5 rnJanuary 3, 2023 1:32 Konika˙2 13 If γ > 1, the modified heat capacity increases and if γ < 1, the modified heat capacity is smaller than that of original heat capacity near the event horizon of stationary KdS black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' The change in Bekenstein-Hawking entropy near the event horizon of KdS black hole increases or decreases if γ′ > 1 or γ′ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' When λ = 0 and ct = cr, λ ̸= 0, the Lorentz violation has been cancelled and the original change in Bekenstein-Hawking entropy near the event horizon of KdS black hole is recovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' The Hawking temperature of rotating nonstationary KdS black hole is also inves- tigated using Klein-Gordon equation, generalized tortoise coordinate transformation and L’Hopital rule in Lorentz violation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' According to Damour and Ruffini6 and Sannan7, the traditional coordinate transformation is given by r∗ = r + 1 2κ1 ln(r − rh(u, θ, φ)), u∗ = u − u0, θ∗ = θ − θ0, φ∗ = φ − φ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (54) Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (54) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (41), the modified surface gravity and Hawking temperature of nonstationary rotating KdS black hole are κ1 = rh(1 + 2Ξrh,u) − M − 2 3Λr3 h − Λ 3 a2rh {Ξ(r2 h + a2) + a2Ξ2 sin2 θrh,u ∆θ }(1 + 2rh,u) + Z + λY (55) and T1 = 1 2π rh(1 + 2Ξrh,u) − M − 2 3Λr3 h − Λ 3 a2rh {Ξ(r2 h + a2) + a2Ξ2 sin2 θrh,u ∆θ }(1 + 2rh,u) + Z + λY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (56) In Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (51) and (56) , if we put λ = pθ = pφ = 0 and kuρ2rh,uΞ+∆θa sin θ(a sin θkr− kθrh,θ) = 0, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (51) and (56) are concordant with the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content='46 Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (54) gives another chemical potential of nonstationary rotating KdS black hole in Lorentz violation theory as wp = 1 g01 − g00rh,u − g03rh,φ − λuuuurh,u + λurur − λuuuφrh,φ − λuuuθrh,θ ×[g13pφ − g03rh,upφ − g22rh,θpθ − g33rh,φpφ − λuurh,u(uφpφ + uθpθ) −λ(uθrh,θ + uφrh,φ)(uθpθ + uφpφ) + λur(uθpθ + uφpφ)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (57) It is observed that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (57) is consistent with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' The surface gravities derived from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (44) and (55) can be combined as κ = κ1 + Ξ1, (58) where κ and κ1 represent the surface gravities of nonstationary KdS black hole due to tortoise coordinate transformations (36) and (54).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' Ξ1 indicates the constant term due to tortoise coordinate transformation (36) and its expression is Ξ1 = − r−1 h {∆λ + Ξ(r2 h + a2)rh,u + aΞrh,φ} {Ξ(r2 h + a2) + a2Ξ2 sin2 θrh,u ∆θ }(1 + 2rh,u) + Z + λY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (59) January 3, 2023 1:32 Konika˙2 14 Authors’ names The rate of correction for Hawking temperature near the event horizon of nonsta- tionary KdS black hole is given by δ = − r−1 h {∆λ + Ξ(r2 h + a2)rh,u + aΞrh,φ} rh(1 + 2Ξrh,u) − M − 2 3Λr3 h − Λ 3 a2rh .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (60) It is worth mentioning that the correction rate is independent of correction term λ and the ether like vectors uα but depends on black hole mass M, rotational param- eter a, angular coordinate θ, cosmological constant Λ and generalized momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' For stationary KdS black hole in the absence of Lorentz violation theory, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (51) and (56) reduce to the same Hawking temperature as TH = T1 = 1 2π � rh − M − 2 3Λr3 h − Λ 3 a2rh Ξ(r2 h + a2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (61) It is noted that for the stationary KdS space time, the different tortoise coordinate transformations give the same Hawking temperature in the absence of Lorentz vio- lation theory which is exactly equal to the actual calculation given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (58), κ1 approaches to zero, the Hawking temperature T does not tend to zero due to extra term Ξ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' If Ξ1 approaches to zero in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (58), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (44) is consistent with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (55).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' From Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (47) and (57), we observe that the chemical potential derived from different tortoise coordinate transformations are equal near the event horizon of black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' It can be concluded that the tortoise coordinate transformation given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (36) is more suitable and accurate in the study of modified surface gravity near the event horizon of nonstationary KdS space time in Lorentz violation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' From Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (44) and (55), the different surface gravities are obtained using the dif- ferent tortoise coordinate transformations near the event horizon of nonstationary KdS black hole in Lorentz violation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' Conclusion In this paper, the tunneling of scalar particle near the event horizon of stationary KdS black hole is investigated using Klein-Gordon equation in Lorentz violation theory, Feynman prescription and WKB approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' Then the corresponding Hawking temperature, heat capacity and change in Bekenstein-Hawking entropy near the event horizon are derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' The Hawking temperature, heat capacity and change in entropy are modified due to presence of correction term λ and ether like vectors uα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' The modified surface gravities of nonstationary rotating black hole are also stud- ied using different tortoise coordinate transformations in Lorentz violation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' Using null surface equation and tortoise coordinate transformation, the horizon equation of nonstationary KdS black hole is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' The modified surface grav- ities and the modified Hawking temperatures are derived with the help of event horizon equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' It is known that the modified Hawking temperature depends not only on the correction term λ but also on the ether like vectors uα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' If we use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' January 3, 2023 1:32 Konika˙2 15 (36) in the study of surface gravity and Hawking radiation of black hole, a constant term Ξ1 is seen to be appeared in the expressions of surface gravity and Hawking temperature near the event horizon of KdS black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' If Ξ1 tends to zero, the two modified surface gravities and Hawking temperatures are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' If λ and Ξ1 ap- proach to zero, the original surface gravities near the event horizon of nonstationary black hole are recovered and are concordant with the earlier literatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content='8,9,54 It is also seen that the correction rate δ of Hawking temperature in Lorentz violation theory does not depend on correction term λ and the ether like vectors uα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' The different tortoise coordinate transformations yield the same chemical potential at the event horizon of black hole but the values of surface gravities and Hawking temperatures are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' This shows that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' (36) is more reliable and accurate in the study of modified surface gravity near the event horizon of nonstationary of KdS black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' For the stationary space time, the different methods yield the same Hawking temperature and the change in Bekenstein-Hawking entropy in the absence of Lorentz violation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' Acknowledgments The first author acknowledges the DST INSPIRE, New Delhi, India for providing financial support (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' IF190759).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' W.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' Majhi, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' D 79, 044005 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' January 3, 2023 1:32 Konika˙2 16 Authors’ names 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' S.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' 121, 404 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' Sakalli and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' Ovgun, 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' Ibungochouba, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' Kenedy, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' Ablu and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' Yugindro, Ind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' 94, 2061 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' Horava, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' Rev.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' Tan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' Zhang and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' Yang, Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQfsPkA/content/2301.00571v1.pdf'} +page_content=' 73, 045402 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file mode 100644 index 0000000000000000000000000000000000000000..ac570140972df61968fe8163e4c5ef89cdb4871f --- /dev/null +++ b/K9E2T4oBgHgl3EQfAwYv/content/tmp_files/2301.03594v1.pdf.txt @@ -0,0 +1,1463 @@ +RingAuth: Wearable Authentication using a Smart Ring +Jack Sturgess, Simon Birnbach, Simon Eberz, and Ivan Martinovic +Department of Computer Science, University of Oxford, Oxford, UK +{firstname.lastname}@cs.ox.ac.uk +Abstract—In this paper, we show that by using inertial sensor +data generated by a smart ring, worn on the finger, the user +can be authenticated when making mobile payments or when +knocking on a door (for access control). The proposed system +can be deployed purely in software and does not require up- +dates to existing payment terminals or infrastructure. We also +demonstrate that smart ring data can authenticate smartwatch +gestures, and vice versa, allowing either device to act as an +implicit second factor for the other. To validate the system, we +conduct a user study (n=21) to collect inertial sensor data from +users as they perform gestures, and we evaluate the system +against an active impersonation attacker. Based on this data, +we develop payment and access control authentication models +for which we achieve EERs of 0.04 and 0.02, respectively. +Index Terms—wearable authentication, mobile payment, smart +ring, smartwatch, tap gesture, knock gesture, authentication +1. Introduction +Mobile payment systems (also known as tap-and-pay), +such as Google Pay, have become pervasive in recent years. +These systems enable the user to provision payment cards to +a virtual wallet on a smartphone and then facilitate cashless +and contactless payments with NFC-enabled point-of-sale +terminals. The functionality of mobile payment systems has +been extended to wearable devices, such as smartwatches. +When paired with a smartphone, a smartwatch can access +and store the same virtual wallet and make payments even +when the smartphone is not present. However, the options +for authenticating the user to a smartwatch are limited, +given its size, and so the smartphone is still regarded as +the primary device for the system. Recently, we are begin- +ning to see commercial smart rings enter the market that +offer payment capabilities, such as the Mastercard K-Ring1, +which have even fewer options for authentication. In order +to protect payment transactions in these emerging systems, +new authentication factors that operate on wearable devices +are needed. +Given that wearable devices are often designed with +continuous healthcare or fitness monitoring use-cases in +mind, they tend to have an inertial measurement unit (IMU) +consisting of (at least) an accelerometer and gyroscope. +1https://thepaymentring.com +Works in behavioural biometrics have shown that inertial +sensor data in smartphones and smartwatches can be used +to infer gait or gestures. These systems require some initial +calibration (i.e., a cumbersome enrolment phase), but can +then continuously or discretely authenticate the user with +reasonable effect. +In this work, we show that inertial sensor data generated +by a smart ring can be used to authenticate the user when +making certain gestures, such as tapping a payment terminal +or knocking on a door. +Contributions. +• We propose a novel, smart ring-based authentication +system. Using only inertial sensor data, we show that +a single tap gesture performed by a user while making a +payment with a smart ring can implicitly authenticate +the user. We also show that smart ring data can be +used to authenticate a user making payments with a +smartwatch, and vice versa, allowing either device to +act as an implicit second factor for the other. +• We show that our approach can also be applied to an +access control context, in which a single knock gesture +against a door can explicitly authenticate the user. +• We demonstrate that our authentication models are +resistant against an active impersonation attacker. +• We make the code and data required to reproduce our +results available at http://github.com/jacksturgess. +Paper Structure. The rest of this paper is organised +in the following way. Section 2 states our objectives and +details our system and threat models. Section 3 describes the +design of our data collection user study. Section 4 explains +the methods that we employ to process and classify our data. +Section 5 presents and analyses our results. Section 6 dis- +cusses peripheral topics. Section 7 compares our approach +to related work. Section 8 considers the limitations of our +approach. Section 9 concludes the paper. +2. Objectives and Assumptions +2.1. Design Goals +In this paper, we investigate the use of a smart ring for +authentication purposes, both implicit and explicit. +For implicit authentication, we select the context of +mobile payments in which we consider the tap gesture made +arXiv:2301.03594v1 [cs.CR] 9 Dec 2022 + +when the user taps an NFC-enabled device against a point- +of-sale terminal to provide an implicit gesture biometric and +we pose the following questions: +• can tap gestures made with a smart ring, as measured +by the inertial sensors of the smart ring, be used to +implicitly authenticate the user during a transaction? +• can tap gestures made with a smart ring, as measured +by the inertial sensors on a smartwatch worn on the +same arm, be used to implicitly authenticate the user +(in cases where the smart ring does not have inertial +sensors of its own) during a transaction? +• can implicit authentication in a smartwatch system be +improved by incorporating (the inertial sensor data of) +a smart ring as a second factor? +For explicit authentication, we choose the act of knock- +ing on a door. A knock gesture could be used implicitly for +identification purposes or it could be performed solely for +access control purposes, in which case it would be treated +as an explicit authentication gesture. We pose the question: +can (explicit) knock gestures, as measured by the inertial +sensors of a smart ring, a smartwatch, or directly mounted +on the door, be used to authenticate the user? We consider +both unrhythmic knocks and user-chosen secret knocks. +2.2. System Model +For our payment model, we consider a system model in +which the user is wearing both a smart ring and a smartwatch +on the same arm and is using them to make NFC-enabled +payments at point-of-sale terminals in a typical setting (e.g., +in a shop). To make a payment, we assume that the user +performs a tap gesture by moving one of the devices towards +the terminal until it is near enough to exchange data via +NFC. The NFC contact point is when the payment provider +would decide whether to approve the payment, so we assume +that this marks the end of the tap gesture. +For our access control model, we consider a system +model in which the user is wearing both a smart ring and a +smartwatch on the same arm and is knocking on a door with +that hand as a means to authenticate to an access control +system. We assume that each knock gesture is bounded by +button-presses on one of the devices. +We assume that the devices have an accelerometer and +gyroscope and that we have access to their data. We use data +from the inertial sensors only. We assume that the user’s +biometric templates are stored securely on the wearable +devices. When we combine data from multiple devices into +a single sample for classification, as we do in some of +our models, we assume that one device shares its inertial +sensor data wirelessly with the other in a trusted fashion. +We assume that the devices will be running a trusted app +that is able to communicate its authentication decisions with +the payment provider or access control system. +2.3. Threat Model +We consider an adversary that has possession of a +legitimate user’s smart ring (or smartwatch, or both, as +Figure 1: The equipment used in our experiments: six fixed termi- +nals (labelled, see Table 1 for details), an NFC reader (affixed to +Terminal 1), a Raspberry Pi for timestamp collection, a Raspberry +Pi attached to a door, and a fixed smartphone for video recording. +Inset: our smart ring and smartwatch worn on the left arm with the +sensor axes shown (the z-axis points upwards through the screen). +Terminal +Height (cm) +Tilt (◦) +Distance (cm) +1 +100 +0 +5 +2 +120 +60 +25 +3 +95 +45 +-10 +4 +105 +30 +15 +5 +110 +15 +10 +6 +115 +90 +30 +F +picked up from centre of platform +Table 1: Details of the terminals used in our experiment; the indices +match those labelled in Figure 1 and ‘F’ is the freestyle terminal. +Height is measured from the floor to the lowest point of the +terminal; Tilt is the inclination at the lowest point of the terminal; +and Distance is measured from the front of the stand to the point +of the terminal that is closest to the user. Terminals 2 and 6 match +terminals on self-service checkouts at supermarket chains. +Figure 2: The attacker’s restricted view of a victim interacting with +Terminal 2 (left), Terminal 3 (centre), and the door (right). +appropriate), has unlocked it, is wearing it, and is attempting +to use it to make a payment at a terminal or to gain entry to +a locked door via an access control system. The adversary +may have (maliciously) stolen the device(s) or (benignly) +borrowed it. Our goal is to authenticate the legitimate user +and to reject the adversary by using only inertial sensor data. +We consider the following two types of attack: +• zero-effort attack: for any given victim, all other users +are considered to be passive, zero-effort attackers; +• observation attack: an active attacker who watches the +victim perform gestures (e.g., via a hidden camera) and +then attempts to mimic them. +2 + +2 +4 +6 +3 +y +5 +1 +z +7In this work, we concentrate on the extent to which +gesture biometrics can be used to defend against these +attacks. We do not consider threats to other components +in the system, tampering of devices or biometric templates, +malware, or denial of service attacks. +3. Experimental Design +3.1. Experiment Overview +To evaluate the extent to which finger and wrist motion +data can be used to authenticate users, and to compare the +two, we designed and conducted a user study to collect data. +Our experiments consisted of six point-of-sale terminals on +an adjustable stand fixed at a height of 100 cm, an ACR122U +NFC reader connected to a Raspberry Pi for timestamp col- +lection, a second Raspberry Pi with an accelerometer and a +gyroscope attached to a closed door, and a smartphone fixed +in position for video recording. For our wearable devices, +we used a smart ring and a smartwatch (detailed below), +both commercial off-the-shelf devices and worn together on +the same arm by the user. We collected motion data from the +wearable devices and the door-mounted Raspberry Pi (with +all clocks synchronised) as the user performed gestures. +Figure 1 shows our apparatus. +For our payment experiment, we affixed an NFC tag to +the front of each wearable device and we affixed the NFC +reader to the front of each point-of-sale terminal in turn. +For each terminal, the user stood in front of the stand and +performed tap gestures on the terminal using a wearable +device, as if making mobile payments, with a short spacing +delay between each tap gesture. The use of NFC tags and +the NFC reader ensured consistent NFC communication +between wearable devices and terminals. Each NFC tag +stored the name of the wearable device to which it was +affixed; each time an NFC contact point was made during +a tap gesture, the Raspberry Pi captured its timestamp and +the name of the triggering device. This was later used to +segment the inertial sensor data collected from the wearable +devices to retrieve the data for each tap gesture. +For our access control experiment, the user stood in front +of the door and performed knock gestures against it using +his device-wearing hand. The user pressed the button on the +smart ring before and after each knock gesture to capture +bounding timestamps. These timestamps were later used to +segment the inertial sensor data collected from all devices +to retrieve the data for each knock gesture. +3.2. Point-of-Sale Terminals +To emulate a real-world mobile payment setting as much +as possible, we captured tap gestures using seven terminals: +six in fixed positions (as detailed in Table 1) and one +‘freestyle’. For five of the six fixed terminals, we surveyed +prominent supermarket and restaurant chains to find popular +or standardised terminal positions (in terms of height, tilt +angle, and distance from the user) and set our terminals to +match common configurations. We set the other fixed termi- +nal (Terminal 3) to match the position of the terminal on a +train station barrier in the UK, which is widely standardised. +For the freestyle terminal, the user picked up the NFC reader +with his other hand and performed a tap gesture against it, +returning it after each interaction, as if a shopkeeper had +handed an unmounted terminal to a customer. +The six fixed terminals remained deactivated throughout +the experiment because their functionality was not required. +For consistent data collection, we affixed the NFC reader to +each terminal when using it. As such, the terminals should +be regarded only as fixtures that enforced positions, as well +as a tool for immersing the user in a payment scenario. +3.3. Wearable Devices and Sensor Modules +For our smart ring, we used a Genki Wave2. This ring is +worn on the index finger and has a button that can be pressed +with the thumb—we utilised this for timestamp collection +in our access control experiment. We wrote a data collection +script in Python using the open source library3 provided by +the developers to interface with the ring over Bluetooth LE. +For our smartwatch, we used a Samsung Galaxy Watch4 +running the Tizen 4.0 operating system. We built a data +collection app and installed it on the smartwatch using the +Tizen Studio IDE. +From each of these wearable devices, we collected +timestamped data from four inertial sensors directly or +derived from their MEMS sensors. The accelerometer mea- +sures change in velocity. The gyroscope measures angular +velocity. The linear accelerometer is derived from the ac- +celerometer with the effects of gravity excluded. The gyro- +scope rotation vector (GRV) is a fusion of sensor readings +to compute the orientation of the device. We collected this +data with sampling rates of 100 Hz for the smart ring (which +we downsampled to 50 Hz), 50 Hz for the smartwatch, and +30 Hz for the door-mounted Raspberry Pi. +The inertial sensor axes are fixed relative to the frame of +each device (as shown in Figure 1). Motion along the x-axis +corresponds with arm extension or withdrawal; the y-axis, +with side-to-side arm waving; and the z-axis, directly up- +and downwards through the screen. +3.4. Tap and Knock Gestures +To collect tap gestures for our payment experiment, we +had each user perform sets of ten of each of the following +tap gestures on each of the seven point-of-sale terminals: +• ring tap: the user tapped the smart ring against the +terminal and moved it around, if necessary, until NFC +contact was made to simulate a ring-based payment; +• watch tap: the user tapped the smartwatch against the +terminal and moved it around, if necessary, until NFC +contact was made to simulate a watch-based payment. +2https://genkiinstruments.com/products/wave +3https://github.com/genkiinstruments/genki-wave +4https://www.samsung.com/uk/watches/galaxy-watch +3 + +To collect knock gestures for our access control experi- +ment, we had each user perform sets of ten of each of the +following knock gestures on the closed door: +• 3-knock: the user knocked on the door three times; +• 5-knock: the user knocked on the door five times; +• secret knock: the user created a knock pattern consisting +of between three and six knocks performed in a manner +of his choosing (some users chose to knock to a rhythm, +some included a twist or jolt of the wrist). +3.5. User Study +To collect our data, we conducted a user study that +was reviewed and approved by the relevant research ethics +committee at our university. We recruited 21 participants, +including staff, students, and members of the public. Each +participant attended two data collection sessions on separate +days. We collected 30 of each of the gestures detailed in +Section 3.4 (510 gestures in total) from each participant. +In each experiment, the participant was asked to stand +facing the terminals or the door; aside from this, we did not +prescribe any constraints on positioning as we wanted the +user to interact comfortably as though acting in a real-world +setting. The first three gestures of any type were performed +in silence, to familiarise, then the reseacher engaged the par- +ticipant in light conversation to simulate the distractions of a +real-world environment. This sometimes elicited additional +hand and body movements if the user gesticulated naturally. +Impersonation. To evaluate the robustness of our ap- +proach against an observation attacker, all 21 participants +consented to having their gestures recorded and participated +in an impersonation exercise. The first 3 were recorded as +victims and the latter 18 impersonated them; then, at the end +of the study, the first 3 were invited back to impersonate the +other 18. This design also enabled us to compare the suscep- +tibility of different victims and the skill of different attackers +(also known as wolf and lamb analysis; see Section 5.2). +We recorded the following six gestures of each participant: +smart ring and smartwatch tap gestures on Terminals 2 and +3, 5-knock gestures, and secret knock gestures. The camera +was fixed in position, as if hidden, and so the amount of +observable information was controlled; Figure 2 shows the +attacker’s view of the terminals and the different information +observable for each type of gesture. For each of the six +attacks, the attacking participant watched a short video of +the victim performing the gesture three times and then made +three attempts to mimic him. The attacker wore the wearable +devices on the same arm as the victim during this exercise. +User Statistics. Of our 21 participants: 15 were male, +17 wore the devices on the left arm (the decision was led by +the smartwatch; everyone who wore them on the right arm +was female), 15 regularly wore a watch of some kind (7 of +which wore a smartwatch), 13 had paid with a smartphone +before, and 5 had paid with a smartwatch (i.e., 71% of those +who regularly wore one). 16 participants (76%) remembered +their secret knock, with an average of 4 days between their +sessions (those who did not remember had an average of +4.2 days between sessions). +4. Methods +4.1. Data Processing +We collect time-series data from the four inertial sensors +on our smart ring and smartwatch worn by the user and from +the accelerometer attached to the door. Each accelerometer, +gyroscope, or linear accelerometer sample is given in the +form (t, x, y, z) and represents the change in velocity or +angular velocity along each axis at time t. Each GRV +sample is given as a quaternion in the form (t, x, y, z, w) +and approximates the orientation of the device at time t. +We express a tap or knock gesture using a series of +inertial sensor data samples within a time window. In our +payment experiment, we retrieve the tap gestures for each +user by segmenting 4-second blocks of sensor data using +the NFC contact point timestamps as the endpoint of each +window. We found that a 4-second maximum window size +was sufficient to encapsulate the entirety of each tap gesture. +To investigate optimum tap gesture parameters, we compare +(in Section 5) the performances of gestures bounded by +various window sizes and offsets, where the offset is the +time between the NFC contact point and the end of the +window. For an NFC contact point timestamp T0, a window +size s, and an offset o, we retrieve a tap gesture with +start time TS and end time TE, where TE = T0 − o +and TS = TE − s. In our access control experiment, we +retrieve the knock gestures for each user using the bounding +timestamps captured when the user pressed the button on the +ring before and after each gesture. +4.2. Feature Extraction +Whenever a gesture is retrieved, we apply a low pass +filter to the data to reduce noise and then process the fol- +lowing five dimensions for each accelerometer, gyroscope, +or linear accelerometer sample: the filtered x-, y-, and z- +values, the energy of those filtered values, and the energy +of the unfiltered (raw) values, where the energy of {x, y, z} +is given by +� +x2 + y2 + z2. As GRV samples are expressed +as quaternions, for those we process only the four filtered +values (since the Euclidean norm of a quaternion is always +1). In total, we process each gesture in 19 dimensions. +For each gesture, we extract the following ten statistical +features in each dimension: minimum, maximum, mean, me- +dian, standard deviation, variance, inter-quartile range, kur- +tosis, skewness, and peak count. We also calculate the mean +and maximum velocities along each axis, the displacement +along each axis, and the Euclidean displacement from each +of its accelerometer, gyroscope, and linear accelerometer +vectors, adding another 30 features. Ultimately, we reduce +each gesture to a feature vector containing 220 members. +In previous work [14, 15], we found this feature set to be +ideal by starting with a larger set and pruning it down using +normalised Gini importances to reject the least informative +features, so we use it again here on similar grounds. This +approach also yielded promising preliminary results in our +4 + +access control model, so we used it for both models to allow +for comparability and consistency throughout the paper. +In some of our models, we combine inertial sensor data +from multiple devices. In these cases, the above features +are extracted for each separate source and then concatenated +together to form a linearly-larger feature vector. +4.3. Classification +We use random forest classifiers in each of our payment +authentication and access control models. Similar works +[1, 15] have shown that random forests are efficient, able +to estimate the importance of features, and robust against +noise. To balance relevance with learning time, we include +100 trees in each forest. To reduce the impact of random +generation on our results, and to ensure that our results are +fair and unselective, we train and test each classifier ten +times with different forest randomisation seeds and average +the outcomes. +Terminal-agnostic Model. To evaluate the zero-effort +attacker, we train a set of classifiers that are user-dependent +and terminal-agnostic. This means that a separate template +(and decision threshold) is generated for each user and that, +for each tap gesture under test, the tap gesture samples +used to train the classifier came from tap gestures made +only against other terminals (i.e., a leave-one-out approach). +The user’s tap gestures form the positive class and all +other user’s tap gestures form the negative class. As this +is an authentication scenario, we ensure that the training +data precedes the testing data by taking the tap gestures +collected in users’ first data collection session as training +data (analogous to the enrolment phase, where the user +template is created) and those collected in the second session +as testing data (analogous to an authentication phase). +Terminal-known Model. To evaluate the observation +attacker, we apply a similar design, except that we do not +exclude tap gestures based on the terminal. We assume that, +since the attacker has already observed the victim using the +terminal in question, that the system has knowledge of the +victim’s tap gestures made against that terminal. We pair up +every user as a victim with every other user as an attacker, +one at a time, and train the classifiers, excluding all of +the attacker’s tap gestures. This enables us to generate the +victim’s decision threshold for that pairing, tuned to the EER +(which we take as our baseline FAR, the base-FAR), with +no knowledge of the attacker, and then we test the attacker’s +impersonation samples against that tuned classifier to find +his attack success rate (the observation-FAR). We compare +the two FARs to measure the success of the observation +attack (see Sections 4.4 and 5.2 for more details). +Terminal-specific Model. For investigative purposes, +we also train a set of classifiers in which each is trained and +tested on tap gestures performed only on a single terminal. +This model enables us to measure the effectiveness of our +approach if implemented on standardised, single-terminal +systems, such as public transport systems. +Access Control Model. For knock gestures, to evaluate +each of the attackers, we use a similar approach as with +each respective payment authentication model above, except +that we do not need to generalise the model over multiple +terminals. We train separate models for each of the three +knock gestures. +4.4. Performance Metrics +In each model, the true positives is the number of times +that the positive class (i.e., the legitimate user) is correctly +accepted; the true negatives is the number of times that the +negative class (i.e., the adversary) is correctly rejected; the +false positives is the number of times that the negative class +is wrongly accepted; and the false negatives is the number +of times that the positive class is wrongly rejected. The +decision threshold, θ, is the score at which the classifier +chooses to assign to a sample the positive class rather than +the negative. To tune a classifier, we adjust θ to modify +the trade-off between security and usability; a larger θ is +more resilient to false positives and thus favours security, a +smaller θ favours usability. +To quantify the performance of our models, we find the +optimum decision threshold such that the false acceptance +rate (FAR) equals the false rejection rate (FRR); this point +is called the equal error rate (EER). The FAR inversely indi- +cates security, by measuring the likelihood that the negative +class will be wrongly accepted. The FRR inversely indicates +usability, by measuring the likelihood that the positive class +will be wrongly rejected. The FAR and FRR are antagonistic +insofar as setting θ to favour one will disfavour the other, +so there will always be a point at which they cross over; the +EER is a measure of system performance when considera- +tion is balanced evenly between security and usability and +is commonly used as a metric in authentication systems. +5. Results +5.1. Zero-effort Attack +Tap Gestures. Figure 3 shows the average EERs for +our terminal-agnostic payment authentication models by +window size and offset. With 21 users and 6 fixed terminals, +each score is the average of scores from 21×6×10 = 1, 260 +classifiers (see Section 4.3 for details). +For ring tap gestures, Figure 3a shows that when using +data from the smart ring only our model achieves EERs as +low as 0.06. Figure 3c shows that when using data from +the smartwatch only, we get 0.08. This suggests that a +smartwatch could perform as a reasonable authenticator for a +smart ring, in a scenario where the smart ring does not have +any inertial sensors of its own (such as the K-Ring). If the +data from both devices are combined, Figure 3e shows that +we achieve EERs as low as 0.04. The optimum parameters +for ring tap gestures, by considering EERs and favouring a +smaller window size for usability, are {s = 2.5, o = 0}. +When we use data from the smartwatch only, we see a +pattern radiating from the bottom left corner of the heatmap. +This shows that, when training only on data where the tap- +ping device is near to the terminal (small window, just before +5 + +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +window size, s (s) +1.0 +0.5 +0.0 +offset, o (s) +0.12 0.11 0.10 0.09 0.08 0.07 1.00 1.00 +0.13 0.11 0.09 0.08 0.08 0.07 0.06 1.00 +0.13 0.10 0.09 0.08 0.07 0.07 0.06 0.06 +(a) ring tap; ring data +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +window size, s (s) +1.0 +0.5 +0.0 +offset, o (s) +0.11 0.10 0.09 0.08 0.07 0.07 1.00 1.00 +0.12 0.10 0.08 0.07 0.07 0.07 0.07 1.00 +0.14 0.12 0.08 0.08 0.06 0.06 0.06 0.07 +(b) watch tap; ring data +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +window size, s (s) +1.0 +0.5 +0.0 +offset, o (s) +0.14 0.10 0.09 0.08 0.08 0.08 1.00 1.00 +0.18 0.12 0.09 0.08 0.09 0.08 0.08 1.00 +0.21 0.16 0.12 0.09 0.08 0.09 0.08 0.08 +(c) ring tap; watch data +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +window size, s (s) +1.0 +0.5 +0.0 +offset, o (s) +0.18 0.12 0.09 0.07 0.06 0.06 1.00 1.00 +0.23 0.17 0.12 0.09 0.07 0.07 0.07 1.00 +0.24 0.21 0.16 0.12 0.09 0.08 0.07 0.07 +(d) watch tap; watch data +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +window size, s (s) +1.0 +0.5 +0.0 +offset, o (s) +0.08 0.06 0.05 0.05 0.05 0.05 1.00 1.00 +0.11 0.08 0.05 0.05 0.05 0.05 0.04 1.00 +0.12 0.09 0.07 0.05 0.04 0.05 0.05 0.04 +(e) ring tap; combined data +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +window size, s (s) +1.0 +0.5 +0.0 +offset, o (s) +0.09 0.07 0.05 0.04 0.04 0.04 1.00 1.00 +0.12 0.09 0.07 0.05 0.04 0.04 0.04 1.00 +0.14 0.12 0.08 0.06 0.05 0.04 0.04 0.04 +(f) watch tap; combined data +Figure 3: Average EERs for our payment authentication models by window size and offset, for tap gestures made with the smart ring +(left) and smartwatch (right), using data from the smart ring only (top), the smartwatch only (middle), and both combined (bottom). (This +figure is based on downsampled smart ring data; Figure A.1 in the Appendix is based on undownsampled smart ring data for comparison.) +the NFC contact point is found), the movements of the wrist +are not discriminative between users. The same is not true +when using data from the smart ring only; this suggests that +the finger remains active during that time, even for watch +tap gestures. Figure 3f shows that including a smart ring as +an additional factor can improve the authentication of a user +making watch-based payments (cf. Figure 3d). +Knock Gestures. Across all of the participants in our +user study, the average 3-knock gesture lasted 2.81 seconds, +the average 5-knock, 3.32 seconds, and the average secret +knock, 3.78 seconds. Table 2 shows the average EERs for +our access control models. Each score is the average of +scores from 21 × 10 = 210 classifiers. Unexpectedly, our +3-knock model performed best, achieving an EER of 0.02 +using data from the smartwatch only. This might be due +to some participants sometimes miscounting the number of +knocks when performing the 5-knock gesture, leading to +messier data, whereas the 3-knock gestures were performed +effortlessly and more consistently. +When we use data from the smartwatch only, we achieve +the best results across all models, including those based on +combined data. This suggests that wrist movements are a +key discriminator in knocking, to such an extent that other +factors act as pollutants. When we use data from the door +only, we achieve the poorest results; the lower sampling rate +of the door-mounted sensors may have had an impact, but +the magnitude of the difference in results is likely explained +by those sensors lacking knowledge of user movements. +Knock +Gesture +Door +Ring +Watch +Combined +3-knock +0.17 +0.06 +0.02 +0.03 +5-knock +0.21 +0.12 +0.04 +0.05 +secret knock +0.19 +0.09 +0.05 +0.05 +Table 2: Average EERs for our access control models, using data +from (i) the door-mounted sensors only, (ii) the smart ring only, +(iii) the smartwatch only, and (iv) all three combined. +5.2. Observation Attack +Figure 4 shows the results of our observation at- +tack against our terminal-known payment authentication +model and our access control model. Each of the first +3 users was the victim of 1,080 ring tap impersonation +attempts (18 attackers × ring tap gestures on 2 terminals × +3 attempts at each gesture×10), 540 5-knock attempts, and +540 secret knock attempts; each of the other 18 users, +separated in the figures by a red line, was the victim of +180 (3 attackers), 90, and 90 attempts, respectively. +For ring tap gestures, the base model achieves average +EERs (base-FARs) of 0.03 when using data from the smart +ring only and of 0.01 when using data from the smart ring +and smartwatch combined; when attacked, the average suc- +cess rates (observation-FARs) are 0.06 and 0.05, respectively +(and we see similar results for watch tap gestures). Figures +4a and 4b show that a small number of our users are lambs +6 + +0.05 +0.10 +0.15 +0.20 +R +0.25 +0.30 +0.350.0 +0.2 +0.4 +0.6 +0.8 +FAR +(a) ring tap; ring data +0.0 +0.2 +0.4 +0.6 +0.8 +(b) ring tap; combined data +0.0 +0.2 +0.4 +0.6 +0.8 +(c) 5-knock; combined data +0.0 +0.2 +0.4 +0.6 +0.8 +(d) secret knock; combined data +0.0 +0.2 +0.4 +FAR +(e) ring tap; ring data +0.0 +0.2 +0.4 +(f) ring tap; combined data +0.0 +0.2 +0.4 +(g) 5-knock; combined data +0.0 +0.2 +0.4 +(h) secret knock; combined data +Figure 4: Results of our observation attack against our terminal-known payment authentication model in optimum window {s = 2.5, o = 0} +and our access control model. The top row shows for each user as a victim the average FAR of the user-specific base model (flat line) and +the average FAR when attacked (circle); if the latter is greater, then the victim’s line is coloured blue, otherwise it is orange. The bottom +row shows for each user as an attacker the average FAR achieved when attempting to impersonate other users. The red line separates the +first 3 users from the other 18, indicating the two groups of users that attacked each other (see Section 3.5 for details). +(i.e., users that are especially susceptible to impersonation), +where their observation-FAR is significantly larger than +their base-FAR. Figure 4b shows that the addition of the +smartwatch data helps to reduce the largest FAR deltas, and +the overall average observation-FAR, but also opens a new +vector that increases the susceptibility of some users. Figures +4e and 4f show that none of our users are wolves (i.e., users +that are especially skilled at impersonation); when using data +from the smart ring only, some attempts got lucky against +a random spattering of users, but when the smartwatch +data was combined, the success rate dropped. This is an +important result, as it suggests that our system provides +resistance against wolves, reducing the likelihood that an +attacker could predictably impersonate a given victim (and +so, in the wider system model, this may act as a deterrent). +For knock gestures, when using data from the door- +mounted sensors, smart ring, and smartwatch combined, we +have average base-FARs of 0.05 for both the 5-knock and +secret knock gestures and average observation-FARs of 0.08 +and 0.09, respectively. Figures 4c and 4d show that we +have a number of lambs, this time with greater FAR deltas. +Knock gestures are notably weak against impersonation if +they are loud and slow. For the secret knock, the fourth and +sixth users after the red line have high base-FARs, because +those users chose common gesture fragments in their secret +knocks. For the former (and the seventh), the gesture was +loud and slow, as evidenced by the huge observation-FAR. +For the latter, curiously, the observation-FAR is far lower +than the base-FAR, suggesting that the gesture was difficult +to mimic intentionally despite having commonality with +other gestures. This gesture contained three fast knocks in +the middle, which captured the attention of attackers only +for them not to match the surrounding knocks. Figures +4g and 4h show that attentive attackers can achieve good +attack success rates, but this is due to the lambs being more +vulnerable rather than wolf-like behaviour. +5.3. Feature Informativeness +To investigate which features are most informative to +our models, we sum the top five features, sorted by Gini +importance, of each classifier. (Note that, w.r.t. the counts, +there are six times more classifiers for the payment models.) +For ring tap gestures, Table 3a shows that our models +favour GRV-derived features when using data from the smart +ring only, but accelerometer- and gyroscope-derived features +when using data from the smartwatch only (similar to the +features favoured in [15]), indicating that the finger moves +to a position faster and remains in a position longer than +the wrist, whose movements are smoother. The combined +model, which achieved stronger results than the others as +we saw in Figure 3, had twice as many members in its +feature vector and ended up favouring a similar set, echoing +the relative importance of these features. +For 3-knock gestures, Table 3b shows the dominance of +features derived from the y- and z-axes of the wearable de- +vices, which is to be expected as these measure the sideways +and forward movements of the hand, respectively. Aside +from these, two notable exceptions show greater importance: +the x-axis of the smartwatch yields a single important fea- +ture, representing the maximum acceleration of the arm as +it extends towards the door initially, and the median impact +experienced by the door-mounted accelerometer, indicating +that each user struck the door with consistent force. +5.4. Sensor Selection +We collected motion data from all of the inertial sensors +available on our devices. Some devices are more limited in +their offering—the accelerometer is the commonest sensor, +as it is the smallest and cheapest. To assess the feasibility +of our approach on devices with fewer sensors, we trained +7 + +Ring +Watch +Combined +Feature +# +Feature +# +Feature +# +GRV-y-med +309 +Acc-x-min +297 +w-Acc-x-min +185 +GRV-y-mean +269 +Acc-y-max +208 +w-Acc-y-max +151 +GRV-x-mean +239 +Acc-x-max +208 +r-Gyr-z-mean +145 +GRV-x-med +223 +Acc-z-max +189 +r-Acc-x-velomax +141 +GRV-x-max +222 Gyr-z-velomean +138 +r-GRV-x-max +135 +GRV-y-max +217 +Gyr-z-mean +126 +r-Acc-x-mean +128 +Gyr-z-mean +192 +Gyr-z-min +124 +r-GRV-y-mean +122 +Acc-x-mean +162 +Gyr-z-disp +119 +r-Acc-x-med +122 +GRV-z-max +157 +Acc-x-mean +117 +r-GRV-x-med +116 +Acc-x-velomax 155 +Acc-x-velomax +116 +r-GRV-y-med +116 +(a) payment model in {s = 2.5, o = 0}, ring tap gesture +Ring +Combined +Feature +# +Feature +# +GRV-w-med +24 +d-Acc-y-med +22 +Acc-y-disp +22 +w-Acc-x-max +20 +GRV-y-max +20 +w-Gyr-z-var +17 +GRV-w-mean +19 +w-Gyr-y-med +17 +GRV-z-med +19 +w-Gyr-z-velomax 15 +Acc-z-mean +18 +w-Gyr-z-stdev +15 +Acc-z-med +18 +w-LAc-z-disp +15 +Gyr-y-velomax +18 +w-GRV-y-iqr +15 +GRV-x-mean +18 +w-GRV-y-med +14 +LAc-x-disp +17 +w-Gyr-y-min +12 +(b) access control model, 3-knock gesture +Table 3: Modal top-five features by Gini importance summed over classifiers for our payment authentication and access control models, +using data from (i) the smart ring only, (ii) the smartwatch only, and (iii) both combined for the former and from (i) the smart ring only and +(ii) the door-mounted sensors, smart ring, and smartwatch combined for the latter. Features are given in the format sensor-axis-statistic; +for combined models, the leading character indicates the device to which the sensor belongs (door, ring, or watch). +a set of sensor-specific models in which each classifier is +trained and tested on data from a subset of sensors. +For models that use data from the smart ring only, we +see distinctly poorer results whenever we remove the GRV +sensor. With just an accelerometer, we see an increase of +approximately 0.04 in every EER. Conversely, for models +that use data from the smartwatch only, Figure A.2 in the +Appendix shows that we achieve better results when using +only the accelerometer and gyroscope; this suggests that +the other sensors pollute the smartwatch classifiers, echoing +similar findings in related work [15]. The combined models +remain roughly unchanged, favouring ring features when the +GRV is included and watch features when not. +5.5. Terminal Positions +We collected tap gestures performed against a range of +terminals. Some systems, such as public transport systems, +have highly standardised terminals (i.e., dedicated terminals +that can be found set at the same position in many instances). +To compare the effectiveness of our approach in a general +setting against a standardised setting, we trained a set of +terminal-specific models in which each classifier is trained +and tested on data from a single terminal. +Table 4 shows the average EERs for our terminal- +specific payment authentication models when trained and +tested on tap gestures from a single terminal. In general, +we gain little improvement from restricting our system to a +single terminal. We find that user comfort has a beneficial +impact on authentication results. Terminals 5 and 6 show +slightly improved results and these were the most comfort- +ably positioned terminals for the majority of participants, +who wore the devices on their left arm (indeed, if we +reconstruct our models using data only from those users +wearing the devices on their left arm, the relative gains +are greater still). Likewise, the freestyle terminal shows +improved results for watch tap gestures, as the watch was +more awkward to tap than the ring and this terminal accom- +modated smoother movements when using it—however, it +Terminal +Ring +Combined +1 +0.07 +0.04 +2 +0.09 +0.07 +3 +0.09 +0.07 +4 +0.08 +0.03 +5 +0.06 +0.02 +6 +0.07 +0.03 +F +0.10 +0.09 +agnostic +0.07 +0.04 +(a) ring tap gesture +Terminal +Watch Combined +1 +0.09 +0.05 +2 +0.09 +0.07 +3 +0.09 +0.07 +4 +0.06 +0.05 +5 +0.06 +0.03 +6 +0.04 +0.02 +F +0.05 +0.04 +agnostic +0.09 +0.05 +(b) watch tap gesture +Table 4: Average EERs for our terminal-specific payment authen- +tication models in optimum window {s = 2.5, o = 0}, using data +from the smart ring only, the smartwatch only, and both combined. +Our terminal-agnostic results are included for comparison. +had the opposite effect for ring tap gestures, perhaps because +most users tilted and moved the terminal a shorter distance +when interacting with it with the ring than with the watch, +eliciting a simpler punch gesture. +5.6. Enrolment Parameters +Behavioural biometric systems typically entail a bur- +densome enrolment phase, where the user must perform +the measured characteristic repeatedly to create the initial +template. To evaluate the extent to which we can expedite +the enrolment phase, we compare the average EERs of our +authentication models when the classifiers are trained on a +smaller positive class (i.e., fewer user samples). +Figure 5a shows that our payment models can authen- +ticate the user with EERs as low as 0.12 when trained on +just twelve of the user’s tap gestures (spread evenly over six +terminals), which can be performed in less than a minute. +Figure 5b shows that, when including watch data, our 3- +knock access control model can authenticate the user with +EERs as low as 0.12 when trained on just two 3-knock +gestures (the access control models for the other knock +8 + +gestures show a similar pattern), which can be performed +in a few seconds. In both cases, we see that the EERs im- +prove as more samples are included in the training set; this +suggests that an update mechanism might benefit the models +over time, relaxing upfront requirements and incorporating +subsequent gestures as the system is used. +6. Discussion +Power Consumption. Wearable devices are designed +to facilitate always-on sensing (e.g., in health monitoring +applications). To measure the impact of our data collection +in practical terms, we wore two of each wearable device +in an identical state, but only collecting data from one of +each. For the smart rings, there was no noticable difference +in power consumption over 6 hours. For the smartwatches, +without any effort put into performance optimisation, our +app caused the smartwatch running it to consume an addi- +tional 1.5% of battery capacity per hour. While we did not +implement the random forest classifier on the devices, we +argue that its energy consumption would be negligible due +to the limited number of inferences that would be required +per day (only when the user needs to authenticate, such as +to make a payment). +Response Time. We calculated the computation time for +classifying a single watch tap gesture, averaged over 10,000, +to be 7.11 ms for authentication on a desktop computer +with an Intel Core i5-6500 processor. Using a benchmarking +tool5, we found that a Samsung Exynos W920 (a modern +smartwatch processor) performs 26 times slower, so we +would expect an authentication decision to be made in +roughly 185 ms on a smartwatch. Robust benchmarking is +not yet publicly available for smart ring CPUs. +User Feedback. All of the participants in our user study +found ring tap gestures to be easier and more comfortable to +perform than watch tap gestures, due to the manoeuvrability +of the hand compared to that of the wrist; some terminal +positions were more awkward than others, depending on the +height of the user and the wrist upon which the smartwatch +was worn. One female participant, who wore the devices +on her right arm, commented that she considered the smart- +watch used in this study to be a men’s watch due to its +bulk and that, while she would normally wear a women’s +(smaller) watch on her right (dominant) wrist, she would +have to wear this one on her left wrist for daily use because +she would find it obstructive otherwise. +7. Related Work +Tap Gestures. The use of inertial sensor data to authen- +ticate tap gestures in tap-and-pay systems was proposed by +Shrestha et al. [13] for smartphone-based systems, achieving +F-measure scores of 0.93, and by Sturgess et al. [14, 15] +for smartwatch-based systems, achieving F-measure scores +of 0.87 and EERs of 0.08. The use of a smartwatch, due +to the physiology of the arm, introduced the challenge that +5https://www.notebookcheck.net +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +number of training samples per terminal +0.06 +0.08 +0.10 +0.12 +0.14 +0.16 +EER +ring tap, ring data +watch tap, ring data +ring tap, watch data +watch tap, watch data +ring tap, combined data +watch tap, combined data +(a) payment model in {s = 2.5, o = 0} +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +number of training samples +0.05 +0.10 +0.15 +0.20 +0.25 +EER +ring data +watch data +door data +combined data +(b) access control model, 3-knock gesture +Figure 5: Average EERs for our payment and access control models +if trained on different numbers of enrolment samples. For the +payment model, each classifier is trained on six terminals. +the sensor axes frequently change reference frames because +the device changes orientation during the tap gesture. We +found that the use of a smart ring sits between the two +in terms of complexity: the smartphone requires no major +change in orientation, the smart ring requires only a single +change because the finger is easily manoeuvred towards +the terminal, and the smartwatch orientation is changed +frequently during the tap gesture. We found similar results +in our smartwatch models and improved results with our +smart ring and combined models. +Smartwatches. The use of inertial sensors on smart- +watches have been used in a variety of authentication cases. +Johnston et al. [6] showed that wrist motion data can be used +to both identify and authenticate a user while walking with +10-second windows of data. Nassi et al. [11] showed that +wrist motion data can be used to authenticate handwritten +signatures and other authors [2, 3, 4, 16] applied a similar +approach to freestyle handwriting with 5- to 60-second win- +dows of data. We found optimum results with 2.5 seconds +of data for tap gestures and 2.8 seconds for knock gestures. +A number of works have used inertial sensor data from a +smartwatch to support the authentication of a user on another +device. Mare et al. [9] showed that wrist motion data can +be used to infer a sequence of interactions from a user and +9 + +correlated against inputs on his workstation, such that he +can be de-authenticated if the correlation stops (however, +the system was found to have vulnerabilities due to design +flaws [5]). Acar et al. [1] showed that wrist motion data can +be correlated with keystrokes to continuously authenticate +the user of the workstation. Other works [7, 10] have +correlated wrist motion data with smartphone interactions +to authenticate the user of the smartphone. +Smart Rings. Few authors have considered the use +of smart rings in authentication use-cases. Sen et al. [12] +proposed the use of a smart ring that is capable of pro- +ducing a vibration to bootstrap a communication channel +with another device held in the same hand that has an +accelerometer to detect the vibration. Liang et al. [8] showed +that inertial sensor data from a smart ring can be correlated +with mouse movements to continuously authenticate the user +of a workstation. To the best of our knowledge, we are the +first to propose the use of inertial sensors on a smart ring to +authenticate a user via implicit or explicit gesture biometrics. +8. Limitations and Future Work +3-knock +Impersonation. +Table +2 +shows +that +we +achieved our best overall results from the 3-knock gesture. +In our preliminary experiments, this was not the case, so we +chose not to include it in the impersonation exercise of our +user study. This is regrettable, as the observation attack may +have yielded more interesting results for 3-knock gestures +than for 5-knock gestures. +Sampling Rate. By comparing Figure 3 with Figure +A.1 in the Appendix, we see that downsampling our smart +ring data from 100 Hz to 50 Hz imposed only a slight cost +in performance (at most, a difference of 0.01 in average +EERs). Nonetheless, future work should endeavour to use +a smartwatch and Raspberry Pi IMU with higher sampling +rates, to remove any reason for downsampling, and may see +improved scores across the board. +9. Conclusion +In this paper, we showed that inertial sensor data from +a smart ring can be used to authenticate the wearer. In a +mobile payment context, we showed that a smart ring user +can be implicitly authenticated with a single tap gesture +with an EER of 0.04. We also showed that inertial sensor +data from a smart ring can be used to authenticate the +user when making a smartwatch payment, and vice versa, +opening the possibility for either device to be used as an +implicit second factor for the other. In an access control +context, we showed that a smart ring user can be (explicitly) +authenticated with a single knock gesture with an EER of +0.06 (or 0.02 with a smartwatch). We demonstrated that our +authentication models provide resistance against an active +impersonation attacker who observed the victim’s gestures +and we showed that successful attacks were more likely the +result of luck than of a skilled attacker. +Acknowledgement +This work was supported financially by Mastercard; +the Engineering and Physical Sciences Research Council +[grant number EP/P00881X/1]; and the PETRAS National +Centre of Excellence for IoT Systems Cybersecurity [grant +number EP/S035362/1]. The authors would like to thank +these organisations for their support, Genki Instruments for +their collaboration and technical support, and the anonymous +reviewers for their feedback. +References +[1] +A. Acar, H. Aksu, A. S. Uluagac, and K. Akkaya. “A Usable and Ro- +bust Continuous Authentication Framework using Wearables”, IEEE +Transactions on Mobile Computing (TMC), 2020. +[2] +F. Ciuffo and G. M. 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Lee. “Implicit Sensor-based Authentication of +Smartphone Users with Smartwatch”, ACM Hardware and Architec- +tural Support for Security and Privacy (HASP), 2016. +[8] +X. Liang and D. Kotz. “AuthoRing: Wearable User-presence Au- +thentication”, ACM Workshop on Wearable Systems and Applications +(WearSys), 2017. +[9] +S. Mare, A. M. Markham, C. Cornelius, R. Peterson, and D. Kotz. +“ZEBRA: Zero-Effort Bilateral Recurring Authentication”, IEEE +Symposium on Security and Privacy (S&P), 2014. +[10] S. Mare, R. Rawassizadeh, R. Peterson, and D. Kotz. “Continuous +Smartphone Authentication using Wristbands”, Workshop on Usable +Security and Privacy (USEC), 2019. +[11] B. Nassi, A. Levy, Y. Elovici, and E. Shmueli. “Handwritten Signature +Verification using Hand-worn Devices”, ACM Interactive, Mobile, +Wearable and Ubiquitous Technologies (IMWUT), Vol. 2, 2016. +[12] S. Sen and D. Kotz. “VibeRing: Using Vibrations from a Smart Ring +as an Out-of-band Channel for Sharing Secret Keys”, Journal of +Pervasive and Mobile Computing, Vol. 78, 2020. +[13] B. Shrestha, M. Mohamed, S. Tamrakar, and N. Saxena. “Theft- +Resilient Mobile Wallets: Transrgessparently Authenticating NFC +Users with Tapping Gesture Biometrics”, Annual Conference on +Computer Security Applications (ACSAC), 2016. +[14] J. Sturgess, S. Eberz, I. Sluganovic, and I. Martinovic. “Inferring User +Height and Improving Impersonation Attacks in Mobile Payments +using a Smartwatch”, IEEE International Conference on Pervasive +Computing and Communication Workshops (PerCom Workshops), +2022. +10 + +[15] J. Sturgess, S. Eberz, I. Sluganovic, and I. Martinovic. “WatchAuth: +User Authentication and Intent Recognition in Mobile Payments using +a Smartwatch”, IEEE European Symposium on Security and Privacy +(EuroS&P), 2022. +[16] R. Wijewickrama, A. Maiti, and M. Jadliwala. “Write to Know: +On the Feasibility of Wrist Motion Based User-Authentication from +Handwriting”, ACM Conference on Security and Privacy in Wireless +and Mobile Networks (WiSec), 2021. +Appendix +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +window size, s (s) +1.0 +0.5 +0.0 +offset, o (s) +0.13 0.10 0.09 0.09 0.07 0.07 1.00 1.00 +0.13 0.11 0.09 0.08 0.07 0.06 0.06 1.00 +0.13 0.10 0.09 0.08 0.07 0.07 0.06 0.06 +(a) ring tap; ring data +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +window size, s (s) +1.0 +0.5 +0.0 +offset, o (s) +0.12 0.10 0.09 0.08 0.07 0.07 1.00 1.00 +0.14 0.10 0.08 0.07 0.07 0.06 0.06 1.00 +0.16 0.12 0.09 0.07 0.06 0.06 0.06 0.06 +(b) watch tap; ring data +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +window size, s (s) +1.0 +0.5 +0.0 +offset, o (s) +0.09 0.06 0.05 0.05 0.05 0.04 1.00 1.00 +0.11 0.08 0.06 0.05 0.05 0.04 0.04 1.00 +0.12 0.09 0.07 0.05 0.05 0.05 0.04 0.04 +(c) ring tap; combined data +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +window size, s (s) +1.0 +0.5 +0.0 +offset, o (s) +0.10 0.07 0.05 0.04 0.04 0.04 1.00 1.00 +0.14 0.09 0.07 0.05 0.04 0.04 0.04 1.00 +0.16 0.12 0.08 0.06 0.05 0.04 0.04 0.04 +(d) watch tap; combined data +Figure A.1: Average EERs for our payment authentication models +by window size and offset, for tap gestures made with the smart +ring (left) and smartwatch (right), using data from the smart ring +only (top) and the smart ring and smartwatch combined (bottom), +in all cases based on undownsampled data from the smart ring. (cf. +Figure 3 to compare with the downsampled smart ring data.) +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +window size, s (s) +1.0 +0.5 +0.0 +offset, o (s) +0.12 0.08 0.06 0.06 0.05 0.05 1.00 1.00 +0.17 0.11 0.07 0.06 0.06 0.05 0.05 1.00 +0.21 0.14 0.10 0.07 0.06 0.06 0.05 0.05 +(a) ring tap; watch data +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +window size, s (s) +1.0 +0.5 +0.0 +offset, o (s) +0.15 0.11 0.08 0.06 0.06 0.05 1.00 1.00 +0.21 0.14 0.10 0.07 0.06 0.06 0.05 1.00 +0.24 0.19 0.14 0.10 0.07 0.07 0.06 0.06 +(b) watch tap; watch data +Figure A.2: Average EERs for our payment authentication models +by window size and offset, for tap gestures made with the smart +ring (left) and smartwatch (right), using only accelerometer and +gyroscope data from the smartwatch. (cf. Figure 3 to compare with +the all-sensor data.) +11 + +0.05 +0.10 +0.15 +0.20 +R +0.25 +0.30 +0.35 \ No newline at end of file diff --git a/K9E2T4oBgHgl3EQfAwYv/content/tmp_files/load_file.txt b/K9E2T4oBgHgl3EQfAwYv/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7fcbdaaf5554cc5a37007fb1e5a2e288d3ae19d9 --- /dev/null +++ b/K9E2T4oBgHgl3EQfAwYv/content/tmp_files/load_file.txt @@ -0,0 +1,1184 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf,len=1183 +page_content='RingAuth: Wearable Authentication using a Smart Ring Jack Sturgess, Simon Birnbach, Simon Eberz, and Ivan Martinovic Department of Computer Science, University of Oxford, Oxford, UK {firstname.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='lastname}@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='ox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='uk Abstract—In this paper, we show that by using inertial sensor data generated by a smart ring, worn on the finger, the user can be authenticated when making mobile payments or when knocking on a door (for access control).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' The proposed system can be deployed purely in software and does not require up- dates to existing payment terminals or infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' We also demonstrate that smart ring data can authenticate smartwatch gestures, and vice versa, allowing either device to act as an implicit second factor for the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' To validate the system, we conduct a user study (n=21) to collect inertial sensor data from users as they perform gestures, and we evaluate the system against an active impersonation attacker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Based on this data, we develop payment and access control authentication models for which we achieve EERs of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='04 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='02, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Index Terms—wearable authentication, mobile payment, smart ring, smartwatch, tap gesture, knock gesture, authentication 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Introduction Mobile payment systems (also known as tap-and-pay), such as Google Pay, have become pervasive in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' These systems enable the user to provision payment cards to a virtual wallet on a smartphone and then facilitate cashless and contactless payments with NFC-enabled point-of-sale terminals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' The functionality of mobile payment systems has been extended to wearable devices, such as smartwatches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' When paired with a smartphone, a smartwatch can access and store the same virtual wallet and make payments even when the smartphone is not present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' However, the options for authenticating the user to a smartwatch are limited, given its size, and so the smartphone is still regarded as the primary device for the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Recently, we are begin- ning to see commercial smart rings enter the market that offer payment capabilities, such as the Mastercard K-Ring1, which have even fewer options for authentication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' In order to protect payment transactions in these emerging systems, new authentication factors that operate on wearable devices are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Given that wearable devices are often designed with continuous healthcare or fitness monitoring use-cases in mind, they tend to have an inertial measurement unit (IMU) consisting of (at least) an accelerometer and gyroscope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' 1https://thepaymentring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='com Works in behavioural biometrics have shown that inertial sensor data in smartphones and smartwatches can be used to infer gait or gestures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' These systems require some initial calibration (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=', a cumbersome enrolment phase), but can then continuously or discretely authenticate the user with reasonable effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' In this work, we show that inertial sensor data generated by a smart ring can be used to authenticate the user when making certain gestures, such as tapping a payment terminal or knocking on a door.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' We propose a novel, smart ring-based authentication system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Using only inertial sensor data, we show that a single tap gesture performed by a user while making a payment with a smart ring can implicitly authenticate the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' We also show that smart ring data can be used to authenticate a user making payments with a smartwatch, and vice versa, allowing either device to act as an implicit second factor for the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' We show that our approach can also be applied to an access control context, in which a single knock gesture against a door can explicitly authenticate the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' We demonstrate that our authentication models are resistant against an active impersonation attacker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' We make the code and data required to reproduce our results available at http://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='com/jacksturgess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Paper Structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' The rest of this paper is organised in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Section 2 states our objectives and details our system and threat models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Section 3 describes the design of our data collection user study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Section 4 explains the methods that we employ to process and classify our data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Section 5 presents and analyses our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Section 6 dis- cusses peripheral topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Section 7 compares our approach to related work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Section 8 considers the limitations of our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Section 9 concludes the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Objectives and Assumptions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Design Goals In this paper, we investigate the use of a smart ring for authentication purposes, both implicit and explicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' For implicit authentication, we select the context of mobile payments in which we consider the tap gesture made arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='03594v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='CR] 9 Dec 2022 when the user taps an NFC-enabled device against a point- of-sale terminal to provide an implicit gesture biometric and we pose the following questions: can tap gestures made with a smart ring, as measured by the inertial sensors of the smart ring, be used to implicitly authenticate the user during a transaction?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' can tap gestures made with a smart ring, as measured by the inertial sensors on a smartwatch worn on the same arm, be used to implicitly authenticate the user (in cases where the smart ring does not have inertial sensors of its own) during a transaction?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' can implicit authentication in a smartwatch system be improved by incorporating (the inertial sensor data of) a smart ring as a second factor?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' For explicit authentication, we choose the act of knock- ing on a door.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' A knock gesture could be used implicitly for identification purposes or it could be performed solely for access control purposes, in which case it would be treated as an explicit authentication gesture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' We pose the question: can (explicit) knock gestures, as measured by the inertial sensors of a smart ring, a smartwatch, or directly mounted on the door, be used to authenticate the user?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' We consider both unrhythmic knocks and user-chosen secret knocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' System Model For our payment model, we consider a system model in which the user is wearing both a smart ring and a smartwatch on the same arm and is using them to make NFC-enabled payments at point-of-sale terminals in a typical setting (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=', in a shop).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' To make a payment, we assume that the user performs a tap gesture by moving one of the devices towards the terminal until it is near enough to exchange data via NFC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' The NFC contact point is when the payment provider would decide whether to approve the payment, so we assume that this marks the end of the tap gesture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' For our access control model, we consider a system model in which the user is wearing both a smart ring and a smartwatch on the same arm and is knocking on a door with that hand as a means to authenticate to an access control system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' We assume that each knock gesture is bounded by button-presses on one of the devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' We assume that the devices have an accelerometer and gyroscope and that we have access to their data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' We use data from the inertial sensors only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' We assume that the user’s biometric templates are stored securely on the wearable devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' When we combine data from multiple devices into a single sample for classification, as we do in some of our models, we assume that one device shares its inertial sensor data wirelessly with the other in a trusted fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' We assume that the devices will be running a trusted app that is able to communicate its authentication decisions with the payment provider or access control system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Threat Model We consider an adversary that has possession of a legitimate user’s smart ring (or smartwatch, or both, as Figure 1: The equipment used in our experiments: six fixed termi- nals (labelled, see Table 1 for details), an NFC reader (affixed to Terminal 1), a Raspberry Pi for timestamp collection, a Raspberry Pi attached to a door, and a fixed smartphone for video recording.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Inset: our smart ring and smartwatch worn on the left arm with the sensor axes shown (the z-axis points upwards through the screen).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Terminal Height (cm) Tilt (◦) Distance (cm) 1 100 0 5 2 120 60 25 3 95 45 10 4 105 30 15 5 110 15 10 6 115 90 30 F picked up from centre of platform Table 1: Details of the terminals used in our experiment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' the indices match those labelled in Figure 1 and ‘F’ is the freestyle terminal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Height is measured from the floor to the lowest point of the terminal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Tilt is the inclination at the lowest point of the terminal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' and Distance is measured from the front of the stand to the point of the terminal that is closest to the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Terminals 2 and 6 match terminals on self-service checkouts at supermarket chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Figure 2: The attacker’s restricted view of a victim interacting with Terminal 2 (left), Terminal 3 (centre), and the door (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' appropriate), has unlocked it, is wearing it, and is attempting to use it to make a payment at a terminal or to gain entry to a locked door via an access control system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' The adversary may have (maliciously) stolen the device(s) or (benignly) borrowed it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Our goal is to authenticate the legitimate user and to reject the adversary by using only inertial sensor data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' We consider the following two types of attack: zero-effort attack: for any given victim, all other users are considered to be passive, zero-effort attackers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' observation attack: an active attacker who watches the victim perform gestures (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=', via a hidden camera) and then attempts to mimic them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' 2 2 4 6 3 y 5 1 z 7In this work, we concentrate on the extent to which gesture biometrics can be used to defend against these attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' We do not consider threats to other components in the system, tampering of devices or biometric templates, malware, or denial of service attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Experimental Design 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Experiment Overview To evaluate the extent to which finger and wrist motion data can be used to authenticate users, and to compare the two, we designed and conducted a user study to collect data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Our experiments consisted of six point-of-sale terminals on an adjustable stand fixed at a height of 100 cm, an ACR122U NFC reader connected to a Raspberry Pi for timestamp col- lection, a second Raspberry Pi with an accelerometer and a gyroscope attached to a closed door, and a smartphone fixed in position for video recording.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' For our wearable devices, we used a smart ring and a smartwatch (detailed below), both commercial off-the-shelf devices and worn together on the same arm by the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' We collected motion data from the wearable devices and the door-mounted Raspberry Pi (with all clocks synchronised) as the user performed gestures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Figure 1 shows our apparatus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' For our payment experiment, we affixed an NFC tag to the front of each wearable device and we affixed the NFC reader to the front of each point-of-sale terminal in turn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' For each terminal, the user stood in front of the stand and performed tap gestures on the terminal using a wearable device, as if making mobile payments, with a short spacing delay between each tap gesture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' The use of NFC tags and the NFC reader ensured consistent NFC communication between wearable devices and terminals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Each NFC tag stored the name of the wearable device to which it was affixed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' each time an NFC contact point was made during a tap gesture, the Raspberry Pi captured its timestamp and the name of the triggering device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' This was later used to segment the inertial sensor data collected from the wearable devices to retrieve the data for each tap gesture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' For our access control experiment, the user stood in front of the door and performed knock gestures against it using his device-wearing hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' The user pressed the button on the smart ring before and after each knock gesture to capture bounding timestamps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' These timestamps were later used to segment the inertial sensor data collected from all devices to retrieve the data for each knock gesture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Point-of-Sale Terminals To emulate a real-world mobile payment setting as much as possible, we captured tap gestures using seven terminals: six in fixed positions (as detailed in Table 1) and one ‘freestyle’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' For five of the six fixed terminals, we surveyed prominent supermarket and restaurant chains to find popular or standardised terminal positions (in terms of height, tilt angle, and distance from the user) and set our terminals to match common configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' We set the other fixed termi- nal (Terminal 3) to match the position of the terminal on a train station barrier in the UK, which is widely standardised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' For the freestyle terminal, the user picked up the NFC reader with his other hand and performed a tap gesture against it, returning it after each interaction, as if a shopkeeper had handed an unmounted terminal to a customer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' The six fixed terminals remained deactivated throughout the experiment because their functionality was not required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' For consistent data collection, we affixed the NFC reader to each terminal when using it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' As such, the terminals should be regarded only as fixtures that enforced positions, as well as a tool for immersing the user in a payment scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Wearable Devices and Sensor Modules For our smart ring, we used a Genki Wave2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' This ring is worn on the index finger and has a button that can be pressed with the thumb—we utilised this for timestamp collection in our access control experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' We wrote a data collection script in Python using the open source library3 provided by the developers to interface with the ring over Bluetooth LE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' For our smartwatch, we used a Samsung Galaxy Watch4 running the Tizen 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='0 operating system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' We built a data collection app and installed it on the smartwatch using the Tizen Studio IDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' From each of these wearable devices, we collected timestamped data from four inertial sensors directly or derived from their MEMS sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' The accelerometer mea- sures change in velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' The gyroscope measures angular velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' The linear accelerometer is derived from the ac- celerometer with the effects of gravity excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' The gyro- scope rotation vector (GRV) is a fusion of sensor readings to compute the orientation of the device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' We collected this data with sampling rates of 100 Hz for the smart ring (which we downsampled to 50 Hz), 50 Hz for the smartwatch, and 30 Hz for the door-mounted Raspberry Pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' The inertial sensor axes are fixed relative to the frame of each device (as shown in Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Motion along the x-axis corresponds with arm extension or withdrawal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' the y-axis, with side-to-side arm waving;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' and the z-axis, directly up- and downwards through the screen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Tap and Knock Gestures To collect tap gestures for our payment experiment, we had each user perform sets of ten of each of the following tap gestures on each of the seven point-of-sale terminals: ring tap: the user tapped the smart ring against the terminal and moved it around, if necessary, until NFC contact was made to simulate a ring-based payment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' watch tap: the user tapped the smartwatch against the terminal and moved it around, if necessary, until NFC contact was made to simulate a watch-based payment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' 2https://genkiinstruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='com/products/wave 3https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='com/genkiinstruments/genki-wave 4https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='samsung.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='com/uk/watches/galaxy-watch 3 To collect knock gestures for our access control experi- ment, we had each user perform sets of ten of each of the following knock gestures on the closed door: 3-knock: the user knocked on the door three times;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' 5-knock: the user knocked on the door five times;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' secret knock: the user created a knock pattern consisting of between three and six knocks performed in a manner of his choosing (some users chose to knock to a rhythm, some included a twist or jolt of the wrist).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' User Study To collect our data, we conducted a user study that was reviewed and approved by the relevant research ethics committee at our university.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' We recruited 21 participants, including staff, students, and members of the public.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Each participant attended two data collection sessions on separate days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' We collected 30 of each of the gestures detailed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='4 (510 gestures in total) from each participant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' In each experiment, the participant was asked to stand facing the terminals or the door;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' aside from this, we did not prescribe any constraints on positioning as we wanted the user to interact comfortably as though acting in a real-world setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' The first three gestures of any type were performed in silence, to familiarise, then the reseacher engaged the par- ticipant in light conversation to simulate the distractions of a real-world environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' This sometimes elicited additional hand and body movements if the user gesticulated naturally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Impersonation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' To evaluate the robustness of our ap- proach against an observation attacker, all 21 participants consented to having their gestures recorded and participated in an impersonation exercise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' The first 3 were recorded as victims and the latter 18 impersonated them;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' then, at the end of the study, the first 3 were invited back to impersonate the other 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' This design also enabled us to compare the suscep- tibility of different victims and the skill of different attackers (also known as wolf and lamb analysis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' We recorded the following six gestures of each participant: smart ring and smartwatch tap gestures on Terminals 2 and 3, 5-knock gestures, and secret knock gestures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' The camera was fixed in position, as if hidden, and so the amount of observable information was controlled;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Figure 2 shows the attacker’s view of the terminals and the different information observable for each type of gesture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' For each of the six attacks, the attacking participant watched a short video of the victim performing the gesture three times and then made three attempts to mimic him.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' The attacker wore the wearable devices on the same arm as the victim during this exercise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' User Statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Of our 21 participants: 15 were male, 17 wore the devices on the left arm (the decision was led by the smartwatch;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' everyone who wore them on the right arm was female), 15 regularly wore a watch of some kind (7 of which wore a smartwatch), 13 had paid with a smartphone before, and 5 had paid with a smartwatch (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=', 71% of those who regularly wore one).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' 16 participants (76%) remembered their secret knock, with an average of 4 days between their sessions (those who did not remember had an average of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='2 days between sessions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Methods 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Data Processing We collect time-series data from the four inertial sensors on our smart ring and smartwatch worn by the user and from the accelerometer attached to the door.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Each accelerometer, gyroscope, or linear accelerometer sample is given in the form (t, x, y, z) and represents the change in velocity or angular velocity along each axis at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Each GRV sample is given as a quaternion in the form (t, x, y, z, w) and approximates the orientation of the device at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' We express a tap or knock gesture using a series of inertial sensor data samples within a time window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' In our payment experiment, we retrieve the tap gestures for each user by segmenting 4-second blocks of sensor data using the NFC contact point timestamps as the endpoint of each window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' We found that a 4-second maximum window size was sufficient to encapsulate the entirety of each tap gesture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' To investigate optimum tap gesture parameters, we compare (in Section 5) the performances of gestures bounded by various window sizes and offsets, where the offset is the time between the NFC contact point and the end of the window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' For an NFC contact point timestamp T0, a window size s, and an offset o, we retrieve a tap gesture with start time TS and end time TE, where TE = T0 − o and TS = TE − s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' In our access control experiment, we retrieve the knock gestures for each user using the bounding timestamps captured when the user pressed the button on the ring before and after each gesture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Feature Extraction Whenever a gesture is retrieved, we apply a low pass filter to the data to reduce noise and then process the fol- lowing five dimensions for each accelerometer, gyroscope, or linear accelerometer sample: the filtered x-, y-, and z- values, the energy of those filtered values, and the energy of the unfiltered (raw) values, where the energy of {x, y, z} is given by � x2 + y2 + z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' As GRV samples are expressed as quaternions, for those we process only the four filtered values (since the Euclidean norm of a quaternion is always 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' In total, we process each gesture in 19 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' For each gesture, we extract the following ten statistical features in each dimension: minimum, maximum, mean, me- dian, standard deviation, variance, inter-quartile range, kur- tosis, skewness, and peak count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' We also calculate the mean and maximum velocities along each axis, the displacement along each axis, and the Euclidean displacement from each of its accelerometer, gyroscope, and linear accelerometer vectors, adding another 30 features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Ultimately, we reduce each gesture to a feature vector containing 220 members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' In previous work [14, 15], we found this feature set to be ideal by starting with a larger set and pruning it down using normalised Gini importances to reject the least informative features, so we use it again here on similar grounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' This approach also yielded promising preliminary results in our 4 access control model, so we used it for both models to allow for comparability and consistency throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' In some of our models, we combine inertial sensor data from multiple devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' In these cases, the above features are extracted for each separate source and then concatenated together to form a linearly-larger feature vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Classification We use random forest classifiers in each of our payment authentication and access control models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Similar works [1, 15] have shown that random forests are efficient, able to estimate the importance of features, and robust against noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' To balance relevance with learning time, we include 100 trees in each forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' To reduce the impact of random generation on our results, and to ensure that our results are fair and unselective, we train and test each classifier ten times with different forest randomisation seeds and average the outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Terminal-agnostic Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' To evaluate the zero-effort attacker, we train a set of classifiers that are user-dependent and terminal-agnostic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' This means that a separate template (and decision threshold) is generated for each user and that, for each tap gesture under test, the tap gesture samples used to train the classifier came from tap gestures made only against other terminals (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=', a leave-one-out approach).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' The user’s tap gestures form the positive class and all other user’s tap gestures form the negative class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' As this is an authentication scenario, we ensure that the training data precedes the testing data by taking the tap gestures collected in users’ first data collection session as training data (analogous to the enrolment phase, where the user template is created) and those collected in the second session as testing data (analogous to an authentication phase).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Terminal-known Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' To evaluate the observation attacker, we apply a similar design, except that we do not exclude tap gestures based on the terminal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' We assume that, since the attacker has already observed the victim using the terminal in question, that the system has knowledge of the victim’s tap gestures made against that terminal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' We pair up every user as a victim with every other user as an attacker, one at a time, and train the classifiers, excluding all of the attacker’s tap gestures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' This enables us to generate the victim’s decision threshold for that pairing, tuned to the EER (which we take as our baseline FAR, the base-FAR), with no knowledge of the attacker, and then we test the attacker’s impersonation samples against that tuned classifier to find his attack success rate (the observation-FAR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' We compare the two FARs to measure the success of the observation attack (see Sections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='2 for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Terminal-specific Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' For investigative purposes, we also train a set of classifiers in which each is trained and tested on tap gestures performed only on a single terminal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' This model enables us to measure the effectiveness of our approach if implemented on standardised, single-terminal systems, such as public transport systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Access Control Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' For knock gestures, to evaluate each of the attackers, we use a similar approach as with each respective payment authentication model above, except that we do not need to generalise the model over multiple terminals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' We train separate models for each of the three knock gestures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Performance Metrics In each model, the true positives is the number of times that the positive class (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=', the legitimate user) is correctly accepted;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' the true negatives is the number of times that the negative class (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=', the adversary) is correctly rejected;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' the false positives is the number of times that the negative class is wrongly accepted;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' and the false negatives is the number of times that the positive class is wrongly rejected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' The decision threshold, θ, is the score at which the classifier chooses to assign to a sample the positive class rather than the negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' To tune a classifier, we adjust θ to modify the trade-off between security and usability;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' a larger θ is more resilient to false positives and thus favours security, a smaller θ favours usability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' To quantify the performance of our models, we find the optimum decision threshold such that the false acceptance rate (FAR) equals the false rejection rate (FRR);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' this point is called the equal error rate (EER).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' The FAR inversely indi- cates security, by measuring the likelihood that the negative class will be wrongly accepted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' The FRR inversely indicates usability, by measuring the likelihood that the positive class will be wrongly rejected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' The FAR and FRR are antagonistic insofar as setting θ to favour one will disfavour the other, so there will always be a point at which they cross over;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' the EER is a measure of system performance when considera- tion is balanced evenly between security and usability and is commonly used as a metric in authentication systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Results 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Zero-effort Attack Tap Gestures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Figure 3 shows the average EERs for our terminal-agnostic payment authentication models by window size and offset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' With 21 users and 6 fixed terminals, each score is the average of scores from 21×6×10 = 1, 260 classifiers (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='3 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' For ring tap gestures, Figure 3a shows that when using data from the smart ring only our model achieves EERs as low as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Figure 3c shows that when using data from the smartwatch only, we get 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' This suggests that a smartwatch could perform as a reasonable authenticator for a smart ring, in a scenario where the smart ring does not have any inertial sensors of its own (such as the K-Ring).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' If the data from both devices are combined, Figure 3e shows that we achieve EERs as low as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' The optimum parameters for ring tap gestures, by considering EERs and favouring a smaller window size for usability, are {s = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='5, o = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' When we use data from the smartwatch only, we see a pattern radiating from the bottom left corner of the heatmap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' This shows that, when training only on data where the tap- ping device is near to the terminal (small window, just before 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='0 window size, s (s) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='5 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='04 (f) watch tap;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' combined data Figure 3: Average EERs for our payment authentication models by window size and offset, for tap gestures made with the smart ring (left) and smartwatch (right), using data from the smart ring only (top), the smartwatch only (middle), and both combined (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' (This figure is based on downsampled smart ring data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='1 in the Appendix is based on undownsampled smart ring data for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=') the NFC contact point is found), the movements of the wrist are not discriminative between users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' The same is not true when using data from the smart ring only;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' this suggests that the finger remains active during that time, even for watch tap gestures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Figure 3f shows that including a smart ring as an additional factor can improve the authentication of a user making watch-based payments (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Figure 3d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Knock Gestures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Across all of the participants in our user study, the average 3-knock gesture lasted 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='81 seconds, the average 5-knock, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='32 seconds, and the average secret knock, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='78 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Table 2 shows the average EERs for our access control models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Each score is the average of scores from 21 × 10 = 210 classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Unexpectedly, our 3-knock model performed best, achieving an EER of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='02 using data from the smartwatch only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' This might be due to some participants sometimes miscounting the number of knocks when performing the 5-knock gesture, leading to messier data, whereas the 3-knock gestures were performed effortlessly and more consistently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' When we use data from the smartwatch only, we achieve the best results across all models, including those based on combined data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' This suggests that wrist movements are a key discriminator in knocking, to such an extent that other factors act as pollutants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' When we use data from the door only, we achieve the poorest results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' the lower sampling rate of the door-mounted sensors may have had an impact, but the magnitude of the difference in results is likely explained by those sensors lacking knowledge of user movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Knock Gesture Door Ring Watch Combined 3-knock 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='03 5-knock 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='05 secret knock 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='05 Table 2: Average EERs for our access control models, using data from (i) the door-mounted sensors only, (ii) the smart ring only, (iii) the smartwatch only, and (iv) all three combined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Observation Attack Figure 4 shows the results of our observation at- tack against our terminal-known payment authentication model and our access control model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Each of the first 3 users was the victim of 1,080 ring tap impersonation attempts (18 attackers × ring tap gestures on 2 terminals × 3 attempts at each gesture×10), 540 5-knock attempts, and 540 secret knock attempts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' each of the other 18 users, separated in the figures by a red line, was the victim of 180 (3 attackers), 90, and 90 attempts, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' For ring tap gestures, the base model achieves average EERs (base-FARs) of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='03 when using data from the smart ring only and of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='01 when using data from the smart ring and smartwatch combined;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' when attacked, the average suc- cess rates (observation-FARs) are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='06 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='05, respectively (and we see similar results for watch tap gestures).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Figures 4a and 4b show that a small number of our users are lambs 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='20 R 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='350.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='8 FAR (a) ring tap;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' ring data 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='8 (b) ring tap;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' combined data 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='8 (c) 5-knock;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' combined data 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='8 (d) secret knock;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' combined data 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='4 FAR (e) ring tap;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' ring data 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='4 (f) ring tap;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' combined data 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='4 (g) 5-knock;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' combined data 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='4 (h) secret knock;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' combined data Figure 4: Results of our observation attack against our terminal-known payment authentication model in optimum window {s = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='5, o = 0} and our access control model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' The top row shows for each user as a victim the average FAR of the user-specific base model (flat line) and the average FAR when attacked (circle);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' if the latter is greater, then the victim’s line is coloured blue, otherwise it is orange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' The bottom row shows for each user as an attacker the average FAR achieved when attempting to impersonate other users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' The red line separates the first 3 users from the other 18, indicating the two groups of users that attacked each other (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='5 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=', users that are especially susceptible to impersonation), where their observation-FAR is significantly larger than their base-FAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Figure 4b shows that the addition of the smartwatch data helps to reduce the largest FAR deltas, and the overall average observation-FAR, but also opens a new vector that increases the susceptibility of some users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Figures 4e and 4f show that none of our users are wolves (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=', users that are especially skilled at impersonation);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' when using data from the smart ring only, some attempts got lucky against a random spattering of users, but when the smartwatch data was combined, the success rate dropped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' This is an important result, as it suggests that our system provides resistance against wolves, reducing the likelihood that an attacker could predictably impersonate a given victim (and so, in the wider system model, this may act as a deterrent).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' For knock gestures, when using data from the door- mounted sensors, smart ring, and smartwatch combined, we have average base-FARs of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='05 for both the 5-knock and secret knock gestures and average observation-FARs of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='08 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='09, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Figures 4c and 4d show that we have a number of lambs, this time with greater FAR deltas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Knock gestures are notably weak against impersonation if they are loud and slow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' For the secret knock, the fourth and sixth users after the red line have high base-FARs, because those users chose common gesture fragments in their secret knocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' For the former (and the seventh), the gesture was loud and slow, as evidenced by the huge observation-FAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' For the latter, curiously, the observation-FAR is far lower than the base-FAR, suggesting that the gesture was difficult to mimic intentionally despite having commonality with other gestures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' This gesture contained three fast knocks in the middle, which captured the attention of attackers only for them not to match the surrounding knocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Figures 4g and 4h show that attentive attackers can achieve good attack success rates, but this is due to the lambs being more vulnerable rather than wolf-like behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Feature Informativeness To investigate which features are most informative to our models, we sum the top five features, sorted by Gini importance, of each classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' (Note that, w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' the counts, there are six times more classifiers for the payment models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=') For ring tap gestures, Table 3a shows that our models favour GRV-derived features when using data from the smart ring only, but accelerometer- and gyroscope-derived features when using data from the smartwatch only (similar to the features favoured in [15]), indicating that the finger moves to a position faster and remains in a position longer than the wrist, whose movements are smoother.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' The combined model, which achieved stronger results than the others as we saw in Figure 3, had twice as many members in its feature vector and ended up favouring a similar set, echoing the relative importance of these features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' For 3-knock gestures, Table 3b shows the dominance of features derived from the y- and z-axes of the wearable de- vices, which is to be expected as these measure the sideways and forward movements of the hand, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Aside from these, two notable exceptions show greater importance: the x-axis of the smartwatch yields a single important fea- ture, representing the maximum acceleration of the arm as it extends towards the door initially, and the median impact experienced by the door-mounted accelerometer, indicating that each user struck the door with consistent force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Sensor Selection We collected motion data from all of the inertial sensors available on our devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Some devices are more limited in their offering—the accelerometer is the commonest sensor, as it is the smallest and cheapest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' To assess the feasibility of our approach on devices with fewer sensors,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' we trained ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='Ring ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='Watch ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='Combined ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='Feature ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='# ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='Feature ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='# ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='Feature ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='# ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='GRV-y-med ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='309 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='Acc-x-min ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='297 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='w-Acc-x-min ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='185 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='GRV-y-mean ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='269 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='Acc-y-max ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='208 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='w-Acc-y-max ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='151 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='GRV-x-mean ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='239 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='Acc-x-max ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='208 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='r-Gyr-z-mean ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='145 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='GRV-x-med ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='223 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='Acc-z-max ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='189 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='r-Acc-x-velomax ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='141 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='GRV-x-max ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='222 Gyr-z-velomean ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='138 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='r-GRV-x-max ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='135 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='GRV-y-max ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='217 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='Gyr-z-mean ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='126 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='r-Acc-x-mean ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='128 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='Gyr-z-mean ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='192 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='Gyr-z-min ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='124 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='r-GRV-y-mean ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='122 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='Acc-x-mean ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='162 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='Gyr-z-disp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='119 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='r-Acc-x-med ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='122 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='GRV-z-max ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='157 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='Acc-x-mean ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='117 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='r-GRV-x-med ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='116 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='Acc-x-velomax 155 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='Acc-x-velomax ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='116 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='r-GRV-y-med ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='116 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='(a) payment model in {s = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' o = 0},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' ring tap gesture Ring Combined Feature # Feature # GRV-w-med 24 d-Acc-y-med 22 Acc-y-disp 22 w-Acc-x-max 20 GRV-y-max 20 w-Gyr-z-var 17 GRV-w-mean 19 w-Gyr-y-med 17 GRV-z-med 19 w-Gyr-z-velomax 15 Acc-z-mean 18 w-Gyr-z-stdev 15 Acc-z-med 18 w-LAc-z-disp 15 Gyr-y-velomax 18 w-GRV-y-iqr 15 GRV-x-mean 18 w-GRV-y-med 14 LAc-x-disp 17 w-Gyr-y-min 12 (b) access control model,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' 3-knock gesture Table 3: Modal top-five features by Gini importance summed over classifiers for our payment authentication and access control models,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' using data from (i) the smart ring only,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' (ii) the smartwatch only,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' and (iii) both combined for the former and from (i) the smart ring only and (ii) the door-mounted sensors,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' smart ring,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' and smartwatch combined for the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Features are given in the format sensor-axis-statistic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' for combined models, the leading character indicates the device to which the sensor belongs (door, ring, or watch).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' a set of sensor-specific models in which each classifier is trained and tested on data from a subset of sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' For models that use data from the smart ring only, we see distinctly poorer results whenever we remove the GRV sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' With just an accelerometer, we see an increase of approximately 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='04 in every EER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Conversely, for models that use data from the smartwatch only, Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='2 in the Appendix shows that we achieve better results when using only the accelerometer and gyroscope;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' this suggests that the other sensors pollute the smartwatch classifiers, echoing similar findings in related work [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' The combined models remain roughly unchanged, favouring ring features when the GRV is included and watch features when not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Terminal Positions We collected tap gestures performed against a range of terminals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Some systems, such as public transport systems, have highly standardised terminals (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=', dedicated terminals that can be found set at the same position in many instances).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' To compare the effectiveness of our approach in a general setting against a standardised setting, we trained a set of terminal-specific models in which each classifier is trained and tested on data from a single terminal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Table 4 shows the average EERs for our terminal- specific payment authentication models when trained and tested on tap gestures from a single terminal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' In general, we gain little improvement from restricting our system to a single terminal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' We find that user comfort has a beneficial impact on authentication results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Terminals 5 and 6 show slightly improved results and these were the most comfort- ably positioned terminals for the majority of participants, who wore the devices on their left arm (indeed, if we reconstruct our models using data only from those users wearing the devices on their left arm, the relative gains are greater still).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Likewise, the freestyle terminal shows improved results for watch tap gestures, as the watch was more awkward to tap than the ring and this terminal accom- modated smoother movements when using it—however, it Terminal Ring Combined 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='04 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='07 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='07 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='03 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='02 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='03 F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='09 agnostic 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='04 (a) ring tap gesture Terminal Watch Combined 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='05 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='07 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='07 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='05 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='03 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='02 F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='04 agnostic 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='05 (b) watch tap gesture Table 4: Average EERs for our terminal-specific payment authen- tication models in optimum window {s = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='5, o = 0}, using data from the smart ring only, the smartwatch only, and both combined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Our terminal-agnostic results are included for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' had the opposite effect for ring tap gestures, perhaps because most users tilted and moved the terminal a shorter distance when interacting with it with the ring than with the watch, eliciting a simpler punch gesture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Enrolment Parameters Behavioural biometric systems typically entail a bur- densome enrolment phase, where the user must perform the measured characteristic repeatedly to create the initial template.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' To evaluate the extent to which we can expedite the enrolment phase, we compare the average EERs of our authentication models when the classifiers are trained on a smaller positive class (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=', fewer user samples).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Figure 5a shows that our payment models can authen- ticate the user with EERs as low as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='12 when trained on just twelve of the user’s tap gestures (spread evenly over six terminals), which can be performed in less than a minute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Figure 5b shows that, when including watch data, our 3- knock access control model can authenticate the user with EERs as low as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='12 when trained on just two 3-knock gestures (the access control models for the other knock 8 gestures show a similar pattern), which can be performed in a few seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' In both cases, we see that the EERs im- prove as more samples are included in the training set;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' this suggests that an update mechanism might benefit the models over time, relaxing upfront requirements and incorporating subsequent gestures as the system is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Discussion Power Consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Wearable devices are designed to facilitate always-on sensing (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=', in health monitoring applications).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' To measure the impact of our data collection in practical terms, we wore two of each wearable device in an identical state, but only collecting data from one of each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' For the smart rings, there was no noticable difference in power consumption over 6 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' For the smartwatches, without any effort put into performance optimisation, our app caused the smartwatch running it to consume an addi- tional 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='5% of battery capacity per hour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' While we did not implement the random forest classifier on the devices, we argue that its energy consumption would be negligible due to the limited number of inferences that would be required per day (only when the user needs to authenticate, such as to make a payment).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Response Time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' We calculated the computation time for classifying a single watch tap gesture, averaged over 10,000, to be 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='11 ms for authentication on a desktop computer with an Intel Core i5-6500 processor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Using a benchmarking tool5, we found that a Samsung Exynos W920 (a modern smartwatch processor) performs 26 times slower, so we would expect an authentication decision to be made in roughly 185 ms on a smartwatch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Robust benchmarking is not yet publicly available for smart ring CPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' User Feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' All of the participants in our user study found ring tap gestures to be easier and more comfortable to perform than watch tap gestures, due to the manoeuvrability of the hand compared to that of the wrist;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' some terminal positions were more awkward than others, depending on the height of the user and the wrist upon which the smartwatch was worn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' One female participant, who wore the devices on her right arm, commented that she considered the smart- watch used in this study to be a men’s watch due to its bulk and that, while she would normally wear a women’s (smaller) watch on her right (dominant) wrist, she would have to wear this one on her left wrist for daily use because she would find it obstructive otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Related Work Tap Gestures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' The use of inertial sensor data to authen- ticate tap gestures in tap-and-pay systems was proposed by Shrestha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' [13] for smartphone-based systems, achieving F-measure scores of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='93, and by Sturgess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' [14, 15] for smartwatch-based systems, achieving F-measure scores of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='87 and EERs of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' The use of a smartwatch, due to the physiology of the arm, introduced the challenge that 5https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='notebookcheck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='net 2 4 6 8 10 12 14 16 18 20 number of training samples per terminal 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='16 EER ring tap, ring data watch tap, ring data ring tap, watch data watch tap, watch data ring tap, combined data watch tap, combined data (a) payment model in {s = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='5, o = 0} 2 4 6 8 10 12 14 16 18 20 number of training samples 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='25 EER ring data watch data door data combined data (b) access control model, 3-knock gesture Figure 5: Average EERs for our payment and access control models if trained on different numbers of enrolment samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' For the payment model, each classifier is trained on six terminals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' the sensor axes frequently change reference frames because the device changes orientation during the tap gesture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' We found that the use of a smart ring sits between the two in terms of complexity: the smartphone requires no major change in orientation, the smart ring requires only a single change because the finger is easily manoeuvred towards the terminal, and the smartwatch orientation is changed frequently during the tap gesture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' We found similar results in our smartwatch models and improved results with our smart ring and combined models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Smartwatches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' The use of inertial sensors on smart- watches have been used in a variety of authentication cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Johnston et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' [6] showed that wrist motion data can be used to both identify and authenticate a user while walking with 10-second windows of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Nassi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' [11] showed that wrist motion data can be used to authenticate handwritten signatures and other authors [2, 3, 4, 16] applied a similar approach to freestyle handwriting with 5- to 60-second win- dows of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' We found optimum results with 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='5 seconds of data for tap gestures and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='8 seconds for knock gestures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' A number of works have used inertial sensor data from a smartwatch to support the authentication of a user on another device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Mare et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' [9] showed that wrist motion data can be used to infer a sequence of interactions from a user and 9 correlated against inputs on his workstation, such that he can be de-authenticated if the correlation stops (however, the system was found to have vulnerabilities due to design flaws [5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Acar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' [1] showed that wrist motion data can be correlated with keystrokes to continuously authenticate the user of the workstation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Other works [7, 10] have correlated wrist motion data with smartphone interactions to authenticate the user of the smartphone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Smart Rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Few authors have considered the use of smart rings in authentication use-cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Sen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' [12] proposed the use of a smart ring that is capable of pro- ducing a vibration to bootstrap a communication channel with another device held in the same hand that has an accelerometer to detect the vibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Liang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' [8] showed that inertial sensor data from a smart ring can be correlated with mouse movements to continuously authenticate the user of a workstation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' To the best of our knowledge, we are the first to propose the use of inertial sensors on a smart ring to authenticate a user via implicit or explicit gesture biometrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Limitations and Future Work 3-knock Impersonation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Table 2 shows that we achieved our best overall results from the 3-knock gesture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' In our preliminary experiments, this was not the case, so we chose not to include it in the impersonation exercise of our user study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' This is regrettable, as the observation attack may have yielded more interesting results for 3-knock gestures than for 5-knock gestures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Sampling Rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' By comparing Figure 3 with Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='1 in the Appendix, we see that downsampling our smart ring data from 100 Hz to 50 Hz imposed only a slight cost in performance (at most, a difference of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='01 in average EERs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Nonetheless, future work should endeavour to use a smartwatch and Raspberry Pi IMU with higher sampling rates, to remove any reason for downsampling, and may see improved scores across the board.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Conclusion In this paper, we showed that inertial sensor data from a smart ring can be used to authenticate the wearer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' In a mobile payment context, we showed that a smart ring user can be implicitly authenticated with a single tap gesture with an EER of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' We also showed that inertial sensor data from a smart ring can be used to authenticate the user when making a smartwatch payment, and vice versa, opening the possibility for either device to be used as an implicit second factor for the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' In an access control context, we showed that a smart ring user can be (explicitly) authenticated with a single knock gesture with an EER of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='06 (or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='02 with a smartwatch).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' We demonstrated that our authentication models provide resistance against an active impersonation attacker who observed the victim’s gestures and we showed that successful attacks were more likely the result of luck than of a skilled attacker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Acknowledgement This work was supported financially by Mastercard;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' the Engineering and Physical Sciences Research Council [grant number EP/P00881X/1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' and the PETRAS National Centre of Excellence for IoT Systems Cybersecurity [grant number EP/S035362/1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' The authors would like to thank these organisations for their support, Genki Instruments for their collaboration and technical support, and the anonymous reviewers for their feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' References [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Acar, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Aksu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' S.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Peterson, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Kotz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' “Continuous Smartphone Authentication using Wristbands”, Workshop on Usable Security and Privacy (USEC), 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' [11] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Nassi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Levy, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Elovici, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Shmueli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' “Handwritten Signature Verification using Hand-worn Devices”, ACM Interactive, Mobile, Wearable and Ubiquitous Technologies (IMWUT), Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' 2, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' [12] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Sen and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Kotz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' “VibeRing: Using Vibrations from a Smart Ring as an Out-of-band Channel for Sharing Secret Keys”, Journal of Pervasive and Mobile Computing, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' 78, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' [13] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Shrestha, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Mohamed, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Tamrakar, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Saxena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' “Theft- Resilient Mobile Wallets: Transrgessparently Authenticating NFC Users with Tapping Gesture Biometrics”, Annual Conference on Computer Security Applications (ACSAC), 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' [14] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Sturgess, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Eberz, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Sluganovic, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Martinovic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' “Inferring User Height and Improving Impersonation Attacks in Mobile Payments using a Smartwatch”, IEEE International Conference on Pervasive Computing and Communication Workshops (PerCom Workshops), 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' 10 [15] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Sturgess, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Eberz, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Sluganovic, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Martinovic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' “WatchAuth: User Authentication and Intent Recognition in Mobile Payments using a Smartwatch”, IEEE European Symposium on Security and Privacy (EuroS&P), 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' [16] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Wijewickrama, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Maiti, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Jadliwala.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' “Write to Know: On the Feasibility of Wrist Motion Based User-Authentication from Handwriting”, ACM Conference on Security and Privacy in Wireless and Mobile Networks (WiSec), 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Appendix 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='5 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='06 (a) ring tap;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' ring data 0.' metadata={'source': 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accelerometer and gyroscope data from the smartwatch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=' Figure 3 to compare with the all-sensor data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content=') 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/K9E2T4oBgHgl3EQfAwYv/content/2301.03594v1.pdf'} +page_content='15 0.' metadata={'source': 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University College London, United +Kingdom +Abstract +The visual dimension of cities has been a fundamental subject in urban studies, since the +pioneering work of scholars such as Sitte, Lynch, Arnheim and Jacobs. Several decades later, +big data and artificial intelligence (AI) are revolutionizing how people move, sense, and interact +with cities. This paper reviews the literature on the appearance and function of cities to illustrate +how visual information has been used to understand them. A conceptual framework, Urban +Visual Intelligence, is introduced to systematically elaborate on how new image data sources +and AI techniques are reshaping the way researchers perceive and measure cities, enabling +the study of the physical environment and its interactions with socioeconomic environment at +various scales. The paper argues that these new approaches enable researchers to revisit the +classic urban theories and themes, and potentially help cities create environments that are more +in line with human behaviors and aspirations in the digital age. +Keywords: +Urban visual intelligence, physical environment, place, street-level imagery, deep learning, +human-environment interactions +1. Introduction +Images have been central to how urbanists understand cities. As a crucial medium, images +inform the methodologies to measure the characteristics of the physical environment. The im- +portance of images dates back to the 19th century when normative theories asserted the aesthetic +∗Corresponding author email: cefzhang@ust.hk +Preprint submitted to Elsevier +January 3, 2023 +arXiv:2301.00580v1 [cs.CV] 2 Jan 2023 + +value of cities. The aesthetic experience of urban spaces was emphasized as a leading factor of +urban planning (Sitte, 1889; Arnheim, 1965). The consideration of making cities beautiful per- +sisted as an important thread throughout the development of planning approaches into the 1920s +(Freestone, 2011). Leading designers and theorists, such as Frederick Law Olmsted Sr., Phillip +Mackintosh, and F. W. Fitzpatrick, argued the societal role of beauty that would elicit citizens’ +satisfaction, comfort, and pride (Wilson et al., 1990; Mackintosh, 2005; Mulford, 1899; Nasar, +1994; Ahlfeldt and Mastro, 2012). +Then, in the 20th century, the tradition of using images to understand cities shifted the +focus with the development of urban design theories. By the late 1950s, urban designers were +concerned with the inhospitable urban places that modernism urban forms had produced. In +response, designers advocated a broader approach that could account for the performance and +use of space. This epistemological shift resulted from the desire to move away from normative +theories of how the city “should be” to how people use urban space by incorporating empirical +observations. These empirical observations largely relied on visual information obtained from +photos and videos to assess how physical environments affect individuals’ use of urban space +(Jacobs, 2011; Whyte, 1980; Appleyard et al., 1981; Jacobs and Appleyard, 1987). For example, +the early work by Kevin Lynch (Lynch, 1960) introduced the concept of imageability to explain +the vast differences in the mental impressions of individuals produced in Boston, Jersey City, +and Los Angeles. Using photos and interviews with residents, Lynch solicited, assembled, and +analyzed perceptual maps to identify areas that captured citizens’ attention and made a lasting +impression. This work was later extended using surveys and visual audits to show that citizens +recall places when they evoke strong feelings (Nasar, 1998), highlighting memory and emotion +as the key determinants (Clay, 1980). +Traditional data collection methods that rely on visual information, such as images, videos, +and direct observation, provide rich insights into the intersection of human activity and city +form. Nevertheless, these methods are very time-consuming and labor-intensive. It is difficult +to scale over large spatial regions or extended time periods. Today, access to new sources of +geotagged data and advances in sensing technologies allow for a more detailed and larger-scale +examination of cities. These changes open up the possibility of comparisons across regions +and over time. With the rich and extensive sources of visual data, what remains unclear is +the different characteristics of various types of visual data, and how visual information can be +effectively derived in a standardized manner. Despite the large body of work dedicated to the +application of visual data in analyzing the appearance of neighborhoods, it’s not yet evident how +the physical environment of a place can be conceptually quantified, and how such a quantified +representation of a place can contribute to a systematic understanding of the human-place +relationship and revisit classical theories and practices. +2 + +This paper reviews the theories and recent empirical work on the use of visual informa- +tion to understand cities. We introduce a conceptual framework, Urban Visual Intelligence, to +illustrate how images and Artificial Intelligence can be paired to observe, measure, and repre- +sent the features of the physical environment, as well as its interplay with the socioeconomic +environment. +2. Historical overview of visual information in urban studies +The tradition of incorporating visual information runs through the history of modern urban +studies. Since the early days of modern city planning, planners have documented and measured +physical environmental attributes that could be extracted directly from photographs or sketches +taken along the street sidewalk. Planners describe the urban forms through shape, proportion, +rhythm, scale, complexity, color, order, elements, and hierarchy (Wohlwill, 1976). This tra- +dition of using the formal attributes of the physical environment to elicit a pleasant sensory +experience for citizens goes back to Camilo Sitte. He argued that the city should be interpreted +through visual art and architecture. In particular, Sitte was highly critical of rigid symmetry +and highlighted the value of irregularity in urban form, proposing that the aesthetic component +of cities should be a leading factor in their design (Sitte, 1889). This focus on the design of +the physical environment as a way to influence the behavior of citizens culminated in planning +utopias, such as the Garden City and the City Beautiful Movement. Advocates of these philoso- +phies believe that the beauty, order and cleanliness of the public realm can influence civic spirit +and improve the quality of life (Talen and Ellis, 2002). Although there is a consensus that cities +should be aesthetically pleasing and beautiful places, the disagreement on what makes a city or +a space beautiful persists. Is beauty in the eye of the beholder? Alternatively, could aesthetics +be measured so that designers could apply that measure to design spaces that appeal to many? +With these questions remaining unanswered, in the 20th century, planners’ focus started +shifting from the formal attributes of the physical environment (aesthetics) to the subjective +experiences they induce. At its core, this approach towards urban design sought a better un- +derstanding of how humans sense and evaluate an urban scene visually. The supporting studies +emphasized the importance of understanding cities using people’s visual perceptions (Arnheim, +1965). In parallel, researchers attempted to capture how a city’s physical environment can trig- +ger emotions that help inform our understanding of appealing and unappealing environments. +Nasar (1998), for example, proposed a model that explains how aesthetic responses arise from +human interaction with the surrounding environment. Similarly, Rapoport (1990) identified +36 characteristics related to the size and shape of typical aesthetically pleasing urban environ- +ments. Overall, these studies on perception focused on how people shape their environment and +how the physical environment, in turn, affects them. However, these studies had only a limited +3 + +impact on the theories and practice of urban studies because they lack approaches to quantify +and represent the physical environment on a large scale. +To quantify and represent the physical environment of a place, Lynch (1960) introduced +“imageability” as a new criterion (Lynch, 1984). Building on ideas about perception but focused +on human cognition instead of just aesthetics, Lynch identified the importance of meaning to +explain how people understand and navigate urban environments. His study showed that as +people move through an environment, they accumulate spatial knowledge obtained through +observation and translate it into mental maps. In the Image of the City, Lynch proposed three +categories to summarize the physical environment: Identity (distinct visual objects), Structure +(recognizable patterns with relationships between objects), and Meaning (emotional values and +character of a place) (Lynch, 1960). Initially, Lynch assessed these three dimensions using +some of the traditional approaches adopted by urbanists to observe the city: sketch maps, field +surveys, and interviews collected for a small number of neighborhoods and participants. In a +similar vein, Milgram (1970) proposed drawing collective maps of New York City measuring +how recognizable it was using several small-scale experiments. +Work as such paved the philosophy shift towards people-centered and place-based urban +design in the late 1900s. Designers and planners started to embrace the emphasis on the per- +formance, vitality, and use of spaces as an alternative way to measure the quality of urban +design(Gehl, 1971). Typically, scholars gather information on how people use urban spaces +using simple recording techniques with pen and paper, augmented by photographic images. A +classic example of this approach is William H. Whyte’s seminal study on the social life of pub- +lic spaces, “The Street Life project,” who used a combination of conversations, photographs, +and a careful analysis of video recordings to observe how people use public spaces (Whyte, +1980). Similarly, in his influential book Life between Buildings, Gehl (1971) employed exten- +sive field observations to document the elements that make public spaces lively and contribute +to people’s social interactions. Overall, scholars studying human-centered urban design applied +observation and video to measure human behavior and people’s appropriation of public areas +(Pushkarev, 1976). These studies have profoundly informed 21st-century’s urban design prac- +tice. Their methods became standard practices to document and understand the interactions +between physical and socioeconomic environments. +These pioneers offered path-breaking implications for urban studies and design. However, +we today live in a world changing much more rapidly and those studies would need to be re- +peated in different contexts and time periods to answer the questions we have today. In addition, +researchers today also question the small sample size and bias in selecting subjects in the afore- +mentioned studies, which makes them subject to the variability of preferences across time and +different populations (Nasar, 1998). +4 + +3. Framework of Urban Visual Intelligence +Today, hybridized sensing techniques - crowdsensing or ad hoc sensor deployment - offer +researchers diverse data to analyze city life. Urban big data and AI-driven approaches provide +tools to quantify the physical environment, socioeconomic conditions, and human dynamics +with unprecedented performance. With these tools, researchers can observe the interaction +between human behavior and the physical environment across spatial and temporal scales. +In view of these opportunities, we propose a framework, Urban Visual Intelligence to re- +view the method and data that researchers adopt today that are different from what was used +historically. In particular, our framework elaborates on how existing visual intelligence tech- +nologies are being used to observe, measure, and represent the urban physical environment +and study its interaction with human dynamics and socioeconomic environments. Importantly, +this framework centers around studies using AI-based tools to analyze street-level imagery. We +hope to illustrate the key issues and complementary approaches to these studies, weave together +different technologies, and think from the existing literature. +Figure 1 shows the framework. There are four hierarchical levels that describe four issues: +How can urban physical environments be observed at a human scale? How can semantic in- +formation be interpreted from street-level imagery? How can the physical environment of a +place be quantified? And how to understand the fine-grained interactions between the physical +environment, human activities, and their socioeconomic environment? The top level of the dia- +gram focuses on the visual data sources (e.g., Google Street View and crowdsourced platforms). +Studies associated with this level focus on using street-level imagery to observe physical envi- +ronments at the human scale. One street-level image can be generally considered a vista. At +the second level of the diagram, a street view image depicts a scene at a certain location. The +second level focuses on computer vision and deep learning techniques to derive semantic infor- +mation about scenes in the physical environment, for example, measuring tree or sky coverage +in the scene depicted in a street-level image. The third level shows how studies can use a collec- +tion of images to characterize and create quantitative representations of a place. For instance, +to study place structure and place perception. Finally, the fourth level goes one step further to +show how studies can also apply the measurements of places or physical environments to study +their interactions with human dynamics and socioeconomic characteristics. +5 + +Figure 1: Framework of Urban Visual Intelligence. The framework shows the key issues identified in the urban +physical environment and the corresponding studies. Accordingly, the framework elaborates on using visual in- +telligence technologies to observe, measure, and represent physical environments and study their interaction with +human dynamics and socioeconomic dimensions at different levels and scales. +From top to bottom, the four levels (observing, interpreting, measuring, and discovering) +increase complexity. They constitute a list of the process to apply images to understand urban +environments. The first level shows the data sources available to observe the physical environ- +ment. The second level shows the techniques used to derive and interpret semantic information +from the urban scene depicted in street-level imagery. The third level shows how this informa- +tion can be synthesized to characterize a place. The fourth level shows how it can be used to +study fine-grained interactions between the physical and socioeconomic environments. These +four levels also indicate the four scales (i.e., vista, scene, place, and city) in which street-level +imagery is used. The following sections unpack each of the four levels of the Urban Visual +Intelligence in more detail and outline relevant works. +6 + +Observing +Map service image +How can urban physical +Street-level +1 +environments be observed at +crowdsourced photos +Imagery +Vista +human-scale +Custom collection +How can +Interpreting +Deep learning +Image +2 +semantic information +LAI +Computer vision +Scene +be derived from +understanding +Image dataset +street-level imagery +Measuring +Place identity +How can physical +Place +3 +Place structure +environments of a place be +Place +characterization +quantified +Place perception + How to understand the +Discovering : +Public health +fine-grained interactions +Human-place +4 +Transportation +between +City +relationship +Socioeconomics +physical and socioeconomic +environment +Urban Visual Intelligence +Urban Physical Environment4. Data—How to observe large-scale urban physical environments? +The first level in the Urban Visual Intelligence framework introduces street-level imagery +as a key data source to study urban environments. According to Keypoint Intelligence, 1 1.43 +trillion photos were taken from digital cameras of mobile phones in 2020. The adoption of +mobile internet technologies and the fast development of web mapping services and crowd- +sourcing platforms has resulted in geotagged images being produced rapidly, blanketing every +corner of cities (Goodchild, 2007). This new data source is commonly known as “street-level +imagery,” it has extensive spatial coverage and has been used widely to observe large-scale +urban environments (Ibrahim et al., 2020; Biljecki and Ito, 2021; Duarte and Ratti, 2021). +Figure 2 shows three common sources of street-level imagery available to analyze cities: +1) map service images (such as Google Street View), 2) crowdsourced photos obtained from +crowdsourcing platforms (such as Flickr and Mapillary), and 3) custom collection of images +(through e.g., Dash Cams and GoPros). For the first category, the images are generated with +a stable update frequency, the widest coverage (more than 200 countries worldwide), and a +uniform and consistent standard to facilitate comparative analysis between places (Anguelov +et al., 2010; Goel et al., 2018). The second category, crowdsourced photos, is essentially a type +of Volunteered Geographic Information (VGI) (Goodchild, 2007). As the volume of data grows +and becomes denser in terms of spatiotemporal coverage, crowdsourced photos are expected to +replace map service as the primary source of street-level imagery. For the third category, the +custom collection features images taken by individuals or researchers for specific topics. For +example, by collecting a time series of images of a place, researchers can record changes in the +physical environment and individual activities. Custom collection can complement mapping +services and crowdsourced imagery. +1https://keypointintelligence.com/ +7 + +Figure 2: Three sources of street-level imagery +Compared to the traditional data sources used to learn about cities, such as personal inter- +views and direct observations, street-level imagery has advantages including easy access, high +spatiotemporal coverage, and objective and standardized views of the physical environment +collected from embedded vantage points (Rzotkiewicz et al., 2018; Ibrahim et al., 2020; Kang +et al., 2020a). +In comparison to satellite imagery, in particular, street view imagery offers a complemen- +tary perspective. The major difference is that street view imagery depicts the world from a +street level with a human-like perspective, instead of approaching it from an aerial view which +is an approach that predominated in mid-20th-century modernism studies (Clay, 1980).2 Such +an approach enables researchers to study visual cues, at the human scale that can directly re- +late to people’s perceptions and the use of cities to inform planning and design. This is only +possible until very recently as images have been standardized across cities and visual analytic +methodologies have been developed to perform the appropriate analysis. +Street-level imagery is now considered one of the most important data sources to study +physical environments (Biljecki and Ito, 2021; Cinnamon and Jahiu, 2021; He and Li, 2021). In +the past few years, we have witnessed its application in a wide range of study areas, including +physical environment auditing (Kang et al., 2018; Li et al., 2018b; Zhang et al., 2020b; Ning +et al., 2021), public health (Nguyen et al., 2018; Keralis et al., 2020; He et al., 2020; Bivand +and Piras, 2015), urban mobility and transportation (Lu et al., 2019; Hong et al., 2020; Mooney +et al., 2020), energy estimation (Liu et al., 2019; Zhang et al., 2022; Sun et al., 2022a), and real +2Historically, urban planners and geographers have been using satellite imagery to understand morphology, +measure urban expansion and densification, and classifying different land uses. +8 + +a) Map service image +b) Crowdsourced photo +c) Custom collection +Google Street View +Mapillary +Dash Cam +Bing Streetside Imagery +Flickr +GoPro +Tencent Street View +Instagram +Mobile Phoneestate (Law et al., 2019; Yang et al., 2020a; Johnson et al., 2020; Kang et al., 2020b), among +others. +5. Technology—how can semantic information be interpreted from street-level imagery +in a scene? +The second level in the Urban Visual Intelligence framework introduces deep learning and +computer vision techniques used to derive and interpret semantic information from street-level +imagery. +5.1. Deep learning and computer vision +Conventionally, photos taken from field surveys are interpreted by eye inspection, making +it a labor-intensive process and imposing an upper limit on the geographic scale of research. +Image processing techniques that enable the analysis of visual information in large groups have +been developed to mitigate these issues, making detailed larger-scale studies possible. How- +ever, conventional image processing-based methods continue to be limited to processing low- +level features, such as color histograms and spectral features. Despite the useful information +they provide, these approaches cannot extract high-level information, such as semantic objects, +styles, and conditions of a scene. As we will explain in more detail, these features are key to +studying the relationship between a city’s physical appearance and human behavior. +Deep learning and computer vision techniques have been developed to learn and extract +high-level information from images. Deep learning refers to a series of computer algorithms +inspired by the human brain’s neural structure, which allows models to mimic human cognitive +functions, such as understanding, learning, planning, and problem-solving. Deep learning has +enabled the advancement of many fields including speech recognition (Hinton et al., 2012), nat- +ural language processing (Sutskever et al., 2014), game problem solving (Silver et al., 2016), +and computer vision (Ren et al., 2015; He et al., 2017), etc. These achievements are attributed +to deep learning models’ outstanding performance in effectively and efficiently extracting high- +level information from images. For city applications, deep learning techniques offer a com- +pelling framework for understanding the content of urban images. +Figure 3 shows how a deep learning model works. In particular, the figure explains how +a deep learning model can be trained for computer vision tasks, e.g., detecting objects in an +image and categorizing an image according to its content. The core model in the figure is a deep +convolutional neural network (DCNN). A DCNN’s ultimate goal is to assign a “true” label to +an image in order to make predictions about the class of scenes or objects in the image. For any +given tasks, the process can be divided into two phases: training and inference. In the training +phase, an image (Figure 3a) is fed into a pre-designed DCNN and processed individually layer +9 + +Label +a) Input Image +c) Output Label +b) Deep Convolutional Neural Network +Predicting +Training +… +Figure 3: +How deep learning works in urban image predictions. a) Input urban image; b) Deep convolutional +neural network; c) Output label +by layer in the DCNN (Figure 3b). The final layer of the DCNN will generate a predicted label +that is then compared with the image’s true label (Figure 3c). The parameters in the DCNN are +optimized in an iterative manner by successively minimizing the difference between predicted +and true labels. In the inference phase, the well-trained model is used to make predictions on +new images. The learning and inference phases are similar to human’s learning procedure— +observing a phenomenon, recognizing a pattern, and allowing feedback to improve the learning +process (LeCun et al., 2015). +5.2. Large-scale image datasets +A key ingredient to successfully train deep learning models is the availability of large +datasets. Deep learning models require a vast amount of labeled images as ground truth to learn +the complex relationships between an input and its label. An ideal image dataset is expected to +be of high coverage and density. The “coverage” requires a quasi-exhaustive representation of +categories and a wide variety of exemplars. The “density” refers to having a sufficiently large +sample of images that cover the diversity of each predicted category (Zhou et al., 2017a). +There are three main methods to construct a deep learning training set: labeling, match- +ing, and synthesizing. “Labeling” entails manually annotating images with categorical labels +(e.g., park or parking lot) or marking the boundaries of object instances (e.g., vehicles or pedes- +trians) by human experts. Several online tools and services have been developed to ease this +labor-intensive and time-consuming process. LabelMe, for example, is one of the first web- +based applications for large-scale image annotations and online sharing (Russell et al., 2008). +Similarly, Amazon Mechanical Turk provides a crowdsourcing platform for on-demand image +labelling tasks (Sorokin and Forsyth, 2008). Other examples use AI to assist in the image la- +10 + +belling process. In this case, a pre-trained AI model annotates the boundaries of object-like +targets even if the actual semantic category of the object is unknown, making it easier for a +human annotator to complete the whole task (Chen et al., 2020b). The “matching” method +refers to the process of matching existing labels with images based on particular associations, +such as co-occurrence and geographical relations. For example, to examine the extent to which +a house’s visual appearance can predict the house’s price, one can collect a large sample of +houses from a real estate market website, each associated with a house photo and its price. To +identify cities based on images, one can use photos from photo-sharing platforms like Flickr (Li +et al., 2013) and Panoramio (Zhang et al., 2019b). On these platforms, a large number of these +photos are labeled with the city in which they were taken. To estimate socioeconomic charac- +teristics from street-level imagery, Google Street View images can be linked to demographics +or human trace data using their geographic location (latitude and longitude) (Suel et al., 2019; +Zhang et al., 2019a; Ilic et al., 2019). Finally, the “synthesizing” method is typically used when +finding suitable image samples for a task that is difficult in practice. Ma et al. (2019), for ex- +ample, train a computer vision model that can recognize and classify the different typefaces +apparent on business signs. Since some types of typefaces are infrequently seen in cities and +the training dataset requires sufficient samples for each typeface category, they employed an +artificial synthesis of text and street images to create a dataset for model training. +The most impactful urban image datasets used in urban studies and geospatial analytics +include the Places2 (Zhou et al., 2017a) and ADE20K (Zhou et al., 2017b). The Places2 dataset, +for instance, contains approximately 10 million images labeled with hundreds of place types, +including residential neighborhoods, highways, and parks, etc. A deep learning model trained +using this dataset can classify a scene type using a street view image as input. Similarly, the +ADE20K dataset contains over 20,000 labeled images with hundreds of visual object categories, +such as plants, sky, vehicles, and buildings. Beyond these two datasets, researchers are also +compiling other sources of data that link images with ground-truth data for specific applications, +such as to describe scene attribute (Patterson and Hays, 2012), to classify architectural styles +(Xu et al., 2014; Sun et al., 2022b), to track neighborhood change (Naik et al., 2017), and to +detect informal settlements (Ibrahim et al., 2021). +5.3. DCNN models for image processing +The emergence of large-scale image datasets has enabled the design of more complex mod- +els with more layers. The DCNN models for urban applications can be broadly divided into +three network architectures: scene classification, object detection, and semantic segmentation. +As shown in Figure 4, the main difference between the three architectures is in the last several +layers of the DCNN. The final layer of a scene classification model is a classifier (Fig. 4a), +which outputs one label out of all the possible ones describing the attributes or categories of +11 + +the scene in an image. Classic DCNN architectures used for classification tasks include ResNet +(He et al., 2016), GoogLeNet (Szegedy et al., 2015), and DenseNet(Huang et al., 2017)), among +others. Object detection determines the objects in an image and can specify the location of the +objects with respect to it. It outputs a pair of results for each object: a predicted class and the +coordinates of the bounding boxes that surround it (Figure 4b). Popular models include Faster +R-CNN (Ren et al., 2015), SSD (Liu et al., 2016b) and YOLO (Bochkovskiy et al., 2020). +Figure 4c presents the output of an image segmentation model, which partitions the image into +various segmented parts and creates a pixel-wise mask of each object in the image. Widely used +models include PSPNet (Zhao et al., 2017), Mask RCNN (He et al., 2017), and HRNet (Wang +et al., 2020), among others. +Object classes +Object location +Pixel-level classes +b) Object Detection +c) Semantic Segmentation +... +a) Scene Classification +Category/attributes +Figure 4: +Three DCNN architectures in urban applications: scene classification, object detection and semantic +segmentation +5.4. Scene measuring and understanding +Scene element extraction is a widely used approach for measuring and analyzing physical +environments. Scene elements can be derived using the object detection DCNN models or ob- +ject segmentation models outlined in the previous section. The object detection models output +the detected objects with bounding boxes, so the number of different objects in an image can be +counted. The scene extraction model predicts the object categories of every pixel in an image, +12 + +Cooajswhich can be further processed to calculate the object proportions of any given scene. Both +models provide a quantitative approach to measure a scene. +A cornerstone example of how DCNN models can be used to measure objective attributes +of the physical environment is the Treepedia project.3 In this project, researchers trained a +deep learning model using images obtained from Google Street View to predict and classify the +tree canopy of streets. Instead of relying on manual audits of the physical environment, this +project introduces a scalable method to analyze the amount of green canopy available when +walking down the street for 30 cities around the world (Seiferling et al., 2017; Li and Ratti, +2018). Other complementary work has used the green canopy measurements combined with +other green indices extracted from satellite imagery to shed light on how perceptions of the +physical environment are affected by the camera’s angle of view (Wang et al., 2019a; Laumer +et al., 2020; Kumakoshi et al., 2020). Beyond the classification of green canopy, deep learning +models have also been used to classify other individual elements of streets, including the sky, +road, buildings, vegetation, vehicles, and pedestrians (Zhang et al., 2018a; Zhou et al., 2019). +Access to high-quality imagery allows for the classification of complex aspects of streets, such +as street signs, abandoned houses, sidewalk cracks, broken windows, and walls in need of repair +(Less et al., 2015; Zou and Wang, 2021). For example, Miranda et al. (2021) demonstrate how +Google Street View can be used to measure objective urban design characteristics of streets +that urban planners have long thought are attractive for pedestrians.4 Using GSV collected in +Boston, they calculate measures of urban furniture, sidewalks, facade complexity (how different +the fronts of buildings are in terms of materials), and visual enclosure (how well streets are +defined by trees, walls, buildings and other vertical elements). These types of measures have +the potential to help urbanists understand which environments are more inviting for pedestrians +(Z¨und and Bettencourt, 2021). +Beyond extracting physical environment elements, DCNN models can be used to classify +images. Using 10 million social media photos labeled with hundreds of semantic categories, +Zhou et al. (2017a) trained a object segmentation model to infer the type of place corresponding +to each image (e.g., residential neighborhood, bus station, and public square) and the attributes +that best describe it (e.g., man-made, messy, and sunny). This model has been recognized as a +benchmark and has been used widely to understand the function of places (Xiao et al., 2020; Zhu +et al., 2020a; Ye et al., 2020). A complementary example of how images can be processed to +draw important insights into the physical environment is classifying street canyon, which is an +attribute typically calculated from a combination of building height and street width measured +3http://senseable.mit.edu/treepedia +4https://senseable.mit.edu/desirable-streets/ +13 + +using sophisticated instruments. Hu et al. (2020a) uses Google Street View images to classify +the street canyon and shows how deep learning models can be used to identify different types +without the need of accurate measurements, reducing costs and saving time. +In addition to interpreting the information that can be directly visible in images, scene in- +ference models can infer scene information that cannot be directly observed from an image, +such as crime (Khosla et al., 2014), real estate values (Law et al., 2019), and changes in human +dynamics over time (Zhang et al., 2019a). Khosla et al. (2014) trained a deep learning model +that can “look beyond the visible scene” and predict objects that are not present in the street +view image. For example, the results show that its feasible to predict the distance to the near- +est grocery or hospital, even when these amenities are far from the specific street view image. +These kinds of models are trained using an end-to-end learning process; the model automati- +cally learns the relationships between the initial input image and the final output labels (crime +rate and house price value, etc.) without having to indicate to the model which visual cues +are most important. The assumption behind these models is that the built and socioeconomic +environment are closely associated. Even though the relationship between them is complex and +non-linear. +6. Representation—how can the physical environment of a place be quantified? +Sections 4 and 5 have focused our attention on studies showing how to objectively charac- +terize the physical environment by interpreting the visual attributes appearing in images. How- +ever, we have yet to note that there is a gap between what computer vision and deep learning +algorithms can measure and the richness of the physical environment of a place that humans +perceive (Tuan, 1979). +The concept of place, which has a long history in geography, is a unit of analysis that +integrates the environmental concepts of the natural and social sciences (Patterson and Williams, +2005; Goodchild, 2011). Due to the inherent complexity and subjectivity of the concept, it was +once considered unquantifiable. The computational representation of a place as a whole seems +to be an insurmountable task. However, the recent argument is that this impossibility does not +really hold (Janowicz et al., 2022). There is now a large volume of work where researchers have +successfully modeled different dimensions of place, such as human activities, cognitive regions, +and semantics (Gao et al., 2017; Purves et al., 2019). Making a formal and computational +representation of place available to all is essential for modern and interdisciplinary research +(Janowicz et al., 2022). +Here, we build on the technical advances outlined in the previous sections, but shift the focus +to show how the physical environment of a place can be quantitatively represented and analyzed +from three perspectives: place identity & similarity, place structure, and place perception (third +14 + +level of the Visual Intelligence framework). We focus on these three dimensions as they have +been proposed as critical to determining whether a place is imageable or not (Lynch, 1960). +More broadly, characterizing places is also important for geography and urban planning studies +that aim to integrate natural and social science concepts to understand cities (Patterson and +Williams, 2005; Morison, 2002). +6.1. Place Identity and Similarity +Quantitatively measuring, assessing, and understanding how humans perceive places is cru- +cial to their study. A place can be represented by a single image or a collection of them. The +visual identity of a place refers to its representativeness, or how similar or distinct it is according +to the ease by which people can identify it. +Deep learning models provide an opportunity to measure the visual identity of places for +many neighborhoods and cities worldwide. In practice, measuring visual identity and similarity +can be formulated as a discriminative classification problem using a DCNN model. First, the +model is trained to predict the place where a given image comes from. Then, one can use the +misclassification rates predicted by the model for each place as a measurement of the similarity +between two places (similar places are more likely to be misclassified because the inherent +sample distributions of the places are similar) and use the model’s accuracy in predicting a +place as a measure of how distinct each place is. A higher accuracy value indicates that the +scenes in the place are not likely to be confused with other places. Finally, one can also rank the +model’s confidence score for each input image to extract the scenes that have the highest place +representative (the confidence score indicates the certainty of the model to predict that a scene +indeed corresponds to the place where it was taken). +Based on the process outlined above, a number of papers have attempted to measure place +identity and similarity at different geographic scales. For instance, Doersch et al. (2012) de- +veloped an automated approach to identify the distinctive architectural elements of a city that +differentiate it from others. They show that visual elements, such as windows, balconies, and +street signs, can distinguish Paris from other cities. On a global scale, Zhang et al. (2019b) +trained a deep learning model to recognize places among 18 cities around the world. They +measured the visual similarity and distinctiveness of the cities and also identified the unique +visual cues of each city (such as landmarks, historical architecture, religious sites, and unique +cityscapes). For indoor spaces, Zhang et al. (2016) analyzed the subtle distinctions of corridors +and spaces in the large interconnected buildings on the MIT campus to understand the visual el- +ements of indoor design and human cognition that facilitate indoor navigation. Similarly, Wang +et al. (2019c) evaluated two train stations’ legibility in Paris and show how a computer vision +model can identify the space from which a given photo was taken. The process through which +the computer vision model identifies space is analogous to the process that pedestrians use to +15 + +navigate spaces and can therefore be used to aid pedestrian routing. Liu et al. (2016a) repro- +duced the Image of the City using two million geotagged photos of 26 cities collected from a +photo-sharing platform. The study yielded a series of cognitive maps of each city, demonstrat- +ing how digital techniques can revisit and enhance our understanding of places across cities. +This digital approach to measuring place identity has also been extended to many other cities in +recent years (Salesses et al., 2013; Zhou et al., 2014; Filomena et al., 2019; Huang et al., 2021). +6.2. Place Structure +The street-level imagery and computer vision techniques discussed in Section 5 outline the +foundation to extract visual elements from images. This process can be used to further under- +stand “place structure.” By place structure, we refer to an understanding of the composition and +hierarchical relationships embedded in the visual elements of an image that might be important +to represent a place quantitatively. +Complementing the handful of structural elements proposed by Lynch (Lynch, 1960), re- +cent papers adopt complementary perspectives to conceptually organize scene elements and +scene types into categories (Patterson and Hays, 2012; Zhou et al., 2017a; Zhang et al., 2018a). +For example, Zhang et al. (2018a), organized hundreds of object categories that commonly +appear in cities into a hierarchical tree based on their conceptual relationship. For example, +“tree,” “flower,” “grass” are sorted into the conceptual category “vegetation.” The “vegetation” +category is combined with “waterbody” and “sky” to form a broader conceptual category “nat- +ural.” This hierarchical semantic tree enables researchers to understand the visual structure of a +neighborhood qualitatively—by understanding the presence of elements of a street at different +levels and quantitatively—by measuring the abundance of scene elements using a pre-trained +deep learning model. With enough images for a place, this hierarchical organization can help +measure the “structure” of any given place. +6.3. Place Perception +Understanding how human perceive their surrounding environment can help assess and eval- +uate the quality of urban design. This topic has long been of interest to a wide variety of fields, +ranging from human geography, and urban planning, to environmental psychology (Kaplan and +Kaplan, 1989; Lynch, 1960; Tuan, 1977; Nasar and Jones, 1997). Street-level imagery and +deep learning techniques are opening up new possibilities to measure human perception. In +particular, access to crowdsourced information collected online allow researchers to measure +preferences and perceptions at an unprecedented scale. A key example of this approach is the +online platform “Place Pulse,” launched to collect online ratings to evaluate human perception +(Salesses et al., 2013). The project collected online volunteers’ ratings on Google Street Views +along six dimensions: “safe”, “lively”, “beautiful,” “wealthy,” “boring” and “depressing.” The +16 + +platform operated for over 5 years and collected around one million ratings on 110,000 street +views from more than 80,000 volunteers. Crowdsourcing platforms as such complement tra- +ditional data collection methods in multiple ways. First, the collected data represent a broad +selection of people from different gender, ages, and diverse racial and cultural backgrounds. +Collecting information from such a wide range of participants was inconceivable using inter- +views or other traditional data collection methods. Second, the evaluation of thousands of street +scenes (56 cities from 28 countries worldwide) allows researchers to account for framing effects +and the consistency of respondents, which small sample questionnaires cannot afford to do. +The rise of crowdsourcing platforms like Place Pulse has enabled a series of studies focused +on how humans visually evaluate their surroundings (Ordonez and Berg, 2014; Dubey et al., +2016). Studies have revisited classic urban theories focused on the relationship between the +physical environment and perceptions that could not be tested before due to small sample sizes +and geographic scale limitations. For example, Zhang et al. (2018b) examined the spatial distri- +bution of human perceptions in Beijing and Shanghai using one million street views and image +segmentation techniques. In particular, the study explores how street features affect human +perceptions and also measures whether the physical disorder of a place (measured using litter, +graffiti, and poorly maintained buildings as proxies) has a negative effect on people’s feelings, +providing an effective tool to evaluate the “sense of place” of large-scale urban areas. Saiz et al. +(2018) uses the ubiquitous posting of millions of photographs online to understand how people +value the aesthetic dimension of the physical environment. They show that street-level imagery +offers a scalable way to measure subjective attractiveness across and within cities, enabling us +to build a more comprehensive understanding of how people perceive their surroundings. +Human perceptions of the physical environment derived from DCNN methods have also +been used to measure cities’ social and economic dynamics. Research on this topic has focused +on using street-level imagery obtained from Google Street View to measure changes in the +neighborhood’s physical appearance. Naik et al. (2017) relate changes in the physical appear- +ance of five US cities with economic and demographic data to document the underlying factors +that predict neighborhood improvement. Zhang et al. (2020a) characterize a place in terms of +physical appearance and popularity, discovering many unassuming but popular restaurants in +Beijing. Locals frequently visit a host of places for social engagements despite their common +location on deep alleys of old neighborhoods with unappealing appearances. +7. Application—How to understand the interactions between the physical environment, +human dynamics, and the socioeconomic environment? +The fourth level in the Urban Visual Intelligence framework shows how images can be used +to study fine-grained interactions between the built and socioeconomic environments. Under- +17 + +standing this relationship is central to geography, environmental science, social science, and +urban studies and planning. In this section, we outline practical applications focused on three +major topics: public health, transportation, and the socioeconomic environment of places. Al- +though these three topics don’t represent the entirety of the work using street view imagery, +they are arguably among the most prevalent, providing fertile ground to illustrate exciting new +applications. +7.1. Public health +Traditional environmental health studies typically employ field surveys and questionnaires +to characterize the physical environment. Researchers and subjects involved in the research +are typically required to record and describe the physical environment of the study areas using +previously designed survey forms (Ball et al., 2001; Takano et al., 2002; Lawlor et al., 2003; +Gull´on et al., 2015). Other studies derive environmental characteristics based on spatial analy- +sis and GIS—for example, using a space syntax approach or measuring accessibility indicators +(Pliakas et al., 2017; Leslie and Cerin, 2008). Street-level imagery and visual intelligence bring +a complementary angle to these previously outlined studies because it allows for cross-country +comparisons (as images are collected across several countries) and provides information cap- +tured from a human perspective (Biljecki and Ito, 2021). +The physical environment impact health outcomes in many ways, ranging from physical +aspects (obesity) to psychological ones (mental health) (Mitchell and Popham, 2008; Ulrich, +1984; Lee et al., 2012; Mehrabian and Russell, 1974). Some of the commonly measured vi- +sual features derived from street-level imagery to study health include: greenness exposure, +visual enclosure, the presence and quality of sidewalks, urban infrastructure and facilities, food +advertisements, and visual cues about the physical disorder. For example, fine-scale green- +ery measurement has been used to understand walking and cycling behaviors (Lu et al., 2018, +2019), its impact on children’s body weight (Yang et al., 2020b), mental health (Svoray et al., +2018; Kang et al., 2019; James et al., 2015), and perceived safety (Li et al., 2015; Kruse et al., +2021). Similarly, greenery metrics measured from street view imagery and remote sensing im- +agery (NDVI–normalized difference vegetation index) are compared in a number of studies, +showing the advantage of using street-level imagery to measure eye-level greenery in streets +(Villeneuve et al., 2018; Lu et al., 2019; Larkin and Hystad, 2019). To conclude, street view +imagery and remote sensing imagery represent different and complementary aspects of natural +environments (Helbich et al., 2019; Larkin and Hystad, 2019; Kang et al., 2020a). +Empirical studies also found the physical aspects measured from street view imagery are +associated with health outcomes. For example, more visually enclosed streets are found con- +nected with higher quality and the presence of sidewalks and crosswalks have been associated +with more walkability and increased mental health (Vargo et al., 2012; Yin and Wang, 2016; +18 + +Nguyen et al., 2018; Li et al., 2018a; Wang et al., 2019b). Features derived from street view +imagery, such as food and beverage advertisements, have been used to identify obesogenic en- +vironments (Feuillet et al., 2016; Roda et al., 2016; Egli et al., 2019), and visual cues, such +as visible utility wire overhead, have been used as proxies for physical disorder (Keralis et al., +2020), and have been associated with diabetes and mental distress (Marco et al., 2017; Chen +et al., 2019; Plascak et al., 2020).5 +7.2. Transportation and mobility +Physical environment features, such as road infrastructure, detected from street-level im- +agery can be directly used to enhance virtual audits (Hong et al., 2020), for example, identify- +ing traffic black spots (Tanprasert et al., 2020) and potential urban congestion spots (Qin et al., +2020). This section focuses on how these traffic and physical environment characteristics de- +rived from street-level imagery can provide insights into their association with transportation +behavior and its consequences. +Beyond virtual audits, studies have also used features from images to study transportation +behavior. For example, studies have found that particular road characteristics, such as traffic +lights, the density of speed bumps and the number of pedestrian crossings are related to traffic +volumes and route choice behavior (Verhoeven et al., 2018; den Braver et al., 2020). Other +road features, such as the number and width of bicycle lanes and the sidewalk and road surface +conditions, have been used to explain the variation in pedestrian crashes and traffic accidents +(Johnson and Gabler, 2015; Isola et al., 2019; Hu et al., 2020b; Kwon and Cho, 2020; Mooney +et al., 2020). These efforts can help plan better cities and aid with testing interventions to +improve the safety of pedestrians and cars. For instance, Miranda et al. (2021), measured +pedestrians deviations from the shortest route to construct a measure of street desirability. The +study then uses computer vision techniques to measure a diverse set of physical environment +characteristics (such as the presence of urban furniture, parks, visual enclosure, and facade +heterogeneity) to see what desirable streets had in common. By measuring these urban design +features and relating them to pedestrian use, researchers can trace how streets change in time, +helping practitioners detect areas affected by blight or perceived as hazardous, thereby focusing +efforts on revitalizing distressed streets. +Deep learning also offers a non-linear modeling approach to studying the associations be- +tween the physical environment and urban mobility. The appearance of the physical environ- +ment perceived from images can reveal information on its function and land use type (Qi et al., +2011; Liu et al., 2012; Yuan et al., 2012; Fan et al., 2021). Deep learning models are able to +5See (Rzotkiewicz et al., 2018; Kang et al., 2020a; Biljecki and Ito, 2021), for other applications in public +health studies. +19 + +capture the non-linear associations between the above aspects through “End-to-End training.” +For instance, Zhang et al. (2019a) inferred hourly human activity intensity at a street level from +street view images—without the need for pedestrians or vehicles to be presented in the images. +The results show the potential of DCNN models to learn high-level street view imagery features +that can explain up to 66.5% of the hourly variation in urban mobility. Similar approaches have +been applied to predict spatial patterns of bicycling and walking using points of interest and +street view images Chen et al. (2020a); Hankey et al. (2021). +Computer vision and deep learning approaches also show great promise for researchers +to understand how the physical environment can be designed to guide people’s use of cities. +For instance, Mirowski et al. (2018) applied deep reinforcement learning to “teach” agents +to navigate cities without a map. By observing only street view images, the agent can learn +physical environment features from the images that help him traverse to destinations that may +be kilometers away. Strategic placement of visual elements and infrastructures can aid the +choice-making process for humans to navigate through cities. +7.3. Socioeconomic characteristics +The physical environment can provide cues about the socioeconomic status of a city. The +fine-grained characterization of the physical environment supported by street-level imagery and +deep learning has recently led to an increased interest in measuring the interactions between +the physical environment and social and economic outcomes, including income, real estate, and +crime, among others (Ibrahim et al., 2020; Biljecki and Ito, 2021). +Crime is one of the most prominent socioeconomic dimensions that has been studied using +street-level imagery and deep learning (Zhou et al., 2021). Its interest is motivated by the idea +that sustainable communities need to be both safe from crime and also be perceived as safe +by their residents. To study whether safe-looking places have indeed lower crime rates, Zhang +et al. (2021) propose a measure of “perception bias,” the mismatch between people’s perception +of safety inferred from Google Street View images and actual violent crime, and explore the +socioeconomic factors associated with the mismatch. +The visual quality of neighborhoods has also been demonstrated to be an effective predictor +of real estate values and housing appreciation (Yang et al., 2020a; Kang et al., 2020b, 2021; +Qiu et al., 2022). The presence of particular elements in images, such as specific vehicle types, +can accurately predict demographics and the political tendency of neighborhoods (Gebru et al., +2017). Similarly, the presence of particular typefaces used in business amenities can proxy for +neighborhood income (Ma et al., 2019). +In addition to the static physical environment features extracted from street view images, +images captured at different times can also be useful to analyze how the physical environment +is changing. Naik et al. (2017) created a metric of physical urban change using images collected +20 + +in different periods, to test theories related to human capital agglomeration and the tipping point +theory of urban change. The results show that infrastructure improvements in neighborhoods +are associated with education and population density, and that neighborhoods with better initial +appearances experience more substantial improvements over time. +Similar to the deep learning “End-2-End training” strategy in transportation, which is used +to model the complex associations between the physical environment and socioeconomic di- +mensions, a DCNN can also be trained using street-level imagery to predict socioeconomic +characteristics. This approach allows researchers to capture complex relationships between the +physical environment and its socioeconomic makeup. Studies in this vein have measured job- +housing patterns (Yao et al., 2021), social and environmental inequalities (Suel et al., 2019) and +income, overcrowding, and environmental deprivation in urban areas (Suel et al., 2021). +8. Discussion +8.1. Towards Urban Visual Intelligence: what it can address and what it misses +Throughout the paper, we show how images and deep learning techniques have been used +to study the visual dimensions of cities and how these are related to broader concerns about +the performance of places. However, there are some dimensions of cities that cannot be under- +stood using images alone. In this section, we discuss the limits of using digitally collected and +processed visual information to understand the city. +Visual detection tasks. One of the pillars of the Urban Visual intelligence framework is +the visual detection of elements from images. As we have shown, advances in deep learning +models allow for the classification of elements appearing in an image (vehicles, buildings, and +vegetation), the human activities within it (walking, talking, and queuing), and the type of scene +it represents (park, parking lot, and residential neighborhood). These physical environment +characteristics can be accurately detected because they are largely consistent across different +geographic contexts and times. Thus, the efficacy of the visual detection tasks depends on +the modeling capabilities of DCNNs, which have already achieved great capacities and are +improving continuously. Hence, visual detection is not likely to be where most of the future +work will focus. +Within-place and between-place inference. Within-place inference refers to how well an +attribute extracted from an image can be used to predict any given outcome for a place with a +geographic scale as small as a block or as large as a region. In an age when the physical envi- +ronment and the social dimensions are deeply intertwined, many aspects of a city is interrelated +(Batty, 2021). Thus, traditional modeling methods can miss the interacted and nonlinear rela- +tionships. Conventional statistical models have been used to measure how a given attribute of a +city (e.g., greenery density) is associated with another one (e.g., neighborhood health outcome). +21 + +DCNN models can accomplish this task with enhanced non-linear modeling capabilities. This +means that by assigning a label to a DCNN model, the model is always capable of finding rela- +tionships (both linear or non-linear) between an input street-level imagery and output variables +(such as the socioeconomic composition of a neighborhood, real estate prices, or the density +of human activity). Hence, “within-place inference” depends on how strong the underlying +relationships are between the physical (visual) environment and its particular social correlates. +“Between-place inference” refers to how well a DCNN model fitted in one place can be +applied to another. The between-place inference is facing three sets of issues, which can be +analyzed from three complementary disciplinary fields: machine learning, urban studies, and +GIScience. +From the perspective of the machine learning field, between-place inference is commonly +challenged by the cross-domain generalizability (Neyshabur et al., 2017). Limitations in the +generalizability of models stem primarily from underlying data distributions—the differences +in the relationships of input image and output label and also between the training domain and +test domain. This issue, termed “domain shift” (Qui˜nonero-Candela et al., 2009), has been ex- +tensively examined in machine learning, and can be addressed through domain adaption (Wang +and Deng, 2018). +Between-place inference from the perspective of urban studies has mainly focused on issues +related to how best to measure a “place.” The heterogeneous uses of places and the fact that they +are perceived differently across cultures make between-place inference difficult. For example, a +DCNN model trained to infer urban mobility patterns using street-view images in China could +fail in cities in Western countries because the human activity patterns vary widely—even when +the streets from the two cities have a similar shape (Zhang et al., 2019a). Places are heteroge- +neous because of differences in culture, geographical context, climate, historical development +and a host of other factors. With these differences, it is unlikely that a fitted inference model +developed in one place can be applied to another place without “domain adaption.” +Finally, from the perspective of GIScience, between-place inference faces the issues of +replicability in spatial analysis (Goodchild et al., 2020; Kedron et al., 2021; Goodchild and +Li, 2021). In particular, challenges with the replicability of models are due to the spatial hetero- +geneity and non-stationarity inherent in spatial data. For example, making inferences between +locations is challenging because the spatial variation of some phenomena/variables make the +results not invariant across locations. Moreover, due to spatial non-stationarity, relationships +between variables across locations might not be consistent, making it impossible to general- +ize across contexts. Together, spatial heterogeneity and non-stationarity explain why a local +DCNN model often does not translate well to other locations. Incorporating these principles +into DCNN models provides an opportunity to improve the generalizability and transferability +22 + +of these models across contexts (Li et al., 2021). +Cultural and subjective meaning. A place is not solely defined by its natural and built +settings, but also by the cultural and subjective meaning that people attribute to it. As a cultural +landscape, a place is given a unique meaning by its occupants, activities, events, and its own +historical evolution. For example, the experience of visiting the Eiffel tower in Paris cannot +be easily grasped by seeing a replica of the tower in, say, Las Vegas or Shenzhen. While a +lot of important social and cultural dimensions are encoded in the physical environment, its +visual expression can also be subtle and unrevealing of its meaning to locals or visitors. A +brick row house in Edinburgh may look similar to its counterpart in Boston, but may have quite +different meanings. This is the premise of special geography (Warntz, 1989) (or idiographic +science), which begins with the assumption that each place is distinct and has unique properties +that cannot be replicated. On the contrary, most machine learning models rely on an inductive +learning process—they try to learn general and replicable rules from existing examples. As +such, it is difficult for a DCNN model to fully interpret complex inhabited cultural landscapes +solely from the elements represented by images. +Differences in the perception of places are also explained by subjective meaning. An indi- +vidual’s sense of place is informed by their own past experiences, their stage in the life-cycle, +and by their taste and preferences (Tuan, 1977). For example, the Temple Mount in Jerusalem +will have different religious significance for Jews, Muslim, and Christians. Besides, a place +may vary dramatically depending on the time of day or day of the week or year. Kim (2015) +develops “spatial ethnography” to reveal the dramatically varying forms of “time-sharing” of +socially, culturally and economically complex sidewalk streetscapes. In summary, while places +are unique for individuals and groups, deep learning techniques can only summarize and infer +the collective knowledge, sacrificing idiosyncratic but important factors. A DCNN model can +easily count different human activities along sidewalks but will miss how people from varying +demographics perceive and use public space. Incorporating individual and group preferences +into AI studies is important for improving the representativeness of future AI models. +8.2. Dealing with uncertainty in street-level imagery +Uncertainty is an inevitable characteristic of spatial data. As a spatial data source, street- +level imagery can be subject to several uncertainty issues, including the Modifiable Areal Unit +Problem (MAUP), ecological fallacy, measurement uncertainty, and temporal uncertainty. +MAUP is the statistical bias resulting from aggregating point-based measures into zones +(Fotheringham and Wong, 1991). Spatially, street-level imagery is not evenly distributed be- +cause the imagery from map services is spatially constrained by the road network, and social +media photos are spatially distributed depending upon the intensity of urban functions and hu- +man activity. Even for a very long street, the characteristics of imagery tend to exhibit high +23 + +internal homogeneity, and may not reflect the fact that a street running in parallel may present a +widely different appearance. Aggregating different combinations of a limited number of image +samples into spatial units can yield entirely different results, exacerbating the MAUP. +The aggregation of street-level imagery can also give rise to the ecological fallacy, which +occurs when conclusions are drawn about individual images based on their aggregation. An +example is inappropriately concluding that all locations in a street are beautiful, just because +the average beauty score of that street is high. +Street-level imagery is also subject to measurement uncertainty, which mainly stems from +the varying distances between the camera and the visual scene being recorded. The position +of the camera has an impact on the proportions of each visual feature appearing in the image, +which may in turn affect how they are analyzed by the computer vision algorithm. For example, +street-level imagery of tall structures may be only represented by the appearance of the parts +closest to the ground level. +The visual features may also be subject to temporal changes that vary across seasons, in- +cluding seasonal changes that affect vegetation, sky view, and the number of pedestrians. These +aspects are overlooked in most current studies that must rely on data that is collected infre- +quently. This is in part due to the fact that street view imagery collected on Google Street View +is updated at most every year. Granular data gathered more frequently is essential for broader +research agendas that measure the physical changes of neighborhoods. The good news is that +access to such datasets are likely to increase as point clouds (LiDAR data) and crowdsourced +initiatives continue to grow (Mapillary). +8.3. Promising avenues of inquiry and future work +Street-level imagery and deep learning techniques can provide more than efficient mea- +surements. These techniques have the potential to support new understandings and knowledge +discovery about cities. Here we discuss several aspects that can be explored in future work. +New knowledge about the functioning of cities can be achieved by studying the hidden +visual cues in images, such as written language or signs of social disorder. For example, writ- +ten language in street names, business names, and ads can be easily acquired from street-level +imagery and used to map points of interest and locations of services such as restaurants, pawn- +brokers and payday loan outlets. Street signage can also help map linguistic or ethnic groups by +providing insights into the composition of the population, social disorder, psychosocial stress +and other important (and frequently less explores) aspects of neighborhoods. +Vehicle types on street-level imagery can also be used to study the social dimensions of +the city. Seminal work by Gebru et al. (2017) uses the vehicle types of a neighborhood mined +from images to infer the demographics and political tendencies of neighborhoods. Other di- +mensions of these vehicles, such as their type (commercial/private), spatiotemporal presence +24 + +patterns (obtained from cameras), and the cost of private vehicles can be used to understand +the socioeconomic characteristics of neighborhoods. One can imagine other visual cues from +street-level imagery as important indicators of neighborhood dynamics. +An entirely new city can be created by combining design criteria with AI scene-generation +techniques. Scene generation is enabled by a special architecture of DCNN called Generative +Adversarial Nets (GAN) (Goodfellow et al., 2014), which can generate entirely new predicted +images after learning what realistic scenes look like from hundreds of thousands of real street +scenes. GAN creates computer-generated urban scenes based on user-generated inputs, such as +objective characteristics extracted from images (e.g., buildings, roads, and vehicles) or encoded +perceptions (e.g., attractiveness, safety, and liveliness of a place). Urban scenes can be also +generated and edited based on some attributes of the physical environment (Bau et al., 2020; +Zhu et al., 2020b; Richter et al., 2021). These types of models are useful for scenario planning +and urban design applications which frequently require that participants envision images of +cities that don’t exist (Wu and Biljecki, 2022). Urban designers are also using GAN to create +vivid high-resolution imagery (Zhao et al., 2021). A pioneering approach looking at this was +introduced by Noyman and Larson (2020), who designed a physical platform 6 that allows +users to generate street scenes by combining a wide range of street elements based on their +preferences, including different land-uses, types of roads, the density of buildings, and presence +of sidewalks. +Designing interpretable and reliable AI models has received increasing attention. We be- +lieve that interpretable models can support the analysis of the urban physical environment in +two ways. For scientific research, machine learning models have stronger fitting and model- +ing capabilities than traditional regression models. They can better predict the human activity +patterns and socioeconomic profiles of cities. For practitioners, research findings can only be +used as a reference by policymakers when there is a clear causal pathway, whereas interpretable +machine learning models can further reveal the confounding factors inherent in causal analysis +for policy formulation and implementation. +Finally, fine-grained characteristics of the physical environment derived from street-level +imagery offer tremendous opportunities to systematically understand the spatial laws of the ur- +ban physical environment. We believe that there are recognizable patterns in the way the phys- +ical features of cities are organized in a city. Following, there may exist a basic spatial unit that +constitutes urban space. Future work could investigate how the computational representation of +physical features changes accordingly as the scales and the organization of spatial units change, +and whether there is a consistent spatial scale to represent the physical urban environment. +6https://www.media.mit.edu/projects/deep-image-of-the-city/ +25 + +9. Conclusion +Using visual information to understand cities has a long tradition in urban studies, city plan- +ning, and design. However, measurements of the physical environment and people’s response +to it have been challenging to evaluate until recently. Today, the emergence of artificial intel- +ligence provides us with more efficient and effective tools to understand the city and how it in +turn affects its citizens. +This paper reviews and compares the traditional and latest literature on the visual analyses of +cities. 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The Urban Book Series. +41 + diff --git a/LNAyT4oBgHgl3EQfsfn-/content/tmp_files/load_file.txt b/LNAyT4oBgHgl3EQfsfn-/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f2fe345d512212c6a9ae466ef05d58b4eaa7164f --- /dev/null +++ b/LNAyT4oBgHgl3EQfsfn-/content/tmp_files/load_file.txt @@ -0,0 +1,2406 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf,len=2405 +page_content='Urban Visual Intelligence: Studying Cities with AI and Street-level Imagery Fan Zhanga,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Arianna Salazar Mirandaa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Fabio Duartea,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Lawrence Valec,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Gary Hackc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Yu Liud,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Michael Battye,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Carlo Rattia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='c aSenseable City Lab,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Massachusetts Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' United States bDepartment of Civil and Environmental Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' The Hong Kong University of Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Hong Kong cDepartment of Urban Studies and Planning,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Massachusetts Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' United States dInstitute of Remote Sensing and Geographical Information System,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Peking University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' China eCentre for Advanced Spatial Analysis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Faculty of the Built Environment,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' University College London,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' United Kingdom Abstract The visual dimension of cities has been a fundamental subject in urban studies,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' since the pioneering work of scholars such as Sitte,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Lynch,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Arnheim and Jacobs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Several decades later, big data and artificial intelligence (AI) are revolutionizing how people move, sense, and interact with cities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' This paper reviews the literature on the appearance and function of cities to illustrate how visual information has been used to understand them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' A conceptual framework, Urban Visual Intelligence, is introduced to systematically elaborate on how new image data sources and AI techniques are reshaping the way researchers perceive and measure cities, enabling the study of the physical environment and its interactions with socioeconomic environment at various scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' The paper argues that these new approaches enable researchers to revisit the classic urban theories and themes, and potentially help cities create environments that are more in line with human behaviors and aspirations in the digital age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Keywords: Urban visual intelligence, physical environment, place, street-level imagery, deep learning, human-environment interactions 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Introduction Images have been central to how urbanists understand cities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' As a crucial medium, images inform the methodologies to measure the characteristics of the physical environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' The im- portance of images dates back to the 19th century when normative theories asserted the aesthetic ∗Corresponding author email: cefzhang@ust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='hk Preprint submitted to Elsevier January 3, 2023 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='00580v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='CV] 2 Jan 2023 value of cities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' The aesthetic experience of urban spaces was emphasized as a leading factor of urban planning (Sitte, 1889;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Arnheim, 1965).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' The consideration of making cities beautiful per- sisted as an important thread throughout the development of planning approaches into the 1920s (Freestone, 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Leading designers and theorists, such as Frederick Law Olmsted Sr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', Phillip Mackintosh, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Fitzpatrick, argued the societal role of beauty that would elicit citizens’ satisfaction, comfort, and pride (Wilson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 1990;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Mackintosh, 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Mulford, 1899;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Nasar, 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Ahlfeldt and Mastro, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Then, in the 20th century, the tradition of using images to understand cities shifted the focus with the development of urban design theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' By the late 1950s, urban designers were concerned with the inhospitable urban places that modernism urban forms had produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' In response, designers advocated a broader approach that could account for the performance and use of space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' This epistemological shift resulted from the desire to move away from normative theories of how the city “should be” to how people use urban space by incorporating empirical observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' These empirical observations largely relied on visual information obtained from photos and videos to assess how physical environments affect individuals’ use of urban space (Jacobs, 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Whyte, 1980;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Appleyard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 1981;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Jacobs and Appleyard, 1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' For example, the early work by Kevin Lynch (Lynch, 1960) introduced the concept of imageability to explain the vast differences in the mental impressions of individuals produced in Boston, Jersey City, and Los Angeles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Using photos and interviews with residents, Lynch solicited, assembled, and analyzed perceptual maps to identify areas that captured citizens’ attention and made a lasting impression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' This work was later extended using surveys and visual audits to show that citizens recall places when they evoke strong feelings (Nasar, 1998), highlighting memory and emotion as the key determinants (Clay, 1980).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Traditional data collection methods that rely on visual information, such as images, videos, and direct observation, provide rich insights into the intersection of human activity and city form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Nevertheless, these methods are very time-consuming and labor-intensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' It is difficult to scale over large spatial regions or extended time periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Today, access to new sources of geotagged data and advances in sensing technologies allow for a more detailed and larger-scale examination of cities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' These changes open up the possibility of comparisons across regions and over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' With the rich and extensive sources of visual data, what remains unclear is the different characteristics of various types of visual data, and how visual information can be effectively derived in a standardized manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Despite the large body of work dedicated to the application of visual data in analyzing the appearance of neighborhoods, it’s not yet evident how the physical environment of a place can be conceptually quantified, and how such a quantified representation of a place can contribute to a systematic understanding of the human-place relationship and revisit classical theories and practices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' 2 This paper reviews the theories and recent empirical work on the use of visual informa- tion to understand cities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' We introduce a conceptual framework, Urban Visual Intelligence, to illustrate how images and Artificial Intelligence can be paired to observe, measure, and repre- sent the features of the physical environment, as well as its interplay with the socioeconomic environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Historical overview of visual information in urban studies The tradition of incorporating visual information runs through the history of modern urban studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Since the early days of modern city planning, planners have documented and measured physical environmental attributes that could be extracted directly from photographs or sketches taken along the street sidewalk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Planners describe the urban forms through shape, proportion, rhythm, scale, complexity, color, order, elements, and hierarchy (Wohlwill, 1976).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' This tra- dition of using the formal attributes of the physical environment to elicit a pleasant sensory experience for citizens goes back to Camilo Sitte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' He argued that the city should be interpreted through visual art and architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' In particular, Sitte was highly critical of rigid symmetry and highlighted the value of irregularity in urban form, proposing that the aesthetic component of cities should be a leading factor in their design (Sitte, 1889).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' This focus on the design of the physical environment as a way to influence the behavior of citizens culminated in planning utopias, such as the Garden City and the City Beautiful Movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Advocates of these philoso- phies believe that the beauty, order and cleanliness of the public realm can influence civic spirit and improve the quality of life (Talen and Ellis, 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Although there is a consensus that cities should be aesthetically pleasing and beautiful places, the disagreement on what makes a city or a space beautiful persists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Is beauty in the eye of the beholder?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Alternatively, could aesthetics be measured so that designers could apply that measure to design spaces that appeal to many?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' With these questions remaining unanswered, in the 20th century, planners’ focus started shifting from the formal attributes of the physical environment (aesthetics) to the subjective experiences they induce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' At its core, this approach towards urban design sought a better un- derstanding of how humans sense and evaluate an urban scene visually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' The supporting studies emphasized the importance of understanding cities using people’s visual perceptions (Arnheim, 1965).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' In parallel, researchers attempted to capture how a city’s physical environment can trig- ger emotions that help inform our understanding of appealing and unappealing environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Nasar (1998), for example, proposed a model that explains how aesthetic responses arise from human interaction with the surrounding environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Similarly, Rapoport (1990) identified 36 characteristics related to the size and shape of typical aesthetically pleasing urban environ- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Overall, these studies on perception focused on how people shape their environment and how the physical environment, in turn, affects them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' However, these studies had only a limited 3 impact on the theories and practice of urban studies because they lack approaches to quantify and represent the physical environment on a large scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' To quantify and represent the physical environment of a place, Lynch (1960) introduced “imageability” as a new criterion (Lynch, 1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Building on ideas about perception but focused on human cognition instead of just aesthetics, Lynch identified the importance of meaning to explain how people understand and navigate urban environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' His study showed that as people move through an environment, they accumulate spatial knowledge obtained through observation and translate it into mental maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' In the Image of the City, Lynch proposed three categories to summarize the physical environment: Identity (distinct visual objects), Structure (recognizable patterns with relationships between objects), and Meaning (emotional values and character of a place) (Lynch, 1960).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Initially, Lynch assessed these three dimensions using some of the traditional approaches adopted by urbanists to observe the city: sketch maps, field surveys, and interviews collected for a small number of neighborhoods and participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' In a similar vein, Milgram (1970) proposed drawing collective maps of New York City measuring how recognizable it was using several small-scale experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Work as such paved the philosophy shift towards people-centered and place-based urban design in the late 1900s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Designers and planners started to embrace the emphasis on the per- formance, vitality, and use of spaces as an alternative way to measure the quality of urban design(Gehl, 1971).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Typically, scholars gather information on how people use urban spaces using simple recording techniques with pen and paper, augmented by photographic images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' A classic example of this approach is William H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Whyte’s seminal study on the social life of pub- lic spaces, “The Street Life project,” who used a combination of conversations, photographs, and a careful analysis of video recordings to observe how people use public spaces (Whyte, 1980).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Similarly, in his influential book Life between Buildings, Gehl (1971) employed exten- sive field observations to document the elements that make public spaces lively and contribute to people’s social interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Overall, scholars studying human-centered urban design applied observation and video to measure human behavior and people’s appropriation of public areas (Pushkarev, 1976).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' These studies have profoundly informed 21st-century’s urban design prac- tice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Their methods became standard practices to document and understand the interactions between physical and socioeconomic environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' These pioneers offered path-breaking implications for urban studies and design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' However, we today live in a world changing much more rapidly and those studies would need to be re- peated in different contexts and time periods to answer the questions we have today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' In addition, researchers today also question the small sample size and bias in selecting subjects in the afore- mentioned studies, which makes them subject to the variability of preferences across time and different populations (Nasar, 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Framework of Urban Visual Intelligence Today, hybridized sensing techniques - crowdsensing or ad hoc sensor deployment - offer researchers diverse data to analyze city life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Urban big data and AI-driven approaches provide tools to quantify the physical environment, socioeconomic conditions, and human dynamics with unprecedented performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' With these tools, researchers can observe the interaction between human behavior and the physical environment across spatial and temporal scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' In view of these opportunities, we propose a framework, Urban Visual Intelligence to re- view the method and data that researchers adopt today that are different from what was used historically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' In particular, our framework elaborates on how existing visual intelligence tech- nologies are being used to observe, measure, and represent the urban physical environment and study its interaction with human dynamics and socioeconomic environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Importantly, this framework centers around studies using AI-based tools to analyze street-level imagery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' We hope to illustrate the key issues and complementary approaches to these studies, weave together different technologies, and think from the existing literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Figure 1 shows the framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' There are four hierarchical levels that describe four issues: How can urban physical environments be observed at a human scale?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' How can semantic in- formation be interpreted from street-level imagery?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' How can the physical environment of a place be quantified?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' And how to understand the fine-grained interactions between the physical environment, human activities, and their socioeconomic environment?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' The top level of the dia- gram focuses on the visual data sources (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', Google Street View and crowdsourced platforms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Studies associated with this level focus on using street-level imagery to observe physical envi- ronments at the human scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' One street-level image can be generally considered a vista.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' At the second level of the diagram, a street view image depicts a scene at a certain location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' The second level focuses on computer vision and deep learning techniques to derive semantic infor- mation about scenes in the physical environment, for example, measuring tree or sky coverage in the scene depicted in a street-level image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' The third level shows how studies can use a collec- tion of images to characterize and create quantitative representations of a place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' For instance, to study place structure and place perception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Finally, the fourth level goes one step further to show how studies can also apply the measurements of places or physical environments to study their interactions with human dynamics and socioeconomic characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' 5 Figure 1: Framework of Urban Visual Intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' The framework shows the key issues identified in the urban physical environment and the corresponding studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Accordingly, the framework elaborates on using visual in- telligence technologies to observe, measure, and represent physical environments and study their interaction with human dynamics and socioeconomic dimensions at different levels and scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' From top to bottom, the four levels (observing, interpreting, measuring, and discovering) increase complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' They constitute a list of the process to apply images to understand urban environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' The first level shows the data sources available to observe the physical environ- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' The second level shows the techniques used to derive and interpret semantic information from the urban scene depicted in street-level imagery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' The third level shows how this informa- tion can be synthesized to characterize a place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' The fourth level shows how it can be used to study fine-grained interactions between the physical and socioeconomic environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' These four levels also indicate the four scales (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', vista, scene, place, and city) in which street-level imagery is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' The following sections unpack each of the four levels of the Urban Visual Intelligence in more detail and outline relevant works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='Observing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='Map service image ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='How can urban physical ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='Street-level ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='environments be observed at ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='crowdsourced photos ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='Imagery ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='Vista ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='human-scale ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='Custom collection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='How can ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='Interpreting ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='Deep learning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='Image ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='semantic information ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='LAI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='Computer vision ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='Scene ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='be derived from ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='understanding ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='Image dataset ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='street-level imagery ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='Measuring ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='Place identity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='How can physical ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='Place ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='Place structure ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='environments of a place be ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='Place ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='characterization ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='quantified ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='Place perception ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='How to understand the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='Discovering : ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='Public health ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='fine-grained interactions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='Human-place ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='Transportation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='between ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='City ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='relationship ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='Socioeconomics ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='physical and socioeconomic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='environment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='Urban Visual Intelligence ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='Urban Physical Environment4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Data—How to observe large-scale urban physical environments?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' The first level in the Urban Visual Intelligence framework introduces street-level imagery as a key data source to study urban environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' According to Keypoint Intelligence, 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='43 trillion photos were taken from digital cameras of mobile phones in 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' The adoption of mobile internet technologies and the fast development of web mapping services and crowd- sourcing platforms has resulted in geotagged images being produced rapidly, blanketing every corner of cities (Goodchild, 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' This new data source is commonly known as “street-level imagery,” it has extensive spatial coverage and has been used widely to observe large-scale urban environments (Ibrahim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Biljecki and Ito, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Duarte and Ratti, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Figure 2 shows three common sources of street-level imagery available to analyze cities: 1) map service images (such as Google Street View), 2) crowdsourced photos obtained from crowdsourcing platforms (such as Flickr and Mapillary), and 3) custom collection of images (through e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', Dash Cams and GoPros).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' For the first category, the images are generated with a stable update frequency, the widest coverage (more than 200 countries worldwide), and a uniform and consistent standard to facilitate comparative analysis between places (Anguelov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Goel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' The second category, crowdsourced photos, is essentially a type of Volunteered Geographic Information (VGI) (Goodchild, 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' As the volume of data grows and becomes denser in terms of spatiotemporal coverage, crowdsourced photos are expected to replace map service as the primary source of street-level imagery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' For the third category, the custom collection features images taken by individuals or researchers for specific topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' For example, by collecting a time series of images of a place, researchers can record changes in the physical environment and individual activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Custom collection can complement mapping services and crowdsourced imagery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' 1https://keypointintelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='com/ 7 Figure 2: Three sources of street-level imagery Compared to the traditional data sources used to learn about cities, such as personal inter- views and direct observations, street-level imagery has advantages including easy access, high spatiotemporal coverage, and objective and standardized views of the physical environment collected from embedded vantage points (Rzotkiewicz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Ibrahim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Kang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2020a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' In comparison to satellite imagery, in particular, street view imagery offers a complemen- tary perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' The major difference is that street view imagery depicts the world from a street level with a human-like perspective, instead of approaching it from an aerial view which is an approach that predominated in mid-20th-century modernism studies (Clay, 1980).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='2 Such an approach enables researchers to study visual cues, at the human scale that can directly re- late to people’s perceptions and the use of cities to inform planning and design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' This is only possible until very recently as images have been standardized across cities and visual analytic methodologies have been developed to perform the appropriate analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Street-level imagery is now considered one of the most important data sources to study physical environments (Biljecki and Ito, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Cinnamon and Jahiu, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' He and Li, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' In the past few years, we have witnessed its application in a wide range of study areas, including physical environment auditing (Kang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2018b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2020b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Ning et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2021), public health (Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Keralis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Bivand and Piras, 2015), urban mobility and transportation (Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Hong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Mooney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2020), energy estimation (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2022a), and real 2Historically, urban planners and geographers have been using satellite imagery to understand morphology, measure urban expansion and densification, and classifying different land uses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' 8 a) Map service image b) Crowdsourced photo c) Custom collection Google Street View Mapillary Dash Cam Bing Streetside Imagery Flickr GoPro Tencent Street View Instagram Mobile Phoneestate (Law et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2020a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Kang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2020b), among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Technology—how can semantic information be interpreted from street-level imagery in a scene?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' The second level in the Urban Visual Intelligence framework introduces deep learning and computer vision techniques used to derive and interpret semantic information from street-level imagery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Deep learning and computer vision Conventionally, photos taken from field surveys are interpreted by eye inspection, making it a labor-intensive process and imposing an upper limit on the geographic scale of research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Image processing techniques that enable the analysis of visual information in large groups have been developed to mitigate these issues, making detailed larger-scale studies possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' How- ever, conventional image processing-based methods continue to be limited to processing low- level features, such as color histograms and spectral features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Despite the useful information they provide, these approaches cannot extract high-level information, such as semantic objects, styles, and conditions of a scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' As we will explain in more detail, these features are key to studying the relationship between a city’s physical appearance and human behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Deep learning and computer vision techniques have been developed to learn and extract high-level information from images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Deep learning refers to a series of computer algorithms inspired by the human brain’s neural structure, which allows models to mimic human cognitive functions, such as understanding, learning, planning, and problem-solving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Deep learning has enabled the advancement of many fields including speech recognition (Hinton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2012), nat- ural language processing (Sutskever et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2014), game problem solving (Silver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2016), and computer vision (Ren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2017), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' These achievements are attributed to deep learning models’ outstanding performance in effectively and efficiently extracting high- level information from images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' For city applications, deep learning techniques offer a com- pelling framework for understanding the content of urban images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Figure 3 shows how a deep learning model works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' In particular, the figure explains how a deep learning model can be trained for computer vision tasks, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', detecting objects in an image and categorizing an image according to its content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' The core model in the figure is a deep convolutional neural network (DCNN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' A DCNN’s ultimate goal is to assign a “true” label to an image in order to make predictions about the class of scenes or objects in the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' For any given tasks, the process can be divided into two phases: training and inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' In the training phase, an image (Figure 3a) is fed into a pre-designed DCNN and processed individually layer 9 Label a) Input Image c) Output Label b) Deep Convolutional Neural Network Predicting Training … Figure 3: How deep learning works in urban image predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' a) Input urban image;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' b) Deep convolutional neural network;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' c) Output label by layer in the DCNN (Figure 3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' The final layer of the DCNN will generate a predicted label that is then compared with the image’s true label (Figure 3c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' The parameters in the DCNN are optimized in an iterative manner by successively minimizing the difference between predicted and true labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' In the inference phase, the well-trained model is used to make predictions on new images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' The learning and inference phases are similar to human’s learning procedure— observing a phenomenon, recognizing a pattern, and allowing feedback to improve the learning process (LeCun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Large-scale image datasets A key ingredient to successfully train deep learning models is the availability of large datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Deep learning models require a vast amount of labeled images as ground truth to learn the complex relationships between an input and its label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' An ideal image dataset is expected to be of high coverage and density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' The “coverage” requires a quasi-exhaustive representation of categories and a wide variety of exemplars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' The “density” refers to having a sufficiently large sample of images that cover the diversity of each predicted category (Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2017a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' There are three main methods to construct a deep learning training set: labeling, match- ing, and synthesizing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' “Labeling” entails manually annotating images with categorical labels (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', park or parking lot) or marking the boundaries of object instances (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', vehicles or pedes- trians) by human experts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Several online tools and services have been developed to ease this labor-intensive and time-consuming process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' LabelMe, for example, is one of the first web- based applications for large-scale image annotations and online sharing (Russell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Similarly, Amazon Mechanical Turk provides a crowdsourcing platform for on-demand image labelling tasks (Sorokin and Forsyth, 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Other examples use AI to assist in the image la- 10 belling process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' In this case, a pre-trained AI model annotates the boundaries of object-like targets even if the actual semantic category of the object is unknown, making it easier for a human annotator to complete the whole task (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2020b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' The “matching” method refers to the process of matching existing labels with images based on particular associations, such as co-occurrence and geographical relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' For example, to examine the extent to which a house’s visual appearance can predict the house’s price, one can collect a large sample of houses from a real estate market website, each associated with a house photo and its price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' To identify cities based on images, one can use photos from photo-sharing platforms like Flickr (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2013) and Panoramio (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2019b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' On these platforms, a large number of these photos are labeled with the city in which they were taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' To estimate socioeconomic charac- teristics from street-level imagery, Google Street View images can be linked to demographics or human trace data using their geographic location (latitude and longitude) (Suel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2019a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Ilic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Finally, the “synthesizing” method is typically used when finding suitable image samples for a task that is difficult in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' (2019), for ex- ample, train a computer vision model that can recognize and classify the different typefaces apparent on business signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Since some types of typefaces are infrequently seen in cities and the training dataset requires sufficient samples for each typeface category, they employed an artificial synthesis of text and street images to create a dataset for model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' The most impactful urban image datasets used in urban studies and geospatial analytics include the Places2 (Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2017a) and ADE20K (Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2017b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' The Places2 dataset, for instance, contains approximately 10 million images labeled with hundreds of place types, including residential neighborhoods, highways, and parks, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' A deep learning model trained using this dataset can classify a scene type using a street view image as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Similarly, the ADE20K dataset contains over 20,000 labeled images with hundreds of visual object categories, such as plants, sky, vehicles, and buildings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Beyond these two datasets, researchers are also compiling other sources of data that link images with ground-truth data for specific applications, such as to describe scene attribute (Patterson and Hays, 2012), to classify architectural styles (Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2022b), to track neighborhood change (Naik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2017), and to detect informal settlements (Ibrahim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' DCNN models for image processing The emergence of large-scale image datasets has enabled the design of more complex mod- els with more layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' The DCNN models for urban applications can be broadly divided into three network architectures: scene classification, object detection, and semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' As shown in Figure 4, the main difference between the three architectures is in the last several layers of the DCNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' The final layer of a scene classification model is a classifier (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' 4a), which outputs one label out of all the possible ones describing the attributes or categories of 11 the scene in an image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Classic DCNN architectures used for classification tasks include ResNet (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2016), GoogLeNet (Szegedy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2015), and DenseNet(Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2017)), among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Object detection determines the objects in an image and can specify the location of the objects with respect to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' It outputs a pair of results for each object: a predicted class and the coordinates of the bounding boxes that surround it (Figure 4b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Popular models include Faster R-CNN (Ren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2015), SSD (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2016b) and YOLO (Bochkovskiy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Figure 4c presents the output of an image segmentation model, which partitions the image into various segmented parts and creates a pixel-wise mask of each object in the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Widely used models include PSPNet (Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2017), Mask RCNN (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2017), and HRNet (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2020), among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Object classes Object location Pixel-level classes b) Object Detection c) Semantic Segmentation .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' a) Scene Classification Category/attributes Figure 4: Three DCNN architectures in urban applications: scene classification, object detection and semantic segmentation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Scene measuring and understanding Scene element extraction is a widely used approach for measuring and analyzing physical environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Scene elements can be derived using the object detection DCNN models or ob- ject segmentation models outlined in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' The object detection models output the detected objects with bounding boxes, so the number of different objects in an image can be counted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' The scene extraction model predicts the object categories of every pixel in an image, 12 Cooajswhich can be further processed to calculate the object proportions of any given scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Both models provide a quantitative approach to measure a scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' A cornerstone example of how DCNN models can be used to measure objective attributes of the physical environment is the Treepedia project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='3 In this project, researchers trained a deep learning model using images obtained from Google Street View to predict and classify the tree canopy of streets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Instead of relying on manual audits of the physical environment, this project introduces a scalable method to analyze the amount of green canopy available when walking down the street for 30 cities around the world (Seiferling et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Li and Ratti, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Other complementary work has used the green canopy measurements combined with other green indices extracted from satellite imagery to shed light on how perceptions of the physical environment are affected by the camera’s angle of view (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2019a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Laumer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Kumakoshi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Beyond the classification of green canopy, deep learning models have also been used to classify other individual elements of streets, including the sky, road, buildings, vegetation, vehicles, and pedestrians (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2018a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Access to high-quality imagery allows for the classification of complex aspects of streets, such as street signs, abandoned houses, sidewalk cracks, broken windows, and walls in need of repair (Less et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Zou and Wang, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' For example, Miranda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' (2021) demonstrate how Google Street View can be used to measure objective urban design characteristics of streets that urban planners have long thought are attractive for pedestrians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='4 Using GSV collected in Boston, they calculate measures of urban furniture, sidewalks, facade complexity (how different the fronts of buildings are in terms of materials), and visual enclosure (how well streets are defined by trees, walls, buildings and other vertical elements).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' These types of measures have the potential to help urbanists understand which environments are more inviting for pedestrians (Z¨und and Bettencourt, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Beyond extracting physical environment elements, DCNN models can be used to classify images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Using 10 million social media photos labeled with hundreds of semantic categories, Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' (2017a) trained a object segmentation model to infer the type of place corresponding to each image (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', residential neighborhood, bus station, and public square) and the attributes that best describe it (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', man-made, messy, and sunny).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' This model has been recognized as a benchmark and has been used widely to understand the function of places (Xiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2020a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Ye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' A complementary example of how images can be processed to draw important insights into the physical environment is classifying street canyon, which is an attribute typically calculated from a combination of building height and street width measured 3http://senseable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='mit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='edu/treepedia 4https://senseable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='mit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='edu/desirable-streets/ 13 using sophisticated instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' (2020a) uses Google Street View images to classify the street canyon and shows how deep learning models can be used to identify different types without the need of accurate measurements, reducing costs and saving time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' In addition to interpreting the information that can be directly visible in images, scene in- ference models can infer scene information that cannot be directly observed from an image, such as crime (Khosla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2014), real estate values (Law et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2019), and changes in human dynamics over time (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2019a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Khosla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' (2014) trained a deep learning model that can “look beyond the visible scene” and predict objects that are not present in the street view image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' For example, the results show that its feasible to predict the distance to the near- est grocery or hospital, even when these amenities are far from the specific street view image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' These kinds of models are trained using an end-to-end learning process;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' the model automati- cally learns the relationships between the initial input image and the final output labels (crime rate and house price value, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=') without having to indicate to the model which visual cues are most important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' The assumption behind these models is that the built and socioeconomic environment are closely associated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Even though the relationship between them is complex and non-linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Representation—how can the physical environment of a place be quantified?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Sections 4 and 5 have focused our attention on studies showing how to objectively charac- terize the physical environment by interpreting the visual attributes appearing in images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' How- ever, we have yet to note that there is a gap between what computer vision and deep learning algorithms can measure and the richness of the physical environment of a place that humans perceive (Tuan, 1979).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' The concept of place, which has a long history in geography, is a unit of analysis that integrates the environmental concepts of the natural and social sciences (Patterson and Williams, 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Goodchild, 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Due to the inherent complexity and subjectivity of the concept, it was once considered unquantifiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' The computational representation of a place as a whole seems to be an insurmountable task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' However, the recent argument is that this impossibility does not really hold (Janowicz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' There is now a large volume of work where researchers have successfully modeled different dimensions of place, such as human activities, cognitive regions, and semantics (Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Purves et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Making a formal and computational representation of place available to all is essential for modern and interdisciplinary research (Janowicz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Here, we build on the technical advances outlined in the previous sections, but shift the focus to show how the physical environment of a place can be quantitatively represented and analyzed from three perspectives: place identity & similarity, place structure, and place perception (third 14 level of the Visual Intelligence framework).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' We focus on these three dimensions as they have been proposed as critical to determining whether a place is imageable or not (Lynch, 1960).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' More broadly, characterizing places is also important for geography and urban planning studies that aim to integrate natural and social science concepts to understand cities (Patterson and Williams, 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Morison, 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Place Identity and Similarity Quantitatively measuring, assessing, and understanding how humans perceive places is cru- cial to their study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' A place can be represented by a single image or a collection of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' The visual identity of a place refers to its representativeness, or how similar or distinct it is according to the ease by which people can identify it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Deep learning models provide an opportunity to measure the visual identity of places for many neighborhoods and cities worldwide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' In practice, measuring visual identity and similarity can be formulated as a discriminative classification problem using a DCNN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' First, the model is trained to predict the place where a given image comes from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Then, one can use the misclassification rates predicted by the model for each place as a measurement of the similarity between two places (similar places are more likely to be misclassified because the inherent sample distributions of the places are similar) and use the model’s accuracy in predicting a place as a measure of how distinct each place is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' A higher accuracy value indicates that the scenes in the place are not likely to be confused with other places.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Finally, one can also rank the model’s confidence score for each input image to extract the scenes that have the highest place representative (the confidence score indicates the certainty of the model to predict that a scene indeed corresponds to the place where it was taken).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Based on the process outlined above, a number of papers have attempted to measure place identity and similarity at different geographic scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' For instance, Doersch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' (2012) de- veloped an automated approach to identify the distinctive architectural elements of a city that differentiate it from others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' They show that visual elements, such as windows, balconies, and street signs, can distinguish Paris from other cities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' On a global scale, Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' (2019b) trained a deep learning model to recognize places among 18 cities around the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' They measured the visual similarity and distinctiveness of the cities and also identified the unique visual cues of each city (such as landmarks, historical architecture, religious sites, and unique cityscapes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' For indoor spaces, Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' (2016) analyzed the subtle distinctions of corridors and spaces in the large interconnected buildings on the MIT campus to understand the visual el- ements of indoor design and human cognition that facilitate indoor navigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Similarly, Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' (2019c) evaluated two train stations’ legibility in Paris and show how a computer vision model can identify the space from which a given photo was taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' The process through which the computer vision model identifies space is analogous to the process that pedestrians use to 15 navigate spaces and can therefore be used to aid pedestrian routing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' (2016a) repro- duced the Image of the City using two million geotagged photos of 26 cities collected from a photo-sharing platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' The study yielded a series of cognitive maps of each city, demonstrat- ing how digital techniques can revisit and enhance our understanding of places across cities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' This digital approach to measuring place identity has also been extended to many other cities in recent years (Salesses et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Filomena et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Place Structure The street-level imagery and computer vision techniques discussed in Section 5 outline the foundation to extract visual elements from images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' This process can be used to further under- stand “place structure.” By place structure, we refer to an understanding of the composition and hierarchical relationships embedded in the visual elements of an image that might be important to represent a place quantitatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Complementing the handful of structural elements proposed by Lynch (Lynch, 1960), re- cent papers adopt complementary perspectives to conceptually organize scene elements and scene types into categories (Patterson and Hays, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2017a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2018a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' For example, Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' (2018a), organized hundreds of object categories that commonly appear in cities into a hierarchical tree based on their conceptual relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' For example,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' “tree,”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' “flower,”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' “grass” are sorted into the conceptual category “vegetation.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' The “vegetation” category is combined with “waterbody” and “sky” to form a broader conceptual category “nat- ural.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' This hierarchical semantic tree enables researchers to understand the visual structure of a neighborhood qualitatively—by understanding the presence of elements of a street at different levels and quantitatively—by measuring the abundance of scene elements using a pre-trained deep learning model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' With enough images for a place, this hierarchical organization can help measure the “structure” of any given place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Place Perception Understanding how human perceive their surrounding environment can help assess and eval- uate the quality of urban design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' This topic has long been of interest to a wide variety of fields, ranging from human geography, and urban planning, to environmental psychology (Kaplan and Kaplan, 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Lynch, 1960;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Tuan, 1977;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Nasar and Jones, 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Street-level imagery and deep learning techniques are opening up new possibilities to measure human perception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' In particular, access to crowdsourced information collected online allow researchers to measure preferences and perceptions at an unprecedented scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' A key example of this approach is the online platform “Place Pulse,” launched to collect online ratings to evaluate human perception (Salesses et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' The project collected online volunteers’ ratings on Google Street Views along six dimensions: “safe”, “lively”, “beautiful,” “wealthy,” “boring” and “depressing.” The 16 platform operated for over 5 years and collected around one million ratings on 110,000 street views from more than 80,000 volunteers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Crowdsourcing platforms as such complement tra- ditional data collection methods in multiple ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' First, the collected data represent a broad selection of people from different gender, ages, and diverse racial and cultural backgrounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Collecting information from such a wide range of participants was inconceivable using inter- views or other traditional data collection methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Second, the evaluation of thousands of street scenes (56 cities from 28 countries worldwide) allows researchers to account for framing effects and the consistency of respondents, which small sample questionnaires cannot afford to do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' The rise of crowdsourcing platforms like Place Pulse has enabled a series of studies focused on how humans visually evaluate their surroundings (Ordonez and Berg, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Dubey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Studies have revisited classic urban theories focused on the relationship between the physical environment and perceptions that could not be tested before due to small sample sizes and geographic scale limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' For example, Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' (2018b) examined the spatial distri- bution of human perceptions in Beijing and Shanghai using one million street views and image segmentation techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' In particular, the study explores how street features affect human perceptions and also measures whether the physical disorder of a place (measured using litter, graffiti, and poorly maintained buildings as proxies) has a negative effect on people’s feelings, providing an effective tool to evaluate the “sense of place” of large-scale urban areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Saiz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' (2018) uses the ubiquitous posting of millions of photographs online to understand how people value the aesthetic dimension of the physical environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' They show that street-level imagery offers a scalable way to measure subjective attractiveness across and within cities, enabling us to build a more comprehensive understanding of how people perceive their surroundings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Human perceptions of the physical environment derived from DCNN methods have also been used to measure cities’ social and economic dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Research on this topic has focused on using street-level imagery obtained from Google Street View to measure changes in the neighborhood’s physical appearance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Naik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' (2017) relate changes in the physical appear- ance of five US cities with economic and demographic data to document the underlying factors that predict neighborhood improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' (2020a) characterize a place in terms of physical appearance and popularity, discovering many unassuming but popular restaurants in Beijing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Locals frequently visit a host of places for social engagements despite their common location on deep alleys of old neighborhoods with unappealing appearances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Application—How to understand the interactions between the physical environment, human dynamics, and the socioeconomic environment?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' The fourth level in the Urban Visual Intelligence framework shows how images can be used to study fine-grained interactions between the built and socioeconomic environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Under- 17 standing this relationship is central to geography, environmental science, social science, and urban studies and planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' In this section, we outline practical applications focused on three major topics: public health, transportation, and the socioeconomic environment of places.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Al- though these three topics don’t represent the entirety of the work using street view imagery, they are arguably among the most prevalent, providing fertile ground to illustrate exciting new applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Public health Traditional environmental health studies typically employ field surveys and questionnaires to characterize the physical environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Researchers and subjects involved in the research are typically required to record and describe the physical environment of the study areas using previously designed survey forms (Ball et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Takano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Lawlor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Gull´on et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Other studies derive environmental characteristics based on spatial analy- sis and GIS—for example, using a space syntax approach or measuring accessibility indicators (Pliakas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Leslie and Cerin, 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Street-level imagery and visual intelligence bring a complementary angle to these previously outlined studies because it allows for cross-country comparisons (as images are collected across several countries) and provides information cap- tured from a human perspective (Biljecki and Ito, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' The physical environment impact health outcomes in many ways, ranging from physical aspects (obesity) to psychological ones (mental health) (Mitchell and Popham, 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Ulrich, 1984;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Mehrabian and Russell, 1974).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Some of the commonly measured vi- sual features derived from street-level imagery to study health include: greenness exposure, visual enclosure, the presence and quality of sidewalks, urban infrastructure and facilities, food advertisements, and visual cues about the physical disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' For example, fine-scale green- ery measurement has been used to understand walking and cycling behaviors (Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2018, 2019), its impact on children’s body weight (Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2020b), mental health (Svoray et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Kang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' James et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2015), and perceived safety (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Kruse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Similarly, greenery metrics measured from street view imagery and remote sensing im- agery (NDVI–normalized difference vegetation index) are compared in a number of studies, showing the advantage of using street-level imagery to measure eye-level greenery in streets (Villeneuve et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Larkin and Hystad, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' To conclude, street view imagery and remote sensing imagery represent different and complementary aspects of natural environments (Helbich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Larkin and Hystad, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Kang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2020a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Empirical studies also found the physical aspects measured from street view imagery are associated with health outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' For example, more visually enclosed streets are found con- nected with higher quality and the presence of sidewalks and crosswalks have been associated with more walkability and increased mental health (Vargo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Yin and Wang, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' 18 Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2018a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2019b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Features derived from street view imagery, such as food and beverage advertisements, have been used to identify obesogenic en- vironments (Feuillet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Roda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Egli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2019), and visual cues, such as visible utility wire overhead, have been used as proxies for physical disorder (Keralis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2020), and have been associated with diabetes and mental distress (Marco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Plascak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Transportation and mobility Physical environment features, such as road infrastructure, detected from street-level im- agery can be directly used to enhance virtual audits (Hong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2020), for example, identify- ing traffic black spots (Tanprasert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2020) and potential urban congestion spots (Qin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' This section focuses on how these traffic and physical environment characteristics de- rived from street-level imagery can provide insights into their association with transportation behavior and its consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Beyond virtual audits, studies have also used features from images to study transportation behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' For example, studies have found that particular road characteristics, such as traffic lights, the density of speed bumps and the number of pedestrian crossings are related to traffic volumes and route choice behavior (Verhoeven et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' den Braver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Other road features, such as the number and width of bicycle lanes and the sidewalk and road surface conditions, have been used to explain the variation in pedestrian crashes and traffic accidents (Johnson and Gabler, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Isola et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2020b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Kwon and Cho, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Mooney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' These efforts can help plan better cities and aid with testing interventions to improve the safety of pedestrians and cars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' For instance, Miranda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' (2021), measured pedestrians deviations from the shortest route to construct a measure of street desirability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' The study then uses computer vision techniques to measure a diverse set of physical environment characteristics (such as the presence of urban furniture, parks, visual enclosure, and facade heterogeneity) to see what desirable streets had in common.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' By measuring these urban design features and relating them to pedestrian use, researchers can trace how streets change in time, helping practitioners detect areas affected by blight or perceived as hazardous, thereby focusing efforts on revitalizing distressed streets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Deep learning also offers a non-linear modeling approach to studying the associations be- tween the physical environment and urban mobility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' The appearance of the physical environ- ment perceived from images can reveal information on its function and land use type (Qi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Yuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Deep learning models are able to 5See (Rzotkiewicz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Kang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2020a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Biljecki and Ito, 2021), for other applications in public health studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' 19 capture the non-linear associations between the above aspects through “End-to-End training.” For instance, Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' (2019a) inferred hourly human activity intensity at a street level from street view images—without the need for pedestrians or vehicles to be presented in the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' The results show the potential of DCNN models to learn high-level street view imagery features that can explain up to 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='5% of the hourly variation in urban mobility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Similar approaches have been applied to predict spatial patterns of bicycling and walking using points of interest and street view images Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' (2020a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Hankey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Computer vision and deep learning approaches also show great promise for researchers to understand how the physical environment can be designed to guide people’s use of cities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' For instance, Mirowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' (2018) applied deep reinforcement learning to “teach” agents to navigate cities without a map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' By observing only street view images, the agent can learn physical environment features from the images that help him traverse to destinations that may be kilometers away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Strategic placement of visual elements and infrastructures can aid the choice-making process for humans to navigate through cities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Socioeconomic characteristics The physical environment can provide cues about the socioeconomic status of a city.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' The fine-grained characterization of the physical environment supported by street-level imagery and deep learning has recently led to an increased interest in measuring the interactions between the physical environment and social and economic outcomes, including income, real estate, and crime, among others (Ibrahim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Biljecki and Ito, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Crime is one of the most prominent socioeconomic dimensions that has been studied using street-level imagery and deep learning (Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Its interest is motivated by the idea that sustainable communities need to be both safe from crime and also be perceived as safe by their residents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' To study whether safe-looking places have indeed lower crime rates, Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' (2021) propose a measure of “perception bias,” the mismatch between people’s perception of safety inferred from Google Street View images and actual violent crime, and explore the socioeconomic factors associated with the mismatch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' The visual quality of neighborhoods has also been demonstrated to be an effective predictor of real estate values and housing appreciation (Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2020a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Kang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2020b, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Qiu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' The presence of particular elements in images, such as specific vehicle types, can accurately predict demographics and the political tendency of neighborhoods (Gebru et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Similarly, the presence of particular typefaces used in business amenities can proxy for neighborhood income (Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' In addition to the static physical environment features extracted from street view images, images captured at different times can also be useful to analyze how the physical environment is changing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Naik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' (2017) created a metric of physical urban change using images collected 20 in different periods, to test theories related to human capital agglomeration and the tipping point theory of urban change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' The results show that infrastructure improvements in neighborhoods are associated with education and population density, and that neighborhoods with better initial appearances experience more substantial improvements over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Similar to the deep learning “End-2-End training” strategy in transportation, which is used to model the complex associations between the physical environment and socioeconomic di- mensions, a DCNN can also be trained using street-level imagery to predict socioeconomic characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' This approach allows researchers to capture complex relationships between the physical environment and its socioeconomic makeup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Studies in this vein have measured job- housing patterns (Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2021), social and environmental inequalities (Suel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2019) and income, overcrowding, and environmental deprivation in urban areas (Suel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Discussion 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Towards Urban Visual Intelligence: what it can address and what it misses Throughout the paper, we show how images and deep learning techniques have been used to study the visual dimensions of cities and how these are related to broader concerns about the performance of places.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' However, there are some dimensions of cities that cannot be under- stood using images alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' In this section, we discuss the limits of using digitally collected and processed visual information to understand the city.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Visual detection tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' One of the pillars of the Urban Visual intelligence framework is the visual detection of elements from images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' As we have shown, advances in deep learning models allow for the classification of elements appearing in an image (vehicles, buildings, and vegetation), the human activities within it (walking, talking, and queuing), and the type of scene it represents (park, parking lot, and residential neighborhood).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' These physical environment characteristics can be accurately detected because they are largely consistent across different geographic contexts and times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Thus, the efficacy of the visual detection tasks depends on the modeling capabilities of DCNNs, which have already achieved great capacities and are improving continuously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Hence, visual detection is not likely to be where most of the future work will focus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Within-place and between-place inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Within-place inference refers to how well an attribute extracted from an image can be used to predict any given outcome for a place with a geographic scale as small as a block or as large as a region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' In an age when the physical envi- ronment and the social dimensions are deeply intertwined, many aspects of a city is interrelated (Batty, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Thus, traditional modeling methods can miss the interacted and nonlinear rela- tionships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Conventional statistical models have been used to measure how a given attribute of a city (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', greenery density) is associated with another one (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', neighborhood health outcome).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' 21 DCNN models can accomplish this task with enhanced non-linear modeling capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' This means that by assigning a label to a DCNN model, the model is always capable of finding rela- tionships (both linear or non-linear) between an input street-level imagery and output variables (such as the socioeconomic composition of a neighborhood, real estate prices, or the density of human activity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Hence, “within-place inference” depends on how strong the underlying relationships are between the physical (visual) environment and its particular social correlates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' “Between-place inference” refers to how well a DCNN model fitted in one place can be applied to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' The between-place inference is facing three sets of issues, which can be analyzed from three complementary disciplinary fields: machine learning, urban studies, and GIScience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' From the perspective of the machine learning field, between-place inference is commonly challenged by the cross-domain generalizability (Neyshabur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Limitations in the generalizability of models stem primarily from underlying data distributions—the differences in the relationships of input image and output label and also between the training domain and test domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' This issue, termed “domain shift” (Qui˜nonero-Candela et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2009), has been ex- tensively examined in machine learning, and can be addressed through domain adaption (Wang and Deng, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Between-place inference from the perspective of urban studies has mainly focused on issues related to how best to measure a “place.” The heterogeneous uses of places and the fact that they are perceived differently across cultures make between-place inference difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' For example, a DCNN model trained to infer urban mobility patterns using street-view images in China could fail in cities in Western countries because the human activity patterns vary widely—even when the streets from the two cities have a similar shape (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2019a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Places are heteroge- neous because of differences in culture, geographical context, climate, historical development and a host of other factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' With these differences, it is unlikely that a fitted inference model developed in one place can be applied to another place without “domain adaption.” Finally, from the perspective of GIScience, between-place inference faces the issues of replicability in spatial analysis (Goodchild et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Kedron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Goodchild and Li, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' In particular, challenges with the replicability of models are due to the spatial hetero- geneity and non-stationarity inherent in spatial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' For example, making inferences between locations is challenging because the spatial variation of some phenomena/variables make the results not invariant across locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Moreover, due to spatial non-stationarity, relationships between variables across locations might not be consistent, making it impossible to general- ize across contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Together, spatial heterogeneity and non-stationarity explain why a local DCNN model often does not translate well to other locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Incorporating these principles into DCNN models provides an opportunity to improve the generalizability and transferability 22 of these models across contexts (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Cultural and subjective meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' A place is not solely defined by its natural and built settings, but also by the cultural and subjective meaning that people attribute to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' As a cultural landscape, a place is given a unique meaning by its occupants, activities, events, and its own historical evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' For example, the experience of visiting the Eiffel tower in Paris cannot be easily grasped by seeing a replica of the tower in, say, Las Vegas or Shenzhen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' While a lot of important social and cultural dimensions are encoded in the physical environment, its visual expression can also be subtle and unrevealing of its meaning to locals or visitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' A brick row house in Edinburgh may look similar to its counterpart in Boston, but may have quite different meanings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' This is the premise of special geography (Warntz, 1989) (or idiographic science), which begins with the assumption that each place is distinct and has unique properties that cannot be replicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' On the contrary, most machine learning models rely on an inductive learning process—they try to learn general and replicable rules from existing examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' As such, it is difficult for a DCNN model to fully interpret complex inhabited cultural landscapes solely from the elements represented by images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Differences in the perception of places are also explained by subjective meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' An indi- vidual’s sense of place is informed by their own past experiences, their stage in the life-cycle, and by their taste and preferences (Tuan, 1977).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' For example, the Temple Mount in Jerusalem will have different religious significance for Jews, Muslim, and Christians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Besides, a place may vary dramatically depending on the time of day or day of the week or year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Kim (2015) develops “spatial ethnography” to reveal the dramatically varying forms of “time-sharing” of socially, culturally and economically complex sidewalk streetscapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' In summary, while places are unique for individuals and groups, deep learning techniques can only summarize and infer the collective knowledge, sacrificing idiosyncratic but important factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' A DCNN model can easily count different human activities along sidewalks but will miss how people from varying demographics perceive and use public space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Incorporating individual and group preferences into AI studies is important for improving the representativeness of future AI models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Dealing with uncertainty in street-level imagery Uncertainty is an inevitable characteristic of spatial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' As a spatial data source, street- level imagery can be subject to several uncertainty issues, including the Modifiable Areal Unit Problem (MAUP), ecological fallacy, measurement uncertainty, and temporal uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' MAUP is the statistical bias resulting from aggregating point-based measures into zones (Fotheringham and Wong, 1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Spatially, street-level imagery is not evenly distributed be- cause the imagery from map services is spatially constrained by the road network, and social media photos are spatially distributed depending upon the intensity of urban functions and hu- man activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Even for a very long street, the characteristics of imagery tend to exhibit high 23 internal homogeneity, and may not reflect the fact that a street running in parallel may present a widely different appearance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Aggregating different combinations of a limited number of image samples into spatial units can yield entirely different results, exacerbating the MAUP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' The aggregation of street-level imagery can also give rise to the ecological fallacy, which occurs when conclusions are drawn about individual images based on their aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' An example is inappropriately concluding that all locations in a street are beautiful, just because the average beauty score of that street is high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Street-level imagery is also subject to measurement uncertainty, which mainly stems from the varying distances between the camera and the visual scene being recorded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' The position of the camera has an impact on the proportions of each visual feature appearing in the image, which may in turn affect how they are analyzed by the computer vision algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' For example, street-level imagery of tall structures may be only represented by the appearance of the parts closest to the ground level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' The visual features may also be subject to temporal changes that vary across seasons, in- cluding seasonal changes that affect vegetation, sky view, and the number of pedestrians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' These aspects are overlooked in most current studies that must rely on data that is collected infre- quently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' This is in part due to the fact that street view imagery collected on Google Street View is updated at most every year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Granular data gathered more frequently is essential for broader research agendas that measure the physical changes of neighborhoods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' The good news is that access to such datasets are likely to increase as point clouds (LiDAR data) and crowdsourced initiatives continue to grow (Mapillary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Promising avenues of inquiry and future work Street-level imagery and deep learning techniques can provide more than efficient mea- surements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' These techniques have the potential to support new understandings and knowledge discovery about cities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Here we discuss several aspects that can be explored in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' New knowledge about the functioning of cities can be achieved by studying the hidden visual cues in images, such as written language or signs of social disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' For example, writ- ten language in street names, business names, and ads can be easily acquired from street-level imagery and used to map points of interest and locations of services such as restaurants, pawn- brokers and payday loan outlets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Street signage can also help map linguistic or ethnic groups by providing insights into the composition of the population, social disorder, psychosocial stress and other important (and frequently less explores) aspects of neighborhoods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Vehicle types on street-level imagery can also be used to study the social dimensions of the city.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Seminal work by Gebru et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' (2017) uses the vehicle types of a neighborhood mined from images to infer the demographics and political tendencies of neighborhoods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Other di- mensions of these vehicles, such as their type (commercial/private), spatiotemporal presence 24 patterns (obtained from cameras), and the cost of private vehicles can be used to understand the socioeconomic characteristics of neighborhoods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' One can imagine other visual cues from street-level imagery as important indicators of neighborhood dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' An entirely new city can be created by combining design criteria with AI scene-generation techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Scene generation is enabled by a special architecture of DCNN called Generative Adversarial Nets (GAN) (Goodfellow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2014), which can generate entirely new predicted images after learning what realistic scenes look like from hundreds of thousands of real street scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' GAN creates computer-generated urban scenes based on user-generated inputs, such as objective characteristics extracted from images (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', buildings, roads, and vehicles) or encoded perceptions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', attractiveness, safety, and liveliness of a place).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Urban scenes can be also generated and edited based on some attributes of the physical environment (Bau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2020b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Richter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' These types of models are useful for scenario planning and urban design applications which frequently require that participants envision images of cities that don’t exist (Wu and Biljecki, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Urban designers are also using GAN to create vivid high-resolution imagery (Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' A pioneering approach looking at this was introduced by Noyman and Larson (2020), who designed a physical platform 6 that allows users to generate street scenes by combining a wide range of street elements based on their preferences, including different land-uses, types of roads, the density of buildings, and presence of sidewalks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Designing interpretable and reliable AI models has received increasing attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' We be- lieve that interpretable models can support the analysis of the urban physical environment in two ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' For scientific research, machine learning models have stronger fitting and model- ing capabilities than traditional regression models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' They can better predict the human activity patterns and socioeconomic profiles of cities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' For practitioners, research findings can only be used as a reference by policymakers when there is a clear causal pathway, whereas interpretable machine learning models can further reveal the confounding factors inherent in causal analysis for policy formulation and implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Finally, fine-grained characteristics of the physical environment derived from street-level imagery offer tremendous opportunities to systematically understand the spatial laws of the ur- ban physical environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' We believe that there are recognizable patterns in the way the phys- ical features of cities are organized in a city.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Following, there may exist a basic spatial unit that constitutes urban space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Future work could investigate how the computational representation of physical features changes accordingly as the scales and the organization of spatial units change, and whether there is a consistent spatial scale to represent the physical urban environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' 6https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='mit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content='edu/projects/deep-image-of-the-city/ 25 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Conclusion Using visual information to understand cities has a long tradition in urban studies, city plan- ning, and design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' However, measurements of the physical environment and people’s response to it have been challenging to evaluate until recently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Today, the emergence of artificial intel- ligence provides us with more efficient and effective tools to understand the city and how it in turn affects its citizens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' This paper reviews and compares the traditional and latest literature on the visual analyses of cities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' We propose a conceptual framework, Urban Visual Intelligence, to summarize and guide our discussion on how digital technology, especially street-level imagery and visual intelligence techniques, is reshaping the way we understand cities and opening up new avenues for research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' Ultimately these new tools will allow us to revisit the classic theories and themes that have guided the understanding and design of cities for over a century, and have the potential to help cities create environments that are more in line with human aspirations and behaviors in the digital age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' References Ahlfeldt, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' and Mastro, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' (2012).' metadata={'source': 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+page_content=' The Urban Book Series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} +page_content=' 41' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNAyT4oBgHgl3EQfsfn-/content/2301.00580v1.pdf'} diff --git a/LdE4T4oBgHgl3EQfig1d/content/tmp_files/2301.05134v1.pdf.txt b/LdE4T4oBgHgl3EQfig1d/content/tmp_files/2301.05134v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..981f8ea7c97c0e8a998ed4efaaa720da8cdacd13 --- /dev/null +++ b/LdE4T4oBgHgl3EQfig1d/content/tmp_files/2301.05134v1.pdf.txt @@ -0,0 +1,722 @@ +arXiv:2301.05134v1 [math.CO] 12 Jan 2023 +A GRID THEOREM FOR STRONG IMMERSIONS OF WALLS +REINHARD DIESTEL, RAPHAEL W. JACOBS, PAUL KNAPPE, AND PAUL WOLLAN +Abstract. We show that a graph contains a large wall as a strong immersion minor if and only if the +graph does not admit a tree-cut decomposition of small ‘width’, which is measured in terms of its adhesion +and the path-likeness of its torsos. +1. Introduction +In their Graph Minors Project [16], Robertson and Seymour investigated the structure of graphs not +containing a fixed graph as a minor. An important example of their structure theorems as well as a result +central to their project is the grid theorem [17] which asserts the equivalence of large tree-width and the +existence of large grid minors. More precisely, every graph of large enough tree-width contains a large grid +as a minor, and large grid minors form an obstruction to small tree-width in that large grids have large +tree-width and every graph has tree-width at least the tree-width of each of its minors. An equivalent +formulation of the grid theorem is that a graph has large tree-width if and only if it contains a large wall +as a topological minor [18]. +Grid theorem-like results, which provide the equivalence of large width with respect to some width- +measure and a large (often grid-like) substructure in terms of a corresponding containment relation, have +since been proven in various other settings. For example, Kreutzer and Kawarabayashi proved a directed +grid theorem [12], which establishes an equivalence of large directed tree-width and the existence of butterfly +minors of large directed grids, and Geelen, Kwon, McCarty and Wollan showed in [9] that a graph has +large rank-width if and only if it contains a vertex-minor isomorphic to a large comparability-grid. +In this paper we are concerned with the structure of graphs not containing a large wall as a strong +immersion minor. As the minor relation, immersion minors generalise the notion of topological minors, but +they are unrelated to minors in that neither the existence of an immersion minor implies the existence of a +minor nor vice-versa. Immersion minors have attracted attention from various algorithmic and structural +viewpoints in recent years, often finding results analogous to those for minors [1,3,6–8,10,13,19,20]. +There are two versions of the immersion relation, a weak one and a strong one. They are defined as +follows. +A weak immersion of a graph H in a graph G is a map α with domain V (H) ∪ E(H) that +embeds V (H) into V (G) and maps every edge uv ∈ H to an α(u)–α(v) path in G which is edge-disjoint +from every other such path. If, additionally, these paths have no internal vertices in α(V (H)), then α is a +strong immersion of H in G. The vertices in α(V (H)) are the branch vertices of this immersion. We say +that H is weakly/strongly immersed in G, or that H is a weak/strong immersion minor of G, if there is a +weak/strong immersion of H in G. +While the (topological) minor relation behaves well with tree-decompositions, the immersion relations +do so with tree-cut decompositions, which form the edge-cut analogue of tree-decompositions. Formally, a +pair (T, X) is a tree-cut decomposition of a graph G if T is a tree and X = (Xt)t∈T a near-partition of V (G), +2020 Mathematics Subject Classification. 05C75, 05C83, 05C40. +Key words and phrases. Strong immersion, wall, grid theorem. +1 + +2 +REINHARD DIESTEL, RAPHAEL W. JACOBS, PAUL KNAPPE, AND PAUL WOLLAN +that is, the Xt are disjoint and their union is V (G) but, in contrast to a partition, the Xt may be empty. +The vertex sets Xt are the parts of (T, X). The torso of (T, X) at a node t ∈ T arises from G by identifying +for every component T ′ of T − t the vertices in � +t′∈T ′ Xt′ to a single vertex, keeping parallel edges as they +arise but omitting loops. These new identification vertices are the peripheral vertices of the torso, while +the ones in Xt are its core vertices. Every edge t1t2 ∈ T induces its adhesion set EG( � +t∈T1 Xt , � +t∈T2 Xt ) +where T1 and T2 are the two components of T − t1t2 with t1 ∈ T1 and t2 ∈ T2. The adhesion of a tree-cut +decomposition then is the maximum size of its adhesion sets. +DeVos, McDonald, Mohar and Scheide [4] and Wollan [20] independently showed that if a graph does not +admit Kℓ, the complete graph of size ℓ, as a weak immersion, then it admits a tree-cut decomposition each +of whose torsos contains at most ℓ vertices of degree at least ℓ2. As a qualitative converse, they proved that +a tree-cut decomposition in which each torso contains at most ℓ vertices of degree at least ℓ precludes the +existence of a weak immersion of Kℓ+1. Introducing the concept of tree-cut width, Wollan [20] also derived +a grid theorem for weak immersions, which establishes the equivalence of large tree-cut width and the +existence of weak immersions of large walls. His proof made use of the classical grid theorem for excluded +wall minors by Robertson and Seymour [17]. +For strong immersions Dvořák and Wollan proved the following general structure theorem describing +the structure of graphs not containing a fixed graph as a strong immersion minor. +Theorem 1.1 ([7, Theorem 4]). For every graph F, there exists an integer α = α(F) > 0 such that if a +graph G does not contain F as a strong immersion minor, then there exists a tree-cut decomposition of G +of adhesion less than α such that each of its torsos is α-basic. +Here, the α-basicness of a graph means, roughly speaking, that it has a path-like decomposition in which +we can accommodate vertices of high degree. +When we restrict F to complete graphs in Theorem 1.1, then there is also a qualitative converse +of Theorem 1.1 in [7]: For every integer α > 0, there exists an integer n = n(α) > 0 such that any +graph with a tree-cut decomposition which has adhesion less than α and whose torsos are α-basic does not +contain Kn as a strong immersion minor. +If we consider strong immersions of walls, as it is the subject of this paper, and we thus look at Theorem 1.1 +with F restricted to walls, then Theorem 1.1 does not have such a qualitative converse. +Indeed, +Figure 1. The wall W4 of size 4 with +the underlying 4 × 8 grid. +for every integer ℓ > 0, the wall Wℓ of size ℓ is formally defined as +the graph arising from the ℓ×2ℓ grid by deleting all edges (i, j)(i′, j′) +with j = j′, i = i′ + 1 and j ≡ i (mod 2) and then removing the +two resulting vertices of degree 1 (see Figure 1). By definition the +wall Wℓ has 2ℓ2 −2 vertices and maximum degree 3. Thus, Wℓ does +not contain K5 as a strong immersion minor. Theorem 1.1 hence +yields an integer α > 0 such that every wall, no matter how large its +size, has a tree-cut decomposition of adhesion less than α such that +each torso is α-basic. In particular, Theorem 1.1 does not admit a +qualitative converse for walls. +In this paper, we prove a grid theorem-like result for the strong immersion relation which says that a +graph contains a large wall as a strong immersion minor if and only if the graph has a tree-cut decompo- +sition of a specific type. These tree-cut decompositions are inspired by those in Theorem 1.1 in that their +torsos still admit a path-like structure, but path-likeness is defined differently, as follows. + +A GRID THEOREM FOR STRONG IMMERSIONS OF WALLS +3 +A graph G is α-thin for an integer α > 0 if there exists an enumeration v1, . . . , vn of its vertices such +that there are at most α edges in G joining {v1, . . . , vi−1} and {vi+1, . . . , vn} for i = 1, . . . , n, and we +call G almost α-thin if it becomes α-thin after removing up to α vertices each of which has at most α +neighbours in G. In our specific tree-cut decompositions, the torsos then have almost α-thin ‘3-centres’. +Following [20], the 3-centre of a torso of a tree-cut decomposition arises from the torso by a maximal +sequence of deleting peripheral vertices of degree at most 1 and suppressing peripheral vertices of degree 2, +removing arising loops. We will show that if a torso of a tree-cut decomposition is itself (almost) α-thin, +then so is its 3-centre, while the converse does not hold. For a further discussion of these notions, we refer +the reader to Section 2. +Our first main result, a version of Theorem 1.1 specifically tailored to walls, then reads as follows: +Theorem 1. For every integer ℓ > 0, there exists an integer α = α(ℓ) > 0 such that if a graph G does +not contain the wall Wℓ as a strong immersion minor, then G has a tree-cut decomposition of adhesion at +most α such that the 3-centres of its torsos are almost α-thin. +We will see that 3-centres are indeed necessary in Theorem 1 in that we cannot consider the torsos them- +selves instead of their 3-centres (see Example 2.6). +In contrast to Theorem 1.1, the notion of path-like torsos as in Theorem 1 now gives the desired equiva- +lence of the existence of strong immersions of large walls and these tree-cut decompositions. More precisely, +our second main results provides a qualitative converse of Theorem 1: +Theorem 2. For every integer α > 0, there exists an integer ℓ = ℓ(α) > 0 such that if a graph G has a +tree-cut decomposition of adhesion at most α such that the 3-centres of its torsos are almost α-thin, then G +does not contain the wall Wℓ as a strong immersion minor. +While the proof of Theorem 2 is self-contained and does not build on any previous results, the proof +of Theorem 1 draws on the grid theorem for excluded wall minors as well as on several previously estab- +lished methods for immersions and tree-cut decompositions [7,17,20]. We emphasise that we do not make +use of any previous structure theorem for weak or strong immersions, but rather use some of their ideas +and combine them in a novel way to fit our specific problem. +This paper is organised as follows. We first discuss the notions of almost α-thin graphs and 3-centres +in Section 2. Then we prove Theorem 2 in Section 3 and Theorem 1 in Section 4. +For basic graph-theoretic terms, we follow [5]. In this paper, we only consider graphs without loops. +But unless called simple, they may contain parallel edges. As a consequence, the degree of a vertex is the +number of its incident edges which may in general differ from the number of its neighbours. Moreover, we +diverge from [5] in that we define the components of a graph not as its maximal connected subgraphs, but +as their vertex sets. +2. Almost α-thin graphs and 3-centres +In this section we take a closer look into our notion of path-likeness, (almost) α-thin graphs, and its +relation to the concept of 3-centres. +Let us first recall the definition of (almost) α-thinness from the +introduction. + +4 +REINHARD DIESTEL, RAPHAEL W. JACOBS, PAUL KNAPPE, AND PAUL WOLLAN +Definition 2.1. Let α > 0 be an integer. A graph G is α-thin if there exists an enumeration v1, . . . , vn +of its vertices satisfying that |EG({v1, . . . , vi−1}, {vi+1, . . . , vn})| ⩽ α for i = 1, . . . , n,1 and the graph G +is almost α-thin if there exists a set X of at most α vertices of G such that each vertex in X has at most α +neighbours in G and G − X is α-thin. +We remark that if a graph is (almost) α-thin, then so are all its subgraphs. +Even though adding vertices of degree 1 to a graph does not change whether it contains a wall as a +strong immersion minor, such vertices affect whether a graph is (almost) α-thin. +Example 2.2. For every integer α > 0, the star K1,n with n = 3α + 3 leaves is not almost α-thin, but +becomes 0-thin after repeatedly removing vertices of degree 1. +Proof. Repeatedly removing its vertices of degree at most 1, the star G := K1,n becomes the graph on one +vertex which is trivially 0-thin. So it remains to argue that G is not almost α-thin, and we suppose for a +contradiction that there exists a set X of at most α vertices of G, each with at most α neighbours in G, +such that G − X is α-thin. +The set X cannot contain the centre c of G since it has n > α many neighbours in G. Thus, X consists +of up to α leaves of G, and G − X is again a star with centre c, now with at least n′ ⩾ 2α + 3 leaves. +Now take an arbitrary enumeration v1, . . . , vn′+1 of V (G − X), and let i ∈ {1, . . . , n′ + 1} with vi = c. +If i ⩾ α + 3, then {v1, . . . , vi−2} contains at least α + 1 leaves of the star, while if i ⩽ (n′ + 1) − (α + 3), +then {vi+2, . . . , vn′+1} contains at least α + 1 leaves. So for j = i − 1 in the first case and j = i + 1 in the +second case, there are at least α + 1 edges joining {v1, . . . , vj−1} and {vj+1, . . . , vn′+1}, which yields the +desired contradiction. +□ +Similar to the addition of vertices of degree 1, subdividing edges has an impact on (almost) α-thinness. +Example 2.3. For every integer α > 0, the graph G that arises from two disjoint stars, each with n = 3α+3 +leaves, by identifying their leaves is not almost α-thin, but becomes 0-thin after repeatedly suppressing +vertices of degree 2. +Proof. If we repeatedly suppress vertices of degree 2, then G turns into the graph consisting of two vertices +which are joined by n edges. This graph is trivially 0-thin. Since G contains K1,n as a subgraph and K1,n +is not almost α-thin by Example 2.2, G cannot be almost α-thin itself. +□ +Just as a graph stays (almost) α-thin when we delete some of its vertices, suppressing vertices of degree 2 +also does not impact the (almost) α-thinness of a graph, as the next lemma demonstrates: +Lemma 2.4. Let G be an (almost) α-thin graph for some integer α > 0. Then suppressing a vertex of G +of degree 2 (and deleting a potentially arising loop) results in an (almost) α-thin graph. +Proof. It is enough to prove the lemma for α-thinness, since the case of almost α-thinness then follows +immediately. So let G be an α-thin graph, let v ∈ G be a vertex of degree 2 and let u and w be the other +endvertices of the two edges of G incident to v. Now let G′ arise from G by suppressing v, omitting a +potentially arising loop. +1Note that the definition of α-thin differs from the similar notion of cutwidth at most α, since cutwidth also takes the edges +joining vi and {vi+1, . . . , vn} into account. While cutwidth bounds thinness from above, there are graphs with unbounded +cutwidth which are 0-thin, as paths with many parallel edges witness. + +A GRID THEOREM FOR STRONG IMMERSIONS OF WALLS +5 +If u = w, then G′ is a subgraph of G which is hence α-thin; so we may assume that u and w are distinct. +Given an enumeration v1, . . . , vn+1 of V (G) witnessing that G is α-thin, let k ∈ {1, . . . , n + 1} with vk = v. +We may assume that v appears in this enumeration in between u and w; otherwise, moving v in between +them only reduces the number of edges joining {v1, . . . , vi−1} and {vi+1, . . . , vn+1} for i = 1, . . . , n + 1. We +then define the enumeration v′ +1, . . . , v′ +n of V (G′) as follows: +v′ +i := + + + +vi +if i < k; +vi+1 +if i > k. +This definition guarantees that for i = 1, . . . , n, the edges of G′ joining {v′ +1, . . . , v′ +i−1} and {v′ +i+1, . . . , v′ +n} +coincide with the edges of G joining {v1, . . . , vj−1} and {vj+1, . . . , vn+1}, where j := i if i < k and j := i+1 +if i ⩾ k, except that the edge of G′ that arose by suppressing v is replaced by one of the two edges incident +to v in G. Thus, G′ is α-thin as witnessed by the enumeration v′ +1, . . . , v′ +n of V (G′). +□ +As we will see below, we need to ignore peripheral vertices of degree at most 2 in torsos to estab- +lish Theorem 1 (see Example 2.6). +Formally, we achieve this with the notion of 3-centres which was +introduced in [20] and is well-defined by [20, Lemma 9]. +Definition 2.5. Let X be a set of vertices of a graph G. The 3-centre of (G, X) arises from G by a +maximal sequence of deleting vertices of degree at most 1 and suppressing vertices of degree 2 (omitting +any resulting loops) where all these vertices are not in X. If (T, X) is a tree-cut decomposition of G, then +the 3-centre of the torso Ht of (T, X) at the node t ∈ T is defined as the 3-centre of (Ht, Xt). +We note that the 3-centre of a pair (G, X) cannot be formed by first repeatedly deleting vertices of +degree 1 and then suppressing all vertices of degree 2. Indeed, the deletion of resulting loops may create +new vertices of degree 1 which have to be deleted afterwards; consider for example the graph G which +consists two triangles joined by an edge and let X be the set of vertices of one of the triangles. +If a graph G is (almost) α-thin, then the 3-centre of (G, X) is also (almost) α-thin since deleting vertices +and suppressing vertices of degree 2 does not affect the (almost) α-thinness of a graph, see Lemma 2.4. In +particular, it is weaker to assume in Theorem 2 that the 3-centres of the torsos are almost α-thin than to +assume that the torsos themselves have this property. +As we have seen in Example 2.2 and Example 2.3, the converse statement is not true: If the 3-centre +of (G, X) is α-thin, then G does not even have to be almost α-thin itself. In fact, the converse holds in an +even stronger form in that we cannot remove 3-centres from Theorem 1 and consider the torsos themselves +instead, as the following example demonstrates. +Example 2.6. For every two integers ℓ ⩾ 2 and α > 0, there exists a graph which does not contain the +wall Wℓ as a strong immersion minor and also does not admit a tree-cut decomposition of adhesion at +most α whose torsos are almost α-thin. +Proof. Let G = K1,n be the star with n = α(3α + 3) leaves and centre c ∈ V (G). Clearly, G does not +contain Wℓ as a strong immersion minor since G contains only a single vertex of degree more than 1 +while Wℓ has several of those because of ℓ ⩾ 2. Thus, it remains to consider an arbitrary tree-cut decom- +position (T, X) of G of adhesion at most α and to show that at least one of the torsos of (T, X) is not +almost α-thin. +Let s be the (unique) node of T such that Xs contains the centre c of the star G. For each neighbour t +of s in T , let Yt := � +t′∈T ′ Xt′ where T ′ is the component of T − st containing t. Since (T, X) has adhesion + +6 +REINHARD DIESTEL, RAPHAEL W. JACOBS, PAUL KNAPPE, AND PAUL WOLLAN +at most α and c ∈ Xs, the set Yt contains at most α leaves of G. But G has α(3α + 3) leaves, so it yields a +star G′ with at least 3α+3 leaves in the torso Hs of (T, X) at s. This star G′, however, is not almost α-thin +by Example 2.2, and hence Hs is not almost α-thin either. +□ +If we adapt Example 2.6 by doubling each edge in G, then this modified version also demonstrates that just +deleting peripheral vertices of degree 1 in each torso instead of taking its 3-centre does not suffice either. +3. Strongly immersed walls are obstructions +In this section, we prove Theorem 2 which we recall here: +Theorem 2. For every integer α > 0, there exists an integer ℓ = ℓ(α) > 0 such that if a graph G has +a tree-cut decomposition of adhesion at most α such that all the 3-centres of its torsos are almost α-thin, +then G does not contain the wall Wℓ as a strong immersion minor. +As a key ingredient to our proof of Theorem 2, we need Lemma 3.1, which asserts that the wall Wℓ +contains a vertex set Z of size ℓ − 2 which is well-linked in Wℓ, that is, for every two disjoint A, B ⊆ Z +with k := |A| = |B|, there exist k vertex-disjoint A–B paths in Wℓ. The existence of such a set Z of size ℓ +follows from the facts that the tree-width of a wall of size ℓ is at least ℓ and the largest size of a well-linked +in a graph is lower bounded by its tree-width (see for example the survey [11]). We include a direct and +constructive proof in the case of walls here which yields a further structural property of the well-linked set. +Lemma 3.1. The wall Wℓ of size ℓ ⩾ 3 contains a set Z of vertices of size ℓ − 2 such that Z is well-linked +in Wℓ and such that every two vertices in Z are joined by three internally vertex-disjoint paths in Wℓ. +Proof. Let Z be the set of all vertices of Wℓ which have the form (ℓ, i) and degree 3. Note that Z has size ℓ−2 +by the definition of Wℓ. It is easy to see that every two vertices of degree 3 are joined by three internally +vertex-disjoint paths in Wℓ as ℓ ⩾ 3. Thus, it remains to show that for every two disjoint subsets A, B of Z +with k := |A| = |B|, there exist k vertex-disjoint A–B paths in Wℓ. By Menger’s theorem [15], it suffices to +prove that for every set X ⊆ V (G) of size at most k − 1, there exists an A–B path in Wℓ which avoids X. +For every j = 1, . . . , ℓ let Vj be the j-th vertical path of the wall Wℓ, that is, Vj is the induced subgraph +of Wℓ on the set of all vertices of the form (2j, i) or (2j − 1, i). Note that these vertical paths are pairwise +vertex-disjoint, that each vertical path meets Z in at most one vertex, and that each vertex in Z is contained +in some vertical path. Thus, the pigeonhole principle yields a vertical path VjA which meets A, but avoids X, +since |A| = k > k − 1 ⩾ |X|; analogously, we obtain a vertical path VjB for B. +Next, for every j = 1, . . . , ℓ let Hj be the j-th horizontal path of the wall Wℓ, that is, the subgraph of Wℓ +induced on the set of all vertices of the form (i, j). Note that these horizontal paths are pairwise vertex- +disjoint. Thus, the pigeonhole principle yields an horizontal path Hj0 which avoids X, since ℓ > |A| > |X|. +By definition, every horizontal path meets every vertical path. Hence, VjA + Hj0 + VjB is a connected +subgraph of Wℓ which meets both A and B, but avoids X. In particular, there exists an A–B path in Wℓ +which avoids X, as desired. +□ +Proof of Theorem 2. We set +ℓ := ℓ(α) := [(α2 + 1)(2(α + 1) + 4) + α2 + α] + [α((α2 + 1)(2(α + 1) + α + 2) + α2 + α)] + 2, +and claim that this ℓ is as desired. So let G be a graph, and let (T, X) be a tree-cut decomposition of G of +adhesion at most α such that the 3-centres of its torsos are almost α-thin. Suppose for a contradiction that +the wall Wℓ is strongly immersed in G, and let U be the set of branch vertices of this strong immersion. + +A GRID THEOREM FOR STRONG IMMERSIONS OF WALLS +7 +By Lemma 3.1, Wℓ contains a vertex set Z of size at least ℓ − 2 such that Z is well-linked in Wℓ and +every two vertices in Z are joined by three internally vertex-disjoint paths in Wℓ. Let ZU ⊆ U be the set of +branch vertices corresponding to Z ⊆ V (Wℓ) in G via the strong immersion. Since Z is well-linked in Wℓ, +the definition of strong immersions yields that this set ZU has the following property: +For every two disjoint sets A, B ⊆ ZU with k := |A| = |B|, there exist k edge-disjoint A–B paths +in G and these k paths meet each vertex of U at most once. +(∗) +Moreover, since every two vertices in Z are joined by three internally vertex-disjoint paths in Wℓ, every +two vertices in ZU are joined by three edge-disjoint paths in G. +Consider an edge t1t2 ∈ T and let Y1 := � +t∈T1 Xt where T1 is the component of T − t1t2 containing t1, +and define Y2 analogously. Since (T, X) has adhesion at most α, the adhesion set EG(Y1, Y2) of G induced +by t1t2 has size at most α. Therefore, precisely one Yi, say Y2, contains at least α + 1 vertices from ZU +by (∗) and since ℓ − 2 ⩾ 2(α + 1). We then orient the edge t1t2 from t1 to t2. In this way, ZU induces +an orientation of the edges of T such that for each node t ∈ T at most one incident edge is oriented away +from t. +Hence, there is a unique sink s of this orientation, that is, a node s ∈ T such that all incident edges are +oriented towards s. Let H be the torso of (T, X) at s. For each neighbour t of s in T , let Yt := � +t′∈T ′ Xt′ +where T ′ is the component of T − st containing t, and let vt be the peripheral vertex of H corresponding +to the identification of the vertices in Yt. +Consider the set ZH of vertices of H corresponding to the vertices in ZU, that is, the set consisting of +the core vertices ZU ∩ Xs together with those peripheral vertices vt with ZU ∩ Yt ̸= ∅. This set ZH is +then contained in the 3-centre ¯H of the torso H. Indeed, for every peripheral vertex vt ∈ ZH, the set ZU +contains a vertex in Yt by the definition of ZH. Since st is oriented towards s, the set ZU also contains +a vertex which is not in Yt. But these two vertices, just as any two vertices in ZU, are joined by three +edge-disjoint paths in G. Thus, vt is never a candidate for deletion or suppression in the construction of ¯H +from H. +For every peripheral vertex vt in ZH, we have |ZU ∩ Yt| ⩽ α, since st was oriented towards s. This +implies that |ZH| ⩾ |ZU ∩Xs|+|ZU ∖Xs|/α. In particular, ZH contains either many core vertices or many +peripheral vertices of H. +We now aim to use the set ZH to derive a contradiction to the 3-centre ¯H of the torso H being almost α- +thin. To do so, consider a set X of at most α vertices of ¯H each of which has at most α neighbours in ¯H and +an enumeration v1, . . . , vn of V ( ¯H − X) which witnesses that ¯H − X is α-thin. Observe that if we remove +the at most α2 neighbours of the set X among the vi, then this partitions the enumeration v1, . . . , vn into +a set I of at most α2 + 1 intervals. +If ZH contains many core vertices of H in that |ZH∩Xs| ⩾ (α2+1)(2(α+1)+4)+α2+α, then {v1, . . . , vn} +contains at least (α2 + 1)(2(α + 1) + 4) core vertices in ZH which are not neighbours of X. Thus, the +pigeonhole principle yields an interval vj, . . . , vk in I containing at least 2(α + 1) + 4 elements of ZH ∩ Xs; +shortening the interval if necessary, we may assume additionally that it contains precisely 2(α + 1) + 4 +elements of ZH ∩ Xs and that its endpoints vj and vk are among those. +Let AH be the set consisting of the 2(α + 1) + 2 elements of ZH ∩ Xs in the interior {vj+1, . . . , vk−1} +of our fixed interval, and let BH be a subset of ZH ∩ Xs of the same size as AH and which is disjoint +from {vj, . . . , vk}; such a set BH exists since ZH ∩ Xs is large enough. Now AH and BH are subsets of ZH +consisting of core vertices of H, and hence AH, BH ⊆ ZU. So we may apply (∗) to AH, BH ⊆ ZU and + +8 +REINHARD DIESTEL, RAPHAEL W. JACOBS, PAUL KNAPPE, AND PAUL WOLLAN +obtain 2(α + 1) + 2 edge-disjoint AH–BH paths in G such that at most two of these paths meet vj or vk, +since vj and vk are contained in ZH ∩ Xs ⊆ ZU ⊆ U. +Since the torso H arises from G by identifications of vertices keeping parallel edges as they arise, +these 2(α + 1) + 2 paths induce 2(α + 1) + 2 edge-disjoint AH–BH walks in H which we can shorten +to 2(α + 1) + 2 edge-disjoint AH–BH paths in H. By the definition of 3-centres, these 2(α + 1) + 2 paths +in the torso H in turn induce 2(α + 1) + 2 edge-disjoint AH–BH paths in the 3-centre ¯H of H. We remark +that throughout this process we maintain the property that at least 2(α+1) of these paths avoid vj and vk. +Since none of the vertices vj, . . . , vk is a neighbour of X, these 2(α + 1) edge-disjoint paths join- +ing {vj+1, . . . , vk−1} and {v1, . . . , vj−1}∪{vk+1, . . . , vn} indeed contain at least 2(α+1) edges in ¯H −X join- +ing {vj+1, . . . , vk−1} and {v1, . . . , vj−1} ∪{vk+1, . . . , vn}. In particular, there are either at least α +1 edges +joining {v1, . . . , vj−1} and {vj+1, . . . , vn} or at least α + 1 edges joining {v1, . . . , vk−1} and {vk+1, . . . , vn}. +In both cases we obtain a contradiction to the choice of the enumeration v1, . . . , vn to witness the α-thinness +of ¯H − X. +If ZH does not contain many core vertices as in the above case, then |ZH| ⩾ |ZU ∩ Xs| + |ZU ∖ Xs|/α +implies that ZH contains many peripheral vertices in that |ZH ∖ Xs| ⩾ (α2 + 1)(2(α + 1) + α + 2) + α2 + α +by our choice of ℓ. We now proceed analogously to the above case of many core vertices in ZH. +By the lower bound on the size of ZH ∖ Xs, there are at least (α2 + 1)(2(α + 1) + α + 2) peripheral +vertices in ZH which are neither neighbours of X nor contained in X. So the pigeonhole principle yields +an interval vj, . . . , vk in I which contains at least 2(α + 1) + α + 2 elements of ZH ∖ Xs. We may assume +additionally, by shortening the interval if necessary, that the interval contains precisely 2(α + 1) + α + 2 +elements of ZH ∖ Xs and that its endpoints vj and vk are among those. +Let AH be the set of the 2(α + 1) + α elements of ZH ∖ Xs in the interior {vj+1, . . . , vk−1} of our fixed +interval, and let BH be a set of 2(α + 1) + α elements of ZH ∖ Xs that are not in {vj, . . . , vk}; such a +set BH exists since ZH ∖ Xs is large enough. Since each peripheral vertex vt in ZH satisfies ZU ∩ Yt ̸= ∅, +we may replace each element vt of AH and BH by an arbitrary vertex in ZU ∩ Yt to obtain subsets AU +and BU of ZU, respectively, which are disjoint and of the same size as AH and BH by the disjointness of +the Yt. Applying (∗) to AU, BU ⊆ ZU then yields 2(α + 1) + α edge-disjoint AU–BU paths in G. +The construction of AU and BU now guarantees that these 2(α + 1) + α many AU–BU paths in G +induce 2(α + 1) + α edge-disjoint AH–BH paths in ¯H as above. Since (T, X) has adhesion at most α +and vj, vk ∈ ZH ∖ Xs are neither contained in AH nor in BH, each of the peripheral vertices vj and vk +lies on at most α/2 of these paths. Thus, there are at least 2(α + 1) + α − 2 · α/2 = 2(α + 1) edge- +disjoint paths in ¯H joining {vj+1, . . . , vk−1} and {v1, . . . , vj−1} ∪ {vk+1, . . . , vn} which avoid vj and vk. +Since {vj, . . . , vk} contains no neighbours of X, these paths contain at least 2(α + 1) edges in ¯H − X +joining {vj+1, . . . , vk−1} and {v1, . . . , vj−1} ∪ {vk+1, . . . , vn}. Thus, there are either at least α + 1 edges +joining {vj+1, . . . , vk−1} and {v1, . . . , vj−1} or at least α + 1 joining {vj+1, . . . , vk−1} and {vk+1, . . . , vn}. +In particular, there are either at least α+1 edges joining {vj+1, . . . , vn} and {v1, . . . , vj−1} or at least α+1 +edges joining {v1, . . . , vk−1} and {vk+1, . . . , vn}, a contradiction. +□ + +A GRID THEOREM FOR STRONG IMMERSIONS OF WALLS +9 +4. Excluding strongly immersed walls +This section is devoted to the proof of Theorem 1, recalled here. +Theorem 1. For every integer ℓ > 0, there exists some integer α = α(ℓ) > 0 such that if a graph G does +not contain the wall Wℓ as a strong immersion minor, then G has a tree-cut decomposition of adhesion at +most α such that the 3-centres of its torsos are almost α-thin. +For the proof of Theorem 1, we first reduce the problem to 3-edge-connected graphs in Section 4.1. +In Section 4.2 we collect some tools from previous research dealing with immersions and tree-cut decompo- +sitions. Then we finally prove Theorem 1 in Section 4.3. +4.1. Reduction to 3-edge-connected graphs. We begin by reducing the proof of Theorem 1 to 3-edge- +connected graphs. This reduction will be done separately for each integer ℓ > 0; so throughout this section, +let ℓ > 0 be an arbitrary, but fixed integer. We first reduce our attention to graphs with minimum degree 3 +which will then facilitate the reduction to 3-edge-connected graphs. +Lemma 4.1. If Theorem 1 holds for all graphs with minimum degree 3, then it holds for all graphs. +Proof. Let α := α(ℓ) ⩾ 2 be an integer such that Theorem 1 holds for all graphs with minimum degree 3. +We now show that Theorem 1 holds with the same α for all graphs G, and we will do so by induction +on |G|. The case |G| = 1 is trivial; so suppose |G| > 1, and assume that G does not contain the wall Wℓ +as a strong immersion minor. +Let v ∈ G be a vertex with degree dG(v) at most 2 in G, and let G′ arise from G by deleting v if dG(v) ⩽ 1 +and suppressing v (omitting any loops which arise) if dG(v) = 2. Clearly, G′ does not contain the wall Wℓ +as a strong immersion minor, as G does not do so. Since |G′| < |G|, we can apply the induction hypothesis +to G′ to obtain a tree-cut decomposition (T ′, X ′) of G′ of adhesion at most α such that the 3-centres of its +torsos are almost α-thin. +If dG(v) ⩾ 1, let u be a neighbour of v in G, and otherwise, let u be an arbitrary vertex in G other +than v. Then there exists a (unique) node tu ∈ T ′ with u ∈ X′ +tu. We then obtain a tree-cut decomposi- +tion (T, X) of G from (T ′, X ′) in that T arises from T ′ by adding a vertex tv adjacent to tu and in that we +set Xtv := {v} while all other parts remain unchanged. We claim that (T, X) is as desired. +We first prove that (T, X) has adhesion at most α. The edge tutv ∈ T induces an adhesion set of size +at most 2 ⩽ α. Every other edge e of T is also an edge of T ′, and the adhesion set of (T, X) induced by e +contains the same edges as the one of (T ′, X ′) induced by e – except that if v has a second neighbour w ̸= u +in G, then the edge of G′ that arose by suppressing v is replaced by the edge joining v and w in G. This +replacement, however, leaves the size of the adhesion set unchanged. Hence, the adhesion of (T, X) is at +most α since (T ′, X ′) has adhesion at most α. +It remains to show that the 3-centres of the torsos of (T, X) are almost α-thin. Since v has degree at +most 2 in G, the 3-centre of the torso of (T, X) at tv consists only of the vertex v and is hence almost α-thin. +For all other nodes t ∈ T , the 3-centre of the torso of (T, X) at t equals the 3-centre of the torso of (T ′, X ′) +at t, since the new peripheral vertex v of the torso of (T, X) at t = tu has degree at most 2 and is hence +deleted or suppressed in the construction of the 3-centre from the torso. +□ +Lemma 4.2. If Theorem 1 holds for all 3-edge-connected graphs, then it holds for all graphs. +Proof. Let α := α(ℓ) ⩾ 2 be an integer such that Theorem 1 holds for all 3-edge-connected graphs. +By Lemma 4.1, it is enough to prove the statement for all graphs G of minimum degree 3, and we will do + +10 +REINHARD DIESTEL, RAPHAEL W. JACOBS, PAUL KNAPPE, AND PAUL WOLLAN +so by induction on |G|. The case |G| = 1 is trivial; so suppose that |G| > 1, that G has minimum degree 3, +that G is not 3-edge-connected, and that G does not contain the wall Wℓ as a strong immersion minor. +Let {A, B} be a bipartition of V (G) such that the size of the cut EG(A, B) of G is minimal among +all bipartitions of V (G). Since G is not 3-edge-connected, the cut EG(A, B) has size at most 2. Hence, +both A and B have size at least 2, since G has minimum degree 3. Let GA arise from G by contracting B +to a single vertex b, keeping parallel edges as they arise, but omitting loops; analogously define GB by +contracting A to a. Then both GA and GB strictly smaller than G, and they do not contain Wℓ as a +strong immersion minor, since G does not do so. By applying the induction hypothesis, we obtain tree-cut +decompositions (T A, X A) of GA and (T B, X B) of GB which both have adhesion at most α and such that +the 3-centres of its torsos are almost α-thin. +Using (T A, X A) and (T B, X B) we now construct a tree-cut decomposition (T, X) of G which, as we will +show afterwards, has the desired properties.2 Let tA ∈ T A and tB ∈ T B be the (unique) nodes with b ∈ XA +tA +and a ∈ XB +tB. We then define the tree T as the disjoint union of T A and T B together with an edge joining tA +and tB. We set XtA := XA +tA ∖ {b} and XtB := XB +tB ∖ {a}, and take Xt as the corresponding XA +t or XB +t for +all other nodes t ∈ T . +Let us first check that (T, X) has adhesion at most α. By construction, every adhesion set of (T, X) that +is induced by an edge of T other than tAtB has the same size as the adhesion set of (T A, X A) or (T B, X B) +induced by the corresponding edge of T A or T B. The adhesion set of (T, X) induced by tAtB is precisely +the cut EG(A, B) and hence of size at most 2 ⩽ α. +Next, we verify that the 3-centres of the torsos of (T, X) are almost α-thin. +For any node t ∈ T +with t ̸= tA, tB, the torso of (T, X) at t equals the corresponding torso of (T A, X A) or (T B, X B) and hence, +its 3-centre is almost α-thin. The torso H of (T, X) at tA differs from the torso HA of (T A, X A) at tA only +in that b is not a core vertex any more, but a peripheral vertex now. So the 3-centre of H arises from the +one of HA by deleting b if d(b) ⩽ 1 and suppressing b if d(b) = 2. But then the 3-centre of the torso H +is almost α-thin since the 3-centre of HA is, because a graph remains almost α-thin after the deletion of +a vertex by definition and the suppression of a vertex of degree 2 does not affect the almost α-thinness +by Lemma 2.4. The analysis of the torso at tB is symmetrical. +□ +4.2. Immersions and tree-cut decompositions. In this section we collect some tools and auxiliary +results for the proof of Theorem 1. +The first lemma asserts that the absence of a large wall as a strong immersion minor implies a bound on +the number of neighbours of every vertex. This is the lemma for whose application in the proof of Theorem 1 +we reduced to 3-edge-connected graphs in Section 4.1. +Lemma 4.3 ([6, Corollary 3]). For every integer ℓ > 0, there is an integer g(ℓ) > 0 such that if a 3-edge- +connected graph G contains a vertex with at least g(ℓ) neighbours, then the wall Wℓ is a strong immersion +minor of G. +The next two lemmas again exclude specific substructures in the absence of a large wall as a strong +immersion minor. First, for every two integers k, n > 0, let Sk,n be the graph on n+1 vertices x, v1, . . . , vn +with k parallel edges joining x and vi for i = 1, . . . , n. For every graph G on n vertices which has maximum +degree k, one can greedily construct a strong immersion of G in Sk,n (cf. [20, Observation 1]). +2This construction follows the one in terms of edge-sums given in [20, Lemma 5]. + +A GRID THEOREM FOR STRONG IMMERSIONS OF WALLS +11 +Lemma 4.4. For every two integers k, n > 0, the graph Sk,n contains every graph on n vertices with +maximum degree k as a strong immersion minor. In particular, S3,2ℓ2 contains the wall Wℓ as a strong +immersion minor for every integer ℓ > 0. +□ +Secondly, let Pn ∗ v denote the graph arising from the path Pn of length n by adding a vertex v that is +adjacent to every vertex of Pn. +Lemma 4.5. For every integer k > 0, the graph P3k−1 ∗ v contains a subdivision of S3,k. In particular, +the wall Wℓ is a strong immersion minor of P6ℓ2−1 ∗ v for every integer ℓ > 0. +Proof. Deleting every third edge on the path P3k−1, the graph P3k−1 ∗ v becomes a subdivision of S3,k. +The observation on Wℓ follows immediately from Lemma 4.4. +□ +The following results together show that the absence of a subdivision of the wall Wℓ, and thus in +particular the absence of Wℓ as a strong immersion minor, implies the existence of a tree-cut decomposi- +tion whose adhesion and torso size are both bounded in terms of the maximum degree and the size ℓ of the +wall. +Theorem 4.6 (Grid Theorem, [17, (2.1)]). For every integer ℓ > 0, there exists an integer w(ℓ) > 0 such +that if a graph G does not contain a subdivision of the wall Wℓ, then G has tree-width at most w(ℓ). +Lemma 4.7 ([20, Lemma 12]). Let w, d > 0 be integers, and let G be a graph with maximum degree at most d +and tree-width at most w. Then there exists a tree-cut decomposition of G of adhesion at most (2w + 2)d +such that every torso has at most (d + 1)(w + 1) vertices. +The next lemma says that a large enough set U of vertices in a connected graph is either reflected in a +subdivision of a star with many leaves in U or a comb with many teeth in U. Here, a comb is a union of +a path P with disjoint, possibly trivial, paths which have precisely their first vertex on P, and we call the +last vertices of these paths their teeth. Note that if a graph has small maximum degree, then the graph +does not contain any large subdivided star. +Lemma 4.8 ([2, Lemma 6.2]). For every two integers s, t > 0, there exists an integer h(s, t) > 0 such that +the following holds: Whenever a set U of vertices of a connected graph G has size at least h(s, t), then there +is either some subdivision of a star in G with all its s leaves in U or a comb in G with all its t teeth in U. +As a final ingredient to our proof of Theorem 1, the following lemma asserts that if a simple graph does +not contain a star as a minor, then it is ‘close’ to a union of vertex-disjoint paths. +Lemma 4.9 ([14, Lemma 20]). Let k > 0 be an integer. If a simple connected graph G does not contain K1,k +as a minor, then G − X is a vertex-disjoint union of paths for some set X of at most 4k vertices of G. +4.3. Proof of Theorem 1. +Proof of Theorem 1. We will define the integer α(ℓ) > 0 explicitly in terms of ℓ in what follows. Let G be +a graph in which the wall Wℓ is not strongly immersed. By Lemma 4.2, we may assume that G is 3-edge- +connected. +To construct the desired tree-cut decomposition (T, X) of G, we will first apply Lemma 4.7 to an appro- +priate minor G′ of G in order to obtain an auxiliary tree-cut decomposition (T, X ′) of G′ from which we +then construct (T, X). This minor G′ of G is defined as follows: Let A be the simple graph on V (G) with + +12 +REINHARD DIESTEL, RAPHAEL W. JACOBS, PAUL KNAPPE, AND PAUL WOLLAN +an edge joining two vertices if there are more than p(ℓ) := 6ℓ2 edges joining them in G. The graph G′ then +arises from G by contracting the components of A, keeping parallel edges as they arise (omitting potentially +arising loops). +Now G′ inherits two key properties of G, which we will need in what follows. First, G′ is still 3-edge-con- +nected as it is a minor of the 3-edge-connected graph G. Secondly, G′ does still not contain the wall Wℓ as +a strong immersion minor. Indeed, G does not contain Wℓ as a strong immersion minor, and since p(ℓ) ⩾ 3 +and Wℓ has maximum degree 3, one could greedily construct a strong immersion of Wℓ in G from a strong +immersion of Wℓ in G′. +In order to apply Lemma 4.7 to G′, we next bound its tree-width and its maximum degree in terms of ℓ. +Claim 1. The tree-width of G′ is at most w(ℓ), the integer given by Theorem 4.6. +Proof. Since G′ does not contain Wℓ as a strong immersion minor, in particular it does not contain a +subdivision of Wℓ. Therefore, Theorem 4.6 yields the desired upper bound w(ℓ) on the tree-width of G′. ■ +Claim 2. There exists an integer d(ℓ) such that G′ has maximum degree d(ℓ). +Proof. We prove that G′ has maximum degree at most d(ℓ) := g(ℓ) · p(ℓ) · h(2ℓ2, 6ℓ2)2, where h(2ℓ2, 6ℓ2) +and g(ℓ) are the integers given by Lemma 4.8 and Lemma 4.3, respectively. Since G′ is 3-edge-connected +and does not contain Wℓ as strong immersion minor, Lemma 4.3 implies that every vertex of G′ has less +than g(ℓ) neighbours in G′. Thus, it is enough to show that there are at most d(ℓ)/g(ℓ) many parallel edges +joining any fixed pair of vertices. By the construction of G′, this is the case if there are at most d(ℓ)/g(ℓ) +edges in G joining any two components of A. +Suppose for a contradiction that there are two components C and C′ of A such that there are more +than d(ℓ)/g(ℓ) = p(ℓ)·h(2ℓ2, 6ℓ2)2 edges of G joining them. By the definition of A, each vertex in C is joined +to each vertex in C′ by at most p(ℓ) parallel edges in G. Thus, at least one of NG(C) ∩ C′ and NG(C′) ∩ C +is of size at least h(2ℓ2, 6ℓ2), and we may assume without loss of generality that it is NG(C′) ∩ C. +Now the simple graph A has maximum degree less than 2ℓ2. Indeed, since p(ℓ) ⩾ 3, a vertex of degree +at least 2ℓ2 in A would immediately imply the existence of a strong immersion of Wℓ in G by Lemma 4.4. +Thus, by Lemma 4.8, A[C] contains a comb with at least 6ℓ2 teeth in NG(C′) ∩ C. +Hence, the wall Wℓ is strongly immersed in G by Lemma 4.5, since G contains P6ℓ2−1 ∗ v as a strong +immersion minor: Fix a vertex v in C′. Since p(ℓ) ⩾ 6ℓ2, we may pick greedily edge-disjoint paths from v +to each tooth in NG(C′) ∩ C such that all edges of each path except its last edge are contained in G[C′]. +These paths together with the comb form a strong immersion of P6ℓ2−1 ∗ v in G. +All in all, the maximum degree of G′ is at most d(ℓ), as desired. +■ +By Claim 1 and Claim 2, we can now apply Lemma 4.7 to G′ and obtain a tree-cut decomposition (T, X ′) +of G′ of adhesion at most a(ℓ) := (2w(ℓ) + 2)d(ℓ) such that every torso of (T, X ′) has size at most k(ℓ) := +(d(ℓ) + 1)(w(ℓ) + 1). From the tree-cut decomposition (T, X ′) of the minor G′ of G, we then construct the +tree-cut decomposition (T, X) of G by defining, for each node t ∈ T , its corresponding part Xt to be the +union of the branch sets of the vertices in X′ +t. +Now let +α = α(ℓ) := max{k(ℓ)(8ℓ2 + 1 + 6ℓ2 · 16ℓ4), p(ℓ) · (6ℓ2)2 + +�16ℓ4 +2 +� +· p(ℓ) · (6ℓ2)2 + k(ℓ) · d(ℓ)/2 + a(ℓ) · k(ℓ)}. +We show that (T, X) has the desired properties in that it has adhesion at most α and the 3-centres of its +torsos are almost α-thin. In doing so, the first term in the maximum defining α will be used to bound the + +A GRID THEOREM FOR STRONG IMMERSIONS OF WALLS +13 +size of the set of vertices that we will delete from the 3-centre of a torso to make it α-thinas well as the +number of its neighbours, while the second term in the maximum will be used to show that the remainder +of the 3-centre is α-thin. +Claim 3. The tree-cut decomposition (T, X) has adhesion at most a(ℓ) ⩽ α. +Proof. +By the construction of (T, X) from (T, X ′), the adhesion sets of (T, X) are the same as those +of (T, X ′). Thus, (T, X) has the same adhesion as (T, X ′) and, in particular, adhesion at most a(ℓ). +■ +Claim 4. The 3-centres of the torsos of (T, X) are almost α-thin. +Proof. We first observe that the torsos of (T, X) equal their 3-centres: Every adhesion set of (T, X) has size +at least 3, since G is 3-edge-connected. Therefore, all peripheral vertices of torsos of (T, X) have degree at +least 3 in the respective torso. This implies that the 3-centre of a torso of (T, X) equals the torso itself. +Given any node t ∈ T , let Ht be the torso of (T, X) at t. +We first find a set X of vertices of Ht +such that A[Xt] − X is a disjoint union of paths, and we show that both the size of X and the size of +its neighbourhood in Ht are bounded by our suitably chosen α = α(ℓ). In a next step, we then use the +structure of A[Xt] − X to exhibit a natural enumeration of V (Ht − X), and we finally show that this +enumeration witnesses the α-thinness of Ht − X. +For every component C of A[Xt], and more generally for every component C of A, the graph A[C] does +not contain the star K1,2ℓ2 as a minor. To prove this, we show that G[C] otherwise contains Wℓ as a strong +immersion minor, contradicting our assumptions on G. By Lemma 4.4, it is enough to show that S3,2ℓ2 +is strongly immersed in G[C]. So let U ⊆ C be a set of vertices containing precisely one vertex of each +branch set of the K1,2ℓ2-minor of A[C]. We then map the centre of S3,2ℓ2 to the vertex of U in the branch +set of the centre of K1,2ℓ2, and map the rest of the vertices of S3,2ℓ2 bijectively to the remaining ones in U. +Since p(ℓ) ⩾ 6ℓ2, we can now greedily embed the edges of S3,2ℓ2 as edge-disjoint paths in G[C] joining the +vertices in U along the K1,2ℓ2-minor. Hence, we have found our desired strong immersion of S3,2ℓ2 in G[C]. +We are now ready to find the desired set X of vertices of the torso Ht of (T, X) at t. +For every +component C of A[Xt], the graph A[C] does not contain K1,2ℓ2 as a minor. +So Lemma 4.9 yields a +set XC ⊆ C of size at most 8ℓ2 such that A[C] − XC is a disjoint union of paths. We then let the set X +be the union of these XC. The construction of X then implies that A[Xt] − X is indeed a disjoint union +of paths. +We first show that the size of X is bounded by α. Note that since A has maximum degree at most 2ℓ2 +(as in the proof of Claim 2), the number of components of A[C]−XC is at most |XC|·2ℓ2 ⩽ 16ℓ4. Since X′ +t +has size at most k(ℓ), the graph A[Xt] has at most k(ℓ) components and thus |X| ⩽ 8ℓ2 · k(ℓ) ⩽ α. +Next, we claim that each vertex x in X has at most 8ℓ2 · k(ℓ) + k(ℓ) + k(ℓ) · 6ℓ2 · 16ℓ4 ⩽ α neighbours +in Ht. Indeed, suppose for a contradiction that x has more than k(ℓ)+k(ℓ)·6ℓ2 ·16ℓ4 neighbours in Ht −X. +We first note that x is adjacent to at most k(ℓ) peripheral vertices of Ht, since there are at most k(ℓ) +many peripheral vertices of Ht: They are in one-to-one correspondence to the peripheral vertices of the +torso of (T, X ′) at t which has size at most k(ℓ). Thus, x has more than k(ℓ) · 6ℓ2 · 16ℓ4 many neighbours +which are core vertices of Ht and not contained in X. By the pigeonhole principle, there then exists one +component C of the at most k(ℓ) components of A[Xt] such that x has at least 6ℓ2 · 16ℓ4 neighbours +in C ∖ XC. Applying the pigeonhole principle a second time, we find one component P of the at most 16ℓ4 +components of A[C] − XC that contains at least 6ℓ2 neighbours of x. Since A[P] is a path by the choice +of XC, we have hence found a subdivision and thus a strong immersion of P6ℓ2−1 ∗ v in Ht. As all the + +14 +REINHARD DIESTEL, RAPHAEL W. JACOBS, PAUL KNAPPE, AND PAUL WOLLAN +vertices in C as well as x are core vertices of the torso Ht, this also is a strong immersion of P6ℓ2−1 ∗ v in G. +So by Lemma 4.5, the wall Wℓ is contained in G as a strong immersion minor – a contradiction. +Since X has size at most α and each element of X has at most α neighbours in Ht, it remains to prove +that Ht − X is α-thin. To do so, we now give an enumeration of V (Ht − X) which we will then show to +witness the α-thinness of Ht − X. +Let c1, . . . , cn be an arbitrary enumeration of the part X′ +t of tree-cut decomposition (T, X ′) of G′, +let Ci be the component of A corresponding to the vertex ci of G′, and write Xi := XCi. Then take an +enumeration v1, . . . , vN of Xt ∖ X such that for i = 1, . . . , n, the set Ci − Xi forms an interval of this +enumeration and within such an interval, every component of A[Ci] − Xi forms an interval with the same +linear order which is given by the path that it induces in A by the choice of Xi. For the peripheral vertices +of Ht, we finally pick an arbitrary enumeration vN+1, . . . , vM. +To show that this enumeration v1, . . . , vM of V (Ht−X) is as desired, first pick an arbitrary i ∈ {1, . . . , N}. +Then there is a component Cj of A[Xt] with vi ∈ Cj, and a component of A[Cj] − Xj containing vi; this +component of A[Cj] − Xj then induces a path P in A by the choice of Xj (which might have length 0). +Every edge in Ht joining {v1, . . . , vi−1} and {vi+1, . . . , vM} belongs to one of the following classes: +(1) edges joining vertices of the path P, +(2) edges joining the distinct components in A[Cj] − Xj, +(3) edges joining distinct components of A[X′ +t], and +(4) edges incident to peripheral vertices of Ht. +We will now separately bound the size of each of these classes (1) to (4) in terms of ℓ. Note that only (1) +depends on the choice of the enumeration; in particular, we will indeed bound the absolute number of all +such edges in Ht in terms of ℓ for (2) to (4). +For (1), the edges joining vertices of P, note that V (P) is a component of A[Cj]−Xj. So there are no edges +in A joining the vertices of P except the ones contained in P itself. Hence, each two non-adjacent vertices +of P are joined by at most p(ℓ) parallel edges. Let P1 and P2 be the two components of P − vi, and note +that both P1 and P2 consist of vertices of G since all vertices on P are in Xt. If |EG(P1, P2)| > p(ℓ)·(6ℓ2)2, +then one of Pi, say P1, has more than 6ℓ2 neighbours in P2−i. Fix any v ∈ P1. Since p(ℓ) ⩾ 6ℓ2, we can +greedily find edge-disjoint paths in G[P1 ∪ P2] joining v and 6ℓ2 of these neighbours in P2, and we thus +find a strong immersion of P6ℓ2−1 ∗ v in G[P1 ∪ P2]. By Lemma 4.5, this graph contains the wall Wℓ as a +strong immersion minor which yields the desired contradiction. +With the same type of argument, one can show that there are at most p(ℓ) · (6ℓ2)2 edges in G joining +any two components of A[Cj] − Xj. Recall that the number of components of A[Cj] − Xj is at most 16ℓ4. +Thus, there are at most +�16ℓ4 +2 +� +·p(ℓ)·(6ℓ2)2 edges in G as in (2), i.e. edges joining components of A[Cj]−Xj. +Next, we turn to (3) and bound the number of edges in G joining distinct components of A[X′ +t]. Since G′ +arises from G by contracting the components of A, these edges are in one-to-one correspondence with the +edges in G′[X′ +t]. Since X′ +t has size at most k(ℓ) and G′ has maximum degree at most d(ℓ), there are at +most k(ℓ) · d(ℓ)/2 edges in G′[X′ +t] and hence joining distinct components of A[X′ +t]. +Along similar lines, we can bound the number of edges in (4), i.e. the edges incident to peripheral +vertices of Ht, in terms of ℓ. Indeed, the edges incident to peripheral vertices of Ht are in one-to-one +correspondence to the edges incident to the peripheral vertices of the torso of (T, X ′) at t. Since (T, X ′) +has adhesion at most a(ℓ), each peripheral vertex has degree at most a(ℓ). Moreover, since X′ +t has size at + +A GRID THEOREM FOR STRONG IMMERSIONS OF WALLS +15 +most k(ℓ), there are at most k(ℓ) peripheral vertices of Ht. Altogether, there are at most a(ℓ) · k(ℓ) edges +incident to peripheral vertices of Ht. +All the above bounds show that by the choice of α = α(ℓ), there are at most α edges joining {v1, . . . , vi−1} +and {vi+1, . . . , vM} in Ht for i = 1, . . . , N. By the choice of our enumeration, the vertices vN+1, . . . , vM +are precisely the peripheral vertices of Ht. +So for i = N + 1, . . . , M, all edges joining {v1, . . . , vi−1} +and {vi+1, . . . , vM} in Ht are incident to peripheral vertices of Ht. As shown above there are at most a(ℓ) · +k(ℓ) ⩽ α(ℓ) such edges. This completes the proof of this claim. +■ +By Claim 3 and Claim 4, our constructed tree-cut decomposition (T, X) of G now is as desired, completing +the proof of Theorem 1. +□ +Acknowledgement +We are very grateful to Nathan Bowler for the suggestion to bound the size of the neighbourhood of +the vertices which we delete in the definition of almost α-thinness. This enabled us to obtain the actual +equivalence given by Theorem 1 and Theorem 2 instead of an approximate one. +References +[1] F. N. Abu-Khzam and M. A. Langston, Graph coloring and the immersion order, International computing and combina- +torics conference, 2003, pp. 394–403. +[2] C. Bürger and J. Kurkofka, Duality theorems for stars and combs IV: Undominating stars, Journal of Graph Theory +100 (2022), no. 1, 140–162. +[3] M. DeVos, Z. Dvořák, J. Fox, J. McDonald, B. Mohar, and D. Scheide, A minimum degree condition forcing complete +graph immersion, Combinatorica 34 (2014), no. 3, 279–298. +[4] M. DeVos, J. McDonald, B. Mohar, and D. Scheide, A note on forbidding clique immersions, The Electronic Journal of +Combinatorics 20 (2013), no. 3. #P55. +[5] R. Diestel, Graph theory (5th edition), Springer-Verlag, 2017. +Electronic edition available at http://diestel-graph-theory.com/. +[6] Z. Dvořák and T. Klimosova, Strong immersions and maximum degree, SIAM Journal on Discrete Mathematics 28 (2014), +no. 1, 177–187. +[7] Z. Dvořák and P. Wollan, A structure theorem for strong immersions, Journal of Graph Theory 83 (2016), no. 2, 152–163. +[8] M. Ferrara, R. J. Gould, G. Tansey, and T. Whalen, On H-immersions, Journal of Graph Theory 57 (2008), no. 3, +245–254. +[9] J. Geelen, O-J. Kwon, R. McCarty, and P. Wollan, The grid theorem for vertex minors, Journal of Combinatorial Theory, +Series B 158 (2023), 93–116. +[10] M. Grohe, K. Kawarabayashi, D. Marx, and P. Wollan, Finding topological subgraphs is fixed parameter tractable, Pro- +ceedings of the forty-third annual ACM symposium on Theory of Computing (2011), 479–488. +[11] D. J. Harvey and D. R. Wood, Parameters tied to treewidth, Journal of Graph Theory 84 (2017), no. 4, 364–385. +[12] K.-i. Kawarabayashi and S. Kreutzer, The directed grid theorem, Proceedings of the forty-seventh annual ACM symposium +on Theory of Computing, 2015, pp. 655–664. +[13] P. Knappe and J. Kurkofka, The immersion-minimal infinitely edge-connected graph, 2022. +[14] D. Marx and P. Wollan, Immersions in highly edge connected graphs, SIAM Journal on Discrete Mathematics 28 (2014), +no. 1, 503–520. +[15] K. Menger, Zur allgemeinen Kurventheorie, Fundamenta mathematicae 10 (1927), no. 1, 96–115. +[16] N. Robertson and P. D. Seymour, Graph minors I–XX, Journal of Combinatorial Theory, Series B (1983). +[17] +, Graph minors. V. Excluding a planar graph, Journal of Combinatorial Theory, Series B 41 (1986), 92–114. +[18] +, Graph minors. XIII. The disjoint paths problem, Journal of Combinatorial Theory, Series B 63 (1995), 65–110. +[19] +, Graph minors. XXIII. Nash-Williams’ immersion conjecture, Journal of Combinatorial Theory, Series B 100 +(2010), 181–205. + +16 +REINHARD DIESTEL, RAPHAEL W. JACOBS, PAUL KNAPPE, AND PAUL WOLLAN +[20] P. Wollan, The structure of graphs not admitting a fixed immersion, Journal of Combinatorial Theory, Series B 110 +(2015), 47–66. + diff --git a/LdE4T4oBgHgl3EQfig1d/content/tmp_files/load_file.txt b/LdE4T4oBgHgl3EQfig1d/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..44545bb1bfc6235fbcc5e7d68e4860f90d305505 --- /dev/null +++ b/LdE4T4oBgHgl3EQfig1d/content/tmp_files/load_file.txt @@ -0,0 +1,850 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf,len=849 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content='05134v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content='CO] 12 Jan 2023 A GRID THEOREM FOR STRONG IMMERSIONS OF WALLS REINHARD DIESTEL, RAPHAEL W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' JACOBS, PAUL KNAPPE, AND PAUL WOLLAN Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' We show that a graph contains a large wall as a strong immersion minor if and only if the graph does not admit a tree-cut decomposition of small ‘width’, which is measured in terms of its adhesion and the path-likeness of its torsos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Introduction In their Graph Minors Project [16], Robertson and Seymour investigated the structure of graphs not containing a fixed graph as a minor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' An important example of their structure theorems as well as a result central to their project is the grid theorem [17] which asserts the equivalence of large tree-width and the existence of large grid minors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' More precisely, every graph of large enough tree-width contains a large grid as a minor, and large grid minors form an obstruction to small tree-width in that large grids have large tree-width and every graph has tree-width at least the tree-width of each of its minors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' An equivalent formulation of the grid theorem is that a graph has large tree-width if and only if it contains a large wall as a topological minor [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Grid theorem-like results, which provide the equivalence of large width with respect to some width- measure and a large (often grid-like) substructure in terms of a corresponding containment relation, have since been proven in various other settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' For example, Kreutzer and Kawarabayashi proved a directed grid theorem [12], which establishes an equivalence of large directed tree-width and the existence of butterfly minors of large directed grids, and Geelen, Kwon, McCarty and Wollan showed in [9] that a graph has large rank-width if and only if it contains a vertex-minor isomorphic to a large comparability-grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' In this paper we are concerned with the structure of graphs not containing a large wall as a strong immersion minor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' As the minor relation, immersion minors generalise the notion of topological minors, but they are unrelated to minors in that neither the existence of an immersion minor implies the existence of a minor nor vice-versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Immersion minors have attracted attention from various algorithmic and structural viewpoints in recent years, often finding results analogous to those for minors [1,3,6–8,10,13,19,20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' There are two versions of the immersion relation, a weak one and a strong one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' They are defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' A weak immersion of a graph H in a graph G is a map α with domain V (H) ∪ E(H) that embeds V (H) into V (G) and maps every edge uv ∈ H to an α(u)–α(v) path in G which is edge-disjoint from every other such path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' If, additionally, these paths have no internal vertices in α(V (H)), then α is a strong immersion of H in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' The vertices in α(V (H)) are the branch vertices of this immersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' We say that H is weakly/strongly immersed in G, or that H is a weak/strong immersion minor of G, if there is a weak/strong immersion of H in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' While the (topological) minor relation behaves well with tree-decompositions, the immersion relations do so with tree-cut decompositions, which form the edge-cut analogue of tree-decompositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Formally, a pair (T, X) is a tree-cut decomposition of a graph G if T is a tree and X = (Xt)t∈T a near-partition of V (G), 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' 05C75, 05C83, 05C40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Strong immersion, wall, grid theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' 1 2 REINHARD DIESTEL, RAPHAEL W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' JACOBS, PAUL KNAPPE, AND PAUL WOLLAN that is, the Xt are disjoint and their union is V (G) but, in contrast to a partition, the Xt may be empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' The vertex sets Xt are the parts of (T, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' The torso of (T, X) at a node t ∈ T arises from G by identifying for every component T ′ of T − t the vertices in � t′∈T ′ Xt′ to a single vertex, keeping parallel edges as they arise but omitting loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' These new identification vertices are the peripheral vertices of the torso, while the ones in Xt are its core vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Every edge t1t2 ∈ T induces its adhesion set EG( � t∈T1 Xt , � t∈T2 Xt ) where T1 and T2 are the two components of T − t1t2 with t1 ∈ T1 and t2 ∈ T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' The adhesion of a tree-cut decomposition then is the maximum size of its adhesion sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' DeVos, McDonald, Mohar and Scheide [4] and Wollan [20] independently showed that if a graph does not admit Kℓ, the complete graph of size ℓ, as a weak immersion, then it admits a tree-cut decomposition each of whose torsos contains at most ℓ vertices of degree at least ℓ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' As a qualitative converse, they proved that a tree-cut decomposition in which each torso contains at most ℓ vertices of degree at least ℓ precludes the existence of a weak immersion of Kℓ+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Introducing the concept of tree-cut width, Wollan [20] also derived a grid theorem for weak immersions, which establishes the equivalence of large tree-cut width and the existence of weak immersions of large walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' His proof made use of the classical grid theorem for excluded wall minors by Robertson and Seymour [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' For strong immersions Dvořák and Wollan proved the following general structure theorem describing the structure of graphs not containing a fixed graph as a strong immersion minor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content='1 ([7, Theorem 4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' For every graph F, there exists an integer α = α(F) > 0 such that if a graph G does not contain F as a strong immersion minor, then there exists a tree-cut decomposition of G of adhesion less than α such that each of its torsos is α-basic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Here, the α-basicness of a graph means, roughly speaking, that it has a path-like decomposition in which we can accommodate vertices of high degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' When we restrict F to complete graphs in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content='1, then there is also a qualitative converse of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content='1 in [7]: For every integer α > 0, there exists an integer n = n(α) > 0 such that any graph with a tree-cut decomposition which has adhesion less than α and whose torsos are α-basic does not contain Kn as a strong immersion minor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' If we consider strong immersions of walls, as it is the subject of this paper, and we thus look at Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content='1 with F restricted to walls, then Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content='1 does not have such a qualitative converse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Indeed, Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' The wall W4 of size 4 with the underlying 4 × 8 grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' for every integer ℓ > 0, the wall Wℓ of size ℓ is formally defined as the graph arising from the ℓ×2ℓ grid by deleting all edges (i, j)(i′, j′) with j = j′, i = i′ + 1 and j ≡ i (mod 2) and then removing the two resulting vertices of degree 1 (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' By definition the wall Wℓ has 2ℓ2 −2 vertices and maximum degree 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Thus, Wℓ does not contain K5 as a strong immersion minor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content='1 hence yields an integer α > 0 such that every wall, no matter how large its size, has a tree-cut decomposition of adhesion less than α such that each torso is α-basic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' In particular, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content='1 does not admit a qualitative converse for walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' In this paper, we prove a grid theorem-like result for the strong immersion relation which says that a graph contains a large wall as a strong immersion minor if and only if the graph has a tree-cut decompo- sition of a specific type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' These tree-cut decompositions are inspired by those in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content='1 in that their torsos still admit a path-like structure, but path-likeness is defined differently, as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' A GRID THEOREM FOR STRONG IMMERSIONS OF WALLS 3 A graph G is α-thin for an integer α > 0 if there exists an enumeration v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' , vn of its vertices such that there are at most α edges in G joining {v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' , vi−1} and {vi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' , vn} for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' , n, and we call G almost α-thin if it becomes α-thin after removing up to α vertices each of which has at most α neighbours in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' In our specific tree-cut decompositions, the torsos then have almost α-thin ‘3-centres’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Following [20], the 3-centre of a torso of a tree-cut decomposition arises from the torso by a maximal sequence of deleting peripheral vertices of degree at most 1 and suppressing peripheral vertices of degree 2, removing arising loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' We will show that if a torso of a tree-cut decomposition is itself (almost) α-thin, then so is its 3-centre, while the converse does not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' For a further discussion of these notions, we refer the reader to Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Our first main result, a version of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content='1 specifically tailored to walls, then reads as follows: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' For every integer ℓ > 0, there exists an integer α = α(ℓ) > 0 such that if a graph G does not contain the wall Wℓ as a strong immersion minor, then G has a tree-cut decomposition of adhesion at most α such that the 3-centres of its torsos are almost α-thin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' We will see that 3-centres are indeed necessary in Theorem 1 in that we cannot consider the torsos them- selves instead of their 3-centres (see Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' In contrast to Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content='1, the notion of path-like torsos as in Theorem 1 now gives the desired equiva- lence of the existence of strong immersions of large walls and these tree-cut decompositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' More precisely, our second main results provides a qualitative converse of Theorem 1: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' For every integer α > 0, there exists an integer ℓ = ℓ(α) > 0 such that if a graph G has a tree-cut decomposition of adhesion at most α such that the 3-centres of its torsos are almost α-thin, then G does not contain the wall Wℓ as a strong immersion minor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' While the proof of Theorem 2 is self-contained and does not build on any previous results, the proof of Theorem 1 draws on the grid theorem for excluded wall minors as well as on several previously estab- lished methods for immersions and tree-cut decompositions [7,17,20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' We emphasise that we do not make use of any previous structure theorem for weak or strong immersions, but rather use some of their ideas and combine them in a novel way to fit our specific problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' This paper is organised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' We first discuss the notions of almost α-thin graphs and 3-centres in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Then we prove Theorem 2 in Section 3 and Theorem 1 in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' For basic graph-theoretic terms, we follow [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' In this paper, we only consider graphs without loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' But unless called simple, they may contain parallel edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' As a consequence, the degree of a vertex is the number of its incident edges which may in general differ from the number of its neighbours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Moreover, we diverge from [5] in that we define the components of a graph not as its maximal connected subgraphs, but as their vertex sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Almost α-thin graphs and 3-centres In this section we take a closer look into our notion of path-likeness, (almost) α-thin graphs, and its relation to the concept of 3-centres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Let us first recall the definition of (almost) α-thinness from the introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' 4 REINHARD DIESTEL, RAPHAEL W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' JACOBS, PAUL KNAPPE, AND PAUL WOLLAN Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Let α > 0 be an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' A graph G is α-thin if there exists an enumeration v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' , vn of its vertices satisfying that |EG({v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' , vi−1}, {vi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' , vn})| ⩽ α for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' , n,1 and the graph G is almost α-thin if there exists a set X of at most α vertices of G such that each vertex in X has at most α neighbours in G and G − X is α-thin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' We remark that if a graph is (almost) α-thin, then so are all its subgraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Even though adding vertices of degree 1 to a graph does not change whether it contains a wall as a strong immersion minor, such vertices affect whether a graph is (almost) α-thin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' For every integer α > 0, the star K1,n with n = 3α + 3 leaves is not almost α-thin, but becomes 0-thin after repeatedly removing vertices of degree 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Repeatedly removing its vertices of degree at most 1, the star G := K1,n becomes the graph on one vertex which is trivially 0-thin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' So it remains to argue that G is not almost α-thin, and we suppose for a contradiction that there exists a set X of at most α vertices of G, each with at most α neighbours in G, such that G − X is α-thin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' The set X cannot contain the centre c of G since it has n > α many neighbours in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Thus, X consists of up to α leaves of G, and G − X is again a star with centre c, now with at least n′ ⩾ 2α + 3 leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Now take an arbitrary enumeration v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' , vn′+1 of V (G − X), and let i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' , n′ + 1} with vi = c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' If i ⩾ α + 3, then {v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' , vi−2} contains at least α + 1 leaves of the star, while if i ⩽ (n′ + 1) − (α + 3), then {vi+2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' , vn′+1} contains at least α + 1 leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' So for j = i − 1 in the first case and j = i + 1 in the second case, there are at least α + 1 edges joining {v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' , vj−1} and {vj+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' , vn′+1}, which yields the desired contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' □ Similar to the addition of vertices of degree 1, subdividing edges has an impact on (almost) α-thinness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' For every integer α > 0, the graph G that arises from two disjoint stars, each with n = 3α+3 leaves, by identifying their leaves is not almost α-thin, but becomes 0-thin after repeatedly suppressing vertices of degree 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' If we repeatedly suppress vertices of degree 2, then G turns into the graph consisting of two vertices which are joined by n edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' This graph is trivially 0-thin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Since G contains K1,n as a subgraph and K1,n is not almost α-thin by Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content='2, G cannot be almost α-thin itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' □ Just as a graph stays (almost) α-thin when we delete some of its vertices, suppressing vertices of degree 2 also does not impact the (almost) α-thinness of a graph, as the next lemma demonstrates: Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Let G be an (almost) α-thin graph for some integer α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Then suppressing a vertex of G of degree 2 (and deleting a potentially arising loop) results in an (almost) α-thin graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' It is enough to prove the lemma for α-thinness, since the case of almost α-thinness then follows immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' So let G be an α-thin graph, let v ∈ G be a vertex of degree 2 and let u and w be the other endvertices of the two edges of G incident to v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Now let G′ arise from G by suppressing v, omitting a potentially arising loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' 1Note that the definition of α-thin differs from the similar notion of cutwidth at most α, since cutwidth also takes the edges joining vi and {vi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' , vn} into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' While cutwidth bounds thinness from above, there are graphs with unbounded cutwidth which are 0-thin, as paths with many parallel edges witness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' A GRID THEOREM FOR STRONG IMMERSIONS OF WALLS 5 If u = w, then G′ is a subgraph of G which is hence α-thin;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' so we may assume that u and w are distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Given an enumeration v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' , vn+1 of V (G) witnessing that G is α-thin, let k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' , n + 1} with vk = v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' We may assume that v appears in this enumeration in between u and w;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' otherwise, moving v in between them only reduces the number of edges joining {v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' , vi−1} and {vi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' , vn+1} for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' , n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' We then define the enumeration v′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' , v′ n of V (G′) as follows: v′ i := \uf8f1 \uf8f2 \uf8f3 vi if i < k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' vi+1 if i > k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' This definition guarantees that for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' , n, the edges of G′ joining {v′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' , v′ i−1} and {v′ i+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' , v′ n} coincide with the edges of G joining {v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' , vj−1} and {vj+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' , vn+1}, where j := i if i < k and j := i+1 if i ⩾ k, except that the edge of G′ that arose by suppressing v is replaced by one of the two edges incident to v in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Thus, G′ is α-thin as witnessed by the enumeration v′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' , v′ n of V (G′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' □ As we will see below, we need to ignore peripheral vertices of degree at most 2 in torsos to estab- lish Theorem 1 (see Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Formally, we achieve this with the notion of 3-centres which was introduced in [20] and is well-defined by [20, Lemma 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Let X be a set of vertices of a graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' The 3-centre of (G, X) arises from G by a maximal sequence of deleting vertices of degree at most 1 and suppressing vertices of degree 2 (omitting any resulting loops) where all these vertices are not in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' If (T, X) is a tree-cut decomposition of G, then the 3-centre of the torso Ht of (T, X) at the node t ∈ T is defined as the 3-centre of (Ht, Xt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' We note that the 3-centre of a pair (G, X) cannot be formed by first repeatedly deleting vertices of degree 1 and then suppressing all vertices of degree 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Indeed, the deletion of resulting loops may create new vertices of degree 1 which have to be deleted afterwards;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' consider for example the graph G which consists two triangles joined by an edge and let X be the set of vertices of one of the triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' If a graph G is (almost) α-thin, then the 3-centre of (G, X) is also (almost) α-thin since deleting vertices and suppressing vertices of degree 2 does not affect the (almost) α-thinness of a graph, see Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' In particular, it is weaker to assume in Theorem 2 that the 3-centres of the torsos are almost α-thin than to assume that the torsos themselves have this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' As we have seen in Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content='2 and Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content='3, the converse statement is not true: If the 3-centre of (G, X) is α-thin, then G does not even have to be almost α-thin itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' In fact, the converse holds in an even stronger form in that we cannot remove 3-centres from Theorem 1 and consider the torsos themselves instead, as the following example demonstrates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' For every two integers ℓ ⩾ 2 and α > 0, there exists a graph which does not contain the wall Wℓ as a strong immersion minor and also does not admit a tree-cut decomposition of adhesion at most α whose torsos are almost α-thin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Let G = K1,n be the star with n = α(3α + 3) leaves and centre c ∈ V (G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Clearly, G does not contain Wℓ as a strong immersion minor since G contains only a single vertex of degree more than 1 while Wℓ has several of those because of ℓ ⩾ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Thus, it remains to consider an arbitrary tree-cut decom- position (T, X) of G of adhesion at most α and to show that at least one of the torsos of (T, X) is not almost α-thin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Let s be the (unique) node of T such that Xs contains the centre c of the star G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' For each neighbour t of s in T , let Yt := � t′∈T ′ Xt′ where T ′ is the component of T − st containing t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Since (T, X) has adhesion 6 REINHARD DIESTEL, RAPHAEL W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' JACOBS, PAUL KNAPPE, AND PAUL WOLLAN at most α and c ∈ Xs, the set Yt contains at most α leaves of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' But G has α(3α + 3) leaves, so it yields a star G′ with at least 3α+3 leaves in the torso Hs of (T, X) at s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' This star G′, however, is not almost α-thin by Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content='2, and hence Hs is not almost α-thin either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' □ If we adapt Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content='6 by doubling each edge in G, then this modified version also demonstrates that just deleting peripheral vertices of degree 1 in each torso instead of taking its 3-centre does not suffice either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Strongly immersed walls are obstructions In this section, we prove Theorem 2 which we recall here: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' For every integer α > 0, there exists an integer ℓ = ℓ(α) > 0 such that if a graph G has a tree-cut decomposition of adhesion at most α such that all the 3-centres of its torsos are almost α-thin, then G does not contain the wall Wℓ as a strong immersion minor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' As a key ingredient to our proof of Theorem 2, we need Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content='1, which asserts that the wall Wℓ contains a vertex set Z of size ℓ − 2 which is well-linked in Wℓ, that is, for every two disjoint A, B ⊆ Z with k := |A| = |B|, there exist k vertex-disjoint A–B paths in Wℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' The existence of such a set Z of size ℓ follows from the facts that the tree-width of a wall of size ℓ is at least ℓ and the largest size of a well-linked in a graph is lower bounded by its tree-width (see for example the survey [11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' We include a direct and constructive proof in the case of walls here which yields a further structural property of the well-linked set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' The wall Wℓ of size ℓ ⩾ 3 contains a set Z of vertices of size ℓ − 2 such that Z is well-linked in Wℓ and such that every two vertices in Z are joined by three internally vertex-disjoint paths in Wℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Let Z be the set of all vertices of Wℓ which have the form (ℓ, i) and degree 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Note that Z has size ℓ−2 by the definition of Wℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' It is easy to see that every two vertices of degree 3 are joined by three internally vertex-disjoint paths in Wℓ as ℓ ⩾ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Thus, it remains to show that for every two disjoint subsets A, B of Z with k := |A| = |B|, there exist k vertex-disjoint A–B paths in Wℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' By Menger’s theorem [15], it suffices to prove that for every set X ⊆ V (G) of size at most k − 1, there exists an A–B path in Wℓ which avoids X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' For every j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' , ℓ let Vj be the j-th vertical path of the wall Wℓ, that is, Vj is the induced subgraph of Wℓ on the set of all vertices of the form (2j, i) or (2j − 1, i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Note that these vertical paths are pairwise vertex-disjoint, that each vertical path meets Z in at most one vertex, and that each vertex in Z is contained in some vertical path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Thus, the pigeonhole principle yields a vertical path VjA which meets A, but avoids X, since |A| = k > k − 1 ⩾ |X|;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' analogously, we obtain a vertical path VjB for B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Next, for every j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' , ℓ let Hj be the j-th horizontal path of the wall Wℓ, that is, the subgraph of Wℓ induced on the set of all vertices of the form (i, j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Note that these horizontal paths are pairwise vertex- disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Thus, the pigeonhole principle yields an horizontal path Hj0 which avoids X, since ℓ > |A| > |X|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' By definition, every horizontal path meets every vertical path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Hence, VjA + Hj0 + VjB is a connected subgraph of Wℓ which meets both A and B, but avoids X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' In particular, there exists an A–B path in Wℓ which avoids X, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' □ Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' We set ℓ := ℓ(α) := [(α2 + 1)(2(α + 1) + 4) + α2 + α] + [α((α2 + 1)(2(α + 1) + α + 2) + α2 + α)] + 2, and claim that this ℓ is as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' So let G be a graph, and let (T, X) be a tree-cut decomposition of G of adhesion at most α such that the 3-centres of its torsos are almost α-thin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Suppose for a contradiction that the wall Wℓ is strongly immersed in G, and let U be the set of branch vertices of this strong immersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' A GRID THEOREM FOR STRONG IMMERSIONS OF WALLS 7 By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content='1, Wℓ contains a vertex set Z of size at least ℓ − 2 such that Z is well-linked in Wℓ and every two vertices in Z are joined by three internally vertex-disjoint paths in Wℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Let ZU ⊆ U be the set of branch vertices corresponding to Z ⊆ V (Wℓ) in G via the strong immersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Since Z is well-linked in Wℓ, the definition of strong immersions yields that this set ZU has the following property: For every two disjoint sets A, B ⊆ ZU with k := |A| = |B|, there exist k edge-disjoint A–B paths in G and these k paths meet each vertex of U at most once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' (∗) Moreover, since every two vertices in Z are joined by three internally vertex-disjoint paths in Wℓ, every two vertices in ZU are joined by three edge-disjoint paths in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Consider an edge t1t2 ∈ T and let Y1 := � t∈T1 Xt where T1 is the component of T − t1t2 containing t1, and define Y2 analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Since (T, X) has adhesion at most α, the adhesion set EG(Y1, Y2) of G induced by t1t2 has size at most α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Therefore, precisely one Yi, say Y2, contains at least α + 1 vertices from ZU by (∗) and since ℓ − 2 ⩾ 2(α + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' We then orient the edge t1t2 from t1 to t2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' In this way, ZU induces an orientation of the edges of T such that for each node t ∈ T at most one incident edge is oriented away from t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Hence, there is a unique sink s of this orientation, that is, a node s ∈ T such that all incident edges are oriented towards s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Let H be the torso of (T, X) at s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' For each neighbour t of s in T , let Yt := � t′∈T ′ Xt′ where T ′ is the component of T − st containing t, and let vt be the peripheral vertex of H corresponding to the identification of the vertices in Yt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Consider the set ZH of vertices of H corresponding to the vertices in ZU, that is, the set consisting of the core vertices ZU ∩ Xs together with those peripheral vertices vt with ZU ∩ Yt ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' This set ZH is then contained in the 3-centre ¯H of the torso H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Indeed, for every peripheral vertex vt ∈ ZH, the set ZU contains a vertex in Yt by the definition of ZH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Since st is oriented towards s, the set ZU also contains a vertex which is not in Yt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' But these two vertices, just as any two vertices in ZU, are joined by three edge-disjoint paths in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Thus, vt is never a candidate for deletion or suppression in the construction of ¯H from H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' For every peripheral vertex vt in ZH, we have |ZU ∩ Yt| ⩽ α, since st was oriented towards s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' This implies that |ZH| ⩾ |ZU ∩Xs|+|ZU ∖Xs|/α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' In particular, ZH contains either many core vertices or many peripheral vertices of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' We now aim to use the set ZH to derive a contradiction to the 3-centre ¯H of the torso H being almost α- thin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' To do so, consider a set X of at most α vertices of ¯H each of which has at most α neighbours in ¯H and an enumeration v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' , vn of V ( ¯H − X) which witnesses that ¯H − X is α-thin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Observe that if we remove the at most α2 neighbours of the set X among the vi, then this partitions the enumeration v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' , vn into a set I of at most α2 + 1 intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' If ZH contains many core vertices of H in that |ZH∩Xs| ⩾ (α2+1)(2(α+1)+4)+α2+α, then {v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' , vn} contains at least (α2 + 1)(2(α + 1) + 4) core vertices in ZH which are not neighbours of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Thus, the pigeonhole principle yields an interval vj, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' , vk in I containing at least 2(α + 1) + 4 elements of ZH ∩ Xs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' shortening the interval if necessary, we may assume additionally that it contains precisely 2(α + 1) + 4 elements of ZH ∩ Xs and that its endpoints vj and vk are among those.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Let AH be the set consisting of the 2(α + 1) + 2 elements of ZH ∩ Xs in the interior {vj+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' , vk−1} of our fixed interval, and let BH be a subset of ZH ∩ Xs of the same size as AH and which is disjoint from {vj, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' , vk};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' such a set BH exists since ZH ∩ Xs is large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Now AH and BH are subsets of ZH consisting of core vertices of H, and hence AH, BH ⊆ ZU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' So we may apply (∗) to AH, BH ⊆ ZU and 8 REINHARD DIESTEL, RAPHAEL W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' JACOBS, PAUL KNAPPE, AND PAUL WOLLAN obtain 2(α + 1) + 2 edge-disjoint AH–BH paths in G such that at most two of these paths meet vj or vk, since vj and vk are contained in ZH ∩ Xs ⊆ ZU ⊆ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Since the torso H arises from G by identifications of vertices keeping parallel edges as they arise, these 2(α + 1) + 2 paths induce 2(α + 1) + 2 edge-disjoint AH–BH walks in H which we can shorten to 2(α + 1) + 2 edge-disjoint AH–BH paths in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' By the definition of 3-centres, these 2(α + 1) + 2 paths in the torso H in turn induce 2(α + 1) + 2 edge-disjoint AH–BH paths in the 3-centre ¯H of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' We remark that throughout this process we maintain the property that at least 2(α+1) of these paths avoid vj and vk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Since none of the vertices vj, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' , vk is a neighbour of X, these 2(α + 1) edge-disjoint paths join- ing {vj+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' , vk−1} and {v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' , vj−1}∪{vk+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' , vn} indeed contain at least 2(α+1) edges in ¯H −X join- ing {vj+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' , vk−1} and {v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' , vj−1} ∪{vk+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' , vn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' In particular, there are either at least α +1 edges joining {v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' , vj−1} and {vj+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' , vn} or at least α + 1 edges joining {v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' , vk−1} and {vk+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' , vn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' In both cases we obtain a contradiction to the choice of the enumeration v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' , vn to witness the α-thinness of ¯H − X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' If ZH does not contain many core vertices as in the above case, then |ZH| ⩾ |ZU ∩ Xs| + |ZU ∖ Xs|/α implies that ZH contains many peripheral vertices in that |ZH ∖ Xs| ⩾ (α2 + 1)(2(α + 1) + α + 2) + α2 + α by our choice of ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' We now proceed analogously to the above case of many core vertices in ZH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' By the lower bound on the size of ZH ∖ Xs, there are at least (α2 + 1)(2(α + 1) + α + 2) peripheral vertices in ZH which are neither neighbours of X nor contained in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' So the pigeonhole principle yields an interval vj, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' , vk in I which contains at least 2(α + 1) + α + 2 elements of ZH ∖ Xs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' We may assume additionally, by shortening the interval if necessary, that the interval contains precisely 2(α + 1) + α + 2 elements of ZH ∖ Xs and that its endpoints vj and vk are among those.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Let AH be the set of the 2(α + 1) + α elements of ZH ∖ Xs in the interior {vj+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' , vk−1} of our fixed interval, and let BH be a set of 2(α + 1) + α elements of ZH ∖ Xs that are not in {vj, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' , vk};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' such a set BH exists since ZH ∖ Xs is large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Since each peripheral vertex vt in ZH satisfies ZU ∩ Yt ̸= ∅, we may replace each element vt of AH and BH by an arbitrary vertex in ZU ∩ Yt to obtain subsets AU and BU of ZU, respectively, which are disjoint and of the same size as AH and BH by the disjointness of the Yt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Applying (∗) to AU, BU ⊆ ZU then yields 2(α + 1) + α edge-disjoint AU–BU paths in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' The construction of AU and BU now guarantees that these 2(α + 1) + α many AU–BU paths in G induce 2(α + 1) + α edge-disjoint AH–BH paths in ¯H as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Since (T, X) has adhesion at most α and vj, vk ∈ ZH ∖ Xs are neither contained in AH nor in BH, each of the peripheral vertices vj and vk lies on at most α/2 of these paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Thus, there are at least 2(α + 1) + α − 2 · α/2 = 2(α + 1) edge- disjoint paths in ¯H joining {vj+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' , vk−1} and {v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' , vj−1} ∪ {vk+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' , vn} which avoid vj and vk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Since {vj, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' , vk} contains no neighbours of X, these paths contain at least 2(α + 1) edges in ¯H − X joining {vj+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' , vk−1} and {v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' , vj−1} ∪ {vk+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' , vn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Thus, there are either at least α + 1 edges joining {vj+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' , vk−1} and {v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' , vj−1} or at least α + 1 joining {vj+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' , vk−1} and {vk+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' , vn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' In particular, there are either at least α+1 edges joining {vj+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' , vn} and {v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' , vj−1} or at least α+1 edges joining {v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' , vk−1} and {vk+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' , vn}, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' □ A GRID THEOREM FOR STRONG IMMERSIONS OF WALLS 9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Excluding strongly immersed walls This section is devoted to the proof of Theorem 1, recalled here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' For every integer ℓ > 0, there exists some integer α = α(ℓ) > 0 such that if a graph G does not contain the wall Wℓ as a strong immersion minor, then G has a tree-cut decomposition of adhesion at most α such that the 3-centres of its torsos are almost α-thin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' For the proof of Theorem 1, we first reduce the problem to 3-edge-connected graphs in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content='2 we collect some tools from previous research dealing with immersions and tree-cut decompo- sitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Then we finally prove Theorem 1 in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Reduction to 3-edge-connected graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' We begin by reducing the proof of Theorem 1 to 3-edge- connected graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' This reduction will be done separately for each integer ℓ > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' so throughout this section, let ℓ > 0 be an arbitrary, but fixed integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' We first reduce our attention to graphs with minimum degree 3 which will then facilitate the reduction to 3-edge-connected graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' If Theorem 1 holds for all graphs with minimum degree 3, then it holds for all graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Let α := α(ℓ) ⩾ 2 be an integer such that Theorem 1 holds for all graphs with minimum degree 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' We now show that Theorem 1 holds with the same α for all graphs G, and we will do so by induction on |G|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' The case |G| = 1 is trivial;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' so suppose |G| > 1, and assume that G does not contain the wall Wℓ as a strong immersion minor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Let v ∈ G be a vertex with degree dG(v) at most 2 in G, and let G′ arise from G by deleting v if dG(v) ⩽ 1 and suppressing v (omitting any loops which arise) if dG(v) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Clearly, G′ does not contain the wall Wℓ as a strong immersion minor, as G does not do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Since |G′| < |G|, we can apply the induction hypothesis to G′ to obtain a tree-cut decomposition (T ′, X ′) of G′ of adhesion at most α such that the 3-centres of its torsos are almost α-thin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' If dG(v) ⩾ 1, let u be a neighbour of v in G, and otherwise, let u be an arbitrary vertex in G other than v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Then there exists a (unique) node tu ∈ T ′ with u ∈ X′ tu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' We then obtain a tree-cut decomposi- tion (T, X) of G from (T ′, X ′) in that T arises from T ′ by adding a vertex tv adjacent to tu and in that we set Xtv := {v} while all other parts remain unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' We claim that (T, X) is as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' We first prove that (T, X) has adhesion at most α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' The edge tutv ∈ T induces an adhesion set of size at most 2 ⩽ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Every other edge e of T is also an edge of T ′, and the adhesion set of (T, X) induced by e contains the same edges as the one of (T ′, X ′) induced by e – except that if v has a second neighbour w ̸= u in G, then the edge of G′ that arose by suppressing v is replaced by the edge joining v and w in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' This replacement, however, leaves the size of the adhesion set unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Hence, the adhesion of (T, X) is at most α since (T ′, X ′) has adhesion at most α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' It remains to show that the 3-centres of the torsos of (T, X) are almost α-thin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Since v has degree at most 2 in G, the 3-centre of the torso of (T, X) at tv consists only of the vertex v and is hence almost α-thin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' For all other nodes t ∈ T , the 3-centre of the torso of (T, X) at t equals the 3-centre of the torso of (T ′, X ′) at t, since the new peripheral vertex v of the torso of (T, X) at t = tu has degree at most 2 and is hence deleted or suppressed in the construction of the 3-centre from the torso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' If Theorem 1 holds for all 3-edge-connected graphs, then it holds for all graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Let α := α(ℓ) ⩾ 2 be an integer such that Theorem 1 holds for all 3-edge-connected graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content='1, it is enough to prove the statement for all graphs G of minimum degree 3, and we will do 10 REINHARD DIESTEL, RAPHAEL W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' JACOBS, PAUL KNAPPE, AND PAUL WOLLAN so by induction on |G|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' The case |G| = 1 is trivial;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' so suppose that |G| > 1, that G has minimum degree 3, that G is not 3-edge-connected, and that G does not contain the wall Wℓ as a strong immersion minor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Let {A, B} be a bipartition of V (G) such that the size of the cut EG(A, B) of G is minimal among all bipartitions of V (G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Since G is not 3-edge-connected, the cut EG(A, B) has size at most 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Hence, both A and B have size at least 2, since G has minimum degree 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Let GA arise from G by contracting B to a single vertex b, keeping parallel edges as they arise, but omitting loops;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' analogously define GB by contracting A to a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Then both GA and GB strictly smaller than G, and they do not contain Wℓ as a strong immersion minor, since G does not do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' By applying the induction hypothesis, we obtain tree-cut decompositions (T A, X A) of GA and (T B, X B) of GB which both have adhesion at most α and such that the 3-centres of its torsos are almost α-thin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Using (T A, X A) and (T B, X B) we now construct a tree-cut decomposition (T, X) of G which, as we will show afterwards, has the desired properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content='2 Let tA ∈ T A and tB ∈ T B be the (unique) nodes with b ∈ XA tA and a ∈ XB tB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' We then define the tree T as the disjoint union of T A and T B together with an edge joining tA and tB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' We set XtA := XA tA ∖ {b} and XtB := XB tB ∖ {a}, and take Xt as the corresponding XA t or XB t for all other nodes t ∈ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Let us first check that (T, X) has adhesion at most α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' By construction, every adhesion set of (T, X) that is induced by an edge of T other than tAtB has the same size as the adhesion set of (T A, X A) or (T B, X B) induced by the corresponding edge of T A or T B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' The adhesion set of (T, X) induced by tAtB is precisely the cut EG(A, B) and hence of size at most 2 ⩽ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Next, we verify that the 3-centres of the torsos of (T, X) are almost α-thin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' For any node t ∈ T with t ̸= tA, tB, the torso of (T, X) at t equals the corresponding torso of (T A, X A) or (T B, X B) and hence, its 3-centre is almost α-thin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' The torso H of (T, X) at tA differs from the torso HA of (T A, X A) at tA only in that b is not a core vertex any more, but a peripheral vertex now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' So the 3-centre of H arises from the one of HA by deleting b if d(b) ⩽ 1 and suppressing b if d(b) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' But then the 3-centre of the torso H is almost α-thin since the 3-centre of HA is, because a graph remains almost α-thin after the deletion of a vertex by definition and the suppression of a vertex of degree 2 does not affect the almost α-thinness by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' The analysis of the torso at tB is symmetrical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Immersions and tree-cut decompositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' In this section we collect some tools and auxiliary results for the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' The first lemma asserts that the absence of a large wall as a strong immersion minor implies a bound on the number of neighbours of every vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' This is the lemma for whose application in the proof of Theorem 1 we reduced to 3-edge-connected graphs in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content='3 ([6, Corollary 3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' For every integer ℓ > 0, there is an integer g(ℓ) > 0 such that if a 3-edge- connected graph G contains a vertex with at least g(ℓ) neighbours, then the wall Wℓ is a strong immersion minor of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' The next two lemmas again exclude specific substructures in the absence of a large wall as a strong immersion minor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' First, for every two integers k, n > 0, let Sk,n be the graph on n+1 vertices x, v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' , vn with k parallel edges joining x and vi for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' For every graph G on n vertices which has maximum degree k, one can greedily construct a strong immersion of G in Sk,n (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' [20, Observation 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' 2This construction follows the one in terms of edge-sums given in [20, Lemma 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' A GRID THEOREM FOR STRONG IMMERSIONS OF WALLS 11 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' For every two integers k, n > 0, the graph Sk,n contains every graph on n vertices with maximum degree k as a strong immersion minor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' In particular, S3,2ℓ2 contains the wall Wℓ as a strong immersion minor for every integer ℓ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' □ Secondly, let Pn ∗ v denote the graph arising from the path Pn of length n by adding a vertex v that is adjacent to every vertex of Pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' For every integer k > 0, the graph P3k−1 ∗ v contains a subdivision of S3,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' In particular, the wall Wℓ is a strong immersion minor of P6ℓ2−1 ∗ v for every integer ℓ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Deleting every third edge on the path P3k−1, the graph P3k−1 ∗ v becomes a subdivision of S3,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' The observation on Wℓ follows immediately from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' □ The following results together show that the absence of a subdivision of the wall Wℓ, and thus in particular the absence of Wℓ as a strong immersion minor, implies the existence of a tree-cut decomposi- tion whose adhesion and torso size are both bounded in terms of the maximum degree and the size ℓ of the wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content='6 (Grid Theorem, [17, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content='1)]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' For every integer ℓ > 0, there exists an integer w(ℓ) > 0 such that if a graph G does not contain a subdivision of the wall Wℓ, then G has tree-width at most w(ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content='7 ([20, Lemma 12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Let w, d > 0 be integers, and let G be a graph with maximum degree at most d and tree-width at most w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Then there exists a tree-cut decomposition of G of adhesion at most (2w + 2)d such that every torso has at most (d + 1)(w + 1) vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' The next lemma says that a large enough set U of vertices in a connected graph is either reflected in a subdivision of a star with many leaves in U or a comb with many teeth in U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Here, a comb is a union of a path P with disjoint, possibly trivial, paths which have precisely their first vertex on P, and we call the last vertices of these paths their teeth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Note that if a graph has small maximum degree, then the graph does not contain any large subdivided star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content='8 ([2, Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' For every two integers s, t > 0, there exists an integer h(s, t) > 0 such that the following holds: Whenever a set U of vertices of a connected graph G has size at least h(s, t), then there is either some subdivision of a star in G with all its s leaves in U or a comb in G with all its t teeth in U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' As a final ingredient to our proof of Theorem 1, the following lemma asserts that if a simple graph does not contain a star as a minor, then it is ‘close’ to a union of vertex-disjoint paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content='9 ([14, Lemma 20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Let k > 0 be an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' If a simple connected graph G does not contain K1,k as a minor, then G − X is a vertex-disjoint union of paths for some set X of at most 4k vertices of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' We will define the integer α(ℓ) > 0 explicitly in terms of ℓ in what follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Let G be a graph in which the wall Wℓ is not strongly immersed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content='2, we may assume that G is 3-edge- connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' To construct the desired tree-cut decomposition (T, X) of G, we will first apply Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content='7 to an appro- priate minor G′ of G in order to obtain an auxiliary tree-cut decomposition (T, X ′) of G′ from which we then construct (T, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' This minor G′ of G is defined as follows: Let A be the simple graph on V (G) with 12 REINHARD DIESTEL, RAPHAEL W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' JACOBS, PAUL KNAPPE, AND PAUL WOLLAN an edge joining two vertices if there are more than p(ℓ) := 6ℓ2 edges joining them in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' The graph G′ then arises from G by contracting the components of A, keeping parallel edges as they arise (omitting potentially arising loops).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Now G′ inherits two key properties of G, which we will need in what follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' First, G′ is still 3-edge-con- nected as it is a minor of the 3-edge-connected graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Secondly, G′ does still not contain the wall Wℓ as a strong immersion minor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Indeed, G does not contain Wℓ as a strong immersion minor, and since p(ℓ) ⩾ 3 and Wℓ has maximum degree 3, one could greedily construct a strong immersion of Wℓ in G from a strong immersion of Wℓ in G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' In order to apply Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content='7 to G′, we next bound its tree-width and its maximum degree in terms of ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Claim 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' The tree-width of G′ is at most w(ℓ), the integer given by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Since G′ does not contain Wℓ as a strong immersion minor, in particular it does not contain a subdivision of Wℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Therefore, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content='6 yields the desired upper bound w(ℓ) on the tree-width of G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' ■ Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' There exists an integer d(ℓ) such that G′ has maximum degree d(ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' We prove that G′ has maximum degree at most d(ℓ) := g(ℓ) · p(ℓ) · h(2ℓ2, 6ℓ2)2, where h(2ℓ2, 6ℓ2) and g(ℓ) are the integers given by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content='8 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content='3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Since G′ is 3-edge-connected and does not contain Wℓ as strong immersion minor, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content='3 implies that every vertex of G′ has less than g(ℓ) neighbours in G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Thus, it is enough to show that there are at most d(ℓ)/g(ℓ) many parallel edges joining any fixed pair of vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' By the construction of G′, this is the case if there are at most d(ℓ)/g(ℓ) edges in G joining any two components of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Suppose for a contradiction that there are two components C and C′ of A such that there are more than d(ℓ)/g(ℓ) = p(ℓ)·h(2ℓ2, 6ℓ2)2 edges of G joining them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' By the definition of A, each vertex in C is joined to each vertex in C′ by at most p(ℓ) parallel edges in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Thus, at least one of NG(C) ∩ C′ and NG(C′) ∩ C is of size at least h(2ℓ2, 6ℓ2), and we may assume without loss of generality that it is NG(C′) ∩ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Now the simple graph A has maximum degree less than 2ℓ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Indeed, since p(ℓ) ⩾ 3, a vertex of degree at least 2ℓ2 in A would immediately imply the existence of a strong immersion of Wℓ in G by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Thus, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content='8, A[C] contains a comb with at least 6ℓ2 teeth in NG(C′) ∩ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Hence, the wall Wℓ is strongly immersed in G by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content='5, since G contains P6ℓ2−1 ∗ v as a strong immersion minor: Fix a vertex v in C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Since p(ℓ) ⩾ 6ℓ2, we may pick greedily edge-disjoint paths from v to each tooth in NG(C′) ∩ C such that all edges of each path except its last edge are contained in G[C′].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' These paths together with the comb form a strong immersion of P6ℓ2−1 ∗ v in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' All in all, the maximum degree of G′ is at most d(ℓ), as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' ■ By Claim 1 and Claim 2, we can now apply Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content='7 to G′ and obtain a tree-cut decomposition (T, X ′) of G′ of adhesion at most a(ℓ) := (2w(ℓ) + 2)d(ℓ) such that every torso of (T, X ′) has size at most k(ℓ) := (d(ℓ) + 1)(w(ℓ) + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' From the tree-cut decomposition (T, X ′) of the minor G′ of G, we then construct the tree-cut decomposition (T, X) of G by defining, for each node t ∈ T , its corresponding part Xt to be the union of the branch sets of the vertices in X′ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Now let α = α(ℓ) := max{k(ℓ)(8ℓ2 + 1 + 6ℓ2 · 16ℓ4), p(ℓ) · (6ℓ2)2 + �16ℓ4 2 � p(ℓ) · (6ℓ2)2 + k(ℓ) · d(ℓ)/2 + a(ℓ) · k(ℓ)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' We show that (T, X) has the desired properties in that it has adhesion at most α and the 3-centres of its torsos are almost α-thin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' In doing so, the first term in the maximum defining α will be used to bound the A GRID THEOREM FOR STRONG IMMERSIONS OF WALLS 13 size of the set of vertices that we will delete from the 3-centre of a torso to make it α-thinas well as the number of its neighbours, while the second term in the maximum will be used to show that the remainder of the 3-centre is α-thin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' The tree-cut decomposition (T, X) has adhesion at most a(ℓ) ⩽ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' By the construction of (T, X) from (T, X ′), the adhesion sets of (T, X) are the same as those of (T, X ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Thus, (T, X) has the same adhesion as (T, X ′) and, in particular, adhesion at most a(ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' ■ Claim 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' The 3-centres of the torsos of (T, X) are almost α-thin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' We first observe that the torsos of (T, X) equal their 3-centres: Every adhesion set of (T, X) has size at least 3, since G is 3-edge-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Therefore, all peripheral vertices of torsos of (T, X) have degree at least 3 in the respective torso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' This implies that the 3-centre of a torso of (T, X) equals the torso itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Given any node t ∈ T , let Ht be the torso of (T, X) at t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' We first find a set X of vertices of Ht such that A[Xt] − X is a disjoint union of paths, and we show that both the size of X and the size of its neighbourhood in Ht are bounded by our suitably chosen α = α(ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' In a next step, we then use the structure of A[Xt] − X to exhibit a natural enumeration of V (Ht − X), and we finally show that this enumeration witnesses the α-thinness of Ht − X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' For every component C of A[Xt], and more generally for every component C of A, the graph A[C] does not contain the star K1,2ℓ2 as a minor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' To prove this, we show that G[C] otherwise contains Wℓ as a strong immersion minor, contradicting our assumptions on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content='4, it is enough to show that S3,2ℓ2 is strongly immersed in G[C].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' So let U ⊆ C be a set of vertices containing precisely one vertex of each branch set of the K1,2ℓ2-minor of A[C].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' We then map the centre of S3,2ℓ2 to the vertex of U in the branch set of the centre of K1,2ℓ2, and map the rest of the vertices of S3,2ℓ2 bijectively to the remaining ones in U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Since p(ℓ) ⩾ 6ℓ2, we can now greedily embed the edges of S3,2ℓ2 as edge-disjoint paths in G[C] joining the vertices in U along the K1,2ℓ2-minor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Hence, we have found our desired strong immersion of S3,2ℓ2 in G[C].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' We are now ready to find the desired set X of vertices of the torso Ht of (T, X) at t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' For every component C of A[Xt], the graph A[C] does not contain K1,2ℓ2 as a minor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' So Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content='9 yields a set XC ⊆ C of size at most 8ℓ2 such that A[C] − XC is a disjoint union of paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' We then let the set X be the union of these XC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' The construction of X then implies that A[Xt] − X is indeed a disjoint union of paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' We first show that the size of X is bounded by α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Note that since A has maximum degree at most 2ℓ2 (as in the proof of Claim 2), the number of components of A[C]−XC is at most |XC|·2ℓ2 ⩽ 16ℓ4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Since X′ t has size at most k(ℓ), the graph A[Xt] has at most k(ℓ) components and thus |X| ⩽ 8ℓ2 · k(ℓ) ⩽ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Next, we claim that each vertex x in X has at most 8ℓ2 · k(ℓ) + k(ℓ) + k(ℓ) · 6ℓ2 · 16ℓ4 ⩽ α neighbours in Ht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Indeed, suppose for a contradiction that x has more than k(ℓ)+k(ℓ)·6ℓ2 ·16ℓ4 neighbours in Ht −X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' We first note that x is adjacent to at most k(ℓ) peripheral vertices of Ht, since there are at most k(ℓ) many peripheral vertices of Ht: They are in one-to-one correspondence to the peripheral vertices of the torso of (T, X ′) at t which has size at most k(ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Thus, x has more than k(ℓ) · 6ℓ2 · 16ℓ4 many neighbours which are core vertices of Ht and not contained in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' By the pigeonhole principle, there then exists one component C of the at most k(ℓ) components of A[Xt] such that x has at least 6ℓ2 · 16ℓ4 neighbours in C ∖ XC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Applying the pigeonhole principle a second time, we find one component P of the at most 16ℓ4 components of A[C] − XC that contains at least 6ℓ2 neighbours of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Since A[P] is a path by the choice of XC, we have hence found a subdivision and thus a strong immersion of P6ℓ2−1 ∗ v in Ht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' As all the 14 REINHARD DIESTEL, RAPHAEL W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' JACOBS, PAUL KNAPPE, AND PAUL WOLLAN vertices in C as well as x are core vertices of the torso Ht, this also is a strong immersion of P6ℓ2−1 ∗ v in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' So by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content='5, the wall Wℓ is contained in G as a strong immersion minor – a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Since X has size at most α and each element of X has at most α neighbours in Ht, it remains to prove that Ht − X is α-thin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' To do so, we now give an enumeration of V (Ht − X) which we will then show to witness the α-thinness of Ht − X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Let c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' , cn be an arbitrary enumeration of the part X′ t of tree-cut decomposition (T, X ′) of G′, let Ci be the component of A corresponding to the vertex ci of G′, and write Xi := XCi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Then take an enumeration v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' , vN of Xt ∖ X such that for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' , n, the set Ci − Xi forms an interval of this enumeration and within such an interval, every component of A[Ci] − Xi forms an interval with the same linear order which is given by the path that it induces in A by the choice of Xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' For the peripheral vertices of Ht, we finally pick an arbitrary enumeration vN+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' , vM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' To show that this enumeration v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' , vM of V (Ht−X) is as desired, first pick an arbitrary i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' , N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Then there is a component Cj of A[Xt] with vi ∈ Cj, and a component of A[Cj] − Xj containing vi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' this component of A[Cj] − Xj then induces a path P in A by the choice of Xj (which might have length 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Every edge in Ht joining {v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' , vi−1} and {vi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' , vM} belongs to one of the following classes: (1) edges joining vertices of the path P, (2) edges joining the distinct components in A[Cj] − Xj, (3) edges joining distinct components of A[X′ t], and (4) edges incident to peripheral vertices of Ht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' We will now separately bound the size of each of these classes (1) to (4) in terms of ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Note that only (1) depends on the choice of the enumeration;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' in particular, we will indeed bound the absolute number of all such edges in Ht in terms of ℓ for (2) to (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' For (1), the edges joining vertices of P, note that V (P) is a component of A[Cj]−Xj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' So there are no edges in A joining the vertices of P except the ones contained in P itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Hence, each two non-adjacent vertices of P are joined by at most p(ℓ) parallel edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Let P1 and P2 be the two components of P − vi, and note that both P1 and P2 consist of vertices of G since all vertices on P are in Xt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' If |EG(P1, P2)| > p(ℓ)·(6ℓ2)2, then one of Pi, say P1, has more than 6ℓ2 neighbours in P2−i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Fix any v ∈ P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Since p(ℓ) ⩾ 6ℓ2, we can greedily find edge-disjoint paths in G[P1 ∪ P2] joining v and 6ℓ2 of these neighbours in P2, and we thus find a strong immersion of P6ℓ2−1 ∗ v in G[P1 ∪ P2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content='5, this graph contains the wall Wℓ as a strong immersion minor which yields the desired contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' With the same type of argument, one can show that there are at most p(ℓ) · (6ℓ2)2 edges in G joining any two components of A[Cj] − Xj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Recall that the number of components of A[Cj] − Xj is at most 16ℓ4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Thus, there are at most �16ℓ4 2 � p(ℓ)·(6ℓ2)2 edges in G as in (2), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' edges joining components of A[Cj]−Xj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Next, we turn to (3) and bound the number of edges in G joining distinct components of A[X′ t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Since G′ arises from G by contracting the components of A, these edges are in one-to-one correspondence with the edges in G′[X′ t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Since X′ t has size at most k(ℓ) and G′ has maximum degree at most d(ℓ), there are at most k(ℓ) · d(ℓ)/2 edges in G′[X′ t] and hence joining distinct components of A[X′ t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Along similar lines, we can bound the number of edges in (4), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' the edges incident to peripheral vertices of Ht, in terms of ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Indeed, the edges incident to peripheral vertices of Ht are in one-to-one correspondence to the edges incident to the peripheral vertices of the torso of (T, X ′) at t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Since (T, X ′) has adhesion at most a(ℓ), each peripheral vertex has degree at most a(ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Moreover, since X′ t has size at A GRID THEOREM FOR STRONG IMMERSIONS OF WALLS 15 most k(ℓ), there are at most k(ℓ) peripheral vertices of Ht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Altogether, there are at most a(ℓ) · k(ℓ) edges incident to peripheral vertices of Ht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' All the above bounds show that by the choice of α = α(ℓ), there are at most α edges joining {v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' , vi−1} and {vi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' , vM} in Ht for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' By the choice of our enumeration, the vertices vN+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' , vM are precisely the peripheral vertices of Ht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' So for i = N + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' , M, all edges joining {v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' , vi−1} and {vi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' , vM} in Ht are incident to peripheral vertices of Ht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' As shown above there are at most a(ℓ) · k(ℓ) ⩽ α(ℓ) such edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' This completes the proof of this claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' ■ By Claim 3 and Claim 4, our constructed tree-cut decomposition (T, X) of G now is as desired, completing the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' □ Acknowledgement We are very grateful to Nathan Bowler for the suggestion to bound the size of the neighbourhood of the vertices which we delete in the definition of almost α-thinness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' This enabled us to obtain the actual equivalence given by Theorem 1 and Theorem 2 instead of an approximate one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' References [1] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Abu-Khzam and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdE4T4oBgHgl3EQfig1d/content/2301.05134v1.pdf'} +page_content=' Langston, Graph coloring and the immersion order, International 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+biwendong20g@ict.ac.cn +Bingbing Xu∗ +Institute of Computing Technology, +Chinese Academy of Sciences +Beijing, China +xubingbing@ict.ac.cn +Xiaoqian Sun∗ +Institute of Computing Technology, +Chinese Academy of Sciences +Beijing, China +sunxiaoqian@ict.ac.cn +Zidong Wang +Institute of Computing Technology, +Chinese Academy of Sciences +Beijing, China +wangzidong@ict.ac.cn +Huawei Shen +Institute of Computing Technology, +Chinese Academy of Sciences +Beijing, China +shenhuawei@ict.ac.cn +Xueqi Cheng∗ +Institute of Computing Technology, +Chinese Academy of Sciences +Beijing, China +cxq@ict.ac.cn +ABSTRACT +Company financial risk is ubiquitous and early risk assessment for +listed companies can avoid considerable losses. Traditional meth- +ods mainly focus on the financial statements of companies and +lack the complex relationships among them. However, the finan- +cial statements are often biased and lagged, making it difficult to +identify risks accurately and timely. To address the challenges, we +redefine the problem as company financial risk assessment on +tribe-style graph by taking each listed company and its share- +holders as a tribe and leveraging financial news to build inter-tribe +connections. Such tribe-style graphs present different patterns to +distinguish risky companies from normal ones. However, most +nodes in the tribe-style graph lack attributes, making it difficult to +directly adopt existing graph learning methods (e.g., Graph Neural +Networks(GNNs)). In this paper, we propose a novel Hierarchical +Graph Neural Network (TH-GNN) for Tribe-style graphs via two +levels, with the first level to encode the structure pattern of the +tribes with contrastive learning, and the second level to diffuse +information based on the inter-tribe relations, achieving effective +and efficient risk assessment. Extensive experiments on the real- +world company dataset show that our method achieves significant +improvements on financial risk assessment over previous compet- +ing methods. Also, the extensive ablation studies and visualization +comprehensively show the effectiveness of our method. +CCS CONCEPTS +• Computing methodologies → Neural networks; • Informa- +tion systems → Social networks. +∗Corresponding authors +This work is licensed under a Creative Commons Attribution +International 4.0 License. +KDD ’22, August 14–18, 2022, Washington, DC, USA +© 2022 Copyright held by the owner/author(s). +ACM ISBN 978-1-4503-9385-0/22/08. +https://doi.org/10.1145/3534678.3539129 +KEYWORDS +company financial risk assessment, tribe-style graph, graph neural +network +ACM Reference Format: +Wendong Bi, Bingbing Xu, Xiaoqian Sun, Zidong Wang, Huawei Shen, +and Xueqi Cheng. 2022. Company-as-Tribe: Company Financial Risk As- +sessment on Tribe-Style Graph with Hierarchical Graph Neural Networks. +In Proceedings of the 28th ACM SIGKDD Conference on Knowledge Discovery +and Data Mining (KDD ’22), August 14–18, 2022, Washington, DC, USA. ACM, +New York, NY, USA, 9 pages. https://doi.org/10.1145/3534678.3539129 +1 +INTRODUCTION +Company financial risk is ubiquitous in the real-world financial +market. Early assessment of risks for the listed company can provide +decision support for company managers and investment institu- +tions, thereby avoiding considerable losses. +Traditional methods [7, 18], such as financial probability meth- +ods, decision tree methods, and Deep Neural Networks (DNNs), +treat each company individually and solely leverage the financial +statements to assess risk. However, financial statements are often +biased and lagged. As Fig. 1 (a) shows, most companies beautify +their published financial data, and some companies even commit +financial fraud. Besides, these traditional methods ignore the inter- +actions among companies, which is critical because risks can passed +between companies. The above limitations make the traditional +methods difficult to identify risks accurately and early. +To effectively assess company financial risks, we found there +exist two other types of valuable information: 1) the investment- +graph of listed companies, e.g., CATL1 has more than 200 investors +(companies or individuals), which form an investment-graph. As +Fig. 1 (b) shows, risky and normal companies often have different +investment patterns; 2) The news-graph among listed companies, +i.e., there exists an edge between two companies if they concur- +rent in at least one piece of news. As Fig. 1 (b) shows, two listed +companies connected usually have strong correlations, and risks +can spread over this graph. Superior to other information, financial +1CATL (Contemporary Amperex Technology Co., Limited) is a typical listed company +in China, which is a global leader of new energy innovative technologies and committed +to providing premier solutions and services for new energy applications worldwide. +arXiv:2301.13492v1 [cs.LG] 31 Jan 2023 + +BYKDD ’22, August 14–18, 2022, Washington, DC, USA +Wendong Bi et al. +Tribe A +Normal +central company +Risky +central company +Declining performance +Rising performance +Company shareholder +Individual shareholder +(b) Tribe-style +(a) Individual-style +Node +Attribute +✓ +✘ +Normal +Risky +B +A +C +Risky +Fraud +News relationship +Financial report +Investment-Graph +Tribe B +Tribe C +Figure 1: Company financial risk assessment on individual-style vs. tribe-style graph. Each company with its investment-graph +in a dotted oval box can be seen as a tribe, and they are further connected by the global relationship (e.g., news relationship). +news is objective and can timely reflect risks. Extensive statistical +analyses are provided in Sec. 2.1 to demonstrate the benefit of these +data for distinguishing risky companies from the normal ones. +Based on the above findings, we redefine the problem as com- +pany financial risk assessment on tribe-style graphs. As illus- +trated in Fig. 1 (b), we take the investment-graph consisting of a +central listed company and its shareholders as a tribe, and leverage +the news-graph to construct inter-tribe edges, the financial state- +ments of listed companies are regarded as initial attributes of tribes. +However, it is challenging to directly adopt existing graph methods +(e.g., Graph Neural Networks (GNNs)) to such tribe-style graphs +due to the following serious issues: 1) only the listed companies in +a tribe have attributes, and other individuals or companies have +no disclosure obligation and therefore do not have attributes, mak- +ing it difficult to conduct message passing in GNNs; 2) The whole +tribe-style graph including both intra-tribe and inter-tribe relation- +ships is large-scale and contains millions of edges, which makes +the GNNs inefficient, and traditional node sampling techniques +[4, 13, 14] cause the loss of information. +In this paper, we propose a novel Hierarchical Graph Neural +Network for the financial risk assessment on the tribe-style graph, +namely TH-GNN. Specifically, for the first challenge that the indi- +viduals and non-central companies in a tribe have no attributes, we +find the structure patterns can reflect the company’s risks. There- +fore, we design a tribe structure encoder (TSE) based on contrastive +learning that learns structural patterns for each tribe (including +the scale of the tribe and its investment structure, etc.) without +relying on node attributes. For the second challenge, although the +whole graph is huge, Fig. 1 (b) shows an important property of +the tribe-style graph that the intra-tribe connections (investment- +graph) are dense while the inter-tribal connections (news-graph) +are sparse. Inspired by this property, TH-GNN encodes the tribe- +style graph through a hierarchical manner, with the first level to +encode tribes defined by the investment graphs via the tribe struc- +ture encoder and the second level to diffuse the information among +tribes based on the news-graph and learn the global representa- +tions. Unlike the traditional GNNs that diffuse information over +edges on the whole graph, TH-GNN converts a tribe-style graph +into parallelly computable local graphs and a smaller global graph. +Extensive experiments on the real-world dataset for company finan- +cial risk assessment show that our approach achieves significant +improvement over previous competing methods. Meanwhile, the +ablation studies and visualizations also comprehensively show the +effectiveness of our method. +The main contributions of this work are summarized as follows: +(1) We redefine the previous individual risk assessment problem +as company financial risk assessment on tribe-style +graph and further design a tribe-style graph consisting of +financial statements, investment-graphs, and news-graphs +rather than solely utilizing financial statements. +(2) We propose a novel Hierarchical Graph Neural Network +named TH-GNN to model the tribe-style graph. To the best of +our knowledge, this is the first graph representation learning +method for company financial risk assessment on the tribe- +style graph. +(3) We conduct extensive experiments on a real-world company +graph dataset with 0.88 million nodes and 1.31 million edges. +The results demonstrate the superiority of the proposed +model over state-of-the-art methods. The code is avaliable 2. +2Our source code is available at https://github.com/wendongbi/TH-GNN + +晶團1晶團用 +用用由晶團用晶團晶團晶團IEWS::用Company-as-Tribe: Company Financial Risk Assessment on Tribe-Style Graph with Hierarchical Graph Neural Networks +KDD ’22, August 14–18, 2022, Washington, DC, USA +2 +PRELIMINARY +In this section, we present the detailed statistical analysis of tribe- +style graph and the formalized definition of company financial risk +assessment on tribe-style graphs. +2.1 +Data Analysis +As illustrated in Fig. 1 (b), the tribe-style graph consists of investment- +graph (tribe) and news-graph, where the investment-graph presents +the intra-tribe connections and the news-graph presents the inter- +tribe connections. We then analyze the investment-graphs and +news-graph comprehensively to verify their benefits for financial +risk assessment. +2.1.1 +investment-graph analysis. An investment-graph usually con- +sists of one central listed company and others (companies or in- +dividuals) which have investment relationships with the central +company. Only the listed company publishes its financial state- +ments as attributes, and the other nodes do not have attributes. +Therefore we mainly focus on the structure patterns of tribes. Then +we first give a case study of investment-graphs and then present the +statistical analysis on all listed companies’ investment-graphs. As +(a) A risky company example. +(b) A normal company example +Figure 2: Examples of investment-graphs3. (The bigger or- +ange point denotes listed company, blue points denote un- +listed companies and red points denote individuals. And the +edges in the graphs represent investment relationships.) +illustrated in Fig. 2, the investment-graphs for risky company and +normal company show different patterns. The investment-graph of +the risky company shown in Fig. 2 (a) is more similar to a star-like +graph, where the single listed company can be viewed as the central +node, and its neighboring nodes (investors) consist of more indi- +viduals or companies that tend to be disconnected from each other. +Different from the risky company, the neighboring nodes of the +normal company in Fig. 2 (b) are often popular companies which +have more neighbors (investors), and there exist dense connections +among these neighbors. Such a pattern with more reliable investors +tends to be more stable, and thus the central listed company is +less likely to have financial risks. This motivates us to leverage the +investment pattern of listed company to identify financial risks. +To further verify this finding, we conduct centrality analysis for +the investment-graphs of all listed companies. Five typical metrics +are used [2, 22], including degree centrality, eigenvector centrality, +clustering centrality, number of bridge, and the central node degree. +We calculate the above metrics on each investment-graph and then +3These two examples are from two real-world companies named LongYuan Construc- +tion and ShangHai Dragon Corporation repectively. +take the average of risky and normal companies respectively to +reflect the differences of centrality between the two classes. +Table 1: Statistical centrality analysis of the investment- +graphs +Statistical metric +risky company +normal company +Degree centrality +0.264 +0.224 +Eigenvector centrality +0.4161 +0.3604 +Clustering coefficient +0.2102 +0.1907 +Number of bridge (avg) +125.4 +112.3 +Central node degree (avg) +58.43 +46.71 +The results of the centrality analysis are presented in Table 1, +which demonstrate that the investment-graphs of risky companies +usually have larger graph centrality compared with that of normal +companies. And this pattern motivates us to take the structure +encoding of investment-graph into consideration for financial risk +assessment. We consider the structural pattern of a investment- +graph as a tribe individually, where the nodes within a tribe are +centered on the centrally-located listed company. We further ex- +plain the benefits of tribe-style graphs in Sec. 3.1.2. +2.1.2 +news-graph analysis. Besides the investment relationship in +tribes, we also use financial news to model the interactions between +different tribes. The financial news has evident timeliness and ob- +jective authenticity, reporting company financial risks promptly. +Specifically, different companies may appear in the same news, +reflecting strong correlation among them, e.g., news reported that +Company A and Company B jointly invested in a failed project, +reflecting that they may have potential financial risks at the same +time (news used for graph construction in this paper all describes +similarity among companies). Then for companies co-existed in +one piece of news, we connect them to construct news-graphs, +indicating the risky associations among them. +As illustrated in Fig. 3, we also conduct statistical analysis to +validate the benefits of the news-graph. We calculate the proportion +of risky companies in neighbors of each node for risky companies +and normal companies respectively. For example, the rightmost +column in Fig. 3 indicates there are nearly 30% of risky companies +(the red bar) with more than 80% of neighbors at-risk, while no +more than 10% of normal companies (the blue bar) with more than +80% of neighbors at-risk. The results show that the probability of +risky companies with high-proportion risky neighbors is much +larger than that of normal companies, that is, the risky nodes in the +news-graph have higher-proportional-risk neighbors. This finding +also suggests that news-graph can benefit the company financial +risk assessment. As a result, we use the news-graph to construct +inter-tribe edges, modeling the relations among listed companies. +2.2 +Problem Definition +Our company financial risk assessment problem is defined on the +tribe-style graph, which consists of a set of tribes and a global +news-graph connecting different tribes. For company financial risk +assessment, we need to classify each company into binary classes: + +KDD ’22, August 14–18, 2022, Washington, DC, USA +Wendong Bi et al. +0.0~0.2 +0.2~0.4 +0.4~0.6 +0.6~0.8 +0.8~1.0 +Proportion of risky neighbors +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +Proportion of nodes (risky/normal) +Company Type +Normal Company +Risky Company +Figure 3: Analysis on proportion of risky neighbors.The x- +axis is the proportion of risky companies in all neighbors of +each node, and the y-axis is the corresponding proportion +of nodes in a certain node type (risky/normal) . Note that we +ignore the 0-degree nodes when calculating these results. +risky or normal. We give a formal definition of the task as fol- +lows. Let GT = {G𝑔𝑙𝑜𝑏𝑎𝑙, G𝑡𝑟𝑖𝑏𝑒} denote a tribe-style graph, where +G𝑔𝑙𝑜𝑏𝑎𝑙 = (𝑉𝐺, 𝐸𝐺,𝑋𝐺) is the global graph which taking all the +listed companies as nodes. Considering that each listed company +is the center of a tribe, we name the listed company node as the +central node for shortly. 𝑉𝐺 denotes the central node set, 𝑁𝑐 is +the number of central nodes, and 𝐸𝐺 is the set of edges connecting +central nodes. 𝑋𝐺 ∈ R𝑁 𝑐×𝐷 is the attribute matrix, and the i-th +row of 𝑋𝐺 denoted by 𝑋𝐺 +𝑖 +is the 𝐷-dimensional attribute vector +of central node 𝑣𝐺 +𝑖 . For each central node 𝑣𝐺 +𝑖 ∈ 𝑉𝐺, there exists a +tribe 𝑔𝑇 +𝑖 ∈ G𝑡𝑟𝑖𝑏𝑒 corresponding to 𝑣𝐺 +𝑖 , where 𝑔𝑇 +𝑖 = (𝑉𝑇 +𝑖 , 𝐸𝑇 +𝑖 ) and +G𝑡𝑟𝑖𝑏𝑒 = {𝑔𝑇 +𝑖 | 𝑖 = 1 · · · 𝑁𝑐}. 𝑉𝑇 +𝑖 and 𝐸𝑇 +𝑖 are the node set and edge +set of tribe 𝑔𝑇 +𝑖 . Each tribe have one central node. Each central node +𝑣𝐺 +𝑖 is associated with a binary label 𝑦𝑖 = {0, 1}, using 1 for risky +companies and 0 for normal companies. Then the listed company’s +financial risks assessment problem on tribe-style graph can be de- +scribed as: given a tribe-style graph GT, and the goal is to classify +each listed company node into binary classes: risky or normal. +3 +METHODS +In this section, we first explain how and why we design the tribe- +style graph. Then we introduce the proposed TH-GNN model for +company financial risk assessment on tribe-style graphs in detail. +3.1 +Tribe-style Graph Construction +3.1.1 +How to construct the tribe-style graph? As illustrated in +Fig. 1, the tribe-style graph consists of company financial state- +ments, investment-graphs, and financial news. Specifically, listed +companies are viewed as central target nodes in the tribe-style +graph, and the investment-graph of each listed company is viewed +as a tribe (i.e., a super node on the tribe-style graph), where only +the central listed company nodes have attributes extracted from +their financial statements. The global graph connecting central com- +panies is constructed by financial news, where two central listed +companies connected if they co-existed in at least one piece of +financial news. Note that we only use news that describes the risky +linkages among listed companies in China. If there are multiple +companies appearing in the same news, we connect all possible +pairs of companies. More details about the dataset information and +preprocessing are presented in Sec. 4.1 and Sec. 4.2. +3.1.2 +Why we construct the tribe-style graph? We summarize the +following advantages of constructing a tribe-style graph. +(1) Based on the analysis in Sec. 2.1, it is the structural pattern of +tribes that benefits the identification of risky companies. +(2) Considering that only the central node of a tribe has attributes, +the central and non-central nodes should be treated separately. +Therefore we make the tribe-style graph a hierarchical graph. +(3) To improve model efficiency, we treat each tribe independently +and obtain the representation of each tribe by graph pooling(regardless +of overlap among tribes) instead of merging them into one graph, +which actually truncates the original large-scale graph. +3.2 +Model Overview +We give an overview of TH-GNN in Fig. 4, TH-GNN includes two +main components, including the Tribe Structure Encoder (TSE) +and Global Graph Representation Learning (GGRL) module. TH- +GNN encodes the tribe-style graph in bottom-up order. TH-GNN +first learns the structural representation for each tribe with the +TSE. Then the learned structural representation of tribes and the +financial statements are fused into the embedding of central node +(listed company) by an attention-based fusion module. Next, the +embedding is diffused over the global news-graph to learn the final +representation of central nodes for financial risks assessment. +3.3 +Tribe Structure Encoder (TSE) +The Tribe Structure Encoder (TSE) is used to learn the structural +representation for each tribe based on contrastive learning, includ- +ing a structure embedding module and a graph encoder module. +Considering that nodes in the tribe have no attributes, we first +initialize the node attributes according to their position in the tribe. +Then we transform the structural attributes into learnable embed- +ding with a structure embedding module for each node in the tribe. +Finally, with the tribe (investment graph) and the node structure +embedding, a GIN model is used to get the representation of tribes. +Inspired by the importance of centrality patterns in our scenario +discussed in Sec. 2.1, we also consider the encoding of centrality +when designing the TSE. For a tribe without node attributes, we first +assign each node with a structure embedding as its initial attributes. +Specifically, each node 𝑣𝑇 +𝑗 ∈ 𝑉𝑇 +𝑖 +on the investment-graph 𝑔𝑇 +𝑖 has +the three properties: (1) node degree (in-degree denoted by 𝑑𝑒𝑔+ +𝑗 +and out-degree denoted by 𝑑𝑒𝑔− +𝑗 for a directed graph); (2) node +type denoted by 𝜙𝑗 (listed company, unlisted company or human); +(3) distance of shortest path (SPD) to the central node denoted by +𝑆𝑃𝐷𝑗. With these structure attributes, we further transform them +into learnable embedding by an Embedding Layer: +𝐸𝑚𝑏(𝑣𝑇 +𝑗 ) = 𝐸𝑚𝑏𝑒𝑑𝑑𝑖𝑛𝑔_𝐿𝑎𝑦𝑒𝑟 +� +𝑑𝑒𝑔+ +𝑗 ,𝑑𝑒𝑔− +𝑗 ,𝜙𝑗,𝑆𝑃𝐷𝑗 +� +(1) +Following [9, 25], we also use the top Laplacian eigenvector with +the largest eigenvalue of each tribe as the nodes positional encoding + +Company-as-Tribe: Company Financial Risk Assessment on Tribe-Style Graph with Hierarchical Graph Neural Networks +KDD ’22, August 14–18, 2022, Washington, DC, USA +2 +1 +3 +Local Message Passing +TSE: Tribe +Structure Encoder +(1) Position +Embedding +… +… +… +(3) Node Type +Embedding +(2) Degree +Embedding +(4) Shortest Path +Distance Embedding +Tribe Structure +Embedding +2 +TSE +Tribe Structure +Embedding +Financial Report +Tribe-style Graph +Attention-based +Fusion Module +𝒉𝟏 +𝑳 +𝒉𝟐 +𝑳 +Contrastive Loss +𝑳𝑪𝑳 +Central Node +Embedding +1 +2 +3 +Global Graph +Representation Learning +Normal +Risky +𝑳𝑩𝑪𝑬 +1 +Figure 4: Overview of our proposed TH-GNN model. TH-GNN includes two main components, including the Tribe Structure +Encoder (TSE) and Global Graph Represenation Learning (GGRL) module. +besides the learnable structure embedding, and the 𝑗-th value of +the top Laplacian eigenvector is exactly the eigenvector centrality +[2] of node 𝑣𝑇 +𝑗 ∈ 𝑉𝑇 +𝑖 . Then the final structure mebedding for each +node on the tribe can be represented as: +𝑍 (𝑔𝑇 +𝑖 ) = +� +𝐸𝑚𝑏(𝑣𝑇 +𝑗 ) || 𝑢0(𝑔𝑇 +𝑖 ) +� +(2) +where 𝑢0(𝑔𝑇 +𝑖 ) is the top Laplacian eigenvector of 𝑔𝑇 +𝑖 . Next, we use +the structure embedding (Eq. 2) for local message passing on tribes. +In this paper, we use GIN with SUM pooling as the graph encoder +for tribes. The GIN updates the node representations by: +ℎ(𝑙) +𝑣𝑇 +𝑖 += 𝑀𝐿𝑃 (𝑙) ��� +� +(1 + 𝜖 (𝑙)) · ℎ(𝑙−1) +𝑣𝑇 +𝑖 ++ +∑︁ +𝑣𝑇 +𝑗 ∈𝒩(𝑣𝑇 +𝑖 ) +ℎ(𝑙−1) +𝑣𝑇 +𝑗 +��� +� +(3) +where 𝜖 is a learnabel parameter, ℎ(𝑙) +𝑣𝑇 +𝑖 +is the learned representation +of node 𝑣𝑇 +𝑖 at the 𝑙-th layer of GIN. Then we get the global graph +representation by performing SUM Pooling for all nodes in each +tribe and taking an average over all layers of the model: +ℎ𝑔𝑇 +𝑖 = +1 +𝐿 + 1 +𝐿 +∑︁ +𝑙=0 +SUM +� +{ℎ(𝐿) +𝑣𝑇 +𝑖 +|𝑣𝑇 +𝑖 ∈ 𝑔𝑇 +𝑖 } +� +(4) +Then the final tribe representation is 𝐻G𝑡𝑟𝑖𝑏𝑒 = {ℎ𝑔𝑇 +𝑖 |𝑔𝑇 +𝑖 ∈ G𝑡𝑟𝑖𝑏𝑒}. +Considering the high cost of data labeling, we introduce con- +trastive learning to guide the training of TSE. Specifically, we design +a graph instance discrimination tasks and use InfoNCE [24] as the +objective to optimize the model parameters. The contrastive task +treats each tribe as a distinct class and leans to discriminate between +different tribes through a self-supervised way, and help the TSE +module learn the structural dissimilarity between different tribes. +Specifically, we first prepare one positive sample pair and 𝑁 − 1 +negative sample pairs for each training batch with N samples and +the tribes of each batch are fed into the TSE twice to obtain the +query representation 𝐻𝑞 +G𝑡𝑟𝑖𝑏𝑒 and key representation 𝐻𝑘 +G𝑡𝑟𝑖𝑏𝑒 of +tribes. Due to the randomness of dropout, there is a certain dif- +ference between the obtained two sets of tribe representations. +Then we use the query and key representations of the same tribe +as positive pairs denoted by {⟨𝑞𝑖,𝑘+ +𝑖 ⟩, 𝑔𝑇 +𝑖 ∈ G𝑡𝑟𝑖𝑏𝑒}, and use the +representations of different tribes to construct negative pairs de- +noted by {⟨𝑞𝑖,𝑘− +𝑗 ⟩, 𝑗 ≠ 𝑖 and 𝑔𝑇 +𝑖 ,𝑔𝑇 +𝑗 ∈ G𝑡𝑟𝑖𝑏𝑒}. Then we compute +InfoNCE Loss as a regular term besides the supervised classification +loss to optimize parameters of the Tribe Structure Encoder module: +L𝐶𝐿 = − 1 +𝑁 +𝑁 +∑︁ +𝑖=1 +𝑙𝑜𝑔 +𝑞𝑇 +𝑖 · 𝑘+ +𝑖 +𝑞𝑇 +𝑖 · 𝑘+ +𝑖 + �𝑁 −1 +𝑗=1 𝑞𝑇 +𝑖 · 𝑘− +𝑗 +(5) +3.4 +Global-Graph Representation Learning +(GGRL) +With TSE, we obtain the representations of tribes. And then for each +central company, its node features come from two parts: the tribe +representation and financial statements, which can be represented +as {(ℎ𝑔𝑇 +𝑖 ,𝑋𝐺 +𝑖 ) | 𝑣𝐺 +𝑖 ∈ 𝑉𝐺 }. We further use an attention-based fusion +module to integrate the tribe representationsℎ𝑔𝑇 +𝑖 and financial state- +ments feature 𝑋𝐺 +𝑖 +into one central node embedding ℎ(0) +𝑖 +. Finally, +the fused central node embedding is used for message passing on +the global news-graph to learn the final representations of central +companies. +To better integrate the node features from financial statements +and tribes on the global graph, we design an attention-based fusion +module to fuse the two features to common space. We first calculate + +晶晶田KDD ’22, August 14–18, 2022, Washington, DC, USA +Wendong Bi et al. +the weights for the financial statements and tribe representations: + + +𝑒𝑔 +𝑖 = 𝜎([ℎ𝑔𝑇 +𝑖 ·𝑊 𝑔||𝑋𝐺 +𝑖 ·𝑊 𝑋 ] · 𝑎𝑇 +𝑔 ), +𝑎𝑔 ∈ R1×2𝐷 +𝑒𝑋 +𝑖 = 𝜎([ℎ𝑔𝑇 +𝑖 ·𝑊 𝑔||𝑋𝐺 +𝑖 ·𝑊 𝑋 ] · 𝑎𝑇 +𝑥 ), +𝑎𝑥 ∈ R1×2𝐷 +(6) +where 𝜎 is the LeakyReLU activation, 𝑊 𝑔 and 𝑊 𝑋 are transforma- +tion matrix to project ℎ𝑔𝑇 +𝑖 and 𝑋𝐺 +𝑖 into common hidden dimension. +Then we normalize them by the softmax function: +𝛼𝑔 +𝑖 = +𝑒𝑥𝑝(𝑒𝑔 +𝑖 ) +𝑒𝑥𝑝(𝑒𝑔 +𝑖 ) + 𝑒𝑥𝑝(𝑒𝑋 +𝑖 ) +, 𝛼𝑋 +𝑖 = +𝑒𝑥𝑝(𝑒𝑋 +𝑖 ) +𝑒𝑥𝑝(𝑒𝑔 +𝑖 ) + 𝑒𝑥𝑝(𝑒𝑋 +𝑖 ) +(7) +Next, we can get the fused central node embedding: +ℎ(0) +𝑖 += 𝛼𝑔 +𝑖 · (ℎ𝑔𝑇 +𝑖 ·𝑊 𝑔) + 𝛼𝑋 +𝑖 · (𝑋𝐺 +𝑖 ·𝑊 𝑋 ) +(8) +Then we use the fused central node embedding as node features +and further perform message passing on the global news-graph to +learn the final representations of each central node: +ℎ(𝑙) +𝑖 += 𝜎 +��� +� +� +∑︁ +𝑗 ∈{𝒩(𝑣𝐺 +𝑖 )∪𝑣𝐺 +𝑖 } +1 +𝑑𝑖 +· ℎ(𝑙−1) +𝑗 +� ·𝑊 𝑙��� +� +(9) +where 𝑑𝑖 is the in-degree of 𝑣𝐺 +𝑖 (including the self-loop). And the +learned representations can be further used for the company finan- +cial risk assessment task. +3.5 +Model Optimization Methods +After aggregating the information from neighbors on the graph, +the obtained representation ˆℎ(𝐿) +𝑖 +is fed into a final fully connected +neural network with a sigmoid activation function, as follows: +𝑝𝑖 = 𝑆𝑖𝑔𝑚𝑜𝑖𝑑(ℎ(𝐿) +𝑖 +·𝑊𝑝 + 𝑏𝑝) +(10) +where 𝑝𝑖 is the probability of company node 𝑣𝑖 suffering risks in +the further. Then we compute binary cross entropy (BCE) loss to +utilize the supervised information of labels: +L𝐵𝐶𝐸 = 1 +𝑁 +𝑁 +∑︁ +𝑖=1 +𝑦𝑖 · log ˆ𝑦𝑖 + (1 − 𝑦𝑖) · log(1 − ˆ𝑦𝑖) +(11) +Then, the final loss function is composed of L𝐵𝐶𝐸 and L𝐶𝐿 (Eq. 5): +L = L𝐵𝐶𝐸 + 𝛼 · L𝐶𝐿 +(12) +where 𝛼 is a hyper-parameter to control the weight of L𝐶𝐿 . +4 +EXPERIMENTS +In this section, we compare TH-GNN with other state-of-the-art +methods on a real-world dataset for company risk assessment. +4.1 +Dataset +The company dataset used in this paper comes from the real-world +data of 4040 listed companies in China from 2019 to 2020, i.e., the +listed company’s financial statements, investment-graph, and finan- +cial news related to these companies. The financial statements and +the company’s investment-graph data are provided by TianYanCha +(an authority enterprise credit institute for company information +inquiry in China). The annual financial statements reflect a listed +company’s industry information and its financial and business sit- +uation in a year. The investment-graph of a company describes +Table 2: Information of the Hierarchical Graph +Graph +#Nodes +#Edges +Whole graph GT +879252 +1311364 +News-graph G𝑔𝑙𝑜𝑏𝑎𝑙 +4040 +16330 +Investment-graphs (total) G𝑙𝑜𝑐𝑎𝑙 +879252 +1295034 +Investment-graphs (average) G𝑙𝑜𝑐𝑎𝑙 +217.6 +320.5 +the relationship between the central company and its shareholders, +including other companies and humans. The financial news data are +provided by Wind ( an authority China finance database), and these +news are obtained from more than 800 authority news websites +in China, which have extremely wide coverage and timeliness to +capture the risk information of companies. Note that the financial +news used in this paper, which has already been preprocessed by +Wind, all describes the risk linkages among companies. Then we +construct the tribe-style graph and more specific information of +this graph is illustrated in Table 2. Based on the real-world risk +events of companies happened in 2020 provided by Wind, the posi- +tive (risky) and negative (normal) labels can be naturally generated, +and we use all companies marked as high-risk as positive samples +and others as negative samples. To prevent information leakage, +the part of the dataset in 2019, including financial and operating +information, investment-graph and news-graph, is used as training +data . There are 1698 positive samples and 2342 negative samples +among all listed companies. +4.2 +Experimental Setup +We conduct experiments on the real-world dataset with different +configurations. We design experiments with different training ra- +tios (percent of nodes in the training set) ranging from 20% to 40%. +And for each training ratio, we use three different random parti- +tions of the dataset and ten random seeds for the model parameter +initialization, a total of 30 trials for each model. For all attributes +of the dataset used in this paper, we preprocess the categorical or +discrete attributes into one-hot vectors, and then we use binning +methods to divide continuous numerical attributes into 50 bins and +use the index of bin as their feature. For fairness, we perform a +hyper-parameter search for all models, and the size of searching +space for each model is the same. The hidden dimension of all +models are searched in {32, 64, 128} and we choose the number of +training epoch from {100, 200, 300}. We use the Adam optimizer for +all experiments and the learning rate is searched in {1e-2, 1e-3, 1e-4}, +weight decay is searched in {1e-4, 1e-3, 5e-3}, and 𝛼 (the coefficient +of L𝐶𝐿) is searched in {0.01, 0.05, 0.1, 0.5, 1.0} for all experiments. +The number of layers for GNN models, including TH-GNN and +other baseline GNN models except for GCNII and DAGNN, are set +to be two layers in this paper. The number of layers for GCNII and +DAGNN, which are designed with deeper depth, are set to 64 and +20 respectively according to their papers [5, 16]. All models used +in this paper were trained on Nvidia Tesla V100 (32G) GPU. +4.3 +Compared Methods +4.3.1 +Baseline methods. We compare our model with two classical +machine learning models (XGBoost [6], DNN), three baseline GNN + +Company-as-Tribe: Company Financial Risk Assessment on Tribe-Style Graph with Hierarchical Graph Neural Networks +KDD ’22, August 14–18, 2022, Washington, DC, USA +Table 3: Binary node classification results (%) of models on our dataset. The model with ∗ means using the concatenation of +tribe structural embedding and financial statements data as inputs (see Sec. 4.3.2). Otherwise, only financial statement data is +used. +Evaluation Metric +binary F1-score +AUC score +Training ratio +60% +40% +20% +60% +40% +20% +XGBoost +57.4 ± 1.63 +56.1 ± 1.55 +55.9 ± 1.05 +65.3 ± 2.23 +64.3 ± 2.07 +62.7 ± 2.23 +DNN +55.8 ± 2.77 +54.9 ± 2.56 +53.5 ± 2.89 +65.1 ± 2.77 +65.6 ± 2.67 +65.3 ± 2.86 +GCN +58.9 ± 3.54 +58.3 ± 1.74 +56.8 ± 2.87 +68.1 ± 1.64 +67.2 ± 1.81 +67.8 ± 1.63 +GAT +57.6 ± 3.86 +57.2 ± 3.75 +57.2 ± 3.96 +67.6 ± 3.11 +67.2 ± 3.52 +66.5 ± 4.21 +GraphSAGE +59.4 ± 2.36 +59.5 ± 2.36 +58.5 ± 2.38 +68.48 ± 3.84 +67.14 ± 3.25 +65.6 ± 1.78 +GCNII +59.9 ± 2.04 +59.7 ± 1.94 +59.1 ± 1.88 +69.6 ± 1.81 +69.1 ± 1.72 +68.8 ± 2.01 +DAGNN +60.6 ± 1.96 +60.7 ± 1.30 +60.0 ± 1.13 +70.6 ± 1.17 +70.4 ± 1.17 +69.7 ± 1.17 +XGBoost∗ +58.2 ± 2.19 +56.8 ± 2.05 +56.2 ± 1.92 +67.2 ± 2.56 +64.9 ± 2.25 +64.7 ± 2.33 +DNN∗ +57.2 ± 2.11 +56.8 ± 2.32 +55.1 ± 2.13 +66.3 ± 1.66 +66.4 ± 1.88 +65.3 ± 2.15 +GCN∗ +59.2 ± 1.30 +58.6 ± 2.01 +57.3 ± 2.01 +70.3 ± 0.95 +68.8 ± 1.31 +69.1 ± 1.55 +GAT∗ +58.4 ± 2.01 +57.4 ± 2.15 +56.8 ± 2.34 +67.8 ± 2.26 +67.0 ± 2.34 +66.1 ± 3.12 +GraphSAGE∗ +59.7 ± 2.08 +58.3 ± 2.0 +57.3 ± 3.76 +70.6 ± 2.15 +69.8 ± 1.04 +68.2 ± 1.23 +GCNII∗ +60.5 ± 1.23 +60.6 ± 1.30 +60.0 ± 1.17 +70.9 ± 1.73 +70.6 ± 1.85 +70.9 ± 2.09 +DAGNN∗ +61.1 ± 1.06 +60.9 ± 1.12 +59.9 ± 3.69 +71.1 ± 1.74 +70.6 ± 1.38 +70.2 ± 0.84 +TH-GNN +63.2 ± 0.75 +62.8 ± 0.95 +62.2 ± 1.13 +73.5 ± 0.54 +72.8 ± 0.54 +72.5 ± 0.54 +models (GCN [15], GAT [30], GraphSAGE [13]), and two state-of- +the-art GNN models (GCNII [5], DAGNN [16]) to demonstrate the +superiority of our TH-GNN model. Furthermore, six variants of TH- +GNN are designed for ablation studies: TH-GNN\Attribute indicates +removing node attributes (financial statements) from TH-GNN and +only using the graph structure information; TH-GNN\TSE indicates +removing Tribe Structure Encoder from TH-GNN, without lever- +aging tribes; TH-GNN\GGRL indicates removing the GGRL module +from TH-GNN, without leveraging the news-graph; TH-GNN\Fusion +indicates removing attention-based fusion module from TH-GNN; +TH-GNN\Emb indicates removing the structure embedding used +in TSE; TH-GNN\CL indicates removing the contrastive loss term +used to optimize the Tribe Structure Encoder from TH-GNN. In our +experiments, we select two widely used metrics as performance +measurement, i.e., AUC (the area enclosed by the coordinate axis +under the ROC curve) and F1-score (the harmonic average of the +precision and recall) on the test set. +4.3.2 +Two implementations for base GNN models. Note that GNN +models except TH-GNN cannot directly encode the tribe-style graph +with hierarchical structures. For fair comparisons, we design two +implementations for each baseline GNN model. +(1) A basic implementation is to directly train the vanilla base- +line GNNs on the news-graph, because nodes except the central +node (listed company) in a tribe do not have attributes (financial +statements), which are necessary for vanilla GNN models. +(2) A two-stage implementation is to learn structure repre- +sentation of tribes (investment-graphs) first and then train baseline +GNNs on the news-graph. In this paper, we use GCC [25] , a graph +contrastive learning based method, to learn structure representa- +tion for tribes, which are concatenated with the financial statement +feature as attributes of nodes on the news-graph. +4.4 +Main Results +The main results of different models are presented in Table 3 and +the major findings are summarized as follows: +(1) We observe that our TH-GNN model significantly outper- +forms other competing methods. Its binary F1-score, with the re- +ported value of 63.2, is at least 5% higher than the tree-based model +and traditional DNN model, and AUC gets 6.3% higher at the same +time. Furthermore, TH-GNN is more advanced than the state-of- +the-art GNN-based methods, i.e., GCNII and DAGNN, with about +2.1% increased F1-score and 2.4% increased AUC. Besides, the lower +standard deviations of the results of TH-GNN indicate that our +proposed model is more robust with different dataset splits. +(2) The results on the upper and lower sides of the middle line in +Tab. 3 show the comparison of the two implementations of baselilne +models (see Sec. 4.3.2). Generally, the performance of base models +with tribe structure representations as additional input (the model +with ∗) are improved with varying degrees, which demonstrate +the effectiveness of tribe structure encoding. Besides, TH-GNN +outperforms the state-of-the-art GNN methods with tribe struc- +ture encoding as additional input. These improvements are mainly +brought by the TSE module of TH-GNN that considers encoding +of graph centrality and the end-to-end training strategy under the +guidance of the supervised loss and the contrastive loss jointly. +4.5 +Ablation Study +Then, we perform ablation studies to demonstrate the effectiveness +of every component in our model. As shown in Table 4, the binary +F-1 and AUC of all variants deteriorate to some extent. +4.5.1 +The effects of tribe-style graph. First, we demonstrate the +effects of the tribe-style graph by removing information of certain +type (i.e., the financial statements, global news-graph or local tribes) + +KDD ’22, August 14–18, 2022, Washington, DC, USA +Wendong Bi et al. +Table 4: Binary node classification results (%) of TH-GNN +and its variants (see Sec. 4.3.1) for ablation study. +Graph +binary F1-score +AUC +TH-GNN\Attribute +56.7\−6.5 +62.4\−10.5 +TH-GNN\GGRL +58.2\−5.0 +67.4\−6.1 +TH-GNN\TSE +60.2\−3.0 +69.5\−4.0 +TH-GNN\CL +62.0\−1.2 +71.9\−1.6 +TH-GNN\Emb +62.6\−0.6 +72.1\−1.4 +TH-GNN\Fusion +62.5\−0.7 +71.6\−1.9 +TH-GNN +63.2\0.0 +73.5\0.0 +respectively. TH-GNN\Attribute, as a variant without financial state- +ments as node attributes, obtains the worst performance among all +the variants with 6.5% decreased in F1-score and 10.5% decreased +in AUC. Moreover, the reasonable score shows that even without +the guidance of financial statements, only the graph structure infor- +mation can still guarantee a certain classification accuracy, which +demonstrates the effectiveness of our proposed tribe-style struc- +ture. Moreover, the performance degradation of TH-GNN\GGRL and +TH-GNN\TSE indicates that the global news-graph and local tribes +(investment-graphs) both benefit the downstream task. +4.5.2 +The effects of different model components. To further verify +the importance of different components in our model, another three +variants of TH-GNN are designed for ablation study. The perfor- +mance degradation of TH-GNN\CL, which removes the contrastive +loss term, shows that the contrastive loss 𝐿𝐶𝐿 plays an essential +role in learning the discriminative structural information of tribes. +Besides, the whole TH-GNN model yields a good performance boost +in comparison to TH-GNN\Emb, a variant without the structure em- +bedding in TSE and using randomly initialized attributes for nodes +on tribes, which indicates that the proposed structure embedding +is beneficial for modeling graphs without node attributes. From the +results of TH-GNN\Fusion, we find that the fusion module, which +integrates the information from financial statements and tribes +comprehensively, brings significant improvements. Overall, the +whole TH-GNN achieves the best result compared with all variants. +4.6 +Visualization of Learned Representations +We analyse the model’s interpretability by visualizing the learned +representations. We first extract the hidden outputs of the penulti- +mate layer for different GNN models. Next, we reduce the hidden +vectors to 2-dimensional vectors with T-SNE algorithm and plot +their the scatter plot . As shown in Fig. 5, the representations learned +by GCN, GAT and GraphSAGE have weak discrimination between +risky and normal nodes. Despite the representations learned by +GCNII and DAGNN have improved node classification ability, our +TH-GNN has the best discriminative power and provides inter- +pretability that other GNN models do not have. Fig. 5 (f) illustrates +the visualization of the representations learned by TH-GNN which +have evident outlier clusters, and most of points in these clusters +are risky companies. This unique phenomenon indicates that some +risky companies have similar local investment-patterns (i.e., the +example presented in Fig. 2 (a)), which is easily affected by their +(a) GCN +(b) GAT +(c) GraphSAGE +(d) GCNII +(e) DAGNN +(f) TH-GNN +Figure 5: T-SNE visualization of the representations learned +by models, where red dots represent samples of risky compa- +nies and blue dots represent samples of normal companies. +risky neighbors. And this observation further demonstrate the in- +dispensable effects of the Tribe Structure Encoder (TSE). +5 +RELATED WORK +In this paper, we mainly focus on using graph learning methods to +solve the company financial risk assessment problem. +Graph Representation Learning. +Graph Neural Networks +(GNNs) [13, 15, 30, 31, 33, 34] have shown excellent performance +on graph representation learning. Graph Convolution Network +(GCN) [15] was first proposed with aggregation-based graph con- +volution operation. After that, its improved models have been pro- +posed. GAT [30] introduced an attention mechanism to distinguish +the importance of neighbors. GIN [36] investigate the effect of differ- +ent aggregation functions for graph representation ability. Graph- +SAGE [13] proposed to use graph sampling for inductive learning +on graphs. Recently, some studies [5, 16] also focus on deeper GCN +with larger receptive filed. GCNII [5] is an extension of vanilla GCN +with initial residual and identity mapping. DAGNN [16] is a deeper +GNN model that decouples the transformation and aggregation of +message passing. However, most of them [12, 19, 20, 32, 33] are +designed without taking the unique domain knowledge into con- +sideration and are not aimed at tackling the company financial risk +assessment on financial networks. +Company Financial Risk Assessment. +Financial risk has +been a major concern for financial companies and governments, +and extensive works have been studied to assess company financial +risks. With the rapid development of machine learning methods in +recent years, researchers have developed machine learning-based +company financial risk prediction models using financial state- +ments or privately-owned data provided by financial institutions +[7, 10, 18, 38]. After the global financial crisis in 2008, the crisis +brought by the collapse of Lehman Brothers spread rapidly and +widely to related companies, and graph theory began to attract +the attention of researchers [1, 3, 17, 21, 23, 26–29, 35]. Recently, +some studies attempt to construct different financial graphs and +design domain-specific GNNs for company financial risk assess- +ment [8, 11, 37, 39]. For example, [8] proposed a High-order Graph +Attention Networks to assess risks for companies on guarantee + +60 +40 +20 +0 +20 +40 +100-75 +-50 +25 +25 +50 +75 +10075 +50 +25 +0 +25 +50 +75 +40 +20 +20 +4040 +20 +-20 +-40 +-60 +-100 +75 +50 +25 +0 +25 +5080 +60 +40 +20 +0 +20 +40 +60 +60 +40 +20 +20 +40 +60 +8060 +40 +20 +0 +20 +40 +60 +80 +80 +60 +40 +20 +0 +20 +60 +8060 +40 +20 +0 +20 +40 +60 +60 +40 +-20 +0 +20 +40 +60Company-as-Tribe: Company Financial Risk Assessment on Tribe-Style Graph with Hierarchical Graph Neural Networks +KDD ’22, August 14–18, 2022, Washington, DC, USA +loans networks considering higher-order neighbors. Yang et al. [37] +proposed a dynamic GNN for supply chain mining. Zheng et al. [39] +designed a heterogeneous graph attention network to predict the +bankruptcy risks of small and medium-sized companies. However, +all these works only consider one type of financial graph, and the +data is specific and private. Models trained on such domain-specific +data may have bias and have poor generalization ability. +6 +CONCLUSION AND FURTHER WORK +In this paper, we investigate the listed company’s financial data in +real words and find that it is far from sufficient to assess the risk +of listed companies solely through financial statements. To pro- +vide more comprehensive representations of companies, we design +a tribe-style network and propose TH-GNN for company finan- +cial risk assessment task. Experiments on a real-world large-scale +dataset show that our proposed model is effective in company finan- +cial risk assessment task and can provide brilliant interpretability of +results. Furthermore, much future work is focused on the scalability +of the method to investigate problems such as credit evaluation, +risk transmission and so on. In addition, TH-GNN provides valuable +tools to analyse the information of companies and their risks. As +future work, we will try more complicated GNN backbones rather +than simply GIN and GCN, and combine real-time information with +tribe-style graph to further improve the dynamic risk identification. +ACKNOWLEDGMENTS +This work was supported by the National Natural Science Founda- +tion of China (Grant No.61902380, No.61802370, No.U21B2046 and +No.U1911401) and the Beijing Nova Program (No. Z201100006820061). +REFERENCES +[1] Franklin Allen and Ana Babus. 2009. Networks in finance. The network challenge: +strategy, profit, and risk in an interlinked world 367 (2009). +[2] Phillip Bonacich. 1987. Power and centrality: A family of measures. 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In Pacific-Asia Conference on Knowledge Discovery and Data Mining. +Springer, 140–151. + diff --git a/PNFRT4oBgHgl3EQfIjco/content/tmp_files/load_file.txt b/PNFRT4oBgHgl3EQfIjco/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8df56753986986bc959bd386e38f1f753d40259a --- /dev/null +++ b/PNFRT4oBgHgl3EQfIjco/content/tmp_files/load_file.txt @@ -0,0 +1,962 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf,len=961 +page_content='Company-as-Tribe: Company Financial Risk Assessment on Tribe-Style Graph with Hierarchical Graph Neural Networks Wendong Bi Institute of Computing Technology, University of Chinese Academy of Sciences Beijing, China biwendong20g@ict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='cn Bingbing Xu∗ Institute of Computing Technology, Chinese Academy of Sciences Beijing, China xubingbing@ict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='cn Xiaoqian Sun∗ Institute of Computing Technology, Chinese Academy of Sciences Beijing, China sunxiaoqian@ict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='cn Zidong Wang Institute of Computing Technology, Chinese Academy of Sciences Beijing, China wangzidong@ict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='cn Huawei Shen Institute of Computing Technology, Chinese Academy of Sciences Beijing, China shenhuawei@ict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='cn Xueqi Cheng∗ Institute of Computing Technology, Chinese Academy of Sciences Beijing, China cxq@ict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='cn ABSTRACT Company financial risk is ubiquitous and early risk assessment for listed companies can avoid considerable losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Traditional meth- ods mainly focus on the financial statements of companies and lack the complex relationships among them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' However, the finan- cial statements are often biased and lagged, making it difficult to identify risks accurately and timely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' To address the challenges, we redefine the problem as company financial risk assessment on tribe-style graph by taking each listed company and its share- holders as a tribe and leveraging financial news to build inter-tribe connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Such tribe-style graphs present different patterns to distinguish risky companies from normal ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' However, most nodes in the tribe-style graph lack attributes, making it difficult to directly adopt existing graph learning methods (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=', Graph Neural Networks(GNNs)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' In this paper, we propose a novel Hierarchical Graph Neural Network (TH-GNN) for Tribe-style graphs via two levels, with the first level to encode the structure pattern of the tribes with contrastive learning, and the second level to diffuse information based on the inter-tribe relations, achieving effective and efficient risk assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Extensive experiments on the real- world company dataset show that our method achieves significant improvements on financial risk assessment over previous compet- ing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Also, the extensive ablation studies and visualization comprehensively show the effectiveness of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' CCS CONCEPTS Computing methodologies → Neural networks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' • Informa- tion systems → Social networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' ∗Corresponding authors This work is licensed under a Creative Commons Attribution International 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='0 License.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' KDD ’22, August 14–18, 2022, Washington, DC, USA © 2022 Copyright held by the owner/author(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' ACM ISBN 978-1-4503-9385-0/22/08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='1145/3534678.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='3539129 KEYWORDS company financial risk assessment, tribe-style graph, graph neural network ACM Reference Format: Wendong Bi, Bingbing Xu, Xiaoqian Sun, Zidong Wang, Huawei Shen, and Xueqi Cheng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Company-as-Tribe: Company Financial Risk As- sessment on Tribe-Style Graph with Hierarchical Graph Neural Networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' In Proceedings of the 28th ACM SIGKDD Conference on Knowledge Discovery and Data Mining (KDD ’22), August 14–18, 2022, Washington, DC, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' ACM, New York, NY, USA, 9 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='1145/3534678.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='3539129 1 INTRODUCTION Company financial risk is ubiquitous in the real-world financial market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Early assessment of risks for the listed company can provide decision support for company managers and investment institu- tions, thereby avoiding considerable losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Traditional methods [7, 18], such as financial probability meth- ods, decision tree methods, and Deep Neural Networks (DNNs), treat each company individually and solely leverage the financial statements to assess risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' However, financial statements are often biased and lagged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' As Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' 1 (a) shows, most companies beautify their published financial data, and some companies even commit financial fraud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Besides, these traditional methods ignore the inter- actions among companies, which is critical because risks can passed between companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' The above limitations make the traditional methods difficult to identify risks accurately and early.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' To effectively assess company financial risks, we found there exist two other types of valuable information: 1) the investment- graph of listed companies, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=', CATL1 has more than 200 investors (companies or individuals), which form an investment-graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' As Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' 1 (b) shows, risky and normal companies often have different investment patterns;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' 2) The news-graph among listed companies, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=', there exists an edge between two companies if they concur- rent in at least one piece of news.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' As Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' 1 (b) shows, two listed companies connected usually have strong correlations, and risks can spread over this graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Superior to other information, financial 1CATL (Contemporary Amperex Technology Co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=', Limited) is a typical listed company in China, which is a global leader of new energy innovative technologies and committed to providing premier solutions and services for new energy applications worldwide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='13492v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='LG] 31 Jan 2023 BYKDD ’22, August 14–18, 2022, Washington, DC, USA Wendong Bi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Tribe A Normal central company Risky central company Declining performance Rising performance Company shareholder Individual shareholder (b) Tribe-style (a) Individual-style Node Attribute ✓ ✘ Normal Risky B A C Risky Fraud News relationship Financial report Investment-Graph Tribe B Tribe C Figure 1: Company financial risk assessment on individual-style vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' tribe-style graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Each company with its investment-graph in a dotted oval box can be seen as a tribe, and they are further connected by the global relationship (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=', news relationship).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' news is objective and can timely reflect risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Extensive statistical analyses are provided in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='1 to demonstrate the benefit of these data for distinguishing risky companies from the normal ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Based on the above findings, we redefine the problem as com- pany financial risk assessment on tribe-style graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' As illus- trated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' 1 (b), we take the investment-graph consisting of a central listed company and its shareholders as a tribe, and leverage the news-graph to construct inter-tribe edges, the financial state- ments of listed companies are regarded as initial attributes of tribes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' However, it is challenging to directly adopt existing graph methods (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=', Graph Neural Networks (GNNs)) to such tribe-style graphs due to the following serious issues: 1) only the listed companies in a tribe have attributes, and other individuals or companies have no disclosure obligation and therefore do not have attributes, mak- ing it difficult to conduct message passing in GNNs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' 2) The whole tribe-style graph including both intra-tribe and inter-tribe relation- ships is large-scale and contains millions of edges, which makes the GNNs inefficient, and traditional node sampling techniques [4, 13, 14] cause the loss of information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' In this paper, we propose a novel Hierarchical Graph Neural Network for the financial risk assessment on the tribe-style graph, namely TH-GNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Specifically, for the first challenge that the indi- viduals and non-central companies in a tribe have no attributes, we find the structure patterns can reflect the company’s risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' There- fore, we design a tribe structure encoder (TSE) based on contrastive learning that learns structural patterns for each tribe (including the scale of the tribe and its investment structure, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=') without relying on node attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' For the second challenge, although the whole graph is huge, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' 1 (b) shows an important property of the tribe-style graph that the intra-tribe connections (investment- graph) are dense while the inter-tribal connections (news-graph) are sparse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Inspired by this property, TH-GNN encodes the tribe- style graph through a hierarchical manner, with the first level to encode tribes defined by the investment graphs via the tribe struc- ture encoder and the second level to diffuse the information among tribes based on the news-graph and learn the global representa- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Unlike the traditional GNNs that diffuse information over edges on the whole graph, TH-GNN converts a tribe-style graph into parallelly computable local graphs and a smaller global graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Extensive experiments on the real-world dataset for company finan- cial risk assessment show that our approach achieves significant improvement over previous competing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Meanwhile, the ablation studies and visualizations also comprehensively show the effectiveness of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' The main contributions of this work are summarized as follows: (1) We redefine the previous individual risk assessment problem as company financial risk assessment on tribe-style graph and further design a tribe-style graph consisting of financial statements, investment-graphs, and news-graphs rather than solely utilizing financial statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' (2) We propose a novel Hierarchical Graph Neural Network named TH-GNN to model the tribe-style graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' To the best of our knowledge, this is the first graph representation learning method for company financial risk assessment on the tribe- style graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' (3) We conduct extensive experiments on a real-world company graph dataset with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='88 million nodes and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='31 million edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' The results demonstrate the superiority of the proposed model over state-of-the-art methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' The code is avaliable 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' 2Our source code is available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='com/wendongbi/TH-GNN 晶團1晶團用 用用由晶團用晶團晶團晶團IEWS::用Company-as-Tribe: Company Financial Risk Assessment on Tribe-Style Graph with Hierarchical Graph Neural Networks KDD ’22, August 14–18, 2022, Washington, DC, USA 2 PRELIMINARY In this section, we present the detailed statistical analysis of tribe- style graph and the formalized definition of company financial risk assessment on tribe-style graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='1 Data Analysis As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' 1 (b), the tribe-style graph consists of investment- graph (tribe) and news-graph, where the investment-graph presents the intra-tribe connections and the news-graph presents the inter- tribe connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' We then analyze the investment-graphs and news-graph comprehensively to verify their benefits for financial risk assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='1 investment-graph analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' An investment-graph usually con- sists of one central listed company and others (companies or in- dividuals) which have investment relationships with the central company.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Only the listed company publishes its financial state- ments as attributes, and the other nodes do not have attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Therefore we mainly focus on the structure patterns of tribes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Then we first give a case study of investment-graphs and then present the statistical analysis on all listed companies’ investment-graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' As (a) A risky company example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' (b) A normal company example Figure 2: Examples of investment-graphs3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' (The bigger or- ange point denotes listed company, blue points denote un- listed companies and red points denote individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' And the edges in the graphs represent investment relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=') illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' 2, the investment-graphs for risky company and normal company show different patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' The investment-graph of the risky company shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' 2 (a) is more similar to a star-like graph, where the single listed company can be viewed as the central node, and its neighboring nodes (investors) consist of more indi- viduals or companies that tend to be disconnected from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Different from the risky company, the neighboring nodes of the normal company in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' 2 (b) are often popular companies which have more neighbors (investors), and there exist dense connections among these neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Such a pattern with more reliable investors tends to be more stable, and thus the central listed company is less likely to have financial risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' This motivates us to leverage the investment pattern of listed company to identify financial risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' To further verify this finding, we conduct centrality analysis for the investment-graphs of all listed companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Five typical metrics are used [2, 22], including degree centrality, eigenvector centrality, clustering centrality, number of bridge, and the central node degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' We calculate the above metrics on each investment-graph and then 3These two examples are from two real-world companies named LongYuan Construc- tion and ShangHai Dragon Corporation repectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' take the average of risky and normal companies respectively to reflect the differences of centrality between the two classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Table 1: Statistical centrality analysis of the investment- graphs Statistical metric risky company normal company Degree centrality 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='264 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='224 Eigenvector centrality 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='4161 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='3604 Clustering coefficient 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='2102 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='1907 Number of bridge (avg) 125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='4 112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='3 Central node degree (avg) 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='43 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='71 The results of the centrality analysis are presented in Table 1, which demonstrate that the investment-graphs of risky companies usually have larger graph centrality compared with that of normal companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' And this pattern motivates us to take the structure encoding of investment-graph into consideration for financial risk assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' We consider the structural pattern of a investment- graph as a tribe individually, where the nodes within a tribe are centered on the centrally-located listed company.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' We further ex- plain the benefits of tribe-style graphs in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='2 news-graph analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Besides the investment relationship in tribes, we also use financial news to model the interactions between different tribes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' The financial news has evident timeliness and ob- jective authenticity, reporting company financial risks promptly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Specifically, different companies may appear in the same news, reflecting strong correlation among them, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=', news reported that Company A and Company B jointly invested in a failed project, reflecting that they may have potential financial risks at the same time (news used for graph construction in this paper all describes similarity among companies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Then for companies co-existed in one piece of news, we connect them to construct news-graphs, indicating the risky associations among them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' 3, we also conduct statistical analysis to validate the benefits of the news-graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' We calculate the proportion of risky companies in neighbors of each node for risky companies and normal companies respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' For example, the rightmost column in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' 3 indicates there are nearly 30% of risky companies (the red bar) with more than 80% of neighbors at-risk, while no more than 10% of normal companies (the blue bar) with more than 80% of neighbors at-risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' The results show that the probability of risky companies with high-proportion risky neighbors is much larger than that of normal companies, that is, the risky nodes in the news-graph have higher-proportional-risk neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' This finding also suggests that news-graph can benefit the company financial risk assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' As a result, we use the news-graph to construct inter-tribe edges, modeling the relations among listed companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='2 Problem Definition Our company financial risk assessment problem is defined on the tribe-style graph, which consists of a set of tribes and a global news-graph connecting different tribes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' For company financial risk assessment, we need to classify each company into binary classes: KDD ’22, August 14–18, 2022, Washington, DC, USA Wendong Bi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='0~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='2~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='4~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='6~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='8~1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='0 Proportion of risky neighbors 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='30 Proportion of nodes (risky/normal) Company Type Normal Company Risky Company Figure 3: Analysis on proportion of risky neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='The x- axis is the proportion of risky companies in all neighbors of each node, and the y-axis is the corresponding proportion of nodes in a certain node type (risky/normal) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Note that we ignore the 0-degree nodes when calculating these results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' risky or normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' We give a formal definition of the task as fol- lows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Let GT = {G𝑔𝑙𝑜𝑏𝑎𝑙, G𝑡𝑟𝑖𝑏𝑒} denote a tribe-style graph, where G𝑔𝑙𝑜𝑏𝑎𝑙 = (𝑉𝐺, 𝐸𝐺,𝑋𝐺) is the global graph which taking all the listed companies as nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Considering that each listed company is the center of a tribe, we name the listed company node as the central node for shortly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' 𝑉𝐺 denotes the central node set, 𝑁𝑐 is the number of central nodes, and 𝐸𝐺 is the set of edges connecting central nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' 𝑋𝐺 ∈ R𝑁 𝑐×𝐷 is the attribute matrix, and the i-th row of 𝑋𝐺 denoted by 𝑋𝐺 𝑖 is the 𝐷-dimensional attribute vector of central node 𝑣𝐺 𝑖 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' For each central node 𝑣𝐺 𝑖 ∈ 𝑉𝐺, there exists a tribe 𝑔𝑇 𝑖 ∈ G𝑡𝑟𝑖𝑏𝑒 corresponding to 𝑣𝐺 𝑖 , where 𝑔𝑇 𝑖 = (𝑉𝑇 𝑖 , 𝐸𝑇 𝑖 ) and G𝑡𝑟𝑖𝑏𝑒 = {𝑔𝑇 𝑖 | 𝑖 = 1 · · · 𝑁𝑐}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' 𝑉𝑇 𝑖 and 𝐸𝑇 𝑖 are the node set and edge set of tribe 𝑔𝑇 𝑖 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Each tribe have one central node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Each central node 𝑣𝐺 𝑖 is associated with a binary label 𝑦𝑖 = {0, 1}, using 1 for risky companies and 0 for normal companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Then the listed company’s financial risks assessment problem on tribe-style graph can be de- scribed as: given a tribe-style graph GT, and the goal is to classify each listed company node into binary classes: risky or normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' 3 METHODS In this section, we first explain how and why we design the tribe- style graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Then we introduce the proposed TH-GNN model for company financial risk assessment on tribe-style graphs in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='1 Tribe-style Graph Construction 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='1 How to construct the tribe-style graph?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' 1, the tribe-style graph consists of company financial state- ments, investment-graphs, and financial news.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Specifically, listed companies are viewed as central target nodes in the tribe-style graph, and the investment-graph of each listed company is viewed as a tribe (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=', a super node on the tribe-style graph), where only the central listed company nodes have attributes extracted from their financial statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' The global graph connecting central com- panies is constructed by financial news, where two central listed companies connected if they co-existed in at least one piece of financial news.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Note that we only use news that describes the risky linkages among listed companies in China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' If there are multiple companies appearing in the same news, we connect all possible pairs of companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' More details about the dataset information and preprocessing are presented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='1 and Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='2 Why we construct the tribe-style graph?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' We summarize the following advantages of constructing a tribe-style graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' (1) Based on the analysis in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='1, it is the structural pattern of tribes that benefits the identification of risky companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' (2) Considering that only the central node of a tribe has attributes, the central and non-central nodes should be treated separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Therefore we make the tribe-style graph a hierarchical graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' (3) To improve model efficiency, we treat each tribe independently and obtain the representation of each tribe by graph pooling(regardless of overlap among tribes) instead of merging them into one graph, which actually truncates the original large-scale graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='2 Model Overview We give an overview of TH-GNN in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' 4, TH-GNN includes two main components, including the Tribe Structure Encoder (TSE) and Global Graph Representation Learning (GGRL) module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' TH- GNN encodes the tribe-style graph in bottom-up order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' TH-GNN first learns the structural representation for each tribe with the TSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Then the learned structural representation of tribes and the financial statements are fused into the embedding of central node (listed company) by an attention-based fusion module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Next, the embedding is diffused over the global news-graph to learn the final representation of central nodes for financial risks assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='3 Tribe Structure Encoder (TSE) The Tribe Structure Encoder (TSE) is used to learn the structural representation for each tribe based on contrastive learning, includ- ing a structure embedding module and a graph encoder module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Considering that nodes in the tribe have no attributes, we first initialize the node attributes according to their position in the tribe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Then we transform the structural attributes into learnable embed- ding with a structure embedding module for each node in the tribe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Finally, with the tribe (investment graph) and the node structure embedding, a GIN model is used to get the representation of tribes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Inspired by the importance of centrality patterns in our scenario discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='1, we also consider the encoding of centrality when designing the TSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' For a tribe without node attributes, we first assign each node with a structure embedding as its initial attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Specifically, each node 𝑣𝑇 𝑗 ∈ 𝑉𝑇 𝑖 on the investment-graph 𝑔𝑇 𝑖 has the three properties: (1) node degree (in-degree denoted by 𝑑𝑒𝑔+ 𝑗 and out-degree denoted by 𝑑𝑒𝑔− 𝑗 for a directed graph);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' (2) node type denoted by 𝜙𝑗 (listed company, unlisted company or human);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' (3) distance of shortest path (SPD) to the central node denoted by 𝑆𝑃𝐷𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' With these structure attributes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' we further transform them into learnable embedding by an Embedding Layer: 𝐸𝑚𝑏(𝑣𝑇 𝑗 ) = 𝐸𝑚𝑏𝑒𝑑𝑑𝑖𝑛𝑔_𝐿𝑎𝑦𝑒𝑟 � 𝑑𝑒𝑔+ 𝑗 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='𝑑𝑒𝑔− 𝑗 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='𝜙𝑗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='𝑆𝑃𝐷𝑗 � (1) Following [9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' 25],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' we also use the top Laplacian eigenvector with the largest eigenvalue of each tribe as the nodes positional encoding Company-as-Tribe: Company Financial Risk Assessment on Tribe-Style Graph with Hierarchical Graph Neural Networks KDD ’22,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' August 14–18,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' 2022,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Washington,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' DC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' USA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='Local Message Passing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='TSE: Tribe ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='Structure Encoder ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='(1) Position ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='Embedding ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='… ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='… ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='… ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='(3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='Node Type ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='Embedding ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='(2) Degree ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='Embedding ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='(4) Shortest Path ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='Distance Embedding ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='Tribe Structure ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='Embedding ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='TSE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='Tribe Structure ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='Embedding ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='Financial Report ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='Tribe-style ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='Graph ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='Attention-based ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='Fusion Module ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='𝒉𝟏 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='𝑳 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='𝒉𝟐 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='𝑳 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='Contrastive Loss ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='𝑳𝑪𝑳 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='Central Node ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='Embedding ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='Global Graph ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='Representation Learning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='Normal ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='Risky ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='𝑳𝑩𝑪𝑬 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='Figure 4: Overview of our proposed TH-GNN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' TH-GNN includes two main components, including the Tribe Structure Encoder (TSE) and Global Graph Represenation Learning (GGRL) module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' besides the learnable structure embedding, and the 𝑗-th value of the top Laplacian eigenvector is exactly the eigenvector centrality [2] of node 𝑣𝑇 𝑗 ∈ 𝑉𝑇 𝑖 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Then the final structure mebedding for each node on the tribe can be represented as: 𝑍 (𝑔𝑇 𝑖 ) = � 𝐸𝑚𝑏(𝑣𝑇 𝑗 ) || 𝑢0(𝑔𝑇 𝑖 ) � (2) where 𝑢0(𝑔𝑇 𝑖 ) is the top Laplacian eigenvector of 𝑔𝑇 𝑖 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Next, we use the structure embedding (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' 2) for local message passing on tribes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' In this paper, we use GIN with SUM pooling as the graph encoder for tribes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' The GIN updates the node representations by: ℎ(𝑙) 𝑣𝑇 𝑖 = 𝑀𝐿𝑃 (𝑙) ��� � (1 + 𝜖 (𝑙)) · ℎ(𝑙−1) 𝑣𝑇 𝑖 + ∑︁ 𝑣𝑇 𝑗 ∈𝒩(𝑣𝑇 𝑖 ) ℎ(𝑙−1) 𝑣𝑇 𝑗 ��� � (3) where 𝜖 is a learnabel parameter, ℎ(𝑙) 𝑣𝑇 𝑖 is the learned representation of node 𝑣𝑇 𝑖 at the 𝑙-th layer of GIN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Then we get the global graph representation by performing SUM Pooling for all nodes in each tribe and taking an average over all layers of the model: ℎ𝑔𝑇 𝑖 = 1 𝐿 + 1 𝐿 ∑︁ 𝑙=0 SUM � {ℎ(𝐿) 𝑣𝑇 𝑖 |𝑣𝑇 𝑖 ∈ 𝑔𝑇 𝑖 } � (4) Then the final tribe representation is 𝐻G𝑡𝑟𝑖𝑏𝑒 = {ℎ𝑔𝑇 𝑖 |𝑔𝑇 𝑖 ∈ G𝑡𝑟𝑖𝑏𝑒}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Considering the high cost of data labeling, we introduce con- trastive learning to guide the training of TSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Specifically, we design a graph instance discrimination tasks and use InfoNCE [24] as the objective to optimize the model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' The contrastive task treats each tribe as a distinct class and leans to discriminate between different tribes through a self-supervised way, and help the TSE module learn the structural dissimilarity between different tribes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Specifically, we first prepare one positive sample pair and 𝑁 − 1 negative sample pairs for each training batch with N samples and the tribes of each batch are fed into the TSE twice to obtain the query representation 𝐻𝑞 G𝑡𝑟𝑖𝑏𝑒 and key representation 𝐻𝑘 G𝑡𝑟𝑖𝑏𝑒 of tribes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Due to the randomness of dropout, there is a certain dif- ference between the obtained two sets of tribe representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Then we use the query and key representations of the same tribe as positive pairs denoted by {⟨𝑞𝑖,𝑘+ 𝑖 ⟩, 𝑔𝑇 𝑖 ∈ G𝑡𝑟𝑖𝑏𝑒}, and use the representations of different tribes to construct negative pairs de- noted by {⟨𝑞𝑖,𝑘− 𝑗 ⟩, 𝑗 ≠ 𝑖 and 𝑔𝑇 𝑖 ,𝑔𝑇 𝑗 ∈ G𝑡𝑟𝑖𝑏𝑒}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Then we compute InfoNCE Loss as a regular term besides the supervised classification loss to optimize parameters of the Tribe Structure Encoder module: L𝐶𝐿 = − 1 𝑁 𝑁 ∑︁ 𝑖=1 𝑙𝑜𝑔 𝑞𝑇 𝑖 · 𝑘+ 𝑖 𝑞𝑇 𝑖 · 𝑘+ 𝑖 + �𝑁 −1 𝑗=1 𝑞𝑇 𝑖 · 𝑘− 𝑗 (5) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='4 Global-Graph Representation Learning (GGRL) With TSE, we obtain the representations of tribes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' And then for each central company, its node features come from two parts: the tribe representation and financial statements, which can be represented as {(ℎ𝑔𝑇 𝑖 ,𝑋𝐺 𝑖 ) | 𝑣𝐺 𝑖 ∈ 𝑉𝐺 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' We further use an attention-based fusion module to integrate the tribe representationsℎ𝑔𝑇 𝑖 and financial state- ments feature 𝑋𝐺 𝑖 into one central node embedding ℎ(0) 𝑖 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Finally, the fused central node embedding is used for message passing on the global news-graph to learn the final representations of central companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' To better integrate the node features from financial statements and tribes on the global graph, we design an attention-based fusion module to fuse the two features to common space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' We first calculate 晶晶田KDD ’22, August 14–18, 2022, Washington, DC, USA Wendong Bi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' the weights for the financial statements and tribe representations: \uf8f1\uf8f4\uf8f4\uf8f2 \uf8f4\uf8f4\uf8f3 𝑒𝑔 𝑖 = 𝜎([ℎ𝑔𝑇 𝑖 ·𝑊 𝑔||𝑋𝐺 𝑖 ·𝑊 𝑋 ] · 𝑎𝑇 𝑔 ), 𝑎𝑔 ∈ R1×2𝐷 𝑒𝑋 𝑖 = 𝜎([ℎ𝑔𝑇 𝑖 ·𝑊 𝑔||𝑋𝐺 𝑖 ·𝑊 𝑋 ] · 𝑎𝑇 𝑥 ), 𝑎𝑥 ∈ R1×2𝐷 (6) where 𝜎 is the LeakyReLU activation, 𝑊 𝑔 and 𝑊 𝑋 are transforma- tion matrix to project ℎ𝑔𝑇 𝑖 and 𝑋𝐺 𝑖 into common hidden dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Then we normalize them by the softmax function: 𝛼𝑔 𝑖 = 𝑒𝑥𝑝(𝑒𝑔 𝑖 ) 𝑒𝑥𝑝(𝑒𝑔 𝑖 ) + 𝑒𝑥𝑝(𝑒𝑋 𝑖 ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' 𝛼𝑋 𝑖 = 𝑒𝑥𝑝(𝑒𝑋 𝑖 ) 𝑒𝑥𝑝(𝑒𝑔 𝑖 ) + 𝑒𝑥𝑝(𝑒𝑋 𝑖 ) (7) Next,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' we can get the fused central node embedding: ℎ(0) 𝑖 = 𝛼𝑔 𝑖 · (ℎ𝑔𝑇 𝑖 ·𝑊 𝑔) + 𝛼𝑋 𝑖 · (𝑋𝐺 𝑖 ·𝑊 𝑋 ) (8) Then we use the fused central node embedding as node features and further perform message passing on the global news-graph to learn the final representations of each central node: ℎ(𝑙) 𝑖 = 𝜎 ��� � � ∑︁ 𝑗 ∈{𝒩(𝑣𝐺 𝑖 )∪𝑣𝐺 𝑖 } 1 𝑑𝑖 ℎ(𝑙−1) 𝑗 � ·𝑊 𝑙��� � (9) where 𝑑𝑖 is the in-degree of 𝑣𝐺 𝑖 (including the self-loop).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' And the learned representations can be further used for the company finan- cial risk assessment task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='5 Model Optimization Methods After aggregating the information from neighbors on the graph, the obtained representation ˆℎ(𝐿) 𝑖 is fed into a final fully connected neural network with a sigmoid activation function, as follows: 𝑝𝑖 = 𝑆𝑖𝑔𝑚𝑜𝑖𝑑(ℎ(𝐿) 𝑖 𝑊𝑝 + 𝑏𝑝) (10) where 𝑝𝑖 is the probability of company node 𝑣𝑖 suffering risks in the further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Then we compute binary cross entropy (BCE) loss to utilize the supervised information of labels: L𝐵𝐶𝐸 = 1 𝑁 𝑁 ∑︁ 𝑖=1 𝑦𝑖 · log ˆ𝑦𝑖 + (1 − 𝑦𝑖) · log(1 − ˆ𝑦𝑖) (11) Then, the final loss function is composed of L𝐵𝐶𝐸 and L𝐶𝐿 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' 5): L = L𝐵𝐶𝐸 + 𝛼 · L𝐶𝐿 (12) where 𝛼 is a hyper-parameter to control the weight of L𝐶𝐿 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' 4 EXPERIMENTS In this section, we compare TH-GNN with other state-of-the-art methods on a real-world dataset for company risk assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='1 Dataset The company dataset used in this paper comes from the real-world data of 4040 listed companies in China from 2019 to 2020, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=', the listed company’s financial statements, investment-graph, and finan- cial news related to these companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' The financial statements and the company’s investment-graph data are provided by TianYanCha (an authority enterprise credit institute for company information inquiry in China).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' The annual financial statements reflect a listed company’s industry information and its financial and business sit- uation in a year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' The investment-graph of a company describes Table 2: Information of the Hierarchical Graph Graph #Nodes #Edges Whole graph GT 879252 1311364 News-graph G𝑔𝑙𝑜𝑏𝑎𝑙 4040 16330 Investment-graphs (total) G𝑙𝑜𝑐𝑎𝑙 879252 1295034 Investment-graphs (average) G𝑙𝑜𝑐𝑎𝑙 217.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='6 320.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='5 the relationship between the central company and its shareholders, including other companies and humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' The financial news data are provided by Wind ( an authority China finance database), and these news are obtained from more than 800 authority news websites in China, which have extremely wide coverage and timeliness to capture the risk information of companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Note that the financial news used in this paper, which has already been preprocessed by Wind, all describes the risk linkages among companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Then we construct the tribe-style graph and more specific information of this graph is illustrated in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Based on the real-world risk events of companies happened in 2020 provided by Wind, the posi- tive (risky) and negative (normal) labels can be naturally generated, and we use all companies marked as high-risk as positive samples and others as negative samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' To prevent information leakage, the part of the dataset in 2019, including financial and operating information, investment-graph and news-graph, is used as training data .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' There are 1698 positive samples and 2342 negative samples among all listed companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='2 Experimental Setup We conduct experiments on the real-world dataset with different configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' We design experiments with different training ra- tios (percent of nodes in the training set) ranging from 20% to 40%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' And for each training ratio, we use three different random parti- tions of the dataset and ten random seeds for the model parameter initialization, a total of 30 trials for each model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' For all attributes of the dataset used in this paper, we preprocess the categorical or discrete attributes into one-hot vectors, and then we use binning methods to divide continuous numerical attributes into 50 bins and use the index of bin as their feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' For fairness, we perform a hyper-parameter search for all models, and the size of searching space for each model is the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' The hidden dimension of all models are searched in {32, 64, 128} and we choose the number of training epoch from {100, 200, 300}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' We use the Adam optimizer for all experiments and the learning rate is searched in {1e-2, 1e-3, 1e-4}, weight decay is searched in {1e-4, 1e-3, 5e-3}, and 𝛼 (the coefficient of L𝐶𝐿) is searched in {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='0} for all experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' The number of layers for GNN models, including TH-GNN and other baseline GNN models except for GCNII and DAGNN, are set to be two layers in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' The number of layers for GCNII and DAGNN, which are designed with deeper depth, are set to 64 and 20 respectively according to their papers [5, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' All models used in this paper were trained on Nvidia Tesla V100 (32G) GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='3 Compared Methods 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='1 Baseline methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' We compare our model with two classical machine learning models (XGBoost [6], DNN), three baseline GNN Company-as-Tribe: Company Financial Risk Assessment on Tribe-Style Graph with Hierarchical Graph Neural Networks KDD ’22, August 14–18, 2022, Washington, DC, USA Table 3: Binary node classification results (%) of models on our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' The model with ∗ means using the concatenation of tribe structural embedding and financial statements data as inputs (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Otherwise, only financial statement data is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Evaluation Metric binary F1-score AUC score Training ratio 60% 40% 20% 60% 40% 20% XGBoost 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='4 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='63 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='1 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='55 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='9 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='05 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='3 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='23 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='3 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='07 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='7 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='23 DNN 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='8 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='77 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='9 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='56 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='5 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='89 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='1 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='77 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='6 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='67 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='3 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='86 GCN 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='9 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='54 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='3 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='74 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='8 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='87 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='1 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='64 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='2 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='81 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='8 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='63 GAT 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='6 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='86 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='2 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='75 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='2 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='96 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='6 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='11 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='2 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='52 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='5 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='21 GraphSAGE 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='4 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='36 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='5 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='36 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='5 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='38 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='48 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='84 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='14 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='25 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='6 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='78 GCNII 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='9 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='04 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='7 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='94 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='1 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='88 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='6 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='81 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='1 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='72 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='8 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='01 DAGNN 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='6 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='96 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='7 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='30 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='0 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='13 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='6 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='17 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='4 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='17 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='7 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='17 XGBoost∗ 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='2 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='19 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='8 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='05 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='2 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='92 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='2 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='56 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='9 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='25 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='7 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='33 DNN∗ 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='2 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='11 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='8 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='32 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='1 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='13 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='3 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='66 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='4 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='88 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='3 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='15 GCN∗ 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='2 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='30 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='6 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='01 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='3 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='01 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='95 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='8 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='31 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='1 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='55 GAT∗ 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='4 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='01 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='4 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='15 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='8 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='34 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='8 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='26 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='0 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='34 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='1 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='12 GraphSAGE∗ 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='7 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='08 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='3 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='0 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='3 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='76 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='6 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='15 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='8 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='04 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='2 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='23 GCNII∗ 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='5 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='23 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='6 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='30 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='0 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='17 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='9 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='73 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='6 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='85 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='9 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='09 DAGNN∗ 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='1 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='06 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='9 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='12 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='9 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='69 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='1 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='74 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='6 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='38 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='84 TH-GNN 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='75 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='95 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='2 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='13 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='54 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='54 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='54 models (GCN [15], GAT [30], GraphSAGE [13]), and two state-of- the-art GNN models (GCNII [5], DAGNN [16]) to demonstrate the superiority of our TH-GNN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Furthermore, six variants of TH- GNN are designed for ablation studies: TH-GNN\\Attribute indicates removing node attributes (financial statements) from TH-GNN and only using the graph structure information;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' TH-GNN\\TSE indicates removing Tribe Structure Encoder from TH-GNN, without lever- aging tribes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' TH-GNN\\GGRL indicates removing the GGRL module from TH-GNN, without leveraging the news-graph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' TH-GNN\\Fusion indicates removing attention-based fusion module from TH-GNN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' TH-GNN\\Emb indicates removing the structure embedding used in TSE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' TH-GNN\\CL indicates removing the contrastive loss term used to optimize the Tribe Structure Encoder from TH-GNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' In our experiments, we select two widely used metrics as performance measurement, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=', AUC (the area enclosed by the coordinate axis under the ROC curve) and F1-score (the harmonic average of the precision and recall) on the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='2 Two implementations for base GNN models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Note that GNN models except TH-GNN cannot directly encode the tribe-style graph with hierarchical structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' For fair comparisons, we design two implementations for each baseline GNN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' (1) A basic implementation is to directly train the vanilla base- line GNNs on the news-graph, because nodes except the central node (listed company) in a tribe do not have attributes (financial statements), which are necessary for vanilla GNN models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' (2) A two-stage implementation is to learn structure repre- sentation of tribes (investment-graphs) first and then train baseline GNNs on the news-graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' In this paper, we use GCC [25] , a graph contrastive learning based method, to learn structure representa- tion for tribes, which are concatenated with the financial statement feature as attributes of nodes on the news-graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='4 Main Results The main results of different models are presented in Table 3 and the major findings are summarized as follows: (1) We observe that our TH-GNN model significantly outper- forms other competing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Its binary F1-score, with the re- ported value of 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='2, is at least 5% higher than the tree-based model and traditional DNN model, and AUC gets 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='3% higher at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Furthermore, TH-GNN is more advanced than the state-of- the-art GNN-based methods, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=', GCNII and DAGNN, with about 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='1% increased F1-score and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='4% increased AUC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Besides, the lower standard deviations of the results of TH-GNN indicate that our proposed model is more robust with different dataset splits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' (2) The results on the upper and lower sides of the middle line in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' 3 show the comparison of the two implementations of baselilne models (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Generally, the performance of base models with tribe structure representations as additional input (the model with ∗) are improved with varying degrees, which demonstrate the effectiveness of tribe structure encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Besides, TH-GNN outperforms the state-of-the-art GNN methods with tribe struc- ture encoding as additional input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' These improvements are mainly brought by the TSE module of TH-GNN that considers encoding of graph centrality and the end-to-end training strategy under the guidance of the supervised loss and the contrastive loss jointly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='5 Ablation Study Then, we perform ablation studies to demonstrate the effectiveness of every component in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' As shown in Table 4, the binary F-1 and AUC of all variants deteriorate to some extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='1 The effects of tribe-style graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' First, we demonstrate the effects of the tribe-style graph by removing information of certain type (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=', the financial statements, global news-graph or local tribes) KDD ’22, August 14–18, 2022, Washington, DC, USA Wendong Bi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Table 4: Binary node classification results (%) of TH-GNN and its variants (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='1) for ablation study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Graph binary F1-score AUC TH-GNN\\Attribute 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='7\\−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='5 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='4\\−10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='5 TH-GNN\\GGRL 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='2\\−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='0 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='4\\−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='1 TH-GNN\\TSE 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='2\\−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='0 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='5\\−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='0 TH-GNN\\CL 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='0\\−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='2 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='9\\−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='6 TH-GNN\\Emb 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='6\\−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='6 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='1\\−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='4 TH-GNN\\Fusion 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='5\\−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='7 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='6\\−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='9 TH-GNN 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='2\\0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='0 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='5\\0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='0 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' TH-GNN\\Attribute, as a variant without financial state- ments as node attributes, obtains the worst performance among all the variants with 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='5% decreased in F1-score and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='5% decreased in AUC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Moreover, the reasonable score shows that even without the guidance of financial statements, only the graph structure infor- mation can still guarantee a certain classification accuracy, which demonstrates the effectiveness of our proposed tribe-style struc- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Moreover, the performance degradation of TH-GNN\\GGRL and TH-GNN\\TSE indicates that the global news-graph and local tribes (investment-graphs) both benefit the downstream task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='2 The effects of different model components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' To further verify the importance of different components in our model, another three variants of TH-GNN are designed for ablation study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' The perfor- mance degradation of TH-GNN\\CL, which removes the contrastive loss term, shows that the contrastive loss 𝐿𝐶𝐿 plays an essential role in learning the discriminative structural information of tribes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Besides, the whole TH-GNN model yields a good performance boost in comparison to TH-GNN\\Emb, a variant without the structure em- bedding in TSE and using randomly initialized attributes for nodes on tribes, which indicates that the proposed structure embedding is beneficial for modeling graphs without node attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' From the results of TH-GNN\\Fusion, we find that the fusion module, which integrates the information from financial statements and tribes comprehensively, brings significant improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Overall, the whole TH-GNN achieves the best result compared with all variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='6 Visualization of Learned Representations We analyse the model’s interpretability by visualizing the learned representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' We first extract the hidden outputs of the penulti- mate layer for different GNN models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Next, we reduce the hidden vectors to 2-dimensional vectors with T-SNE algorithm and plot their the scatter plot .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' 5, the representations learned by GCN, GAT and GraphSAGE have weak discrimination between risky and normal nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Despite the representations learned by GCNII and DAGNN have improved node classification ability, our TH-GNN has the best discriminative power and provides inter- pretability that other GNN models do not have.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' 5 (f) illustrates the visualization of the representations learned by TH-GNN which have evident outlier clusters, and most of points in these clusters are risky companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' This unique phenomenon indicates that some risky companies have similar local investment-patterns (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=', the example presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' 2 (a)), which is easily affected by their (a) GCN (b) GAT (c) GraphSAGE (d) GCNII (e) DAGNN (f) TH-GNN Figure 5: T-SNE visualization of the representations learned by models, where red dots represent samples of risky compa- nies and blue dots represent samples of normal companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' risky neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' And this observation further demonstrate the in- dispensable effects of the Tribe Structure Encoder (TSE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' 5 RELATED WORK In this paper, we mainly focus on using graph learning methods to solve the company financial risk assessment problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Graph Representation Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Graph Neural Networks (GNNs) [13, 15, 30, 31, 33, 34] have shown excellent performance on graph representation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Graph Convolution Network (GCN) [15] was first proposed with aggregation-based graph con- volution operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' After that, its improved models have been pro- posed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' GAT [30] introduced an attention mechanism to distinguish the importance of neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' GIN [36] investigate the effect of differ- ent aggregation functions for graph representation ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Graph- SAGE [13] proposed to use graph sampling for inductive learning on graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Recently, some studies [5, 16] also focus on deeper GCN with larger receptive filed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' GCNII [5] is an extension of vanilla GCN with initial residual and identity mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' DAGNN [16] is a deeper GNN model that decouples the transformation and aggregation of message passing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' However, most of them [12, 19, 20, 32, 33] are designed without taking the unique domain knowledge into con- sideration and are not aimed at tackling the company financial risk assessment on financial networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Company Financial Risk Assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Financial risk has been a major concern for financial companies and governments, and extensive works have been studied to assess company financial risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' With the rapid development of machine learning methods in recent years, researchers have developed machine learning-based company financial risk prediction models using financial state- ments or privately-owned data provided by financial institutions [7, 10, 18, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' After the global financial crisis in 2008, the crisis brought by the collapse of Lehman Brothers spread rapidly and widely to related companies, and graph theory began to attract the attention of researchers [1, 3, 17, 21, 23, 26–29, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Recently, some studies attempt to construct different financial graphs and design domain-specific GNNs for company financial risk assess- ment [8, 11, 37, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' For example,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' [8] proposed a High-order Graph ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='Attention Networks to assess risks for companies on guarantee ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='60Company-as-Tribe: Company Financial Risk Assessment on Tribe-Style Graph with Hierarchical Graph Neural Networks ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='KDD ’22,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' August 14–18,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' 2022,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Washington,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' DC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' USA loans networks considering higher-order neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' [37] proposed a dynamic GNN for supply chain mining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' [39] designed a heterogeneous graph attention network to predict the bankruptcy risks of small and medium-sized companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' However, all these works only consider one type of financial graph, and the data is specific and private.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Models trained on such domain-specific data may have bias and have poor generalization ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' 6 CONCLUSION AND FURTHER WORK In this paper, we investigate the listed company’s financial data in real words and find that it is far from sufficient to assess the risk of listed companies solely through financial statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' To pro- vide more comprehensive representations of companies, we design a tribe-style network and propose TH-GNN for company finan- cial risk assessment task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Experiments on a real-world large-scale dataset show that our proposed model is effective in company finan- cial risk assessment task and can provide brilliant interpretability of results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Furthermore, much future work is focused on the scalability of the method to investigate problems such as credit evaluation, risk transmission and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' In addition, TH-GNN provides valuable tools to analyse the information of companies and their risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' As future work, we will try more complicated GNN backbones rather than simply GIN and GCN, and combine real-time information with tribe-style graph to further improve the dynamic risk identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' ACKNOWLEDGMENTS This work was supported by the National Natural Science Founda- tion of China (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='61902380, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='61802370, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='U21B2046 and No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content='U1911401) and the Beijing Nova Program (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFRT4oBgHgl3EQfIjco/content/2301.13492v1.pdf'} +page_content=' Z201100006820061).' 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Asano2 +Fatih Porikli1, Amirhossein Habibian1 +1Qualcomm AI Research‡ , 2QUVA Lab, University of Amsterdam +shashanka.venkataramanan@inria.fr +ghodrati@qti.qualcomm.com +Tiny +Small +Base +1 +2 +3 +4 +5 +6 +7 +72 +74 +76 +78 +80 +82 +84 +ViT +ViT+SkipAT +Loading [MathJax]/extensions/MathMenu.js +Tiny +Small +Base +28 +30 +32 +34 +36 +38 +ViT +ViT+SkipAT +Loading [MathJax]/extensions/MathMenu.js +Tiny +Small +Base +10 +15 +20 +25 +30 +35 +38 +40 +42 +44 +46 +48 +ViT +ViT+SkipAT +Loading [MathJax]/extensions/MathMenu.js +top-1 accuracy +Throughput (images/sec ×10!) +top-1 accuracy +GPU-hours +mIoU +Throughput (images/sec) +Image classification +(ImageNet-1k) +Self-supervised learning +(ImageNet-1K) +Semantic segmentation +(ADE20K) +Tiny +Small +Base +20 +40 +60 +80 +100 +39.65 +39.7 +39.75 +39.8 +39.85 +39.9 +39.95 +40 +UFormer +UFormer+SkipAT +Loading [MathJax]/extensions/MathMenu.js +Ep. 20 +Ep. 40 +Ep. 60 +Ep. 80 Ep. 100 +40 +80 +120 +60 +64 +68 +72 +76 +DINO +DINO+SkipAT +Loading [MathJax]/extensions/MathMenu.js +Unsupervised segmentation +(Pascal VOC2012) +Jaccard similarity +Model variant +Image Denoising +(SIDD) +PSNR +FLOPs (G) +Figure 1. Performance of SKIPAT across 5 different tasks. Our novel SKIPAT parametric function achieves superior accuracy vs. +efficiency trade-off over the baseline transformer on a wide array of tasks. +Abstract +This work aims to improve the efficiency of vision trans- +formers (ViT). While ViTs use computationally expensive +self-attention operations in every layer, we identify that +these operations are highly correlated across layers – a key +redundancy that causes unnecessary computations. Based +on this observation, we propose SKIPAT, a method to reuse +self-attention computation from preceding layers to approx- +imate attention at one or more subsequent layers. To ensure +that reusing self-attention blocks across layers does not de- +grade the performance, we introduce a simple parametric +function, which outperforms the baseline transformer’s per- +formance while running computationally faster. We show +the effectiveness of our method in image classification and +self-supervised learning on ImageNet-1K, semantic seg- +mentation on ADE20K, image denoising on SIDD, and +video denoising on DAVIS. We achieve improved through- +put at the same-or-higher accuracy levels in all these tasks. +*equal contribution +†Work done during internship at Qualcomm AI Research +‡Qualcomm AI Research is an initiative of Qualcomm Technologies, +Inc +1. Introduction +The transformer architecture [50] has become an important +and highly influential model family, due to its simplicity, +scalability, and its wide range of applications. While orig- +inally stemming from the domain of natural language pro- +cessing (NLP), with the advent of the Vision transformer +(ViT) [15], this has become a standard architecture in com- +puter vision, setting various state-of-the-art (SoTA) per- +formances on tasks ranging from representation learning, +semantic segmentation, object detection and video under- +standing [4, 5, 18, 30, 31]. +However, the original formulation of the transformer in- +cludes a quadratic computational complexity with respect +to the number of input tokens. Given that this number typi- +cally ranges from 142 for image classification all the way to +1282 = 16K for image denoising, this constraint on mem- +ory and compute severely limits its applicability. To tackle +this problem, there have been three sets of approaches. The +first leverages redundancies across input tokens and simply +reduces computation by efficient sampling, e.g., dropping +or merging redundant tokens [17, 46, 63]. This, however, +means that the final output of the ViT is not spatially contin- +uous and can thus not be used beyond image-level applica- +tions such as semantic segmentation or object localization. +The second set of approaches aims to cheaply estimate the +1 +arXiv:2301.02240v1 [cs.CV] 5 Jan 2023 + +attention computation, but generally at the cost of reduced +performances [10, 65]. Finally, another line of works aims +to merge convolutional architectures with the transformer, +yielding hybrid architectures [29, 29, 39]. While these in- +crease speed, they do not tackle the fundamental problem of +the quadratic complexity, and often introduce an exorbitant +number of design choices (essentially a union of those of +the transformer and CNNs). +In this work, we propose a novel, so far unexplored ap- +proach to solving this problem: simply approximating the +computationally expensive blocks of the transformer with a +much faster, simpler parametric function. To arrive at this +solution, we first thoroughly analyse the crucial multi-head +self-attention (MSA) block of the ViT. Through this analy- +sis, we find that the attention of the CLS tokens to the spatial +patches has a very high correlation across the transformer’s +blocks, thus leading to unnecessary computations. This mo- +tivates our approach to leverage attention from an early part +of the model and simply reuse it for deeper blocks – basi- +cally “skipping” subsequent SA calculations instead of re- +computing them at every layer. +Based on this, we go one step further and explore if the +entire MSA block of a layer can be skipped by reusing +the representation from previous layers. +We find that a +simple parametric function inspired from ResneXt’s depth- +wise convolutions [62] can outperform the baseline per- +formance – while being computationally faster in terms of +throughput and FLOPs. Our method is general-purpose and +can be applied to a ViT in any context: Figure 1 shows +that our novel parametric function for Skipping Attention +(SKIPAT) achieves superior accuracy vs. efficiency trade- +off compared to the baseline transformer on a wide variety +of tasks, datasets and model sizes. +In summary, our main contributions are as follows: +1. We propose a novel plug-in module that can be placed +in any ViT architecture for reducing the costly O(n2) +Self-Attention computations (subsection 4.3) +2. We achieve state-of-the-art performances in terms of +throughput at same-or-better accuracies for ImageNet, +Pascal-VOC2012, SIDD, DAVIS and ADE20K (in the +latter of which we obtain 40% speedup) (section 5) +3. We further demonstrate the generality of our method +by obtaining a 26% reduction in self-supervised pre- +training time (at no downstream accuracy loss) and +by demonstrating superior on-device latency (subsec- +tion 5.2, subsection 5.1) +4. Finally, we analyse the sources of performance gains +and extensively ablate our method to provide a model +family which can be used for trading off accuracy and +throughput (subsection 5.6) +2. Related Work +There has been great effort made to improve the efficiency +of vision transformers (ViT) [15] from multiple aspects: +Token sampling +improves the efficiency either by re- +structuring images during the tokenization step [21, 66], +pruning the redundant tokens over training [26, 46] or dy- +namically at inference [7, 17, 43, 63]. Despite their effec- +tiveness in reducing the computational cost in image clas- +sification, token sampling methods are hardly applicable to +dense prediction tasks, e.g. semantic segmentation and im- +age denoising, where the output image should be spatially +continuous. Our approach is complementary to these lines +of work and performs favorably against them as validated +experimentally. Moreover, given that we keep representing +all tokens throughout the network, our approach is applica- +ble to both classification and dense prediction tasks. +Hybrid architectures +integrate efficient convolutional +modules into vision transformers [32, 36, 39] by adoption of +MobileNet blocks in Uniformer [29], MobileNetV2 blocks +in MobileViT [35] or using stacks of convolutions in the +image tokenization step [19, 59]. Similarly, we use convo- +lutions to speed up vision transformers, however, instead of +crafting customized blocks as in [29, 35, 36, 39], we adhere +to the original transformer architecture and approximate en- +tire MSA computations through convolutions. +Efficient attentions +address the quadratic cost of the self- +attention operation in vision transformers by global down- +sampling of key and value embeddings [54, 59], performing +self-attention in local windows [31], alternating between lo- +cal and global self-attentions [10, 35, 39], or replacing self- +attention with a simple pooling [65]. However, reducing the +self-attention to a local neighborhood hinders their ability to +model the long range dependencies and leads to a significant +performance drop with moderate speed up [69]. Moreover, +some of the introduced operations come with no efficient +support, e.g. cyclic shift in Swin [31], limiting their actual +efficiency gains in terms of latency. Different to this, our +method relies on the strong, yet inefficient self-attention op- +erator at a few blocks and lighter, accurate attention estima- +tors in other blocks. As the estimators only rely on standard +convolutional operations, our method translates to actual la- +tency gains. Related to this paper, [55, 60, 64] observed the +redundancies in attention maps, for NLP tasks. However, +instead of simply copying attention maps [60, 64], we pro- +pose an efficient parametric function that, as we show, are +critical to achieve a high throughput whilst retaining high +model performance in vision tasks. +2 + +0.48 +0.63 +0.84 +0.95 +0.94 +0.78 +0.95 +0.93 +0.97 +0.62 +0.91 +0.44 +0.72 +0.87 +0.92 +0.96 +0.85 +0.91 +0.97 +0.97 +0.84 +0.94 +A9 +[CLS] +A10 +[CLS] +A11 +[CLS] +A12 +[CLS] +A5 +[CLS] +A6 +[CLS] +A7 +[CLS] +A8 +[CLS] +A1 +[CLS] +A2 +[CLS] +A3 +[CLS] +A4 +[CLS] +Figure 2. Attention correlation. Mean of the attention heads from the CLS token of a pretrained ViT-T/16 at different layers from the +validation set of ImageNet-1K. Numbers below each attention map indicates the cosine similarity of A[CLS] +l +with A[CLS] +l−1 . +(a) CKA of 𝐴[CLS] +(b) CKA of 𝑍[MSA] +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +Figure 3. CKA analysis of A[CLS] and ZMSA across different lay- +ers of pretrained ViT-T/16 on the validation set of Imagenet-1K. +Vanilla ViT-T/16 has high correlation across both attention maps +(layer 3 to 10) and ZMSA (layer 2 to 8) +Hierarchical architectures +introduce hierarchical repre- +sentations, as a long-standing principle in computer vi- +sion, to vision transformers [19, 31, 40, 54, 69]. Using a +multi-scale representation significantly improves the mem- +ory and computational cost of the isotropic architectures, +such as ViT. More recently, the idea has been extended +to more complex architectures with U-Net [57] or multi- +branch structures [20]. Our work is complementary to these +works, as they do not tackle the fundamental problem of +reducing the quadratic complexity of the self-attention op- +erator. We experimentally validate the effectiveness of our +method on such isotropic and hierarchical architectures. +3. Related Work +There has been great effort made to improve the efficiency +of vision transformers (ViT) [15] from multiple aspects: +Token sampling +improves the efficiency either by re- +structuring images during the tokenization step [21, 66], +pruning the redundant tokens over training [26, 46] or dy- +namically at inference [7, 17, 43, 63]. Despite their effec- +tiveness in reducing the computational cost in image clas- +sification, token sampling methods are hardly applicable to +dense prediction tasks, e.g. semantic segmentation and im- +age denoising, where the output image should be spatially +continuous. Our approach is complementary to these lines +of work and performs favorably against them as validated +experimentally. Moreover, given that we keep representing +all tokens throughout the network, our approach is applica- +ble to both classification and dense prediction tasks. +Hybrid architectures +integrate efficient convolutional +modules into vision transformers [32, 36, 39] by adoption of +MobileNet blocks in Uniformer [29], MobileNetV2 blocks +in MobileViT [35] or using stacks of convolutions in the +image tokenization step [19, 59]. Similarly, we use convo- +lutions to speed up vision transformers, however, instead of +crafting customized blocks as in [29, 35, 36, 39], we adhere +to the original transformer architecture and approximate en- +tire MSA computations through convolutions. +Efficient attentions +address the quadratic cost of the self- +attention operation in vision transformers by global down- +sampling of key and value embeddings [54, 59], performing +self-attention in local windows [31], alternating between lo- +cal and global self-attentions [10, 35, 39], or replacing self- +attention with a simple pooling [65]. However, reducing the +self-attention to a local neighborhood hinders their ability to +model the long range dependencies and leads to a significant +performance drop with moderate speed up [69]. Moreover, +some of the introduced operations come with no efficient +support, e.g. cyclic shift in Swin [31], limiting their actual +efficiency gains in terms of latency. Different to this, our +method relies on the strong, yet inefficient self-attention op- +erator at a few blocks and lighter, accurate attention estima- +tors in other blocks. As the estimators only rely on standard +convolutional operations, our method translates to actual la- +tency gains. Related to this paper, [55, 60, 64] observed the +redundancies in attention maps, for NLP tasks. However, +instead of simply copying attention maps [60, 64], we pro- +pose an efficient parametric function that, as we show, are +3 + +1 +0.84 +0.86 +0.81 +0.76 +0.72 +0.53 +0.63 +0.63 +0.56 +0.5 +0.44 +0.84 +1 +0.91 +60 +9870 +0.84 +0.66 +0.74 +0.73 +0.64 +0.51 +0.41 +0.86 +0.91 +1 +0.96 +0.92 +68'0 +0.74 +0.8 +6L'0 +0.71 +0.55 +0.45 +0.81 +0.9 +0.96 +1 +0.96 +0.94 +0.78 +E8:0 +0.81 +0.56 +0.44 +0.76 +0.86 +0.92 +0.96 +1 +0.95 +0.8 +0.82 +TB'O +0.75 +0.55 +0.43 +0.72 +0.84 +680 +0.94 +0.95 +1 +580 +58'0 +EB'O +0.77 +0.55 +0.44 +. +0.53 +0.66 +0.78 +0.8 +590 +1 +0.79 +0.81 +0.79 +0.52 +6E'0 +0.63 +0.74 +0.8 +E8:0 +0.82 +0.79 +1 +99'0 +0.79 +0.57 +0.42 +0.63 +0.73 +0.79 +0.81 +0.81 +0.83 +0.81 +0.86 +1 +0.82 +0.6 +0.45 +0.56 +0.64 +0.71 +0.74 +0.75 +0.77 +0.79 +0.79 +Z80 +1 +0.72 +0.56 +0.5 +0.51 +0.55 +0.56 +0.55 +0.55 +0.52 +0.57 +0.6 +0.72 +1 +0.92 +0.44 +0.41 +0.45 +0.44 +0.43 +0.44 +6E'0 +0.42 +0.45 +0.56 +0.92 +11 +0.32 +0.29 +0.31 +0.25 +0.25 +0.12 +0.13 +0.2 +0.078 +0.072 +0.073 +0.32 +1 +0.85 +0.8 +0.75 +0.76 +0.62 +0.57 +0.35 +0.46 +0.3 +0.41 +0.29 +0.85 +1 +0.87 +0.74 +0.79 +0.67 +0.59 +0.35 +0.48 +0.27 +0.44 +0.31 +0.8 +0.87 +1 +0.81 +0.82 +0.68 +0.63 +0.39 +0.51 +0.32 +0.44 +0.25 +0.75 +0.74 +0.81 +1 +0.8 +0.71 +0.74 +0.49 +0.6 +0.37 +0.52 +0.25 +0.76 +0.79 +0.82 +0.8 +1 +0.7 +0.66 +0.44 +0.52 +0.34 +0.44 +0.12 +0.62 +0.67 +0.68 +0.71 +0.7 +1 +0.83 +0.38 +0.77 +0.47 +0.66 +0.13 +0.57 +0.59 +0.63 +0.74 +0.66 +0.83 +1 +0.63 +0.78 +0.49 +0.65 +0.2 +0.35 +0.35 +0.39 +0.49 +0.44 +0.38 +0.63 +1 +0.48 +0.41 +0.38 +0.078 +0.46 +0.48 +0.51 +0.6 +0.52 +0.77 +0.78 +0.48 +1 +0.5 +0.69 +0.072 +0.3 +0.27 +0.32 +0.37 +0.34 +0.47 +0.49 +0.41 +0.5 +1 +0.42 +0.073 +0.41 +0.44 +0.44 +0.52 +0.44 +0.66 +0.65 +0.38 +0.69 +0.42 +1critical to achieve a high throughput whilst retaining high +model performance in vision tasks. +Hierarchical architectures +introduce hierarchical repre- +sentations, as a long-standing principle in computer vi- +sion, to vision transformers [19, 31, 40, 54, 69]. Using a +multi-scale representation significantly improves the mem- +ory and computational cost of the isotropic architectures, +such as ViT. More recently, the idea has been extended +to more complex architectures with U-Net [57] or multi- +branch structures [20]. Our work is complementary to these +works, as they do not tackle the fundamental problem of +reducing the quadratic complexity of the self-attention op- +erator. We experimentally validate the effectiveness of our +method on such isotropic and hierarchical architectures. +4. Skip-Attention +4.1. Preliminaries +Vision Transformer. +Let x ∈ Rh×w×c be an input image, +where h × w is the spatial resolution and c is the number of +channels. The image is first tokenized into n = hw/p2 non- +overlapping patches, where p × p is patch size. Each patch +is projected into an embedding zi ∈ Rd using a linear layer +to obtain the tokenized image: +Z0 = (z1; . . . ; zn) ∈ Rn×d +(1) +Here, “; ” denotes row-wise stacking. Positional embed- +dings are added to Z0 to retain positional information. The +token embeddings are then input to a L = {1, . . . , L} layer +transformer whose output is denoted as ZL. In the super- +vised setting, a learnable token z[CLS] ∈ Rd is prepended +to the tokenized image in +(1) as Z0 := (z[CLS]; Z0) ∈ +R(n+1)×d. +Transformer Layer. Every layer of the transformer con- +sists of a multi-head self attention (MSA) block followed by +a multi-layer perceptron (MLP) block. In the MSA block, +the input, Zl−1 ∈ Rn×d, for l ∈ L, is first projected into +three learnable embeddings {Q, K, V } ∈ Rn×d. The atten- +tion matrix A, is calculated as +A := σ +�QKT +√ +d +� +∈ Rn×n +(2) +where σ(.) denotes the row-wise softmax operation. The +“multi-head” in MSA is defined by considering h attention +heads where each head is a sequence of n × d +h matrix. The +attention heads are reprojected back to n × d using a linear +layer which is combined with the value matrix as +ZMSA := AV ∈ Rn×d +(3) +The output representations from the MSA block is then in- +put to the MLP block which comprises two linear layers +separated by a GeLU activation [24]. At a given layer l, +the computational flow of representations through a trans- +former block is denoted as +Zl ← ZMSA +l ++ Zl−1, +(4) +Zl ← MLP(Zl) + Zl. +(5) +Both the MSA and MLP blocks have residual connections +with layer normalization (LN) [3]. While MSA blocks in +each layer of the transformer learn representations inde- +pendently, in the next subsection, we show that empirically +there exist high correlation across these layers. +4.2. Motivation: Layer Correlation Analysis +Attention-map correlation. +The MSA block in ViT en- +codes the similarity of each patch to every other patch as +an n × n attention matrix. This operator is computation- +ally expensive with O(n2) complexity (2). As ViTs scale +up, i.e., as n increases, the complexity grows quadrati- +cally and this operation becomes a bottleneck. Recent NLP +works [51, 52] have shown that self-attention across adja- +cent layers in SoTA language models exhibit very high cor- +relation. This raises the question – is it worth to compute +self-attention at every layer of a vision transformer? +To address this question, we analyze the correlation of the +self-attention maps across different layers of ViT. As shown +in Figure 2, the self-attention maps from the class token, +A[CLS], exhibit high correlation especially in the interme- +diate layers. +The cosine similarity between A[CLS] +l−1 +and +A[CLS] +l +can be as high as 0.97, as indicated in the bottom +of each attention map in Figure 2. Similar behavior is ob- +served from other token embeddings, which we analyze in +the supplementary material. We quantitatively analyze this +correlation across all the samples of the validation set of +ImageNet-1K, by computing the Centered Kernel Align- +ment (CKA) [12, 27] between A[CLS] +i +and A[CLS] +j +for every +i, j ∈ L. CKA measures the similarity between represen- +tations obtained from intermediate layers of the network, +where a high value of CKA indicates high correlation be- +tween the representations. From Figure 3 (a), we observe +that ViT-T has a high correlation across A[CLS] especially +from layer 3 through 10. +Feature correlation. +In ViTs, the high correlation is not +just limited to A[CLS], but the representation from MSA +blocks, ZMSA, also show high correlation throughout the +model [42]. To analyze the similarity across these represen- +tations, we compute the CKA between ZMSA +i +and ZMSA +j +for +every i, j ∈ L. We observe from Figure 3 (b), that ZMSA +also have high similarity across adjacent layers of the model +especially in the earlier layers, i.e., from layer 2 through 8. +4 + +𝑍!"# +$%& +𝑛×𝑑 +𝑛×2𝑑 +spatial +flatten +𝑛× 𝑛×2𝑑 +𝑛×2𝑑 +reshape +DwC +FC +𝑍"! +$%& +𝑟×𝑟 +𝑛× 𝑛×2𝑑 +𝑛×𝑑 +FC & +ECA +𝑛×𝑑 +SKIPAT parametric function Φ +MLP +⨁ +MSA +⨁ +MSA +⨁ +MLP +⨁ +⨁ +MLP +⨁ +… +MSA +𝑍!"# +$%& +𝑍"! +$%& +skip +skip +MSA +⨁ +skip +MLP +⨁ +Φ!"# +Φ!"$ +… +Φ! +Figure 4. SKIPAT framework We illustrate SKIPAT on ViT [15]. The SKIPAT parametric function (Φ) uses representations of the MSA +block (in solid color) ZMSA +l−1 as input, which undergoes a series of transformations. An element-wise summation (�) with the output of the +MLP block from layer l − 1 and ˆZMSA +l +is used as input to the MLP block at layer l. The MSA operation (crossed out) is thus not computed +and is discarded from the computational graph. With SKIPAT the total number of layers remains unchanged. +4.3. Improving Efficiency by Skipping Attention +Based on our observation of high representation similarity +across MSA blocks of a transformer (subsection 4.2), we +propose to leverage the correlation across both the atten- +tion matrix and the representations from the MSA block to +improve the efficiency of vision transformers. Instead of +computing the MSA operation (3) independently at every +layer, we explore a simple and effective strategy to utilize +dependencies across the features from these layers. +In particular, we propose to skip MSA computation in one +or more layers of a transformer by reusing representations +from its adjacent layers. We term this operation as Skip +Attention or SKIPAT. As the compute and memory benefit +from skipping the entire MSA block is greater than skipping +just the self-attention operation (O(n2d+nd2) vs. O(n2d)), +in this paper we focus on former. However, instead of di- +rectly re-using features, i.e., copying the features from the +source MSA block to one or more adjacent MSA blocks, +we introduce a parametric function. The parametric func- +tion ensures that directly reusing features does not affect +the translation invariance and equivariance in these MSA +blocks and acts as a strong regularizer to improve model +generalization. +SKIPAT parametric function +Let Φ : Rn×d → Rn×d +denote the parametric function that maps output of the MSA +block from l − 1 to l as ˆZMSA +l +:= Φ(ZMSA +l−1 ). Here, ˆZMSA +l +is the approximation of ZMSA +l +. The parametric function can +be as simple as an identity function, where ZMSA +l−1 is directly +reused. Instead of computing MSA operation at l, we use +ZMSA +l−1 as the input to the MLP block at l. When using an +identity function, due to the absence of MSA operation at +l, the relation across tokens is no longer encoded in the at- +tention matrix, which affects representation learning. To +mitigate this, we introduce the SKIPAT parametric function +inspired from ResNeXt [62] as shown in Figure 4, to en- +code local relations among tokens. The SKIPAT parametric +function consists of two linear layers and a depth-wise con- +volution (DwC) [9] in between, as follows: +ˆZMSA +l +:= ECA +� +FC2 +� +DwC +� +FC1(ZMSA +l−1 ) +��� +(6) +In the case of supervised learning, we first separate the CLS +embeddings from ZMSA ∈ R(n+1)×d into class embeddings +ZMSA +C +∈ Rd and the patch embeddings to ZMSA +P +∈ Rn×d. +The patch embeddings are then input to the first linear layer +FC1 : Rn×d → Rn×2d, which expands the channel di- +mension. +This is followed by DwC : R +√n×√n×2d → +R +√n×√n×2d with kernel r × r to capture cross-token re- +lations. Note that before the DwC operation, we spatially +reshape the input matrix to a feature tensor. The output of +the DwC is then flattened back to a vector and fed to the +last FC layer FC2 : Rn×2d → Rn×d which reduces the +channel dimension back to its initial dimension d. We use +GeLU activations after FC1 and DwC. Following [53], we +use efficient channel attention module (ECA) after FC2 to +enhance the cross-channel dependencies. The ECA module +first aggregates the features along the channel dimension +using global average pooling (GAP). A 1 × 1 convolution +with adaptive kernel size proportional to channel dimension +is applied followed by sigmoid activation. This operation +of the ECA module enhances cross-channel dependencies. +We then concatenate the embedding of the class-token with +the output of the ECA to obtain ˆZMSA +l +. +SKIPAT +framework. +The +overall +framework +of +SKIPAT is illustrated in Figure 4. +SKIPAT can be in- +corporated into any transformer architecture which we +empirically show in subsection 5.4. +Depending on the +architecture, one can skip the MSA operation in one or +more layers of the transformer. In ViT, as we empirically +observe that representations from the MSA block, ZMSA, +have high correlations from layer 2 through 7 (subsec- +tion 4.2), we employ the SKIPAT parametric function in +these layers. This means that we use the ZMSA +2 +as input to +the SKIPAT parametric function and skip MSA operations +in layers 3-8. Instead, the features from the output of the +SKIPAT parametric function is used as input to the MLP +block. +The computation flow of representations is now +5 + +modified to +Zl ← Φ(ZMSA +l−1 ) + Zl−1 +(7) +Zl ← MLP(Zl) + Zl +(8) +Due to the presence of residual connections in the MSA and +MLP blocks, which is standard in ViT [15], the MLP blocks +at layer 3 through 8 learn representations independently and +cannot be discarded from the computational graph. It is im- +portant to note that, with SKIPAT the total number of layers +in ViT remain unchanged, but there are fewer MSA blocks. +Complexity: MSA vs. SKIPAT +The self-attention oper- +ation involves three operations. Firstly, the token embed- +dings are projected into query, key and value embeddings, +secondly, attention matrix A is computed as dot product be- +tween Q and K and finally, the output representations are +computed as dot product between A and V . This results in +a complexity of O(4nd2 + n2d). Since d ≪ n, the com- +plexity of MSA block can be reduced to O(n2d). +The SKIPAT parametric function consists of two linear lay- +ers and one depth-wise convolution operation, which re- +sults in a O(2nd2 + r2nd) complexity, where r × r is the +kernel size of the DwC operation. The overall complex- +ity of SKIPAT can be reduced to O(nd2) since r2 ≪ d. +Thus, SKIPAT has fewer FLOPs than the MSA block as +O(nd2) < O(n2d) when n increases as transformers scale +up. +5. Experiments +5.1. Image Classification +We use ViT-T/16 [15], ViT-S/16 [15] and ViT-B/16 [15] +as our backbone on ImageNet-1K. For fair comparisons, +we follow the experimental settings in [48] and evaluate +SKIPAT against SoTA methods: A-ViT [63], Dynamic- +ViT [38], SViTE [7], SPViT [26], ATS [17], PS-ViT [46], +HVT [40] and Rev-Vit [34]. To the best of our knowledge, +these are all the works that improve the efficiency of ViT +without modifying its underlying architecture. +From Table 1, we observe that SKIPAT achieves the best ac- +curacy vs. efficiency trade-off compared to all SoTA meth- +ods on different variants of ViT. Notably, we outperform +baseline ViT-T, ViT-S and ViT-B by 0.1%, 0.4% and 0.4% +respectively, while SoTA methods achieve lower accuracy +or are on-par with the baseline. Since SKIPAT uses a para- +metric function to skip computing MSA blocks, our reduc- +tion in number of parameters and in FLOPs is comparable +to the SoTA. In terms of throughput, SKIPAT is 19%, 21% +and 25% faster than the baseline ViT-T, ViT-S and ViT-B re- +spectively. Dehghani et al. [13] highlight the significance of +using throughput as a metric to measure model efficiency: +as the reduction in FLOPs does not necessarily correspond +BACKBONE METHOD +TOP-1↑ PARAM↓ GFLOPS↓ THROUGHPUT↑ +(%) +(×106) +(IM/S ×103) +ViT [15] +72.8 +5.7 +1.2 +5.8 +A-ViT [63] +71.0 +5.7 +0.8 +6.3 +Dynamic ViT [43] +70.9 +– +0.9 +6.1 +SViTE [7] +71.7 +4.0 +0.9 +6.2 +ViT-T/16 +SPViT [26] +72.7 +5.7 +0.9 +6.7 +ATS [17] +72.7 +5.7 +0.9 +6.1 +PS-ViT [46] +72.6 +– +0.7 +6.6 +HVT [40] +70.2 +5.7 +0.7 +7.2 +SKIPAT +72.9 +5.8 +1.1 +6.9 +ViT [15] +79.8 +22.4 +4.6 +3.2 +A-ViT [63] +78.6 +22.4 +3.6 +3.4 +Dynamic ViT [43] +78.3 +23.1 +3.4 +3.6 +SViTE [7] +80.2 +13.1 +2.7 +3.5 +ViT-S/16 +ATS [17] +79.7 +22.4 +2.9 +3.3 +PS-ViT [46] +79.4 +– +2.6 +3.9 +SPViT [26] +79.3 +22.1 +2.7 +3.5 +Rev-ViT [34] +79.8 +22.4 +4.6 +3.6 +HVT[40] +78.0 +22.5 +2.4 +4.1 +SKIPAT +80.2 +22.1 +4.0 +3.8 +ViT [15] +81.8 +87.3 +17.6 +1.2 +SViTE [7] +81.6 +52.0 +11.5 +1.3 +ViT-B/16 +Rev-ViT [34] +81.5 +87.3 +17.6 +1.2 +PS-ViT [46] +81.5 +– +9.8 +1.6 +SKIPAT +82.2 +86.7 +15.2 +1.5 +Table 1. Image classification on ImageNet-1K. Accuracy vs. ef- +ficiency comparison of SKIPAT with SoTA methods for image res- +olution 224 × 224. For all the methods, we measure throughput +(image/sec) with a batch size of 1024 on a single NVIDIA A100 +GPU, averaged over the validation set of ImageNet-1K. +to improvements in latency, as it does not take into account +the degree of parallelism or other hardware details. In line +with this argument, we observe that while SoTA methods +such as ATS [17] and SPViT [26] achieve large reduction +in FLOPs, they actually have lower throughput when com- +pared to SKIPAT. Furthermore, HVT [40] while achieving +a higher gain in both throughput and FLOPs has poor top- +1 accuracy (2.6% drop in ViT-T and 1.8% drop in ViT-S). +Thus, SKIPAT demonstrates the ability to simultaneously +improve both accuracy and throughput over SoTA methods. +Visualizing attention maps and ZMSA correlation. We +analyze the effect of the SKIPAT parametric function by vi- +sualizing the mean of attention heads of the CLS token from +the last four layers of ViT-T/16. From Figure 5, we observe +that while attention maps from vanilla ViT (last two lay- +ers) do not solely attend to the object, the attention maps +from SKIPAT accurately focuses on the object. It is inter- +esting to note that, the attention maps from SKIPAT are also +capable of attending to multiple objects in the image (Fig- +ure 5: second example). We further analyze the CKA of +the representations from MSA block across all the layers +of ViT-T/16. From Figure 6, we observe that ZMSA has +lower correlation across layers except between the layers +where the MSA operation is skipped (layer 3 to 8). How- +ever, unlike vanilla ViT (Figure 3 (b)) the correlation from +each layer to every other layer is quite low. This shows that +6 + +A! +[#$%] +A'( +[#$%] +A'' +[#$%] +A') +[#$%] +baseline +baseline +SKIPAT +SKIPAT +Figure 5. Visualizing attention maps. Mean of the attention of +different heads from A[CLS] from last four layers of ViT-T/16 on +the validation set of ImageNet-1K. Attention maps from last four +blocks show that SKIPAT localizes the object better than vanilla +ViT. +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +Figure 6. CKA analysis of SKIPAT shows that ZMSA has lower +correlation between layers. The high correlation is only between +consecutive layers 2 through 8, where the MSA operation is +skipped. +our SKIPAT parametric function acts as a strong regularizer +and thus improves the representations of the model. +METHOD +JACCARD↑ +CORLOC↑ +ViT-T [15] +32.2 +39.5 +ViT-T + SKIPAT +38.0 +41.5 +ViT-S [15] +29.0 +40.6 +ViT-S + SKIPAT +34.0 +41.2 +ViT-B [15] +33.6 +36.4 +ViT-B + SKIPAT +36.8 +37.2 +Table 2. +Unsupervised Segmentation and Object Localiza- +tion using Jaccard similarity [5] and Correct Localization (Cor- +Loc) [37], on the validation set of Pascal VOC2012. All models +have been pretrained on ImageNet-1K in a supervised setting. +Figure 7. +Visualization of segmentation masks using vanilla +ViT-S/16 (top) and ViT-S + SKIPAT (bottom) pretrained supervis- +edly on ImageNet-1K. We visualize masks obtained by threshold- +ing the self-attention maps to keep 80% of the mass. +Probing self-attention maps in ViTs. +We further analyze +whether pretrained ViTs can attend to semantically mean- +ingful regions of the image when evaluated on a different +dataset without fine-tuning it. We follow the evaluation pro- +tocol in [5], and visualize the segmentation masks produced +from the final layer of the pretrained SKIPAT on the Pascal- +VOC12 [16] validation set. From Figure 7, 1 we observe +that while vanilla ViT-S/16 does not accurately attend to +the object, SKIPAT is able to localize objects quite accu- +rately without any fine-tuning. To quantify this observa- +tion, we follow [5] and use the Jaccard similarity between +predicted segmentation mask and ground truth mask. As +shown in Table 2, SKIPAT outperforms different variants of +vanilla ViT with a significant gap in terms of Jaccard sim- +ilarity. Additionally, we measure the quality of the gener- +ated maps for unsupervised object localization using Cor- +Loc [37] as the evaluation metric. From Table 2, we ob- +serve that SKIPAT achieves notable gains across all variants +of ViT. +Performance on mobile device. To verify the efficiency +of SKIPAT on low-power devices, we measure its inference +time (averaged over 20 iterations) on a Samsung Galaxy +S22 device powered by Qualcomm “Snapdragon® 8 Gen. +1 Mobile Platform” with a Qualcomm® HexagonTM proces- +sor2, for image resolutions of 224×224 and 384×384 using +ViT-T/16. The inference is performed on Neural Processing +Unit in 8-bit precision. As shown in Table 3, SKIPAT im- +proves the runtime by 19% for image size of 224 × 224. +The gain is even larger at 34% for image resolution 384 × +1The original image sources, before masking, from left to right: +Kangra valley train (CC BY-SA 4.0) +Ecuadorian fishing boat (CC BY-SA 2.0) +Sheep near Snowshill (CC BY-SA 2.0) +2Snapdragon and Qualcomm Hexagon are products of Qualcomm +Technologies, Inc. and/or its subsidiaries. +7 + +1 +0.6 +0.48 +0.41 +0.29 +0.3 +0.3 +0.29 +0.28 +0.34 +0.25 +0.2 +0.6 +1 +0.77 +0.59 +0.38 +0.43 +0.45 +0.45 +0.47 +0.57 +0.41 +0.3 +0.48 +0.77 +1 +0.78 +0.52 +0.53 +0.51 +0.5 +0.47 +0.59 +0.44 +0.34 +0.41 +0.59 +0.78 +1 +0.75 +0.68 +0.6 +0.55 +0.47 +0.51 +0.4 +0.31 +0.29 +0.38 +0.52 +0.75 +1 +0.78 +0.66 +0.56 +0.4 +0.36 +0.31 +0.24 +0.3 +0.43 +0.53 +0.68 +0.78 +1 +0.83 +0.73 +0.56 +0.44 +0.37 +0.27 +0.3 +0.45 +0.51 +0.6 +0.66 +0.83 +1 +0.84 +0.67 +0.47 +0.4 +0.28 +0.29 +0.45 +0.5 +0.55 +0.56 +0.73 +0.84 +1 +0.78 +0.48 +0.41 +0.29 +0.28 +0.47 +0.47 +0.47 +0.4 +0.56 +0.67 +0.78 +1 +0.5 +0.4 +0.27 +0.34 +0.57 +0.59 +0.51 +0.36 +0.44 +0.47 +0.48 +0.5 +1 +0.68 +0.42 +0.25 +0.41 +0.44 +0.4 +0.31 +0.37 +0.4 +0.41 +0.4 +0.68 +1 +0.4 +0.2 +0.3 +0.34 +0.31 +0.24 +0.27 +0.28 +0.29 +0.27 +0.42 +0.4 +1384, since the number of token increases. Thus, skipping +computationally-heavy MSA blocks increases throughput +by large margins and is confirmed even on mobile hardware. +METHOD +224 × 224 +384 × 384 +ViT-T/16 +5.65 +20.49 +ViT-T/16 + SKIPAT +4.76 +15.22 +Table 3. +On-device latency (in msec) of vanilla ViT vs. +SKIPAT for different image resolutions on a Samsung Galaxy S22 +powered by Qualcomm Snapdragon 8 Gen. 1. +5.2. Self-Supervised Learning with DINO +Next, we show the generality of SKIPAT as its use in the +backbone for self-supervised representation learning (SSL), +using DINO [5]. Since, SSL methods are quite expensive in +the pretraining stage in terms of compute and training time, +we illustrate that SKIPAT achieves comparable performance +to using a ViT but with shorter training time. Following the +experimental settings of DINO [5], we use ViT-S/16 [15] as +our student and teacher networks with SKIPAT parametric +function. We pretrain both baseline and ours using DINO +for 100 epochs. We observe that SKIPAT achieves almost +the same performance as fully trained DINO with around +26% less training time (73.3% in 96 GPU-hours vs. 73.6% +in 131 GPU-hours). When trained on 100 epochs, we ob- +serve that SKIPAT outperforms DINO by 0.5% (74.1% vs. +73.6%). We show the performance of SKIPAT to down- +stream tasks in the supplementary material. +5.3. Semantic Segmentation on ADE20K +We go beyond classification and show the performance +of SKIPAT to dense prediction tasks such as seman- +METHOD +BACKBONE +MIOU↑ GFLOPS↓ THROUGHPUT↑ +ResNet-101 [65] +40.7 +261 +24.1 +Semantic FPN [25] PoolFormer-S36 [65] +42.0 +191 +8.4 +PoolFormer-M36 [65] +42.4 +271 +5.4 +ResNet-18 [23] +39.9 +886 +17.1 +ResNet-101 [23] +44.9 +1031 +12.0 +Swin-T [31] +45.8 +945 +14.2 +ConvNeXt-T [32] +46.7 +939 +15.7 +UperNet [61] +ViT-T [15] +37.3 +212 +24.1 +ViT-T + SKIPAT +40.6 +173 +34.7 +ViT-S [15] +44.4 +360 +19.5 +ViT-S + SKIPAT +45.3 +283 +27.2 +ViT-B [15] +45.6 +787 +11.1 +ViT-B + SKIPAT +46.3 +633 +15.5 +Table 4. Semantic Segmentation results on ADE20K. All mod- +els are pretrained on ImageNet-1k and fine-tuned on ADE20K. +Following Swin [31] and ConvNeXt [32], we report mIoU with +multi-scale testing. FLOPs and throughput are calculated on the +input size of 2048 × 512. Throughput of all models are measured +with a batch size of 1 on a single NVIDIA A100 GPU, averaged +over 100 forward passes. +METHOD +PSNR↑ +SSIM↑ +GFLOPS↓ +THROUGHPUT↑ +UNet [45] +39.65 +- +35 +– +DAGL [38] +38.94 +0.953 +255 +– +DeamNet [44] +39.47 +0.957 +145 +– +MPRNet [68] +39.71 +0.958 +573 +– +NBNet [8] +39.75 +0.959 +91 +– +Restormer [67] +40.02 +0.960 +140 +– +Uformer-T [57] +39.66 +– +12 +17.6 +Uformer-T + SKIPAT +39.69 +0.959 +11 +22.2 +Uformer-S [57] +39.77 +0.959 +44 +15.1 +Uformer-S + SKIPAT +39.84 +0.960 +39 +18.9 +Uformer-B [57] +39.89 +0.960 +89 +9.2 +Uformer-B + SKIPAT +39.94 +0.960 +77 +10.9 +Table 5. +Image denoising on SIDD dataset using PSNR and +SSIM [56] as the evaluation metrics in the RGB space. FLOPs +and throughput are calculated on the input size of 256 × 256, on a +single NVIDIA V100 GPU, averaged over the test set of SIDD. +tic segmentation. +We follow the experimental settings +in [31, 32] and use MMSegmentation [11] to evaluate +SKIPAT on ADE20K [70]. We observe from Table 4, that +SKIPAT consistently outperforms all variants of ViT with +15% fewer FLOPs and 25% improved throughput. Inter- +estingly, SKIPAT-S (ViT-S + SKIPAT) achieves 8% higher +mIoU while being faster than ViT-T. Furthermore, SKIPAT- +S has comparable mIoU with Swin-T [31] whilst having +3× fewer FLOPs and being 1.7× faster. +Comparing to +fully convolution-based architectures, SKIPAT-T (ViT-T + +SKIPAT) is on par with ResNet-18 in mIoU while having +4.7× fewer FLOPs and being 1.8× faster. +5.4. Image Denoising +SKIPAT can also generalize to low-level tasks such as im- +age denoising on SIDD [1], which consists of images with +real-world noise. +We also demonstrate that SKIPAT can +generalize to other transformer architectures. +In partic- +ular, we apply it on Uformer [57], a SoTA image de- +noising model. +Uformer is a U-shaped hierarchical net- +work with Swin transformer blocks as the encoder and de- +coder, and skip connections between them. +In SKIPAT, +we skip window self-attention (WSA) block in each de- +coder block by reusing attention of the corresponding en- +coder block via SKIPAT parametric function. Detailed im- +plementation is in the supplementary material. +Follow- +ing the experimental settings in [57], we observe in Ta- +ble 5 that SKIPAT outperforms the baseline Uformer vari- +ants with the 25% higher throughput on average. Further- +more, we observe that SKIPAT-B (Uformer-B + SKIPAT) +achieves comparable performance with Restormer [67], in +terms of PSNR and SSIM, which is the SoTA image denois- +ing method while having 2× fewer FLOPs. Thus, we show +the ability of SKIPAT to generalize to different tasks and +also across architectures. +8 + +METHOD +FastDVDNet +PaCNet +VRT +UniFormer +UniFormer+ +[47] +[49] +[30] +[28] +SKIPAT +PSNR↑ +34.04 +34.79 +36.52 +35.24 +35.16 +GFLOPS↓ +41.9 +34.8 +708.8 +93.2 +77.1 +Table 6. Video denoising Quantitative comparison (average RGB +channel PSNR) with state-of-the-art methods for video denoising +on DAVIS, with additive noise level σ = 30. FLOPs are calculated +per frame per patch size of 256 × 256. +FUNCTION +KERNEL +CHANNEL +TOP-1↑ +THROUGHPUT↑ +Φ +EXPANSION +(%) +(img/sec ×103) +ViT-T +- +- +65.8 +5.8 +IDENTITY +- +- +61.1 +8.5 +CONV +5 × 5 +- +65.4 +5.2 +DWC +5 × 5 +- +65.6 +7.8 +3 × 3 +67.1 +7.3 +SKIPAT +5 × 5 +2 +67.7 +6.9 +7 × 7 +67.4 +6.6 +0.5 +64.4 +7.4 +SKIPAT +5 × 5 +1 +65.9 +7.2 +2 +67.7 +6.9 +Table 7. +Ablations using ViT-T/16 on ImageNet-1K for 100 +epochs. We measure throughput (image/sec) with a batch size of +1024 on a single NVIDIA A100 GPU, averaged over the validation +set of ImageNet-1K. +5.5. Video Denoising +We further apply our model to the temporal task of video +denoising. As encoder and decoder backbone, we use Uni- +Former [28], a U-shaped hybrid encoder-decoder architec- +ture with 3D convolutions and spatio-temporal global self- +attention blocks. Detailed implementation is provided in +the supplementary material. Similar to image denoising, we +skip MSA blocks in the decoder, however, simply adopt a +naive SKIPAT, where we reuse window self-attention ma- +trix, A, of the corresponding encoder block using an Iden- +tity function. We empirically observe that reusing atten- +tion works better in this task, and shows the ability of +our method to be applied for different scenarios. We fol- +low the experimental settings in [47] and train SKIPAT on +DAVIS [41] dataset. We train using Charbonnier loss [6] on +patches of 7 × 128 × 128 using a multiple-input, multiple- +output (MIMO) paradigm (i.e. the model outputs 7 recon- +structed frames from 7 input frames) for noise level σ = 30. +From Table 6, we observe that SKIPAT performs on par +with baseline Uniformer, while having 17% fewer FLOPs. +This shows that SKIPAT can generalize to temporal tasks. +5.6. Ablations +All ablations are performed using ViT-T/16 on ImageNet- +1K for 100 epochs to reduce the training time. Unless spec- +ified, following SKIPAT we skip the MSA blocks from layer +3 through 8 for all ablations. +Parametric function Φ. +We study the effect of different +parametric functions in terms of accuracy and throughput. +As discussed in subsection 4.3, Φ can be as simple as an +identity function, where we directly reuse representations +from a previous MSA block into one of more subsequent +MSA blocks. From Table 7, using an identity function re- +sults in a 4.7% drop in top-1 accuracy while being 47% +faster than baseline ViT. Using a convolution or DwC [9] +with kernel size 5 × 5 as a parametric function leads to the +same performance as the baseline. However, DwC is 0.2% +better and 50% faster than convolution, and 34% faster than +the baseline. SKIPAT parametric function outperforms all. +Kernel size. +By default SKIPAT uses a DwC with kernel +size of 5 × 5. As shown in Table 7, while using a 3 × 3 ker- +nel is faster than default SKIPAT by 6%, it is 0.6% worse +in terms of accuracy. A larger kernel size has poor accu- +racy and lower throughout. However, irrespective of the +kernel size, SKIPAT outperforms the baseline ViT-T by at +least 1.4%, showing its ability to encode cross-token inter- +actions. +Channel expansion. In the SKIPAT , the first linear layer +FC1, expands the channel dimension from d → 2d. Ta- +ble 7 shows the impact of channel dimension, i.e., when +the channel expansion ratio of FC1 is 1.0 (d → d) and 0.5 +(d → d/2). We observe that while the lower channel expan- +sion ratio improves the throughput, it performs worse than +default SKIPAT. This could be due to sub-optimal represen- +tations encoded by the DwC due to fewer filters. +Skipping MSA in alternate configuration. +Instead of +skipping the MSA operation in the layers 3 − 8, we study +the effect of skipping MSA operation at l ∈ {3, 5, 7, 9}. +We observe the latter configuration outperforms the base- +line ViT by 2.7% (65.8 vs. 67.5%). However, it performs +0.2% lower and is 8% slower than our default SKIPAT con- +figuration. +6. Conclusion +We proposed SKIPAT, a plug-in module that can be placed +in any ViT architecture for reducing the costly Self- +Attention computations. SKIPAT leverages the dependency +across MSA blocks and bypasses attention computation by +re-using attention from previous MSA blocks. To ensure +that the metaphorical sharing is caring we introduced a sim- +ple and light parametric function that does not affect the +inductive bias encoded in MSA. 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We train +baseline ViT and SKIPAT for 300 epochs from scratch on 4 +NVIDIA A100 GPUs using batch sizes of 2048 for ViT-T +and 1024 for ViT-S and ViT-B. +ImageNet-1K: Self-supervised learning. +We follow the +experimental settings of DINO [5] and pre-train DINO and +SKIPAT on ImageNet-1K using ViT-S/16 as the backbone. +While likely the hyperparameters could be tuned further +for our proposed SKIPAT +method, we use same hyper- +parameters for both the baseline and ours, yielding a con- +servative estimate of our model’s performance. We pre-train +both methods from scratch for 100 epochs using 4 NVIDIA +A100 GPUs. For linear-probing, we freeze the backbone +from the pre-training stage and fine-tune the classifier for +100 epochs, exactly as done in [5]. +Pascal-VOC2012: +Unsupervised object segmentation. +We use the Pascal VOC 2012 [16] validation set for this +experiment, containing 1449 images. +We follow DINO +and obtain unsupervised segmentation masks by threshold- +ing the averaged self-attention map (extracted from the last +layer of a pretrained ViT/SKIPAT model) to keep 80% of the +mass. The Jaccard similarity J between a predicted mask, +P, and ground-truth mask, G, is defined as: +J(P, G) = G ∩ P +G ∪ P +We report Jaccard similarity, averaged over all the samples. +ADE20K: +Semantic +segmentation. +We +evalu- +ate SKIPAT on ADE20K [70], a widely-used semantic +segmentation dataset, covering 150 semantic categories. +The dataset includes 20K and 2K images in the train- +ing and validation set, respectively. +Different variants +of SKIPAT are evaluated using UperNet [61] as the +backbone. We use our ImageNet-1K pretrained model to +initialize the backbone and Kaiming [22] initialization for +other layers. We use AdamW [33], with an initial learning +rate of 6e−5, weight decay of 1e−2, and linear warmup of +1500 iterations. All models are trained for 160K iterations +with a batch size of 16 using MMSegmentation repo [11]. +We keep the same hyper-parameters for SKIPAT and ViT. +SIDD: Image denoising. +We follow the experimental set- +tings in Uformer [57] and train SKIPAT on the Smartphone +Image Denoising Dataset (SIDD) [2] which consists of +real-world noise. The training samples are first randomly +cropped to 128×128 patches and input to the model, which +is trained for 250 epochs using batch size 32. The model is +then evaluated on images of size 256 × 256. +DAVIS: Video denoising. +We further apply our model to +the temporal task of video denoising. We adopt the same U- +shape encoder-decoder based architecture of UFormer. As +the encoder and decoder backbone, we use UniFormer [28]. +We train the model on noise level σ = 30 using Charbonnier +loss [6] on patches of 7 × 128 × 128 using a multiple-input, +multiple-output (MIMO) paradigm [30] (i.e., the model out- +puts 7 reconstructed frames from 7 input frames). During +inference, a video is divided into 3D patches of 7×128×128 +with an overlap of 10 pixels. Each patch is fed to the model +and the outputs are merged to obtain the final denoised +video. Following [47], PSNR is calculated as averaged over +videos. We use the same training hyper-parameters as im- +age denoising. +A.2. Architecture +Image Classification. +All baseline ViT variants have 12 +layers in total, which remains unchanged with SKIPAT. Fol- +lowing the CKA analysis of ZMSA in Figure 3(b) of our +main paper, we skip computing the MSA blocks in layer +3 through 8 for all ViT variants and retrain it from scratch. +Image Denoising. +We apply SKIPAT to Uformer [29] a +SoTA image denoising model. Uformer is a U-shaped hi- +erarchical network with Swin transformer blocks as the en- +coder and decoder, and skip connections between them. In +SKIPAT, we skip window self-attention (WSA) block in +each decoder block by reusing attention of the correspond- +ing encoder block via SKIPAT parametric function. +Let +ZWSAe +l +∈ Rn×c denote the output of the WSA block at layer +l from the encoder and Zd +l−1 ∈ Rn×c denote the output of +the layer l − 1 from the decoder of Uformer. The input to +the WSA block (which is skipped) at layer l of the decoder +is given by +ˆZWSAd +l += Φ(ZWSAe +l +; Zd +l−1) ∈ Rn×2c +(9) +Here, “;” denotes concatenation along the channel dimen- +sion. We show the framework of SKIPAT on Uformer in +Figure 8 +Video Denoising. +we apply SKIPAT to UniFormer [28], +a U-shaped hybrid encoder-decoder architecture with +3D convolutions and spatio-temporal global self-attention +blocks. +The encoder of UniFormer comprises two 3D +12 + +𝑥 +MLP +WSA +MLP +WSA +MLP +WSA +skip +MLP +WSA +skip +𝑍𝑙 +WSA𝑒 +መ𝑍𝑙 +WSA𝑑 +Φ𝑖+1 +Φ𝑖 +… +⨁ +⨁ +⨀ +⨁ +⨁ +MLP +WSA +⨁ +⨁ +MLP +WSA +⨁ +⨁ +⨁ +⨁ +⨁ +⨁ +⨁ +… +MLP +WSA +⨁ +⨁ +⨀ +ො𝑥 +Input +image +Denoised +image +Figure 8. Framework of SKIPAT on Uformer Instead of standard +MSA block in ViT, Uformer uses window self-attention (WSA) +block similar to Swin Transformer. We skip WSA block in the +layers close to the bottleneck. +convolution layers followed by two spatio-temporal trans- +former layers with global self-attention (MSA) blocks. A +downsampling operation is used after every layer in the en- +coder. The decoder is symmetric to the encoder with two +transformer layers followed by two 3D convolution layers +with an upsampling operation between each layer. Similar +to Uformer, skip connections are used between encoder and +decoder. Similar to image denoising, we skip MSA blocks +in the decoder, however, simply adopt a naive SKAT, where +we reuse global self-attention matrix Al, from the encoder +at layer l in the corresponding decoder stage at the same +layer using an Identity function. Let Ae +l ∈ Rn×n denote +the self-attention matrix at layer l from the encoder. The +self-attention in the decoder stage at layer l is given by +Ad +l = I(Ae +l ) ∈ Rn×n, where I(.) is the identity function. +We observe that skipping attention A using an identity +function works better than skipping MSA blocks using the +SKIPAT parametric function. +This shows the generality +of SKIPAT, regardless of approximating attention or MSA +blocks. +B. Additional experiments +Image classification. +Here we extend our SoTA compar- +ison with methods that go beyond vanilla ViT architec- +tures. +These methods include hierarchical (Swin, PVT, +Poolformer, MobileViT, Twins-SVT) and Hybrid (Con- +vNext, CoAT) architectures. We provide the complete set +of SoTA methods that improve the efficiency of ViT either +by token sampling (extending Table 1 in our main paper), +using hybrid architectures or window self-attention blocks +in Table 8. Apart from methods that perform efficient token +sampling, none of the other methods are directly compara- +ble because they modify the underlying architecture of ViT, +either by using window self-attention blocks or reducing the +overall number of transformer layers. +BACKBONE +METHOD +TOP-1 +PARAM +GFLOPS +THROUGHPUT +(%) +(×106) +(img/sec ×103) +T2T-ViT [66] +71.7 +5.8 +1.1 +– +ConvNeXt (iso) [32] +72.7 +5.7 +1.1 +5.8 +ViT [15] +72.8 +5.7 +1.2 +5.8 +A-ViT [63] +71.0 +5.7 +0.8 +6.3 +Dynamic ViT [43] +70.9 +– +0.9 +6.1 +ViT-T/16 +SViTE [7] +71.7 +4.0 +0.9 +6.2 +SPViT [26] +72.7 +5.7 +0.9 +6.7 +ATS [17] +72.7 +5.7 +0.9 +6.1 +PS-ViT [46] +72.6 +– +0.7 +6.6 +HVT [40] +70.2 +5.7 +0.7 +7.2 +SKIPAT +72.9 +5.8 +1.1 +6.9 +ConvNext-T [32] +82.1 +29.0 +4.5 +2.6 +ConvNeXt (iso) [32] +79.7 +22.4 +4.3 +3.3 +Swin-T [31] +81.3 +28.3 +4.5 +2.5 +T2T-ViT [66] +80.7 +21.5 +5.2 +– +CoaT-Lite-Small +81.9 +20.0 +4.0 +– +Poolformer-S24 [65] +80.3 +21.0 +3.4 +– +Twins-SVT-S [10] +81.7 +24.0 +2.8 +– +MobileViT-S [35] +78.4 +5.6 +2.0 +– +PVT [54] +79.8 +24.5 +3.8 +– +ViT-S/16 +ViT [15] +79.8 +22.4 +4.6 +3.2 +A-ViT [63] +78.6 +22.4 +3.6 +3.4 +Dynamic ViT [43] +78.3 +23.1 +3.4 +3.6 +SViTE [7] +80.2 +13.1 +2.7 +3.5 +ATS [17] +79.7 +22.4 +2.9 +3.3 +PS-ViT [46] +79.4 +– +2.6 +3.9 +SPViT [26] +79.3 +22.1 +2.7 +3.5 +Rev-ViT [34] +79.8 +22.4 +4.6 +3.6 +HVT[40] +78.0 +22.5 +2.4 +4.1 +SKIPAT +80.2 +22.1 +4.0 +3.8 +Swin-S [31] +83.5 +88.0 +15.4 +1.0 +Twins-SVT-B [10] +83.2 +56.0 +8.6 +– +PVT [54] +81.7 +61.4 +9.8 +– +ConvNeXt (iso) [32] +82.0 +87.3 +16.9 +1.3 +ViT-B/16 +ViT [15] +81.8 +87.3 +17.6 +1.2 +SViTE [7] +81.6 +52.0 +11.5 +1.3 +Rev-ViT [34] +81.5 +87.3 +17.6 +1.2 +PS-ViT [46] +81.5 +– +9.8 +1.6 +SKIPAT +82.2 +86.7 +15.2 +1.5 +Table 8. Image classification on ImageNet-1K. Accuracy vs. ef- +ficiency comparison of SKIPAT with SoTA methods for image res- +olution 224 × 224. For all the methods, we measure throughput +(image/sec) with a batch size of 1024 on a single NVIDIA A100 +GPU, averaged over the validation set of ImageNet-1K. +Unsupervised +segmentation +of +DINO. +We +follow +DINO [5] and evaluate the performance of baseline DINO +vs. SKIPAT on unsupervised object segmentation on +Pascal-VOC2012 [16] dataset. We follow the experimental +setting as discussed in Appendix A and observe that +baseline DINO has a Jaccard similarity of 45.3 while +SKIPAT achieves 44.7. While SKIPAT outperforms DINO +on image classification by 0.5%, we achieve comparable +performance in terms of unsupervised object segmentation. +C. Additional ablations +Reusing self-attention. +As mentioned in Subsection 3.3, +we skip the ZMSA in SKIPAT as the compute and memory +benefit from skipping the entire MSA block is greater than +13 + +skipping just the self-attention operation. Here we study +the effect of skipping just the self-attention operation. Let +Al−1 denote the self-attention matrix at layer l −1, then the +self-attention matrix at layer l is given by ˆAl = I(Al−1). +Similar to SKIPAT we skip computing the self-attention ma- +trix from layers 3 through 8. As parametric function Φ, we +use an identity mapping and train ViT-T/16 from scratch +for 100 epochs on ImageNet-1K. We observe from Table 9, +that skipping the self-attention matrix results in a top-1 ac- +curacy of 63.2% which is 2.1% higher than the skipping +ZMSA with an identity function (61.1% - Table 7 of main +paper). However, skipping self-attention matrix results in +20% decrease in throughput (8500 → 6800 images/sec) as +compared to using an identity function to skip MSA block. +It is interesting to note that skipping self-attention matrix +results in a lower drop in performance as compared to skip- +ping MSA block. However, applying a parametric function +to skip self-attention can be challenging due to the proper- +ties of the self-attention matrix, and we leave this to future +work. +SKIPAT in pretrained model. +As mentioned in subsec- +tion A.2, we train SKIPAT with all variants of ViT from +scratch. For completeness, we also study the effect of skip- +ping the self-attention matrix and the MSA block on a pre- +trained ViT-T using an Identity function, without retraining. +We observe from Table 9 that skipping the self-attention +computation in layers 3 through 8, results in a top-1 accu- +racy of 53.9%, while skipping MSA blocks results in top-1 +accuracy of 47.8%. It is interesting to note that the drop in +top-1 accuracy from skipping self-attention is merely 19% +(72.8 → 53.9) on average and does not result in an extremely +large drop as one might expect. This shows that there in- +deed exists high correlation across self-attention and ZMSA, +which SKIPAT utilizes to improve the efficiency of the ViTs. +METHOD +TRAINING +TOP-1 (%) +THROUGHPUT +A +✓ +63.2 +6800 +ZMSA +✓ +61.1 +8500 +A +✗ +53.9 +6800 +ZMSA +✗ +47.8 +8500 +Table 9. Ablations on the effect of skipping the self-attention, +A, and the MSA block, ZMSA. In the first two rows, models are +trained for 100 epochs. In the last two rows we use a pretrained +ViT-T/16 and simply skip computations in blocks 3-8 during in- +ference. For all the experiments with use Identity function as Φ. +D. CKA analysis of attention from ViT-T +As discussed in Section 3.2 of our main paper, we analyze +the CKA of the self-attention matrix for all tokens between +different layers of ViT-T/16 pretrained on ImageNet-1K. +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +Figure 9. CKA analysis of A for all tokens from pretrained vanilla +ViT-T/16 on the validation set of ImageNet-1K. We observe a high +correlation for all tokens in A from layers 1 to 8. +Since in the supervised setting A ∈ R(n+1)×(n+1), we first +remove the CLS token to obtain AP ∈ Rn×n. We then com- +pute the CKA of AP +l for l ∈ L. We observe from Figure 9, +that there exists a high correlation across all the tokens from +the self-attention matrix. Thus, reusing self-attention from +different layers of the ViT can improve the overall through- +put while yielding comparable accuracy as the baseline ViT. +14 + +1 +0.82 +0.61 +0.53 +0.47 +0.54 +0.46 +0.43 +0.31 +0.29 +0.23 +0.45 +0.82 +1 +0.79 +0.77 +0.59 +0.72 +0.57 +0.53 +0.38 +0.36 +0.29 +0.66 +0.61 +0.79 +1 +0.86 +0.74 +0.79 +0.64 +0.59 +0.42 +0.4 +0.34 +0.71 +0.53 +0.77 +0.86 +1 +0.75 +0.85 +0.61 +0.56 +0.39 +0.37 +0.31 +0.78 +0.47 +0.59 +0.74 +0.75 +1 +0.84 +0.73 +0.63 +0.42 +0.39 +0.31 +0.64 +0.54 +0.72 +0.79 +0.85 +0.84 +1 +0.77 +0.68 +0.51 +0.46 +0.38 +0.77 +0.46 +0.57 +0.64 +0.61 +0.73 +0.77 +1 +0.81 +0.66 +0.56 +0.48 +0.6 +0.43 +0.53 +0.59 +0.56 +0.63 +0.68 +0.81 +1 +0.58 +0.61 +0.64 +0.57 +0.31 +0.38 +0.42 +0.39 +0.42 +0.51 +0.66 +0.58 +1 +0.61 +0.54 +0.46 +0.29 +0.36 +0.4 +0.37 +0.39 +0.46 +0.56 +0.61 +0.61 +1 +0.55 +0.47 +0.23 +0.29 +0.34 +0.31 +0.31 +0.38 +0.48 +0.64 +0.54 +0.55 +1 +0.47 +0.45 +0.66 +0.71 +0.78 +0.64 +0.77 +0.6 +0.57 +0.46 +0.47 +0.47 +1 \ No newline 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+page_content=' 40 Ep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' 60 Ep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' 80 Ep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' 100 40 80 120 60 64 68 72 76 DINO DINO+SkipAT Loading [MathJax]/extensions/MathMenu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='js Unsupervised segmentation (Pascal VOC2012) Jaccard similarity Model variant Image Denoising (SIDD) PSNR FLOPs (G) Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Performance of SKIPAT across 5 different tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Our novel SKIPAT parametric function achieves superior accuracy vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' efficiency trade-off over the baseline transformer on a wide array of tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Abstract This work aims to improve the efficiency of vision trans- formers (ViT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' While ViTs use computationally expensive self-attention operations in every layer, we identify that these operations are highly correlated across layers – a key redundancy that causes unnecessary computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Based on this observation, we propose SKIPAT, a method to reuse self-attention computation from preceding layers to approx- imate attention at one or more subsequent layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' To ensure that reusing self-attention blocks across layers does not de- grade the performance, we introduce a simple parametric function, which outperforms the baseline transformer’s per- formance while running computationally faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' We show the effectiveness of our method in image classification and self-supervised learning on ImageNet-1K, semantic seg- mentation on ADE20K, image denoising on SIDD, and video denoising on DAVIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' We achieve improved through- put at the same-or-higher accuracy levels in all these tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' equal contribution †Work done during internship at Qualcomm AI Research ‡Qualcomm AI Research is an initiative of Qualcomm Technologies, Inc 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Introduction The transformer architecture [50] has become an important and highly influential model family, due to its simplicity, scalability, and its wide range of applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' While orig- inally stemming from the domain of natural language pro- cessing (NLP), with the advent of the Vision transformer (ViT) [15], this has become a standard architecture in com- puter vision, setting various state-of-the-art (SoTA) per- formances on tasks ranging from representation learning, semantic segmentation, object detection and video under- standing [4, 5, 18, 30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' However, the original formulation of the transformer in- cludes a quadratic computational complexity with respect to the number of input tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Given that this number typi- cally ranges from 142 for image classification all the way to 1282 = 16K for image denoising, this constraint on mem- ory and compute severely limits its applicability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' To tackle this problem, there have been three sets of approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' The first leverages redundancies across input tokens and simply reduces computation by efficient sampling, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=', dropping or merging redundant tokens [17, 46, 63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' This, however, means that the final output of the ViT is not spatially contin- uous and can thus not be used beyond image-level applica- tions such as semantic segmentation or object localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' The second set of approaches aims to cheaply estimate the 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='02240v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='CV] 5 Jan 2023 attention computation, but generally at the cost of reduced performances [10, 65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Finally, another line of works aims to merge convolutional architectures with the transformer, yielding hybrid architectures [29, 29, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' While these in- crease speed, they do not tackle the fundamental problem of the quadratic complexity, and often introduce an exorbitant number of design choices (essentially a union of those of the transformer and CNNs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' In this work, we propose a novel, so far unexplored ap- proach to solving this problem: simply approximating the computationally expensive blocks of the transformer with a much faster, simpler parametric function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' To arrive at this solution, we first thoroughly analyse the crucial multi-head self-attention (MSA) block of the ViT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Through this analy- sis, we find that the attention of the CLS tokens to the spatial patches has a very high correlation across the transformer’s blocks, thus leading to unnecessary computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' This mo- tivates our approach to leverage attention from an early part of the model and simply reuse it for deeper blocks – basi- cally “skipping” subsequent SA calculations instead of re- computing them at every layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Based on this, we go one step further and explore if the entire MSA block of a layer can be skipped by reusing the representation from previous layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' We find that a simple parametric function inspired from ResneXt’s depth- wise convolutions [62] can outperform the baseline per- formance – while being computationally faster in terms of throughput and FLOPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Our method is general-purpose and can be applied to a ViT in any context: Figure 1 shows that our novel parametric function for Skipping Attention (SKIPAT) achieves superior accuracy vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' efficiency trade- off compared to the baseline transformer on a wide variety of tasks, datasets and model sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' In summary, our main contributions are as follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' We propose a novel plug-in module that can be placed in any ViT architecture for reducing the costly O(n2) Self-Attention computations (subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='3) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' We achieve state-of-the-art performances in terms of throughput at same-or-better accuracies for ImageNet, Pascal-VOC2012, SIDD, DAVIS and ADE20K (in the latter of which we obtain 40% speedup) (section 5) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' We further demonstrate the generality of our method by obtaining a 26% reduction in self-supervised pre- training time (at no downstream accuracy loss) and by demonstrating superior on-device latency (subsec- tion 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='2, subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='1) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Finally, we analyse the sources of performance gains and extensively ablate our method to provide a model family which can be used for trading off accuracy and throughput (subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='6) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Related Work There has been great effort made to improve the efficiency of vision transformers (ViT) [15] from multiple aspects: Token sampling improves the efficiency either by re- structuring images during the tokenization step [21, 66], pruning the redundant tokens over training [26, 46] or dy- namically at inference [7, 17, 43, 63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Despite their effec- tiveness in reducing the computational cost in image clas- sification, token sampling methods are hardly applicable to dense prediction tasks, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' semantic segmentation and im- age denoising, where the output image should be spatially continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Our approach is complementary to these lines of work and performs favorably against them as validated experimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Moreover, given that we keep representing all tokens throughout the network, our approach is applica- ble to both classification and dense prediction tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Hybrid architectures integrate efficient convolutional modules into vision transformers [32, 36, 39] by adoption of MobileNet blocks in Uniformer [29], MobileNetV2 blocks in MobileViT [35] or using stacks of convolutions in the image tokenization step [19, 59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Similarly, we use convo- lutions to speed up vision transformers, however, instead of crafting customized blocks as in [29, 35, 36, 39], we adhere to the original transformer architecture and approximate en- tire MSA computations through convolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Efficient attentions address the quadratic cost of the self- attention operation in vision transformers by global down- sampling of key and value embeddings [54, 59], performing self-attention in local windows [31], alternating between lo- cal and global self-attentions [10, 35, 39], or replacing self- attention with a simple pooling [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' However, reducing the self-attention to a local neighborhood hinders their ability to model the long range dependencies and leads to a significant performance drop with moderate speed up [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Moreover, some of the introduced operations come with no efficient support, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' cyclic shift in Swin [31], limiting their actual efficiency gains in terms of latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Different to this, our method relies on the strong, yet inefficient self-attention op- erator at a few blocks and lighter, accurate attention estima- tors in other blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' As the estimators only rely on standard convolutional operations, our method translates to actual la- tency gains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Related to this paper, [55, 60, 64] observed the redundancies in attention maps, for NLP tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' However, instead of simply copying attention maps [60, 64], we pro- pose an efficient parametric function that, as we show, are critical to achieve a high throughput whilst retaining high model performance in vision tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='63 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='84 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Mean of the attention heads from the CLS token of a pretrained ViT-T/16 at different layers from the validation set of ImageNet-1K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Numbers below each attention map indicates the cosine similarity of A[CLS] l with A[CLS] l−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' (a) CKA of 𝐴[CLS] (b) CKA of 𝑍[MSA] 1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' CKA analysis of A[CLS] and ZMSA across different lay- ers of pretrained ViT-T/16 on the validation set of Imagenet-1K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Vanilla ViT-T/16 has high correlation across both attention maps (layer 3 to 10) and ZMSA (layer 2 to 8) Hierarchical architectures introduce hierarchical repre- sentations, as a long-standing principle in computer vi- sion, to vision transformers [19, 31, 40, 54, 69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Using a multi-scale representation significantly improves the mem- ory and computational cost of the isotropic architectures, such as ViT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' More recently, the idea has been extended to more complex architectures with U-Net [57] or multi- branch structures [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Our work is complementary to these works, as they do not tackle the fundamental problem of reducing the quadratic complexity of the self-attention op- erator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' We experimentally validate the effectiveness of our method on such isotropic and hierarchical architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Related Work There has been great effort made to improve the efficiency of vision transformers (ViT) [15] from multiple aspects: Token sampling improves the efficiency either by re- structuring images during the tokenization step [21, 66], pruning the redundant tokens over training [26, 46] or dy- namically at inference [7, 17, 43, 63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Despite their effec- tiveness in reducing the computational cost in image clas- sification, token sampling methods are hardly applicable to dense prediction tasks, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' semantic segmentation and im- age denoising, where the output image should be spatially continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Our approach is complementary to these lines of work and performs favorably against them as validated experimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Moreover, given that we keep representing all tokens throughout the network, our approach is applica- ble to both classification and dense prediction tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Hybrid architectures integrate efficient convolutional modules into vision transformers [32, 36, 39] by adoption of MobileNet blocks in Uniformer [29], MobileNetV2 blocks in MobileViT [35] or using stacks of convolutions in the image tokenization step [19, 59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Similarly, we use convo- lutions to speed up vision transformers, however, instead of crafting customized blocks as in [29, 35, 36, 39], we adhere to the original transformer architecture and approximate en- tire MSA computations through convolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Efficient attentions address the quadratic cost of the self- attention operation in vision transformers by global down- sampling of key and value embeddings [54, 59], performing self-attention in local windows [31], alternating between lo- cal and global self-attentions [10, 35, 39], or replacing self- attention with a simple pooling [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' However, reducing the self-attention to a local neighborhood hinders their ability to model the long range dependencies and leads to a significant performance drop with moderate speed up [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Moreover, some of the introduced operations come with no efficient support, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' cyclic shift in Swin [31], limiting their actual efficiency gains in terms of latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Different to this, our method relies on the strong, yet inefficient self-attention op- erator at a few blocks and lighter, accurate attention estima- tors in other blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' As the 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='073 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='69 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='42 1critical to achieve a high throughput whilst retaining high model performance in vision tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Hierarchical architectures introduce hierarchical repre- sentations, as a long-standing principle in computer vi- sion, to vision transformers [19, 31, 40, 54, 69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Using a multi-scale representation significantly improves the mem- ory and computational cost of the isotropic architectures, such as ViT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' More recently, the idea has been extended to more complex architectures with U-Net [57] or multi- branch structures [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Our work is complementary to these works, as they do not tackle the fundamental problem of reducing the quadratic complexity of the self-attention op- erator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' We experimentally validate the effectiveness of our method on such isotropic and hierarchical architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Skip-Attention 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Preliminaries Vision Transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Let x ∈ Rh×w×c be an input image, where h × w is the spatial resolution and c is the number of channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' The image is first tokenized into n = hw/p2 non- overlapping patches, where p × p is patch size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Each patch is projected into an embedding zi ∈ Rd using a linear layer to obtain the tokenized image: Z0 = (z1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' zn) ∈ Rn×d (1) Here, “;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' ” denotes row-wise stacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Positional embed- dings are added to Z0 to retain positional information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' The token embeddings are then input to a L = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' , L} layer transformer whose output is denoted as ZL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' In the super- vised setting, a learnable token z[CLS] ∈ Rd is prepended to the tokenized image in (1) as Z0 := (z[CLS];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Z0) ∈ R(n+1)×d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Transformer Layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Every layer of the transformer con- sists of a multi-head self attention (MSA) block followed by a multi-layer perceptron (MLP) block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' In the MSA block, the input, Zl−1 ∈ Rn×d, for l ∈ L, is first projected into three learnable embeddings {Q, K, V } ∈ Rn×d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' The atten- tion matrix A, is calculated as A := σ �QKT √ d � ∈ Rn×n (2) where σ(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=') denotes the row-wise softmax operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' The “multi-head” in MSA is defined by considering h attention heads where each head is a sequence of n × d h matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' The attention heads are reprojected back to n × d using a linear layer which is combined with the value matrix as ZMSA := AV ∈ Rn×d (3) The output representations from the MSA block is then in- put to the MLP block which comprises two linear layers separated by a GeLU activation [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' At a given layer l, the computational flow of representations through a trans- former block is denoted as Zl ← ZMSA l + Zl−1, (4) Zl ← MLP(Zl) + Zl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' (5) Both the MSA and MLP blocks have residual connections with layer normalization (LN) [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' While MSA blocks in each layer of the transformer learn representations inde- pendently, in the next subsection, we show that empirically there exist high correlation across these layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Motivation: Layer Correlation Analysis Attention-map correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' The MSA block in ViT en- codes the similarity of each patch to every other patch as an n × n attention matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' This operator is computation- ally expensive with O(n2) complexity (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' As ViTs scale up, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=', as n increases, the complexity grows quadrati- cally and this operation becomes a bottleneck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Recent NLP works [51, 52] have shown that self-attention across adja- cent layers in SoTA language models exhibit very high cor- relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' This raises the question – is it worth to compute self-attention at every layer of a vision transformer?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' To address this question, we analyze the correlation of the self-attention maps across different layers of ViT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' As shown in Figure 2, the self-attention maps from the class token, A[CLS], exhibit high correlation especially in the interme- diate layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' The cosine similarity between A[CLS] l−1 and A[CLS] l can be as high as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='97, as indicated in the bottom of each attention map in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Similar behavior is ob- served from other token embeddings, which we analyze in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' We quantitatively analyze this correlation across all the samples of the validation set of ImageNet-1K, by computing the Centered Kernel Align- ment (CKA) [12, 27] between A[CLS] i and A[CLS] j for every i, j ∈ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' CKA measures the similarity between represen- tations obtained from intermediate layers of the network, where a high value of CKA indicates high correlation be- tween the representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' From Figure 3 (a), we observe that ViT-T has a high correlation across A[CLS] especially from layer 3 through 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Feature correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' In ViTs, the high correlation is not just limited to A[CLS], but the representation from MSA blocks, ZMSA, also show high correlation throughout the model [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' To analyze the similarity across these represen- tations, we compute the CKA between ZMSA i and ZMSA j for every i, j ∈ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' We observe from Figure 3 (b), that ZMSA also have high similarity across adjacent layers of the model especially in the earlier layers, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=', from layer 2 through 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' 4 𝑍!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' "# $%& 𝑛×𝑑 𝑛×2𝑑 spatial flatten 𝑛× 𝑛×2𝑑 𝑛×2𝑑 reshape DwC FC 𝑍"!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' $%& 𝑟×𝑟 𝑛× 𝑛×2𝑑 𝑛×𝑑 FC & ECA 𝑛×𝑑 SKIPAT parametric function Φ MLP ⨁ MSA ⨁ MSA ⨁ MLP ⨁ ⨁ MLP ⨁ … MSA 𝑍!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' "# $%& 𝑍"!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' $%& skip skip MSA ⨁ skip MLP ⨁ Φ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' "# Φ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' "$ … Φ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' SKIPAT framework We illustrate SKIPAT on ViT [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' The SKIPAT parametric function (Φ) uses representations of the MSA block (in solid color) ZMSA l−1 as input, which undergoes a series of transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' An element-wise summation (�) with the output of the MLP block from layer l − 1 and ˆZMSA l is used as input to the MLP block at layer l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' The MSA operation (crossed out) is thus not computed and is discarded from the computational graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' With SKIPAT the total number of layers remains unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Improving Efficiency by Skipping Attention Based on our observation of high representation similarity across MSA blocks of a transformer (subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='2), we propose to leverage the correlation across both the atten- tion matrix and the representations from the MSA block to improve the efficiency of vision transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Instead of computing the MSA operation (3) independently at every layer, we explore a simple and effective strategy to utilize dependencies across the features from these layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' In particular, we propose to skip MSA computation in one or more layers of a transformer by reusing representations from its adjacent layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' We term this operation as Skip Attention or SKIPAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' As the compute and memory benefit from skipping the entire MSA block is greater than skipping just the self-attention operation (O(n2d+nd2) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' O(n2d)), in this paper we focus on former.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' However, instead of di- rectly re-using features, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=', copying the features from the source MSA block to one or more adjacent MSA blocks, we introduce a parametric function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' The parametric func- tion ensures that directly reusing features does not affect the translation invariance and equivariance in these MSA blocks and acts as a strong regularizer to improve model generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' SKIPAT parametric function Let Φ : Rn×d → Rn×d denote the parametric function that maps output of the MSA block from l − 1 to l as ˆZMSA l := Φ(ZMSA l−1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Here, ˆZMSA l is the approximation of ZMSA l .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' The parametric function can be as simple as an identity function, where ZMSA l−1 is directly reused.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Instead of computing MSA operation at l, we use ZMSA l−1 as the input to the MLP block at l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' When using an identity function, due to the absence of MSA operation at l, the relation across tokens is no longer encoded in the at- tention matrix, which affects representation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' To mitigate this, we introduce the SKIPAT parametric function inspired from ResNeXt [62] as shown in Figure 4, to en- code local relations among tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' The SKIPAT parametric function consists of two linear layers and a depth-wise con- volution (DwC) [9] in between, as follows: ˆZMSA l := ECA � FC2 � DwC � FC1(ZMSA l−1 ) ��� (6) In the case of supervised learning, we first separate the CLS embeddings from ZMSA ∈ R(n+1)×d into class embeddings ZMSA C ∈ Rd and the patch embeddings to ZMSA P ∈ Rn×d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' The patch embeddings are then input to the first linear layer FC1 : Rn×d → Rn×2d, which expands the channel di- mension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' This is followed by DwC : R √n×√n×2d → R √n×√n×2d with kernel r × r to capture cross-token re- lations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Note that before the DwC operation, we spatially reshape the input matrix to a feature tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' The output of the DwC is then flattened back to a vector and fed to the last FC layer FC2 : Rn×2d → Rn×d which reduces the channel dimension back to its initial dimension d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' We use GeLU activations after FC1 and DwC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Following [53], we use efficient channel attention module (ECA) after FC2 to enhance the cross-channel dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' The ECA module first aggregates the features along the channel dimension using global average pooling (GAP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' A 1 × 1 convolution with adaptive kernel size proportional to channel dimension is applied followed by sigmoid activation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' This operation of the ECA module enhances cross-channel dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' We then concatenate the embedding of the class-token with the output of the ECA to obtain ˆZMSA l .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' SKIPAT framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' The overall framework of SKIPAT is illustrated in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' SKIPAT can be in- corporated into any transformer architecture which we empirically show in subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Depending on the architecture, one can skip the MSA operation in one or more layers of the transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' In ViT, as we empirically observe that representations from the MSA block, ZMSA, have high correlations from layer 2 through 7 (subsec- tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='2), we employ the SKIPAT parametric function in these layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' This means that we use the ZMSA 2 as input to the SKIPAT parametric function and skip MSA operations in layers 3-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Instead, the features from the output of the SKIPAT parametric function is used as input to the MLP block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' The computation flow of representations is now 5 modified to Zl ← Φ(ZMSA l−1 ) + Zl−1 (7) Zl ← MLP(Zl) + Zl (8) Due to the presence of residual connections in the MSA and MLP blocks, which is standard in ViT [15], the MLP blocks at layer 3 through 8 learn representations independently and cannot be discarded from the computational graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' It is im- portant to note that, with SKIPAT the total number of layers in ViT remain unchanged, but there are fewer MSA blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Complexity: MSA vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' SKIPAT The self-attention oper- ation involves three operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Firstly, the token embed- dings are projected into query, key and value embeddings, secondly, attention matrix A is computed as dot product be- tween Q and K and finally, the output representations are computed as dot product between A and V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' This results in a complexity of O(4nd2 + n2d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Since d ≪ n, the com- plexity of MSA block can be reduced to O(n2d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' The SKIPAT parametric function consists of two linear lay- ers and one depth-wise convolution operation, which re- sults in a O(2nd2 + r2nd) complexity, where r × r is the kernel size of the DwC operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' The overall complex- ity of SKIPAT can be reduced to O(nd2) since r2 ≪ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Thus, SKIPAT has fewer FLOPs than the MSA block as O(nd2) < O(n2d) when n increases as transformers scale up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Experiments 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Image Classification We use ViT-T/16 [15], ViT-S/16 [15] and ViT-B/16 [15] as our backbone on ImageNet-1K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' For fair comparisons, we follow the experimental settings in [48] and evaluate SKIPAT against SoTA methods: A-ViT [63], Dynamic- ViT [38], SViTE [7], SPViT [26], ATS [17], PS-ViT [46], HVT [40] and Rev-Vit [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' To the best of our knowledge, these are all the works that improve the efficiency of ViT without modifying its underlying architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' From Table 1, we observe that SKIPAT achieves the best ac- curacy vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' efficiency trade-off compared to all SoTA meth- ods on different variants of ViT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Notably, we outperform baseline ViT-T, ViT-S and ViT-B by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='1%, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='4% and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='4% respectively, while SoTA methods achieve lower accuracy or are on-par with the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Since SKIPAT uses a para- metric function to skip computing MSA blocks, our reduc- tion in number of parameters and in FLOPs is comparable to the SoTA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' In terms of throughput, SKIPAT is 19%, 21% and 25% faster than the baseline ViT-T, ViT-S and ViT-B re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Dehghani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' [13] highlight the significance of using throughput as a metric to measure model efficiency: as the reduction in FLOPs does not necessarily correspond BACKBONE METHOD TOP-1↑ PARAM↓ GFLOPS↓ THROUGHPUT↑ (%) (×106) (IM/S ×103) ViT [15] 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='8 A-ViT [63] 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='8 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='3 Dynamic ViT [43] 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='9 – 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='9 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='1 SViTE [7] 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='9 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='2 ViT-T/16 SPViT [26] 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='9 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='7 ATS [17] 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='9 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='1 PS-ViT [46] 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='6 – 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='7 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='6 HVT [40] 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='7 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='2 SKIPAT 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='9 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='9 ViT [15] 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='8 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='2 A-ViT [63] 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='6 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='4 Dynamic ViT [43] 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='3 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='6 SViTE [7] 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='2 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='5 ViT-S/16 ATS [17] 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='7 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='3 PS-ViT [46] 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='4 – 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='9 SPViT [26] 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='3 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='5 Rev-ViT [34] 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='8 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='6 HVT[40] 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='0 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='1 SKIPAT 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='2 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='8 ViT [15] 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='8 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='3 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='2 SViTE [7] 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='6 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='0 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='3 ViT-B/16 Rev-ViT [34] 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='5 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='3 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='2 PS-ViT [46] 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='5 – 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='6 SKIPAT 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='2 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='7 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='5 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Image classification on ImageNet-1K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Accuracy vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' ef- ficiency comparison of SKIPAT with SoTA methods for image res- olution 224 × 224.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' For all the methods, we measure throughput (image/sec) with a batch size of 1024 on a single NVIDIA A100 GPU, averaged over the validation set of ImageNet-1K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' to improvements in latency, as it does not take into account the degree of parallelism or other hardware details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' In line with this argument, we observe that while SoTA methods such as ATS [17] and SPViT [26] achieve large reduction in FLOPs, they actually have lower throughput when com- pared to SKIPAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Furthermore, HVT [40] while achieving a higher gain in both throughput and FLOPs has poor top- 1 accuracy (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='6% drop in ViT-T and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='8% drop in ViT-S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Thus, SKIPAT demonstrates the ability to simultaneously improve both accuracy and throughput over SoTA methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Visualizing attention maps and ZMSA correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' We analyze the effect of the SKIPAT parametric function by vi- sualizing the mean of attention heads of the CLS token from the last four layers of ViT-T/16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' From Figure 5, we observe that while attention maps from vanilla ViT (last two lay- ers) do not solely attend to the object, the attention maps from SKIPAT accurately focuses on the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' It is inter- esting to note that, the attention maps from SKIPAT are also capable of attending to multiple objects in the image (Fig- ure 5: second example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' We further analyze the CKA of the representations from MSA block across all the layers of ViT-T/16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' From Figure 6, we observe that ZMSA has lower correlation across layers except between the layers where the MSA operation is skipped (layer 3 to 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' How- ever, unlike vanilla ViT (Figure 3 (b)) the correlation from each layer to every other layer is quite low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' This shows that 6 A!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=" [#$%] A'( [#$%] A'' [#$%] A') [#$%] baseline baseline SKIPAT SKIPAT Figure 5." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Visualizing attention maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Mean of the attention of different heads from A[CLS] from last four layers of ViT-T/16 on the validation set of ImageNet-1K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Attention maps from last four blocks show that SKIPAT localizes the object better than vanilla ViT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' 1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' CKA analysis of SKIPAT shows that ZMSA has lower correlation between layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' The high correlation is only between consecutive layers 2 through 8, where the MSA operation is skipped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' our SKIPAT parametric function acts as a strong regularizer and thus improves the representations of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' METHOD JACCARD↑ CORLOC↑ ViT-T [15] 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='2 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='5 ViT-T + SKIPAT 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='0 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='5 ViT-S [15] 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='0 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='6 ViT-S + SKIPAT 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='0 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='2 ViT-B [15] 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='6 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='4 ViT-B + SKIPAT 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='8 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='2 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Unsupervised Segmentation and Object Localiza- tion using Jaccard similarity [5] and Correct Localization (Cor- Loc) [37], on the validation set of Pascal VOC2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' All models have been pretrained on ImageNet-1K in a supervised setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Visualization of segmentation masks using vanilla ViT-S/16 (top) and ViT-S + SKIPAT (bottom) pretrained supervis- edly on ImageNet-1K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' We visualize masks obtained by threshold- ing the self-attention maps to keep 80% of the mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Probing self-attention maps in ViTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' We further analyze whether pretrained ViTs can attend to semantically mean- ingful regions of the image when evaluated on a different dataset without fine-tuning it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' We follow the evaluation pro- tocol in [5], and visualize the segmentation masks produced from the final layer of the pretrained SKIPAT on the Pascal- VOC12 [16] validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' From Figure 7, 1 we observe that while vanilla ViT-S/16 does not accurately attend to the object, SKIPAT is able to localize objects quite accu- rately without any fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' To quantify this observa- tion, we follow [5] and use the Jaccard similarity between predicted segmentation mask and ground truth mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' As shown in Table 2, SKIPAT outperforms different variants of vanilla ViT with a significant gap in terms of Jaccard sim- ilarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Additionally, we measure the quality of the gener- ated maps for unsupervised object localization using Cor- Loc [37] as the evaluation metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' From Table 2, we ob- serve that SKIPAT achieves notable gains across all variants of ViT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Performance on mobile device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' To verify the efficiency of SKIPAT on low-power devices, we measure its inference time (averaged over 20 iterations) on a Samsung Galaxy S22 device powered by Qualcomm “Snapdragon® 8 Gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' 1 Mobile Platform” with a Qualcomm® HexagonTM proces- sor2, for image resolutions of 224×224 and 384×384 using ViT-T/16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' The inference is performed on Neural Processing Unit in 8-bit precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' As shown in Table 3, SKIPAT im- proves the runtime by 19% for image size of 224 × 224.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' The gain is even larger at 34% for image resolution 384 × 1The original image sources, before masking, from left to right: Kangra valley train (CC BY-SA 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='0) Ecuadorian fishing boat (CC BY-SA 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='0) Sheep near Snowshill (CC BY-SA 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='0) 2Snapdragon and Qualcomm Hexagon are products of Qualcomm Technologies, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' and/or its subsidiaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' 7 1 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='4 1384, since the number of token increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Thus, skipping computationally-heavy MSA blocks increases throughput by large margins and is confirmed even on mobile hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' METHOD 224 × 224 384 × 384 ViT-T/16 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='65 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='49 ViT-T/16 + SKIPAT 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='76 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='22 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' On-device latency (in msec) of vanilla ViT vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' SKIPAT for different image resolutions on a Samsung Galaxy S22 powered by Qualcomm Snapdragon 8 Gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Self-Supervised Learning with DINO Next, we show the generality of SKIPAT as its use in the backbone for self-supervised representation learning (SSL), using DINO [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Since, SSL methods are quite expensive in the pretraining stage in terms of compute and training time, we illustrate that SKIPAT achieves comparable performance to using a ViT but with shorter training time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Following the experimental settings of DINO [5], we use ViT-S/16 [15] as our student and teacher networks with SKIPAT parametric function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' We pretrain both baseline and ours using DINO for 100 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' We observe that SKIPAT achieves almost the same performance as fully trained DINO with around 26% less training time (73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='3% in 96 GPU-hours vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='6% in 131 GPU-hours).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' When trained on 100 epochs, we ob- serve that SKIPAT outperforms DINO by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='5% (74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='1% vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='6%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' We show the performance of SKIPAT to down- stream tasks in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Semantic Segmentation on ADE20K We go beyond classification and show the performance of SKIPAT to dense prediction tasks such as seman- METHOD BACKBONE MIOU↑ GFLOPS↓ THROUGHPUT↑ ResNet-101 [65] 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='7 261 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='1 Semantic FPN [25] PoolFormer-S36 [65] 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='0 191 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='4 PoolFormer-M36 [65] 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='4 271 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='4 ResNet-18 [23] 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='9 886 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='1 ResNet-101 [23] 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='9 1031 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='0 Swin-T [31] 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='8 945 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='2 ConvNeXt-T [32] 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='7 939 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='7 UperNet [61] ViT-T [15] 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='3 212 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='1 ViT-T + SKIPAT 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='6 173 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='7 ViT-S [15] 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='4 360 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='5 ViT-S + SKIPAT 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='3 283 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='2 ViT-B [15] 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='6 787 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='1 ViT-B + SKIPAT 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='3 633 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='5 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Semantic Segmentation results on ADE20K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' All mod- els are pretrained on ImageNet-1k and fine-tuned on ADE20K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Following Swin [31] and ConvNeXt [32], we report mIoU with multi-scale testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' FLOPs and throughput are calculated on the input size of 2048 × 512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Throughput of all models are measured with a batch size of 1 on a single NVIDIA A100 GPU, averaged over 100 forward passes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' METHOD PSNR↑ SSIM↑ GFLOPS↓ THROUGHPUT↑ UNet [45] 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='65 35 – DAGL [38] 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='953 255 – DeamNet [44] 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='957 145 – MPRNet [68] 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='71 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='958 573 – NBNet [8] 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='959 91 – Restormer [67] 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='960 140 – Uformer-T [57] 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='66 – 12 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='6 Uformer-T + SKIPAT 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='69 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='959 11 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='2 Uformer-S [57] 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='959 44 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='1 Uformer-S + SKIPAT 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='84 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='960 39 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='9 Uformer-B [57] 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='89 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='960 89 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='2 Uformer-B + SKIPAT 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='960 77 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='9 Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Image denoising on SIDD dataset using PSNR and SSIM [56] as the evaluation metrics in the RGB space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' FLOPs and throughput are calculated on the input size of 256 × 256, on a single NVIDIA V100 GPU, averaged over the test set of SIDD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' tic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' We follow the experimental settings in [31, 32] and use MMSegmentation [11] to evaluate SKIPAT on ADE20K [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' We observe from Table 4, that SKIPAT consistently outperforms all variants of ViT with 15% fewer FLOPs and 25% improved throughput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Inter- estingly, SKIPAT-S (ViT-S + SKIPAT) achieves 8% higher mIoU while being faster than ViT-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Furthermore, SKIPAT- S has comparable mIoU with Swin-T [31] whilst having 3× fewer FLOPs and being 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='7× faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Comparing to fully convolution-based architectures, SKIPAT-T (ViT-T + SKIPAT) is on par with ResNet-18 in mIoU while having 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='7× fewer FLOPs and being 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='8× faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Image Denoising SKIPAT can also generalize to low-level tasks such as im- age denoising on SIDD [1], which consists of images with real-world noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' We also demonstrate that SKIPAT can generalize to other transformer architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' In partic- ular, we apply it on Uformer [57], a SoTA image de- noising model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Uformer is a U-shaped hierarchical net- work with Swin transformer blocks as the encoder and de- coder, and skip connections between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' In SKIPAT, we skip window self-attention (WSA) block in each de- coder block by reusing attention of the corresponding en- coder block via SKIPAT parametric function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Detailed im- plementation is in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Follow- ing the experimental settings in [57], we observe in Ta- ble 5 that SKIPAT outperforms the baseline Uformer vari- ants with the 25% higher throughput on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Further- more, we observe that SKIPAT-B (Uformer-B + SKIPAT) achieves comparable performance with Restormer [67], in terms of PSNR and SSIM, which is the SoTA image denois- ing method while having 2× fewer FLOPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Thus, we show the ability of SKIPAT to generalize to different tasks and also across architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' 8 METHOD FastDVDNet PaCNet VRT UniFormer UniFormer+ [47] [49] [30] [28] SKIPAT PSNR↑ 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='04 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='79 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='52 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='24 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='16 GFLOPS↓ 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='9 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='8 708.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='8 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='2 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='1 Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Video denoising Quantitative comparison (average RGB channel PSNR) with state-of-the-art methods for video denoising on DAVIS, with additive noise level σ = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' FLOPs are calculated per frame per patch size of 256 × 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' FUNCTION KERNEL CHANNEL TOP-1↑ THROUGHPUT↑ Φ EXPANSION (%) (img/sec ×103) ViT-T 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='8 IDENTITY 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='1 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='5 CONV 5 × 5 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='2 DWC 5 × 5 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='6 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='8 3 × 3 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='1 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='3 SKIPAT 5 × 5 2 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='7 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='9 7 × 7 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='4 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='5 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='4 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='4 SKIPAT 5 × 5 1 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='9 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='2 2 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='7 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='9 Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Ablations using ViT-T/16 on ImageNet-1K for 100 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' We measure throughput (image/sec) with a batch size of 1024 on a single NVIDIA A100 GPU, averaged over the validation set of ImageNet-1K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Video Denoising We further apply our model to the temporal task of video denoising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' As encoder and decoder backbone, we use Uni- Former [28], a U-shaped hybrid encoder-decoder architec- ture with 3D convolutions and spatio-temporal global self- attention blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Detailed implementation is provided in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Similar to image denoising, we skip MSA blocks in the decoder, however, simply adopt a naive SKIPAT, where we reuse window self-attention ma- trix, A, of the corresponding encoder block using an Iden- tity function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' We empirically observe that reusing atten- tion works better in this task, and shows the ability of our method to be applied for different scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' We fol- low the experimental settings in [47] and train SKIPAT on DAVIS [41] dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' We train using Charbonnier loss [6] on patches of 7 × 128 × 128 using a multiple-input, multiple- output (MIMO) paradigm (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' the model outputs 7 recon- structed frames from 7 input frames) for noise level σ = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' From Table 6, we observe that SKIPAT performs on par with baseline Uniformer, while having 17% fewer FLOPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' This shows that SKIPAT can generalize to temporal tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Ablations All ablations are performed using ViT-T/16 on ImageNet- 1K for 100 epochs to reduce the training time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Unless spec- ified, following SKIPAT we skip the MSA blocks from layer 3 through 8 for all ablations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Parametric function Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' We study the effect of different parametric functions in terms of accuracy and throughput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' As discussed in subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='3, Φ can be as simple as an identity function, where we directly reuse representations from a previous MSA block into one of more subsequent MSA blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' From Table 7, using an identity function re- sults in a 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='7% drop in top-1 accuracy while being 47% faster than baseline ViT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Using a convolution or DwC [9] with kernel size 5 × 5 as a parametric function leads to the same performance as the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' However, DwC is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='2% better and 50% faster than convolution, and 34% faster than the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' SKIPAT parametric function outperforms all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Kernel size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' By default SKIPAT uses a DwC with kernel size of 5 × 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' As shown in Table 7, while using a 3 × 3 ker- nel is faster than default SKIPAT by 6%, it is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='6% worse in terms of accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' A larger kernel size has poor accu- racy and lower throughout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' However, irrespective of the kernel size, SKIPAT outperforms the baseline ViT-T by at least 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='4%, showing its ability to encode cross-token inter- actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Channel expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' In the SKIPAT , the first linear layer FC1, expands the channel dimension from d → 2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Ta- ble 7 shows the impact of channel dimension, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=', when the channel expansion ratio of FC1 is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='0 (d → d) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='5 (d → d/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' We observe that while the lower channel expan- sion ratio improves the throughput, it performs worse than default SKIPAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' This could be due to sub-optimal represen- tations encoded by the DwC due to fewer filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Skipping MSA in alternate configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Instead of skipping the MSA operation in the layers 3 − 8, we study the effect of skipping MSA operation at l ∈ {3, 5, 7, 9}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' We observe the latter configuration outperforms the base- line ViT by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='7% (65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='8 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='5%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' However, it performs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='2% lower and is 8% slower than our default SKIPAT con- figuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Conclusion We proposed SKIPAT, a plug-in module that can be placed in any ViT architecture for reducing the costly Self- Attention computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' SKIPAT leverages the dependency across MSA blocks and bypasses attention computation by re-using attention from previous MSA blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' To ensure that the metaphorical sharing is caring we introduced a sim- ple and light parametric function that does not affect the inductive bias encoded in MSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' The SKIPAT function is able capture cross-token relations and outperforms the base- line while being computationally faster in terms of through- put and FLOPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' We plugged SKIPAT into different trans- former architectures and showed its effectiveness on 7 dif- ferent tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' References [1] Abdelrahman Abdelhamed, Stephen Lin, and Michael S Brown.' metadata={'source': 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In CVPR, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' 1, 2, 3, 6, 13 [64] Chengxuan Ying, Guolin Ke, Di He, and Tie-Yan Liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Lazy- former: Self attention with lazy update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' arXiv preprint arXiv:2102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='12702, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' 2, 3 [65] Weihao Yu, Mi Luo, Pan Zhou, Chenyang Si, Yichen Zhou, Xinchao Wang, Jiashi Feng, and Shuicheng Yan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Metaformer is actually what you need for vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' In CVPR, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' 2, 3, 8, 13 [66] Li Yuan, Yunpeng Chen, Tao Wang, Weihao Yu, Yujun Shi, Zi-Hang Jiang, Francis EH Tay, Jiashi Feng, and Shuicheng Yan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Tokens-to-token vit: Training vision transformers from scratch on imagenet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' In ICCV, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' 2, 3, 13 [67] Syed Waqas Zamir, Aditya Arora, Salman Khan, Mu- nawar Hayat, Fahad Shahbaz Khan, and Ming-Hsuan Yang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Restormer: Efficient transformer for high-resolution image restoration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' In CVPR, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' 8 [68] Syed Waqas Zamir, Aditya Arora, Salman Khan, Munawar Hayat, Fahad Shahbaz Khan, Ming-Hsuan Yang, and Ling Shao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Multi-stage progressive image restoration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' In CVPR, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' 8 [69] Pengchuan Zhang, Xiyang Dai, Jianwei Yang, Bin Xiao, Lu Yuan, Lei Zhang, and Jianfeng Gao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Multi-scale vision long- former: A new vision transformer for high-resolution image encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' In ICCV, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' 2, 3, 4 [70] Bolei Zhou, Hang Zhao, Xavier Puig, Sanja Fidler, Adela Barriuso, and Antonio Torralba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Scene parsing through ade20k dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' In CVPR, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' 8, 12 11 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Implementation details A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Hyper-parameters ImageNet-1K: Image classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' We train SKIPAT on the ILSVRC-2012 dataset [14] with 1000 classes (referred as ImageNet-1K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' We follow the experi- mental settings of DeIT [48] and use the codebase from the timm library [58] to train ViT-T, ViT-S and ViT-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' We use the default 16 × 16 patch size, using an image resolution of 224 × 224 with total number of tokens n = 196.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' We train baseline ViT and SKIPAT for 300 epochs from scratch on 4 NVIDIA A100 GPUs using batch sizes of 2048 for ViT-T and 1024 for ViT-S and ViT-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' ImageNet-1K: Self-supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' We follow the experimental settings of DINO [5] and pre-train DINO and SKIPAT on ImageNet-1K using ViT-S/16 as the backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' While likely the hyperparameters could be tuned further for our proposed SKIPAT method, we use same hyper- parameters for both the baseline and ours, yielding a con- servative estimate of our model’s performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' We pre-train both methods from scratch for 100 epochs using 4 NVIDIA A100 GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' For linear-probing, we freeze the backbone from the pre-training stage and fine-tune the classifier for 100 epochs, exactly as done in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Pascal-VOC2012: Unsupervised object segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' We use the Pascal VOC 2012 [16] validation set for this experiment, containing 1449 images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' We follow DINO and obtain unsupervised segmentation masks by threshold- ing the averaged self-attention map (extracted from the last layer of a pretrained ViT/SKIPAT model) to keep 80% of the mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' The Jaccard similarity J between a predicted mask, P, and ground-truth mask, G, is defined as: J(P, G) = G ∩ P G ∪ P We report Jaccard similarity, averaged over all the samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' ADE20K: Semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' We evalu- ate SKIPAT on ADE20K [70], a widely-used semantic segmentation dataset, covering 150 semantic categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' The dataset includes 20K and 2K images in the train- ing and validation set, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Different variants of SKIPAT are evaluated using UperNet [61] as the backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' We use our ImageNet-1K pretrained model to initialize the backbone and Kaiming [22] initialization for other layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' We use AdamW [33], with an initial learning rate of 6e−5, weight decay of 1e−2, and linear warmup of 1500 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' All models are trained for 160K iterations with a batch size of 16 using MMSegmentation repo [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' We keep the same hyper-parameters for SKIPAT and ViT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' SIDD: Image denoising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' We follow the experimental set- tings in Uformer [57] and train SKIPAT on the Smartphone Image Denoising Dataset (SIDD) [2] which consists of real-world noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' The training samples are first randomly cropped to 128×128 patches and input to the model, which is trained for 250 epochs using batch size 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' The model is then evaluated on images of size 256 × 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' DAVIS: Video denoising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' We further apply our model to the temporal task of video denoising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' We adopt the same U- shape encoder-decoder based architecture of UFormer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' As the encoder and decoder backbone, we use UniFormer [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' We train the model on noise level σ = 30 using Charbonnier loss [6] on patches of 7 × 128 × 128 using a multiple-input, multiple-output (MIMO) paradigm [30] (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=', the model out- puts 7 reconstructed frames from 7 input frames).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' During inference, a video is divided into 3D patches of 7×128×128 with an overlap of 10 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Each patch is fed to the model and the outputs are merged to obtain the final denoised video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Following [47], PSNR is calculated as averaged over videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' We use the same training hyper-parameters as im- age denoising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Architecture Image Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' All baseline ViT variants have 12 layers in total, which remains unchanged with SKIPAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Fol- lowing the CKA analysis of ZMSA in Figure 3(b) of our main paper, we skip computing the MSA blocks in layer 3 through 8 for all ViT variants and retrain it from scratch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Image Denoising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' We apply SKIPAT to Uformer [29] a SoTA image denoising model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Uformer is a U-shaped hi- erarchical network with Swin transformer blocks as the en- coder and decoder, and skip connections between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' In SKIPAT, we skip window self-attention (WSA) block in each decoder block by reusing attention of the correspond- ing encoder block via SKIPAT parametric function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Let ZWSAe l ∈ Rn×c denote the output of the WSA block at layer l from the encoder and Zd l−1 ∈ Rn×c denote the output of the layer l − 1 from the decoder of Uformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' The input to the WSA block (which is skipped) at layer l of the decoder is given by ˆZWSAd l = Φ(ZWSAe l ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Zd l−1) ∈ Rn×2c (9) Here, “;”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' denotes concatenation along the channel dimen- sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' We show the framework of SKIPAT on Uformer in Figure 8 Video Denoising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' we apply SKIPAT to UniFormer [28], a U-shaped hybrid encoder-decoder architecture with 3D convolutions and spatio-temporal global self-attention blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' The encoder of UniFormer comprises two 3D 12 𝑥 MLP WSA MLP WSA MLP WSA skip MLP WSA skip 𝑍𝑙 WSA𝑒 መ𝑍𝑙 WSA𝑑 Φ𝑖+1 Φ𝑖 … ⨁ ⨁ ⨀ ⨁ ⨁ MLP WSA ⨁ ⨁ MLP WSA ⨁ ⨁ ⨁ ⨁ ⨁ ⨁ ⨁ … MLP WSA ⨁ ⨁ ⨀ ො𝑥 Input image Denoised image Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Framework of SKIPAT on Uformer Instead of standard MSA block in ViT, Uformer uses window self-attention (WSA) block similar to Swin Transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' We skip WSA block in the layers close to the bottleneck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' convolution layers followed by two spatio-temporal trans- former layers with global self-attention (MSA) blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' A downsampling operation is used after every layer in the en- coder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' The decoder is symmetric to the encoder with two transformer layers followed by two 3D convolution layers with an upsampling operation between each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Similar to Uformer, skip connections are used between encoder and decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Similar to image denoising, we skip MSA blocks in the decoder, however, simply adopt a naive SKAT, where we reuse global self-attention matrix Al, from the encoder at layer l in the corresponding decoder stage at the same layer using an Identity function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Let Ae l ∈ Rn×n denote the self-attention matrix at layer l from the encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' The self-attention in the decoder stage at layer l is given by Ad l = I(Ae l ) ∈ Rn×n, where I(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=') is the identity function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' We observe that skipping attention A using an identity function works better than skipping MSA blocks using the SKIPAT parametric function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' This shows the generality of SKIPAT, regardless of approximating attention or MSA blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Additional experiments Image classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Here we extend our SoTA compar- ison with methods that go beyond vanilla ViT architec- tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' These methods include hierarchical (Swin, PVT, Poolformer, MobileViT, Twins-SVT) and Hybrid (Con- vNext, CoAT) architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' We provide the complete set of SoTA methods that improve the efficiency of ViT either by token sampling (extending Table 1 in our main paper), using hybrid architectures or window self-attention blocks in Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Apart from methods that perform efficient token sampling, none of the other methods are directly compara- ble because they modify the underlying architecture of ViT, either by using window self-attention blocks or reducing the overall number of transformer layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' BACKBONE METHOD TOP-1 PARAM GFLOPS THROUGHPUT (%) (×106) (img/sec ×103) T2T-ViT [66] 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='1 – ConvNeXt (iso) [32] 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='8 ViT [15] 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='8 A-ViT [63] 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='8 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='3 Dynamic ViT [43] 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='9 – 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='9 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='1 ViT-T/16 SViTE [7] 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='9 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='2 SPViT [26] 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='9 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='7 ATS [17] 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='9 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='1 PS-ViT [46] 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='6 – 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='7 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='6 HVT [40] 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='7 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='2 SKIPAT 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='9 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='9 ConvNext-T [32] 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='1 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='6 ConvNeXt (iso) [32] 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='7 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='3 Swin-T [31] 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='3 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='5 T2T-ViT [66] 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='7 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='2 – CoaT-Lite-Small 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='9 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='0 – Poolformer-S24 [65] 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='3 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='4 – Twins-SVT-S [10] 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='7 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='8 – MobileViT-S [35] 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='0 – PVT [54] 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='8 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='8 – ViT-S/16 ViT [15] 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='8 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='2 A-ViT [63] 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='6 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='4 Dynamic ViT [43] 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='3 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='6 SViTE [7] 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='2 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='5 ATS [17] 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='7 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='3 PS-ViT [46] 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='4 – 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='9 SPViT [26] 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='3 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='5 Rev-ViT [34] 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='8 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='6 HVT[40] 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='0 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='1 SKIPAT 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='2 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='8 Swin-S [31] 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='5 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='0 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='0 Twins-SVT-B [10] 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='2 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='6 – PVT [54] 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='7 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='4 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='8 – ConvNeXt (iso) [32] 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='0 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='3 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='3 ViT-B/16 ViT [15] 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='8 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='3 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='2 SViTE [7] 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='6 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='0 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='3 Rev-ViT [34] 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='5 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='3 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='2 PS-ViT [46] 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='5 – 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='6 SKIPAT 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='2 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='7 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='5 Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Image classification on ImageNet-1K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Accuracy vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' ef- ficiency comparison of SKIPAT with SoTA methods for image res- olution 224 × 224.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' For all the methods, we measure throughput (image/sec) with a batch size of 1024 on a single NVIDIA A100 GPU, averaged over the validation set of ImageNet-1K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Unsupervised segmentation of DINO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' We follow DINO [5] and evaluate the performance of baseline DINO vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' SKIPAT on unsupervised object segmentation on Pascal-VOC2012 [16] dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' We follow the experimental setting as discussed in Appendix A and observe that baseline DINO has a Jaccard similarity of 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='3 while SKIPAT achieves 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' While SKIPAT outperforms DINO on image classification by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='5%, we achieve comparable performance in terms of unsupervised object segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Additional ablations Reusing self-attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' As mentioned in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='3, we skip the ZMSA in SKIPAT as the compute and memory benefit from skipping the entire MSA block is greater than 13 skipping just the self-attention operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Here we study the effect of skipping just the self-attention operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Let Al−1 denote the self-attention matrix at layer l −1, then the self-attention matrix at layer l is given by ˆAl = I(Al−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Similar to SKIPAT we skip computing the self-attention ma- trix from layers 3 through 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' As parametric function Φ, we use an identity mapping and train ViT-T/16 from scratch for 100 epochs on ImageNet-1K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' We observe from Table 9, that skipping the self-attention matrix results in a top-1 ac- curacy of 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='2% which is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='1% higher than the skipping ZMSA with an identity function (61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='1% - Table 7 of main paper).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' However, skipping self-attention matrix results in 20% decrease in throughput (8500 → 6800 images/sec) as compared to using an identity function to skip MSA block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' It is interesting to note that skipping self-attention matrix results in a lower drop in performance as compared to skip- ping MSA block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' However, applying a parametric function to skip self-attention can be challenging due to the proper- ties of the self-attention matrix, and we leave this to future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' SKIPAT in pretrained model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' As mentioned in subsec- tion A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='2, we train SKIPAT with all variants of ViT from scratch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' For completeness, we also study the effect of skip- ping the self-attention matrix and the MSA block on a pre- trained ViT-T using an Identity function, without retraining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' We observe from Table 9 that skipping the self-attention computation in layers 3 through 8, results in a top-1 accu- racy of 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='9%, while skipping MSA blocks results in top-1 accuracy of 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='8%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' It is interesting to note that the drop in top-1 accuracy from skipping self-attention is merely 19% (72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='8 → 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='9) on average and does not result in an extremely large drop as one might expect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' This shows that there in- deed exists high correlation across self-attention and ZMSA, which SKIPAT utilizes to improve the efficiency of the ViTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' METHOD TRAINING TOP-1 (%) THROUGHPUT A ✓ 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='2 6800 ZMSA ✓ 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='1 8500 A ✗ 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='9 6800 ZMSA ✗ 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='8 8500 Table 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Ablations on the effect of skipping the self-attention, A, and the MSA block, ZMSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' In the first two rows, models are trained for 100 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' In the last two rows we use a pretrained ViT-T/16 and simply skip computations in blocks 3-8 during in- ference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' For all the experiments with use Identity function as Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' CKA analysis of attention from ViT-T As discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content='2 of our main paper, we analyze the CKA of the self-attention matrix for all tokens between different layers of ViT-T/16 pretrained on ImageNet-1K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' 1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12 Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' CKA analysis of A for all tokens from pretrained vanilla ViT-T/16 on the validation set of ImageNet-1K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' We observe a high correlation for all tokens in A from layers 1 to 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' Since in the supervised setting A ∈ R(n+1)×(n+1), we first remove the CLS token to obtain AP ∈ Rn×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQfTwBe/content/2301.02240v1.pdf'} +page_content=' We then com- 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a/RdAzT4oBgHgl3EQfI_vE/content/tmp_files/2301.01073v1.pdf.txt b/RdAzT4oBgHgl3EQfI_vE/content/tmp_files/2301.01073v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..55446414940d2a88c173a91b491172d7b668928d --- /dev/null +++ b/RdAzT4oBgHgl3EQfI_vE/content/tmp_files/2301.01073v1.pdf.txt @@ -0,0 +1,3571 @@ +arXiv:2301.01073v1 [math.AP] 3 Jan 2023 +A FREE DISCONTINUITY APPROACH TO OPTIMAL PROFILES IN STOKES +FLOWS +DORIN BUCUR, ANTONIN CHAMBOLLE, ALESSANDRO GIACOMINI, AND MICKA¨EL NAHON +Abstract. In this paper we study obstacles immerged in a Stokes flow with Navier boundary condi- +tions. We prove the existence and regularity of an obstacle with minimal drag, among all shapes of +prescribed volume and controlled surface area, taking into account that these shapes may naturally +develop geometric features of codimension 1. The existence is carried out in the framework of free +discontinuity problems and leads to a relaxed solution in the space of special functions of bounded +deformation (SBD). In dimension 2, we prove that the solution is classical. +Contents +1. +Introduction +1 +2. +Notations and Preliminaries +5 +2.1. +Basic notation. +5 +2.2. +Functions of bounded variation and sets of finite perimeter +6 +2.3. +Functions of bounded deformation +6 +3. +Obstacles in Stokes fluids and drag minimization +7 +3.1. +The flow around the obstacle +7 +3.2. +The drag force +8 +3.3. +The optimization problem +9 +4. +A relaxed formulation of the shape optimization problem and statements of the main results 10 +5. +Some technical results in SBD +12 +5.1. +An immersion result +12 +5.2. +Closure of the non-penetration constraint +13 +5.3. +A lower semicontinuity result for surface energies in SBD +15 +6. +Existence of minimizers: proof of Theorem 4.8 +21 +7. +Regularity of two-dimensional minimizers: proof of Theorem 4.10 +23 +7.1. +The smoothing lemma +24 +7.2. +Regularity for quasi minimizers of the Griffith energy +30 +7.3. +Proof of Theorem 4.10 +36 +7.4. +Some remarks on a “strong” formulation of the problem +37 +References +39 +1. Introduction +Consider an obstacle E ⊂ Rd (d = 2, 3 in real applications) contained in a (finite) channel Ω in which +a fluid with viscosity coefficient µ > 0 is flowing. Assume that the flow is stationary and incompressible, +and that the associated velocity field u is equal to a constant vector V∞ on the walls of the channel. +The obstacle E experiences a force, whose component in direction of V∞ will be denoted by Drag(E) +2020 Mathematics Subject Classification. +49Q10, 76D07, 76D55, 35R35. +Key words and phrases. Free discontinuity problems, Stokes flow, Navier boundary conditions, drag. +1 + +2 +D. BUCUR, A. CHAMBOLLE, A. GIACOMINI, AND M. NAHON +and is usually called the drag force. If we further assume that the velocity of the fluid satisfies the +Stokes equation in Ω \ E and obeys to Navier boundary conditions on ∂E, the expression of the drag +force turns out to be given (up to a multiplicative constant) by +(1.1) +Drag(E) = 2µ +� +Ω\E +|e(u)|2 dx + β +� +∂E +|u|2 dHd−1, +where e(u) := 1 +2(Du+(Du)∗) denotes the symmetrized gradient of u and β > 0 is the friction coefficient +(we refer to Subsection 3.2 for details). +We are interested in minimizing the drag force among all obstacles E with a prescribed volume +and controlled surface area. Precisely we look for the existence of such an optimal obstacle and for +its qualitative properties. The existence question is not very relevant as soon as one imposes strong +geometric constraints on the admissible obstacles (e.g. convexity, uniform cone conditions, etc.) since +this may hide some specific features which would naturally occur. Indeed, letting the geometry of +the obstacle to be completely free, some qualitative behavior (blocked by rigid geometric constraints) +can be observed. This is the case of our problem, where the optimal obstacle (that we prove to exist +without imposing any geometric or topological constraint) may be composed, roughly speaking as a +union of a body with the prescribed volume and pieces of surfaces of dimension d−1. Those surfaces do +not have volume, but count for the total surface area Hd−1(∂E) and of course have a strong influence +on the flow. +Penalizing the surface area and the volume, the model problem we are interested in can be written +as +min +E +� +Drag(E) + cHd−1(∂E) + f(|E|) +� +, +where c > 0 and f : (0, |Ω|) → R ∪ {+∞} is a lower semicontinuous function. Roughly speaking, the +terms involving perimeter and volume can be thought as a price to pay in order to build the obstacle +E, and we can give the two relevant choices of function f: +f(m) = +∞1{m̸=m0} for some m0 ∈ (0, |Ω|), or f(m) = −λm for some λ > 0. +Many similar optimisation problems have been considered under the “no-slip” boundary condition, +meaning flows for which u = 0 at ∂E. Under volume constraint and an a priori symmetry hypothesis +around an axis parallel to the flow, the minimal drag question has been studied in [33] on smooth +surfaces. In [28], still under symmetry hypotheses, it was conjectured that the optimal profile in three +dimensions is a prolate spheroid with sharp ends of angle of 120 degrees. +In the same symmetry +context, let us also mention the slender body approximation of [31]. +We also refer the reader to +the paper by ˘Sver´ak [32] who, in two dimensions, proves the existence of an optimal obstacle under +topological hypotheses, namely that the obstacle has at most a given number of connected components +(in particular this number can be equal to 1). The proof is genuinely two dimensional and can not be +extended to higher dimensions. +The Navier boundary condition gives many new challenges, namely the possible apparition of lower +dimensional structures in the obstacle that minimize the drag, something which was absent under the +no-slip condition. The Navier boundary condition may be seen as a partial adherence to the boundary +of the obstacle, and it may be asymptotically obtained as a limit of flows with perfect slip on an obstacle +with rough boundary. More precisely, a periodic microstructure with the right scaling on the boundary +is modelled at the limit by a Navier boundary condition, as was proved in [13]. In dimension higher +than two it is also necessary to take into account more complex geometries for the microstructure, +which at the limit produce an anisotropic factor that favors certain directions of the flow. Moreover, +infinitesimal boundary perturbations can dramatically modify the solution of the Stokes equation with +Navier boundary conditions, while in presence of no-slip boundary conditions the solution remains + +A FREE DISCONTINUITY APPROACH TO OPTIMAL PROFILES IN STOKES FLOWS +3 +stable. We refer the reader to [8] for an analysis of those phenomena and for a discussion on the +pertinence of the Navier boundary conditions in physical models. +For a fixed obstacle E, the minimization of the drag with respect to the friction parameter β of +the Navier conditions (meaning, from a physical point of view, with respect to the microstructure +on the boundary) has been studied in [5], for both Stokes and Navier-Stokes flows. While for Stokes +flows the drag is increasing with the friction parameter, an important observation which occurs for the +Navier-Stokes equation is that the monotonicity of the drag with respect to the parameter β does not +hold. This is a reason for which the results we give in this paper for the Stokes flows are not expected +to hold, as such, for the Navier-Stokes equation. +Since the stationary velocity field associated to a Lipschitz obstacle E turns out to be characterized +variationally as the minimizer of the right hand side of (1.1) in the class of admissible velocities +Vreg +E,V∞(Ω) = +� +u ∈ H1(Ω \ E) : divu = 0, u|∂E · νE = 0, u|∂Ω = V∞ +� +(see (3.4) in Subsection 3.1 for more details), we can conveniently rephrase the minimization problem +by letting also the velocity fields intervene explicitely in the form +(1.2) +min +E,u∈Vreg +E,V∞(Ω) +� +2µ +� +Ω\E +|e(u)|2 dx + β +� +∂E +|u|2 dHd−1 + cHd−1(∂E) + f(|E|) +� +. +The first main goal of the paper is to find suitable relaxations of problem (1.2) for which we can prove +the existence of minimizers without any a priori constraint on the regularity or the topology of the +sets E. +In order to avoid unnatural geometric restrictions on the obstacle E, it is natural in view of the +third term appearing in (1.2) to let it vary within the class of sets of finite perimeter (see Subsection +2.2), and replace the topological boundary with reduced one ∂∗E. +In order to describe properly obstacles with very narrow spikes which in the limit degenerate to +(d−1)-surfaces and that cannot be taken into account through the reduced boundary, it is convenient to +consider admissible velocity fields which can be discontinuous outside E (see Subsection 3.3). Since the +symmetrized gradient e(u) is involved explicitly in (1.2), a natural family for the admissible velocities +is given by the space of functions of bounded deformation SBD. The natural relaxation of the energy +takes the form +J (E, u) :=2µ +� +Ω\E +|e(u)|2 dx + β +� +∂∗E +|u+|2 dHd−1 + β +� +Ju\∂∗E +[|u+|2 + |u−|2] dHd−1 ++ cHd−1(∂∗E) + 2cHd−1(Ju \ ∂∗E) + f(|E|), +(1.3) +where u is set equal to zero a.e. in E, while Ju denotes the discontinuity set of u and u± are the traces +of u on ∂∗E and Ju (the trace u− vanishes on ∂∗E by the choice of orientation, while u+ is on the +outward side). +Within this framework the global obstacle is given by E∪Ju, so that it contains also lower dimensional +parts, namely Ju \∂∗E: roughly speaking, for the optimal velocity these discontinuous regions generate +(d − 1)-surfaces which correspond to volumeless, lower dimensional subsets of the optimal obstacle. +Admissible velocities must be tangent to the obstacles, meaning that not only u is tangent to ∂∗E, +but also the two traces u± are orthogonal to the normal νu along the jump set Ju. The contribution +of the Navier surface term takes naturally into account the contribution from both sides given by u±. +Concerning the perimeter term, we count twice the lower dimensional parts because we see the relaxed +obstacle as a limit of regular obstacles, such that points of Ju \ ∂∗E correspond to thin parts of the +regular obstacle that collapse on a lower-dimensional structure. We could also see the perimeter term + +4 +D. BUCUR, A. CHAMBOLLE, A. GIACOMINI, AND M. NAHON +as a price to pay in order to construct the obstacle and just keep Hd−1(∂∗E ∪ Ju) instead, and the +main results of the paper would not be affected. +The relaxed optimization problem can be seen as a minimization problem on the pairs (E, u) which +has the features of classical geometrical problems for E coupled with a free discontinuity problem for +u, with a surface term depending on the traces which are subject to suitable tangency constraints and +boundary conditions. +The first main results of the paper (Theorem 4.8) concerns the existence of minimizers for the relaxed +functional J in (1.3) among the class of admissible configurations (see Definition 4.1 for the precise +definition). +The main difficulties we have to face in order to prove that the problem is well posed are the +following: +(a) the closure of the non-penetration constraint for the velocity on ∂∗E ∪ Ju under the natural +weak convergence of the problem; +(b) the lower semicontinuity of energies of the form +(1.4) +� +Ju +[|u+|2 + |u−|2] dHd−1 +associated to the Navier conditions. +Point (a) is a consequence of a lower semicontinuity result for the energy +� +Ju +� +|u+ · νu| + |u− · νu| +� +dHd−1 +which is proved in Theorem 5.2, by resorting to recent lower semicontinuity results for functionals on +SBD by Friedrich, Perugini and Solombrino [26]. +The energy of point (b) naturally appears in a scalar setting when dealing with shape optimization +problems involving Robin boundary conditions (see e.g. [7, 11, 10, 12]), and it is easily seen to enjoy +lower semicontinuity properties by working with sections. +The lower semicontinuity result in the +vectorial SBD setting is given by Theorem 5.4 and cannot rely on an easy argument by sections, which +instead would yield the lower semicontinuity of an energy of the form +� +Ju +� +|u+ · ξ|2 + |u− · ξ|2� +|ξ · νu| dHd−1 +with ξ ∈ Rd with |ξ| = 1: the optimization in ξ in order to recover (1.4) does not seem feasible in +dimension d ≥ 3. +We thus follow a different strategy based on a blow up argument in which we +reconstruct the vector quantities u± by controlling them along a sufficiently high number of directions +(see Subsection 5.3 for details): in this way we can deal with more general energy densities of the form +φ(u+) + φ(u−), where φ is a lower semicontinuous function. +The second main result of the paper (see Theorem 4.10) concerns the regularity of the relaxed +minimizers of (1.3). Provided that the volume penalization function f is Lipschitz and that we are in +two dimensions, we prove that for a minimizer (E, u) of J , the optimal obstacle E ∪ Ju is a closed set, +while the optimal velocity u is a smooth Sobolev function outside the obstacle, recovering somehow +the classical setting of the problem. More precisely we show that +(1.5) +H1(Ω ∩ ∂∗E ∪ Ju \ (∂∗E ∪ Ju)) = 0, +so that the optimal obstacle can be described as the closed set obtained by the complement of the +connected components of Ω \ ∂∗E ∪ Ju on which u does not vanish identically. +The technical ideas to prove (1.5) stem from the pioneering result of De Giorgi, Carriero and Leaci +on the Mumford-Shah problem [22], where the key of the proof is a decay estimate obtained by a + +A FREE DISCONTINUITY APPROACH TO OPTIMAL PROFILES IN STOKES FLOWS +5 +contradiction/compactness argument. For vectorial problems, a similar strategy, but definitely more +involved, was used for the Griffith fracture problem in [18] (for the two-dimensional case) and in [15] +(for higher dimension). In the fracture problem, the key compactness result relies on the possibility to +approximate a field u ∈ SBD([−1, 1]d) with a small jump set by a Sobolev function which is locally +controlled in H1 (via the classical Korn inequality). +In our case, we follow a similar approximation procedure, but we have to handle two additional +constraints: incompressibility and non-penetration at the jumps. From a technical point of view, this +is problematic since the bound in [18] in not strong enough to stay in divergence-free vector fields and +the method in [15] creates new jumps on which the non-penetration constraint is not a priori verified. +However, when restricted to two dimensions, the method of [15] leads to a stronger result, so that both +constraints can be handled. +The paper is organized as follows. In Section 2 we recall fix the notation and recall some basic +facts concerning sets of finite perimeter, functions of bounded deformation and Hausdorff convergence +of compact sets. Section 3 is devoted to the precise exposition of the drag optimization problem. In +Section 4 we detail the relaxation of the problem in the family of obstacle of finite perimeter and with +velocities of bounded deformation, and formulate the main results of the paper concerning the existence +of minimizers (in any dimension) and their regularity in dimension two. The proof of the existence of +minimizers is given in Section 6, and it is based on some technical results for SBD functions collected +in Section 5, while the regularity result is proved in Section 7. +2. Notations and Preliminaries +2.1. Basic notation. If E ⊆ Rd, we will denote with |E| its d-dimensional Lebesgue measure, and by +Hd−1(E) its (d−1)-dimensional Hausdorff measure: we refer to [23, Chapter 2] for a precise definition, +recalling that for sufficiently regular sets Hd−1 coincides with the usual area measure. Moreover, we +denote by Ec the complementary set of E, and by 1E its characteristic function, i.e., 1E(x) = 1 if +x ∈ E, 1E(x) = 0 otherwise. In addition we will say that E1 ⋐ E2 if E1 ⊂ E2. Finally we will denote +with Qx,r ⊆ Rd the cube of center x and side r: when x = 0, we will simply write Qr. +If A ⊆ Rd is open and 1 ≤ p ≤ +∞, we denote by Lp(A) the usual space of p-summable functions on +A with norm indicated by ∥ · ∥p. W 1,p(A) will stand for the Sobolev space of functions in Lp(A) whose +gradient in the sense of distributions belongs to Lp(A; Rd). Finally, given a finite dimensional unitary +space Y , we will denote by Mb(A; Y ) will denote the space of Y -valued Radon measures on A, which +can be identified with the dual of Y -valued continuous functions on A vanishing at the boundary. +We will denote by Md×m the set of d × m matrices with values in R: when d = m we will denote +by Md×d +sym the subspace of d × d symmetric matrices. For a ∈ Rd and b ∈ Rm we will denote with a ⊗ b +the element of Md×m such that +(a ⊗ b)ij = aibj, +while if a, b ∈ Rd we denote with a ⊙ b the matrix in Md×d +sym such that +(a ⊙ b)ij = aibj + ajbi +2 +. +Given ξ ∈ Rd with |ξ| = 1, we denote with ξ⊥ the hyperplane through the origin orthogonal to ξ. If +E ⊆ Rd, we set +(2.1) +Eξ := πξ⊥(E), +where π denotes the orthogonal projection, and for y ∈ ξ⊥ we set +(2.2) +Eξ +y := {t ∈ R : y + tξ ∈ E}. + +6 +D. BUCUR, A. CHAMBOLLE, A. GIACOMINI, AND M. NAHON +2.2. Functions of bounded variation and sets of finite perimeter. If Ω ⊆ Rd is open, we say +that u ∈ BV (Ω; Rm) if u ∈ L1(Ω; Rm) and its derivative in the sense of distributions is a finite +Radon measure on Ω, i.e., Du ∈ Mb(Ω; Md×m). +BV (Ω; Rm) is called the space of functions of +bounded variation on Ω with values in Rm and it is a Banach space under the norm ∥u∥BV (Ω;Rm) := +∥u∥L1(Ω;Rm) + ∥Du∥Mb(Ω;Md×m). We call |Du|(Ω) := ∥Du∥Mb(Ω;Md×m) the total variation of u. We +refer the reader to [1] for an exhaustive treatment of the space BV . +We say that u ∈ SBV (Ω; Rm) if u ∈ BV (Ω; Rm) and its distributional derivative can be written in +the form +Du = ∇u dx + (u+ − u−) ⊗ νuHd−1⌊Ju, +where ∇u ∈ L1(Ω; Md×m) denotes the approximate gradient of u, Ju denotes the set of approximate +jumps of u, u+ and u− are the traces of u on Ju, and νu(x) is the normal to Ju at x. +Note that if u ∈ SBV (Ω; Rm), then the singular part of Du is concentrated on Ju which is a +countably Hd−1-rectifiable set: there exists a set E with Hd−1(E) = 0 and a sequence (Mi)i∈N of +C1-submanifolds of Rd such that Ju ⊆ E ∪ � +i∈N Mi. +We will say that E ⊆ Rd with |E| < +∞ has finite perimeter if 1E ∈ BV (Rd). The perimeter of E +is defined as +Per(E) = |D1E|(Rd). +It turns out that +D1E = νEHd−1⌊∂∗E, +Per(E) = Hd−1(∂∗E), +where ∂∗E is called the reduced boundary of E, and νE is the associated inner approximate normal (see +[1, Section 3.5]). We have that ∂∗E ⊆ ∂E, but the topological boundary can in in general be much +larger than the reduced one. If A ⊆ Rd is open and bounded with Hd−1(A) < +∞, then A has finite +perimeter with Per(A) ≤ Hd−1(∂A). +2.3. Functions of bounded deformation. If Ω ⊆ Rd is open, we say that u ∈ BD(Ω) if u ∈ +L1(Ω; Rd) and its symmetric gradient Eu := Du+(Du)∗ +2 +in the sense of distributions is a finite Radon +measure on Ω, i.e., Eu ∈ Mb(Ω; Md×d +sym). BD(Ω) is called the space of functions of bounded deformation +on Ω. We refer the reader to [30, 29] for the main properties of the space BD. +We will make use of a subspace of BD(Ω) called the space of special functions of bounded deformation +introduced in [2]. We say that u ∈ SBD(Ω) if u ∈ BD(Ω) and its symmetrized distributional derivative +can be written in the form +Eu = e(u) dx + (u+ − u−) ⊙ νuHd−1⌊Ju, +where e(u) ∈ L1(Ω; Md×d +sym) denotes the approximate symmetrized gradient of u, Ju denotes the set of +approximate jumps of u, u+ and u− are the traces of u on Ju, and νu(x) is the normal to Ju at x. As +in the case of functions of bounded variation, Ju is a Hd−1-countably rectifiable set. +We will use the following compactness and lower semicontinuity result proved in [3]. +Theorem 2.1. Let Ω ⊆ Rd be open, bounded and with a Lipschitz boundary, and let (un)n∈N be a +sequence in SBD(Ω) such that +sup +n +� +|Eun|(Ω) + ∥un∥L1(Ω;Rd) + ∥e(un)∥Lp(Ω;Md×d +sym) + Hd−1(Jun) +� +< +∞ +for some p > 1. Then there exists u ∈ SBD(Ω) and a subsequence (unk)k∈N such that +unk → u +strongly in L1(Ω; Rd), +e(unk) ⇀ e(u) +weakly in Lp(Ω; Md×d +sym), +and +Hd−1(Ju) ≤ lim inf +k→+∞ Hd−1(Junk ). + +A FREE DISCONTINUITY APPROACH TO OPTIMAL PROFILES IN STOKES FLOWS +7 +We will need also some properties of the sections of SBD-functions. If Ω ⊆ Rd is open and u ∈ +SBD(Ω), let us consider the scalar function on Ωξ +y given by +(2.3) +ˆuξ +y(t) := u(y + tξ) · ξ +and the set +(2.4) +Jξ +u := {x ∈ Ju : (u+(x) − u−(x)) · ξ ̸= 0} +The following result holds true (see [2]). +Theorem 2.2 (One dimensional sections of SBD-functions). Let Ω ⊆ Rd be open, ξ ∈ Rd with +|ξ| = 1 and let u ∈ SBD(Ω). Then for Hd−1-a.e. y ∈ Ωξ we have +ˆuξ +y ∈ SBV (Ωξ +y) +with +(ˆuξ +y)′(t) = (e(u)ξ · ξ)(y + tξ) +for a.e. t ∈ Ωξ +y +and +Jˆuξ +y = (Jξ +u)ξ +y. +3. Obstacles in Stokes fluids and drag minimization +In this section we explain the drag problem for an obstacle immersed in a stationary flow. +3.1. The flow around the obstacle. Let Ω ⊂ Rd be an open bounded set with Lipschitz boundary, +and let V ∈ C1(Rd; Rd) be a divergence free vector field. Given E ⋐ Ω open and with a Lipschitz +boundary, let us consider the stationary flow for a viscous incompressible fluid around E with boundary +conditions on ∂Ω given by V , and with Navier boundary conditions on ∂E. More precisely, if u : Ω\E → +Rd is the velocity field, we require that the following items hold true. +(a) Incompressibility: div u = 0 in Ω \ E. +(b) Boundary conditions: we have +u = V on ∂Ω +and +the non-penetration condition u · ν = 0 on ∂E, +where ν denotes the exterior normal to E. +(c) Equilibrium: considering the stress +(3.1) +σ := −pId + 2µe(u), +where µ > 0 is a viscosity parameter, e(u) the symmetrized gradient of u (also denoted by +D(u)) and p is the pressure, we require +(3.2) +div σ = 0 +in Ω \ E. +(d) Navier conditions on the obstacle: we have +(σν)τ = βu +on ∂E, +where β > 0 is a friction parameter, and (σν)τ denotes the tangential component of force σν. +The stationary flow has the following variational characterization: u is the minimizer of the energy +(3.3) +E(u) := 2µ +� +Ω\E +|e(u)|2 dx + β +� +∂E +|u|2 dHd−1 +among the class of (sufficiently regular) admissible fields +(3.4) +Vreg +E,V (Ω) := {v ∈ H1(Ω \ E; Rd) : v satisfies points (a) and (b)}, + +8 +D. BUCUR, A. CHAMBOLLE, A. GIACOMINI, AND M. NAHON +where Hd−1 stands for the (d − 1)-dimensional Hausdorff measures, which reduces to the area measure +on sufficiently regular sets. Indeed if u is a minimizer, and ϕ is an admissible variation, so that ϕ = 0 +on ∂Ω, we get +0 = 2µ +� +Ω\E +e(u) : e(ϕ) dx + β +� +∂E +u · ϕ dHd−1 += 2µ +� +Ω\E +e(u) : ∇ϕ dx + β +� +∂E +u · ϕ dHd−1 += −2µ +� +Ω\E +div e(u) · ϕ dx + +� +∂E +[−2µe(u)ν + βu] · ϕ dHd−1 +In particular, choosing ϕ with compact support in Ω \ E we have +2µdiv e(u) = ∇p +for some pressure field p: as a consequence σ := −pId + 2µe(u) satisfies (3.2) of condition (c). +Since the admissible functions ϕ are tangent to ∂E, the optimality condition reduces to +(3.5) +0 = +� +∂E +[−2µe(u)ν + βu] · ϕ dHd−1 = +� +∂E +[−σν + βu] · ϕ dHd−1. +Notice that every tangential vector field η on ∂E can be extended to a divergence free vector field on +Ω\E which vanishes on ∂Ω, hence it is the trace of an admissible variation ϕ: indeed any extension W +which vanishes on ∂Ω has a divergence with zero mean, so that considering W1 with div W1 = div W +with W1 = 0 on ∂Ω and on ∂E (whose existence is guaranteed, for example by [6, Theorem IV.3.1])), +the required extension is given by W − W1. We conclude that the optimality condition (3.5) yields the +Navier condition of point (b). +3.2. The drag force. Assume now that the external vector field V is equal to a constant V∞ ∈ Rd\{0}, +i.e. the obstacle E is immersed in a uniform flow. The flow is perturbed near E, assuming the value +u, and the obstacle experiences a force which has a component in the direction V∞ which is given by +Drag(E) := +� +∂E +σν · V∞ +|V∞| dHd−1, +which is called the drag force on E in the direction of the flow. +We claim that +(3.6) +Drag(E) = +1 +|V∞|E(u), +where E(u) is the energy defined in (3.3). Using the facts that σ is symmetric and with zero divergence +(so that also the vector field σV∞ is divergence free), and that u = V∞ on ∂Ω, we may write +� +∂E +σν · V∞ dHd−1 = +� +∂E +σV∞ · ν dHd−1 = +� +∂Ω +σV∞ · ν dHd−1 = +� +∂Ω +σu · ν dHd−1 += +� +Ω\E +div (σu) dx + +� +∂E +σu · ν dHd−1 = +� +Ω\E +σ : ∇u dx + +� +∂E +σν · u dHd−1. +(3.7) +Using again that σ is symmetric and that u is divergence free, together with the constitutive equation +(3.1), we have +� +Ω\E +σ : ∇u dx = +� +Ω\E +σ : e(u) dx = +� +Ω\E +(−p Id + 2µe(u)) : e(u) dx += +� +Ω\E +(−p div u + 2µ|e(u|2) dx = 2µ +� +Ω\E +|e(u)|2 dx, + +A FREE DISCONTINUITY APPROACH TO OPTIMAL PROFILES IN STOKES FLOWS +9 +while in view of the Navier conditions on ∂E and the fact that u is tangent to the obstacle +� +∂E +σν · u dHd−1 = +� +∂E +(σν)τ · u dHd−1 = β +� +∂E +|u|2 dHd−1. +Inserting into (3.7), we get that (3.6) follows. +3.3. The optimization problem. Let c > 0 and let f : (0, |Ω|) → R ∪ {+∞} be a lower semicontin- +uous functions that is not identically equal to +∞. We are concerned with the following optimization +problem: +min +E +� +Drag(E) + cHd−1(∂E) + f(|E|) +� +. +We are thus interested in finding the optimal shape of an obstacle which minimizes the drag force, +under a penalization involving its perimeter and its volume. +In view of the energetic characterization of the drag force established in Subsection 3.2, we can +formulate the problem as a minimization problem among the pairs (E, u), where u is a velocity field +belonging to the family Vreg +E,V∞(Ω) defined in (3.4): +min +E,u∈Vreg +E,V∞(Ω) +� +2µ +|V∞| +� +Ω\E +|e(u)|2 dx + +β +|V∞| +� +∂E +|u|2 dHd−1 + cHd−1(∂E) + f(|E|) +� +. +Setting all the constants equal to 1, and replacing V∞ by a given divergence free velocity vector +field V as in Subsection 3.1, the drag minimization problem above is a particular case of the following +shape optimization problem +(3.8) +min +E,u∈Vreg +E,V (Ω) +�� +Ω\E +|e(u)|2 dx + +� +∂E +|u|2 dHd−1 + Hd−1(∂E) + f(|E|) +� +. +If we want to apply the direct method of the calculus of variations to the problem, i.e., if we want to +recover a minimizer by looking at minimizing sequences (En, un)n∈N, the following considerations are +quite natural. +(a) Since the problem involves the perimeter of E, the sequence (En)n∈N is relatively compact in +the family of sets of finite perimeter (see Section 2). +(b) Concerning the velocities, it turns out naturally that it is convenient to consider also dis- +continuous vector fields. Indeed if un → u in some sense, and ∂En collapses in some parts +generating a surface Γ outside the limit set E, the limit velocity field u can present, in general, +discontinuities across Γ. +En +E +Γ +We thus expect an extra term in the surface integral related to the Navier conditions, which +amounts at least to +� +Γ\∂E +[|u+|2 + |u−|2] dHd−1, +where u± are the two traces from both sides of Γ. + +10 +D. BUCUR, A. CHAMBOLLE, A. GIACOMINI, AND M. NAHON +The previous considerations yield to formulate a relaxed version of problem (3.8) in which E varies +among the family of sets of finite perimeter contained in Ω, while the family of associated admissible +velocity fields u is naturally contained in the space of special functions of bounded deformation SBD(Ω) +(see Section 2). +In Section 4, we will give a precise formulation of problem in this weak setting, which guarantees +existence of optimal solutions, describing in particular how the boundary conditions on ∂Ω and on the +obstacle have to be rephrased in this context. +4. A relaxed formulation of the shape optimization problem and statements of the +main results +Let Ω ⊆ Rd be open, bounded and with a Lipschitz boundary, and let V ∈ C1(Rd; Rd) be a divergence +free vector field. In order to deal conveniently with the boundary conditions, let us consider Ω′ ⊆ Rd +open and bounded such that Ω ⋐ Ω′. +The following definition deals with the family of admissible configurations in the relaxed setting. +Definition 4.1 (The class A(V ) of admissible obstacle-velocity configurations). We say that +(E, u) is an admissible configuration for the external velocity V , and we will write (E, u) ∈ A(V ), if +E ⊆ Ω is a set of finite perimeter, while +u ∈ SBD(Ω′) ∩ L2(Ω′; Rd) +is such that u = 0 a.e. on E and the following conditions are satisfied. +(a) The flow is divergence free: div u = 0 in the sense of distributions in Ω′. +(b) External boundary conditions: u = V a.e. on Ω′ \ Ω. +(c) Non-penetration condition on the obstacle: +u± · ν = 0 on ∂∗E ∪ Ju, +where ν denotes the normal to the rectifiable set ∂∗E ∪ Ju. +Remark 4.2. The crucial difference between admissible velocities in the present framework and those +of the family Vreg +E,V (Ω) introduced before (see (3.4)) is that they may have discontinuities outside of E. +Within the new setting, the global obstacle is given by +E ∪ Ju +i.e. it may contain (d − 1) dimensional parts. +Given (E, u) ∈ A(V ), concerning the traces of u on ∂∗E, we will denote with u+ the trace in the +direction of the external normal νE, so that u− = 0 Hd−1-a.e. on ∂∗E. +Concerning the non-penetration constraint, notice that it suffices to require it only on Ju, since it is +then automatically verified also on ∂∗E. Indeed for Hd−1-a.e. x ∈ ∂∗E\Ju, we have u−(x) = u+(x) = 0 +and the constraint is verified, while for Hd−1-a.e. x ∈ Ju ∩ ∂∗E the two rectifiable sets Ju and ∂∗E +share the same normal vector. +Remark 4.3. The space SBD(Ω′) is naturally a subspace of L1(Ω′; Rd): we require for admissibility +that u ∈ L2(Ω′; Rd) to ensure that the velocity field has finite kinetic energy. It will turn out that +velocities in SBD(Ω′) which are interesting for our problem (i.e., with finite energy) are automatically +elements of L2(Ω′; Rd) (see Theorem 5.1). +Remark 4.4 (On the boundary condition). If (E, u) ∈ A(V ), then u ∈ SBD(Ω′) with u = V a.e. +on Ω′ \ Ω, so that +Ju ∩ ∂Ω = {x ∈ ∂Ω : γ(u)(x) ̸= V (x)}, +where γ(u) is the trace of u on ∂Ω coming from Ω (i.e., the usual trace of u seen as an element +of SBD(Ω)). We conclude that within the present framework, the boundary condition is somehow + +A FREE DISCONTINUITY APPROACH TO OPTIMAL PROFILES IN STOKES FLOWS +11 +relaxed: a possible mismatch between u and V on ∂Ω is admitted, but then the zone is counted +as a jump part of the velocity field, and consequently as a part of the obstacle ∂∗E ∪ Ju, and will +carry a contribution for the energy (see (4.2) below). Such a relaxation of the boundary condition is +a feature which is common to several applications of functions of bounded variation to problems in +continuum mechanics (see for example [25, 21] in connection to fracture mechanics or [20] for problems +in plasticity). +Remark 4.5. Given (E, u) ∈ A(V ), the obstacle E ∪ Ju may touch ∂Ω only on those part where V is +tangent to Ω: this is due to the fact that on (∂∗E ∪ Ju) ∩ ∂Ω, the two sets share Hd−1-a.e. the same +normal, and u+ = V (if the orientation is suitably chosen). +Remark 4.6. Let E ⋐ Ω be open and with a Lipschitz boundary. Then we can find W ∈ H1(Ω\E; Rd) +such that W = V on ∂Ω, W = 0 on ∂E and div W = 0. Indeed if ϕ ∈ C∞(Rd) is such that ϕ = 1 on a +neighborhood of Rd \ Ω and ϕ = 0 on a neighborhood of E, we can consider the vector field V1 := ϕV , +whose divergence has zero mean on Ω \ E (by Gauss theorem). Then we can find V2 ∈ H1 +0(Ω \ E; Rd) +such that div V = div V1 (see [6, Theorem IV.3.1]), so that the field W := V1 − V2 is an admissible +choice. In particular we get that (E, W) ∈ A(V ), so that the class of admissible configurations is not +empty. +Let +(4.1) +f : [0, |Ω|] → [0, +∞] be lower semicontinuous, not identically equal to +∞. +For every (E, u) ∈ A(V ), let us set (normalizing to 1 the constants involved in the drag force problem) +J (E, u) := +� +Ω′ |e(u)|2 dx + +� +∂∗E +|u+|2 dHd−1 + +� +Ju\∂∗E +[|u+|2 + |u−|2] dHd−1 ++ Hd−1(∂∗E) + 2Hd−1(Ju \ ∂∗E) + f(|E|). +(4.2) +Remark 4.7. Concerning the volume integral in J (E, u), the density e(u) is equal to e(V ) a.e. on +Ω′ \Ω and equal to 0 a.e. on E: as a consequence we could replace it with an integral on Ω\E without +affecting the minimization of J . +Concerning the Navier energy and the surface penalization for ∂∗E∪Ju, notice that it counts also for +the possible mismatch at the boundary between u and V as pointed out in Remark 4.4: the mismatch +is thus “penalized” by the energy of the problem. +The previous observations show that the larger domain Ω′ plays only an instrumental role for the +problem, as it can be replaced by any open domain strictly containing Ω. +The first main result of the paper is the following +Theorem 4.8 (Existence of optimal obstacles). Let Ω ⊆ Rd be a bounded open set with Lipschitz +boundary, V ∈ C1(Rd; Rd) a divergence-free vector field, and f a function satisfying (4.1). Let the +family of admissible configurations A(V ) be given by Definition 4.1 and let J be the functional defined +in (4.2). Then the problem +(4.3) +min +(E,u)∈A(V ) J (E, u) +admits a solution. +Remark 4.9. We recover the original drag minimization problem when V is a constant nonzero +vector V∞, and we restore properly in the functional the physical constants µ and β, together with the +perimeter penalization constant c. +The second main result of the paper concerns the regularity of minimizers in the two dimensional +setting. + +12 +D. BUCUR, A. CHAMBOLLE, A. GIACOMINI, AND M. NAHON +Theorem 4.10 (Regularity in dimension two). Let Ω ⊆ R2 be a bounded open set with Lipschitz +boundary, V ∈ C1(R2; R2) a divergence-free vector field, and f : [0, |Ω|] → [0, +∞[ a Lipschitz function. +Let (E, u) ∈ A(V ) be a solution to (4.3) according to Theorem 4.8. Then +H1 � +Ω ∩ (Ju ∪ ∂∗E \ (Ju ∪ ∂∗E)) +� += 0, +and u ∈ C∞(Ω \ Ju ∪ ∂∗E; R2). +Theorem 4.8 will be proved in Section 6, on the basis of some technical results established in 5. The +proof of Theorem 4.10 will be addressed in Section 7. +5. Some technical results in SBD +In this section we collect some technical properties concerning the space SBD that will be funda- +mental in the proof of Theorem 4.8. In particular in Theorem 5.1 we will prove that admissible velocity +vector fields enjoy higher summability properties (indeed they belong to L +2d +d−1 ). In Theorem 5.3 we +will prove that velocity fields u with u± tangent to the discontinuity set Ju form a closed set under the +natural convergence of minimizing sequences for the main optimization problem. Finally in Theorem +5.4 we will prove a lower semicontinuity result for surface energies depending on the traces, which +entails in particular the lower semicontinuity of the term associated to the Navier conditions. +5.1. An immersion result. The following embedding result holds true. +Theorem 5.1. Let Ω ⊆ Rd be a bounded open set, and let u ∈ SBD(Rd) be supported in Ω such that +E(u) := +� +Ω +|e(u)|2 dx + +� +Ju +� +|u+|2 + |u−|2� +dHd−1 < +∞. +Then u ∈ L +2d +d−1 (Ω) with +∥u∥ 2d +d−1 ≤ C +� +E(u), +where C depends on d and diam(Ω) only. +Proof. It suffices to follow the strategy of the proof of the classical embedding of BD into Ld/d−1 +explained in [29], but concentrating on the square of the components. +Let us consider the unit vector +ξ := +1 +√ +d +(1, 1, . . . , 1) ∈ Rd. +Employing the characterization by sections recalled in Section 2, for Hd−1-a.e. y ∈ ξ⊥ we have +ˆuξ +y ∈ SBV (Ωξ +y) +with +� +Ωξ +y +|(ˆuξ +y)′|2 dt + +� +t∈Jˆuξ +y +� +|(ˆuξ +y)+(t)|2 + |(ˆuξ +y)−(t)|2� +< +∞. +Then we can write for a.e. t ∈ R +∥ˆuξ +y∥2 +L∞(Ωξ +y) ≤ +���D(ˆuξ +y)2��� (Ωξ +y) = +� +Ωξ +y +2|ˆuξ +y(ˆuξ +y)′|dt + +� +t∈Jˆuξ +y +���|(ˆuξ +y)+(t)|2 − |(ˆuξ +y)−(t)|2��� +≤ 1 +2∥ˆuξ +y∥2 +L∞(Ωξ +y) + 2|Ωξ +y| +� +Ωξ +y +���(ˆuξ +y)′��� +2 +dt + +� +t∈Jˆuξ +y +����(ˆuξ +y)+(t) +��� +2 ++ +���(ˆuξ +y)−(t) +��� +2� +, +(5.1) + +A FREE DISCONTINUITY APPROACH TO OPTIMAL PROFILES IN STOKES FLOWS +13 +Let us set +gξ(x) := +� +Ωξ +y +|(ˆuξ +y)′|2 dt + +� +t∈Jˆuξ +y +� +|(ˆuξ +y)+(t)|2 + |(ˆuξ +y)−(t)|2� +, +where y := πξ⊥(x), i.e., the projection of x on the hyperplane ξ⊥. gξ(x) only depends on the projection +of x on ξ⊥ and +� +ξ⊥ gξdHd−1 = +� +Ω +|e(u)ξ · ξ|2 dx + +� +Ju +� +|u+|2 + |u−|2� +|ξ · ν| dHd−1 +≤ C +�� +Ω +|e(u)|2 dx + +� +Ju +� +|u+|2 + |u−|2� +dHd−1 +� +where C depends only on d. Thanks to (5.1) we have +(5.2) +|ξ · u|2 ≤ Cgξ +a.e. on Ω, +where C depends on the diameter of Ω, and from now on all the constants C that appear depend on +n, diam(Ω). For every k = 1, . . . , d − 1, we can write +ξ = +1 +√ +d +ek + +� +d − 1 +d +hk, +where ek is the k-th vector of the canonical base, and hk is the unit vector in the direction +√ +dξ − ek. +Reasoning as above on the decomposition +ξ · u = +� +d − 1 +d +hk · u + 1 +√ +d +ek · u +we obtain a similar estimate +(5.3) +|ξ · u|2 ≤ C (ghk + gek) , +Multiplying inequality (5.2) with inequalities (5.3) for k = 1, . . . , d − 1, we obtain reasoning as in [29, +Chapter II, Theorem 1.2] +∥(ξ · u)2∥ +d +d−1 ≤ C +�� +Ω +|e(u)|2 dx + +� +Ju +� +|u+|2 + |u−|2� +dHd−1 +� +. +Since this estimate does not depend on the particular choice of the basis and hence holds for any ξ +with norm one, the theorem is proved. +□ +5.2. Closure of the non-penetration constraint. In the context of equi-Lipschitz boundaries, the +preservation of the non-penetration property for a sequence of Sobolev functions converging weakly, +comes rather directly via the divergence theorem (we refer the reader, for instance, to [8]). However, +in the case of collapsing boundaries, so that the limit function lives on both sides of a surface and +in absence of any smoothness of the limit set, this technique does not work. The proof of the non- +penetration preservation requires different technical arguments that we handle in the SBD context. +Let us start with the following lower semicontinuity result. +Theorem 5.2. Let Ω ⊆ Rd be a bounded open set, and let (un)n∈N be a sequence in SBD(Ω) such +that +sup +n +�� +Ω +|e(un)|2 dx + Hd−1(Jun) +� +< +∞ +with +un → u +in measure + +14 +D. BUCUR, A. CHAMBOLLE, A. GIACOMINI, AND M. NAHON +for some u ∈ SBD(Ω). Then +� +Ju +� +|u+ · νu| + |u− · νu| +� +dHd−1 ≤ lim inf +n→+∞ +� +Jun +� +|u+ +n · νun| + |u− · νun| +� +dHd−1. +Proof. Let us consider a countable set of functions {ϕh : h ∈ N} which is dense with respect to ∥ · ∥∞ +inside the set +� +f ∈ C0 +c (]0, +∞[) : +� +∞ +0 +f dt = 0 and ∥f∥∞ ≤ 1 +� +. +Given ε > 0, let us consider +gh,k(x) := +� +1 +2|x−xk|2 +0 +ϕh(t) dt, +where {xk : k ∈ N} is a countable and dense set in Bε(0) ⊂ Rd with x0 = 0. Clearly gh,k ∈ C1 +c (Rd) +with +Gh,k(x) := ∇gh,k(x) = ϕh +�1 +2|x − xk|2 +� +(x − xk). +We have that Gh,k is a continuous conservative vector field with compact support on Rd. +Let us set for (i, j) ∈ Rd × Rd and ν ∈ Rd with |ν| = 1 +fε(i, j, ν) := sup +h,k +(Gh,k(i) − Gh,k(j)) · ν. +By construction fε is a symmetric jointly convex function according to [26, Definition 3.1]. We claim +that for i ̸= j +(5.4) +|i · ν| + |j · ν| ≤ fε(i, j, ν) ≤ |i · ν| + |j · ν| + 2ε. +In view of the lower semicontinuity result [26, Theorem 5.1] we have +lim inf +n→+∞ +� +Jun +fε(u+ +n , u− +n , νun) dHd−1 ≥ +� +Ju +fε(u+, u−, νu) dHd−1. +We can thus write +lim inf +n→+∞ +�� +Jun +� +|u+ +n · νun| + |u− +n · νun| +� +dHd−1 + 2εHd−1(Jun) +� +≥ lim inf +n→+∞ +� +Jun +fε(u+ +n , u− +n , νun) dHd−1 ≥ +� +Ju +fε(u+, u−, νu) dHd−1 +≥ +� +Ju +� +|u+ · νu| + |u− · νu| +� +dHd−1, +so that the result follows taking into account the bound on Hd−1(Jun) and letting ε → 0. +In order to complete the proof, we need to show claim (5.4). The estimate from above follows from +[Gh,k(i) − Gh,k(j)] · ν ≤ |(i − xk) · ν| + |(j − xk) · ν| ≤ |i · ν| + |j · ν| + 2ε +since ∥ϕh∥∞ ≤ 1 and |xk| < ε. Let us prove the estimate from below. We select xkn → 0 such that +|i − xkn| ̸= |j − xkn| (which is always possibile in view of the density of {xk : k ∈ N} inside Bε(0) and +since i ̸= j) and then ϕhn such that for n → +∞ +ϕhn +�1 +2|i − xkn|2 +� +→ +i · ν +|i · ν| + η +and +ϕhn +�1 +2|j − xkn|2 +� +→ − +j · ν +|j · ν| + η, +where η > 0. By definition of fε we infer that +fε(i, j, ν) ≥ |i · ν| + |j · ν| − 2η, + +A FREE DISCONTINUITY APPROACH TO OPTIMAL PROFILES IN STOKES FLOWS +15 +so that the estimate from below follows by sending η → 0. +□ +We are now in a position to prove the main result of the section. +Theorem 5.3 (Closure of the non-penetration constraint on the jump set). Let Ω ⊆ Rd be a +bounded open set, and let (un)n∈N be a sequence in SBD(Ω) such that +sup +n +�� +Ω +|e(un)|2 dx + Hd−1(Jun) +� +< +∞ +and +un → u +in measure +for some u ∈ SBD(Ω). If +u± +n · νun = 0 +Hd−1-a.e. on Jun, +then +u± · νu = 0 +Hd−1-a.e. on Ju. +Proof. By Theorem 5.2 we may write +� +Ju +� +|u+ · νu| + |u− · νu| +� +dHd−1 ≤ lim inf +n→+∞ +� +Jun +� +|u+ +n · νun| + |u− · νun| +� +dHd−1 = 0, +so that the result follows. +□ +5.3. A lower semicontinuity result for surface energies in SBD. In this section we deal with +the lower semicontinuity of the surface term of the functional J in (4.2) connected with the Navier +conditions on the obstacle. The following lower semicontinuity result holds true. +Theorem 5.4. Let Ω ⊆ Rd be an open set, un, u ∈ SBD(Ω) such that +un → u +strongly in L1(Ω; Rd) +and +sup +n +�� +Ω +|e(un)|2 dx + Hd−1(Jun) +� +< +∞. +Then if φ : Rd → [0, +∞] is a lower semicontinuous function, we have +� +Ju +[φ(u+) + φ(u−)] dHd−1 ≤ lim inf +n→+∞ +� +Jun +[φ(u+ +n ) + φ(u− +n )] dHd−1. +This applies in particular to φ(u) = |u|2 and φ(u) = 1{u̸=0}, which will be of interest to us. +Proof. Notice first that φ may be supposed to be continuous. Indeed for any lower-semicontinuous +nonnegative φ, by considering a sequence of continuous nonnegative functions φk ր φ we get +� +Ju +[φ(u+) + φ(u−)] dHd−1 = lim inf +k→∞ +� +Ju +[φk(u+) + φk(u−)] dHd−1 +≤ lim inf +k→∞ lim inf +n→+∞ +� +Jun +[φk(u+ +n ) + φk(u− +n )] dHd−1 +≤ lim inf +n→+∞ +� +Jun +[φ(u+ +n ) + φ(u− +n )] dHd−1 +Through a by now standard blow-up argument ( see Remark 5.6), we can reduce the problem to the +following lower semicontinuity result. Let Q1 ⊆ Rd be the unit square centred at 0, and let us set +H := Q1 ∩ {xd = 0} +and +Q± +1 := Q1 ∩ {xd ≷ 0}. + +16 +D. BUCUR, A. CHAMBOLLE, A. GIACOMINI, AND M. NAHON +Given u± ∈ Rd with u+ ̸= u− and un ∈ SBD(Q1) with +(5.5) +un → u := u+1Q+ +1 + u−1Q− +1 +strongly in L1(Q1; Rd), +(5.6) +sup +n Hd−1(Jun) < +∞ +and +(5.7) +e(un) → 0 +strongly in L1(Q1; Md×d +sym), +then +(5.8) +φ(u+) + φ(u−) ≤ lim inf +n→+∞ +� +Jun +[φ(u+ +n ) + φ(u− +n )] dHd−1. +We now divide the proof in several steps, and we will employ the characterization by sections of SBD +functions explained in Section 2. +Step 1. Let ε > 0 be given. We fix δ > 0 and N ∈ N with N > d: these numbers will be subject to +several constraints that will appear during the proof. +Let us fix N unit vectors {ξi}1≤i≤N such that +(5.9) +|ed · ξi − 1| < δ +and such that any subset of d of them forms a basis of Rd. Moreover, we may assume in addition that +(5.10) +(u+ − u−) · ξi ̸= 0 +for every i = 1, . . . , N. +Thanks to (5.5) and (5.6), we can fix a > 0 small such that setting H± := H × {±a} = H ± aed, we +have +(un)|H± → u± +strongly in L1(H±; Rd) +and +(5.11) +∀n ∈ N : Hd−1(Jun ∩ H±) = 0. +Step 2. We claim that, up to a subsequence, we can find H− +ε ⊂ H− with +(5.12) +Hd−1(H− \ H− +ε ) < ε +such that for every i = 1, . . . , N, for every y ∈ H− +ε and for every n ∈ N +(5.13) +H− +ε ∩ Jun = ∅, +and +(5.14) +H0((Jun)ξi +y ) < +∞, +H0((Jun)ξi +y ∩R+) ≥ 1 . +Moreover setting +� +(un) +ξi +y := un(y + tξi) · ξi, +for every y ∈ H− +ε we have +� +(un) +ξi +y ∈ SBV ((Q1)ξi +y ), +(5.15) +J � +(un) +ξi +y += (Jun)ξi +y +(cf notation (2.4)), +(5.16) +∥[ � +(un) +ξi +y ]′∥L1 → 0 +uniformly for y ∈ H− +ε , + +A FREE DISCONTINUITY APPROACH TO OPTIMAL PROFILES IN STOKES FLOWS +17 +and +(5.17) +(un)|H− → u− +uniformly on H− +ε . +Indeed, if the number δ appearing in (5.9) is small enough, we can find A− +ε ⊆ H− with +(5.18) +Hd−1(H− \ A− +ε ) < ε +2 +and such that for every y ∈ A− +ε the lines {y + tξi : t ∈ R} intersect H+ for every i = 1, . . . , N. In view +of (5.5), (5.6) and (5.7), and since pointwise convergence implies almost uniform convergence, we can +find Nε ⊂ A− +ε with +(5.19) +Hd−1(Nε) < ε +2 +and such that, up to a subsequence +(5.20) +∥ � +(un) +ξi +y − �uξi +y ∥L1 → 0 +uniformly for y ∈ A− +ε \ Nε +(5.21) +∥[ � +(un) +ξi +y ]′∥L1 → 0 +uniformly for y ∈ A− +ε \ Nε +(5.22) +(un)|H− → u− +uniformly on A− +ε \ Nε, +and for every y ∈ A− +ε \ Nε +(5.23) +H0((Jun)ξi +y ) < +∞. +Notice that for n large enough and for every y ∈ A− +ε \ Nε we have +(5.24) +(Jun)ξi +y ̸= ∅. +Indeed otherwise, we would get for nk → +∞ the existence of yk ∈ A− +ε \Nε with � +(unk) +ξi +yk ∈ W 1,1((Q1)ξi +yk), +and (5.22) together with (5.21) would yield +∥ � +(unk) +ξi +yk − u−∥1 → 0 +against (5.20) (recall that by the choice (5.10) of the ξi, the functions �uξi +y have a jump). The claim +follows by setting +H− +ε := Aε \ +� +Nε ∪ +� +n +(Jun ∩ H−) +� +. +Indeed (5.12) follows from (5.18), (5.19) and (5.11), while (5.13) is clearly satisfied. Relation (5.14) +follows by (5.23) and (5.24), while relation (5.16) follows from (5.21). Finally relation (5.17) follows +from (5.22). +Step 3. For every i = 1, . . . , N, let us consider the set Ji,− +n +given by the first point of intersection +(with t > 0) of the line {y + tξi : t ∈ R} with the jump set Jun as y varies in the set H− +ε defined in +Step 2 (recall (5.14) and (5.15)). In view of (5.16) and (5.17), we can find ηn → 0 such that for every +x ∈ Ji,− +n +with νun · ξi > 0 +(5.25) +|u− +n (x) · ξi − u− · ξi| < ηn. +Step 4. We claim that, for δ small enough and N large enough, up to a subsequence, we can find +˜J− +n ⊆ Jun with +(5.26) +Hd−1( ˜J− +n ) ≥ 1 − cε, + +18 +D. BUCUR, A. CHAMBOLLE, A. GIACOMINI, AND M. NAHON +where cε → 0 as ε → 0, and such that for every x ∈ ˜J− +n +(5.27) +x ∈ Ji,− +n +for d different indices i ∈ {1, . . . , N}, +where Ji,− +n +is defined in Step 3. Moreover, we can orient νun on ˜J− +n in such a way that +(5.28) +ed · νun > 0 +and +ξi · νun > 0 for every i = 1, . . . , N. +Intuitively speaking, the points in ˜J− +n are seen from H− +ε under d different directions: moreover the +associated lines cut the jump transversaly, from the “lower” to the “upper” part. +Indeed, in view of the definition of ξi (which form a very small angle with ed as δ → 0) and of the +area formula (cf for instance [24, Sec. 3.2]), we can assume that δ is so small that for every i = 1, . . . , N +(5.29) +Hd−1(Ji,− +n ) ≥ +� +Ji,− +n +|νun · ξi| dHd−1 = Hd−1((H− +ε )ξi) = +1 +1 + ˆcδ +Hd−1(H− +ε ), +where the notation (H− +ε )ξi is defined in (2.1) and where ˆcδ → 0, so that, taking into account (5.12), +for small δ we have +(5.30) +Hd−1(Ji,− +n ) ≥ 1 − 2ε. +By Lemma 5.5 below (with X = Jun, µ = Hd−1, and M given by the family of Borel sets) if N is large +enough we can find an index ¯i such that +(5.31) +Hd−1 + + + +J¯i,− +n +\ +� +i1 ε} +and +Bn,ε := ˜J− +n \ Gn,ε, +coming back to (5.29) we have +Hd−1(Gn,ε) + ε2Hd−1(Bn,ε) > 1 − 3ε, +so that +Hd−1(Gn,ε) > 1 − 3ε − ε2C, +where C := supn Hd−1(Jun) < +∞. Finally we orient the normal νun on Gn,ε in such a way that +νun · ξ¯i > ε. +The inequalities (5.28) then also hold true on Gn,ε if δ is small enough thanks to (5.9). Reducing ˜J− +n +to Gn,ε if necessary, the full claim follows taking into account (5.32) and (5.33). +Step 5. Let ˜J− +n ⊆ Jun be the set given by Step 4. Since the points of this set are seen from H− +ε under +d different directions, in view of (5.25) we infer that there exists ˜ηn → 0 such that for every x ∈ ˜J− +n +|u− +n (x) − u−| < ˜ηn. + +A FREE DISCONTINUITY APPROACH TO OPTIMAL PROFILES IN STOKES FLOWS +19 +Reasoning in a similar way starting from the upper part H+ +ε , and employing the opposite directions +{−ξi : i = 1, . . . , N}, we can construct ˜J+ +n ⊆ Jun with νun oriented such that again +ed · νun > 0 +and +ξi · νun > 0 for every i = 1, . . . , N, +such that +(5.34) +Hd−1( ˜J+ +n ) ≥ 1 − cε +with cε → 0 as ε → 0, and such that for every x ∈ ˜J+ +n +|u+ +n (x) − u+| < ˜ηn. +Notice that for x ∈ ˜J− +n ∩ ˜J+ +n , the orientation chosen is compatible with that of (5.28), so that indeed +u− +n (x) and u+ +n (x) are the two traces of un at x. +We can thus write, in view of the continuity of φ +� +Jun +[φ(u+ +n ) + φ(u− +n )] dHd−1 ≥ +� +˜J+ +n ∩ ˜J− +n +[φ(u+ +n ) + φ(u− +n )] dHd−1 + +� +˜J+ +n ∆ ˜J− +n +[φ(u+ +n ) + φ(u− +n )] dHd−1 +≥ +� +˜J+ +n ∩ ˜J− +n +[φ(u+ +n ) + φ(u− +n )] dHd−1 + +� +˜J+ +n \ ˜J− +n +φ(u+ +n ) dHd−1 + +� +˜J− +n \ ˜J+ +n +φ(u− +n ) dHd−1 +≥ +� +˜J+ +n +φ(u+ +n ) dHd−1 + +� +˜J− +n +φ(u− +n ) dHd−1 +≥ [φ(u+) − ˜ηn]Hd−1( ˜J+ +n ) + [φ(u−) − ˜ηn]Hd−1( ˜J− +n ) +where ˜ηn → 0, so that, taking into account (5.26) and (5.34) +lim inf +n→+∞ +� +Jun +[φ(u+ +n ) + φ(u− +n )] dHd−1 ≥ [φ(u+) + φ(u−)](1 − 2cε). +The conclusion follows by letting ε → 0. +□ +In the proof of Theorem 5.4 we made use of the following abstract lemma. +Lemma 5.5. Let (X, M, µ) be a finite measure space. Let ε > 0 and d ≥ 2. Then there exists N ∈ N +that only depends on µ(X), ε, d such that if {Ei}i=1,...,N is a family of sets in M, we can find ¯i such +that +µ + +E¯i \ +� +j1 0, there is some N(d, ε) ≥ 1 such that any family of N ≥ N(d, ε) of sets +(Ei)1≤i≤N there is some i that verifies +µ + +Ei \ +� +J⊂[1,N]\{i},|J|=d−1 +� +j∈J +Ej + + < ε, +meaning that there is some i such that every point of Ei outside a set of measure less than ε is in (at +least) d − 1 other sets Ej (for j ̸= i). +We prove it by recursion. If d = 2, let N := +� 1 +ε +� +, where [·] denotes the integer part. Given (Ei)1≤i≤N, +let us consider the sets +� +Ei \ � +1≤j≤N,j̸=i Ej +� +1≤i≤N. These are disjoint and µ(X) = 1, so there is some +i such that +µ + +Ei \ +� +1≤j≤N,j̸=i +Ej + + ≤ 1 +N ≤ ε, + +20 +D. BUCUR, A. CHAMBOLLE, A. GIACOMINI, AND M. NAHON +which proves the initialisation. +Assume now that the result is true for d and let us check it for d + 1. Let +N := N +� +d, ε +2 +� +and +M := +�2 +ε +� +, +and let us consider N × M sets that we classify into N groups of M sets, written (Ek,i)1≤k≤N,1≤i≤M. +For every k ∈ [1, N], the sets +� +Ek,i \ � +1≤j≤M,j̸=i Ek,j +� +1≤i≤M are disjoints so there is some ik such that +µ + +Ek,ik \ +� +1≤i≤M,i̸=ik +Ek,i + + ≤ 1 +M ≤ ε +2. +Considering the sets (Ek,ik)1≤k≤N, since N = N +� +d, ε +2 +� +we find some k such that +µ + +Ek,ik \ +� +K⊂[1,N]\{k},|K|=d−1 +� +k∈K +Ek,ik + + ≤ ε +2. +This means that outside a set of measure at most ε +2, every point of Ek,ik is in d − 1 sets of the form +Ek,ik for k ̸= k, and similarly every point outside a set of measure at most ε +2 is also in one set of the +form Ek,i for some i ̸= ik. We conclude that outside of measure at most ε, every point of Ek,ik belongs +to d other sets, meaning N(d + 1, ε) is well-defined and N(d + 1, ε) ≤ N +� +d, ε +2 +� � 2 +ε +� +. +□ +Remark 5.6. Let us detail the blow up argument used in the proof of Theorem 5.4. If we set +µn := [φ(u+ +n ) + φ(u− +n )]Hd−1⌊Jun +and assume that (up to a subsequence) +µn +∗⇀ µ +weakly* in Mb(Ω) +for some Radon measure µ on Ω, the conclusion follows if we show that +µ ≥ [φ(u+) + φ(u−)]Hd−1⌊Ju +as measures on Ω. +With this aim is sufficient to show that +(5.35) +dµ +dHd−1 (x) ≥ [φ(u+(x)) + φ(u−(x))] +for Hd−1-a.e. x ∈ Ju, +where +dµ +dHd−1 denotes the Radon-Nykodim derivative of µ with respect to Hd−1 (restricted to Ju). +Let us assume (up to subsequences) that +λn := Hd−1⌊Jun +∗⇀ λ +weakly* in Mb(Ω), +and that +|e(un)| dx +∗⇀ f dx +weakly* in Mb(Ω), +where f ∈ L1(Ω) (this is possible since (e(un))n∈N is bounded in L2). +Let x ∈ Ju be such that +dµ +dHd−1 (x) = lim +r→0 +µ(Qx,r) +rd−1 +, +lim +r→0 +λ(Qx,r) +rd−1 +< +∞, +lim +r→0 +1 +rd−1 +� +Qr(x) +|f| dx = 0, +and (having choosen the axis so that νu(x) = ed), for r → 0+ +u(x + r·) → u+(x)1Q+ +1 + u−(x)1Q− +1 +strongly in L1(Q1; Rd). +Since Hd−1-a.e. x ∈ Ju satisfies these properties, it suffices to concentrate on such points to prove +inequality (5.35). + +A FREE DISCONTINUITY APPROACH TO OPTIMAL PROFILES IN STOKES FLOWS +21 +Let rk → 0 be such that +µ(∂Qx,rk) = λ(∂Qx,rk) = 0. +Since by weak convergence and the relation above we have µn(Qx,rk) → µ(Qx,rk), and similarly for λ, +we can choose nk ր +∞ such that +µ(Qx,rk) ≥ µnk(Qx,rk) − rd−1 +k +k +, +λ(Qx,rk) ≥ λnk(Qx,rk) − rd−1 +k +k +, +and +� +Qx,rk +|f| dx ≥ +� +Qx,rk +|e(unk| dx − rd−1 +k +k +. +Moreover, setting vk(y) := unk(x + rky) we can assume also +vk → u+(x)1Q+ +1 + u−(x)1Q− +1 +strongly in L1(Q1; Rd). +We get +� +Q1 +|e(vk)| dx = +1 +rd−1 +k +� +Qx,rk +|e(unk| dx ≤ +1 +rd−1 +k +� +Qx,rk +|f| dx + 1 +k → 0 +and +Hd−1(Jvk) = +1 +rd−1 +k +Hd−1(Junk ∩ Qx,rk) = λnk(Qx,rk) +rd−1 +k +≤ λ(Qx,rk) +rd−1 +k ++ 1 +k → c < +∞, +so that, using the lower semicontinuity (5.8) concerning functions on the unit square (and to which +the proof of the Theorem has been reduced) +dµ +dHd−1 (x) = +lim +k→+∞ +µ(Qx,rk) +rd−1 +k +≥ lim inf +k→+∞ +µnk(Qx,rk) +rd−1 +k += lim inf +k→+∞ +� +Jvk +[φ(v+ +k ) + φ(v− +k )] dHd−1 ≥ φ(u+(x)) + φ(u−(x)) +and (5.35) follows. +6. Existence of minimizers: proof of Theorem 4.8 +We are now in a position to prove the first main result of the paper. +Proof of Theorem 4.8. Let (En, un)n∈N be a minimizing sequence: since the function f is not identically +equal to +∞, and in view of Remark 4.6, there exists C > 0 such that +J (En, un) ≤ C. +Since un = 0 a.e. on En we may write +� +∂∗En +|u+ +n |2 dHd−1 + +� +Jun\∂∗En +[|u+ +n |2 + |u− +n |2] dHd−1 = +� +Jun +[|u+ +n |2 + |u− +n |2] dHd−1 +so that we infer +Hd−1(∂∗En) ≤ C +and +� +Ω +|e(un)|2 dx + Hd−1(Jun) + +� +Jun +[|u+ +n |2 + |u− +n |2] dHd−1 ≤ C. + +22 +D. BUCUR, A. CHAMBOLLE, A. GIACOMINI, AND M. NAHON +Notice that +|E(un)|(Ω′) = +� +Ω′ |e(un)| dx + +� +Jun +|u+ +n − u− +n | dHd−1 +≤ +� +Ω′\Ω +|e(V )| dx + +� +Ω +|e(un)| dx + +� +Jun +[|u+ +n | + |u− +n |] dHd−1 +≤ +� +Ω′\Ω +|e(V )| dx + 1 +2 +� +|Ω| + +� +Ω +|e(un)|2 dx + 2Hd−1(Jun) + +� +Jun +[|u+ +n |2 + |u− +n |2] dHd−1 +� +≤ ˜C, +for some ˜C > 0. Moreover, thanks to Theorem 5.1 applied to u − V we may assume also that +(6.1) +∥un∥ +L +2d +d−1 (Ω′) ≤ ˜C. +By the compactness result in SBD (see Theorem 2.1), there exist a subsequence (unk)k∈N and u ∈ +SBD(Ω′) with u = V on Ω′ \ Ω and such that +(6.2) +unk → u +strongly in L1(Ω′; Rd), +(6.3) +e(unk) ⇀ e(u) +weakly in L2(Ω′; Md×d +sym), +and +Hd−1(Ju) ≤ lim inf +k→+∞ Hd−1(Junk ). +Concerning the sets Enk, we may assume, up to a further subsequence if necessary, that there exists a +set of fine perimeter E ⊆ Ω such that +(6.4) +1Enk → 1E +strongly in L1(Rd) +with +Hd−1(∂∗E) ≤ lim inf +k→+∞ Hd−1(∂∗Enk). +In particular we get +(6.5) +f(|E|) ≤ lim inf +n→+∞ f(|En|). +Let us prove that +(6.6) +(E, u) ∈ A(V ). +In view of (6.1) we infer that u ∈ L +2d +d−1 (Ω′; Rd) so that in particular u ∈ L2(Ω′; Rd). Moreover u = V +on Ω′ \ Ω, while u = 0 a.e. on E thanks to (6.2) and (6.4). +Since the divergence constraint is intended in the sense of distributions on Ω, this passes easily to +the limit thanks to (6.2). Moreover, in view of Theorem 5.3 we deduce +u± ⊥ νu +on Ju. +In particular this entails +u+ ⊥ νE +on ∂∗E ∩ Ω, +since for x ∈ ∂∗E we have either x ∈ Ju or u+(x) = 0. We conclude that the non-penetration constraint +for the velocity field holds on ∂∗E and on Ju \ ∂∗E, so that (6.6) holds true. +Let us prove the pair (E, u) is a minimizer for the problem. Thanks to (6.3) we get +� +Ω′ |e(u)|2 dx ≤ lim inf +k→+∞ +� +Ω′ |e(unk)|2 dx, +while in view of Theorem 5.4 we have that +� +Ju +[|u+|2 + |u−|2] dHd−1 ≤ lim inf +k→+∞ +� +Junk +[|u+ +nk|2 + |u− +nk|2] dHd−1, + +A FREE DISCONTINUITY APPROACH TO OPTIMAL PROFILES IN STOKES FLOWS +23 +which entails +� +∂∗E +|u+|2 dHd−1 + +� +Ju\∂∗E +[|u+|2 + |u−|2] dHd−1 +≤ lim inf +k→+∞ +�� +∂∗Enk +[u+ +nk|2 dHd−1 + +� +Junk \∂∗Enk +[|u+ +nk|2 + |u− +nk|2] dHd−1 +� +(6.7) +since u = 0 a.e. on E and unk = 0 a.e. on Enk. +Let us prove that +(6.8) +2Hd−1(Ju \ ∂∗E) + Hd−1(∂∗E) ≤ lim inf +k→+∞ +� +2Hd−1(Junk \ ∂∗Enk) + Hd−1(∂∗Enk) +� +. +Let us choose h ∈ Rd such that +Hd−1({x ∈ ∂∗E ∪ Ju : u+(x) = h}) = Hd−1({x ∈ ∂∗E ∪ Ju : u−(x) = h}) += Hd−1({x ∈ ∂∗Enk ∪ Junk : u+ +nk(x) = h}) = Hd−1({x ∈ ∂∗Enk ∪ Junk : u− +nk(x) = h}) = 0. +This is possible because for example the sets {x ∈ ∂∗E ∪ Ju : u+(x) = h} are disjoint as h varies, and +similarly for the other sets. In particular, setting +vh := u + h1E +and +vh +nk := unk + h1Enk +we have +Jvh = Ju ∪ J1E = ∂∗E ∪ Ju +and +Jvhnk = Junk ∪ J1Enk = ∂∗Enk ∪ Junk +up to Hd−1-negligible sets. If we apply Theorem 5.4 with the choice φh(s) = 1{s̸=h} to the sequence +(vh +nk)k∈N we get +Hd−1(∂∗E) + 2Hd−1(Ju \ ∂∗E) = +� +Jvh +[φh((vh)+) + φh((vh)−)]dHd−1 +≤ lim inf +k→+∞ +� +Jvhnk +[φh((vh +nk)+) + φh((vh +nk)−)]dHd−1 += lim inf +k→+∞ +� +Hd−1(∂∗Enk) + 2Hd−1(Junk \ ∂∗Enk) +� +(6.9) +so that (6.8) holds true. +Gathering (6.3), (6.7), (6.5) and (6.8), we deduce +J (E, u) ≤ lim inf +k→+∞ J (Enk, unk) +so that, taking into account (6.6), the pair (E, u) is a minimizer of the main problem (4.3), and the +proof is concluded. +□ +7. Regularity of two-dimensional minimizers: proof of Theorem 4.10 +This section is devoted to the proof Theorem 4.10 concerning the regularity of minimizers in dimen- +sion two. +As mentioned in the Introduction, the general strategy used by De Giorgi, Carriero and Leaci for +the Mumford-Shah problem in [22] faces the new difficulties given by the vectorial context, considered +in [18, 15] in connection to the Griffith fracture problem, and also by extra conditions proper to our +problem, that is incompressibility and non-penetration for the velocity fields. We follow the main lines +of [18, 15]: however technical difficulties allow us to deal only with dimension 2 (see point (a) below). +Since our drag problem involves pairs (E, u) as admissible configurations, and some points of ∂∗E +may not be jump points of u, it will be useful to deal with pairs (J, u), where J is a rectifiable set and + +24 +D. BUCUR, A. CHAMBOLLE, A. GIACOMINI, AND M. NAHON +u is a function whose jumps are contained (up to H1-negligible sets) in J and satisfy the constraints +of zero divergence and non-penetration. More precisely we formulate the following definition. +Definition 7.1 (The class V). Let Ω ⊆ R2 be an open set. We say that (J, u) ∈ V(Ω) if J ⊆ Ω is a +rectifiable set, and u ∈ SBD(Ω) is such that div u = 0 in the sense of distributions in Ω, H1(Ju\J) = 0 +and u± +|J · νJ = 0 H1-a.e. on J. +The structure of the section is the following. +(a) In Section 7.1 we prove a fundamental approximation lemma (Smoothing Lemma 7.2), which +allows us to approximate every (J, u) ∈ V(Q1) with H1(J) small by a configuration (J \Qr, v) ∈ +V(Q1), where v is a Sobolev function in the slightly smaller square Qr with a control on the +energy. The idea is that the jumps of u in Qr are “smoothed out”, giving rise to the function +v which preserves the divergence free constraint together with the non-penetration condition. +This result is inspired by [15], and it is here that the dimension two is fundamental. +(b) In Section 7.2 we prove regularity for local minimizers of a Griffith functional +G(J, u) := +� +Ω +|e(u)|2dx + H1(J), +defined on pairs (J, u) ∈ V(Ω). The kind of local minimality considered is very weak, and +inspired by the kind of competitors that can be constructed thanks to the Smoothing Lemma +7.2. The key result to get regularity is given by the decay estimate contained in Proposition +7.7. +Regularity for minimizers of the Griffith energy is then used in Section 7.3 to prove Theorem +4.10, that is to show the regularity of minimizers of the drag problem. +(c) Finally, motivated by the regularity result of Theorem 4.10, in Section 7.4 we describe a differ- +ent relaxation of the drag problem which involves topologically closed obstacles and Sobolev +velocities: the regularity result can be used to prove that such a formulation is well posed in +dimension two. +7.1. The smoothing lemma. We fix a standard radial, smooth, nonnegative mollifier ρ with support +in a disc of radius 1/8 and denote +ρδ(x) := δ−2ρ +�x +δ +� +. +The main result of the section is the following smoothing lemma which is in the spirit of [15]. +Lemma 7.2 (Smoothing Lemma). There exist C, η > 0 such that for any (J, u) ∈ V(Q1) with +H1(J) < η, then letting δ := H1(J) +1 +2 there exist r ∈]1 − δ +1 +2 , 1[ and v ∈ SBD(Q1) ∩ H1(Qr) such that +the following items hold true. +(a) H0(J ∩ ∂Qr) = 0 and for every 0 < s < r +H1(J ∩ (Qr \ Qr−s)) ≤ Cδ +3 +2s. +(b) {v ̸= u} ⊆ Qr and (J \ Qr, v) ∈ V(Q1). +(c) It holds +∥e(v)∥L2(Q1) ≤ (1 + Cδ +1 +6 )∥e(u)∥L2(Q1). +(d) There exists a cut-off function ϕ ∈ C∞(Qr, [0, 1]) with ϕ = 0 on Qr \ Qr−δ, ϕ = 1 on Qr−4δ, +and such that +∥e(v) − ϕρδ ∗ e(u)∥L2(Qr) ≤ Cδ +1 +6 ∥e(u)∥L2(Q1). +Proof. The proof follows the strategy introduced in [15], and some parts will be referred directly to +that paper. However, since our conclusion is slightly different, we prefer to develop some computations +in detail. We will use the notation a ≲ b when a ≤ Cb for some dimensional constant C. + +A FREE DISCONTINUITY APPROACH TO OPTIMAL PROFILES IN STOKES FLOWS +25 +We divide the proof in several steps. +Step 1: Subdivision in small squares. Let us set +N := 1 + +� +H1(J)− 1 +2 +� +, +where [·] denotes the integer part. In the following we will assume that H1(J) is arbitrary small, so +that N is arbitrarily large. For convenience in the construction, we will set δ = 1/N≤ H1(J) +1 +2 , which +(mildly) differs from the choice of the statement: +yet since δ is asymptotically equivalent to H1(J) +1 +2, +the mismatch does not affect the validity of the conclusion. +For r ∈]1 − δ +1 +2 , 1[ and each k ≥ −2, let us set +δk := δr +2k +and +rk = +� +N − 1 +2k +� +δ . +Then we consider a partition +(up to a negligible set) of Qr into cubes obtained by filling Qr0 with +cubes of side δ0 and denoted by (˜q0,j)j, and then each Qrk \ Qrk−1 with cubes of side δk and denoted +(˜qk,j)j (note that there is only one way to do this). +For any square q = z + [−t, t]2, we write +q′ := z + +� +−8 +7t, 8 +7t +�2 +and +q′′ := (q′)′. +We will set +qk,j := (˜qk,j)′. +We may notice that with our choices +(7.1) +∀k ≥ 1 : q′′ +k,j ⋐ Qrk+1 \ Qrk−2, +and {q′′ +k,j}k,j is a covering of Qr with a fixed finite number of overlapping: indeed each q′′ +k,j meets at +most 8 neighbours q′′ +p,i, and they all verify |k − p| ≤ 1, meaning δk/δp ∈ +� 1 +2, 1, 2 +� +. This is because the +factor 8 +7 above is chosen such that +� 8 +7 +�3 < 3 +2. +Step 2: Choice of the square Qr. We now make a convenient choice of r such that the density of +J near ∂Qr is small, following an approach similar to [17, Theorem 2.1]. +We claim that there exist C, η > 0 such that for δ < η we can choose r ∈]1− +√ +δ, 1[ with H0(J∩∂Qr) = +0, +(7.2) +∀ s ∈]0, r[ : H1(J ∩ (Qr \ Qr−s)) ≤ Cδ +3 +2 s +and +(7.3) +� +Qr\Qr−2 +|e(u)|2 dx < Cδ +1 +2 +� +Q1 +|e(u)|2 dx. +Consider indeed the measure µ on [0, 1] defined as +µ(E) := H1(J ∩ QE) +H1(J) ++ +� +QE |e(u)|2 dx +� +Q1 |e(u)|2 dx , +where QE := ∪r∈E∂Qr is the cubic shell associated to E ⊂ [0, 1]. It suffices to prove that we can find +r ∈]1 − δ +1 +2, 1[ such that +(7.4) +H0(J ∩ ∂Qr) = 0, +and, denoting Is +r := [r − s, r[ for 0 < s < r, +(7.5) +µ(Is +r) ≤ ˆCδ− 1 +2 s, + +26 +D. BUCUR, A. CHAMBOLLE, A. GIACOMINI, AND M. NAHON +where ˆC > 0 is a suitable constant which we fix below. Indeed, if δ is small enough this implies that +(recall that H1(J) behaves like δ2) +H1(J ∩ (Qr \ Qr−s)) ≤ H1(J)µ(Is +r) ≤ ˆCδ +3 +2 s +and +� +Qr\Qr−4δr +|e(u)|2 dx ≤ ˆCδ− 1 +2 (4δr) +� +Q1 +|e(u)|2 dx ≤ 4 ˆCδ +1 +2 +� +Q1 +|e(u)|2 dx, +so that (7.2) and (7.3) follow by choosing C := 4 ˆC. +Let I1 be the union of all intervals that do not satisfy (7.5). If (Isi +ri ) is a Vitali covering of I, then +2 = µ([0, 1]) ≥ +� +i +µ(Isi +ri ) > ˆCδ− 1 +2 +� +i +|Isi +ri | = +ˆCδ− 1 +2 +5 +� +i +|5Isi +ri | ≥ +ˆCδ− 1 +2 +5 +|I1| +hence |I1| < 10 +ˆC δ +1 +2 . +Let I2 := πx(J) ∪ πy(J), where πx, πy denote the projection on the coordinate axis: we have asymp- +totically |I2| ≤ 2δ2. If C > 10, this implies that for δ small enough +]1 − +√ +δ, 1[\(I1 ∪ I2) ̸= ∅, +which yields the existence of r which verifies claims (7.4) and (7.5). +Step 3: A first approximation. In view of (7.2) and of (7.1), for every k ≥ 1 we have +H1(Ju ∩ q′′ +k,j) ≲ δ +3 +2δk, +while if δ is small enough (recall that H1(J) behaves like δ2 and r ∈]1 − δ +1 +2, 1[) +H1(Ju ∩ q′′ +0,j) ≤ H1(Ju) ≲ δδ0. +This means that the jump set of u in every cube of the constructed subdivision is arbitrarily small +compared to its sides. +Thanks to [14, Proposition 3], and taking into account the preceding inequalities , for every (k, j) +there is a set ωk,j ⊂ q′ +k,j and an affine function ak,j with e(ak,j) = 0, such that +(7.6) +|ωk,j| ≲ δkH1(Ju ∩ q′′ +k,j) ≲ δδ2 +k +(7.7) +� +q′ +k,j\ωk,j +|u − ak,j|4 dx ≲ +� +δk +� +q′′ +k,j +|e(u)|2 dx +�2 +, +and the function vk,j := u + (ak,j − u)1ωk,j verifies +� +qk,j +|e(ρδk ∗ vk,j) − ρδk ∗ e(u)|2 dx ≲ +� +H1(Ju ∩ q′′ +k,j) +δk +� 1 +3 � +q′′ +k,j +|e(u)|2 dx +≲ δ +1 +3 +� +q′′ +j,k +|e(u)|2 dx, +(7.8) +(see [14, p. 1389]) where ρ is the mollifier defined at the beginning of the section. +Notice that in view of our construction (namely the choice of r), we have +(7.9) +|ωk,j| ≪ |qk,j|, +and this is where we most use the fact that we are in two dimensions. + +A FREE DISCONTINUITY APPROACH TO OPTIMAL PROFILES IN STOKES FLOWS +27 +We now let (ϕk,j) be a partition of unity associated to the covering (qk,j) of Qr and such that +|∇ϕk,j| ≲ 1 +δk . Let us set +w := 1Q1\Qru + 1Qr +� +k,j +ϕk,jwk,j +where +wk,j := ρδk ∗ vk,j. +We claim that +(7.10) +w ∈ SBD(Q1) ∩ H1(Qr), +{w ̸= u} ⊂ Qr, +H1(Jw \ J) = 0, +(7.11) +������ +e(w) − +� +k,j +ϕk,jρδk ∗ e(u) +������ +L2(Qr) +≲ δ +1 +6∥e(u)∥L2(Q1), +and +(7.12) +the trace of w and u on ∂Qr coincide. +We postpone the proof of these claims to Step 5. +Let us set +ϕ := +� +(0,j)∈K +ϕ0,j, +where K denotes the set of indices such that q0,j has a distance greater than 2δr from ∂Qr. Since +r ∈]1 − δ +1 +2 , 1[, in view of the definition of the set of indices K, we get that the function ϕ vanishes on +Q \ Qr−δ and it is equal to 1 on Qr−4δ. +We can write +e(w) − +� +k,j +ϕk,jρδk ∗ e(u) = +� +e(w) − ϕρδ ∗ e(u) +� +− +� +(k,j)̸∈K +ϕk,jρδk ∗ e(u). +Thanks to (7.3) we have +������ +� +(k,j)̸∈K +ϕk,jρδk ∗ e(u) +������ +2 +L2(Qr) += +������ +� +(k,j)̸∈K +ϕk,jρδk ∗ e(u) +������ +2 +L2(Qr\Qr−2δr) +≲ +� +(k,j)̸∈K +∥ϕj,kρδk ∗ e(u)∥2 +L2(Q1\Qr−2δr) +≲ ∥e(u)∥2 +L2(Q1\Qr−3δr) ≲ δ +1 +2 ∥e(u)∥2 +L2(Q1), +so that in view of (7.11) we conclude +(7.13) +∥e(w) − ϕρδ ∗ e(u)∥L2(Qr) ≲ δ +1 +6∥e(u)∥L2(Q1). +Moreover we may write +∥e(w)∥L2(Qr) ≤ ∥ϕρδ ∗ e(u)∥L2(Qr) + ∥e(w) − ϕρδ ∗ e(u)∥L2(Qr) += ∥ϕρδ ∗ e(u)∥L2(Qr−δ) + ∥e(w) − ϕρδ ∗ e(u)∥L2(Qr) +≤ ∥e(u)∥L2(Q1) + ∥e(w) − ϕρδ ∗ e(u)∥L2(Qr), +so that taking into account (7.13) we deduce +(7.14) +∥e(w)∥L2(Q1) ≤ (1 + Cδ +1 +6 )∥e(u)∥L2(Q1), +where C > 0. + +28 +D. BUCUR, A. CHAMBOLLE, A. GIACOMINI, AND M. NAHON +Step 4: Enforcing the divergence free constraint. By admissibility, u is divergence free in the +sense of distributions in Q1, so that the trace of e(u) is zero in Q1, while +(7.15) +� +∂Qr +u · ν dH1 = 0, +where ν is the outward normal vector of Qr, and u denotes the trace on ∂Qr (J does not intersect ∂Qr +by construction). +Recalling that w ∈ H1(Qr), we may write thanks to (7.13) +∥div w∥L2(Qr) = ∥Tr(e(w))∥L2(Qr) = ∥Tr(e(w) − ϕρδ ∗ e(u))∥L2(Qr) ≲ δ +1 +6 ∥e(u)∥L2(Q1). +By (7.12) the trace of u on ∂Qr coincides with that of w, so that from (7.15) we deduce +� +Qr +div w dx = 0. +Using a classical result (recorded at the end of this proof in Lemma 7.3), there exists a vector field +q ∈ H1 +0(Qr) such that +(7.16) +div q = div w +and +∥∇q∥L2(Qr) ≲ ∥div w∥L2(Qr) ≲ δ +1 +6 ∥e(u)∥L2(Q1). +Let +v := +� +w − q +in Qr +u +in Q1 \ Qr, +and let us check that v satisfies the conclusions of the lemma. +The choice of r given by Step 2 yields immediately point (a). Clearly v ∈ SBD(Q1) ∩ H1(Qr) with +{v ̸= u} ⊆ Qr. Moreover, since the trace of w − q and u coincide on ∂Qr, we get div v = 0 in the sense +of distributions in Q1, so that point (b) is proved. Points (c) and (d) follow from the corresponding +properties for w (see (7.13) and (7.14)) taking into account that the correction term q has a small +gradient norm of the order δ +1 +6 as estimated in (7.16). +Step 5: Proof of the claims (7.10), (7.11) and (7.12). In order to conclude the proof, we need to +check the claims on the function w contained in Step 3. +Let us start by noticing that the oscillation of the maps ak,j on intersecting squares can be estimated. +Indeed as soon as qk,j and qp,i intersects, then +|qk,j ∩ qp,i| ≳ max(|qk,j|, |qp,i|), +and since (see (7.9)) +|(q′ +k,j ∩ q′ +p,i) ∩ (ωk,j ∪ ωp,i)| ≪ |q′ +k,j ∩ q′ +p,i| +and aj,k, ai,p are affine, then using [15, Lemma 3.4] and (7.7) we deduce +(7.17) +∥ak,j − ap,i∥L4(q′ +k,j∩q′ +p,i) ≲ ∥ak,j − ap,i∥L4((q′ +k,j∩q′ +p,i)\(ωk,j∪ωp,i)) +≤ ∥ak,j − u∥L4(q′ +k,j\ωk,j) + ∥ap,i − u∥L4(q′ +p,i\ωp,i) ≲ δ +1 +2 +k ∥e(u)∥L2(q′′ +k,j) + δ +1 +2p ∥e(u)∥L2(q′′ +p,i) +≲ δ +1 +2 +k ∥e(u)∥L2(q′′ +k,j∪q′′ +p,i), +as δk and δp are comparable. +Let us come to the claims. Clearly +e(w) = +� +k,j +ϕk,je(wk,j) + +� +k,j +∇ϕk,j ⊙ wk,j, + +A FREE DISCONTINUITY APPROACH TO OPTIMAL PROFILES IN STOKES FLOWS +29 +so that +(7.18) +e(w) − +� +k,j +ϕk,jρδk ∗ e(u) = +� +k,j +ϕk,j +� +e(wk,j) − ρδk ∗ e(u) +� ++ +� +k,j +∇ϕk,j ⊙ wk,j. +For the first term of the right hand side, we have thanks to (7.8) +������ +� +k,j +ϕk,j +� +e(wk,j) − ρδk ∗ e(u) +� +������ +2 +L2(Qr) +≲ +� +k,j +���ϕk,j +� +e(wk,j) − ρδk ∗ e(u) +���� +2 +L2(Qr) +≤ +� +k,j +∥e(wk,j) − ρδk ∗ e(u)∥2 +L2(qk,j) ≤ δ +1 +3 +� +k,j +∥e(u)∥2 +L2(q′′ +k,j) ≲ δ +1 +3 ∥e(u)∥2 +L2(Qr), +(7.19) +where we used the finite overlapping of the squares q′′ +k,j for the first and last estimates. +Let us estimate the second term on the right hand side of (7.18). Notice that we may write +� +k,j +∇ϕk,j ⊙ wk,j = +� +qk,j∩qp,i̸=∅ +∇ϕk,j ⊙ (wk,j − wp,i) +on qp,i +since � +k,j ∇ϕk,j = 0. +(a1) If q′′ +p,i ⋐ Qr−1, then qj,k ∩ qi,p ̸= ∅ means that δk = δp = δ, k = p = 0, and we may rewrite the +term as +� +q0,j∩q0,i̸=∅ +∇ϕ0,j ⊙ (w0,j − w0,i) +We get +(7.20) +������ +� +q0,j∩q0,i̸=∅ +∇ϕ0,j ⊙ (w0,j − w0,i) +������ +2 +L2(q0,i) +≲ +� +q0,j∩q0,i̸=∅ +1 +δ2 ∥w0,j − w0,i∥2 +L2(q0,j∩q0,i). +Now +∥w0,j − w0,i∥L2(qo,j∩q0,i) = ∥ρδ ∗ (v0,j − v0,i)∥L2(q0,j∩q0,i) ≤ ∥v0,j − v0,i∥L2(q′ +0,j∩q′ +0,i). +Since +∥v0,j − v0,i∥L2(q′ +0,j∩q′ +0,i) ≤ ∥(a0,j − a0,i)1ω0,j∪ω0,i∥L2(q′ +0,j∩q′ +0,i) + ∥(u − a0,j)1ω0,i∥L2(q′ +0,j\ω0,j) ++ ∥(u − a0,i)1ω0,j∥L2(q′ +0,i\ω0,i) +≤ ∥(a0,j − a0,i)∥L4(q′ +0,j∩q′ +0,i)|ω0,j ∪ ω0,i| +1 +4 + ∥(u − a0,j)∥L4(q′ +0,j\ω0,j)|ω0,i| +1 +4 ++ ∥(u − a0,i)∥L4(q′ +0,i\ω0,i)|ω0,j| +1 +4 , +recalling (7.6), (7.7) and (7.17) we get +∥w0,j − w0,i∥L2(qo,j∩q0,i) ≤ ∥v0,j − v0,i∥L2(q′ +0,j∩q′ +0,i) ≤ δ1+ 1 +4∥e(u)∥L2(q′′ +0,j∪q′′ +0,i). +Coming back to (7.20) we infer +������ +� +k,j +∇ϕk,j ⊙ wk,j +������ +2 +L2(q0,i) +≤ +������ +� +q0,j∩q0,i̸=∅ +∇ϕ0,j ⊙ (w0,j − w0,i) +������ +2 +L2(q0,i) +≲ δ +1 +2 +� +q0,j∩q0,i̸=∅ +∥e(u)∥2 +L2(q′′ +0,j∪q′′ +0,i). +(7.21) + +30 +D. BUCUR, A. CHAMBOLLE, A. GIACOMINI, AND M. NAHON +(a2) If qp,i ⊈ Qr−1, then for qk,j ∩ qp,i ̸= ∅, we decompose +wp,i − wk,j = ρδp ∗ (vp,i − ap,i) − ρδk ∗ (wk,j − ak,j) + (ap,i − ak,j). +Notice the crucial step that ρδk ∗ ak,j = ak,j due to the fact that ak,j is harmonic (since it is +affine). Then we have thanks to (7.7) and (7.17) +∥ρδk ∗ (vp,i − ap,i)∥L2(qk,j∩qp,i) ≤ ∥vp,i − ap,i∥L2(q′ +p,i) ≲ δp∥e(u)∥L2(q′′ +i,p) +∥ρδk ∗ (vk,j − ak,j)∥L2(qk,j∩qp,i) ≤ ∥vk,j − ak,j∥L2(q′ +k,j) ≲ δp∥e(u)∥L2(q′′ +k,j) +∥ap,i − ak,j∥L2(qk,j∩qp,i) ≲ δ +1+ 1 +4 +p +∥e(u)∥L2(q′′ +k,j∪q′′ +p,i), +where we also used the fact that δp and δk differ from at most a factor 2. And so we obtain +with the same computations as the previous point that +(7.22) +������ +� +k,j +∇ϕk,j ⊙ wk,j +������ +2 +L2(qp,i) +≤ +� +qk,j∩qp,i̸=∅ +∥e(u)∥2 +L2(q′′ +k,j∪q′′ +p,i). +Gathering (7.21) and (7.22), and in view of the choice of r which satisfies (7.3), we deduce +������ +� +k,j +∇ϕk,j ⊙ wk,j +������ +2 +L2(Qr) +≤ +� +p,i +������ +� +k,j +∇ϕk,j ⊙ wk,j +������ +2 +L2(qp,i) +≲ δ +1 +2 ∥e(u)∥2 +L2(Qr1) + ∥e(u)∥2 +L2(Qr\Qr−2) ≲ δ +1 +2∥e(u)∥2 +L2(Q1). +(7.23) +Coming back to (7.18), in view of (7.19) and (7.23) we deduce that +������ +e(w) − +� +k,j +ϕk,jρδk ∗ e(u) +������ +L2(Qr) +≲ δ +1 +6∥e(u)∥L2(Q1), +so that claim (7.11) follows. +In particular we get also that w ∈ H1(Qr). +Claim (7.12) concerning the traces follows by the +construction which involves convolutions whose radius becomes finer and finer as we approach ∂Qr as +detailed in [15]. Finally we deduce that w ∈ SBD(Q1), and that claim (7.10) holds true. +□ +In the proof of Proposition 7.2 we made use of the following lemma due to Neˇcas ( see [6, Theorem +IV.3.1], or also [4]). +Lemma 7.3. Let Ω be a bounded, connected open set with Lipschitz boundary, and let L2 +0(Ω) be the +set of zero-average L2-functions. Then there is a continuous linear map Φ : L2 +0(Ω) → H1 +0(Ω; Rd) such +that div ◦ Φ = IdL2 +0(Ω). +7.2. Regularity for quasi minimizers of the Griffith energy. Let Ω ⊆ R2 be an open set. In all +the following, we will consider the Griffith functional +G(J, u, B) := +� +B +|e(u)|2 dx + H1(J ∩ B), +where B ⊆ Ω is a Borel set. +We consider the following (very weak) notion of local minimality. + +A FREE DISCONTINUITY APPROACH TO OPTIMAL PROFILES IN STOKES FLOWS +31 +Definition 7.4 (Quasi minimizers). Let Λ, r > 0. We say that (J, u) ∈ V(Ω) (recall Definition 7.1) +is a (Λ, r) quasi minimizer of G on V(Ω) if G(J, u, ω) < +∞ for any open set ω ⋐ Ω, and for any +square Qx,r ⋐ Ω with r ∈ (0, r), H0(J ∩ ∂Qx,r) = 0 and +lim sup +s→0+ +1 +sH1 (J ∩ (Qx,r \ Qx,r−s)) < 1, +and for any function v ∈ H1(Qx,r; R2) with div v = 0 and v = u on ∂Qx,r, we have +(7.24) +� +Qx,r +|e(u)|2 dx + H1(J ∩ Qx,r) ≤ +� +Qx,r +|e(v)|2 dx + Λr2. +Remark 7.5. Notice that under the assumption of the previous definition, we have (J\Qx,r, v) ∈ V(Ω), +where we extended v to the entire Ω by setting v = u in Ω \Qx,r, and inequality (7.24) may be written +as +G(J, u, Qx,r) ≤ G(J \ Qx,r, v, Qx,r) + Λr2. +The local minimality property involves thus a comparison between (J, u) and very special competi- +tors: the Sobolev function v is obtained by “smoothing out” the jumps of u inside suitable squares Qx,r, +so that it can be paired with the rectifiable set J \Qx,r, yielding the admissible pair (J \Qx,r, v). Such +competitors are provided by the Smoothing Lemma 7.2, for which the dimension two is essential. A +somehow related weak notion of minimality involving Sobolev competitors, still in dimension two, has +been investigated in [9] (minimality with respect to its own jump set) for the (scalar) Mumford-Shah +functional. +Remark 7.6. The notion of minimality is weak enough to include any local minimizer of a functional +of the form +F(u, A) := +� +A +|e(u)|2 dx + +� +Ju∩A +Θ(νu, u+, u−)dH1 +where Θ is a measurable function such that inf(Θ) ≥ 1 (or, inf(Θ) > 0 up to scaling). +The following result is the key ingredient for obtaining regularity. +Proposition 7.7 (Decay estimate). Let Λ > 0. There exists a universal constant τ ∈ (0, 1) such +that for every τ ∈ (0, ¯τ) there exist ε = ε(τ) and ¯r = ¯r(τ) with the property that for any (Λ, r)-quasi +minimizer (J, u) of G on V(Ω), if for r < ¯r +G(J, u, Qr) ≥ r3/2 +and +H1(J ∩ Qr) ≤ εr, +then +G(J, u, Qτr) ≤ τ 3/2G(J, u, Qr). +Proof. By contradiction assume that for τ sufficiently small there exist εn → 0, ¯rn → 0, 0 < rn < ¯rn, +and a sequence (Kn, wn) of (Λ, rn)-minimizers for such that for every n +G(Kn, wn, Qrn) ≥ r3/2 +n , +H1(Kn ∩ Qrn) ≤ εnrn, +and +G(Kn, wn, Qτrn) > τ 3/2G(Kn, wn, Qrn). +Let +gn := G(Kn, wn, Qrn), +Jn := Kn +rn +and +un(x) := wn(rnx) +√gn +. +Then (Jn, un) is a (Λ√rn, 1)-minimizer of Gn(·, ·, Q1), where +Gn(J, u, A) := +� +A +|e(u)|2 dx + rn +gn +H1(J ∩ A), +with +(7.25) +Gn(Jn, un, Q1) = 1, +Gn(Jn, un, Qτ) > τ 3/2 +and +H1(Jn ∩ Q1) = εn. + +32 +D. BUCUR, A. CHAMBOLLE, A. GIACOMINI, AND M. NAHON +Let us apply the Smoothing Lemma 7.2: if δn = ε +1 +2n, let Qsn with 1 − δ +1 +2n < sn < 1 be the square on +which the jumps of un are smoothed out giving raise to the function vn, associated to an admissible +pair (J \ Qsn, vn) ∈ V(Q1). In particular +(7.26) +∥e(vn)∥L2(Q1) ≤ (1 + Cδ +1 +6n )∥e(un)∥L2(Q1) +with +∥e(un)∥L2(Q1) ≤ 1, +and +(7.27) +∥e(vn) − ϕnρδn ∗ e(u)∥L2(Qsn) ≤ Cδ +1 +6n ∥e(un)∥L2(Q1), +where C > 0 is independent of n and ϕn ∈ C∞(Qsn, [0, 1]) is such that ϕn = 0 on Qsn \Qsn−δn, ϕn = 1 +on Qsn−4δn. Since vn is divergence free and Sobolev on Qsn we have +(7.28) +� +∂Qsn +vn · ν dH1 = 0. +By the classical Korn inequality on Qsn there is an antisymmetric affine function an such that +� +Qsn(vn − an) dx = 0 and +� +Qsn +|∇(vn − an)|2 dx ≤ C1 +� +Qsn +|e(vn)|2 dx +for some C1 > 0 independent of n. We infer that (vn − an) is bounded in H1(Qsn). Since sn → 1, we +can assume, up to extracting a further subsequence, +(7.29) +vn − an ⇀ w +weakly in H1 +loc(Q1; R2) +for some w ∈ H1(Q1). Since every vn−an has zero divergence, then so does w. Moreover ∥e(w)∥L2(Q1) ≤ +1. +Let ψ ∈ C∞ +c (Q1; R2) have zero divergence, and let η ∈ C∞ +c (Q1, [0, 1]) be a cut-off function such that +{ψ ̸= 0} ⋐ {η = 1}. Let us consider +zn := +� +PQsn +� +(1 − η)vn + η(an + w + ψ) +� +in Qsn +un +in Q1 \ Qsn, +where PQsn denotes the projection on divergence free H1(Qsn) vector fields which preserves the trace +obtained according to Lemma 7.3 by considering +PQsn(u) := u − ΦQsn(div u) +for any u ∈ H1(Qsn; R2) with a zero mean divergence. Note that zn is well defined as +(1 − η)vn + η(an + w + ψ) = vn +on ∂Qsn +for n large enough, and so its divergence has zero mean thanks to (7.28). +Since (Jn \ ∂Qsn, zn) is an admissible competitor for (Jn, un) according to Definition 7.4, we obtain +Gn(Jn, un, Qsn) +≤ +���e +� +PQsn +� +(1 − η)vn + η(an + w + ψ) +����� +2 +L2(Qsn) + Λ√rn +≤ +� +∥e ((1 − η)vn + η(an + w + ψ)) ∥L2(Qsn) + C∥div((1 − η)vn + η(an + w + ψ))∥L2(Qsn) +�2 + Λ√rn +≤ +� +∥e ((1 − η)vn + η(an + w + ψ)) ∥L2(Qsn) + C∥∇η · (w + an − vn)∥L2(Qsn) +�2 + Λ√rn. +Since +∥∇η · (w + an − vn)∥L2(Qsn) → 0 + +A FREE DISCONTINUITY APPROACH TO OPTIMAL PROFILES IN STOKES FLOWS +33 +and (recall that {ψ ̸= 0} ⋐ {η = 1}) +∥e ((1 − η)vn + η(an + w + ψ)) ∥L2(Qsn) = ∥(1 − η)e(vn) + ηe(w + ψ) + ∇η ⊙ (w + an − vn)∥L2(Qsn) +≤ ∥(1 − η)e(vn) + ηe(w + ψ)∥L2(Qsn) + on +where on → 0, we infer thanks to (7.26) (and since e(an) = 0) +(7.30) +Gn(Jn, un, Qsn) ≤ ∥(1 − η)e(vn − an) + ηe(w + ψ)∥2 +L2(Qsn) + on. +Now, still using (7.26) we may write +� +Qsn +|e(vn − an)|2 dx ≤ (1 + Cδ +1 +6n )2 +� +Qsn +|e(un)|2 dx ≤ (1 + Cδ +1 +6n )2Gn(Jn, un, Qsn) + on, +and so coming back to (7.30) we deduce +∥e(vn − an)∥2 +L2(Qsn) ≤ ∥(1 − η)e(vn − an) + ηe(w + ψ)∥2 +L2(Qsn) + on. +This yields +� +Qsn +� +1 − (1 − η)2� +|e(vn − an)|2 dx ≤ +� +Qsn +� +2η(1 − η)e(vn − an) : e(w) + η2|e(w + ψ)|2� +dx + on +so that in view of (7.29) +� +Q1 +� +1 − (1 − η)2� +|e(w)|2 dx ≤ lim sup +n→∞ +� +Q1 +� +1 − (1 − η)2� +|e(vn − an)|2 dx +≤ +� +Q1 +� +2η(1 − η)e(w) : e(w) + η2|e(w + ψ)|2� +dx. +Notice that by choosing ψ = 0 and letting η localize on characteristic functions of open sets, we infer +that +(7.31) +e(vn − an) → e(w) +strongly in L2 +loc(Q1; M2×2 +sym). +In particular we get +� +Q1 +|e(w)|2 dx ≤ +� +Q1 +|e(w + ψ)|2 dx, +which means that w is a local minimizer of the energy z �→ ∥e(z)∥2 +L2(Q1) on H1 functions with zero +divergence. This yields ∆w = ∇p for some p ∈ L2(Q1). Using the Lemma 7.8 below, we have +� +Qτ +|e(w)|2 dx ≤ 1 +2τ +3 +2 +� +Q1 +|e(w)|2 dx ≤ 1 +2τ +3 +2. +Taking into account (7.31) we deduce +(7.32) +∥e(vn)∥2 +L2(Qτ+δn) ≤ 1 +2τ +3 +2 + on. +By minimality we have +G(un, Jn, Qsn) ≤ ∥e(vn)∥2 +L2(Qsn) + Λ√rn = ∥e(vn)∥2 +L2(Qτ+δn) + ∥e(vn)∥2 +L2(Qsn\Qτ+δn) + Λ√rn +while thanks to (7.27) +∥e(vn)∥L2(Qsn\Qτ+δn) ≤ ∥e(vn) − ϕnρδn ∗ e(un)∥L2(Qsn\Qτ+δn) + ∥ϕnρδn ∗ e(un)∥L2(Qsn\Qτ+δn) +≤ on + ∥ρδn ∗ e(un)∥L2(Qsn−δn\Qτ+δn) ≤ on + ∥e(un)∥L2(Qsn\Qτ ). + +34 +D. BUCUR, A. CHAMBOLLE, A. GIACOMINI, AND M. NAHON +In view of (7.32) we infer +G(un, Jn, Qsn) ≤ ∥e(vn)∥2 +L2(Qτ+δn)+[on+∥e(un)∥L2(Qsn\Qτ )]2+Λ√rn ≤ 1 +2τ +3 +2+˜on+∥e(un)∥2 +L2(Qsn\Qτ ) +so that +G(un, Jn, Qτ) ≤ 1 +2τ +3 +2 + ˜on. +In conclusion, taking into account (7.25), if n is large enough we get +τ +3 +2 < G(un, Jn, Qτ) ≤ 1 +2τ +3 +2 + ˜on, +which is a contradiction. +□ +In the preceding proof, we made use of the following result. +Lemma 7.8. There exists a constant C0 > 0 such that for any divergence-free vector field u ∈ +H1(Q1; R2) such that ∆u = ∇p for some pressure p ∈ L2(Q1), we have +∀τ ∈ (0, 1/2] : +� +Qτ +|e(u)|2 dx ≤ C0τ 2 +� +Q1 +|e(u)|2 dx. +In particular, for any 0 < τ ≤ τ := +1 +4C2 +0 ∧ 1 +2 we have +� +Qτ +|e(u)|2 dx ≤ 1 +2τ 3/2 +� +Q1 +|e(u)|2 dx. +Proof. Notice that e(u) is invariant by the addition of an asymmetric affine function a. +Up to a +translation by such a function, Korn’s inequality tells us that +� +Q1 +u2 dx ≤ C +� +Q1 +|e(u)|2 dx. +The equations verified by u are equivalent to the existence of ϕ ∈ H2(Q1) such that ϕ(0) = 0, u = ∇⊥ϕ, +and ∆2ϕ = 0. By elliptic regularity there is a constant C′ such that +sup +Q1/2 +��∇2ϕ +��2 ≤ C′ +� +Q1 +|∇ϕ|2 dx +and so for any τ ≤ 1/2, +� +Qτ +|e(u)|2 dx ≤ 4|Qτ| sup +Q1/2 +��∇2ϕ +��2 ≤ 4CC′|Q1|τ 2 +� +Q1 +|e(u)|2 dx. +□ +The decay estimate can be iterated as follows. +Lemma 7.9 (Iteration of the decay). Let Λ > 0 and, according to Proposition 7.7, let τ0 be small +enough such that the decay estimate applies +with ε0 = ε(τ0) and ¯r0 = ¯r(τ0), and let τ1 ∈ (0, ε2 +0) be +small enough that the decay property applies with ε1, ¯r1. Finally, let +¯r := min +� +¯r0, ¯r1, ε2 +0τ 2 +1 , ε2 +0τ 3 +0 +τ1 +� +. +Suppose that (J, u) is a (Λ, ¯r)-quasi minimizer of G on V(Ω) and G(J, u, Qx,r) ≤ ε1r for some r ∈ (0, ¯r). +Then for all k ∈ N, +G(J, u, Qx,τ k +0 τ1r) ≤ ε0τ +3 +2k +0 τ1r. + +A FREE DISCONTINUITY APPROACH TO OPTIMAL PROFILES IN STOKES FLOWS +35 +Proof. Let us prove the statement by induction on k. In the following, we write g(r) = G(J, u, Qx,r), +so that we need to check that if g(r) ≤ ε1r, then for every k ∈ N +(7.33) +g(τ k +0 τ1r) ≤ ε0τ +3 +2k +0 τ1r. +The inequality is true for k = 0. Indeed we have the following alternatives: +(a) If g(r) > r3/2 then g(τ1r) ≤ τ 3/2 +1 +g(r) ≤ √τ1τ1ε1r ≤ ε0τ1r by definition of τ1. +(b) If g(r) ≤ r3/2, then g(τ1r) ≤ g(r) ≤ r3/2 ≤ ε0τ1r by definition of ¯r. +Assume now that (7.33) holds. Notice that by definition of G we have (since τ0 < 1) +H1(J ∩ Qτ k +0 τ1r) ≤ g(τ k +0 τ1r) ≤ ε0τ +3 +2k +0 τ1r ≤ ε0τ k +0 τ1r, +so the decay property of Proposition 7.7 may be applied. Again we have two alternatives. +(a) If g(τ k +0 τ1r) > (τ k +0 τ1r)3/2, by the decay property we have, using (7.33), +g(τ k+1 +0 +τ1r) ≤ τ 3/2 +0 +g(τ k +0 τ1r) ≤ ε0τ +3 +2(k+1) +0 +τ1r. +(b) If g(τ k +0 τ1r) ≤ (τ k +0 τ1r)3/2 then by the definition of ¯r +g(τ k+1 +0 +τ1r) ≤ g(τ k +0 τ1r) ≤ +� τ1r +ε2 +0τ 3 +0 +ε0τ +3 +2(k+1) +0 +τ1r ≤ ε0τ +3 +2 (k+1) +0 +τ1r. +In both cases, (7.33) follows for the choice k + 1, so that the induction step is proved. +□ +If we want to draw some conclusions on the regularity of quasi minimizers (J, u), we need somehow to +bound the freedom connected to the choice of J: notice indeed that any pair (J∆N, u) with H1(N) = 0 +is essentially equivalent to (J, u), where A∆B denotes the symmetric difference of sets. +We set +(7.34) +J+ := +� +x ∈ Ω : lim sup +r→0 +H1(J ∩ Qx,r) +r +> 0 +� +. +J+ is a sort of normalized version of J, where points of density zero have been erased. +By standard properties of rectifiable sets we have +H1(J∆J+) = 0. +As a consequence if (J, u) ∈ V(Ω), then also (J+, u) ∈ V(Ω) with G(J, u, A) = G(J+, u, A) for every +Borel set A ⊆ Ω. +Proposition 7.10. Given Λ > 0, there exist ε, ¯r > 0 such that of any (Λ, r)-quasi minimizer (J, u) of +G on V(Ω), if G(J, u, Qx,r) ≤ εr for some Qx,r ⋐ Ω with r < ¯r, then J+ ∩ Qx, r +2 = ∅. +Proof. Let ε0, ε1, τ1, τ2, ¯r be given according to Lemma 7.9. Notice that if G(J, u, Qx,r) ≤ ε1r with +r < ¯r, then for any ρ ∈ (0, r) +(7.35) +G(J, u, Qx,ρ) ≤ C0r− 1 +2 ρ +3 +2 +where C0 := max +� +ε1τ +− 3 +2 +1 +, ε0τ +− 1 +2 +0 +τ +− 1 +2 +1 +� +. +Let us set ε := 1 +2ε1, and assume G(J, u, Qx,r) ≤ εr. Notice that for any y ∈ Qx, r +2, we have +G(J, u, Qy, r +2) ≤ G(J, u, Qx,r) ≤ εr = ε1 +r +2 +so that from (7.35) +0 = lim +ρ→0+ +G(J, u, Qy,ρ) +ρ +≥ lim sup +ρ→0+ +H1(J ∩ Qy,ρ) +ρ +, +which yields J+ ∩ Qx, r +2 = ∅. + +36 +D. BUCUR, A. CHAMBOLLE, A. GIACOMINI, AND M. NAHON +□ +Proposition 7.11 (Regularity for quasi minimizers). Let Λ, r > 0. Then for any (Λ, r)-quasi +minimizer (J, u) of G on V(Ω) we have that J+ (see (7.34)) is essentially closed in Ω, i.e., +H1 � +Ω ∩ (J+ \ J+) +� += 0, +while u ∈ C∞(Ω \ J+). +Proof. Since the functional G coincides with a volume integral outside J, there exists a H1-negligible +set N ⊂ Ω \ J such that for every x ∈ Ω \ (J ∪ N) we have +lim +ρ→0 +G(J, u, Qx,ρ) +ρ += 0. +Thanks to Proposition 7.10 we infer +Ω ∩ J+ ⊂ J ∪ N ⊂ J+ ∪ (J \ J+) ∪ N. +Since the last two sets are H1-negligible, we infer H1(Ω ∩ (J+ \ J+)) = 0. +Since +H1(Ju \ J+) ≤ H1(J \ J+) = 0, +we get that u is locally H1 on Ω \ J+ (thanks to Korn’s inequality): smoothness then follows from the +regularity theory for solutions to Stokes equation (see e.g. [6, Theorem IV.5.8]). +□ +7.3. Proof of Theorem 4.10. We are now in a position to prove the regularity result given by +Theorem 4.10. +Let (E, u) be a minimizer of J and let us set +Λ := 4Lip(f) +and +J := Ju ∪ ∂∗E. +We also assume (up to multiplying u by c− 1 +2) that the constant c of (4.2) is 1. +We first prove that (J, u) is a (Λ, 1) quasi minimizer of the Griffith functional G on V(Ω) according +to Definition 7.4. Indeed, let Qx,r ⋐ Ω with r < 1 be a square as in Definition 7.4, with associated +competitor (J \ Qx,r, v). We claim that either +(7.36) +H1(∂Qx,r \ E(1)) = 0 +or +H1(∂Qx,r \ E(0)) = 0. +In the first case, from the minimality inequality +J (E, u) ≤ J (E, u1Ω\Qx,r) +we deduce u = 0 a.e. on Qx,r and H1(∂∗E ∩ Qx,r) = 0, so that the inequality to check for quasi +minimality is trivially satisfied. Notice that admissibility of (E, u1Ω\Qx,r) for the main problem follows +from the fact that the trace of u on ∂Qx,r is zero, being that boundary composed of points of density +one of the set E on which u vanishes. +If the second possibility in (7.36) holds true, then the relations (see [27, Theorem 16.3] and recall +that H0(∂Qx,r ∩ ∂∗E) = 0 by the properties of Qx,r) +J (E, u) ≤ J (E \ Qx,r, v) +and +∂∗(E \ Qx,r) = ∂∗E \ Qx,r = ∂∗E \ Qx,r +yield in particular +� +Qx,r +|e(u)|2 dx + H1(J ∩ Qx,r) ≤ +� +Qx,r +|e(v)|2 dx + Λr2, +so that the quasi minimality of (J, u) follows. +By Proposition 7.11, we get that the normalized set J+ (see (7.34)) is essentially closed in Ω, i.e., +H1(Ω ∩ (J+ \ J+)) = 0, + +A FREE DISCONTINUITY APPROACH TO OPTIMAL PROFILES IN STOKES FLOWS +37 +and u is smooth on Ω \ J+, so that +Ω ∩ Ju ⊆ J+. +On the other hand, in view of the general properties of the reduced boundary of sets of finite perime- +ter (see [1, Theorem 3.59] or [27, Theorem 15.5]) we have ∂∗E ⊆ J+. +Taking into account that +H1(J+∆J) = 0 (where ∆ denotes the symmetric difference of sets) we infer +H1(Ω ∩ Ju ∪ ∂∗E \ (Ju ∪ ∂∗E)) ≤ H1(Ω ∩ J+ \ J) ≤ H1(Ω ∩ J+ \ J+) + H1(J+ \ J) = 0 +so that the conclusion follows. +In order to complete the proof, we need to check claim (7.36). +Assume by contradiction that the claim is false. Then there exists p ∈ E(1)∩∂Qx,r and q ∈ E(0)∩∂Qx,r +that are not in one of the corners. Without loss of generality we suppose p, q ∈ {x − re2 + Re1} with +p1 < q1, the case when both are in different sides being analog. We let for s > 0 small +• Cp := p + [−s, 0] × [0, s] and Cq := q + [0, s]2, +• gs : [p1 − s, q1 + s] → [0, 1] be zero at the extremes, affine on [p1 − s, p1] and [q1, q1 + s] and +equal to 1 on [p1, q1], +• fs ∈ C1 +c (]0, s[) with 0 ≤ fs ≤ 1, +• ϕs(x) = gs(x1)fs(x2 + r). +Then +H1(J ∩ (Qx,r \ Qx,r−s)) ≥ +� +∂∗E +ϕs(νE)1 dH1 = +� +E +∂1ϕs dH1 += 1 +s +� +E∩Cp +fs(y2 + r)dy − 1 +s +� +E∩Cq +fs(y2 + r)dy +so that, letting fs ր 1 we get +H1(J ∩ (Qx,r \ Qx,r−s)) +s +≥ |E ∩ Cp| +|Cp| +− |E ∩ Cq| +|Cq| +. +Since as s → 0+, by assumption on the density properties of p and q, we have +|E ∩ Cp| +|Cp| +→ 1 +and +|E ∩ Cq| +|Cq| +→ 0, +we infer +lim sup +s→0 +H1(J ∩ (Qx,r \ Qx,r−s)) +s +≥ 1 +which is against the assumption on r in Definition 7.4 of quasi minimality. The proof is thus concluded. +7.4. Some remarks on a “strong” formulation of the problem. In this section we elaborate on +a different relaxation of the drag minimization problem which involves topologically closed (but not +necessarily regular) obstacles F in the channel Ω and velocity vector fields which are H1 +loc on Ω \ F. +Within this perspective, given Ω ⊂ Rd open and bounded, it is natural to start with pairs (F, u) +such that +(7.37) +F ⊆ Ω is relatively closed, +Ω ∩ ∂F is rectifiable, +Hd−1(Ω ∩ ∂F) < +∞ +and +(7.38) +u ∈ H1 +loc(Ω \ F; Rd), +div u = 0 in Ω \ F, +e(u) ∈ L2(Ω \ F; Md×d +sym). +Notice that, as for the relaxation studied in the previous sections, ∂F may contain “lower dimensional” +parts. The set Ω \ F is open, so that the space H1 +loc(Ω \ F; Rd) is well defined. +It is not clear how to talk about traces on ∂(Ω\F), which are fundamental to formulate the tangency +constraint, as the set is in general not regular. It turns out that velocities admit a well defined trace + +38 +D. BUCUR, A. CHAMBOLLE, A. GIACOMINI, AND M. NAHON +on Hd−1 almost every point ∂F even if this set is not assumed to be only rectifiable and not regular. +This is a consequence of the following result which involves the space GSBD of Generalised Functions +of Bounded Deformations introduced in [19]. Let us set +(7.39) +˜u := +� +u +in Ω \ F +0 +in F. +Lemma 7.12. Let Ω ⊆ Rd be a bounded open set, and assume that the pair (F, u) satisfies (7.37) and +(7.38). Then ˜u ∈ GSBD(Ω) with Hd−1(J˜u \ ∂F) = 0. +Proof. Since Hd−1(Ω ∩ ∂F) < ∞, for every ε > 0 we may find some covering of ∂F through a finite +union of balls of radius less than ε, denoted (Bε +i )1≤i≤Nε, such that +Nε +� +i=1 +�diam(Bε +i ) +2 +�d−1 +≤ C +for some C > 0 that does not depend on ε. Let Bε be the union of these balls - which is a Lipschitz +set up to a small perturbation of the radii - and let uε := u1Ω\Bε. Then uε ∈ SBD(Ω) with +Euε = e(u) dx⌊(Ω \ (F ∪ Bε)) + uHd−1⌊∂Bε. +Moreover +uε → ˜u +a.e. in Ω +with +lim sup +ε→0 +� +Ω +|e(uε)|2 dx + Hd−1(Juε) < +∞. +We apply [16, Theorem 1.1] to (uε): since ˜u is finite almost everywhere, we directly identify ˜u with the +limit that is obtained, and we infer ˜u ∈ GSBD(Ω); moreover up to a Hd−1-negligible set, J˜u ⊂ ∂F by +construction, and the result follows. +□ +Coming back to configurations (F, u) satisfying (7.37) and (7.38), up to a choice of orientation of +the rectifiable set Ω ∩ ∂F, there is no ambiguity in defining the traces u± +|∂F of u on Hd−1-almost all +points of Ω ∩ ∂F. +In addition to the previous items, we thus require also for (F, u) the non-penetration condition +(7.40) +u± +|∂F · ν∂F = 0 +Hd−1-a.e. on Ω ∩ ∂F. +Given an admissible configuration (F, u), we can consider the following energy (all the constants +have been normalized to 1) +J(F, u) := +� +Ω\F +|e(u)|2 dx + +� +Ω∩∂eF +|u+|2 dHd−1 + +� +Ω∩F (0)∩F +[|u+|2 + |u−|2] dHd−1 ++ Hd−1(Ω ∩ ∂eF) + 2Hd−1(Ω ∩ F (0) ∩ F) + f(|F|), +where ∂eF denotes the measure theoretical boundary of F, and f is the penalization function introduced +in the previous sections (see (4.1)). +Configuration with finite energy are linked to admissible configurations of our main relaxed problem +by the following result. +Lemma 7.13. Let Ω ⊆ Rd be open and bounded, and let (F, u) satisfy (7.37) and (7.38) with J(F, u) < ++∞. Then the function ˜u defined in (7.39) is such that ˜u ∈ SBD(Ω). + +A FREE DISCONTINUITY APPROACH TO OPTIMAL PROFILES IN STOKES FLOWS +39 +Proof. It suffices to note that for every direction ξ ∈ Sd−1 we have +J(F, u) ≥ +� +ξ⊥ + + +� +Ωξ +y +|(˜uξ +y)′(t)|2dt + +� +t∈J˜uξ +y +� +1 + |(˜uξ +y)+(t)|2 + |(˜uξ +y)−(t)|2� + + dH1(y). +□ +Dealing with boundary conditions yields to the same problem highlighted in our main relaxation. +Assume Ω has a Lipschitz boundary, and let us write simply u in place of ˜u. We have that u ∈ SBD(Ω) +so that the trace on ∂Ω is well defined. Given a divergence free vector field V ∈ C1(Rd; Rd), we can +deal with the relaxation of the boundary condition by considering the set +Γu,V := {x ∈ ∂Ω : u(x) ̸= V (x)}, +and enforcing the non-penetration constraint leading to +(7.41) +u · ν∂Ω = 0 +and +V · ν∂Ω = 0 +Hd−1-a.e. on ΓV,∂Ω. +So for a configuration (F, u) satisfying (7.37), (7.38), (7.40) and (7.41), we can consider the energy +J strong(F, u) := J(F, u) + 2Hd−1(Γu,V ) + +� +Γu,V +[|V |2 + |u|2] dHd−1. +The minimization of J strong on admissible configurations is a different possible relaxation of the original +drag minimization problem. We clearly have +min +(E,u)∈AV (Ω) J (E, u) ≤ inf +(F,u) J strong(F, u). +Equality is reached in dimension two thanks to the regularity result given by Theorem 4.10. Indeed, +if (E, u) is a minimizer for J , we know that ∂∗E ∪ Ju is essentially closed, so that an admissible +relatively closed set F arises by considering the complement of the union of the connected components +of Ω \ ∂∗E ∪ Ju on which u does not vanish identically. The function u is smooth outside F, so that +the pair (F, u) is strongly admissible with J strong(F, u) = J (E, u). As a consequence, in dimension +two the relaxed problem +min +(F,u) J strong(F, u) +is indeed well posed. +References +[1] L. Ambrosio, N. Fusco, and D. Pallara. Functions of bounded variation and free discontinuity problems. Oxford +Mathematical Monographs. The Clarendon Press Oxford University Press, New York, 2000. +[2] Luigi Ambrosio, Alessandra Coscia, and Gianni Dal Maso. Fine properties of functions with bounded deformation. +Arch. Rational Mech. Anal., 139(3):201–238, 1997. +[3] G. Bellettini, A. Coscia, and G. Dal Maso. Compactness and lower semicontinuity properties in SBD(Ω). Math. Z., +228(2):337–351, 1998. +[4] M. E. Bogovski˘ı. Solution of the first boundary value problem for an equation of continuity of an incompressible +medium. Dokl. Akad. Nauk SSSR, 248(5):1037–1040, 1979. +[5] Matthieu Bonnivard and Dorin Bucur. Microshape control, riblets, and drag minimization. SIAM J. Appl. Math., +73(2):723–740, 2013. +[6] Franck Boyer and Pierre Fabrie. Mathematical tools for the study of the incompressible Navier-Stokes equations and +related models, volume 183 of Applied Mathematical Sciences. Springer, New York, 2013. +[7] D. Bucur and A. Giacomini. A variational approach to the isoperimetric inequality for the Robin eigenvalue problem. +Arch. Ration. Mech. Anal., 198(3):927–961, 2010. +[8] Dorin Bucur, Eduard Feireisl, and ˇS´arka Neˇcasov´a. Boundary behavior of viscous fluids: influence of wall roughness +and friction-driven boundary conditions. Arch. Ration. Mech. Anal., 197(1):117–138, 2010. +[9] Dorin Bucur, Ilaria Fragal`a, and Alessandro Giacomini. Local minimality results for the Mumford-Shah functional +via monotonicity. Anal. PDE, 13(3):865–899, 2020. + +40 +D. BUCUR, A. CHAMBOLLE, A. GIACOMINI, AND M. NAHON +[10] Dorin Bucur and Alessandro Giacomini. Faber-Krahn inequalities for the Robin-Laplacian: a free discontinuity +approach. Arch. Ration. Mech. Anal., 218(2):757–824, 2015. +[11] Dorin Bucur and Alessandro Giacomini. Shape optimization problems with Robin conditions on the free boundary. +Ann. Inst. H. Poincar´e Anal. Non Lin´eaire, 33(6):1539–1568, 2016. +[12] Luis A. Caffarelli and Dennis Kriventsov. A free boundary problem related to thermal insulation. Comm. Partial +Differential Equations, 41(7):1149–1182, 2016. +[13] J. Casado-D´ıaz, M. Luna-Laynez, and F. J. Su´arez-Grau. Asymptotic behavior of a viscous fluid with slip boundary +conditions on a slightly rough wall. Math. Models Methods Appl. Sci., 20(1):121–156, 2010. +[14] Antonin Chambolle, Sergio Conti, and Gilles Francfort. Korn-Poincar´e inequalities for functions with a small jump +set. Indiana Univ. Math. J., 65(4):1373–1399, 2016. +[15] Antonin Chambolle, Sergio Conti, and Flaviana Iurlano. Approximation of functions with small jump sets and +existence of strong minimizers of Griffith’s energy. J. Math. Pures Appl. (9), 128:119–139, 2019. +[16] Antonin Chambolle and Vito Crismale. Compactness and lower semicontinuity in GSBD. J. Eur. Math. Soc. (JEMS), +23(3):701–719, 2021. +[17] Sergio Conti, Matteo Focardi, and Flaviana Iurlano. Integral representation for functionals defined on SBDp in +dimension two. Arch. Ration. Mech. Anal., 223(3):1337–1374, 2017. +[18] Sergio Conti, Matteo Focardi, and Flaviana Iurlano. Existence of strong minimizers for the Griffith static fracture +model in dimension two. Ann. Inst. H. Poincar´e Anal. Non Lin´eaire, 36(2):455–474, 2019. +[19] Gianni Dal Maso. Generalised functions of bounded deformation. J. Eur. Math. Soc. (JEMS), 15(5):1943–1997, 2013. +[20] Gianni Dal Maso, Antonio DeSimone, and Maria Giovanna Mora. Quasistatic evolution problems for linearly elastic- +perfectly plastic materials. Arch. Ration. Mech. Anal., 180(2):237–291, 2006. +[21] Gianni Dal Maso, Gilles A. Francfort, and Rodica Toader. Quasistatic crack growth in nonlinear elasticity. Arch. +Ration. Mech. Anal., 176(2):165–225, 2005. +[22] E. De Giorgi, M. Carriero, and A. Leaci. Existence theorem for a minimum problem with free discontinuity set. Arch. +Rational Mech. Anal., 108(3):195–218, 1989. +[23] L. C. Evans and R. F. Gariepy. Measure theory and fine properties of functions. Studies in Advanced Mathematics. +CRC Press, Boca Raton, FL, 1992. +[24] H. Federer. Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften, Band 153. Springer- +Verlag New York Inc., New York, 1969. +[25] Gilles A. Francfort and Christopher J. Larsen. Existence and convergence for quasi-static evolution in brittle fracture. +Comm. Pure Appl. Math., 56(10):1465–1500, 2003. +[26] Manuel Friedrich, Matteo Perugini, and Francesco Solombrino. Lower semicontinuity for functionals defined on +piecewise rigid functions and on GSBD. J. Funct. Anal., 280(7):108929, 45, 2021. +[27] Francesco Maggi. Sets of finite perimeter and geometric variational problems, volume 135 of Cambridge Studies in +Advanced Mathematics. Cambridge University Press, Cambridge, 2012. An introduction to geometric measure theory. +[28] O. Pironneau. On optimum profiles in Stokes flow. J. Fluid Mech., 59:117–128, 1973. +[29] R. Temam. Variational problems of plasticity. In Trends in applications of pure mathematics to mechanics, Vol. III +(Edinburgh, 1979), volume 11 of Monographs Stud. Math., pages 209–217. Pitman, Boston, Mass.-London, 1981. +[30] Roger Temam and Gilbert Strang. Functions of bounded deformation. Arch. Rational Mech. Anal., 75(1):7–21, +1980/81. +[31] E. O. Tuck. Toward the calculation and minimization of stokes drag on bodies of arbitrary shape. In 3rd Australasian +Conference on Hydraulics and Fluid Mechanics (Institute of Engineers, Australia, 1970), pages 29–32, 1968. +[32] V. ˇSver´ak. On optimal shape design. J. Math. Pures Appl. (9), 72(6):537–551, 1993. +[33] S. R. Watson. Towards the minimum drag on a body of given volume in slow viscous flow. J. Inst. Math. Appl., +7:367–376, 1971. + +A FREE DISCONTINUITY APPROACH TO OPTIMAL PROFILES IN STOKES FLOWS +41 +(Dorin Bucur) Laboratoire de Math´ematiques CNRS UMR 5127 Universit´e de Savoie Mont Blanc Campus +Scientifique 73 376 Le Bourget-Du-Lac, France +Email address, D. Bucur: +dorin.bucur@univ-savoie.fr +(Antonin Chambolle) Ceremade, CNRS and Universit´e de Paris-Dauphine PSL, Place de Lattre de Tas- +signy, 75775 Paris Cedex 16, France +Email address, A. Chambolle: chambolle@ceremade.dauphine.fr +(Alessandro Giacomini) DICATAM, Sezione di Matematica, Universit`a degli Studi di Brescia, Via Branze +43, 25123 Brescia, Italy +Email address, A. Giacomini: alessandro.giacomini@unibs.it +(Micka¨el Nahon) Max-Planck-Institut f¨ur Mathematik in den Naturwissenschaften, 04103 Leipzig, Ger- +many +Email address, M. Nahon: mickael.nahon@mis.mpg.de + diff --git a/RdAzT4oBgHgl3EQfI_vE/content/tmp_files/load_file.txt b/RdAzT4oBgHgl3EQfI_vE/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2f49e3ff14d6bb4fd197e0578c635d1620cc6d7f --- /dev/null +++ b/RdAzT4oBgHgl3EQfI_vE/content/tmp_files/load_file.txt @@ -0,0 +1,1545 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf,len=1544 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='01073v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='AP] 3 Jan 2023 A FREE DISCONTINUITY APPROACH TO OPTIMAL PROFILES IN STOKES FLOWS DORIN BUCUR, ANTONIN CHAMBOLLE, ALESSANDRO GIACOMINI, AND MICKA¨EL NAHON Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' In this paper we study obstacles immerged in a Stokes flow with Navier boundary condi- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' We prove the existence and regularity of an obstacle with minimal drag, among all shapes of prescribed volume and controlled surface area, taking into account that these shapes may naturally develop geometric features of codimension 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' The existence is carried out in the framework of free discontinuity problems and leads to a relaxed solution in the space of special functions of bounded deformation (SBD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' In dimension 2, we prove that the solution is classical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Introduction 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Notations and Preliminaries 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Basic notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Functions of bounded variation and sets of finite perimeter 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Functions of bounded deformation 6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Obstacles in Stokes fluids and drag minimization 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' The flow around the obstacle 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' The drag force 8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' The optimization problem 9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' A relaxed formulation of the shape optimization problem and statements of the main results 10 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Some technical results in SBD 12 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' An immersion result 12 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Closure of the non-penetration constraint 13 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' A lower semicontinuity result for surface energies in SBD 15 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Existence of minimizers: proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='8 21 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Regularity of two-dimensional minimizers: proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='10 23 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' The smoothing lemma 24 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Regularity for quasi minimizers of the Griffith energy 30 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='10 36 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Some remarks on a “strong” formulation of the problem 37 References 39 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Introduction Consider an obstacle E ⊂ Rd (d = 2, 3 in real applications) contained in a (finite) channel Ω in which a fluid with viscosity coefficient µ > 0 is flowing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Assume that the flow is stationary and incompressible, and that the associated velocity field u is equal to a constant vector V∞ on the walls of the channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' The obstacle E experiences a force, whose component in direction of V∞ will be denoted by Drag(E) 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' 49Q10, 76D07, 76D55, 35R35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Free discontinuity problems, Stokes flow, Navier boundary conditions, drag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' 1 2 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' BUCUR, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' CHAMBOLLE, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' GIACOMINI, AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' NAHON and is usually called the drag force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' If we further assume that the velocity of the fluid satisfies the Stokes equation in Ω \\ E and obeys to Navier boundary conditions on ∂E, the expression of the drag force turns out to be given (up to a multiplicative constant) by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='1) Drag(E) = 2µ � Ω\\E |e(u)|2 dx + β � ∂E |u|2 dHd−1, where e(u) := 1 2(Du+(Du)∗) denotes the symmetrized gradient of u and β > 0 is the friction coefficient (we refer to Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='2 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' We are interested in minimizing the drag force among all obstacles E with a prescribed volume and controlled surface area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Precisely we look for the existence of such an optimal obstacle and for its qualitative properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' The existence question is not very relevant as soon as one imposes strong geometric constraints on the admissible obstacles (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' convexity, uniform cone conditions, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=') since this may hide some specific features which would naturally occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Indeed, letting the geometry of the obstacle to be completely free, some qualitative behavior (blocked by rigid geometric constraints) can be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' This is the case of our problem, where the optimal obstacle (that we prove to exist without imposing any geometric or topological constraint) may be composed, roughly speaking as a union of a body with the prescribed volume and pieces of surfaces of dimension d−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Those surfaces do not have volume, but count for the total surface area Hd−1(∂E) and of course have a strong influence on the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Penalizing the surface area and the volume, the model problem we are interested in can be written as min E � Drag(E) + cHd−1(∂E) + f(|E|) � , where c > 0 and f : (0, |Ω|) → R ∪ {+∞} is a lower semicontinuous function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Roughly speaking, the terms involving perimeter and volume can be thought as a price to pay in order to build the obstacle E, and we can give the two relevant choices of function f: f(m) = +∞1{m̸=m0} for some m0 ∈ (0, |Ω|), or f(m) = −λm for some λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Many similar optimisation problems have been considered under the “no-slip” boundary condition, meaning flows for which u = 0 at ∂E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Under volume constraint and an a priori symmetry hypothesis around an axis parallel to the flow, the minimal drag question has been studied in [33] on smooth surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' In [28], still under symmetry hypotheses, it was conjectured that the optimal profile in three dimensions is a prolate spheroid with sharp ends of angle of 120 degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' In the same symmetry context, let us also mention the slender body approximation of [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' We also refer the reader to the paper by ˘Sver´ak [32] who, in two dimensions, proves the existence of an optimal obstacle under topological hypotheses, namely that the obstacle has at most a given number of connected components (in particular this number can be equal to 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' The proof is genuinely two dimensional and can not be extended to higher dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' The Navier boundary condition gives many new challenges, namely the possible apparition of lower dimensional structures in the obstacle that minimize the drag, something which was absent under the no-slip condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' The Navier boundary condition may be seen as a partial adherence to the boundary of the obstacle, and it may be asymptotically obtained as a limit of flows with perfect slip on an obstacle with rough boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' More precisely, a periodic microstructure with the right scaling on the boundary is modelled at the limit by a Navier boundary condition, as was proved in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' In dimension higher than two it is also necessary to take into account more complex geometries for the microstructure, which at the limit produce an anisotropic factor that favors certain directions of the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Moreover, infinitesimal boundary perturbations can dramatically modify the solution of the Stokes equation with Navier boundary conditions, while in presence of no-slip boundary conditions the solution remains A FREE DISCONTINUITY APPROACH TO OPTIMAL PROFILES IN STOKES FLOWS 3 stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' We refer the reader to [8] for an analysis of those phenomena and for a discussion on the pertinence of the Navier boundary conditions in physical models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' For a fixed obstacle E, the minimization of the drag with respect to the friction parameter β of the Navier conditions (meaning, from a physical point of view, with respect to the microstructure on the boundary) has been studied in [5], for both Stokes and Navier-Stokes flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' While for Stokes flows the drag is increasing with the friction parameter, an important observation which occurs for the Navier-Stokes equation is that the monotonicity of the drag with respect to the parameter β does not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' This is a reason for which the results we give in this paper for the Stokes flows are not expected to hold, as such, for the Navier-Stokes equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Since the stationary velocity field associated to a Lipschitz obstacle E turns out to be characterized variationally as the minimizer of the right hand side of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='1) in the class of admissible velocities Vreg E,V∞(Ω) = � u ∈ H1(Ω \\ E) : divu = 0, u|∂E · νE = 0, u|∂Ω = V∞ � (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='4) in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='1 for more details), we can conveniently rephrase the minimization problem by letting also the velocity fields intervene explicitely in the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='2) min E,u∈Vreg E,V∞(Ω) � 2µ � Ω\\E |e(u)|2 dx + β � ∂E |u|2 dHd−1 + cHd−1(∂E) + f(|E|) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' The first main goal of the paper is to find suitable relaxations of problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='2) for which we can prove the existence of minimizers without any a priori constraint on the regularity or the topology of the sets E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' In order to avoid unnatural geometric restrictions on the obstacle E, it is natural in view of the third term appearing in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='2) to let it vary within the class of sets of finite perimeter (see Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='2), and replace the topological boundary with reduced one ∂∗E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' In order to describe properly obstacles with very narrow spikes which in the limit degenerate to (d−1)-surfaces and that cannot be taken into account through the reduced boundary, it is convenient to consider admissible velocity fields which can be discontinuous outside E (see Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Since the symmetrized gradient e(u) is involved explicitly in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='2), a natural family for the admissible velocities is given by the space of functions of bounded deformation SBD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' The natural relaxation of the energy takes the form J (E, u) :=2µ � Ω\\E |e(u)|2 dx + β � ∂∗E |u+|2 dHd−1 + β � Ju\\∂∗E [|u+|2 + |u−|2] dHd−1 + cHd−1(∂∗E) + 2cHd−1(Ju \\ ∂∗E) + f(|E|), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='3) where u is set equal to zero a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' in E, while Ju denotes the discontinuity set of u and u± are the traces of u on ∂∗E and Ju (the trace u− vanishes on ∂∗E by the choice of orientation, while u+ is on the outward side).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Within this framework the global obstacle is given by E∪Ju, so that it contains also lower dimensional parts, namely Ju \\∂∗E: roughly speaking, for the optimal velocity these discontinuous regions generate (d − 1)-surfaces which correspond to volumeless, lower dimensional subsets of the optimal obstacle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Admissible velocities must be tangent to the obstacles, meaning that not only u is tangent to ∂∗E, but also the two traces u± are orthogonal to the normal νu along the jump set Ju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' The contribution of the Navier surface term takes naturally into account the contribution from both sides given by u±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Concerning the perimeter term, we count twice the lower dimensional parts because we see the relaxed obstacle as a limit of regular obstacles, such that points of Ju \\ ∂∗E correspond to thin parts of the regular obstacle that collapse on a lower-dimensional structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' We could also see the perimeter term 4 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' BUCUR, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' CHAMBOLLE, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' GIACOMINI, AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' NAHON as a price to pay in order to construct the obstacle and just keep Hd−1(∂∗E ∪ Ju) instead, and the main results of the paper would not be affected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' The relaxed optimization problem can be seen as a minimization problem on the pairs (E, u) which has the features of classical geometrical problems for E coupled with a free discontinuity problem for u, with a surface term depending on the traces which are subject to suitable tangency constraints and boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' The first main results of the paper (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='8) concerns the existence of minimizers for the relaxed functional J in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='3) among the class of admissible configurations (see Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='1 for the precise definition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' The main difficulties we have to face in order to prove that the problem is well posed are the following: (a) the closure of the non-penetration constraint for the velocity on ∂∗E ∪ Ju under the natural weak convergence of the problem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' (b) the lower semicontinuity of energies of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='4) � Ju [|u+|2 + |u−|2] dHd−1 associated to the Navier conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Point (a) is a consequence of a lower semicontinuity result for the energy � Ju � |u+ · νu| + |u− · νu| � dHd−1 which is proved in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='2, by resorting to recent lower semicontinuity results for functionals on SBD by Friedrich, Perugini and Solombrino [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' The energy of point (b) naturally appears in a scalar setting when dealing with shape optimization problems involving Robin boundary conditions (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' [7, 11, 10, 12]), and it is easily seen to enjoy lower semicontinuity properties by working with sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' The lower semicontinuity result in the vectorial SBD setting is given by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='4 and cannot rely on an easy argument by sections, which instead would yield the lower semicontinuity of an energy of the form � Ju � |u+ · ξ|2 + |u− · ξ|2� |ξ · νu| dHd−1 with ξ ∈ Rd with |ξ| = 1: the optimization in ξ in order to recover (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='4) does not seem feasible in dimension d ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' We thus follow a different strategy based on a blow up argument in which we reconstruct the vector quantities u± by controlling them along a sufficiently high number of directions (see Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='3 for details): in this way we can deal with more general energy densities of the form φ(u+) + φ(u−), where φ is a lower semicontinuous function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' The second main result of the paper (see Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='10) concerns the regularity of the relaxed minimizers of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Provided that the volume penalization function f is Lipschitz and that we are in two dimensions, we prove that for a minimizer (E, u) of J , the optimal obstacle E ∪ Ju is a closed set, while the optimal velocity u is a smooth Sobolev function outside the obstacle, recovering somehow the classical setting of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' More precisely we show that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='5) H1(Ω ∩ ∂∗E ∪ Ju \\ (∂∗E ∪ Ju)) = 0, so that the optimal obstacle can be described as the closed set obtained by the complement of the connected components of Ω \\ ∂∗E ∪ Ju on which u does not vanish identically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' The technical ideas to prove (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='5) stem from the pioneering result of De Giorgi, Carriero and Leaci on the Mumford-Shah problem [22], where the key of the proof is a decay estimate obtained by a A FREE DISCONTINUITY APPROACH TO OPTIMAL PROFILES IN STOKES FLOWS 5 contradiction/compactness argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' For vectorial problems, a similar strategy, but definitely more involved, was used for the Griffith fracture problem in [18] (for the two-dimensional case) and in [15] (for higher dimension).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' In the fracture problem, the key compactness result relies on the possibility to approximate a field u ∈ SBD([−1, 1]d) with a small jump set by a Sobolev function which is locally controlled in H1 (via the classical Korn inequality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' In our case, we follow a similar approximation procedure, but we have to handle two additional constraints: incompressibility and non-penetration at the jumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' From a technical point of view, this is problematic since the bound in [18] in not strong enough to stay in divergence-free vector fields and the method in [15] creates new jumps on which the non-penetration constraint is not a priori verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' However, when restricted to two dimensions, the method of [15] leads to a stronger result, so that both constraints can be handled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' In Section 2 we recall fix the notation and recall some basic facts concerning sets of finite perimeter, functions of bounded deformation and Hausdorff convergence of compact sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Section 3 is devoted to the precise exposition of the drag optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' In Section 4 we detail the relaxation of the problem in the family of obstacle of finite perimeter and with velocities of bounded deformation, and formulate the main results of the paper concerning the existence of minimizers (in any dimension) and their regularity in dimension two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' The proof of the existence of minimizers is given in Section 6, and it is based on some technical results for SBD functions collected in Section 5, while the regularity result is proved in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Notations and Preliminaries 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Basic notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' If E ⊆ Rd, we will denote with |E| its d-dimensional Lebesgue measure, and by Hd−1(E) its (d−1)-dimensional Hausdorff measure: we refer to [23, Chapter 2] for a precise definition, recalling that for sufficiently regular sets Hd−1 coincides with the usual area measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Moreover, we denote by Ec the complementary set of E, and by 1E its characteristic function, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=', 1E(x) = 1 if x ∈ E, 1E(x) = 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' In addition we will say that E1 ⋐ E2 if E1 ⊂ E2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Finally we will denote with Qx,r ⊆ Rd the cube of center x and side r: when x = 0, we will simply write Qr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' If A ⊆ Rd is open and 1 ≤ p ≤ +∞, we denote by Lp(A) the usual space of p-summable functions on A with norm indicated by ∥ · ∥p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' W 1,p(A) will stand for the Sobolev space of functions in Lp(A) whose gradient in the sense of distributions belongs to Lp(A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Finally, given a finite dimensional unitary space Y , we will denote by Mb(A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Y ) will denote the space of Y -valued Radon measures on A, which can be identified with the dual of Y -valued continuous functions on A vanishing at the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' We will denote by Md×m the set of d × m matrices with values in R: when d = m we will denote by Md×d sym the subspace of d × d symmetric matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' For a ∈ Rd and b ∈ Rm we will denote with a ⊗ b the element of Md×m such that (a ⊗ b)ij = aibj, while if a, b ∈ Rd we denote with a ⊙ b the matrix in Md×d sym such that (a ⊙ b)ij = aibj + ajbi 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Given ξ ∈ Rd with |ξ| = 1, we denote with ξ⊥ the hyperplane through the origin orthogonal to ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' If E ⊆ Rd, we set (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='1) Eξ := πξ⊥(E), where π denotes the orthogonal projection, and for y ∈ ξ⊥ we set (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='2) Eξ y := {t ∈ R : y + tξ ∈ E}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' 6 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' BUCUR, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' CHAMBOLLE, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' GIACOMINI, AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' NAHON 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Functions of bounded variation and sets of finite perimeter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' If Ω ⊆ Rd is open, we say that u ∈ BV (Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Rm) if u ∈ L1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Rm) and its derivative in the sense of distributions is a finite Radon measure on Ω, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=', Du ∈ Mb(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Md×m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' BV (Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Rm) is called the space of functions of bounded variation on Ω with values in Rm and it is a Banach space under the norm ∥u∥BV (Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='Rm) := ∥u∥L1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='Rm) + ∥Du∥Mb(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='Md×m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' We call |Du|(Ω) := ∥Du∥Mb(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='Md×m) the total variation of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' We refer the reader to [1] for an exhaustive treatment of the space BV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' We say that u ∈ SBV (Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Rm) if u ∈ BV (Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Rm) and its distributional derivative can be written in the form Du = ∇u dx + (u+ − u−) ⊗ νuHd−1⌊Ju, where ∇u ∈ L1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Md×m) denotes the approximate gradient of u, Ju denotes the set of approximate jumps of u, u+ and u− are the traces of u on Ju, and νu(x) is the normal to Ju at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Note that if u ∈ SBV (Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Rm), then the singular part of Du is concentrated on Ju which is a countably Hd−1-rectifiable set: there exists a set E with Hd−1(E) = 0 and a sequence (Mi)i∈N of C1-submanifolds of Rd such that Ju ⊆ E ∪ � i∈N Mi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' We will say that E ⊆ Rd with |E| < +∞ has finite perimeter if 1E ∈ BV (Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' The perimeter of E is defined as Per(E) = |D1E|(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' It turns out that D1E = νEHd−1⌊∂∗E, Per(E) = Hd−1(∂∗E), where ∂∗E is called the reduced boundary of E, and νE is the associated inner approximate normal (see [1, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' We have that ∂∗E ⊆ ∂E, but the topological boundary can in in general be much larger than the reduced one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' If A ⊆ Rd is open and bounded with Hd−1(A) < +∞, then A has finite perimeter with Per(A) ≤ Hd−1(∂A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Functions of bounded deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' If Ω ⊆ Rd is open, we say that u ∈ BD(Ω) if u ∈ L1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Rd) and its symmetric gradient Eu := Du+(Du)∗ 2 in the sense of distributions is a finite Radon measure on Ω, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=', Eu ∈ Mb(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Md×d sym).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' BD(Ω) is called the space of functions of bounded deformation on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' We refer the reader to [30, 29] for the main properties of the space BD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' We will make use of a subspace of BD(Ω) called the space of special functions of bounded deformation introduced in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' We say that u ∈ SBD(Ω) if u ∈ BD(Ω) and its symmetrized distributional derivative can be written in the form Eu = e(u) dx + (u+ − u−) ⊙ νuHd−1⌊Ju, where e(u) ∈ L1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Md×d sym) denotes the approximate symmetrized gradient of u, Ju denotes the set of approximate jumps of u, u+ and u− are the traces of u on Ju, and νu(x) is the normal to Ju at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' As in the case of functions of bounded variation, Ju is a Hd−1-countably rectifiable set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' We will use the following compactness and lower semicontinuity result proved in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Let Ω ⊆ Rd be open, bounded and with a Lipschitz boundary, and let (un)n∈N be a sequence in SBD(Ω) such that sup n � |Eun|(Ω) + ∥un∥L1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='Rd) + ∥e(un)∥Lp(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='Md×d sym) + Hd−1(Jun) � < +∞ for some p > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Then there exists u ∈ SBD(Ω) and a subsequence (unk)k∈N such that unk → u strongly in L1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Rd), e(unk) ⇀ e(u) weakly in Lp(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Md×d sym), and Hd−1(Ju) ≤ lim inf k→+∞ Hd−1(Junk ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' A FREE DISCONTINUITY APPROACH TO OPTIMAL PROFILES IN STOKES FLOWS 7 We will need also some properties of the sections of SBD-functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' If Ω ⊆ Rd is open and u ∈ SBD(Ω), let us consider the scalar function on Ωξ y given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='3) ˆuξ y(t) := u(y + tξ) · ξ and the set (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='4) Jξ u := {x ∈ Ju : (u+(x) − u−(x)) · ξ ̸= 0} The following result holds true (see [2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='2 (One dimensional sections of SBD-functions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Let Ω ⊆ Rd be open, ξ ∈ Rd with |ξ| = 1 and let u ∈ SBD(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Then for Hd−1-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' y ∈ Ωξ we have ˆuξ y ∈ SBV (Ωξ y) with (ˆuξ y)′(t) = (e(u)ξ · ξ)(y + tξ) for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' t ∈ Ωξ y and Jˆuξ y = (Jξ u)ξ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Obstacles in Stokes fluids and drag minimization In this section we explain the drag problem for an obstacle immersed in a stationary flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' The flow around the obstacle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Let Ω ⊂ Rd be an open bounded set with Lipschitz boundary, and let V ∈ C1(Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Rd) be a divergence free vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Given E ⋐ Ω open and with a Lipschitz boundary, let us consider the stationary flow for a viscous incompressible fluid around E with boundary conditions on ∂Ω given by V , and with Navier boundary conditions on ∂E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' More precisely, if u : Ω\\E → Rd is the velocity field, we require that the following items hold true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' (a) Incompressibility: div u = 0 in Ω \\ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' (b) Boundary conditions: we have u = V on ∂Ω and the non-penetration condition u · ν = 0 on ∂E, where ν denotes the exterior normal to E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' (c) Equilibrium: considering the stress (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='1) σ := −pId + 2µe(u), where µ > 0 is a viscosity parameter, e(u) the symmetrized gradient of u (also denoted by D(u)) and p is the pressure, we require (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='2) div σ = 0 in Ω \\ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' (d) Navier conditions on the obstacle: we have (σν)τ = βu on ∂E, where β > 0 is a friction parameter, and (σν)τ denotes the tangential component of force σν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' The stationary flow has the following variational characterization: u is the minimizer of the energy (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='3) E(u) := 2µ � Ω\\E |e(u)|2 dx + β � ∂E |u|2 dHd−1 among the class of (sufficiently regular) admissible fields (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='4) Vreg E,V (Ω) := {v ∈ H1(Ω \\ E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Rd) : v satisfies points (a) and (b)}, 8 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' BUCUR, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' CHAMBOLLE, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' GIACOMINI, AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' NAHON where Hd−1 stands for the (d − 1)-dimensional Hausdorff measures, which reduces to the area measure on sufficiently regular sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Indeed if u is a minimizer, and ϕ is an admissible variation, so that ϕ = 0 on ∂Ω, we get 0 = 2µ � Ω\\E e(u) : e(ϕ) dx + β � ∂E u · ϕ dHd−1 = 2µ � Ω\\E e(u) : ∇ϕ dx + β � ∂E u · ϕ dHd−1 = −2µ � Ω\\E div e(u) · ϕ dx + � ∂E [−2µe(u)ν + βu] · ϕ dHd−1 In particular, choosing ϕ with compact support in Ω \\ E we have 2µdiv e(u) = ∇p for some pressure field p: as a consequence σ := −pId + 2µe(u) satisfies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='2) of condition (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Since the admissible functions ϕ are tangent to ∂E, the optimality condition reduces to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='5) 0 = � ∂E [−2µe(u)ν + βu] · ϕ dHd−1 = � ∂E [−σν + βu] · ϕ dHd−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Notice that every tangential vector field η on ∂E can be extended to a divergence free vector field on Ω\\E which vanishes on ∂Ω, hence it is the trace of an admissible variation ϕ: indeed any extension W which vanishes on ∂Ω has a divergence with zero mean, so that considering W1 with div W1 = div W with W1 = 0 on ∂Ω and on ∂E (whose existence is guaranteed, for example by [6, Theorem IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='1])), the required extension is given by W − W1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' We conclude that the optimality condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='5) yields the Navier condition of point (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' The drag force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Assume now that the external vector field V is equal to a constant V∞ ∈ Rd\\{0}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' the obstacle E is immersed in a uniform flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' The flow is perturbed near E, assuming the value u, and the obstacle experiences a force which has a component in the direction V∞ which is given by Drag(E) := � ∂E σν · V∞ |V∞| dHd−1, which is called the drag force on E in the direction of the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' We claim that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='6) Drag(E) = 1 |V∞|E(u), where E(u) is the energy defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Using the facts that σ is symmetric and with zero divergence (so that also the vector field σV∞ is divergence free), and that u = V∞ on ∂Ω, we may write � ∂E σν · V∞ dHd−1 = � ∂E σV∞ · ν dHd−1 = � ∂Ω σV∞ · ν dHd−1 = � ∂Ω σu · ν dHd−1 = � Ω\\E div (σu) dx + � ∂E σu · ν dHd−1 = � Ω\\E σ : ∇u dx + � ∂E σν · u dHd−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='7) Using again that σ is symmetric and that u is divergence free, together with the constitutive equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='1), we have � Ω\\E σ : ∇u dx = � Ω\\E σ : e(u) dx = � Ω\\E (−p Id + 2µe(u)) : e(u) dx = � Ω\\E (−p div u + 2µ|e(u|2) dx = 2µ � Ω\\E |e(u)|2 dx, A FREE DISCONTINUITY APPROACH TO OPTIMAL PROFILES IN STOKES FLOWS 9 while in view of the Navier conditions on ∂E and the fact that u is tangent to the obstacle � ∂E σν · u dHd−1 = � ∂E (σν)τ · u dHd−1 = β � ∂E |u|2 dHd−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Inserting into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='7), we get that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='6) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' The optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Let c > 0 and let f : (0, |Ω|) → R ∪ {+∞} be a lower semicontin- uous functions that is not identically equal to +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' We are concerned with the following optimization problem: min E � Drag(E) + cHd−1(∂E) + f(|E|) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' We are thus interested in finding the optimal shape of an obstacle which minimizes the drag force, under a penalization involving its perimeter and its volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' In view of the energetic characterization of the drag force established in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='2, we can formulate the problem as a minimization problem among the pairs (E, u), where u is a velocity field belonging to the family Vreg E,V∞(Ω) defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='4): min E,u∈Vreg E,V∞(Ω) � 2µ |V∞| � Ω\\E |e(u)|2 dx + β |V∞| � ∂E |u|2 dHd−1 + cHd−1(∂E) + f(|E|) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Setting all the constants equal to 1, and replacing V∞ by a given divergence free velocity vector field V as in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='1, the drag minimization problem above is a particular case of the following shape optimization problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='8) min E,u∈Vreg E,V (Ω) �� Ω\\E |e(u)|2 dx + � ∂E |u|2 dHd−1 + Hd−1(∂E) + f(|E|) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' If we want to apply the direct method of the calculus of variations to the problem, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=', if we want to recover a minimizer by looking at minimizing sequences (En, un)n∈N, the following considerations are quite natural.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' (a) Since the problem involves the perimeter of E, the sequence (En)n∈N is relatively compact in the family of sets of finite perimeter (see Section 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' (b) Concerning the velocities, it turns out naturally that it is convenient to consider also dis- continuous vector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Indeed if un → u in some sense, and ∂En collapses in some parts generating a surface Γ outside the limit set E, the limit velocity field u can present, in general, discontinuities across Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' En E Γ We thus expect an extra term in the surface integral related to the Navier conditions, which amounts at least to � Γ\\∂E [|u+|2 + |u−|2] dHd−1, where u± are the two traces from both sides of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' 10 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' BUCUR, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' CHAMBOLLE, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' GIACOMINI, AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' NAHON The previous considerations yield to formulate a relaxed version of problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='8) in which E varies among the family of sets of finite perimeter contained in Ω, while the family of associated admissible velocity fields u is naturally contained in the space of special functions of bounded deformation SBD(Ω) (see Section 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' In Section 4, we will give a precise formulation of problem in this weak setting, which guarantees existence of optimal solutions, describing in particular how the boundary conditions on ∂Ω and on the obstacle have to be rephrased in this context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' A relaxed formulation of the shape optimization problem and statements of the main results Let Ω ⊆ Rd be open, bounded and with a Lipschitz boundary, and let V ∈ C1(Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Rd) be a divergence free vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' In order to deal conveniently with the boundary conditions, let us consider Ω′ ⊆ Rd open and bounded such that Ω ⋐ Ω′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' The following definition deals with the family of admissible configurations in the relaxed setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='1 (The class A(V ) of admissible obstacle-velocity configurations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' We say that (E, u) is an admissible configuration for the external velocity V , and we will write (E, u) ∈ A(V ), if E ⊆ Ω is a set of finite perimeter, while u ∈ SBD(Ω′) ∩ L2(Ω′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Rd) is such that u = 0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' on E and the following conditions are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' (a) The flow is divergence free: div u = 0 in the sense of distributions in Ω′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' (b) External boundary conditions: u = V a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' on Ω′ \\ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' (c) Non-penetration condition on the obstacle: u± · ν = 0 on ∂∗E ∪ Ju, where ν denotes the normal to the rectifiable set ∂∗E ∪ Ju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' The crucial difference between admissible velocities in the present framework and those of the family Vreg E,V (Ω) introduced before (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='4)) is that they may have discontinuities outside of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Within the new setting, the global obstacle is given by E ∪ Ju i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' it may contain (d − 1) dimensional parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Given (E, u) ∈ A(V ), concerning the traces of u on ∂∗E, we will denote with u+ the trace in the direction of the external normal νE, so that u− = 0 Hd−1-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' on ∂∗E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Concerning the non-penetration constraint, notice that it suffices to require it only on Ju, since it is then automatically verified also on ∂∗E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Indeed for Hd−1-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' x ∈ ∂∗E\\Ju, we have u−(x) = u+(x) = 0 and the constraint is verified, while for Hd−1-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' x ∈ Ju ∩ ∂∗E the two rectifiable sets Ju and ∂∗E share the same normal vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' The space SBD(Ω′) is naturally a subspace of L1(Ω′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Rd): we require for admissibility that u ∈ L2(Ω′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Rd) to ensure that the velocity field has finite kinetic energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' It will turn out that velocities in SBD(Ω′) which are interesting for our problem (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=', with finite energy) are automatically elements of L2(Ω′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Rd) (see Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='4 (On the boundary condition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' If (E, u) ∈ A(V ), then u ∈ SBD(Ω′) with u = V a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' on Ω′ \\ Ω, so that Ju ∩ ∂Ω = {x ∈ ∂Ω : γ(u)(x) ̸= V (x)}, where γ(u) is the trace of u on ∂Ω coming from Ω (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=', the usual trace of u seen as an element of SBD(Ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' We conclude that within the present framework, the boundary condition is somehow A FREE DISCONTINUITY APPROACH TO OPTIMAL PROFILES IN STOKES FLOWS 11 relaxed: a possible mismatch between u and V on ∂Ω is admitted, but then the zone is counted as a jump part of the velocity field, and consequently as a part of the obstacle ∂∗E ∪ Ju, and will carry a contribution for the energy (see (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='2) below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Such a relaxation of the boundary condition is a feature which is common to several applications of functions of bounded variation to problems in continuum mechanics (see for example [25, 21] in connection to fracture mechanics or [20] for problems in plasticity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Given (E, u) ∈ A(V ), the obstacle E ∪ Ju may touch ∂Ω only on those part where V is tangent to Ω: this is due to the fact that on (∂∗E ∪ Ju) ∩ ∂Ω, the two sets share Hd−1-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' the same normal, and u+ = V (if the orientation is suitably chosen).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Let E ⋐ Ω be open and with a Lipschitz boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Then we can find W ∈ H1(Ω\\E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Rd) such that W = V on ∂Ω, W = 0 on ∂E and div W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Indeed if ϕ ∈ C∞(Rd) is such that ϕ = 1 on a neighborhood of Rd \\ Ω and ϕ = 0 on a neighborhood of E, we can consider the vector field V1 := ϕV , whose divergence has zero mean on Ω \\ E (by Gauss theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Then we can find V2 ∈ H1 0(Ω \\ E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Rd) such that div V = div V1 (see [6, Theorem IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='1]), so that the field W := V1 − V2 is an admissible choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' In particular we get that (E, W) ∈ A(V ), so that the class of admissible configurations is not empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Let (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='1) f : [0, |Ω|] → [0, +∞] be lower semicontinuous, not identically equal to +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' For every (E, u) ∈ A(V ), let us set (normalizing to 1 the constants involved in the drag force problem) J (E, u) := � Ω′ |e(u)|2 dx + � ∂∗E |u+|2 dHd−1 + � Ju\\∂∗E [|u+|2 + |u−|2] dHd−1 + Hd−1(∂∗E) + 2Hd−1(Ju \\ ∂∗E) + f(|E|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='2) Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Concerning the volume integral in J (E, u), the density e(u) is equal to e(V ) a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' on Ω′ \\Ω and equal to 0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' on E: as a consequence we could replace it with an integral on Ω\\E without affecting the minimization of J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Concerning the Navier energy and the surface penalization for ∂∗E∪Ju, notice that it counts also for the possible mismatch at the boundary between u and V as pointed out in Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='4: the mismatch is thus “penalized” by the energy of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' The previous observations show that the larger domain Ω′ plays only an instrumental role for the problem, as it can be replaced by any open domain strictly containing Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' The first main result of the paper is the following Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='8 (Existence of optimal obstacles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Let Ω ⊆ Rd be a bounded open set with Lipschitz boundary, V ∈ C1(Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Rd) a divergence-free vector field, and f a function satisfying (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Let the family of admissible configurations A(V ) be given by Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='1 and let J be the functional defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Then the problem (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='3) min (E,u)∈A(V ) J (E, u) admits a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' We recover the original drag minimization problem when V is a constant nonzero vector V∞, and we restore properly in the functional the physical constants µ and β, together with the perimeter penalization constant c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' The second main result of the paper concerns the regularity of minimizers in the two dimensional setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' 12 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' BUCUR, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' CHAMBOLLE, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' GIACOMINI, AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' NAHON Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='10 (Regularity in dimension two).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Let Ω ⊆ R2 be a bounded open set with Lipschitz boundary, V ∈ C1(R2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' R2) a divergence-free vector field, and f : [0, |Ω|] → [0, +∞[ a Lipschitz function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Let (E, u) ∈ A(V ) be a solution to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='3) according to Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Then H1 � Ω ∩ (Ju ∪ ∂∗E \\ (Ju ∪ ∂∗E)) � = 0, and u ∈ C∞(Ω \\ Ju ∪ ∂∗E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='8 will be proved in Section 6, on the basis of some technical results established in 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' The proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='10 will be addressed in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Some technical results in SBD In this section we collect some technical properties concerning the space SBD that will be funda- mental in the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' In particular in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='1 we will prove that admissible velocity vector fields enjoy higher summability properties (indeed they belong to L 2d d−1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' In Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='3 we will prove that velocity fields u with u± tangent to the discontinuity set Ju form a closed set under the natural convergence of minimizing sequences for the main optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Finally in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='4 we will prove a lower semicontinuity result for surface energies depending on the traces, which entails in particular the lower semicontinuity of the term associated to the Navier conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' An immersion result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' The following embedding result holds true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Let Ω ⊆ Rd be a bounded open set, and let u ∈ SBD(Rd) be supported in Ω such that E(u) := � Ω |e(u)|2 dx + � Ju � |u+|2 + |u−|2� dHd−1 < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Then u ∈ L 2d d−1 (Ω) with ∥u∥ 2d d−1 ≤ C � E(u), where C depends on d and diam(Ω) only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' It suffices to follow the strategy of the proof of the classical embedding of BD into Ld/d−1 explained in [29], but concentrating on the square of the components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Let us consider the unit vector ξ := 1 √ d (1, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' , 1) ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Employing the characterization by sections recalled in Section 2, for Hd−1-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' y ∈ ξ⊥ we have ˆuξ y ∈ SBV (Ωξ y) with � Ωξ y |(ˆuξ y)′|2 dt + � t∈Jˆuξ y � |(ˆuξ y)+(t)|2 + |(ˆuξ y)−(t)|2� < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Then we can write for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' t ∈ R ∥ˆuξ y∥2 L∞(Ωξ y) ≤ ���D(ˆuξ y)2��� (Ωξ y) = � Ωξ y 2|ˆuξ y(ˆuξ y)′|dt + � t∈Jˆuξ y ���|(ˆuξ y)+(t)|2 − |(ˆuξ y)−(t)|2��� ≤ 1 2∥ˆuξ y∥2 L∞(Ωξ y) + 2|Ωξ y| � Ωξ y ���(ˆuξ y)′��� 2 dt + � t∈Jˆuξ y ����(ˆuξ y)+(t) ��� 2 + ���(ˆuξ y)−(t) ��� 2� , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='1) A FREE DISCONTINUITY APPROACH TO OPTIMAL PROFILES IN STOKES FLOWS 13 Let us set gξ(x) := � Ωξ y |(ˆuξ y)′|2 dt + � t∈Jˆuξ y � |(ˆuξ y)+(t)|2 + |(ˆuξ y)−(t)|2� , where y := πξ⊥(x), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=', the projection of x on the hyperplane ξ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' gξ(x) only depends on the projection of x on ξ⊥ and � ξ⊥ gξdHd−1 = � Ω |e(u)ξ · ξ|2 dx + � Ju � |u+|2 + |u−|2� |ξ · ν| dHd−1 ≤ C �� Ω |e(u)|2 dx + � Ju � |u+|2 + |u−|2� dHd−1 � where C depends only on d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Thanks to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='1) we have (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='2) |ξ · u|2 ≤ Cgξ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' on Ω, where C depends on the diameter of Ω, and from now on all the constants C that appear depend on n, diam(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' For every k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' , d − 1, we can write ξ = 1 √ d ek + � d − 1 d hk, where ek is the k-th vector of the canonical base, and hk is the unit vector in the direction √ dξ − ek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Reasoning as above on the decomposition ξ · u = � d − 1 d hk · u + 1 √ d ek · u we obtain a similar estimate (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='3) |ξ · u|2 ≤ C (ghk + gek) , Multiplying inequality (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='2) with inequalities (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='3) for k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' , d − 1, we obtain reasoning as in [29, Chapter II, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='2] ∥(ξ · u)2∥ d d−1 ≤ C �� Ω |e(u)|2 dx + � Ju � |u+|2 + |u−|2� dHd−1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Since this estimate does not depend on the particular choice of the basis and hence holds for any ξ with norm one, the theorem is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Closure of the non-penetration constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' In the context of equi-Lipschitz boundaries, the preservation of the non-penetration property for a sequence of Sobolev functions converging weakly, comes rather directly via the divergence theorem (we refer the reader, for instance, to [8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' However, in the case of collapsing boundaries, so that the limit function lives on both sides of a surface and in absence of any smoothness of the limit set, this technique does not work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' The proof of the non- penetration preservation requires different technical arguments that we handle in the SBD context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Let us start with the following lower semicontinuity result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Let Ω ⊆ Rd be a bounded open set, and let (un)n∈N be a sequence in SBD(Ω) such that sup n �� Ω |e(un)|2 dx + Hd−1(Jun) � < +∞ with un → u in measure 14 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' BUCUR, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' CHAMBOLLE, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' GIACOMINI, AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' NAHON for some u ∈ SBD(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Then � Ju � |u+ · νu| + |u− · νu| � dHd−1 ≤ lim inf n→+∞ � Jun � |u+ n · νun| + |u− · νun| � dHd−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Let us consider a countable set of functions {ϕh : h ∈ N} which is dense with respect to ∥ · ∥∞ inside the set � f ∈ C0 c (]0, +∞[) : � +∞ 0 f dt = 0 and ∥f∥∞ ≤ 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Given ε > 0, let us consider gh,k(x) := � 1 2|x−xk|2 0 ϕh(t) dt, where {xk : k ∈ N} is a countable and dense set in Bε(0) ⊂ Rd with x0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Clearly gh,k ∈ C1 c (Rd) with Gh,k(x) := ∇gh,k(x) = ϕh �1 2|x − xk|2 � (x − xk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' We have that Gh,k is a continuous conservative vector field with compact support on Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Let us set for (i, j) ∈ Rd × Rd and ν ∈ Rd with |ν| = 1 fε(i, j, ν) := sup h,k (Gh,k(i) − Gh,k(j)) · ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' By construction fε is a symmetric jointly convex function according to [26, Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' We claim that for i ̸= j (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='4) |i · ν| + |j · ν| ≤ fε(i, j, ν) ≤ |i · ν| + |j · ν| + 2ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' In view of the lower semicontinuity result [26, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='1] we have lim inf n→+∞ � Jun fε(u+ n , u− n , νun) dHd−1 ≥ � Ju fε(u+, u−, νu) dHd−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' We can thus write lim inf n→+∞ �� Jun � |u+ n · νun| + |u− n · νun| � dHd−1 + 2εHd−1(Jun) � ≥ lim inf n→+∞ � Jun fε(u+ n , u− n , νun) dHd−1 ≥ � Ju fε(u+, u−, νu) dHd−1 ≥ � Ju � |u+ · νu| + |u− · νu| � dHd−1, so that the result follows taking into account the bound on Hd−1(Jun) and letting ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' In order to complete the proof, we need to show claim (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' The estimate from above follows from [Gh,k(i) − Gh,k(j)] · ν ≤ |(i − xk) · ν| + |(j − xk) · ν| ≤ |i · ν| + |j · ν| + 2ε since ∥ϕh∥∞ ≤ 1 and |xk| < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Let us prove the estimate from below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' We select xkn → 0 such that |i − xkn| ̸= |j − xkn| (which is always possibile in view of the density of {xk : k ∈ N} inside Bε(0) and since i ̸= j) and then ϕhn such that for n → +∞ ϕhn �1 2|i − xkn|2 � → i · ν |i · ν| + η and ϕhn �1 2|j − xkn|2 � → − j · ν |j · ν| + η, where η > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' By definition of fε we infer that fε(i, j, ν) ≥ |i · ν| + |j · ν| − 2η, A FREE DISCONTINUITY APPROACH TO OPTIMAL PROFILES IN STOKES FLOWS 15 so that the estimate from below follows by sending η → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' □ We are now in a position to prove the main result of the section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='3 (Closure of the non-penetration constraint on the jump set).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Let Ω ⊆ Rd be a bounded open set, and let (un)n∈N be a sequence in SBD(Ω) such that sup n �� Ω |e(un)|2 dx + Hd−1(Jun) � < +∞ and un → u in measure for some u ∈ SBD(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' If u± n · νun = 0 Hd−1-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' on Jun, then u± · νu = 0 Hd−1-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' on Ju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' By Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='2 we may write � Ju � |u+ · νu| + |u− · νu| � dHd−1 ≤ lim inf n→+∞ � Jun � |u+ n · νun| + |u− · νun| � dHd−1 = 0, so that the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' A lower semicontinuity result for surface energies in SBD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' In this section we deal with the lower semicontinuity of the surface term of the functional J in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='2) connected with the Navier conditions on the obstacle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' The following lower semicontinuity result holds true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Let Ω ⊆ Rd be an open set, un, u ∈ SBD(Ω) such that un → u strongly in L1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Rd) and sup n �� Ω |e(un)|2 dx + Hd−1(Jun) � < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Then if φ : Rd → [0, +∞] is a lower semicontinuous function, we have � Ju [φ(u+) + φ(u−)] dHd−1 ≤ lim inf n→+∞ � Jun [φ(u+ n ) + φ(u− n )] dHd−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' This applies in particular to φ(u) = |u|2 and φ(u) = 1{u̸=0}, which will be of interest to us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Notice first that φ may be supposed to be continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Indeed for any lower-semicontinuous nonnegative φ, by considering a sequence of continuous nonnegative functions φk ր φ we get � Ju [φ(u+) + φ(u−)] dHd−1 = lim inf k→∞ � Ju [φk(u+) + φk(u−)] dHd−1 ≤ lim inf k→∞ lim inf n→+∞ � Jun [φk(u+ n ) + φk(u− n )] dHd−1 ≤ lim inf n→+∞ � Jun [φ(u+ n ) + φ(u− n )] dHd−1 Through a by now standard blow-up argument ( see Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='6), we can reduce the problem to the following lower semicontinuity result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Let Q1 ⊆ Rd be the unit square centred at 0, and let us set H := Q1 ∩ {xd = 0} and Q± 1 := Q1 ∩ {xd ≷ 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' 16 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' BUCUR, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' CHAMBOLLE, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' GIACOMINI, AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' NAHON Given u± ∈ Rd with u+ ̸= u− and un ∈ SBD(Q1) with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='5) un → u := u+1Q+ 1 + u−1Q− 1 strongly in L1(Q1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Rd), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='6) sup n Hd−1(Jun) < +∞ and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='7) e(un) → 0 strongly in L1(Q1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Md×d sym), then (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='8) φ(u+) + φ(u−) ≤ lim inf n→+∞ � Jun [φ(u+ n ) + φ(u− n )] dHd−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' We now divide the proof in several steps, and we will employ the characterization by sections of SBD functions explained in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Let ε > 0 be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' We fix δ > 0 and N ∈ N with N > d: these numbers will be subject to several constraints that will appear during the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Let us fix N unit vectors {ξi}1≤i≤N such that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='9) |ed · ξi − 1| < δ and such that any subset of d of them forms a basis of Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Moreover, we may assume in addition that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='10) (u+ − u−) · ξi ̸= 0 for every i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Thanks to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='5) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='6), we can fix a > 0 small such that setting H± := H × {±a} = H ± aed, we have (un)|H± → u± strongly in L1(H±;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Rd) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='11) ∀n ∈ N : Hd−1(Jun ∩ H±) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' We claim that, up to a subsequence, we can find H− ε ⊂ H− with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='12) Hd−1(H− \\ H− ε ) < ε such that for every i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' , N, for every y ∈ H− ε and for every n ∈ N (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='13) H− ε ∩ Jun = ∅, and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='14) H0((Jun)ξi y ) < +∞, H0((Jun)ξi y ∩R+) ≥ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Moreover setting � (un) ξi y := un(y + tξi) · ξi, for every y ∈ H− ε we have � (un) ξi y ∈ SBV ((Q1)ξi y ), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='15) J � (un) ξi y = (Jun)ξi y (cf notation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='4)), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='16) ∥[ � (un) ξi y ]′∥L1 → 0 uniformly for y ∈ H− ε , A FREE DISCONTINUITY APPROACH TO OPTIMAL PROFILES IN STOKES FLOWS 17 and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='17) (un)|H− → u− uniformly on H− ε .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Indeed, if the number δ appearing in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='9) is small enough, we can find A− ε ⊆ H− with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='18) Hd−1(H− \\ A− ε ) < ε 2 and such that for every y ∈ A− ε the lines {y + tξi : t ∈ R} intersect H+ for every i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' In view of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='5), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='6) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='7), and since pointwise convergence implies almost uniform convergence, we can find Nε ⊂ A− ε with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='19) Hd−1(Nε) < ε 2 and such that, up to a subsequence (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='20) ∥ � (un) ξi y − �uξi y ∥L1 → 0 uniformly for y ∈ A− ε \\ Nε (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='21) ∥[ � (un) ξi y ]′∥L1 → 0 uniformly for y ∈ A− ε \\ Nε (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='22) (un)|H− → u− uniformly on A− ε \\ Nε, and for every y ∈ A− ε \\ Nε (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='23) H0((Jun)ξi y ) < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Notice that for n large enough and for every y ∈ A− ε \\ Nε we have (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='24) (Jun)ξi y ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Indeed otherwise, we would get for nk → +∞ the existence of yk ∈ A− ε \\Nε with � (unk) ξi yk ∈ W 1,1((Q1)ξi yk), and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='22) together with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='21) would yield ∥ � (unk) ξi yk − u−∥1 → 0 against (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='20) (recall that by the choice (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='10) of the ξi, the functions �uξi y have a jump).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' The claim follows by setting H− ε := Aε \\ � Nε ∪ � n (Jun ∩ H−) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Indeed (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='12) follows from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='18), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='19) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='11), while (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='13) is clearly satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Relation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='14) follows by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='23) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='24), while relation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='16) follows from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Finally relation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='17) follows from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' For every i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' , N, let us consider the set Ji,− n given by the first point of intersection (with t > 0) of the line {y + tξi : t ∈ R} with the jump set Jun as y varies in the set H− ε defined in Step 2 (recall (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='14) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='15)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' In view of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='16) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='17), we can find ηn → 0 such that for every x ∈ Ji,− n with νun · ξi > 0 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='25) |u− n (x) · ξi − u− · ξi| < ηn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Step 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' We claim that, for δ small enough and N large enough, up to a subsequence, we can find ˜J− n ⊆ Jun with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='26) Hd−1( ˜J− n ) ≥ 1 − cε, 18 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' BUCUR, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' CHAMBOLLE, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' GIACOMINI, AND M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' NAHON where cε → 0 as ε → 0, and such that for every x ∈ ˜J− n (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='27) x ∈ Ji,− n for d different indices i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' , N}, where Ji,− n is defined in Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Moreover, we can orient νun on ˜J− n in such a way that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='28) ed · νun > 0 and ξi · νun > 0 for every i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Intuitively speaking, the points in ˜J− n are seen from H− ε under d different directions: moreover the associated lines cut the jump transversaly, from the “lower” to the “upper” part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' Indeed, in view of the definition of ξi (which form a very small angle with ed as δ → 0) and of the area formula (cf for instance [24, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='2]), we can assume that δ is so small that for every i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' , N (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='29) Hd−1(Ji,− n ) ≥ � Ji,− n |νun · ξi| dHd−1 = Hd−1((H− ε )ξi) = 1 1 + ˆcδ Hd−1(H− ε ), where the notation (H− ε )ξi is defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='1) and where ˆcδ → 0, so that, taking into account (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='12), for small δ we have (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='30) Hd−1(Ji,− n ) ≥ 1 − 2ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='5 below (with X = Jun, µ = Hd−1, and M given by the family of Borel sets) if N is large enough we can find an index ¯i such that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAzT4oBgHgl3EQfI_vE/content/2301.01073v1.pdf'} +page_content='31) Hd−1 \uf8eb \uf8ec \uf8ec \uf8edJ¯i,− n \\ � i1 0. The proof has two steps: we first show that for a satisfiable +instance of quantum k-SAT, most subproblems on a constant number of qubits are +satisfiable by a product state. We then show that for an instance of quantum k-SAT +which is far from satisfiable by a product state, most subproblems are unsatisfiable +by a product state. Given the promise, quantum k-SAT may therefore be solved +by checking satisfiability by a product state on randomly chosen subsystems of +constant size. +1 +Introduction +Property testing. +In computer science, one often needs to determine whether a struc- +ture defined by a large quantity of data possesses a certain property which is global, in +the sense that it is defined with reference to all the data. Property testing algorithms aim +to determine with high probability whether a structure possesses a global property by +performing local checks; that is, checking properties of randomly chosen small subsets of +the data defining the structure. Since the time complexity of a local check is smaller than +that of a global check, and one only inspects a small subset of the data, this approach, +when feasible, often leads to algorithms with excellent time and query complexity. +Of course, in general the structure may not possess the global property, but be close +enough to possessing it that one is highly unlikely to perform a local check which witnesses +this fact. For this reason property testing algorithms usually require a promise that the +structure either possesses the property or is far from possessing the property, where +distance from possessing the property is defined by a measure specific to the problem +and is usually quantified by some constant ϵ. +In [AS03], Alon and Shapira showed that the NP-hard problem of classical k-SAT +(satisfiability of Boolean functions on n variables, where each function depends on exactly +k variables) is amenable to a property testing approach. Specifically, with the promise +that an instance of classical k-SAT is either satisfiable or far from satisfiable, satisfiability +may be determined in randomised constant time, by choosing constant-sized subsets of +variables and checking satisfiability of those functions which depend only on variables in +those subsets. Because of the promise, their result is limited to dense instances of k-SAT, +where there are Ω(nk) functions. +∗ashley.montanaro@bristol.ac.uk +†changpeng.shao@bristol.ac.uk +‡dominic.verdon@bristol.ac.uk +1 +arXiv:2301.10699v1 [quant-ph] 25 Jan 2023 + +Testing quantum satisfiability. +In [Bra11] Bravyi defined a quantum analogue of +classical k-SAT, which we will here call quantum k-SAT. An instance of this problem is +defined by n Hilbert spaces of dimension 2 (qubits) and, for every subset s of k qubits, a +projector Πs on the Hilbert space of those qubits; we say that the instance is satisfiable +if there is a state of all n qubits which is in the kernel of all of the projections. While +classical k-SAT is NP-complete whenever k ≥ 3, quantum k-SAT is QMA1-complete +whenever k ≥ 3, where QMA1 is the quantum analogue of the class MA with one- +sided error. In the terminology of physics quantum k-SAT is precisely the problem of +determining whether a k-local Hamiltonian on n qubits is frustration free. +Since the problem seems very difficult in general, previous work on algorithms for +quantum k-SAT has focused on tractable special cases. +There are many interesting +results on random quantum k-SAT, for instance [LLM+10, LMSS10, BMR10, HLL+13]. +It has been shown that quantum 2-SAT can be solved in linear time [dBG16, ASSZ18]; +moreover, in this case there is always a satisfying state which is a product of one- and +two-qubit states [CCD+11]. The case where there are relatively few nontrivial projectors +all of rank 1 is also often tractable [AKS12, AdGS21]. +In this work we also treat a special case: the dense case, where the interaction hyper- +graph (i.e. the k-regular hypergraph on n vertices whose edges are subsets of k qubits +whose associated projection is nontrivial) contains Ω(nk) edges. We show that the re- +sults of [AS03] extend to the quantum setting. +Given the promise that the instance +is either satisfiable or far from satisfiable by a product state, quantum k-SAT may be +solved in randomised polynomial time in n by checking satisfiability by a product state +on constant-sized subsets of qubits. (The polynomial dependence on n is due to the fact +that, in the data specifying an instance of quantum k-SAT, the projectors are given to +poly(n) precision, so all computations involving them will pick up a poly(n) factor.) +Some comments on our results. +Before stating our results formally, we will make +two comments. +Firstly, as we emphasised in the last paragraph, the promise is that +the instance, if unsatisfiable, is far from satisfiable by a product state. Obviously, this +is a less stringent requirement than being far from satisfiable by any state. The reason +we are able to weaken the promise in this way is that an instance of quantum k-SAT +which is satisfiable by any state is locally satisfiable by a product state with high prob- +ability. This result (Theorem 1.5) is essentially a fact about quantum states which may +be of independent interest; there are some connections with monogamy of entangle- +ment [KW04, CW04, BCY11] and de Finetti theorems [BH16, BH13] which we discuss +briefly in Remark 2.12. +Secondly, our proofs are combinatorial in nature; we make no use of the norm. In +particular, we do not require the polynomial lower bound on the ground state energy +in the unsatisfiable case that is usually given as a promise in the definition of quantum +k-SAT; our testing algorithm therefore solves a harder problem than quantum k-SAT as +it is usually defined. +1.1 +Results +We will first state the three basic definitions we will use in this work. Throughout we +write σk(S) for the set of k-element subsets of a set S and [n] for the set {1, . . . , n}. For +any subset x ∈ σk(S) we use the notation x := (S − x) for the complement. +Definition 1.1 (Quantum k-SAT). An instance of quantum k-SAT (without a lower +bound on the ground state energy) is defined by the following data. +• A number of qubits n. +• For every k-element subset s ∈ σk([n]), a projector Πs on (C2)⊗n which acts non- +trivially only on the qubits in s. These projectors are given as matrices whose +entries have precision polynomial in n (i.e. they are specified by poly(n) bits). +2 + +We write the instance as ([n], {Πs}). The problem is to determine whether there exists +some state |x⟩ ∈ (C2)⊗n such that the following equation holds: +� +s∈σk([n]) +Πs |x⟩ = 0. +(1) +If such a state exists we say that the instance is satisfiable; if it does not we say that +the instance is unsatisfiable. If there is a product state |φ⟩ satisfying (1) we say that the +instance is satisfiable by a product state. +The usual definition of quantum k-SAT includes a promise that the ground-state energy +has a polynomial lower bound if the instance is unsatisfiable [Bra11, §1]; Definition 1.1 +is therefore a strictly harder problem. +Definition 1.2 (ϵ-far from satisfiable by a product state). Let Σ ⊂ σk([n]) be such that, +for some product state |φ⟩ ∈ (C2)⊗n: +� +s∈(σk([n])−Σ) +Πs |φ⟩ = 0. +(2) +We say that an instance ([n], {Πs}) of quantum k-SAT is ϵ-far from satisfiable by a +product state if this implies that |Σ| ≥ ϵnk. +Remark 1.3. Our definition of ϵ-far differs slightly from that in [AS03]; whereas they +remove individual clauses (which in our case correspond to rank-one projections) we +remove the whole projection for a given k-subset. This does not make much difference +to the overall theory, since defining ϵ-far in terms of the number of rank-one projections +removed would only alter ϵ by a constant factor. +Definition 1.4 (Local satisfiability). Let ([n], {Πs}s∈σk([n])) be an instance of quantum +k-SAT. Let C ⊂ [n]. +We say that the restriction of the instance to C is the instance of quantum k-SAT +defined by the data (C, {Πs}s∈σk(C)). We say that ([n], {Πs}) is locally satisfiable by a +product state at C if (C, {Πs}s∈σk(C)) is satisfiable by a product state. +Our result is based on the following two theorems, whose proofs are given in the remainder +of the paper. +Theorem 1.5. Let ([n], {Πs}) be a satisfiable instance of quantum k-SAT, and let c ∈ N +be fixed, where c ≥ 3. Let C ∈ σc([n]) be a subset chosen uniformly at random. The +probability that the instance is locally satisfiable by a product state at C is greater than +p ∈ (0, 1) whenever n > Ψ(p, c), where Ψ(p, c) = O(26c/ log(1/p)). +Equivalently, let |ψ⟩ ∈ (C2)⊗n be any state and let C ∈ σc([n]) be a subset chosen +uniformly at random. The probability that the subspace supp(TrC(|ψ⟩ ⟨ψ|)) ⊆ (C2)⊗c +contains a product state is greater than p ∈ (0, 1) whenever n > Ψ(p, c). +Remark 1.6. The theorem applies to the case where c ≥ 3; a similar result holds for +c = 2, but the result for c = 2 is much simpler and stronger and is treated separately in +Proposition 2.11. We give a precise bound on Ψ(p, c) in the proof. The constant factors +in this bound are far from optimal, since we made many approximations to simplify the +proof; if required, the reader could easily improve the bound by tightening the analysis +without changing the structure of the proof. +However, we would be surprised if the +exponential dependence on c could be removed without a different approach. +For the second theorem, we assume that n is large enough (but still finite). +Theorem 1.7. There exists a constant c(k, ϵ) (i.e. not depending on n) such that the +following holds. Let ([n], {Πs}) be any instance of quantum k-SAT which is ϵ-far from +satisfiable by a product state. Then, for a randomly chosen subset C ∈ σc(k,ϵ)([n]), the +instance is locally unsatisfiable by a product state at C with probability p > 0.75. +3 + +Remark 1.8. The main weakness of Theorem 1.7 is that we do not have a good upper +bound for the constant c(k, ϵ). +Our proof in Section 3 does yield an upper bound, +but it is extremely large (bigger than (((3!)!)! · · · )!, where the chain of factorials has +length (5/ϵ)4k−1). The bound has this form because the degree of the m-qubit Segre +variety is the factorial m! (Lemma 3.6). It is an open question whether the upper bound +can be made small enough for the testing result to be practically useful rather than +merely theoretically interesting. Improvements on this scale are not unknown in classical +property testing (see e.g. the discussion in [AS03, § 1.1]). +Remark 1.9. Together, Theorems 1.5 and 1.7 imply that, for large enough n, an instance +([n], {Πs}) of quantum k-SAT cannot be both satisfiable and far from satisfiable by a +product state. +The consequent result about testability of quantum k-SAT is as follows. +Corollary 1.10. With a promise that the instance is either satisfiable or ϵ-far from +satisfiable by a product state, quantum k-SAT may be solved in randomised polynomial +time in n. +Proof. Let ([n], {Πs}) be an instance of quantum k-SAT, and let c be some constant. +Given a subset C ∈ σc([n]), we can check whether or not (n, {Πs}) is locally satisfiable +by a product state at C in time polynomial in n (and exponential in c) using Gr¨obner +basis methods. Indeed, let {Π1, . . . , ΠN} be the set of nontrivial projectors on C. Let +Πi = �ri +j=1 |aij⟩ ⟨aij|, where ri ≤ 2k is the rank of πi and |aij⟩ ∈ (C2)⊗k. Let |φ⟩ = +|φ1⟩ ⊗ · · · ⊗ |φc⟩ be a unknown product state defined on C. We set +|φi⟩ = xi0 |0⟩ + xi1 |1⟩ , +i ∈ {1, 2, . . . , c}. +Let si ∈ σk(C) be the set of qubits on which Πi acts. For each si, we write |φsi⟩ := +� +v∈si |φv⟩. Then (1) implies the following equations: +⟨aij|φsi⟩ = 0, +i ∈ {1, 2, . . . , N}, j ∈ {1, 2, . . . , ri}. +(3) +We also have that |xi0|2 + |xi1|2 = 1. As we can rescale (3) arbitrarily, we can relax this +to |xi0|2 + |xi1|2 ̸= 0; equivalently, xij ̸= 0 for at least one j. To ensure this we introduce +some new variables {yij} with the same index sets as the {xij}, and some new equations: +(xi0yi0 − 1)(xi1yi1 − 1) = 0, +i ∈ {1, . . . , c}. +(4) +In total we obtain c + �N +i=1 ri equations. Each equation has degree at most k or 2, and +there are 4c variables over the field C. We want to determine if there is a solution for +this polynomial system over C. It is known that this polynomial system has no zeros in +C if and only if its Gr¨obner basis is {1}. This is Hilbert’s Nullstellensatz, which is an +EXPSPACE-complete problem [VZGG13, Chapter 21.7]. By [BFS15, Proposition 1], an +upper bound on the time complexity in this case is +D(c + +N +� +i=1 +ri) +�4c + D +D +�ω +poly(n), +where D ≤ 1 + 3c + (k − 1) �N +i=1 ri is the maximal degree of the elements in the reduced +Gr¨obner basis, and ω < 2.373 is the exponent of matrix multiplication. The poly(n) +factor is due to the polynomial precision of the matrix entries specifying the quantum +k-SAT instance (Definition 1.1). +We can therefore straightforwardly check satisfiability of an instance ([n], {Πs}) of +quantum k-SAT with the testing promise. Since we are only interested in scaling w.r.t. n, +we may assume that n is large enough that (by Theorem 1.5) if the instance is satisfiable +then it will be locally satisfiable by a product state on c(k, ϵ)-subsets with probability +p > 0.75, and (by Theorem 1.7) if the instance is ϵ-far from satisfiable by a product state +then it will be locally unsatisfiable by a product state on c(k, ϵ)-subsets with probability +p > 0.75. +4 + +We choose subsets {Ci ∈ σc(k,ϵ)([n])}m +i=1 at random and check whether the instance +is locally satisfiable by a product state on these subsets. +If the majority are locally +satisfiable by a product state we conclude that the instance is satisfiable; otherwise we +conclude that the instance is ϵ-far from satisfiable by a product state. By the Chernoff +bound for the binomial distribution the probability that the conclusion is incorrect tends +exponentially to zero as m increases. +1.2 +Acknowledgements +We thank Niel de Beaudrap and Aram Harrow for useful discussions. This project has +received funding from the European Research Council (ERC) under the European Union’s +Horizon 2020 research and innovation programme (grant agreement No. 817581). We +acknowledge support from EPSRC grant EP/T001062/1. +2 +Proof of Theorem 1.5 +We will make use of some combinatorial lemmas, some of which are well-known. For the +reader’s convenience we prove all these lemmas, but to avoid a long digression we have +relegated the proofs to Appendix A. Throughout we consider the binomial coefficient +�x +k +� +to be the polynomial x(x − 1) · · · (x − k + 1)/k!. It is defined for every real number x, +and is positive and increasing for x ≥ k − 1. +Lemma 2.1 (Scaling binomial coefficients). Let θ, x ∈ R≥0 and c ∈ N, where 0 ≤ θ ≤ 1 +and x ≥ c. Then, for some α ∈ [0, 1), we have: +θ +�x +c +� += +�θ1/cx +c +� ++ α +�θ1/cx +c − 1 +� +≤ +�θ1/cx + 1 +c +� +. +(5) +Definition 2.2. Let G be an l-regular hypergraph on n vertices, and let k ≤ l. We +define the k-shadow G|k of G to be the k-regular hypergraph on n vertices whose edges +are precisely those k-subsets which are contained in an edge of G. +Definition 2.3 (c.f. [BE15, Fit18]). Let G be an l-regular hypergraph on n vertices, let +k ≤ l, and let ω ∈ [0, 1]. We say that an k-regular hypergraph on n vertices G|k,ω is a +partial (k, ω)-shadow of G if it can be obtained by the following construction. For each +edge E of G, choose a subset KE ⊂ σk(E) of size |KE| = ⌊ω +�l +k +� +⌋. The edge set of G|k,ω +is then defined as � +E∈G KE. +Obviously there are many different partial (k, ω)-shadows of an l-regular hypergraph +G; they depend on the choice made of the set KE for each edge E ∈ G. Recall that the +edge density 0 ≤ θ ≤ 1 of a k-regular hypergraph on n vertices with edge set E is defined +by |E| = θ +�n +k +� +. +Lemma 2.4 (Edge density of shadows [Kee08, Thm. 1]). If G is an l-regular hypergraph +with edge density θ, then the k-shadow G|k has edge density θ|k ≥ (θ1/l − 2/n)k. +Lemma 2.5 (Edge density of partial shadows). Let n ∈ N and let c ∈ N. Let l := n/2c+δ +and k := l/2 + ϵ, where −3 ≤ δ < −1 and −1/2 ≤ ϵ ≤ 0, be natural numbers. Let G be +an l-regular hypergraph on n vertices with edge density θ ∈ [0, 1]. +Suppose that n ≥ 3(2c) and n ≥ 3(2c+1) +� +1 +2 + +1 +ln(A) +� +A1/3 +4 +− ln(θ) +�� +for some constant +A > 1. Then the edge density θ|k,1/2 of a partial (k, 1/2)-shadow of G has the following +lower bound: +θ|k,1/2 ≥ +θ +KA +:= +θ +2 +� +A exp(A1/3/2) +. +(6) +Lemma 2.6 (Edge density for a certain construction). Let n ∈ N be odd. We construct +an (n − 1)/2-regular hypergraph G as follows. For every subset X ∈ σ(n−1)/2([n]), we do +precisely one of the following: +5 + +1. Add X as an edge to G. +2. Add all the subsets in σ(n−1)/2(X) as edges to G. +The edge density θ of a hypergraph G constructed in this way satisfies θ ≥ 1/2. +Recall that the degree of a vertex in a hypergraph is the number of edges containing it. +Lemma 2.7 (Degree of a random vertex). Let G be a k-regular hypergraph on n vertices +with edge density θ, where k = βn for some β ∈ [0, 1]. Let N be the degree of a vertex +picked uniformly at random, and let τ := N/ +�n−1 +k−1 +� +. Then, for any ϵ > 0 and α ∈ (0, 1): +Pr [|θ − τ| ≥ min(∆(α, β, n), (1 − θ)(1 − α))] ≤ α, +(7) +where +∆(α, β, n) = +5 exp +� +1 +12α(1−α)n + +1 +12β(1−β)n +� +� +2πα(1 − α)β(1 − β)n +=: 5(Lα,β)1/n +Mα,β +√n . +(8) +The final ingredients for the proof are two facts about entangled subspaces. Recall that +a subspace L ⊂ C2 ⊗ Cm is completely entangled if there is no product state in L. +Lemma 2.8 ([ATL11, Lem. 1 and Lem. 2]). Let L ⊂ C2 ⊗Cm be a completely entangled +subspace, for any m ≥ 2. Then the orthogonal complement L⊥ ⊂ C2 ⊗ Cm contains a +product state. +Corollary 2.9. Let m ∈ N, and let V ⊂ (C2)⊗m be a subspace of dimension ≥ 2. Let +L ⊂ C2 ⊗ (C2)⊗m be a subspace such that: +1. C2 ⊗ V ⊥ ⊂ L. +2. (C2 ⊗ V ) ∩ L is a completely entangled subspace of C2 ⊗ V . +Then L⊥ ⊂ C2 ⊗ Cm contains a product state. +Proof. Since C2 ⊗ V ⊥ ⊂ L, we have L = (C2 ⊗ V ⊥) ⊕ +� +(C2 ⊗ V ⊥)⊥ ∩ L +� += (C2 ⊗ V ⊥) ⊕ +� +(C2 ⊗ V ) ∩ L +� +. Then L⊥ = (C2 ⊗ V ) ∩ +� +(C2 ⊗ V ) ∩ L +�⊥ = +� +(C2 ⊗ V ) ∩ L +��⊥, where �⊥ +is the orthogonal complement in C2 ⊗ V . But (C2 ⊗ V ) ∩ L is a completely entangled +subspace of C2 ⊗V , so by Lemma 2.8 we know that +� +(C2 ⊗ V ) ∩ L +��⊥ contains a product +state; therefore L⊥ contains a product state. +Lemma 2.10. Let |ψ⟩ ∈ (C2)⊗n be a state and let x ∈ [n]. Let S1, S2 ⊂ ([n] − {x}) be +two subsets such that S1 ∩ S2 = ∅ and S1 ⊔ S2 = ([n] − {x}). Then at least one of the +following is true: +1. The subspace supp(TrS2(|ψ⟩ ⟨ψ|)) contains a product state |φx⟩ ⊗ |φS1⟩. +2. The subspace supp(TrS1(|ψ⟩ ⟨ψ|)) contains a product state |φx⟩ ⊗ |φS2⟩. +Proof. To clarify our notation, we first recall the definition of supports and kernels. By +definition, for any subset S ⊂ [n], we have supp(TrS(|ψ⟩ ⟨ψ|)) = ker(TrS(|ψ⟩ ⟨ψ|))⊥, +where the orthogonal complement is taken in the space (C2)⊗|S|. If we write (C2)⊗n = +(C2)⊗|S| ⊗ (C2)⊗(n−|S|), where the first factor corresponds to those qubits in the set S, +then: +ker(TrS(|ψ⟩ ⟨ψ|)) += +span{|v⟩ ∈ (C2)⊗|S| : (⟨v| ⊗ 1) |ψ⟩ = 0}, +(9) +supp(TrS(|ψ⟩ ⟨ψ|))) += +span{(⟨v| ⊗ 1) |ψ⟩ : |v⟩ ∈ (C2)|S|}. +(10) +We now observe that if any of (i) dim(supp(Trx(|ψ⟩ ⟨ψ|))), (ii) dim(supp(TrS1(|ψ⟩ ⟨ψ|))) +or (iii) dim(supp(TrS2(|ψ⟩ ⟨ψ|))) equal 1, then at least one of the two claims in this lemma +is true. Indeed, for (i) observe that if dim(supp(Trx(|ψ⟩ ⟨ψ|))) = 1, then |ψ⟩ = |ψx⟩ ⊗ +|ψ([n]−{x})⟩. To see this, observe that there exists |w⟩ ∈ C2 such that (⟨w| ⊗ 1) |ψ⟩ = 0; +6 + +it follows by the Schmidt decomposition that |ψ⟩ is a product state. The argument for +(ii) and (iii) is similar. +We therefore now assume that those three dimensions are greater than 1. We consider +the subspace +ker(Tr{x}⊔S1(|ψ⟩ ⟨ψ|)) ∩ +� +supp(Tr{x}(|ψ⟩ ⟨ψ|)) ⊗ supp(TrS1(|ψ⟩ ⟨ψ|)) +� +⊆ supp(Tr{x}(|ψ⟩ ⟨ψ|)) ⊗ supp(TrS1(|ψ⟩ ⟨ψ|)). +We have a dichotomy: +(a) The subspace ker(Tr{x}⊔S1(|ψ⟩ ⟨ψ|))∩ +� +supp(Tr{x}(|ψ⟩ ⟨ψ|))⊗supp(TrS1(|ψ⟩ ⟨ψ|)) +� +is a completely entangled subspace of supp(Tr{x}(|ψ⟩ ⟨ψ|)) ⊗ supp(TrS1(|ψ⟩ ⟨ψ|)). +(b) The subspace ker(Tr{x}⊔S1(|ψ⟩ ⟨ψ|))∩ +� +supp(Tr{x}(|ψ⟩ ⟨ψ|))⊗supp(TrS1(|ψ⟩ ⟨ψ|)) +� +contains a product state. +In case (a), we are precisely in the situation of Corollary 2.9, where V = supp(TrS1(|ψ⟩ ⟨ψ|)) +and L = ker(Tr{x}⊔S1(|ψ⟩ ⟨ψ|)); therefore claim 1 is true. +On the other hand, in +case (b) there exists a product state |v⟩⊗|w⟩ ∈ supp(Tr{x}(|ψ⟩ ⟨ψ|))⊗supp(TrS1(|ψ⟩ ⟨ψ|)) +such that (⟨v| ⊗ ⟨w| ⊗ 1) |ψ⟩ = 0. +Consider the state |ψ′⟩ := (1 ⊗ ⟨w| ⊗ 1) |ψ⟩ ∈ +supp(TrS1(|ψ⟩ ⟨ψ|)). Since (⟨v|⊗1) |ψ′⟩ = 0, we have that dim(ker(Tr{x}(|ψ′⟩ ⟨ψ′|))) = 1; +by the Schmidt decomposition this implies that |ψ′⟩ = |ψ′ +x⟩ ⊗ |ψ′ +S2⟩, and so claim 2 is +true. +We can now prove the theorem. To warm up, we prove the case c = 2 first. +Proposition 2.11. Let ([n], {Πs}) be a satisfiable instance of quantum k-SAT, and let +x1 ∈ [n] be any qubit. Then there is at most one qubit x2 ∈ ([n] − {x1}) such that the +instance is not locally satisfiable by a product state at {x1, x2}. +Equivalently, let |ψ⟩ ∈ (C2)⊗n be any state, and let x1 ∈ [n] be any qubit. Then there +is at most one qubit x2 ∈ ([n] − {x1}) such that the subspace supp(Tr{x1,x2}(|ψ⟩ ⟨ψ|))) ⊆ +(C2)⊗2 does not contain a product state. +Proof. The equivalence of the two statements will be shown in the first paragraph of the +proof of Theorem 1.5 below. We therefore need only prove the second statement. Let +x2 ∈ ([n] − {x1}) be any qubit. We now use Lemma 2.10; in the notation of that lemma, +let x := x1, S1 := {x2} and S2 := ([n] − {x1, x2}). Then either supp(Tr{x1,x2}(|ψ⟩ ⟨ψ|))) +contains a product state or supp(Tr{x1,x′}(|ψ⟩ ⟨ψ|))) contains a product state for any +x′ ∈ ([n] − {x1, x2}). +We now move onto the case c ≥ 3. +Theorem (Restatement of Theorem 1.5). Let ([n], {Πs}) be a satisfiable instance of +quantum k-SAT, and let c ∈ N be some constant, where c ≥ 3. Let C ∈ σc([n]) be a +subset chosen uniformly at random. The probability that the instance is locally satisfiable +by a product state at C is greater than p ∈ (0, 1) whenever: +n > Ψ(p, c) := +max +� +(2c − 1)(c − 1) ln(4) +ln(1/p) ++ (2c − 2) , +e2 +�180(c − 2)(2)3(c−2) +√ +2π(3 − 1/(2)c−2) +�2 +, +6(2c+1)(c − 1) +� +. +Equivalently, let |ψ⟩ ∈ (C2)⊗n be any state and let C ∈ σc([n]) be a subset chosen +uniformly at random. The probability that the subspace supp(TrC(|ψ⟩ ⟨ψ|)) ⊆ (C2)⊗c +contains a product state is greater than p ∈ (0, 1) whenever n > Ψ(p, c). +Remark 2.12. Before giving the proof, we remark that the following weaker fact follows +quite straightforwardly from known results: +7 + +Let ([n], {Πs}) be a satisfiable instance of quantum k-SAT, let c ∈ N be some +constant, and let ϵ > 0. Let C ∈ σc([n]) be a subset chosen uniformly at ran- +dom. The probability that there exists a product state |φ⟩ ∈ (C2)⊗c such that +� +s∈σk(C) ⟨φ| Πs |φ⟩ < ϵ is greater than p whenever p > O(nk−c−1/4) +�c +k +� +/ϵ. +To prove this, one can use [BH16, Eq. 91], which implies that there is a product state +|φ⟩ ∈ (C2)⊗n such that � +s∈σk([n]) ⟨φ| Πs |φ⟩ ≤ O(nk−1/4). We can write +� +s∈σk([n]) +⟨φ| Πs |φ⟩ = +� +C∈σc([n]) +� +s∈σk(C) +1 +�c +k +� ⟨φ| Πs |φ⟩ = +� +C∈σc([n]) +SC, +where SC := � +s∈σk(C) +�c +k +�−1 ⟨φ| Πs |φ⟩. Since there are O(nc) c-subsets, the expected +value of SC for a randomly chosen c-subset is O(n(k−c)−1/4); the result then follows by +Markov’s inequality. +An alternative approach to proving this fact, which has some issues but nevertheless +reveals something about the nature of the problem, is to use monogamy of the squashed +entanglement [KW04, Thm. 8]. For this, let |ψ⟩ be a state satisfying ([n], {Πs}), and +let l1, l2 be constants. It follows straightforwardly from monogamy that, for a randomly +chosen C ∈ σc([n]), the expected entanglement across every bipartition of C is O(1/n). +Then by [BCY11, Thm, §II], TrC(|ψ⟩ ⟨ψ|) is close to a biseparable state across every +bipartition of C. If we could conclude that this implied closeness to a fully separable +state, the fact would follow. +Theorem 1.5 is a stronger, exact analogue, where we demand exact local satisfiability +by a product state rather than approximate satisfiability. +We were unable to derive +Theorem 1.5 from the non-exact result, and the proof we are about to provide follows a +different approach, which does not make use of any continuous entanglement measure. +Proof of Theorem 1.5. We first observe the equivalence of the two statements in this +theorem. If ([n], {Πs}) is satisfiable, let |ψ⟩ ∈ (C2)⊗n be a satisfying state; clearly any +state |ψC⟩ ∈ supp(TrC(|ψ⟩ ⟨ψ|)) is a local solution to ([n], {Πs}) at C, so the second +statement implies the first. In the other direction, let |ψ⟩ ∈ (C2)⊗n be a state; then for +all C ∈ σc([n]) let ΠC be the projector onto ker(TrC(|ψ⟩ ⟨ψ|)), and consider the instance +([n], {ΠC}C∈σc([n])) of quantum c-SAT, which is of course satisfied by the state |ψ⟩. +Given this equivalence it is sufficient to show that for any state |ψ⟩ ∈ (C2)⊗n and a +randomly chosen subset C ∈ σc([n]), there exists a product state in supp(TrC(|ψ⟩ ⟨ψ|)) +with high probability. +We will choose qubits {x1, . . . , xc} =: C one by one at random. Let us pick the first +qubit x1. There are two possibilities: +(a) n − 1 is even. +(b) n − 1 is odd. +In case (a), let B([n]−{x1}) be the set of all equal bipartions of the set ([n]−{x1}), where +an equal bipartition is a pair of sets B1, B2 ⊂ ([n] − {x1}) of size |B1| = |B2| = (n − 1)/2 +such that B1 ∪ B2 = ([n] − {x1}). We write such a bipartition as Bi +1|Bi +2, where i ∈ I +is some index for the bipartitions. +We observe that, for any finite set S with even +cardinality, |B(S)| = +� |S| +|S|/2 +� +/2. Now, for any equal bipartition B1|B2 ∈ B([n]−{x1}), we +know by Lemma 2.10 that at least one of the following holds: +(i) The subspace supp(TrB2(|ψ⟩ ⟨ψ|)) contains a product state |φx1⟩ ⊗ |φB1⟩. +(ii) The subspace supp(TrB1(|ψ⟩ ⟨ψ|)) contains a product state |φx1⟩ ⊗ |φB2⟩. +We define an (n − 1)/2-regular hypergraph G1 on the set ([n] − {x1}) as follows. For +each equal bipartition Bi +1|Bi +2 ∈ B([n] − {x1}), if supp(TrBi +j(|ψ⟩ ⟨ψ|)) contains a product +state then add the set Bi +j as an edge to the graph G1. Clearly, the edge density θ1 of G1 +has the lower bound θ1 ≥ 1/2. The physical significance of this hypergraph G1 is that, +8 + +for any edge E of G1, we know that supp(Tr{x1}⊔E(|ψ⟩ ⟨ψ|)) contains a product state, +which we will write as |ψ1⟩ ⊗ |ψE⟩. +This definition of G1 depended on the fact that n − 1 was even. In case (b), we make +a slightly different definition. For any subset X ∈ σn/2−1([n] − {x1}), at least one of the +following holds, by Lemma 2.10: +(i) The subspace supp(Tr{x1}⊔X(|ψ⟩ ⟨ψ|)) contains a product state |φx1⟩ ⊗ |φX⟩. +(ii) The subspace supp(TrX(|ψ⟩ ⟨ψ|)) contains a product state |φx1⟩ ⊗ |φ{x1}⊔X⟩. +We now define an (n/2 − 1)-regular hypergraph G1 on the set ([n] − {x1}) as follows. +For every X ∈ σn/2−1([n] − {x1}), if supp(Tr{x1}⊔X(|ψ⟩ ⟨ψ|)) contains a product state +then we add X as an edge to G1. On the other hand, if supp(TrX(|ψ⟩ ⟨ψ|)) contains +a product state then we add all the subsets in σn/2−1({x1} ⊔ X) as edges to G1. By +Lemma 2.6 the edge density θ1 of G satisfies θ1 ≥ 1/2. Again, the physical significance of +this hypergraph G1 is that, for any edge E of G1, we know that supp(Tr{x1}⊔E(|ψ⟩ ⟨ψ|)) +contains a product state, which we will write as |ψ1⟩ ⊗ |ψE⟩. +Having obtained the hypergraph G1, we now pick the next qubit x2 ∈ ([n] − {x1}) +at random. We define a new hypergraph W2 (W for ‘working’, since this graph is an +intermediate step towards obtaining G2) on the set ([n] − {x1, x2}) with the edges {(E − +{x2}) | E is an edge of G1 and x2 ∈ E}. This hypergraph W2 is k2-regular, where k2 = +((n − 3)/2) if n − 1 was even, and k2 = (n/2 − 2) if n − 1 was odd. Now we define +the graph G2; again the definition will depend on whether k2 is even or odd. If k2 is +even then, for every edge E of W2 and every X ∈ σk2/2(E), we again have the following +dichotomy, by Lemma 2.10: +(i) The subspace supp(Tr{x1,x2}⊔X(|ψ1⟩⊗|ψE⟩) contains a product state |ψ1⟩⊗|ψ2⟩⊗ +|ψX⟩. +(ii) The subspace supp(TrX(|ψ1⟩⊗|ψE⟩) contains a product state |ψ1⟩⊗|ψ2⟩⊗|ψE−(X⊔x2)⟩. +We then define the (k2/2)-regular hypergraph G2 as follows. For every edge E in W2, and +every X ∈ σk2/2(E), if supp(Tr{x1,x2}⊔X(|ψ1⟩ ⊗ |ψE⟩) contains a product state then we +add X as an edge to G2. On the other hand, if supp(TrX(|ψ1⟩⊗|ψE⟩) contains a product +state then we add X as an edge to G2. This graph contains a partial (k2/2, 1/2)-shadow +of W2. +If k2 is odd then we define the (k2 − 1)/2-regular hypergraph G2 as follows: for every +edge E of W2 and every X ∈ σ(k2−1)/2(E), if supp(Tr{x1,x2}⊔X(|ψ1⟩ ⊗ |ψE⟩)) contains a +product state then we add X as an edge to G2. On the other hand, if supp(TrX(|ψ1⟩ ⊗ +|ψE⟩)) contains a product state then we add all the subsets in σ(k2−1)/2(X) as edges to +G2. By Lemma 2.6 applied to each edge E, this graph contains a partial ((k2−1)/2, 1/2)- +shadow of W2. +The physical significance of the hypergraph G2 is that for any edge E of G2, we have +that supp(Tr{x1,x2}⊔E(|ψ⟩ ⟨ψ|)) contains a product state over the qubit x1, the qubit x2 +and the qubits in E, which we will write as |ψ1⟩ ⊗ |ψ2⟩ ⊗ |ψE⟩. +We now iterate this construction for the qubits {x3, . . . , xc−1} and obtain a kc−1- +regular hypergraph Gc−1 on the set ([n] − {x1, . . . , xc−1}); for any edge E of Gc−1, we +know that supp(Tr{x1,...,xc−1}⊔E) contains a product state, which we write as |ψ1⟩⊗· · ·⊗ +|ψc−1⟩ ⊗ |ψE⟩. +Let us obtain bounds on the edge size kc−1 and edge density θc−1 of the hypergraph +Gc−1. We determine the edge size first. The hypergraph G1 was k1-regular, where either +k1 = (n − 1)/2 or k1 = (n/2 − 1), and had edge density θ1 ≥ 1/2. To get G2 from +G1 we defined the graph W2 by picking a single qubit; this graph had edges of size +l2 = k1 − 1. Iterating this construction, we see that in general the graph Gi+1 will have +edges of size ki+1 = li+1/2 or ki+1 = li+1/2 − 1/2, where li+1 = ki − 1. It follows that +ki/2 > ki+1 ≥ ki/2 − 1, and k0 = n. Solving the linear recurrence, we obtain: +n/2i > ki ≥ n/2i + 1/2i−1 − 2 > n/2i − 2, +9 + +n/2i − 1 > li+1 ≥ n/2i + 1/2i−1 − 3 > n/2i − 3. +(11) +In particular, +n/2c−1 > kc−1 ≥ n/2c−1 + 1/2c−2 − 2. +(12) +We now obtain a lower bound on the edge density θc−1. To simplify the calculation +we will make numerous approximations. By (11), Lemma 2.5 will always hold when we +go from Wi to Gi, as long as n is big enough; we at least need n ≥ 3(2c−1) for the +step from Wc−1 to Gc−1, so we assume this from now on. There is a free constant A +in Lemma 2.5; we arbitrarily set it to A = 216W(22/3/6)3 ≃ 2.107, where W(x) is the +Lambert W-function; this implies that KA = 4. +On the other hand, Lemma 2.7 gives us a bound on the change in edge density when +we go from Gi to Wi+1. The bound is a minimum over two functions; we will just consider +the function (8). Let βi be the value of β when we apply Lemma 2.7 to go from Gi to Wi; +by (11), we always have 1/2 > 1/2 +i > βi ≥ 1/2i − 2/n ≥ 1/2i − 2/3(2c−2) ≥ 1/3(2c−2). +Let β := 1/3(2c−2). We also fix some α ∈ (0, 1), which we will determine later. By the +union bound, Lemma 2.5 and Lemma 2.7 we then have +Pr[θc−1 ≥ Φ] ≥ 1 − (c − 2)(1 − α), +(13) +where Φ := +1 +2(4)c−2 − (c − 2)∆(α, β, n). +Finally, we pick the qubit xc. As long as it is contained in an edge of Gc−1, there will +be a product state in supp(Tr{x1,...,xc−1,xc}(|ψ⟩ ⟨ψ|)). The worst case is that the edges +of Gc−1 form a complete hypergraph on a subset C ⊂ ([n] − {x1, . . . , xc−1}), with some +overspill onto a single vertex not in C. Let |C| = σ(n−c+1) for some σ ∈ [0, 1]. By (12) +and Lemma 2.1, we have: +σ ≥ (θc−1) +2c−1 +n+2−2c ≥ (Φ) +2c−1 +n+2−2c . +(14) +Let p be the probability that supp(TrC(|ψ⟩ ⟨ψ|)) contains a product state; by (13) +and (14) we have that +p ≥ (1 − (c − 2)(1 − α))(Φ) +2c−1 +n+2−2c . +(15) +We will assume that p = p is given and find values for the as yet undefined variables +α, n which satisfy (15) while minimising n. (Again, we will make approximations, so the +minimisation will not be tight.) Assuming 1 − (c − 2)(1 − α) > 0, we need: +� +p +(1 − (c − 2)(1 − α)) +� n+2−2c +2c−1 +≤ +1 +2(4)c−2 − (c − 2) +5(Lα,β)1/n +Mα,β +√n . +(16) +We make n big enough that +(c − 2) +5(Lα,β)1/n +Mα,β +√n +≤ +1 +2t(4)c−2 +(17) +for some t > 1; this is to say that +1 +n ln(Lα,β) − 1 +2 ln(n) ≤ ln +� +Mα,β +10(c − 2)(4)c−2t +� +. +(18) +Let us assume that α ≥ 1 − 1/3(2c−2); then +Lα,β ≤ exp +� +1 +6α(1 − α) +� +, +Mα,β ≥ +√ +2πα(1 − α), +and the equation (18) reduces to +1 +6nα(1 − α) − 1 +2 ln(n) ≤ ln +� √ +2πα(1 − α) +10(c − 2)(4)c−2t +� +. +(19) +10 + +Now assume that +ln(n) ≥ 2 +� +ln +�10(c − 2)(4)c−2t +√ +2πα(1 − α) +� ++ 1 +� +⇔ +n ≥ e2 +�10t(c − 2)(4)c−2 +√ +2πα(1 − α) +�2 +, +(20) +so the equation (19) reduces to: +1 +6nα(1 − α) ≤ 1 +⇔ +n ≥ +1 +6α(1 − α) +which is already implied by (20). By (17), the equation (16) then reduces to: +� +p +1 − (c − 2)(1 − α) +� n+2−2c +2c−1 +≤ +t − 1 +2t(4)c−2 . +(21) +Clearly a larger α here will reduce the required size of n in (21), but it will worsen it +in (20); so we set α = 1 − 1/3(2c−2). We then assume that: +p +1 − (c − 2)(1 − α) < 1 +⇔ +p < 1 − +c − 2 +3(2c−2). +We also arbitrarily set t = 2. Given these assumptions, (21) reduces to +n ≥ +(2c − 1)(c − 1) ln(4) +ln(1 − (c − 2)/3(2c−2)) − ln(p) + (2c − 2), +which is implied by the simpler +n ≥ (2c − 1)(c − 1) ln(4) +ln(1/p) ++ (2c − 2). +(22) +In addition to the lower bounds (20) and (22), there is also a lower bound on n coming +from the second lower bound on n in Lemma 2.5. By (13) and (17) we always have +θi ≥ Φ ≥ +1 +4(4)c−2 , so using ln(A) ≥ ln(2) we obtain the following bound: +n ≥ 6(2c+1)(c − 1). +Substituting the chosen values of α and t into (20), we obtain the theorem. +3 +Proof of Theorem 1.7 +We first remark that one idea for a proof of this theorem is to add the usual polynomial +ground state energy bound to the definition of quantum k-SAT, then use the standard +approach with a δ-net to reduce the problem of satisfiability by a product state to an +instance of classical k′-SAT. It is straightforwardly seen that this instance of k′-SAT is +also ϵ-far from satisfiable, so the result from [AS03] can be directly applied to show local +unsatisfiability. The problem is that the constant δ determining the precision of the net, +and therefore also the number k′, depends polynomially on n; this implies that the size +of the subsets to be checked also depends at least polynomially on n. Since the time +taken to check for a product state solution is exponential in the size of the subset, as we +saw in the proof of Corollary 1.10, we end up with time exponential in n. +Instead, we give a proof which is heavily inspired by Alon and Shapira’s proof of the +classical analogue (to the point where we were able to lift most of the constants from +their work), and which is in any case more general than the above approach since it does +not require any lower bound on the ground state energy when an instance of quantum +k-SAT is unsatisfiable. Firstly, we consider the problem of extending a local assignment, +which in our setting is a product subspace. We define the notion of a bad qubit; that is, +a qubit to which either the local assignment cannot be extended, or such that extension +to that qubit greatly reduces the dimension of possible extensions to the other qubits. +11 + +If, in the course of constructing a local assignment by extension, one extends through +5(4)k−1/ϵ bad qubits, then the assignment can no longer be extended (Lemma 3.4). We +show that if an instance of quantum k-SAT is ϵ-far from satisfiable by a product state, +then there are always more than ϵn/5 bad qubits with respect to any local assignment +(Lemma 3.5). +At this point, Alon and Shapira [AS03, Claim 2.4] built a binary tree of possible +assignments to a set of randomly picked variables and showed that each branch of the +tree will run into more than 5(4)k−1/ϵ bad variables with high probability; they then used +the union bound to conclude that there is no local assignment to those variables with high +probability. This approach cannot be applied directly in the quantum setting because +there are in general continuously many assignments to each variable (the space of possible +assignments is a Hilbert space and not a binary set) and it is therefore not possible to +build a finite tree. Instead, we use a backtracking approach which is summarised in the +first paragraph of the proof of the theorem at the end of this section. We still need to +discretise somehow; for this we use Lemma 3.6, which implies that we need only eliminate +a finite number of possible solutions. +Notation 3.1. Throughout this section we write |H| := dim(H) for the dimension of +a Hilbert space H and |X| for the cardinality of a set X. When the context does not +adequately distinguish these two meanings we explicitly use dim(H) for the dimension +and #(X) for the cardinality. +Let V, W be two sets. We write σk(V ⊔ W) for the set of k-element subsets of V ⊔ W +containing all the elements of V ; that is, underlining one of the sets in the union means +all its elements must be contained in the k-element subsets. +For a subset X ⊆ [n] we write HX := (C2)⊗|X| for the Hilbert space of the qubits in +X. +Definition 3.2. Fix a subset S ⊂ [n], and let HS := (C2)⊗|S| be the Hilbert space of +the qubits in S. We say that an assignment to S is a product subspace PS ⊆ HS such +that every state in the subspace is a local solution for ([n], {Πs}) at S; that is, a subspace +PS = V1 ⊗· · ·⊗V|S| ⊆ HS such that every state |φS⟩ ∈ PS satisfies � +s∈σk(S) Πs |φS⟩ = 0. +For every {x1, . . . , xj} ∈ σj([n] − S), where 1 ≤ j ≤ k − 1, we define LS,PS(x1, . . . , xj) ⊆ +Hx1,...,xj to be the space of all states |vx1,...,xj⟩ ∈ Hx1,...,xj such that +� +s∈σk({x1,...,xj}⊔S) +Πs(|vx1,...,xj⟩ ⊗ |φS⟩) = 0 +∀ |φS⟩ ∈ PS. +(23) +We define LS,PS,j := � +{x1,...,xj}∈σj([n]−S) LS,PS(x1, . . . , xj). In words, LS,PS(x1, . . . , xj) +is the space of possible assignments |vx1,...,xj⟩ on the qubits x1, . . . , xj that are compatible +with the assigment (S, PS), in the sense that |vx1,...,xj⟩ ⊗ PS is in the kernel of every +projector on the qubits {x1, . . . , xj} ⊔ S which acts on all of the qubits x1, . . . , xj. The +space LS,PS,j is the direct sum of all these possible j-qubit ‘extensions’ of the assignment +(S, PS), and its dimension measures how many possible j-qubit extensions there are to +the local assignment (S, PS). +In what follows we will consider taking a qubit x /∈ S and extending the assignment +PS on S to an assignment Vx ⊗ PS on {x} ⊔ S, where Vx ⊆ LS,PS(x). We define +δS,PS,x,Vx,j := +� +{x1,...,xj}∈σj([n]−({x}⊔S)) +|LS,PS(x1, . . . , xj)| − |L{x}⊔S,Vx⊗PS(x1, . . . , xj)| +and δS,PS,x,Vx := �k−1 +j=1 δS,PS,x,Vx,j nk−j−1. +In words, δS,PS,x,Vx measures how much +the dimension of the space of possible extensions has been reduced by extending the +assignment (S, PS) to ({x} ⊔ S, VX ⊗ PS). +We say that a qubit x ∈ ([n]−S) conflicts with respect to (w.r.t.) (S, PS) if LS,PS(x) = +∅. If a qubit x ∈ ([n] − S) does not conflict w.r.t. (S, PS), we say that it is heavy w.r.t. +(S, PS) if minVx⊆LS,PS (x) δS,PS,x,Vx > ϵnk−1/5. We say that a qubit x ∈ ([n] − S) is bad +w.r.t. (S, PS) if it is either heavy or conflicting. +12 + +Finally, in the proof of Lemma 3.5 we will define a set of subsets Σ ⊂ σk([n]) and +only consider projectors on subsets not in Σ. For a local assignment (S, PS) and every +{x1, . . . , xj} ∈ σj([n] − S) we therefore define LΣ +S,PS(x1, . . . , xj) ⊂ Hx1,...,xj to be the +space of all states |vx1,...,xj⟩ ∈ Hx1,...,xj such that +� +s∈σk({x1,...,xj}⊔S)−Σ +Πs(|vx1,...,xj⟩ ⊗ |φS⟩) = 0 +∀ |φS⟩ ∈ PS. +This is identical to (3.2), but with the projectors in Σ removed. The other definitions +above can be similarly generalised. +We first show that it is impossible to extend an assignment through more than 5(d2)k−1/ϵ +heavy qubits. +Definition 3.3. We define a chain of length m to be a local assignment ({y1, . . . , ym}, P1⊗ +· · · ⊗ Pm) such that, for each 2 ≤ l ≤ m, each qubit yl is heavy with respect to the as- +signment ({y1, . . . , yl−1}, P1 ⊗ · · · ⊗ Pl−1), and y1 is heavy with respect to the empty +assignment (∅, ∅). +Lemma 3.4. Let ({y1, . . . , yγ}, P1 ⊗ · · · ⊗ Pγ) be a chain of length γ(k, ϵ) := 5(4)k−1/ϵ. +Then every qubit x ∈ ([n] − {y1, . . . , yγ}) is conflicting w.r.t. the chain. +Proof. Let us construct the chain ({y1, . . . , yγ}, P1 ⊗ · · · ⊗ Pγ) starting from the empty +assignment (∅, ∅). We define +Wi := +k−1 +� +j=1 +|L{y1,...,yi},P1⊗···⊗Pi,j| nk−j−1. +The initial size of each |L∅,∅,j| is at most � +{v1,...,vj}∈σj([n]) 2j = +�n +j +� +2j ≤ 2j +j! nj. Therefore +the initial value of W0 = �k−1 +j=1 |L∅,∅,j|nk−j−1 is at most �k−1 +j=1 +2j +j! njnk−j−1 ≤ 2k−1nk−1. +When we extend to the qubit yi from the assignment ({y1, . . . , yi−1}, P1 ⊗ · · · ⊗ Pi−1) +we have +k−1 +� +j=1 +δ{y1,...,yi−1},P1⊗···⊗Pi−1,yi,Pi,j nk−j−1 = δ{y1,...,yi−1},P1⊗···⊗Pi−1,yi,Pi > ϵnk−1/5 +by definition of heaviness, and therefore Wi ≤ Wi−1 − ϵnk−1/5. So by the time we have +made 5dk−1/ϵ extensions we must have W = 0. In particular L{y1,...,yγ},P1⊗···⊗Pγ(x) = 0 +for all x ∈ ([n] − {y1, . . . , yγ}). +We now show that the ϵ-far condition implies that there will always be many bad qubits +with respect to any local assignment. +Lemma 3.5. Let ([n], {Πs}) be an instance of quantum k-SAT which is ϵ-far from sat- +isfiable by a product state. Let S ⊂ [n] and let PS be some local assignment to S. Then +there are at least ϵn/5 bad qubits w.r.t. (S, PS). +Proof. Suppose that there are fewer than ϵn/5 bad qubits w.r.t (S, PS). We will define a +subset Σ ⊂ σk([n]) satisfying (2) and such that |Σ| < ϵnk, contradicting the assumption. +This is a two-step process. +Let Xnb ⊂ ([n] − S) be the set of qubits which are +not bad w.r.t (S, PS). In the first step we will extend the assignment (S, PS) to the +qubits in Xnb one at a time, while adding k-subsets to Σ according to the following +prescription. Let x ∈ Xnb be the first qubit to which we extend; we choose some Vx ⊆ +LS,PS(x) minimising δS,PS,x,Vx, and extend the assignment to ({x} ⊔ S, Vx ⊗ PS). Let +us consider in turn each {x1, . . . , xj} ∈ σj([n] − ({x} ⊔ S)), where 1 ≤ j ≤ k − 1. +Clearly L{x}⊔S,Vx⊗PS(x1, . . . , xj) ⊆ LS,PS(x1, . . . , xj). If this inclusion is an equality we +add no k-subsets to Σ. However, if the inclusion is strict then we add all the subsets +13 + +s ∈ σk({x1, . . . , xj} ⊔ {x} ⊔ S) to Σ. Then for any states |vx1,...,xj⟩ ∈ LS,PS(x1, . . . , xj), +|vx⟩ ∈ Vx and |φS⟩ ∈ PS we have: +� +s∈σk({x1,...,xj}⊔{x}⊔S)−Σ +Πs(|vx1,...,xj⟩ ⊗ |vx⟩ ⊗ |φS⟩) += +� +s∈σk({x1,...,xj}⊔S) +Πs(|vx1,...,xj⟩ ⊗ |vx⟩ ⊗ |φS⟩) ++ +� +s∈σk({x1,...,xj}⊔{x}⊔S) +Πs(|vx1,...,xj⟩ ⊗ |vx⟩ ⊗ |φS⟩) +− +� +s∈σk({x1,...,xj}⊔{x}⊔S)) +Πs(|vx1,...,xj⟩ ⊗ |vx⟩ ⊗ |φS⟩) += +0. +Here the second equality is by definition of LS,PS(x1, . . . , xj). After adding these subsets +to Σ, we therefore have an equality LΣ +{x}⊔S,Vx⊗PS(x1, . . . , xj) = LS,PS(x1, . . . , xj). After +doing this for all the {x1, . . . , xj}, we enlarge the assignment to ({x} ⊔ S, Vx ⊗ PS). We +then repeat this process until we have extended to an assignment on Xnb ⊔ S. +Importantly, we claim that each qubit in Xnb remains not bad throughout this process, +even as the assignment is extended. Indeed, it is clear that none of these qubits will +ever conflict, since we add subsets to Σ so that LΣ(x) is fixed throughout. It remains +to show that none of these qubits will become heavy. +To see this, suppose that we +extend the assignment to ({y1, . . . ym} ⊔ S, Vy1 ⊗ · · · ⊗ Vym ⊗ PS), removing the relevant +subsets as above, and then extend to x ∈ (Xnb − ({y1, . . . ym} ⊔ S)) with subspace +Vx ∈ LΣ +{y1,...ym}⊔S,Vy1⊗···⊗Vym⊗PS(x) = LS,PS(x) (the equality is due to the addition of +subsets to Σ detailed above). Alternatively, we could have extended to x directly from +(S, PS), without extending through {y1, . . . , ym} first. It is sufficient to show that, for +any {x1, . . . , xj} ∈ [n] − ({x, y1, . . . , ym} ⊔ S), there is an inclusion: +L{x}⊔S,Vx⊗PS(x1, . . . , xj) ⊆ LΣ +{x,y1,...ym}⊔S,Vx⊗Vy1⊗···⊗Vym⊗PS(x1, . . . , xj). +(24) +Here the subspaces are taken prior to the removal of subsets following the extension to +qubit x; but on the RHS we have removed all the relevant subsets following extension to +{y1, . . . , ym}. +We will now prove the inclusion. Let |vx1,...,xj⟩ ∈ L{x}⊔S,Vx⊗PS(x1, . . . , xj). We will +consider two cases. The first is where j = k − 1. In the following equations Σ is defined +after extension to {y1, . . . , ym}. For all |vx⟩ ∈ Vx, |vyi⟩ ∈ Vyi and |φS⟩ ∈ PS, we have: +� +s∈{x1,...,xk−1}⊔{x}⊔{y1,...,ym}⊔S−Σ +Πs(|vx1,...,xk−1⟩ ⊗ |vx⟩ ⊗ |vy1⟩ ⊗ · · · ⊗ |vym⟩ ⊗ |φS⟩) += +� +s∈{x1,...,xk−1}⊔{y1,...,ym}⊔S−Σ +Πs(|vx1,...,xk−1⟩ ⊗ |vx⟩ ⊗ |vy1⟩ ⊗ · · · ⊗ |vym⟩ ⊗ |φS⟩) ++Π{x1,...,xk−1,x}(|vx1,...,xk−1⟩ ⊗ |vx⟩ ⊗ |vy1⟩ ⊗ · · · ⊗ |vym⟩ ⊗ |φS⟩) += +0. +Here the first term on the second line is zero since Σ is defined so that +LΣ +{y1,...ym}⊔S,Vy1⊗···⊗Vym⊗PS(x1, . . . , xk−1) = LS,PS(x1, . . . , xk−1); +the second term is zero by definition of L{x}⊔S,Vx⊗PS(x1, . . . , xk−1). Therefore |vx1,...,xj⟩ ∈ +L{x,y1,...ym}⊔S,Vx⊗Vy1⊗···⊗Vym⊗PS(x1, . . . , xj) and we obtain the inclusion (24). +On the other hand, suppose that j < k − 1. Then for all |vx⟩ ∈ Vx, |vyi⟩ ∈ Vyi and +|φS⟩ ∈ PS we have the following equation: +� +s∈{x1,...,xk−1}⊔{x}⊔{y1,...,ym}⊔S−Σ +Πs(|vx1,...,xk−1⟩ ⊗ |vx⟩ ⊗ |vy1⟩ ⊗ · · · ⊗ |vym⟩ ⊗ |φS⟩) +14 + += +� +s∈{x1,...,xk−1}⊔{y1,...,ym}⊔S−Σ +Πs(|vx1,...,xk−1⟩ ⊗ |vx⟩ ⊗ |vy1⟩ ⊗ · · · ⊗ |vym⟩ ⊗ |φS⟩) ++ +� +s∈{x1,...,xk−1,x}⊔{y1,...,ym}⊔S−Σ +Πs(|vx1,...,xk−1⟩ ⊗ |vx⟩ ⊗ |vy1⟩ ⊗ · · · ⊗ |vym⟩ ⊗ |φS⟩) += +0. +Here the terms in the second line are both zero since Σ is defined so that: +LΣ +{y1,...ym}⊔S,Vy1⊗···⊗Vym⊗PS(x1, . . . , xk−1) = LS,PS(x1, . . . , xk−1), +LΣ +{y1,...ym}⊔S,Vy1⊗···⊗Vym⊗PS(x1, . . . , xk−1, x) = LS,PS(x1, . . . , xk−1, x). +Therefore |vx1,...,xj⟩ ∈ LΣ +{x,y1,...ym}⊔S,Vx⊗Vy1⊗···⊗Vym⊗PS(x1, . . . , xj) and we obtain the +inclusion (24). We have shown that a qubit in Xnb cannot become bad upon extension. +The first step ends when we have extended to an assignment on Vnb ⊔ S. In the +second step we add to Σ all subsets containing any qubit in [n] − (Vnb ⊔ S) and then +extend the assignment to the whole of [n]. We thus obtain a product state |φ⟩ ∈ (C2)⊗n +satisfying (2). +To finish we must show that we added less than ϵnk subsets to Σ. In the first step, +after extending to a qubit x we added all the subsets s ∈ σk({x1, . . . , xj} ⊔ x ⊔ S) each +time that |L{x}⊔S,Vx⊗PS(x1, . . . , xj)| was smaller than |LS,PS(x1, . . . , xj)|. This is at most +� +n +k−j−1 +� +≤ nk−j−1 subsets each time, since |S| ≤ n. Since the qubit x was never heavy, +we therefore added at most �k−1 +j=1 δS,PS,x,Vx,jnk−j−1 = δS,PS,x,Vx ≤ ϵnk−1/5 projectors +every time we extended. As there are at most n qubits in Xnb, in the first step we added +less than ϵnk/5 subsets to Σ. +In the second step, we added all the subsets which contained a qubit not in S ⊔ Vnb. +Since by assumption there were ≤ ϵn/5 bad qubits to begin with, we therefore added at +most (ϵn/5) +� n +k−1 +� +≤ ϵnk/5 subsets in the second step. Altogether we added less than ϵnk +subsets, as claimed. +For discretisation we will also need the following lemma. For a state |φ⟩ ∈ C2, we write +⟨|φ⟩⟩ ∈ P1 for the corresponding point on the complex projective line (a.k.a. the Bloch +sphere). +Lemma 3.6. Let L ⊆ (C2)⊗m be a subspace. We define +X := {x ∈ P1 : ∃{|φi⟩ ∈ C2}m +i=1 s.t. |φ1⟩ |φ2⟩ . . . |φm⟩ ∈ L and ⟨|φ1⟩⟩ = x}. +Then either (i) X = P1, or (ii) #(X) ≤ m!. The bound in case (ii) is tight. +Proof. It is straightforward to show that X is either finite or covers the whole space. +Indeed, the subspace L corresponds to a projective linear subspace P(L) ⊆ P2m−1. By the +Segre embedding the product states form a projective subvariety Σm := P1 × · · · × P1 ⊂ +P2m−1. But it is an elementary result in algebraic geometry that the projection map +π1 : P1 × · · · × P1 → P1 is closed in the Zariski topology, since every projective variety is +complete [Har13, §2 Thm. 4.9]. The intersection P(L) ∩ Σm is an algebraic set in Σm, +and therefore by closure of the projection map, the image under the projection will be +an algebraic set in P1. This set will either be zero-dimensional, in which case it is a finite +set of points; or one-dimensional, in which case it must be the whole of P1. +The only thing left is to bound the number of points in the case where the image +is zero-dimensional. +We first note that for any variety S ⊂ Σm, the image π1(S) is +irreducible; indeed, were it not irreducible then by continuity of π1 the fibres would +give a nontrivial decomposition of S. +If π1(S) ̸= P1, it must therefore be a single +point. It follows that all we need to do is bound the number of irreducible components +of P(L) ∩ Σm. We know that P(L) is an intersection of hyperplanes {Pi}2m−|L| +i=1 +. We +use [Har13, §1 Thm. 7.7], which yields the following statement in our special case. For +15 + +any subvariety V ⊂ P2m−1 of dimension ≥ 1, and for any hyperplane P ⊂ P2m−1 not +containing V , let Z1, . . . , Zs be the irreducible components of V ∩ P; then: +s +� +j=1 +deg(Zj) ≤ deg(V ). +(25) +Here deg(Zj), deg(V ) ∈ N>0 are the degrees of the varieties in P2m−1. If P contains V +then the intersection is just V again. We observe that the irreducible components of +P(L) ∩ Σm can be obtained by the following process: take the intersection Σm ∩ P1; then +for each of the irreducible components of that intersection take the intersection with P2; +then for each of the irreducible components of that intersection take the intersection with +P3; etc. Since deg(Σm) = m!, by (25) we finish with at most m! irreducible components +(which would all have degree 1). The bound follows. +To see that the bound is tight, observe that by definition of the degree, the variety +Σm intersects a generic linear subvariety P(V ) ⊂ P2m−1 of dimension 2m − (m + 1) in +precisely m! points, and generically these points will be mapped to m! different points in +P1 by the projection π1. +We can now prove the theorem. +Definition 3.7. Let (S, PS) be some local assignment. For any X ⊆ ([n] − S) we define +LS,PS(X) ⊆ HX to be the space of all states |vX⟩ ∈ HX such that +� +s∈σk(X⊔S) +Πs(|vX⟩ ⊗ |φS⟩) = 0 +∀ |φS⟩ ∈ PS. +(26) +(The difference between this and (23) is that all the subsets of X ⊔ S are included, not +just those containing every element of X.) +Theorem (Restatement of Theorem 1.7). There exists a constant c(k, ϵ) (i.e. not de- +pending on n) such that the following holds. Let ([n], {Πs}) be any instance of quantum +k-SAT which is ϵ-far from satisfiable by a product state. Then, for a randomly chosen +subset C ∈ σc(k,ϵ)([n]), the instance is locally unsatisfiable by a product state at C with +probability p > 0.75. +Proof of Theorem 1.7. The idea of the proof is as follows. We will build a randomly +chosen subset C ⊆ [n] of qubits by picking qubits at random from [n], one by one. For +any local assignment on some subset of the qubits which have already been picked, we +know by Lemma 3.5 that the next qubit we pick has an ϵ/5 chance of being bad w.r.t. +that assignment. By varying the local assignment we consider before picking each qubit, +we will show that, after enough qubits have been picked, there can be no local assignment +to all the qubits in C with high probability. This is essentially a backtracking argument; +we know by Lemma 3.4 that we cannot extend a chain through more than γ(k, ϵ) heavy +variables, so we simply build such a maximal chain and gradually eliminate all the possible +assignments to that chain. We have not endeavoured to optimise this procedure and it +can certainly be improved, although whether one can bring the constant c(k, ϵ) down to +the level in the classical setting [AS03, Thm. 1.1] using such a backtracking approach is +by no means clear. +The key to our backtracking argument is the ability to eliminate possible assignments +to a chain. We formalise this as follows, assuming for the time being that we can pick +bad qubits each time; of course this will not be the case when we pick qubits at random, +but in the end we will simply ignore those we pick that are not bad. Suppose that, +given any chain ({y1, . . . , ym}, P1 ⊗ · · · ⊗ Pm) of length m, there exists an elimination +procedure, which is defined by a rule in the following way. The procedure will define a +set Γ ⊆ ([n] − {y1, . . . , ym}) of picked qubits; at the beginning of the procedure Γ = ∅. +At each step of the procedure, we: +• Choose some local assignment (S, PS) on a subset S ⊂ {y1, . . . , ym} ⊔ Γ, according +to the rule. +16 + +• Pick at random a qubit x ∈ ([n] − ({y1, . . . , ym} ⊔ Γ)) which is bad w.r.t. the +assignment (S, PS). +• Add the qubit x to Γ. +The rule defines an elimination procedure with constant φ(m) ∈ N if, while |Γ| ≤ φ(m), +we can stop the procedure and conclude that there is no local assignment to {y1, . . . , ym}⊔ +Γ which assigns P1 ⊗ · · · ⊗ Pm−1 to the qubits {y1, . . . , ym−1}. +We claim that if an elimination procedure exists for chains of length m with constant +φ(m), then there also exists an elimination procedure for chains of length m − 1 with +constant φ(m − 1) = ((φ(m) + 2)! + 1)(φ(m) + 1). The procedure is defined as follows. +Let ({y1, . . . , ym−1}, P1 ⊗ · · · ⊗ Pm−1) be a chain of length m − 1, and let |pm−1⟩ ∈ Pm−1 +be any state. We pick at random a bad qubit y0 +m ∈ ([n] − {y1, . . . , ym−1}) with respect +to the local assignment ({y1, . . . , ym−1}, P1 ⊗ · · · ⊗ Pm−2 ⊗ |pm−1⟩) and add it to Γ. If it +is conflicting, then the procedure is complete. If it is heavy, then we perform the length +m elimination procedure on the chain ({y1, . . . , ym−1, y0 +m}, P1 ⊗ · · · ⊗ Pm−2 ⊗ |pm−1⟩ ⊗ +L{y1,...,ym−1},P1⊗···⊗Pm−2⊗|pm−1⟩(y0 +m)). This gives us a set Γ0 +m ⊆ ([n] − {y1, . . . , y0 +m}) of +qubits, where |Γ0 +m| ≤ φ(m), such that there is no local assignment to {y1, . . . , y0 +m} ⊔ Γ0 +m +which assigns P1 ⊗ · · · ⊗ Pm−2 ⊗ |pm−1⟩ to the qubits {y1, . . . , ym−1}. We add the qubits +in Γ0 +m to Γ. Now consider the following subspace: +L := L{y1,...,ym−2},P1⊗···⊗Pm−2(ym−1, y0 +m, Γ0 +m) ⊆ Hym−1 ⊗ Hy0m ⊗ HΓ0m ∼= C2 ⊗ C2|Γ0 +m|+1. +Let X = {⟨|φ0⟩⟩ ∈ P1 +: +|φ0⟩ ⊗ |φ1⟩ ⊗ · · · ⊗ |φ|Γ0m|+1⟩ ∈ L}. Since there is no local +assignment to {y1, . . . , y0 +m} ⊔ Γ0 +m which assigns P1 ⊗ · · · ⊗ Pm−2 ⊗ |pm−1⟩ to the qubits +{y1, . . . , ym−1}, we know that X ̸= P1, since ⟨|pm−1⟩⟩ /∈ X. But then, by Lemma 3.6, we +have #(X) ≤ (|Γ0 +m| + 2)!. The worst case is where this is an equality: in this case let the +corresponding states be {|pi +m−1⟩ ∈ Hym−1}(|Γ0 +m|+2)! +i=1 +. Now for each 1 ≤ i ≤ (|Γ0 +m| + 2)! in +turn we pick at random a qubit yi +m ∈ ([n] − Γ) which is bad with respect to the local +assignment ({y1, . . . , ym−1}, P1 ⊗ · · · ⊗ Pm−2 ⊗ |pi +m−1⟩) and add this qubit to Γ. These +qubits {yi +m}(|Γ0 +m|+2)! +i=1 +are either conflicting or heavy with respect to their corresponding +local assignments; in the worst case they are all heavy. In this case, we consider each of +the following chains in turn: +({y1, . . . , ym−1, ym}, P1 ⊗ · · · ⊗ Pm−2 ⊗ |pi +m−1⟩ ⊗ L{y1,...,ym−1},P1⊗···⊗Pm−2⊗|pi +m−1⟩(yi +m)). +For each of these chains in turn we perform the length m elimination procedure, thereby +obtaining a set Γi +m ⊂ ([n] − Γ) of qubits, where |Γi +m| ≤ φ(m), which we add to Γ. +By definition of the length m elimination procedure, for each Γi there is no local as- +signment to {y1, . . . , yi +m} ⊔ Γi +m which assigns P1 ⊗ · · · ⊗ Pm−2 ⊗ |pi +m−1⟩ to the qubits +{y1, . . . , ym−1}. We now stop the procedure and conclude that there is no local assign- +ment to {y1, . . . , ym−1} ⊔ Γ which assigns P1 ⊗ · · · ⊗ Pm−2 to the qubits {y1, . . . , ym−2}. +Since Γ = �(|Γ0 +m|+2)! +i=0 +(Γi +m⊔{yi +m}), we see that |Γ| ≤ ((φ(m)+2)!+1)(φ(m)+1) = φ(m−1). +We now observe that an elimination procedure exists for chains of length γ := γ(k, ϵ), +with constant φ(γ) = 1. Indeed, for any such chain ({y1, . . . , yγ}, P1 ⊗ · · · ⊗ Pγ), as soon +as we add any qubit to Γ we can conclude by Lemma 3.4 that there is no local assignment +to {y1, . . . , yγ} ⊔ Γ which assigns P1 ⊗ · · · ⊗ Pγ−1 to the qubits {y1, . . . , yγ−1}. It follows +that an elimination procedure exists for every 1 ≤ m ≤ γ, with constant φ(m) defined +by the recurrence relation φ(m − 1) = ((φ(m) + 2)! + 1)(φ(m) + 1). +Finally, we observe that this gives rise to an elimination procedure for the empty +assignment (∅, ∅), with constant φ(0) = φ(1) + 1. This procedure is as follows: pick a +qubit x ∈ [n] which is bad w.r.t. (∅, ∅) and add it to Γ. If it conflicts then there is no +local assignment on Γ = {x}. If it is heavy, then use the length 1 elimination procedure +for the chain ({x}, L∅,∅(x)) to obtain a set Γ1 ⊂ ([n]−{x}), where |Γ1| ≤ φ(1), such that +there is no local assignment on {x} ⊔ Γ1. +The proof of the theorem now reduces to showing that if we pick qubits C = +{x1, . . . , xc} ⊆ [n] at random, one at a time, then they will implement the elimina- +tion procedure for the empty assignment with high probability. Indeed, we recall from +17 + +Lemma 3.5 that, with respect to any local assignment (S, PS), there are more than ϵn/5 +bad qubits in ([n] − S). For large enough n, every time we pick a new qubit there is +therefore a roughly ϵ/5 chance of it being bad w.r.t. any local assignment. (The actual +value will generally be slightly lower than ϵ/5 because we cannot pick any of the bad +qubits that lie in ([n] − S) ∩ ˜C, where ˜C is the set of qubits which have already been +picked; this is why we need n to be large enough.) 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Monogamy of quantum entanglement +and other correlations. +Physical Review A, 69(2):022309, 2004. +arXiv: +quant-ph/0310037, doi:10.1103/PhysRevA.69.022309. +[LLM+10] Christopher R Laumann, AM L¨auchli, R Moessner, A Scardicchio, and +Shivaji Lal Sondhi. Product, generic, and random generic quantum satis- +fiability. +Physical Review A, 81(6):062345, 2010. +arXiv:0910.2058, doi: +10.1103/PhysRevA.81.062345. +[LMSS10] C. R. Laumann, R. Moessner, A. Scardicchio, and S. L. Sondhi. Random +quantum satisfiability. Quantum Information and Computation, 10(1):1–15, +Jan 2010. +[VZGG13] Joachim Von Zur Gathen and J¨urgen Gerhard. Modern Computer Algebra. +Cambridge University Press, Cambridge, 2013. +19 + +A +Proofs of combinatorial lemmas +Lemma (Restatement of Lemma 2.1). Let θ, x ∈ R≥0 and c ∈ N, where 0 ≤ θ ≤ 1 and +x ≥ c. Then, for some α ∈ [0, 1), we have: +θ +�x +c +� += +�θ1/cx +c +� ++ α +�θ1/cx +c − 1 +� +< +�θ1/cx + 1 +c +� +. +(27) +Proof. Setting m = x − l for l < x, we have: +θ +�x +c +� +− +�m +c +� +� m +c−1 +� += θx · · · (x − c + 1) +cm · · · (m − c + 2) − m − c + 1 +c +. +Now, if we set m = θ1/cx, we obtain: +α := θ +�x +c +� +− +�θ1/cx +c +� +�θ1/cx +c−1 +� += +1 +c +� +θx · · · (x − c + 1) +θ1/cx · · · (θ1/cx − c + 2) − (θ1/cx − c + 1) +� += +1 +c +� +θ1/cx · · · (x − c + 1) +x(x − θ−1/c) . . . (x − θ−1/c(c − 2)) − (θ1/cx − c + 1) +� +≤ +1 +c (θ1/c(x − c + 1) − (θ1/cx − c + 1)) += +(c − 1)(1 − θ1/c) +c +< 1. +The inequality then follows from the standard binomial recurrence relation +�x +c +� ++ +� x +c−1 +� += +�x+1 +c +� +. +Lemma (Restatement of Lemma 2.4). If G is an l-regular hypergraph with edge density +θ, then the k-shadow G|k has edge density θ|k ≥ (θ1/l − 2/n)k. +Proof. The number of edges in G|k is the minimum number of k-edges in a k-regular hy- +pergraph such that all the edges in G appear as complete l-vertex subgraphs of that graph; +for a given number of k-edges we may therefore ask what is the optimal arrangement +maximising the number of complete l-vertex subgraphs. It follows from [Kee08, Thm. +1] that a complete k-regular hypergraph on a subset S ⊆ [n] is an optimal arrangement. +It then follows from Lemma 2.1 that G|k has at least +�⌊θ1/ln⌋ +k +� +edges. Observing that +⌊θ1/ln⌋ ≥ θ1/ln−1 = (θ1/l−2/n)n+1 and using the inequality in (5) gives the result. +Lemma (Restatement of Lemma 2.5). Let n, c ∈ N be such that l := n/2c + δ and +k := l/2 + ϵ, where −3 ≤ δ < −1 and −1/2 ≤ ϵ ≤ 0, are natural numbers. Let G be an +l-regular hypergraph on n vertices with edge density θ ∈ [0, 1]. +Suppose that n ≥ 3(2c) and n ≥ 3(2c+1) +� +1 +2 + +1 +ln(A) +� +A1/3 +4 +− ln(θ) +�� +for some constant +A > 1. Then the edge density θ|k,1/2 of a partial (k, 1/2)-shadow of G has the following +lower bound: +θ|k,1/2 ≥ +θ +KA +:= +θ +2 +� +A exp(A1/3/2) +Proof. We first fix some notation; let ω = 1/2, α = 1/2c, and β = 1/2c+1; then l = +(α +δ/n)n and k = (β +ϵ/n)n. Let θ′ be the edge density of a k-regular hypergraph H′. +We will find the largest edge density θ of an l-regular hypergraph H which has H′ as a +(k, ω)-shadow. We observe that we can add an l-edge to H precisely when at least ω +�l +k +� +of the k-subsets of that l-edge are k-edges of H′. We thus have the following equation +for the maximum number of edges in H. Let S(H′) ⊆ [n] be the set of vertices which are +contained in some k-edge of H′. For 0 ≤ i ≤ |S(H′)|, let Ni be the number of subsets of +S(H′) of size i such that the induced subgraph on that subset has ≥ ω +�l +k +� +edges. Then: +θ +�n +l +� += +l +� +i=0 +Ni +�n − |S(H′)| +l − i +� +. +(28) +20 + +We now observe that Ni = 0 for all i ̸= l. Indeed, consider Nl−1. The required density of +edges in a subset of H′ of size l−1 which contains ω +�l +k +� +edges is ω +�l +k +� +/ +�l−1 +k +� += ωl/(l−k) = +ω(α+δ/n) +α+δ/n−(β+ϵ/n), which is greater than 1 for all choices of −3 ≤ δ < −1, −1/2 ≤ ϵ ≤ 0. We +therefore have Nl−1 = 0. The sum (28) therefore reduces to a much simpler equation: +θ = Nl +�n +l +�. +(29) +We immediately observe that as soon as θ′ > ω then we can obtain θ = 1 by distributing +the edges uniformly over the n vertices, giving Nl = +�n +l +� +. +Assuming from now on that θ′ < ω, we make some further observations which will +enable us to bound θ. To maximise Nl the best distribution of k-edges is a uniform +distribution over as many vertices as possible, with density ω. If we multiplied the edge +density by +1 +ω, we could form the complete hypergraph on these vertices, and so we see +by Lemma 2.1 that the maximum number of vertices is upper bounded by ⌈(θ′/ω)1/kn⌉. +We therefore obtain the following bound: +Nl ≤ +�⌈(θ′/ω)1/kn⌉ +l +� +≤ +� +(θ′/ω)1/k + 1 +n +�l �n +l +� += +� +(θ′/ω) +1 +βn+ϵ + 1 +n +�αn+δ �n +l +� +. +We now claim that, if n > 3(2c) and furthermore n ≥ 3(2c+1) +ln(A) ln(1/(2θ′)) for some constant +A > 1, then: +Nl +�n +l +� ≤ A(2θ′)2 exp +� +αA1/3� +. +(30) +Indeed, note that 0 < θ′ < ω < 1; thus, since n > 3(2c) implies that αn + δ > 0: +� +(θ′/ω) +1 +βn+ϵ + 1 +n +�αn+δ +≤ +� +(θ′/ω) +1 +βn + 1 +n +�αn+δ += +(θ′/ω) +αn+δ +βn +� +1 + 1 +n(ω/θ′) +1 +βn +�αn+δ +≤ +(θ′/ω) +αn−3 +βn +� +1 + 1 +n(ω/θ′) +1 +βn +�αn +≤ +(θ′/ω) +αn−3 +βn exp +� +α(ω/θ′) +1 +βn +� += +(θ′/ω) +α +β (ω/θ′) +3 +βn exp +� +α(ω/θ′) +1 +βn +� +. +Substituting (30) into (29) and inverting the inequality, we obtain the statement in +the theorem. +Lemma (Restatement of Lemma 2.6). Let n ∈ N be odd. We construct an (n − 1)/2- +regular hypergraph G as follows. For every subset X ∈ σ(n−1)/2([n]), we do precisely one +of the following: +1. Add X as an edge to G. +2. Add all the subsets in σ(n−1)/2(X) as edges to G. +The edge density θ of a hypergraph G constructed in this way satisfies θ ≥ 1/2. +Proof. We call a choice of 1 or 2 for each X ∈ σ(n−1)/2([n]) a configuration. For a given +configuration, let S1 ⊂ σ(n−1)/2([n]) be the subsets for which option 1 is chosen, and let +S2 ⊂ σ(n−1)/2([n]) be the subsets for which option 2 is chosen. We observe that there is +always a configuration minimising the number of edges in G (a minimal configuration) +which satisfies the property that, for every X ∈ S2, σ(n−1)/2(X) ⊂ S1. Indeed, suppose +this is not the case, and there is some Y ∈ σ(n−1)/2(X) ⊂ S2. But then one can change +the configuration by moving Y to S1 without increasing the number of edges in G. If we +21 + +start with a minimal configuration, by doing this for all X ∈ S2, we obtain a minimal +configuration satisfying the desired property. +Let us calculate the minimal number of edges when such a configuration is chosen. +Let C be the (n−1)/2-regular hypergraph on those vertices of G contained in an edge of +S1, whose edges are precisely the subsets in S1. Every Y ∈ S2 corresponds to a induced +complete hypergraph on (n + 1)/2 vertices of C. By Lemma 2.4, the arrangement of +(n−1)/2-edges maximising the number of induced complete hypergraphs is the complete +hypergraph on C. We have Y ⊂ C for every Y ∈ S2, which implies that C ⊂ Y . We +therefore obtain the following inequality giving a lower bound on the size of C: +� +|C| +(n − 1)/2 +� ++ +� +|C| +(n + 1)/2 − (n − |C|) +� +≥ +� +n +(n − 1)/2 +� +. +Setting |C| = n − 2, we find the inequality is violated; but setting |C| = n − 1, we find +the equality is satisfied. The minimal number of edges in G is therefore +� +n−1 +(n−1)/2 +� += +n+1 +2n +� +n +(n−1)/2 +� +, and the result follows. +Lemma (Restatement of Lemma 2.7). Let G be a k-regular hypergraph on n vertices +with edge density θ, where k = βn for some β ∈ [0, 1]. Let N be the degree of a vertex +picked uniformly at random, and let τ := N/ +�n−1 +k−1 +� +. Then, for any ϵ > 0 and α ∈ (0, 1): +Pr [|θ − τ| ≥ min(∆(α, β, n), (1 − θ)(1 − α))] ≤ α, +where +∆(α, β, n) = +5 exp +� +1 +12α(1−α)n + +1 +12β(1−β)n +� +� +2πα(1 − α)β(1 − β)n +=: 5(Lα,β)1/n +Mα,β +√n . +Proof. We prove the case where τ < θ; the case τ > θ follows by considering the com- +plement of the hypergraph. For any hypergraph G with edge density θ, let C ⊂ [n] be +the subset of vertices whose degree is less than or equal to �τ +�n−1 +k−1 +� +, where �τ < θ; clearly +Pr +� +� +N ≤ �τ +�n−1 +k−1 +�� += |C|/n, where � +N is the degree of a vertex of G picked uniformly at +random. Let S := � +v∈C dv, where dv is the degree of the vertex v ∈ C; we must have +S ≤ |C|�τ +�n−1 +k−1 +� +. We will obtain a lower bound for �τ in terms of θ and |C| by defining +a configuration of edges which minimises S. Since we are interested in how the lower +bound changes as n increases, we will consider |C| to be some function of n; but θ will +remain constant. The configuration is defined as follows: we first place all the edges +contained entirely in (n − |C|), which does not increase S. Then we place all the edges +which only contain one vertex in C, at the cost S �→ S + 1. Then we can place all the +edges which only contain two vertices of C, at the cost S �→ S + 2, etc. We will stop +when we have placed all θ +�n +k +� +edges. Let r + 1 be the number of vertices of C in the final +edge we place. The following inequality encodes the fact that we have placed θ +�n +k +� +edges: +r +� +i=0 +�n − |C| +k − i +��|C| +i +� +≤ θ +�n +k +� +≤ +r+1 +� +i=0 +�n − |C| +k − i +��|C| +i +� +⇒ +P(Xn−|C|,|C|,k ≤ r) ≤ θ ≤ P(Xn−|C|,|C|,k ≤ r + 1). +(31) +Here Xn−|C|,|C|,k is a random variable distributed according to the classical hypergeo- +metric distribution with parameters n, |C|, k. The following inequality encodes the fact +that S ≤ |C|�τ +�n−1 +k−1 +� +: +r +� +i=0 +i +|C| +�n − |C| +k − i +��|C| +i +� +≤ �τ +�n − 1 +k − 1 +� +≤ +r+1 +� +i=0 +i +|C| +�n − |C| +k − i +��|C| +i +� +⇒ +P(Xn−|C|,|C|−1,k−1 ≤ r − 1) ≤ �τ ≤ P(Xn−|C|,|C|−1,k−1 ≤ r). +(32) +22 + +Here Xn−|C|,|C|−1,k−1 is a random variable distributed according to the classical hyper- +geometric distribution with parameters n − 1, |C| − 1, k − 1. It follows that: +P(Xn−|C|,|C|,k ≤ r) − P(Xn−|C|,|C|−1,k−1 ≤ r) +≤ θ − �τ +≤ P(Xn−|C|,|C|,k ≤ r + 1) − P(Xn−|C|,|C|−1,k−1 ≤ r − 1). +(33) +For some intuition, we recall the standard model for the classical hypergeometric distri- +bution with parameters n, |C|, k: there are n balls, |C| of which are white and (n − |C|) +of which are black, and we pick k of them at random without replacement; the hyperge- +ometric distribution counts the number of white balls picked. We observe that +P(Xa,b,c ≤ r + 1) = a + b − c +a + b +P(Xa,b−1,c ≤ r + 1) + +c +a + bP(Xa,b−1,c−1 ≤ r). +(34) +This equation is obtained by conditioning on whether a specific white ball is picked. We +further observe: +P(Xa,b,c ≤ r) = P(Xa,b,c−1 ≤ r − 1) + a + r − (c − 1) +a + b − (c − 1)P(Xa,b,c−1 = r). +(35) +This equation is obtained by conditioning on how many of the first c − 1 balls picked +are white. Recall k = βn; let |C| = αn. Then, substituting (34) and (35) into (33), we +obtain: +θ − �τ = P(Xn−|C|,|C|−1,k−1 = r) + n − |C| + r + 2 − k +n +P(Xn−|C|,|C|−1,k−1 = r + 1) +≤ P(Xn−|C|,|C|−1,k−1 = r) + (4 − α − β)P(Xn−|C|,|C|−1,k−1 = r + 1). +For the inequality we used that r < n. The mode of Xn−|C|,|C|−1,k−1 is M := ⌊ k|C| +n+1⌋, so +by Stirling’s approximation we have: +θ − �τ ≤ 5 +�n−|C| +k−M +��|C| +M +� +�n +k +� +≤ +5ef(n) +� +2παβ(1 − α)(1 − β)n +≤ +5 exp +� +1 +12α(1−α)n + +1 +12β(1−β)n +� +� +2πα(1 − α)β(1 − β)n +, +(36) +where +f(n) += +1 +12αn + +1 +12βn + +1 +12(1 − α)n + +1 +12(1 − β)n − +1 +12n + 1 − +1 +12αβn + 1 +− +1 +12(1 − α)βn + 1 − +1 +12(1 − α)(1 − β)n + 1 − +1 +12α(1 − β)n + 1. +The free variables are α and �τ, so we fix α and determine the minimal �τ. Since +Pr +� +� +N ≤ �τ +�n − 1 +k − 1 +�� += |C| +n = α, +and, by taking the sum of the degrees of all the vertices, +θ ≤ (1 − α) + α�τ +⇒ +θ − �τ ≤ (1 − θ)(1 − α), +we obtain the bound in the statement of the lemma. +23 + diff --git a/SNFGT4oBgHgl3EQfgygC/content/tmp_files/load_file.txt b/SNFGT4oBgHgl3EQfgygC/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c83b789b407b20cbbf003ec8bd702e16da417934 --- /dev/null +++ b/SNFGT4oBgHgl3EQfgygC/content/tmp_files/load_file.txt @@ -0,0 +1,1418 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf,len=1417 +page_content='Testing quantum satisfiability Ashley Montanaro∗1,2, Changpeng Shao†1, and Dominic Verdon‡1 1School of Mathematics, University of Bristol, UK 2Phasecraft Ltd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=', UK January 26, 2023 Abstract Quantum k-SAT (the problem of determining whether a k-local Hamiltonian is frustration-free) is known to be QMA1-complete for k ≥ 3, and hence likely hard for quantum computers to solve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Building on a classical result of Alon and Shapira, we show that quantum k-SAT can be solved in randomised polynomial time given the ‘property testing’ promise that the instance is either satisfiable (by any state) or far from satisfiable by a product state;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' by ‘far from satisfiable by a product state’ we mean that ϵnk constraints must be removed before a product state solution exists, for some fixed ϵ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' The proof has two steps: we first show that for a satisfiable instance of quantum k-SAT, most subproblems on a constant number of qubits are satisfiable by a product state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We then show that for an instance of quantum k-SAT which is far from satisfiable by a product state, most subproblems are unsatisfiable by a product state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Given the promise, quantum k-SAT may therefore be solved by checking satisfiability by a product state on randomly chosen subsystems of constant size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' 1 Introduction Property testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' In computer science, one often needs to determine whether a struc- ture defined by a large quantity of data possesses a certain property which is global, in the sense that it is defined with reference to all the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Property testing algorithms aim to determine with high probability whether a structure possesses a global property by performing local checks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' that is, checking properties of randomly chosen small subsets of the data defining the structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Since the time complexity of a local check is smaller than that of a global check, and one only inspects a small subset of the data, this approach, when feasible, often leads to algorithms with excellent time and query complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Of course, in general the structure may not possess the global property, but be close enough to possessing it that one is highly unlikely to perform a local check which witnesses this fact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' For this reason property testing algorithms usually require a promise that the structure either possesses the property or is far from possessing the property, where distance from possessing the property is defined by a measure specific to the problem and is usually quantified by some constant ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' In [AS03], Alon and Shapira showed that the NP-hard problem of classical k-SAT (satisfiability of Boolean functions on n variables, where each function depends on exactly k variables) is amenable to a property testing approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Specifically, with the promise that an instance of classical k-SAT is either satisfiable or far from satisfiable, satisfiability may be determined in randomised constant time, by choosing constant-sized subsets of variables and checking satisfiability of those functions which depend only on variables in those subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Because of the promise, their result is limited to dense instances of k-SAT, where there are Ω(nk) functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' ∗ashley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='montanaro@bristol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='uk †changpeng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='shao@bristol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='uk ‡dominic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='verdon@bristol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='uk 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='10699v1 [quant-ph] 25 Jan 2023 Testing quantum satisfiability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' In [Bra11] Bravyi defined a quantum analogue of classical k-SAT, which we will here call quantum k-SAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' An instance of this problem is defined by n Hilbert spaces of dimension 2 (qubits) and, for every subset s of k qubits, a projector Πs on the Hilbert space of those qubits;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' we say that the instance is satisfiable if there is a state of all n qubits which is in the kernel of all of the projections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' While classical k-SAT is NP-complete whenever k ≥ 3, quantum k-SAT is QMA1-complete whenever k ≥ 3, where QMA1 is the quantum analogue of the class MA with one- sided error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' In the terminology of physics quantum k-SAT is precisely the problem of determining whether a k-local Hamiltonian on n qubits is frustration free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Since the problem seems very difficult in general, previous work on algorithms for quantum k-SAT has focused on tractable special cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' There are many interesting results on random quantum k-SAT, for instance [LLM+10, LMSS10, BMR10, HLL+13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' It has been shown that quantum 2-SAT can be solved in linear time [dBG16, ASSZ18];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' moreover, in this case there is always a satisfying state which is a product of one- and two-qubit states [CCD+11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' The case where there are relatively few nontrivial projectors all of rank 1 is also often tractable [AKS12, AdGS21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' In this work we also treat a special case: the dense case, where the interaction hyper- graph (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' the k-regular hypergraph on n vertices whose edges are subsets of k qubits whose associated projection is nontrivial) contains Ω(nk) edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We show that the re- sults of [AS03] extend to the quantum setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Given the promise that the instance is either satisfiable or far from satisfiable by a product state, quantum k-SAT may be solved in randomised polynomial time in n by checking satisfiability by a product state on constant-sized subsets of qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' (The polynomial dependence on n is due to the fact that, in the data specifying an instance of quantum k-SAT, the projectors are given to poly(n) precision, so all computations involving them will pick up a poly(n) factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=') Some comments on our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Before stating our results formally, we will make two comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Firstly, as we emphasised in the last paragraph, the promise is that the instance, if unsatisfiable, is far from satisfiable by a product state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Obviously, this is a less stringent requirement than being far from satisfiable by any state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' The reason we are able to weaken the promise in this way is that an instance of quantum k-SAT which is satisfiable by any state is locally satisfiable by a product state with high prob- ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' This result (Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='5) is essentially a fact about quantum states which may be of independent interest;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' there are some connections with monogamy of entangle- ment [KW04, CW04, BCY11] and de Finetti theorems [BH16, BH13] which we discuss briefly in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Secondly, our proofs are combinatorial in nature;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' we make no use of the norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' In particular, we do not require the polynomial lower bound on the ground state energy in the unsatisfiable case that is usually given as a promise in the definition of quantum k-SAT;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' our testing algorithm therefore solves a harder problem than quantum k-SAT as it is usually defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='1 Results We will first state the three basic definitions we will use in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Throughout we write σk(S) for the set of k-element subsets of a set S and [n] for the set {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' For any subset x ∈ σk(S) we use the notation x := (S − x) for the complement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='1 (Quantum k-SAT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' An instance of quantum k-SAT (without a lower bound on the ground state energy) is defined by the following data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' A number of qubits n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' For every k-element subset s ∈ σk([n]), a projector Πs on (C2)⊗n which acts non- trivially only on the qubits in s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' These projectors are given as matrices whose entries have precision polynomial in n (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' they are specified by poly(n) bits).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' 2 We write the instance as ([n], {Πs}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' The problem is to determine whether there exists some state |x⟩ ∈ (C2)⊗n such that the following equation holds: � s∈σk([n]) Πs |x⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' (1) If such a state exists we say that the instance is satisfiable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' if it does not we say that the instance is unsatisfiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' If there is a product state |φ⟩ satisfying (1) we say that the instance is satisfiable by a product state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' The usual definition of quantum k-SAT includes a promise that the ground-state energy has a polynomial lower bound if the instance is unsatisfiable [Bra11, §1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='1 is therefore a strictly harder problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='2 (ϵ-far from satisfiable by a product state).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Let Σ ⊂ σk([n]) be such that, for some product state |φ⟩ ∈ (C2)⊗n: � s∈(σk([n])−Σ) Πs |φ⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' (2) We say that an instance ([n], {Πs}) of quantum k-SAT is ϵ-far from satisfiable by a product state if this implies that |Σ| ≥ ϵnk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Our definition of ϵ-far differs slightly from that in [AS03];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' whereas they remove individual clauses (which in our case correspond to rank-one projections) we remove the whole projection for a given k-subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' This does not make much difference to the overall theory, since defining ϵ-far in terms of the number of rank-one projections removed would only alter ϵ by a constant factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='4 (Local satisfiability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Let ([n], {Πs}s∈σk([n])) be an instance of quantum k-SAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Let C ⊂ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We say that the restriction of the instance to C is the instance of quantum k-SAT defined by the data (C, {Πs}s∈σk(C)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We say that ([n], {Πs}) is locally satisfiable by a product state at C if (C, {Πs}s∈σk(C)) is satisfiable by a product state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Our result is based on the following two theorems, whose proofs are given in the remainder of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Let ([n], {Πs}) be a satisfiable instance of quantum k-SAT, and let c ∈ N be fixed, where c ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Let C ∈ σc([n]) be a subset chosen uniformly at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' The probability that the instance is locally satisfiable by a product state at C is greater than p ∈ (0, 1) whenever n > Ψ(p, c), where Ψ(p, c) = O(26c/ log(1/p)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Equivalently, let |ψ⟩ ∈ (C2)⊗n be any state and let C ∈ σc([n]) be a subset chosen uniformly at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' The probability that the subspace supp(TrC(|ψ⟩ ⟨ψ|)) ⊆ (C2)⊗c contains a product state is greater than p ∈ (0, 1) whenever n > Ψ(p, c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' The theorem applies to the case where c ≥ 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' a similar result holds for c = 2, but the result for c = 2 is much simpler and stronger and is treated separately in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We give a precise bound on Ψ(p, c) in the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' The constant factors in this bound are far from optimal, since we made many approximations to simplify the proof;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' if required, the reader could easily improve the bound by tightening the analysis without changing the structure of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' However, we would be surprised if the exponential dependence on c could be removed without a different approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' For the second theorem, we assume that n is large enough (but still finite).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' There exists a constant c(k, ϵ) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' not depending on n) such that the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Let ([n], {Πs}) be any instance of quantum k-SAT which is ϵ-far from satisfiable by a product state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Then, for a randomly chosen subset C ∈ σc(k,ϵ)([n]), the instance is locally unsatisfiable by a product state at C with probability p > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' 3 Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' The main weakness of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='7 is that we do not have a good upper bound for the constant c(k, ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Our proof in Section 3 does yield an upper bound, but it is extremely large (bigger than (((3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=')!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=')!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' · · · )!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=', where the chain of factorials has length (5/ϵ)4k−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' The bound has this form because the degree of the m-qubit Segre variety is the factorial m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' (Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' It is an open question whether the upper bound can be made small enough for the testing result to be practically useful rather than merely theoretically interesting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Improvements on this scale are not unknown in classical property testing (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' the discussion in [AS03, § 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Together, Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='5 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='7 imply that, for large enough n, an instance ([n], {Πs}) of quantum k-SAT cannot be both satisfiable and far from satisfiable by a product state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' The consequent result about testability of quantum k-SAT is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' With a promise that the instance is either satisfiable or ϵ-far from satisfiable by a product state, quantum k-SAT may be solved in randomised polynomial time in n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Let ([n], {Πs}) be an instance of quantum k-SAT, and let c be some constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Given a subset C ∈ σc([n]), we can check whether or not (n, {Πs}) is locally satisfiable by a product state at C in time polynomial in n (and exponential in c) using Gr¨obner basis methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Indeed, let {Π1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , ΠN} be the set of nontrivial projectors on C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Let Πi = �ri j=1 |aij⟩ ⟨aij|, where ri ≤ 2k is the rank of πi and |aij⟩ ∈ (C2)⊗k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Let |φ⟩ = |φ1⟩ ⊗ · · · ⊗ |φc⟩ be a unknown product state defined on C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We set |φi⟩ = xi0 |0⟩ + xi1 |1⟩ , i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , c}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Let si ∈ σk(C) be the set of qubits on which Πi acts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' For each si, we write |φsi⟩ := � v∈si |φv⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Then (1) implies the following equations: ⟨aij|φsi⟩ = 0, i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , N}, j ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , ri}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' (3) We also have that |xi0|2 + |xi1|2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' As we can rescale (3) arbitrarily, we can relax this to |xi0|2 + |xi1|2 ̸= 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' equivalently, xij ̸= 0 for at least one j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' To ensure this we introduce some new variables {yij} with the same index sets as the {xij}, and some new equations: (xi0yi0 − 1)(xi1yi1 − 1) = 0, i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , c}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' (4) In total we obtain c + �N i=1 ri equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Each equation has degree at most k or 2, and there are 4c variables over the field C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We want to determine if there is a solution for this polynomial system over C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' It is known that this polynomial system has no zeros in C if and only if its Gr¨obner basis is {1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' This is Hilbert’s Nullstellensatz, which is an EXPSPACE-complete problem [VZGG13, Chapter 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' By [BFS15, Proposition 1], an upper bound on the time complexity in this case is D(c + N � i=1 ri) �4c + D D �ω poly(n), where D ≤ 1 + 3c + (k − 1) �N i=1 ri is the maximal degree of the elements in the reduced Gr¨obner basis, and ω < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='373 is the exponent of matrix multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' The poly(n) factor is due to the polynomial precision of the matrix entries specifying the quantum k-SAT instance (Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We can therefore straightforwardly check satisfiability of an instance ([n], {Πs}) of quantum k-SAT with the testing promise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Since we are only interested in scaling w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' n, we may assume that n is large enough that (by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='5) if the instance is satisfiable then it will be locally satisfiable by a product state on c(k, ϵ)-subsets with probability p > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='75, and (by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='7) if the instance is ϵ-far from satisfiable by a product state then it will be locally unsatisfiable by a product state on c(k, ϵ)-subsets with probability p > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' 4 We choose subsets {Ci ∈ σc(k,ϵ)([n])}m i=1 at random and check whether the instance is locally satisfiable by a product state on these subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' If the majority are locally satisfiable by a product state we conclude that the instance is satisfiable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' otherwise we conclude that the instance is ϵ-far from satisfiable by a product state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' By the Chernoff bound for the binomial distribution the probability that the conclusion is incorrect tends exponentially to zero as m increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='2 Acknowledgements We thank Niel de Beaudrap and Aram Harrow for useful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' 817581).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We acknowledge support from EPSRC grant EP/T001062/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' 2 Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='5 We will make use of some combinatorial lemmas, some of which are well-known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' For the reader’s convenience we prove all these lemmas, but to avoid a long digression we have relegated the proofs to Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Throughout we consider the binomial coefficient �x k � to be the polynomial x(x − 1) · · · (x − k + 1)/k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='. It is defined for every real number x, and is positive and increasing for x ≥ k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='1 (Scaling binomial coefficients).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Let θ, x ∈ R≥0 and c ∈ N, where 0 ≤ θ ≤ 1 and x ≥ c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Then, for some α ∈ [0, 1), we have: θ �x c � = �θ1/cx c � + α �θ1/cx c − 1 � ≤ �θ1/cx + 1 c � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' (5) Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Let G be an l-regular hypergraph on n vertices, and let k ≤ l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We define the k-shadow G|k of G to be the k-regular hypergraph on n vertices whose edges are precisely those k-subsets which are contained in an edge of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='3 (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' [BE15, Fit18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Let G be an l-regular hypergraph on n vertices, let k ≤ l, and let ω ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We say that an k-regular hypergraph on n vertices G|k,ω is a partial (k, ω)-shadow of G if it can be obtained by the following construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' For each edge E of G, choose a subset KE ⊂ σk(E) of size |KE| = ⌊ω �l k � ⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' The edge set of G|k,ω is then defined as � E∈G KE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Obviously there are many different partial (k, ω)-shadows of an l-regular hypergraph G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' they depend on the choice made of the set KE for each edge E ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Recall that the edge density 0 ≤ θ ≤ 1 of a k-regular hypergraph on n vertices with edge set E is defined by |E| = θ �n k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='4 (Edge density of shadows [Kee08, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' If G is an l-regular hypergraph with edge density θ, then the k-shadow G|k has edge density θ|k ≥ (θ1/l − 2/n)k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='5 (Edge density of partial shadows).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Let n ∈ N and let c ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Let l := n/2c+δ and k := l/2 + ϵ, where −3 ≤ δ < −1 and −1/2 ≤ ϵ ≤ 0, be natural numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Let G be an l-regular hypergraph on n vertices with edge density θ ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Suppose that n ≥ 3(2c) and n ≥ 3(2c+1) � 1 2 + 1 ln(A) � A1/3 4 − ln(θ) �� for some constant A > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Then the edge density θ|k,1/2 of a partial (k, 1/2)-shadow of G has the following lower bound: θ|k,1/2 ≥ θ KA := θ 2 � A exp(A1/3/2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' (6) Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='6 (Edge density for a certain construction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Let n ∈ N be odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We construct an (n − 1)/2-regular hypergraph G as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' For every subset X ∈ σ(n−1)/2([n]), we do precisely one of the following: 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Add X as an edge to G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Add all the subsets in σ(n−1)/2(X) as edges to G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' The edge density θ of a hypergraph G constructed in this way satisfies θ ≥ 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Recall that the degree of a vertex in a hypergraph is the number of edges containing it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='7 (Degree of a random vertex).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Let G be a k-regular hypergraph on n vertices with edge density θ, where k = βn for some β ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Let N be the degree of a vertex picked uniformly at random, and let τ := N/ �n−1 k−1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Then, for any ϵ > 0 and α ∈ (0, 1): Pr [|θ − τ| ≥ min(∆(α, β, n), (1 − θ)(1 − α))] ≤ α, (7) where ∆(α, β, n) = 5 exp � 1 12α(1−α)n + 1 12β(1−β)n � � 2πα(1 − α)β(1 − β)n =: 5(Lα,β)1/n Mα,β √n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' (8) The final ingredients for the proof are two facts about entangled subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Recall that a subspace L ⊂ C2 ⊗ Cm is completely entangled if there is no product state in L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='8 ([ATL11, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' 1 and Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' 2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Let L ⊂ C2 ⊗Cm be a completely entangled subspace, for any m ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Then the orthogonal complement L⊥ ⊂ C2 ⊗ Cm contains a product state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Let m ∈ N, and let V ⊂ (C2)⊗m be a subspace of dimension ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Let L ⊂ C2 ⊗ (C2)⊗m be a subspace such that: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' C2 ⊗ V ⊥ ⊂ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' (C2 ⊗ V ) ∩ L is a completely entangled subspace of C2 ⊗ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Then L⊥ ⊂ C2 ⊗ Cm contains a product state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Since C2 ⊗ V ⊥ ⊂ L, we have L = (C2 ⊗ V ⊥) ⊕ � (C2 ⊗ V ⊥)⊥ ∩ L � = (C2 ⊗ V ⊥) ⊕ � (C2 ⊗ V ) ∩ L � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Then L⊥ = (C2 ⊗ V ) ∩ � (C2 ⊗ V ) ∩ L �⊥ = � (C2 ⊗ V ) ∩ L ��⊥, where �⊥ is the orthogonal complement in C2 ⊗ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' But (C2 ⊗ V ) ∩ L is a completely entangled subspace of C2 ⊗V , so by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='8 we know that � (C2 ⊗ V ) ∩ L ��⊥ contains a product state;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' therefore L⊥ contains a product state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Let |ψ⟩ ∈ (C2)⊗n be a state and let x ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Let S1, S2 ⊂ ([n] − {x}) be two subsets such that S1 ∩ S2 = ∅ and S1 ⊔ S2 = ([n] − {x}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Then at least one of the following is true: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' The subspace supp(TrS2(|ψ⟩ ⟨ψ|)) contains a product state |φx⟩ ⊗ |φS1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' The subspace supp(TrS1(|ψ⟩ ⟨ψ|)) contains a product state |φx⟩ ⊗ |φS2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' To clarify our notation, we first recall the definition of supports and kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' By definition, for any subset S ⊂ [n], we have supp(TrS(|ψ⟩ ⟨ψ|)) = ker(TrS(|ψ⟩ ⟨ψ|))⊥, where the orthogonal complement is taken in the space (C2)⊗|S|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' If we write (C2)⊗n = (C2)⊗|S| ⊗ (C2)⊗(n−|S|), where the first factor corresponds to those qubits in the set S, then: ker(TrS(|ψ⟩ ⟨ψ|)) = span{|v⟩ ∈ (C2)⊗|S| : (⟨v| ⊗ 1) |ψ⟩ = 0}, (9) supp(TrS(|ψ⟩ ⟨ψ|))) = span{(⟨v| ⊗ 1) |ψ⟩ : |v⟩ ∈ (C2)|S|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' (10) We now observe that if any of (i) dim(supp(Trx(|ψ⟩ ⟨ψ|))), (ii) dim(supp(TrS1(|ψ⟩ ⟨ψ|))) or (iii) dim(supp(TrS2(|ψ⟩ ⟨ψ|))) equal 1, then at least one of the two claims in this lemma is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Indeed, for (i) observe that if dim(supp(Trx(|ψ⟩ ⟨ψ|))) = 1, then |ψ⟩ = |ψx⟩ ⊗ |ψ([n]−{x})⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' To see this, observe that there exists |w⟩ ∈ C2 such that (⟨w| ⊗ 1) |ψ⟩ = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' 6 it follows by the Schmidt decomposition that |ψ⟩ is a product state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' The argument for (ii) and (iii) is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We therefore now assume that those three dimensions are greater than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We consider the subspace ker(Tr{x}⊔S1(|ψ⟩ ⟨ψ|)) ∩ � supp(Tr{x}(|ψ⟩ ⟨ψ|)) ⊗ supp(TrS1(|ψ⟩ ⟨ψ|)) � ⊆ supp(Tr{x}(|ψ⟩ ⟨ψ|)) ⊗ supp(TrS1(|ψ⟩ ⟨ψ|)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We have a dichotomy: (a) The subspace ker(Tr{x}⊔S1(|ψ⟩ ⟨ψ|))∩ � supp(Tr{x}(|ψ⟩ ⟨ψ|))⊗supp(TrS1(|ψ⟩ ⟨ψ|)) � is a completely entangled subspace of supp(Tr{x}(|ψ⟩ ⟨ψ|)) ⊗ supp(TrS1(|ψ⟩ ⟨ψ|)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' (b) The subspace ker(Tr{x}⊔S1(|ψ⟩ ⟨ψ|))∩ � supp(Tr{x}(|ψ⟩ ⟨ψ|))⊗supp(TrS1(|ψ⟩ ⟨ψ|)) � contains a product state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' In case (a), we are precisely in the situation of Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='9, where V = supp(TrS1(|ψ⟩ ⟨ψ|)) and L = ker(Tr{x}⊔S1(|ψ⟩ ⟨ψ|));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' therefore claim 1 is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' On the other hand, in case (b) there exists a product state |v⟩⊗|w⟩ ∈ supp(Tr{x}(|ψ⟩ ⟨ψ|))⊗supp(TrS1(|ψ⟩ ⟨ψ|)) such that (⟨v| ⊗ ⟨w| ⊗ 1) |ψ⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Consider the state |ψ′⟩ := (1 ⊗ ⟨w| ⊗ 1) |ψ⟩ ∈ supp(TrS1(|ψ⟩ ⟨ψ|)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Since (⟨v|⊗1) |ψ′⟩ = 0, we have that dim(ker(Tr{x}(|ψ′⟩ ⟨ψ′|))) = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' by the Schmidt decomposition this implies that |ψ′⟩ = |ψ′ x⟩ ⊗ |ψ′ S2⟩, and so claim 2 is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We can now prove the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' To warm up, we prove the case c = 2 first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Let ([n], {Πs}) be a satisfiable instance of quantum k-SAT, and let x1 ∈ [n] be any qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Then there is at most one qubit x2 ∈ ([n] − {x1}) such that the instance is not locally satisfiable by a product state at {x1, x2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Equivalently, let |ψ⟩ ∈ (C2)⊗n be any state, and let x1 ∈ [n] be any qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Then there is at most one qubit x2 ∈ ([n] − {x1}) such that the subspace supp(Tr{x1,x2}(|ψ⟩ ⟨ψ|))) ⊆ (C2)⊗2 does not contain a product state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' The equivalence of the two statements will be shown in the first paragraph of the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='5 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We therefore need only prove the second statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Let x2 ∈ ([n] − {x1}) be any qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We now use Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' in the notation of that lemma, let x := x1, S1 := {x2} and S2 := ([n] − {x1, x2}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Then either supp(Tr{x1,x2}(|ψ⟩ ⟨ψ|))) contains a product state or supp(Tr{x1,x′}(|ψ⟩ ⟨ψ|))) contains a product state for any x′ ∈ ([n] − {x1, x2}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We now move onto the case c ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Theorem (Restatement of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Let ([n], {Πs}) be a satisfiable instance of quantum k-SAT, and let c ∈ N be some constant, where c ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Let C ∈ σc([n]) be a subset chosen uniformly at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' The probability that the instance is locally satisfiable by a product state at C is greater than p ∈ (0, 1) whenever: n > Ψ(p, c) := max � (2c − 1)(c − 1) ln(4) ln(1/p) + (2c − 2) , e2 �180(c − 2)(2)3(c−2) √ 2π(3 − 1/(2)c−2) �2 , 6(2c+1)(c − 1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Equivalently, let |ψ⟩ ∈ (C2)⊗n be any state and let C ∈ σc([n]) be a subset chosen uniformly at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' The probability that the subspace supp(TrC(|ψ⟩ ⟨ψ|)) ⊆ (C2)⊗c contains a product state is greater than p ∈ (0, 1) whenever n > Ψ(p, c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Before giving the proof, we remark that the following weaker fact follows quite straightforwardly from known results: 7 Let ([n], {Πs}) be a satisfiable instance of quantum k-SAT, let c ∈ N be some constant, and let ϵ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Let C ∈ σc([n]) be a subset chosen uniformly at ran- dom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' The probability that there exists a product state |φ⟩ ∈ (C2)⊗c such that � s∈σk(C) ⟨φ| Πs |φ⟩ < ϵ is greater than p whenever p > O(nk−c−1/4) �c k � /ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' To prove this, one can use [BH16, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' 91], which implies that there is a product state |φ⟩ ∈ (C2)⊗n such that � s∈σk([n]) ⟨φ| Πs |φ⟩ ≤ O(nk−1/4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We can write � s∈σk([n]) ⟨φ| Πs |φ⟩ = � C∈σc([n]) � s∈σk(C) 1 �c k � ⟨φ| Πs |φ⟩ = � C∈σc([n]) SC, where SC := � s∈σk(C) �c k �−1 ⟨φ| Πs |φ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Since there are O(nc) c-subsets, the expected value of SC for a randomly chosen c-subset is O(n(k−c)−1/4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' the result then follows by Markov’s inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' An alternative approach to proving this fact, which has some issues but nevertheless reveals something about the nature of the problem, is to use monogamy of the squashed entanglement [KW04, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' For this, let |ψ⟩ be a state satisfying ([n], {Πs}), and let l1, l2 be constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' It follows straightforwardly from monogamy that, for a randomly chosen C ∈ σc([n]), the expected entanglement across every bipartition of C is O(1/n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Then by [BCY11, Thm, §II], TrC(|ψ⟩ ⟨ψ|) is close to a biseparable state across every bipartition of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' If we could conclude that this implied closeness to a fully separable state, the fact would follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='5 is a stronger, exact analogue, where we demand exact local satisfiability by a product state rather than approximate satisfiability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We were unable to derive Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='5 from the non-exact result, and the proof we are about to provide follows a different approach, which does not make use of any continuous entanglement measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We first observe the equivalence of the two statements in this theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' If ([n], {Πs}) is satisfiable, let |ψ⟩ ∈ (C2)⊗n be a satisfying state;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' clearly any state |ψC⟩ ∈ supp(TrC(|ψ⟩ ⟨ψ|)) is a local solution to ([n], {Πs}) at C, so the second statement implies the first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' In the other direction, let |ψ⟩ ∈ (C2)⊗n be a state;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' then for all C ∈ σc([n]) let ΠC be the projector onto ker(TrC(|ψ⟩ ⟨ψ|)), and consider the instance ([n], {ΠC}C∈σc([n])) of quantum c-SAT, which is of course satisfied by the state |ψ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Given this equivalence it is sufficient to show that for any state |ψ⟩ ∈ (C2)⊗n and a randomly chosen subset C ∈ σc([n]), there exists a product state in supp(TrC(|ψ⟩ ⟨ψ|)) with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We will choose qubits {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , xc} =: C one by one at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Let us pick the first qubit x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' There are two possibilities: (a) n − 1 is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' (b) n − 1 is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' In case (a), let B([n]−{x1}) be the set of all equal bipartions of the set ([n]−{x1}), where an equal bipartition is a pair of sets B1, B2 ⊂ ([n] − {x1}) of size |B1| = |B2| = (n − 1)/2 such that B1 ∪ B2 = ([n] − {x1}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We write such a bipartition as Bi 1|Bi 2, where i ∈ I is some index for the bipartitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We observe that, for any finite set S with even cardinality, |B(S)| = � |S| |S|/2 � /2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Now, for any equal bipartition B1|B2 ∈ B([n]−{x1}), we know by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='10 that at least one of the following holds: (i) The subspace supp(TrB2(|ψ⟩ ⟨ψ|)) contains a product state |φx1⟩ ⊗ |φB1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' (ii) The subspace supp(TrB1(|ψ⟩ ⟨ψ|)) contains a product state |φx1⟩ ⊗ |φB2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We define an (n − 1)/2-regular hypergraph G1 on the set ([n] − {x1}) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' For each equal bipartition Bi 1|Bi 2 ∈ B([n] − {x1}), if supp(TrBi j(|ψ⟩ ⟨ψ|)) contains a product state then add the set Bi j as an edge to the graph G1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Clearly, the edge density θ1 of G1 has the lower bound θ1 ≥ 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' The physical significance of this hypergraph G1 is that, 8 for any edge E of G1, we know that supp(Tr{x1}⊔E(|ψ⟩ ⟨ψ|)) contains a product state, which we will write as |ψ1⟩ ⊗ |ψE⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' This definition of G1 depended on the fact that n − 1 was even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' In case (b), we make a slightly different definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' For any subset X ∈ σn/2−1([n] − {x1}), at least one of the following holds, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='10: (i) The subspace supp(Tr{x1}⊔X(|ψ⟩ ⟨ψ|)) contains a product state |φx1⟩ ⊗ |φX⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' (ii) The subspace supp(TrX(|ψ⟩ ⟨ψ|)) contains a product state |φx1⟩ ⊗ |φ{x1}⊔X⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We now define an (n/2 − 1)-regular hypergraph G1 on the set ([n] − {x1}) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' For every X ∈ σn/2−1([n] − {x1}), if supp(Tr{x1}⊔X(|ψ⟩ ⟨ψ|)) contains a product state then we add X as an edge to G1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' On the other hand, if supp(TrX(|ψ⟩ ⟨ψ|)) contains a product state then we add all the subsets in σn/2−1({x1} ⊔ X) as edges to G1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='6 the edge density θ1 of G satisfies θ1 ≥ 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Again, the physical significance of this hypergraph G1 is that, for any edge E of G1, we know that supp(Tr{x1}⊔E(|ψ⟩ ⟨ψ|)) contains a product state, which we will write as |ψ1⟩ ⊗ |ψE⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Having obtained the hypergraph G1, we now pick the next qubit x2 ∈ ([n] − {x1}) at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We define a new hypergraph W2 (W for ‘working’, since this graph is an intermediate step towards obtaining G2) on the set ([n] − {x1, x2}) with the edges {(E − {x2}) | E is an edge of G1 and x2 ∈ E}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' This hypergraph W2 is k2-regular, where k2 = ((n − 3)/2) if n − 1 was even, and k2 = (n/2 − 2) if n − 1 was odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Now we define the graph G2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' again the definition will depend on whether k2 is even or odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' If k2 is even then, for every edge E of W2 and every X ∈ σk2/2(E), we again have the following dichotomy, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='10: (i) The subspace supp(Tr{x1,x2}⊔X(|ψ1⟩⊗|ψE⟩) contains a product state |ψ1⟩⊗|ψ2⟩⊗ |ψX⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' (ii) The subspace supp(TrX(|ψ1⟩⊗|ψE⟩) contains a product state |ψ1⟩⊗|ψ2⟩⊗|ψE−(X⊔x2)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We then define the (k2/2)-regular hypergraph G2 as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' For every edge E in W2, and every X ∈ σk2/2(E), if supp(Tr{x1,x2}⊔X(|ψ1⟩ ⊗ |ψE⟩) contains a product state then we add X as an edge to G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' On the other hand, if supp(TrX(|ψ1⟩⊗|ψE⟩) contains a product state then we add X as an edge to G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' This graph contains a partial (k2/2, 1/2)-shadow of W2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' If k2 is odd then we define the (k2 − 1)/2-regular hypergraph G2 as follows: for every edge E of W2 and every X ∈ σ(k2−1)/2(E), if supp(Tr{x1,x2}⊔X(|ψ1⟩ ⊗ |ψE⟩)) contains a product state then we add X as an edge to G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' On the other hand, if supp(TrX(|ψ1⟩ ⊗ |ψE⟩)) contains a product state then we add all the subsets in σ(k2−1)/2(X) as edges to G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='6 applied to each edge E, this graph contains a partial ((k2−1)/2, 1/2)- shadow of W2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' The physical significance of the hypergraph G2 is that for any edge E of G2, we have that supp(Tr{x1,x2}⊔E(|ψ⟩ ⟨ψ|)) contains a product state over the qubit x1, the qubit x2 and the qubits in E, which we will write as |ψ1⟩ ⊗ |ψ2⟩ ⊗ |ψE⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We now iterate this construction for the qubits {x3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , xc−1} and obtain a kc−1- regular hypergraph Gc−1 on the set ([n] − {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , xc−1});' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' for any edge E of Gc−1, we know that supp(Tr{x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=',xc−1}⊔E) contains a product state, which we write as |ψ1⟩⊗· · ·⊗ |ψc−1⟩ ⊗ |ψE⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Let us obtain bounds on the edge size kc−1 and edge density θc−1 of the hypergraph Gc−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We determine the edge size first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' The hypergraph G1 was k1-regular, where either k1 = (n − 1)/2 or k1 = (n/2 − 1), and had edge density θ1 ≥ 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' To get G2 from G1 we defined the graph W2 by picking a single qubit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' this graph had edges of size l2 = k1 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Iterating this construction, we see that in general the graph Gi+1 will have edges of size ki+1 = li+1/2 or ki+1 = li+1/2 − 1/2, where li+1 = ki − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' It follows that ki/2 > ki+1 ≥ ki/2 − 1, and k0 = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Solving the linear recurrence, we obtain: n/2i > ki ≥ n/2i + 1/2i−1 − 2 > n/2i − 2, 9 n/2i − 1 > li+1 ≥ n/2i + 1/2i−1 − 3 > n/2i − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' (11) In particular, n/2c−1 > kc−1 ≥ n/2c−1 + 1/2c−2 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' (12) We now obtain a lower bound on the edge density θc−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' To simplify the calculation we will make numerous approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' By (11), Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='5 will always hold when we go from Wi to Gi, as long as n is big enough;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' we at least need n ≥ 3(2c−1) for the step from Wc−1 to Gc−1, so we assume this from now on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' There is a free constant A in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' we arbitrarily set it to A = 216W(22/3/6)3 ≃ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='107, where W(x) is the Lambert W-function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' this implies that KA = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' On the other hand, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='7 gives us a bound on the change in edge density when we go from Gi to Wi+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' The bound is a minimum over two functions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' we will just consider the function (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Let βi be the value of β when we apply Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='7 to go from Gi to Wi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' by (11), we always have 1/2 > 1/2 i > βi ≥ 1/2i − 2/n ≥ 1/2i − 2/3(2c−2) ≥ 1/3(2c−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Let β := 1/3(2c−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We also fix some α ∈ (0, 1), which we will determine later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' By the union bound, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='5 and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='7 we then have Pr[θc−1 ≥ Φ] ≥ 1 − (c − 2)(1 − α), (13) where Φ := 1 2(4)c−2 − (c − 2)∆(α, β, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Finally, we pick the qubit xc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' As long as it is contained in an edge of Gc−1, there will be a product state in supp(Tr{x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=',xc−1,xc}(|ψ⟩ ⟨ψ|)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' The worst case is that the edges of Gc−1 form a complete hypergraph on a subset C ⊂ ([n] − {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , xc−1}), with some overspill onto a single vertex not in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Let |C| = σ(n−c+1) for some σ ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' By (12) and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='1, we have: σ ≥ (θc−1) 2c−1 n+2−2c ≥ (Φ) 2c−1 n+2−2c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' (14) Let p be the probability that supp(TrC(|ψ⟩ ⟨ψ|)) contains a product state;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' by (13) and (14) we have that p ≥ (1 − (c − 2)(1 − α))(Φ) 2c−1 n+2−2c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' (15) We will assume that p = p is given and find values for the as yet undefined variables α, n which satisfy (15) while minimising n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' (Again, we will make approximations, so the minimisation will not be tight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=') Assuming 1 − (c − 2)(1 − α) > 0, we need: � p (1 − (c − 2)(1 − α)) � n+2−2c 2c−1 ≤ 1 2(4)c−2 − (c − 2) 5(Lα,β)1/n Mα,β √n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' (16) We make n big enough that (c − 2) 5(Lα,β)1/n Mα,β √n ≤ 1 2t(4)c−2 (17) for some t > 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' this is to say that 1 n ln(Lα,β) − 1 2 ln(n) ≤ ln � Mα,β 10(c − 2)(4)c−2t � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' (18) Let us assume that α ≥ 1 − 1/3(2c−2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' then Lα,β ≤ exp � 1 6α(1 − α) � , Mα,β ≥ √ 2πα(1 − α), and the equation (18) reduces to 1 6nα(1 − α) − 1 2 ln(n) ≤ ln � √ 2πα(1 − α) 10(c − 2)(4)c−2t � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' (19) 10 Now assume that ln(n) ≥ 2 � ln �10(c − 2)(4)c−2t √ 2πα(1 − α) � + 1 � ⇔ n ≥ e2 �10t(c − 2)(4)c−2 √ 2πα(1 − α) �2 , (20) so the equation (19) reduces to: 1 6nα(1 − α) ≤ 1 ⇔ n ≥ 1 6α(1 − α) which is already implied by (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' By (17), the equation (16) then reduces to: � p 1 − (c − 2)(1 − α) � n+2−2c 2c−1 ≤ t − 1 2t(4)c−2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' (21) Clearly a larger α here will reduce the required size of n in (21), but it will worsen it in (20);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' so we set α = 1 − 1/3(2c−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We then assume that: p 1 − (c − 2)(1 − α) < 1 ⇔ p < 1 − c − 2 3(2c−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We also arbitrarily set t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Given these assumptions, (21) reduces to n ≥ (2c − 1)(c − 1) ln(4) ln(1 − (c − 2)/3(2c−2)) − ln(p) + (2c − 2), which is implied by the simpler n ≥ (2c − 1)(c − 1) ln(4) ln(1/p) + (2c − 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' (22) In addition to the lower bounds (20) and (22), there is also a lower bound on n coming from the second lower bound on n in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' By (13) and (17) we always have θi ≥ Φ ≥ 1 4(4)c−2 , so using ln(A) ≥ ln(2) we obtain the following bound: n ≥ 6(2c+1)(c − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Substituting the chosen values of α and t into (20), we obtain the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' 3 Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='7 We first remark that one idea for a proof of this theorem is to add the usual polynomial ground state energy bound to the definition of quantum k-SAT, then use the standard approach with a δ-net to reduce the problem of satisfiability by a product state to an instance of classical k′-SAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' It is straightforwardly seen that this instance of k′-SAT is also ϵ-far from satisfiable, so the result from [AS03] can be directly applied to show local unsatisfiability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' The problem is that the constant δ determining the precision of the net, and therefore also the number k′, depends polynomially on n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' this implies that the size of the subsets to be checked also depends at least polynomially on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Since the time taken to check for a product state solution is exponential in the size of the subset, as we saw in the proof of Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='10, we end up with time exponential in n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Instead, we give a proof which is heavily inspired by Alon and Shapira’s proof of the classical analogue (to the point where we were able to lift most of the constants from their work), and which is in any case more general than the above approach since it does not require any lower bound on the ground state energy when an instance of quantum k-SAT is unsatisfiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Firstly, we consider the problem of extending a local assignment, which in our setting is a product subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We define the notion of a bad qubit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' that is, a qubit to which either the local assignment cannot be extended, or such that extension to that qubit greatly reduces the dimension of possible extensions to the other qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' 11 If, in the course of constructing a local assignment by extension, one extends through 5(4)k−1/ϵ bad qubits, then the assignment can no longer be extended (Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We show that if an instance of quantum k-SAT is ϵ-far from satisfiable by a product state, then there are always more than ϵn/5 bad qubits with respect to any local assignment (Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' At this point, Alon and Shapira [AS03, Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='4] built a binary tree of possible assignments to a set of randomly picked variables and showed that each branch of the tree will run into more than 5(4)k−1/ϵ bad variables with high probability;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' they then used the union bound to conclude that there is no local assignment to those variables with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' This approach cannot be applied directly in the quantum setting because there are in general continuously many assignments to each variable (the space of possible assignments is a Hilbert space and not a binary set) and it is therefore not possible to build a finite tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Instead, we use a backtracking approach which is summarised in the first paragraph of the proof of the theorem at the end of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We still need to discretise somehow;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' for this we use Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='6, which implies that we need only eliminate a finite number of possible solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Notation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Throughout this section we write |H| := dim(H) for the dimension of a Hilbert space H and |X| for the cardinality of a set X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' When the context does not adequately distinguish these two meanings we explicitly use dim(H) for the dimension and #(X) for the cardinality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Let V, W be two sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We write σk(V ⊔ W) for the set of k-element subsets of V ⊔ W containing all the elements of V ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' that is, underlining one of the sets in the union means all its elements must be contained in the k-element subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' For a subset X ⊆ [n] we write HX := (C2)⊗|X| for the Hilbert space of the qubits in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Fix a subset S ⊂ [n], and let HS := (C2)⊗|S| be the Hilbert space of the qubits in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We say that an assignment to S is a product subspace PS ⊆ HS such that every state in the subspace is a local solution for ([n], {Πs}) at S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' that is, a subspace PS = V1 ⊗· · ·⊗V|S| ⊆ HS such that every state |φS⟩ ∈ PS satisfies � s∈σk(S) Πs |φS⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' For every {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , xj} ∈ σj([n] − S), where 1 ≤ j ≤ k − 1, we define LS,PS(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , xj) ⊆ Hx1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=',xj to be the space of all states |vx1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=',xj⟩ ∈ Hx1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=',xj such that � s∈σk({x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=',xj}⊔S) Πs(|vx1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=',xj⟩ ⊗ |φS⟩) = 0 ∀ |φS⟩ ∈ PS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' (23) We define LS,PS,j := � {x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=',xj}∈σj([n]−S) LS,PS(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , xj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' In words, LS,PS(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , xj) is the space of possible assignments |vx1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=',xj⟩ on the qubits x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , xj that are compatible with the assigment (S, PS), in the sense that |vx1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=',xj⟩ ⊗ PS is in the kernel of every projector on the qubits {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , xj} ⊔ S which acts on all of the qubits x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , xj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' The space LS,PS,j is the direct sum of all these possible j-qubit ‘extensions’ of the assignment (S, PS), and its dimension measures how many possible j-qubit extensions there are to the local assignment (S, PS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' In what follows we will consider taking a qubit x /∈ S and extending the assignment PS on S to an assignment Vx ⊗ PS on {x} ⊔ S, where Vx ⊆ LS,PS(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We define δS,PS,x,Vx,j := � {x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=',xj}∈σj([n]−({x}⊔S)) |LS,PS(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , xj)| − |L{x}⊔S,Vx⊗PS(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , xj)| and δS,PS,x,Vx := �k−1 j=1 δS,PS,x,Vx,j nk−j−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' In words, δS,PS,x,Vx measures how much the dimension of the space of possible extensions has been reduced by extending the assignment (S, PS) to ({x} ⊔ S, VX ⊗ PS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We say that a qubit x ∈ ([n]−S) conflicts with respect to (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=') (S, PS) if LS,PS(x) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' If a qubit x ∈ ([n] − S) does not conflict w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' (S, PS), we say that it is heavy w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' (S, PS) if minVx⊆LS,PS (x) δS,PS,x,Vx > ϵnk−1/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We say that a qubit x ∈ ([n] − S) is bad w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' (S, PS) if it is either heavy or conflicting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' 12 Finally, in the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='5 we will define a set of subsets Σ ⊂ σk([n]) and only consider projectors on subsets not in Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' For a local assignment (S, PS) and every {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , xj} ∈ σj([n] − S) we therefore define LΣ S,PS(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , xj) ⊂ Hx1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=',xj to be the space of all states |vx1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=',xj⟩ ∈ Hx1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=',xj such that � s∈σk({x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=',xj}⊔S)−Σ Πs(|vx1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=',xj⟩ ⊗ |φS⟩) = 0 ∀ |φS⟩ ∈ PS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' This is identical to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='2), but with the projectors in Σ removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' The other definitions above can be similarly generalised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We first show that it is impossible to extend an assignment through more than 5(d2)k−1/ϵ heavy qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We define a chain of length m to be a local assignment ({y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , ym}, P1⊗ · · ⊗ Pm) such that, for each 2 ≤ l ≤ m, each qubit yl is heavy with respect to the as- signment ({y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , yl−1}, P1 ⊗ · · · ⊗ Pl−1), and y1 is heavy with respect to the empty assignment (∅, ∅).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Let ({y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , yγ}, P1 ⊗ · · · ⊗ Pγ) be a chain of length γ(k, ϵ) := 5(4)k−1/ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Then every qubit x ∈ ([n] − {y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , yγ}) is conflicting w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' the chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Let us construct the chain ({y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , yγ}, P1 ⊗ · · · ⊗ Pγ) starting from the empty assignment (∅, ∅).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We define Wi := k−1 � j=1 |L{y1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=',yi},P1⊗···⊗Pi,j| nk−j−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' The initial size of each |L∅,∅,j| is at most � {v1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=',vj}∈σj([n]) 2j = �n j � 2j ≤ 2j j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' nj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Therefore the initial value of W0 = �k−1 j=1 |L∅,∅,j|nk−j−1 is at most �k−1 j=1 2j j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' njnk−j−1 ≤ 2k−1nk−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' When we extend to the qubit yi from the assignment ({y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , yi−1}, P1 ⊗ · · · ⊗ Pi−1) we have k−1 � j=1 δ{y1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=',yi−1},P1⊗···⊗Pi−1,yi,Pi,j nk−j−1 = δ{y1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=',yi−1},P1⊗···⊗Pi−1,yi,Pi > ϵnk−1/5 by definition of heaviness, and therefore Wi ≤ Wi−1 − ϵnk−1/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' So by the time we have made 5dk−1/ϵ extensions we must have W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' In particular L{y1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=',yγ},P1⊗···⊗Pγ(x) = 0 for all x ∈ ([n] − {y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , yγ}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We now show that the ϵ-far condition implies that there will always be many bad qubits with respect to any local assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Let ([n], {Πs}) be an instance of quantum k-SAT which is ϵ-far from sat- isfiable by a product state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Let S ⊂ [n] and let PS be some local assignment to S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Then there are at least ϵn/5 bad qubits w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' (S, PS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Suppose that there are fewer than ϵn/5 bad qubits w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='t (S, PS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We will define a subset Σ ⊂ σk([n]) satisfying (2) and such that |Σ| < ϵnk, contradicting the assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' This is a two-step process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Let Xnb ⊂ ([n] − S) be the set of qubits which are not bad w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='t (S, PS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' In the first step we will extend the assignment (S, PS) to the qubits in Xnb one at a time, while adding k-subsets to Σ according to the following prescription.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Let x ∈ Xnb be the first qubit to which we extend;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' we choose some Vx ⊆ LS,PS(x) minimising δS,PS,x,Vx, and extend the assignment to ({x} ⊔ S, Vx ⊗ PS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Let us consider in turn each {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , xj} ∈ σj([n] − ({x} ⊔ S)), where 1 ≤ j ≤ k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Clearly L{x}⊔S,Vx⊗PS(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , xj) ⊆ LS,PS(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , xj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' If this inclusion is an equality we add no k-subsets to Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' However, if the inclusion is strict then we add all the subsets 13 s ∈ σk({x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , xj} ⊔ {x} ⊔ S) to Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Then for any states |vx1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=',xj⟩ ∈ LS,PS(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , xj), |vx⟩ ∈ Vx and |φS⟩ ∈ PS we have: � s∈σk({x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=',xj}⊔{x}⊔S)−Σ Πs(|vx1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=',xj⟩ ⊗ |vx⟩ ⊗ |φS⟩) = � s∈σk({x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=',xj}⊔S) Πs(|vx1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=',xj⟩ ⊗ |vx⟩ ⊗ |φS⟩) + � s∈σk({x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=',xj}⊔{x}⊔S) Πs(|vx1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=',xj⟩ ⊗ |vx⟩ ⊗ |φS⟩) − � s∈σk({x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=',xj}⊔{x}⊔S)) Πs(|vx1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=',xj⟩ ⊗ |vx⟩ ⊗ |φS⟩) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Here the second equality is by definition of LS,PS(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , xj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' After adding these subsets to Σ, we therefore have an equality LΣ {x}⊔S,Vx⊗PS(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , xj) = LS,PS(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , xj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' After doing this for all the {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , xj}, we enlarge the assignment to ({x} ⊔ S, Vx ⊗ PS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We then repeat this process until we have extended to an assignment on Xnb ⊔ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Importantly, we claim that each qubit in Xnb remains not bad throughout this process, even as the assignment is extended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Indeed, it is clear that none of these qubits will ever conflict, since we add subsets to Σ so that LΣ(x) is fixed throughout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' It remains to show that none of these qubits will become heavy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' To see this, suppose that we extend the assignment to ({y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' ym} ⊔ S, Vy1 ⊗ · · · ⊗ Vym ⊗ PS), removing the relevant subsets as above, and then extend to x ∈ (Xnb − ({y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' ym} ⊔ S)) with subspace Vx ∈ LΣ {y1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='ym}⊔S,Vy1⊗···⊗Vym⊗PS(x) = LS,PS(x) (the equality is due to the addition of subsets to Σ detailed above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Alternatively, we could have extended to x directly from (S, PS), without extending through {y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , ym} first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' It is sufficient to show that, for any {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , xj} ∈ [n] − ({x, y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , ym} ⊔ S), there is an inclusion: L{x}⊔S,Vx⊗PS(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , xj) ⊆ LΣ {x,y1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='ym}⊔S,Vx⊗Vy1⊗···⊗Vym⊗PS(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , xj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' (24) Here the subspaces are taken prior to the removal of subsets following the extension to qubit x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' but on the RHS we have removed all the relevant subsets following extension to {y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , ym}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We will now prove the inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Let |vx1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=',xj⟩ ∈ L{x}⊔S,Vx⊗PS(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , xj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We will consider two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' The first is where j = k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' In the following equations Σ is defined after extension to {y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , ym}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' For all |vx⟩ ∈ Vx, |vyi⟩ ∈ Vyi and |φS⟩ ∈ PS, we have: � s∈{x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=',xk−1}⊔{x}⊔{y1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=',ym}⊔S−Σ Πs(|vx1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=',xk−1⟩ ⊗ |vx⟩ ⊗ |vy1⟩ ⊗ · · · ⊗ |vym⟩ ⊗ |φS⟩) = � s∈{x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=',xk−1}⊔{y1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=',ym}⊔S−Σ Πs(|vx1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=',xk−1⟩ ⊗ |vx⟩ ⊗ |vy1⟩ ⊗ · · · ⊗ |vym⟩ ⊗ |φS⟩) +Π{x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=',xk−1,x}(|vx1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=',xk−1⟩ ⊗ |vx⟩ ⊗ |vy1⟩ ⊗ · · · ⊗ |vym⟩ ⊗ |φS⟩) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Here the first term on the second line is zero since Σ is defined so that LΣ {y1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='ym}⊔S,Vy1⊗···⊗Vym⊗PS(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , xk−1) = LS,PS(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , xk−1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' the second term is zero by definition of L{x}⊔S,Vx⊗PS(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , xk−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Therefore |vx1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=',xj⟩ ∈ L{x,y1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='ym}⊔S,Vx⊗Vy1⊗···⊗Vym⊗PS(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , xj) and we obtain the inclusion (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' On the other hand, suppose that j < k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Then for all |vx⟩ ∈ Vx, |vyi⟩ ∈ Vyi and |φS⟩ ∈ PS we have the following equation: � s∈{x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=',xk−1}⊔{x}⊔{y1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=',ym}⊔S−Σ Πs(|vx1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=',xk−1⟩ ⊗ |vx⟩ ⊗ |vy1⟩ ⊗ · · · ⊗ |vym⟩ ⊗ |φS⟩) 14 = � s∈{x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=',xk−1}⊔{y1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=',ym}⊔S−Σ Πs(|vx1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=',xk−1⟩ ⊗ |vx⟩ ⊗ |vy1⟩ ⊗ · · · ⊗ |vym⟩ ⊗ |φS⟩) + � s∈{x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=',xk−1,x}⊔{y1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=',ym}⊔S−Σ Πs(|vx1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=',xk−1⟩ ⊗ |vx⟩ ⊗ |vy1⟩ ⊗ · · · ⊗ |vym⟩ ⊗ |φS⟩) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Here the terms in the second line are both zero since Σ is defined so that: LΣ {y1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='ym}⊔S,Vy1⊗···⊗Vym⊗PS(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , xk−1) = LS,PS(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , xk−1), LΣ {y1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='ym}⊔S,Vy1⊗···⊗Vym⊗PS(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , xk−1, x) = LS,PS(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , xk−1, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Therefore |vx1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=',xj⟩ ∈ LΣ {x,y1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='ym}⊔S,Vx⊗Vy1⊗···⊗Vym⊗PS(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , xj) and we obtain the inclusion (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We have shown that a qubit in Xnb cannot become bad upon extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' The first step ends when we have extended to an assignment on Vnb ⊔ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' In the second step we add to Σ all subsets containing any qubit in [n] − (Vnb ⊔ S) and then extend the assignment to the whole of [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We thus obtain a product state |φ⟩ ∈ (C2)⊗n satisfying (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' To finish we must show that we added less than ϵnk subsets to Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' In the first step, after extending to a qubit x we added all the subsets s ∈ σk({x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , xj} ⊔ x ⊔ S) each time that |L{x}⊔S,Vx⊗PS(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , xj)| was smaller than |LS,PS(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , xj)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' This is at most � n k−j−1 � ≤ nk−j−1 subsets each time, since |S| ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Since the qubit x was never heavy, we therefore added at most �k−1 j=1 δS,PS,x,Vx,jnk−j−1 = δS,PS,x,Vx ≤ ϵnk−1/5 projectors every time we extended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' As there are at most n qubits in Xnb, in the first step we added less than ϵnk/5 subsets to Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' In the second step, we added all the subsets which contained a qubit not in S ⊔ Vnb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Since by assumption there were ≤ ϵn/5 bad qubits to begin with, we therefore added at most (ϵn/5) � n k−1 � ≤ ϵnk/5 subsets in the second step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Altogether we added less than ϵnk subsets, as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' For discretisation we will also need the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' For a state |φ⟩ ∈ C2, we write ⟨|φ⟩⟩ ∈ P1 for the corresponding point on the complex projective line (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' the Bloch sphere).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Let L ⊆ (C2)⊗m be a subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We define X := {x ∈ P1 : ∃{|φi⟩ ∈ C2}m i=1 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' |φ1⟩ |φ2⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' |φm⟩ ∈ L and ⟨|φ1⟩⟩ = x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Then either (i) X = P1, or (ii) #(X) ≤ m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='. The bound in case (ii) is tight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' It is straightforward to show that X is either finite or covers the whole space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Indeed, the subspace L corresponds to a projective linear subspace P(L) ⊆ P2m−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' By the Segre embedding the product states form a projective subvariety Σm := P1 × · · · × P1 ⊂ P2m−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' But it is an elementary result in algebraic geometry that the projection map π1 : P1 × · · · × P1 → P1 is closed in the Zariski topology, since every projective variety is complete [Har13, §2 Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' The intersection P(L) ∩ Σm is an algebraic set in Σm, and therefore by closure of the projection map, the image under the projection will be an algebraic set in P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' This set will either be zero-dimensional, in which case it is a finite set of points;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' or one-dimensional, in which case it must be the whole of P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' The only thing left is to bound the number of points in the case where the image is zero-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We first note that for any variety S ⊂ Σm, the image π1(S) is irreducible;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' indeed, were it not irreducible then by continuity of π1 the fibres would give a nontrivial decomposition of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' If π1(S) ̸= P1, it must therefore be a single point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' It follows that all we need to do is bound the number of irreducible components of P(L) ∩ Σm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We know that P(L) is an intersection of hyperplanes {Pi}2m−|L| i=1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We use [Har13, §1 Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='7], which yields the following statement in our special case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' For 15 any subvariety V ⊂ P2m−1 of dimension ≥ 1, and for any hyperplane P ⊂ P2m−1 not containing V , let Z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , Zs be the irreducible components of V ∩ P;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' then: s � j=1 deg(Zj) ≤ deg(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' (25) Here deg(Zj), deg(V ) ∈ N>0 are the degrees of the varieties in P2m−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' If P contains V then the intersection is just V again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We observe that the irreducible components of P(L) ∩ Σm can be obtained by the following process: take the intersection Σm ∩ P1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' then for each of the irreducible components of that intersection take the intersection with P2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' then for each of the irreducible components of that intersection take the intersection with P3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Since deg(Σm) = m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=', by (25) we finish with at most m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' irreducible components (which would all have degree 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' The bound follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' To see that the bound is tight, observe that by definition of the degree, the variety Σm intersects a generic linear subvariety P(V ) ⊂ P2m−1 of dimension 2m − (m + 1) in precisely m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' points, and generically these points will be mapped to m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' different points in P1 by the projection π1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We can now prove the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Let (S, PS) be some local assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' For any X ⊆ ([n] − S) we define LS,PS(X) ⊆ HX to be the space of all states |vX⟩ ∈ HX such that � s∈σk(X⊔S) Πs(|vX⟩ ⊗ |φS⟩) = 0 ∀ |φS⟩ ∈ PS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' (26) (The difference between this and (23) is that all the subsets of X ⊔ S are included, not just those containing every element of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=') Theorem (Restatement of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' There exists a constant c(k, ϵ) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' not de- pending on n) such that the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Let ([n], {Πs}) be any instance of quantum k-SAT which is ϵ-far from satisfiable by a product state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Then, for a randomly chosen subset C ∈ σc(k,ϵ)([n]), the instance is locally unsatisfiable by a product state at C with probability p > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' The idea of the proof is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We will build a randomly chosen subset C ⊆ [n] of qubits by picking qubits at random from [n], one by one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' For any local assignment on some subset of the qubits which have already been picked, we know by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='5 that the next qubit we pick has an ϵ/5 chance of being bad w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' that assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' By varying the local assignment we consider before picking each qubit, we will show that, after enough qubits have been picked, there can be no local assignment to all the qubits in C with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' This is essentially a backtracking argument;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' we know by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='4 that we cannot extend a chain through more than γ(k, ϵ) heavy variables, so we simply build such a maximal chain and gradually eliminate all the possible assignments to that chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We have not endeavoured to optimise this procedure and it can certainly be improved, although whether one can bring the constant c(k, ϵ) down to the level in the classical setting [AS03, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='1] using such a backtracking approach is by no means clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' The key to our backtracking argument is the ability to eliminate possible assignments to a chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We formalise this as follows, assuming for the time being that we can pick bad qubits each time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' of course this will not be the case when we pick qubits at random, but in the end we will simply ignore those we pick that are not bad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Suppose that, given any chain ({y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , ym}, P1 ⊗ · · · ⊗ Pm) of length m, there exists an elimination procedure, which is defined by a rule in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' The procedure will define a set Γ ⊆ ([n] − {y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , ym}) of picked qubits;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' at the beginning of the procedure Γ = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' At each step of the procedure, we: Choose some local assignment (S, PS) on a subset S ⊂ {y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , ym} ⊔ Γ, according to the rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' 16 Pick at random a qubit x ∈ ([n] − ({y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , ym} ⊔ Γ)) which is bad w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' the assignment (S, PS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Add the qubit x to Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' The rule defines an elimination procedure with constant φ(m) ∈ N if, while |Γ| ≤ φ(m), we can stop the procedure and conclude that there is no local assignment to {y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , ym}⊔ Γ which assigns P1 ⊗ · · · ⊗ Pm−1 to the qubits {y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , ym−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We claim that if an elimination procedure exists for chains of length m with constant φ(m), then there also exists an elimination procedure for chains of length m − 1 with constant φ(m − 1) = ((φ(m) + 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' + 1)(φ(m) + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' The procedure is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Let ({y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , ym−1}, P1 ⊗ · · · ⊗ Pm−1) be a chain of length m − 1, and let |pm−1⟩ ∈ Pm−1 be any state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We pick at random a bad qubit y0 m ∈ ([n] − {y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , ym−1}) with respect to the local assignment ({y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , ym−1}, P1 ⊗ · · · ⊗ Pm−2 ⊗ |pm−1⟩) and add it to Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' If it is conflicting, then the procedure is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' If it is heavy, then we perform the length m elimination procedure on the chain ({y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , ym−1, y0 m}, P1 ⊗ · · · ⊗ Pm−2 ⊗ |pm−1⟩ ⊗ L{y1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=',ym−1},P1⊗···⊗Pm−2⊗|pm−1⟩(y0 m)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' This gives us a set Γ0 m ⊆ ([n] − {y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , y0 m}) of qubits, where |Γ0 m| ≤ φ(m), such that there is no local assignment to {y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , y0 m} ⊔ Γ0 m which assigns P1 ⊗ · · · ⊗ Pm−2 ⊗ |pm−1⟩ to the qubits {y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , ym−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We add the qubits in Γ0 m to Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Now consider the following subspace: L := L{y1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=',ym−2},P1⊗···⊗Pm−2(ym−1, y0 m, Γ0 m) ⊆ Hym−1 ⊗ Hy0m ⊗ HΓ0m ∼= C2 ⊗ C2|Γ0 m|+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Let X = {⟨|φ0⟩⟩ ∈ P1 : |φ0⟩ ⊗ |φ1⟩ ⊗ · · · ⊗ |φ|Γ0m|+1⟩ ∈ L}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Since there is no local assignment to {y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , y0 m} ⊔ Γ0 m which assigns P1 ⊗ · · · ⊗ Pm−2 ⊗ |pm−1⟩ to the qubits {y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , ym−1}, we know that X ̸= P1, since ⟨|pm−1⟩⟩ /∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' But then, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='6, we have #(X) ≤ (|Γ0 m| + 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='. The worst case is where this is an equality: in this case let the corresponding states be {|pi m−1⟩ ∈ Hym−1}(|Γ0 m|+2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' i=1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Now for each 1 ≤ i ≤ (|Γ0 m| + 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' in turn we pick at random a qubit yi m ∈ ([n] − Γ) which is bad with respect to the local assignment ({y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , ym−1}, P1 ⊗ · · · ⊗ Pm−2 ⊗ |pi m−1⟩) and add this qubit to Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' These qubits {yi m}(|Γ0 m|+2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' i=1 are either conflicting or heavy with respect to their corresponding local assignments;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' in the worst case they are all heavy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' In this case, we consider each of the following chains in turn: ({y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , ym−1, ym}, P1 ⊗ · · · ⊗ Pm−2 ⊗ |pi m−1⟩ ⊗ L{y1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=',ym−1},P1⊗···⊗Pm−2⊗|pi m−1⟩(yi m)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' For each of these chains in turn we perform the length m elimination procedure, thereby obtaining a set Γi m ⊂ ([n] − Γ) of qubits, where |Γi m| ≤ φ(m), which we add to Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' By definition of the length m elimination procedure, for each Γi there is no local as- signment to {y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , yi m} ⊔ Γi m which assigns P1 ⊗ · · · ⊗ Pm−2 ⊗ |pi m−1⟩ to the qubits {y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , ym−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We now stop the procedure and conclude that there is no local assign- ment to {y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , ym−1} ⊔ Γ which assigns P1 ⊗ · · · ⊗ Pm−2 to the qubits {y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , ym−2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Since Γ = �(|Γ0 m|+2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' i=0 (Γi m⊔{yi m}), we see that |Γ| ≤ ((φ(m)+2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='+1)(φ(m)+1) = φ(m−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We now observe that an elimination procedure exists for chains of length γ := γ(k, ϵ), with constant φ(γ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Indeed, for any such chain ({y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , yγ}, P1 ⊗ · · · ⊗ Pγ), as soon as we add any qubit to Γ we can conclude by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='4 that there is no local assignment to {y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , yγ} ⊔ Γ which assigns P1 ⊗ · · · ⊗ Pγ−1 to the qubits {y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , yγ−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' It follows that an elimination procedure exists for every 1 ≤ m ≤ γ, with constant φ(m) defined by the recurrence relation φ(m − 1) = ((φ(m) + 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' + 1)(φ(m) + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Finally, we observe that this gives rise to an elimination procedure for the empty assignment (∅, ∅), with constant φ(0) = φ(1) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' This procedure is as follows: pick a qubit x ∈ [n] which is bad w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' (∅, ∅) and add it to Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' If it conflicts then there is no local assignment on Γ = {x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' If it is heavy, then use the length 1 elimination procedure for the chain ({x}, L∅,∅(x)) to obtain a set Γ1 ⊂ ([n]−{x}), where |Γ1| ≤ φ(1), such that there is no local assignment on {x} ⊔ Γ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' The proof of the theorem now reduces to showing that if we pick qubits C = {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' , xc} ⊆ [n] at random, one at a time, then they will implement the elimina- tion procedure for the empty assignment with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Indeed, we recall from 17 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='5 that, with respect to any local assignment (S, PS), there are more than ϵn/5 bad qubits in ([n] − S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' For large enough n, every time we pick a new qubit there is therefore a roughly ϵ/5 chance of it being bad w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' any local assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' (The actual value will generally be slightly lower than ϵ/5 because we cannot pick any of the bad qubits that lie in ([n] − S) ∩ ˜C, where ˜C is the set of qubits which have already been picked;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' this is why we need n to be large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=') The elimination procedure for the empty assignment requires us to pick a bad qubit w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' some local assignment φ(0) times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Using the Chernoff bound for the binomial distribution, we find that the proba- bility that we have not picked φ(0) bad qubits after c(k, ϵ) := 5φ(0)/ϵ + 1 qubits have been picked is less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' References [AdGS21] Marco Aldi, Niel de Beaudrap, Sevag Gharibian, and Seyran Saeedi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' On efficiently solvable cases of 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Physical Review A, 81(6):062345, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' arXiv:0910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='2058, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='1103/PhysRevA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='062345.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' [LMSS10] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Laumann, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Moessner, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Scardicchio, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Sondhi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Random quantum satisfiability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Quantum Information and Computation, 10(1):1–15, Jan 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' [VZGG13] Joachim Von Zur Gathen and J¨urgen Gerhard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Modern Computer Algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Cambridge University Press, Cambridge, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' 19 A Proofs of combinatorial lemmas Lemma (Restatement of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Let θ, x ∈ R≥0 and c ∈ N, where 0 ≤ θ ≤ 1 and x ≥ c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Then, for some α ∈ [0, 1), we have: θ �x c � = �θ1/cx c � + α �θ1/cx c − 1 � < �θ1/cx + 1 c � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' (27) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Setting m = x − l for l < x, we have: θ �x c � − �m c � � m c−1 � = θx · · · (x − c + 1) cm · · · (m − c + 2) − m − c + 1 c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Now, if we set m = θ1/cx, we obtain: α := θ �x c � − �θ1/cx c � �θ1/cx c−1 � = 1 c � θx · · · (x − c + 1) θ1/cx · · · (θ1/cx − c + 2) − (θ1/cx − c + 1) � = 1 c � θ1/cx · · · (x − c + 1) x(x − θ−1/c) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' (x − θ−1/c(c − 2)) − (θ1/cx − c + 1) � ≤ 1 c (θ1/c(x − c + 1) − (θ1/cx − c + 1)) = (c − 1)(1 − θ1/c) c < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' The inequality then follows from the standard binomial recurrence relation �x c � + � x c−1 � = �x+1 c � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Lemma (Restatement of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' If G is an l-regular hypergraph with edge density θ, then the k-shadow G|k has edge density θ|k ≥ (θ1/l − 2/n)k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' The number of edges in G|k is the minimum number of k-edges in a k-regular hy- pergraph such that all the edges in G appear as complete l-vertex subgraphs of that graph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' for a given number of k-edges we may therefore ask what is the optimal arrangement maximising the number of complete l-vertex subgraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' It follows from [Kee08, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' 1] that a complete k-regular hypergraph on a subset S ⊆ [n] is an optimal arrangement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' It then follows from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='1 that G|k has at least �⌊θ1/ln⌋ k � edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Observing that ⌊θ1/ln⌋ ≥ θ1/ln−1 = (θ1/l−2/n)n+1 and using the inequality in (5) gives the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Lemma (Restatement of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Let n, c ∈ N be such that l := n/2c + δ and k := l/2 + ϵ, where −3 ≤ δ < −1 and −1/2 ≤ ϵ ≤ 0, are natural numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Let G be an l-regular hypergraph on n vertices with edge density θ ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Suppose that n ≥ 3(2c) and n ≥ 3(2c+1) � 1 2 + 1 ln(A) � A1/3 4 − ln(θ) �� for some constant A > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Then the edge density θ|k,1/2 of a partial (k, 1/2)-shadow of G has the following lower bound: θ|k,1/2 ≥ θ KA := θ 2 � A exp(A1/3/2) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We first fix some notation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' let ω = 1/2, α = 1/2c, and β = 1/2c+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' then l = (α +δ/n)n and k = (β +ϵ/n)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Let θ′ be the edge density of a k-regular hypergraph H′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We will find the largest edge density θ of an l-regular hypergraph H which has H′ as a (k, ω)-shadow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We observe that we can add an l-edge to H precisely when at least ω �l k � of the k-subsets of that l-edge are k-edges of H′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We thus have the following equation for the maximum number of edges in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Let S(H′) ⊆ [n] be the set of vertices which are contained in some k-edge of H′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' For 0 ≤ i ≤ |S(H′)|, let Ni be the number of subsets of S(H′) of size i such that the induced subgraph on that subset has ≥ ω �l k � edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Then: θ �n l � = l � i=0 Ni �n − |S(H′)| l − i � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' (28) 20 We now observe that Ni = 0 for all i ̸= l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Indeed, consider Nl−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' The required density of edges in a subset of H′ of size l−1 which contains ω �l k � edges is ω �l k � / �l−1 k � = ωl/(l−k) = ω(α+δ/n) α+δ/n−(β+ϵ/n), which is greater than 1 for all choices of −3 ≤ δ < −1, −1/2 ≤ ϵ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We therefore have Nl−1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' The sum (28) therefore reduces to a much simpler equation: θ = Nl �n l �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' (29) We immediately observe that as soon as θ′ > ω then we can obtain θ = 1 by distributing the edges uniformly over the n vertices, giving Nl = �n l � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Assuming from now on that θ′ < ω, we make some further observations which will enable us to bound θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' To maximise Nl the best distribution of k-edges is a uniform distribution over as many vertices as possible, with density ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' If we multiplied the edge density by 1 ω, we could form the complete hypergraph on these vertices, and so we see by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='1 that the maximum number of vertices is upper bounded by ⌈(θ′/ω)1/kn⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We therefore obtain the following bound: Nl ≤ �⌈(θ′/ω)1/kn⌉ l � ≤ � (θ′/ω)1/k + 1 n �l �n l � = � (θ′/ω) 1 βn+ϵ + 1 n �αn+δ �n l � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We now claim that, if n > 3(2c) and furthermore n ≥ 3(2c+1) ln(A) ln(1/(2θ′)) for some constant A > 1, then: Nl �n l � ≤ A(2θ′)2 exp � αA1/3� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' (30) Indeed, note that 0 < θ′ < ω < 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' thus, since n > 3(2c) implies that αn + δ > 0: � (θ′/ω) 1 βn+ϵ + 1 n �αn+δ ≤ � (θ′/ω) 1 βn + 1 n �αn+δ = (θ′/ω) αn+δ βn � 1 + 1 n(ω/θ′) 1 βn �αn+δ ≤ (θ′/ω) αn−3 βn � 1 + 1 n(ω/θ′) 1 βn �αn ≤ (θ′/ω) αn−3 βn exp � α(ω/θ′) 1 βn � = (θ′/ω) α β (ω/θ′) 3 βn exp � α(ω/θ′) 1 βn � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Substituting (30) into (29) and inverting the inequality, we obtain the statement in the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Lemma (Restatement of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Let n ∈ N be odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We construct an (n − 1)/2- regular hypergraph G as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' For every subset X ∈ σ(n−1)/2([n]), we do precisely one of the following: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Add X as an edge to G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Add all the subsets in σ(n−1)/2(X) as edges to G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' The edge density θ of a hypergraph G constructed in this way satisfies θ ≥ 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We call a choice of 1 or 2 for each X ∈ σ(n−1)/2([n]) a configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' For a given configuration, let S1 ⊂ σ(n−1)/2([n]) be the subsets for which option 1 is chosen, and let S2 ⊂ σ(n−1)/2([n]) be the subsets for which option 2 is chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We observe that there is always a configuration minimising the number of edges in G (a minimal configuration) which satisfies the property that, for every X ∈ S2, σ(n−1)/2(X) ⊂ S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Indeed, suppose this is not the case, and there is some Y ∈ σ(n−1)/2(X) ⊂ S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' But then one can change the configuration by moving Y to S1 without increasing the number of edges in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' If we 21 start with a minimal configuration, by doing this for all X ∈ S2, we obtain a minimal configuration satisfying the desired property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Let us calculate the minimal number of edges when such a configuration is chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Let C be the (n−1)/2-regular hypergraph on those vertices of G contained in an edge of S1, whose edges are precisely the subsets in S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Every Y ∈ S2 corresponds to a induced complete hypergraph on (n + 1)/2 vertices of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='4, the arrangement of (n−1)/2-edges maximising the number of induced complete hypergraphs is the complete hypergraph on C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We have Y ⊂ C for every Y ∈ S2, which implies that C ⊂ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We therefore obtain the following inequality giving a lower bound on the size of C: � |C| (n − 1)/2 � + � |C| (n + 1)/2 − (n − |C|) � ≥ � n (n − 1)/2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Setting |C| = n − 2, we find the inequality is violated;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' but setting |C| = n − 1, we find the equality is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' The minimal number of edges in G is therefore � n−1 (n−1)/2 � = n+1 2n � n (n−1)/2 � , and the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Lemma (Restatement of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Let G be a k-regular hypergraph on n vertices with edge density θ, where k = βn for some β ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Let N be the degree of a vertex picked uniformly at random, and let τ := N/ �n−1 k−1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Then, for any ϵ > 0 and α ∈ (0, 1): Pr [|θ − τ| ≥ min(∆(α, β, n), (1 − θ)(1 − α))] ≤ α, where ∆(α, β, n) = 5 exp � 1 12α(1−α)n + 1 12β(1−β)n � � 2πα(1 − α)β(1 − β)n =: 5(Lα,β)1/n Mα,β √n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We prove the case where τ < θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' the case τ > θ follows by considering the com- plement of the hypergraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' For any hypergraph G with edge density θ, let C ⊂ [n] be the subset of vertices whose degree is less than or equal to �τ �n−1 k−1 � , where �τ < θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' clearly Pr � � N ≤ �τ �n−1 k−1 �� = |C|/n, where � N is the degree of a vertex of G picked uniformly at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Let S := � v∈C dv, where dv is the degree of the vertex v ∈ C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' we must have S ≤ |C|�τ �n−1 k−1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We will obtain a lower bound for �τ in terms of θ and |C| by defining a configuration of edges which minimises S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Since we are interested in how the lower bound changes as n increases, we will consider |C| to be some function of n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' but θ will remain constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' The configuration is defined as follows: we first place all the edges contained entirely in (n − |C|), which does not increase S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Then we place all the edges which only contain one vertex in C, at the cost S �→ S + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Then we can place all the edges which only contain two vertices of C, at the cost S �→ S + 2, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We will stop when we have placed all θ �n k � edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Let r + 1 be the number of vertices of C in the final edge we place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' The following inequality encodes the fact that we have placed θ �n k � edges: r � i=0 �n − |C| k − i ��|C| i � ≤ θ �n k � ≤ r+1 � i=0 �n − |C| k − i ��|C| i � ⇒ P(Xn−|C|,|C|,k ≤ r) ≤ θ ≤ P(Xn−|C|,|C|,k ≤ r + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' (31) Here Xn−|C|,|C|,k is a random variable distributed according to the classical hypergeo- metric distribution with parameters n, |C|, k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' The following inequality encodes the fact that S ≤ |C|�τ �n−1 k−1 � : r � i=0 i |C| �n − |C| k − i ��|C| i � ≤ �τ �n − 1 k − 1 � ≤ r+1 � i=0 i |C| �n − |C| k − i ��|C| i � ⇒ P(Xn−|C|,|C|−1,k−1 ≤ r − 1) ≤ �τ ≤ P(Xn−|C|,|C|−1,k−1 ≤ r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' (32) 22 Here Xn−|C|,|C|−1,k−1 is a random variable distributed according to the classical hyper- geometric distribution with parameters n − 1, |C| − 1, k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' It follows that: P(Xn−|C|,|C|,k ≤ r) − P(Xn−|C|,|C|−1,k−1 ≤ r) ≤ θ − �τ ≤ P(Xn−|C|,|C|,k ≤ r + 1) − P(Xn−|C|,|C|−1,k−1 ≤ r − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' (33) For some intuition, we recall the standard model for the classical hypergeometric distri- bution with parameters n, |C|, k: there are n balls, |C| of which are white and (n − |C|) of which are black, and we pick k of them at random without replacement;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' the hyperge- ometric distribution counts the number of white balls picked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We observe that P(Xa,b,c ≤ r + 1) = a + b − c a + b P(Xa,b−1,c ≤ r + 1) + c a + bP(Xa,b−1,c−1 ≤ r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' (34) This equation is obtained by conditioning on whether a specific white ball is picked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' We further observe: P(Xa,b,c ≤ r) = P(Xa,b,c−1 ≤ r − 1) + a + r − (c − 1) a + b − (c − 1)P(Xa,b,c−1 = r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' (35) This equation is obtained by conditioning on how many of the first c − 1 balls picked are white.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Recall k = βn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' let |C| = αn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Then, substituting (34) and (35) into (33), we obtain: θ − �τ = P(Xn−|C|,|C|−1,k−1 = r) + n − |C| + r + 2 − k n P(Xn−|C|,|C|−1,k−1 = r + 1) ≤ P(Xn−|C|,|C|−1,k−1 = r) + (4 − α − β)P(Xn−|C|,|C|−1,k−1 = r + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' For the inequality we used that r < n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' The mode of Xn−|C|,|C|−1,k−1 is M := ⌊ k|C| n+1⌋, so by Stirling’s approximation we have: θ − �τ ≤ 5 �n−|C| k−M ��|C| M � �n k � ≤ 5ef(n) � 2παβ(1 − α)(1 − β)n ≤ 5 exp � 1 12α(1−α)n + 1 12β(1−β)n � � 2πα(1 − α)β(1 − β)n , (36) where f(n) = 1 12αn + 1 12βn + 1 12(1 − α)n + 1 12(1 − β)n − 1 12n + 1 − 1 12αβn + 1 − 1 12(1 − α)βn + 1 − 1 12(1 − α)(1 − β)n + 1 − 1 12α(1 − β)n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' The free variables are α and �τ, so we fix α and determine the minimal �τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' Since Pr � � N ≤ �τ �n − 1 k − 1 �� = |C| n = α, and, by taking the sum of the degrees of all the vertices, θ ≤ (1 − α) + α�τ ⇒ θ − �τ ≤ (1 − θ)(1 − α), we obtain the bound in the statement of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} +page_content=' 23' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFGT4oBgHgl3EQfgygC/content/2301.10699v1.pdf'} diff --git a/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf b/TdAzT4oBgHgl3EQf0v6x/content/2301.01789v1.pdf new file mode 100644 index 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b/U9FJT4oBgHgl3EQfNyw_/content/tmp_files/2301.11479v1.pdf.txt @@ -0,0 +1,4660 @@ +Alien Coding +Thibault Gauthier1, Miroslav Olšák2, and Josef Urban1 +1 Czech Technical University in Prague, Czech Republic +2 Institut des Hautes Etudes Scientifiques Paris, France +Abstract +We introduce a self-learning algorithm for synthesizing programs for OEIS sequences. The +algorithm starts from scratch initially generating programs at random. Then it runs many iterations of +a self-learning loop that interleaves (i) training neural machine translation to learn the correspondence +between sequences and the programs discovered so far, and (ii) proposing many new programs for +each OEIS sequence by the trained neural machine translator. The algorithm discovers on its own +programs for more than 78000 OEIS sequences, sometimes developing unusual programming methods. +We analyze its behavior and the invented programs in several experiments. +1 +Introduction +Galileo once said, "Mathematics is the language of Science." Hence, facing the same laws of the +physical world, alien mathematics must have a good deal of similarity to ours. +– R. Hamming - Mathematics on a Distant Planet [7] +Most of today’s successful coding assistants, e.g. GitHub Copilot [2], are trained on large code +repositories such as GitHub. This makes them quite versatile and capable of coding in multiple program- +ming languages. They can also transfer some of the knowledge acquired in one programming language, +provided there are large training corpora for both programming languages. However, since they are +trained in a supervised way to mimic existing human-written code, they may be biased towards possibly +non-optimal solutions. This may make them incapable of coming up with better solutions on their own. +In order to free themselves from human bias, techniques such as reinforcement learning (RL) can +be used. RL techniques do not rely on training examples but on search and rewards. An early example +of such a system is MENACE [12] which can learn to play noughts and crosses on its own and much +faster than brute-forcing all possible solutions. Recently, with a residual network as a machine learner, +AlphaGoZero [17] has learned playing Go better than professional players using only self-play. In this +process, the system discovered many effective moves that went against 3000 years of human wisdom. +The early 3-3 invasion was considered a bad opening move but is now frequently used by professionals. +The goal of this paper is to develop a self-learning system inspired by [17] capable discovering on its +own interesting and possibly unusual (alien) mathematical formulas/programs from common patterns +found in today’s mathematics. We choose those patterns to be integer sequences taken from the Online +Encyclopedia of Integer Sequences (OEIS) [18]. Formulas for these sequences will be constructed from +a small general programming language. This makes the search space manageable, allowing our system to +bootstrap itself without the need for supervised data. +Our work can also be viewed as an advancement in the field of automated theorem proving (ATP) [16]. +There, finding existential witnesses of the form ∃f.P(f) is a grand challenge. Our task is of this form: +to find a program f satisfying the property “P: f generates the sequence s“. In today’s ATP, this often +turns into brute-force enumeration. Our experience with ATP problems created from OEIS sequences is +that the ATP systems have hard time deriving a witness more complicated than the doubling function. +Recent efforts in automated reasoning go beyond the deduction paradigm. They incorporate feedback +loops between machine learning and theorem proving[22, 8, 10, 9] and neural models for conjecturing +and related synthesis tasks [21, 15, 13]. +arXiv:2301.11479v1 [cs.AI] 27 Jan 2023 + +Alien Coding +Gauthier, Olšák, Urban +Search +Check +Learn +programs +examples +weights +Figure 1: The tree phases of the self-learning loop. +Overview and Contributions +In order to learn how to find programs generating integer sequences, our +approach relies on a self-learning loop that alternates between the three phases represented in Figure 1 . +During the search phase, our machine learning model synthesizes programs for integer sequences. In this +work we predominantly use neural machine translation (NMT) as the machine learning component. For +each OEIS sequence, the NMT trained on previous examples typically creates 240 candidate programs +using beam search. In the first iteration (generation) of this loop, programs are randomly constructed. +Then, during the checking phase, the proposed millions of programs are checked to see if they generate +their target sequence, or any other OEIS sequences. The smallest and fastest programs generating an +OEIS sequence are kept to produce the training examples. In the learning phase, NMT trains on these +examples to translate the “solved” OEIS sequences into the best discovered program(s) generating it. +This updates the weights of the NMT network which influences the next search phase. Each iteration +of the self-learning loop leads to the discovery of more solutions, as well as to the optimization of the +existing solutions. +Our work builds mostly on our work in [6]. Our contributions are the following. The tree neural +network architecture is replaced by a relatively fast encoder-decoder NMT network (bidirectional LSTM +with attention). We replace the previously used MCTS search with a relatively wide beam search during +the NMT decoding. Our objective function lets us collect both the smallest and fastest programs for +each sequence instead of only the smallest programs. We provide many experiments comparing different +parameters (embedding size, programming language, search strategy). In particular, we experiment with +local and global definitions. We analyze the solutions and their evolution (getting faster and smaller) over +many generations. Our longest run finds solutions for more than 78000 OEIS sequences in 190 iterations, +and all our experiments have so far together produced solutions for 84587 OEIS sequences. This is more +than three times the number (27987) invented in our first experiments [6]. +2 +Components +In this section, we give a technical description of the components of our system: the OEIS datasets, the +programming language and it representations, and the checking phase. +2.1 +The OEIS Dataset +The OEIS is a repository created and maintained by Neil Sloane where amateur and professional +mathematicians can contribute integer sequences. There are currently more than 350,000 sequences +in this repository. Each entry contains the terms of the sequence and a short English description. It is +referenced by a A-number. For example, A40 is the reference for prime number. Additional information +may be provided such as: alternative descriptions, links to other entries and to papers where the sequence +was investigated and in about one third of the cases a program for generating the sequence is provided. +These programs are written in many different languages such as PARI, Matlab, Haskell, .... In our +experiments, we ignore the human-written programs. Thus, our problem set consists of all OEIS +sequences (351663 as of March 2022) without any corresponding programs. +2 + +Alien Coding +Gauthier, Olšák, Urban +2.2 +The Programming Language and its Python and NMT Representations +A formal description of the programming language used in this paper is given in [6]. It is minimalistic by +design, to avoid human-informed bias. Since many examples given in this paper require an understanding +of the language, we briefly summarize it here. The language contains two variables x and y, that can take +as values arbitrary-precision integers. It includes the standard operators 0, 1, 2, +, ×, mod, div (integer +division) and the conditional operators cond(a, b, c) := if a ≤ 0 then b else c. These programming +operators follow the standard semantics of most programming languages (including C and Python). +In this programming language, an expression p can either be evaluated to an integer if given specific +values for x and y or can be used to create a binary function f defined by f(x, y) = p. The three +looping operators of this language treat their arguments in these two different manners depending on +their positions. Looping expressions may themselves be used as arguments of looping operators allowing +for arbitrarily nesting of loops. +The loop Operator +This operator takes three arguments: one function and two integers. +loop(f, a, b) := b +if a ≤ 0 +f(loop(f, a − 1, b), a) +otherwise +This definition is almost the same as the one used to define primitive recursion in the standard theory +of primitive recursive functions. For more clarity and portability, we can translate this construction to +Python. We capitalize the variables in a and b, which play a different role than the variables in f, to avoid +undesirable variable capture. Python’s implementation F of the function that can be derived from the +expression loop(f, a, b) is as follows: +def F(X,Y) = +x = b[x/X,y/Y] +for y in range (1,a[x/X,y/Y] + 1) +x = f(x,y) +return x +Some common uses of loop include: 2x written as loop(2 × x, x, 1) and x! written as loop(y × x, x, 1). +The loop2 Operator +This operator takes five arguments: two functions and three integers. +loop2(f, g, a, b, c) := b +if a ≤ 0 +loop2(f, g, a − 1, f(b, c), g(b, c)) +otherwise +This operators starts with the pair of numbers (b, c) and updates their values a times using the functions f +and g before returning b. This is a generalization of the loop operator. Given g such that g(x, y) = y + 1, +we have loop(f, a, b) = loop2(f, g, a, b, 1). Therefore, the loop operator could be removed from +the language without affecting its expressiveness. It is kept in the language as it is a natural and +useful instantiation of loop2. Python’s implementation F of the function derived from the expression +loop2(f, g, a, b, c) is: +def F(X,Y) = +x = b[x/X,y/Y] +y = c[x/X,y/Y] +for _ in range (1,a[x/X,y/Y] + 1) +x = f(x,y) +y = g(x,y) +3 + +Alien Coding +Gauthier, Olšák, Urban +return x +The following constructions have a natural implementation using loop2. They are however difficult to +express using loop and would generally require encodings such as the Cantor pairing function: The +Fibonacci function Fibonacci(x) can be implemented by the program loop2(x + y, x, x, 0, 1), and the +power function xy by the program loop2(x × y, y, y, 1, x). +The compr Operator +The comprehension operator takes two arguments: one function and one integer. +compr(f, a) := +failure +if a < 0 +min{m | m ≥ 0 ∧ f(m, 0) ≤ 0} +if a = 0 +min{m | m > compr(f, a − 1) ∧ f(m, 0) ≤ 0} +otherwise +The comprehension expression finds the a + 1th smallest nonnegative integer m satisfying the predicate +f(m, 0). If the value of a is 0 then this behaves like the minimization operator µ in the theory of general +recursive functions, thus making the language Turing-complete. It gives a natural way of constructing in- +creasing sequences of numbers (i.e., sets) from a predicate. In particular, suppose we have constructed the +function fprime(x, y) = if x is prime then 0 else 1. Then the expression compr(fprime, x) constructs +the sequence of primes as as the value of x increases from 0 to infinity. Note that the operator thus +behaves similarly to the set comprehension operator in set theory. The Python’s implementation F of the +function derived from the expression compr(f, a) is: +def F(X,Y): +x,i = 0,0 +while i <= a[x/X,y/Y]: +if f(x,0) <= 0: +i = i + 1 +x = x + 1 +return x - 1 +Linear Representation of Programs for NMT +We use prefix notation (with argument order re- +versed) to represent a program as a sequence of tokens. The main advantage of this approach is +that the notation does not require the use of parentheses. For example, the prefix notation for the +program loop(x × y, x, 1) is loop 1 x × y x. When using NMT, the 14 operators/actions/tokens +[0, 1, 2, +, −, ×, div, mod, cond, loop, x, y, compr, loop2] are represented by capital letters from A to +N. Thus, the program for factorial is written as J B K F L K. +Definitions +To allow code re-use, human programmers introduce definitions. We allow NMT to +produce definitions in two different settings and experiments: one using local definitions and the other +using global definitions. We have decided that such definitions will be arbitrary sequences of actions. This +is quite a powerful setup since such definitions can represent not just subprograms, but also subprograms +with holes. A sequence of actions is called a macro. A program can be always constructed by a sequence +of actions, but not every sequence of actions is a program. +Local Definitions +In this setting, we add to the programming language ten tokens representing ten +possible local definitions (macros) that may include preceding macros. A special action/token is used as +a separator between the different macros. The macros in the generated programs are unfolded before the +4 + +Alien Coding +Gauthier, Olšák, Urban +Figure 2: Representing local macros. A macro version and expanded version of a program invented for A1813 +(a(n) = (2n)!/n!). Note that the macro here (K D L B) is not a proper program, only a sequence of actions. +checking phase takes place. Note that the naming of such macros is often inconsistent across different +programs,1 possibly making them harder to learn across many examples. Figure 2 shows an example of +the solution found for the sequence A18132 which involved synthesis of a local macro that is then used +three times in the body of the invented program. +Global Definitions +In this setup, we allow for arbitrarily many macros stored in a global array and +shared across all programs. This makes the naming of the macros consistent in all programs, possibly +making them easier to learn. Programs may refer to any macro stored in the global array, by writing its +index in base 10. This again requires 10 additional actions (one for each digit) and a special action to +separate the references to the macros. As in the local definition setup, the global macros may contain +references to macros with lower indices. Figure 3 shows an example of the solution found for the +sequence A141873 which involved three global macros that are altogether used five times in the body of +the invented program. +Introduction and use of local macros for a particular input integer sequence is completely a “local” +decision of the trained NMT that generates the particular program. In the global case, we however need +more coordination to introduce the global macros consistently. This can be done in various ways and +we now use the following method. At every iteration of the overall loop, we add the ten most frequent +sequences of actions to the array of global macros. To force the network to learn to use the global macros, +1E.g., in program P1, a macro called m could be used to define n!, while in P2, m could be used to define 2n. +2https://oeis.org/A1813 - Quadruple factorial numbers: a(n) = (2n)!/n!. +3https://oeis.org/A14187 - Cubes of palindromes. +5 + +Alien Coding +Gauthier, Olšák, Urban +Figure 3: Representing global macros. A macro version and expanded version of a program invented for A14187 +(cubes of palindromes). Note that two macros here (K C and B F F K K) are not proper programs, while the third +one (D F C D C C C) is a program that evaluates to 10. +we greedily (starting from the macros with the lowest indices) recognize sub-sequences of actions that +correspond to the macros, and replace them by the macros’ names (indices) in the programs. +2.3 +The Program Checker +In its most basic form the checker takes a program and a sequence and checks if that program generates +the sequence. Since programs may depend on two variables, we say that the program f(x, y) = p +generates the finite sequence (sx)0≤x≤n if and only if +∀x ∈ Z. 0 ≤ x ≤ n ⇒ f(x, 0) = sx +6 + +Alien Coding +Gauthier, Olšák, Urban +A40 +A5843 +A45 +0 +2 +2 +4 +6 +1 +1 +2 +3 +5 +7 +Figure 4: Tree of OEIS sequences with branches for primes (A40), even numbers (A5843) and Fibonacci numbers +(A45). +We say that a sequence s has a solution if we have found a program p generating it. The number of +OEIS sequences with at least one solution is the number reported in all our experiments under the label +solutions. +Timeout +The first issue when implementing a program checker is to determine the time limit for +running the (generally non-terminating) programs. In particular, to adapt the time limit for longer +sequences, we compute the generated terms in the order f(0, 0), f(1, 0), f(2, 0), ... and stop the program +if it has not generated the n-th term in less that n × tcall abstract time units. This effectively means that +we give a timeout of tcall time units per call with the time unused during previous calls added to the +timeout of the current call. +This abstract time unit is computed to be an approximation of the number of CPU instructions +needed to perform each operation. The cost of an operation is 1 for the +, −, × operations, it is 5 for the +div, mod operations, and it is the number of bits of the result if the result is larger than 64 bits. Using the +abstract time is also important to get accurate and repeatable measurements of the speed of the programs. +Hindsight Experience Replay +In order to augment the training data using a limited form of hindsight +experience replay [1], we check our program against all OEIS sequences at the same time. This can be +done effectively by organizing the sequences into a tree of sequences (Fig. 4) and stopping the checking +as soon as the generated sequence reaches a leaf in that tree or takes a non-existing branch in the tree. All +sequences (typically at most one) found along the path taken by the generated sequence are said to have +a solution. +Objectives +After each iterations we keep only the fastest and the smallest program (which could +be the same) for each sequence s. The speed of a program for s is the total number of abstract time +units necessary to generate s. The size of a program is the number of operators/tokens in its linear +representation. As soon as the checker has found a program that is a solution for a particular OEIS +sequence, we compare it with the existing solutions for that sequence. We use the abstract time to select +the fastest program among the ones that match the solutions. The fastest and smallest programs are also +used as the training examples for the next generation of the loop. +2.4 +Comprehension Limit +Evaluating each term of the sequence compr(f, 0), ,compr(f, 1),. . .,compr(f, n − 1) separately is most +of the time too slow. This computation can be sped up using the fact that each term can be computed from +the preceding term in the sequence. In general, when executing a program containing comprehension +7 + +Alien Coding +Gauthier, Olšák, Urban +operators, we precompute the results of applying compr(f, i) for each f appearing as the first argument +of compr, where the number i ranges from 0 to ncompr −1. The number ncompr is a parameter called the +comprehension limit. The pre-computation times out if it takes more than i × tcall time units to produce +the outputs for compr(f, 0), ..., compr(f, i). When the top program is executed, a call to a compr(f, a) +subprogram times out if no precomputed value exists for the input i created by the subprogram a. +Otherwise, the call returns the precomputed value for compr(f, i) to the top program. +2.5 +Choice of the Timeout Parameters +The two parameters that determine how long a program may be run for is the timer per call tcall and +the comprehension limit ncompr. A program times out if it exceeds the timeout or if one of the compr +expression reaches the comprehension limit or if an integer with an absolute value larger than 10285 is +produced. We may run either a fast check, a slow check or a hybrid check on the set of candidate programs. +The fast check uses as parameters tcall = 1000, ncompr = 20, and the slow check uses tcall = 100000, +ncompr = 200. +Hybrid Check +The hybrid check tries to achieve the performance of the fast check while retaining +most of the additional solutions found by the slow check. The first phase of the hybrid check is the +fast check. After this check, we look at the programs that generated a prefix of an OEIS sequence +but could not complete the full task. At this point, if we were to perform the slow check on all those +prefix-generating programs, the hybrid check would take a time equivalent to the slow check. To get +a gain in performance, we select only the ones that are the smallest for each prefix to be tested for the +longer time. Fast programs implicitly differentiate themselves from others by generating longer prefixes, +therefore are also selected by the same criteria for further checking. +In most of our experiments, we use the hybrid check because it is about 15 times faster than the slow +check. However, since it does not test all programs with the long timeout, it misses out on some solutions +found by the slow check. In the longer NMT runs, we eventually switched from the hybrid check to the +more robust slow check, to discover more solutions. +3 +OEIS Synthesis as an NMT Task +Neural networks have in the last decade become competitive in language modeling and machine transla- +tion tasks, leading to applications in many areas. In particular, recurrent neural networks (RNNs) with +attention [3] and transformers [23] have been recently applied in mathematical and symbolic tasks such +as rewriting [14], autoformalization [24] and synthesis of mathematical conjectures and proof steps. +Many of these tasks are naturally formulated as sequence-to-sequence translation tasks. +NMT Representation +The OEIS program synthesis can also be cast as such task. In this work we +therefore experiment with replacing the TNN architecture with a reasonably fast encoder-decoder neural +machine translation (NMT) system. In particular, we represent the input integer sequence as a series of +digits, separated by an additional token at the integer boundaries. Since the initial integers in a sequence +are typically smaller and may be more informative for the NMT decoding phase, we reverse the input +sequences. The output program is also represented as a sequence of tokens in Polish notation (Fig. 5). +Beam Search +To make full use of the NMT capabilities, we also replace the original MCTS search +with a wide beam search during the NMT decoding. Beam search with width N is an alternative to +greedy decoding. Instead of a single greedily best output, NMT in beam search keeps track of the N +8 + +Alien Coding +Gauthier, Olšák, Urban +Figure 5: Representing sequences and solutions for NMT. +conditionally most probable outputs, updating their ranking after each decoding step. When the NMT +decoding for a particular input OEIS sequence s is finished, the final N best outputs can be used as +NMT’s N alternative suggestions of programs that solve s. +NMT Framework and Hyperparameters +After several initial evaluations we have chosen for the +experiments Luong’s NMT framework [11]. It works efficiently on our hardware both in the training and +wide-beam inference mode, and we were able to find suitable hyperparameters for it [24]. In more detail, +we use for most experiments a 2-layer bidirectional LSTM equipped with the “scaled Luong” attention +and 512 units. In our NMT experiments (Section 5) we start by using one NMT model for training and +inference, using many default NMT hyperparameters. As the iterations progress, we gradually adjust the +parameters and add more models trained differently and on differently selected data. We also experiment +with larger models. +Combining NMT Models +In most NMT runs we train two to four different NMT models in parallel +each on its own GPU. We then run the inference with two of them in parallel, thus using all 4 GPUs on the +server. The rationale behind training and inference with differently trained models is the standard portfolio +argument, used routinely, e.g., in automated theorem proving [19, 8]. A complementary portfolio of +specialists typically outperforms a single general strategy. In feedback loops that alternate between proof +search and learning [22], this also further benefits the learning phase, since each learner can additionally +use the training data accumulated by others. In Section 5 we see that this indeed considerably improves +the performance. +Continuous training +NMT models can be trained either only on the latest version of the solutions or +in a continuous way. The latter re-uses the model trained in the previous iteration and trains it on the +latest data for more steps. This makes such model more stable, being eventually trained for orders of +magnitude more steps. It also makes it different from the models trained from scratch only on the latest +data. Even when only a few solutions arrive in the latest iteration, the network is training further on +the whole latest corpus, thus becoming smarter and hopefully more competent for the next inference +9 + +Alien Coding +Gauthier, Olšák, Urban +Table 1: Solutions at generation 0,5,10,15 and 20. +Embedding size +16 +2005 +14920 +19674 +21163 +22203 +32 +1972 +17608 +23490 +25750 +27351 +64 +2017 +18156 +23737 +26490 +28463 +96 +1993 +16051 +20127 +22890 +24718 +96 l.s. +3771 +20434 +25378 +28361 +30344 +Language (embedding size 64) +minimal +1157 +7096 +8547 +9126 +9965 +default +2017 +18156 +23737 +26490 +28463 +extra +1763 +18690 +23757 +26905 +28794 +Objective (local search) +both +3771 +20434 +25378 +28361 +30344 +small +3725 +20501 +26231 +29124 +31520 +fast +3716 +21021 +26270 +29326 +31421 +phase. The models trained only on the latest corpus are on the other hand less concerned by the old +(slower/longer) solutions and more focused on exploring the latest ones. +4 +Experiments with a Tree Neural Network +In the work of [6], a tree neural network (TNN) serves as a machine learning model. In the following, +we replicate their work and test how varying a range of parameters influence the self-learning process. +To test the limit of the TNN, we run the TNN-guided learning loop for 500 generations instead of the +original 25. This final experiment also provides a baseline for the NMT experiments. +Varying Parameters +We investigate the effect of three different parameters: the TNN embedding size, +the choice of the programming language and the choice of the objectives. Unless specified otherwise, +the default value for those parameters are respectively 96 for the dimension, the previously described +programming language (see Section 2.2) and the selection of the smallest and fastest programs. In +Table 1, we present the results of running experiments with different parameters for 20 generations. In +a first experiment (first block in the table), we observe that the number of solutions increases with the +dimension until dimension 64. Surprisingly, dimension 96 gave worse results, this is mainly due to the +fact that larger networks are more expensive to compute and therefore produce fewer programs. As +a countermeasure, we introduce a local search (l.s.) that tests all programs that are one action away +from being constructed in the search tree. This balances the generation time and checking time better +leading to an increase in the number of solutions. In a second experiment (second block in the table), +we measure how changing the programming language affects the performance. All the experiments +presented in this block use dimension 64. The minimal row runs the self-learning loop with an even +more minimalistic programming language consisting of four operators: 0, successor, predecessor, loop. +This makes the learning much more challenging. One of the reasons is that creating a large number +such as one million in this language requires at least one million steps. Therefore, it is impossible to +10 + +Alien Coding +Gauthier, Olšák, Urban +produce large numbers within the checker’s time limit. Such considerations justify the inclusion in the +default language of operators efficiently computed by current hardware such as + and ×. The extra row +shows what happens when we include the extra constants 3,4,...,10 as primitive operators. This makes +the computation of large numbers more efficient which increases the performance of our system slightly. +However, introducing more and more primitive operators introduces human bias that we would rather +avoid. In a third experiment (third block in the table), the results of learning with different program +objectives are presented. All these experiments were performed with dimension 96 and local search. +The small (resp. fast) row shows the effect of only collecting and learning from the smallest (resp. the +fastest) programs for a given OEIS sequence. The results imply that focusing on one objective at a time +simplifies the work of the machine learner. Yet, we expect more synergy between the two objectives to +occur during longer runs. +Long Run +The result of a long-lasting experiment, running for 500 generations with the default +parameters and local search, is shown in Fig. 6 (tnn). To fit also the NMT runs, we display only the first +190 iterations, however the average increments (Fig. 7, tnn) between iteration 200 and 300 drops below +20. At the end of the run, the TNN seems to have reached its limit and about five new solutions are found +at each generation. As we will see in Section 5, due to its larger embedding size and its more involved +architecture and the introduction of continuous training, the main NMT experiment does not plateau and +reaches a much higher number in an equivalent amount of time. +5 +Experiments with NMT +5.1 +Basic run (nmt0) +In the basic NMT run4 (nmt0), we are running the loop in a way that is most similar to the TNN run. In +particular, the checking phase is interleaved with training a single NMT model0, which is then used for +the search phase, implemented as NMT inference using a wide beam search. As in the TNN experiment, +we start with the initial random generation which yields 3771 solutions. Then we run 100 iterations of +the NMT-based learn-generate-check loop. +Training: We use a batch size of 512 and SGD with a learning rate of 1.0. In each iteration, we train +on the latest set of solutions, initially for 12000 steps, and since iter. 45 for 14000 steps (to adjust for the +growing number of training examples).5 The bleu scores on small test and development sets are typically +between 25 and 30. There is only one memory crash (iter. 97). +Inference: We use two GPUs in parallel (splitting OEIS into two parts), each with a batch size of 32 +and beam width 240. These values are determined experimentally, to load the GPUs efficiently without +memory crashes. The parallelized inference time grows from 2 to 8 hours, increasing as the iterations +invent longer examples, and the trained network and the beam search become less confident about when +to stop decoding. Each inference phase yields 240 ∗ 351663 = 84.4M program candidates that are then +checked. +Checking: We use the hybrid checking mode (Section 2.5) parallelized over 18 CPUs. The checking +time grows from about 2 minutes to about 8-10 minutes. This is negligible compared to the NMT training +4https://github.com/Anon52MI4/oeis-alien/tree/master/run0 +5This corresponds to 438 epochs in iter. 3 where there are about 14k examples, 92 epochs in iter. 44 (67k examples), 107 epochs in +iter. 45 (after switching to 14000 steps), and 76 epochs in iter. 101 (81k examples). On one GPU (GTX1080 Ti, 12G RAM) the +training takes on average 100m with 12000 steps and 130m with 14000 steps. +11 + +Alien Coding +Gauthier, Olšák, Urban +and inference times. The 100 iterations of this loop took about a month of real time, reaching 46707 +solutions (Fig. 6, nmt0). However, the increments drop below 200 and 100 after iteration 37 and 94, +respectively. +5.2 +Long extended run (nmt1) +Since the time taken by one nmt0 iteration reaches about half a day towards the end of the run, we explore +more efficient approaches. This leads to the longest run6 nmt1, which has at the time of submitting this +paper reached 190 iterations and over 78000 solutions (Fig. 6, nmt1).7 Unlike in nmt0, the number of +new solutions produced in each iteration rarely drops below 200, even after many iterations (Fig. 7, nmt1). +This challenges the standard wisdom of “plateauing curves” appearing in the TNN and nmt0 runs. +Combining models: The nmt1 run bootstraps from nmt0, inheriting its first 20 iterations, thus +starting with 34420 solutions. Then we start combining multiple NMT models (Section 3). Since iter. 21, +we add training of model1 to model0. It is trained only on a randomly chosen half of the training set, +however for twice as many steps/epochs. This yields a differently trained (and more focused) specialist +in each iteration. The number of solution candidates produced by the inference phase thus doubles, to +168.8M. After de-duplication, this yields 32.5M unique candidates in iter. 21 compared to 13.4M in iter. +20. The checking (initially also using the hybrid mode) is still fast, taking 5 minutes on 18 CPUs. The +difference to nmt0 is remarkable: 687 new solutions in nmt1 vs 272 in nmt0 in iter. 21. This effect +continues over the next iterations, see Fig. 7. +Further modifications: Since iter. 70, we start training model1 in a continuous way (Section 3). +By iter. 190 model1 is thus trained for over 1.4M steps on the evolving data, making it quite different +from other models. As we invent longer programs, we also allow training on longer sequences, raising +the default NMT values of 50 input and 50 output tokens gradually to 80 and 140, respectively. Since +iter. 156 we also add training of model2, which takes specialization even further. It is trained for four +times as many steps as model0 on a random quarter of the data. The bleu scores of model0 are at this +point low, due to the larger size of the data, while model2 still achieves scores above 25. We then use +model0 only as a backup when model2 diverges. Since iter. 159 we switch from the hybrid check to the +slow (full) check (Section 2.5), raising the checking time from 45m (iter. 158) to 6h, and the number of +solutions from 178 to 860 (Fig. 7, nmt1). This jump is likely due to many programs using comprehension +being newly allowed. It includes sudden solutions for hundreds of problems that combine congruence +operations and primes.8 +Bigger Network Since iter. 170, we add continuous training of a bigger model3 which uses 1024 +units instead of 512. To keep all models and phases in sync, we train model3 for fewer steps (8000), +decreasing also its batch size and inference width to 288 and 120, respectively. Table 2 analyzes the +benefits of using model3, model2 and model0 in addition to model1over 12 iterations (175-186). The +number of unique candidates is much lower for model3 (due to the beam width) than for model2, however +still higher than for model0, which is at this point likely undertrained. In general, model3 performs better +than model0, but is inferior to model2. A faster model, producing twice as many plausible candidates in +the same amount of time, is here better than a slower model which is a bit more precise.9 +6https://github.com/Anon52MI4/oeis-alien/tree/master/run1 +7The 190 nmt1 iterations took 3 months on a 4-GPU server. +8https://bit.ly/3QPkquE +9Also, model3 is trained continuously and may resemble more model1, providing fewer different candidates. And model2 is +trained only on a random quarter of the latest data, making it even more orthogonal to the continuous models that have seen much +more data. +12 + +Alien Coding +Gauthier, Olšák, Urban +Table 2: Influence of using models M3, M2 or M0 for inference in addition to M1. UC are unique candidates +(millions), NS new solutions added, TS total solutions including all found so far, and OS own solutions, i.e., the +sequences covered by the current iteration. +Iter +175 +176 +177 +178 +179 +180 +Model +M2 +M3 +M0 +M2 +M2 +M2 +UC +101.7 +79.7 +64.7 +99.7 +96.0 +106.2 +OS +74746 +74916 +74130 +75196 +75386 +75313 +TS +75471 +75666 +75795 +75985 +76160 +76404 +NS +260 +195 +129 +190 +175 +244 +Iter +181 +182 +183 +184 +185 +186 +Model +M2 +M2 +M2 +M2 +M3 +M3 +UC +105.8 +106.1 +106.7 +101.1 +84.5 +85.7 +OS +75686 +75986 +76271 +76397 +76793 +77007 +TS +76656 +76861 +77055 +77244 +77411 +77577 +NS +252 +205 +194 +189 +167 +166 +5.3 +Runs with Local and Global Macros +As the average program size grows in nmt1 (Fig. 8), the NMT decoding times increase, taking almost 12h +in the latest iterations. Verbatim repetition of blocks of code also feels suboptimal to human programmers. +This motivates later additional runs nmt2 and nmt3, where we experiment with using local and global +macros. Apart for allowing these additional macro mechanisms (Section 2.2), the two runs resemble +run nmt1. In each of them we train the basic model0 together with the continuous model1, and infer +with both of them. Since the sequences are shorter (making model0 easier to train), we have not so far +experimented with model2 here. While the two runs seem superior to nmt1 in Fig. 7, this is mainly due +to the use of both models since iter. 2 (instead of iter. 21 in nmt1), and also an earlier switch to the slow +check (iter. 75 in nmt2 and iter. 67 in nmt3). The two runs also do not significantly complement nmt1: +each of them adds less than 2800 solutions to nmt1. This may mean that the alien system nmt1 has so +far less trouble than humans with expanding everything and the use of definitions is not as critical as in +humans. None of the two runs has however reached iter. 100 yet, making the comparison with nmt1 +only preliminary. Especially the statistics of the use of global macros10 in nmt3 are interesting, and can +be used for analyzing the evolution of its coding trends. +6 +Analysis of the Results +We provide a statistical analysis of the 78118 solutions found during the nmt1 run and discuss the +details of some techniques developed by our system. More information on the nmt runs is available +in our anonymous repository.11 For some sequences, its subdirectory12 contains our analysis of the +10https://bit.ly/3ZNe7fm +11https://bit.ly/3iVIfnX +12https://bit.ly/3XHZsjK +13 + +Alien Coding +Gauthier, Olšák, Urban +0 +20000 +40000 +60000 +80000 +25 +50 +75 +100 +125 +150 +175 +tnn +nmt0 +nmt1 +nmt2 +nmt3 +Figure 6: All solutions. +0 +250 +500 +750 +1000 +25 +50 +75 +100 +125 +150 +175 +tnn +nmt0 +nmt1 +nmt2 +nmt3 +Figure 7: Increments of solutions. +14 + +Alien Coding +Gauthier, Olšák, Urban +Generation +Avrg. Size +0 +20 +40 +60 +25 +50 +75 +100 +125 +150 +175 +small +fast +Figure 8: Avrg. size in iters. +Generation +Avrg. Time +20000 +40000 +60000 +80000 +200000 +400000 +600000 +25 +50 +75 +100 +125 +150 +175 +fast +small +Figure 9: Avrg. time in iters. +15 + +Alien Coding +Gauthier, Olšák, Urban +Generation +Avrg. Size After X Iters. +0 +10 +20 +30 +20 +40 +60 +80 +100 +Figure 10: Avrg. size after X iters. +Generation +Avrg. Time After X Iters. +0 +50000 +100000 +150000 +200000 +250000 +20 +40 +60 +80 +100 +Figure 11: Avrg. time after X iters. +16 + +Alien Coding +Gauthier, Olšák, Urban +evolution13,14 and proliferation15,16 of important programs such as primes and sigma, as well as proofs +that some of the alien programs match the human OEIS intention.17,18,19 See also the appendix for more +details. Noteworthy sequences from this repository include convolution of primes with themselves,20 +showing the proficiency of our system with primes. Motzkin numbers21 [25] are an example where our +synthesized programs relies on a pairing function. This programming technique, re-invented by our +system, packs two variables into one, allowing further programs (see Appendix B for details). And a +solution for the unique monotonic sequence of nonnegative integers satisfying a(a(n)) = 3n is needed +in solving a problem in 27th British Math. Olympiad.22 +Evolution of the Programs +Fig. 8 shows the evolution of the average size and Fig. 9 the speed of the +solutions. We see that as new solutions are found, they become longer and typically also take more time. +However, there are interesting exceptions to the latter rule (Fig. 11), as more efficient code is invented +and propagated by the alien system. We thus also measure the gradual size reduction (Fig. 10) and time +reduction (Fig. 11) for the short and fast solutions (respectively). This is for each sequence computed for +100 iterations after its first solution was found. We see that the iterations induce a remarkable speedup of +the invented fast solutions (Fig. 11). +Generalization of the Solutions to Larger Indices +OEIS provides additional terms for some of the +OEIS entries in b-files. Among the 78118 solutions, 40,577 of them have a b-file that contains 100 +additional terms for their OEIS entry. We evaluate both the small and the fast programs with the slow +check parameters on the 100 additional terms. Here, 14,701 small and 11,056 fast programs time out. +Among the programs that do not timeout, the percentage of generalizing programs (producing matching +additional terms) is 90.57% for the slow programs and 77.51% for the fast programs. A common error is +reliance on an approximation for a real number, such as π. +7 +Related work +The most relevant related work is summarized in [6]. This includes [4], where the general approach is +focused on training a single model using supervised learning techniques on synthetic data. The deep +reinforcement learning system DreamCoder [5] has demonstrated self-improvement from scratch in +several programming tasks. Its main contribution is the use of definitions that compress existing solutions +and facilitate building new solutions on top of the existing ones. Our experiments with local and global +macros are however so far inconclusive. While humans certainly benefit from introducing new names, +concepts and shortcuts, it is so far unclear if it results in a clear improvement in our current alien setting. +While the previous version of our system used MCTS inspired by AlphaGoZero [17], the current +NMT approach uses just straightforward beam search. The general search-verify-learn positive feedback +loop between ML-guided symbolic search and statistical learning from the verified results has been +used for long time in large-theory automated theorem proving, going back at least to the MaLARea +13https://bit.ly/3iJ4oGd +14https://bit.ly/3HfwemI +15https://bit.ly/3ZNExO4 +16https://bit.ly/3IVzrJz +17https://bit.ly/3QPB25o +18https://bit.ly/3WlVHPM +19https://bit.ly/3XJi96j +20https://bit.ly/3XJi96j +21https://bit.ly/3QPB25o +22https://bit.ly/3wjmqSg +17 + +Alien Coding +Gauthier, Olšák, Urban +system [20, 22]. Its recent instances include systems such as rlCoP [10] and ENIGMA [9]. Such systems +can be also seen as synthesis frameworks, where the mathematical object synthesized by the guided search +is however the full proof of a theorem, rather than just a particular interesting witness or a conjecture. +Synthesis of full proofs of mathematical theorems is an all-encompassing task that can be extremely +hard in cases like Fermat’s Last Theorem. Our current setting may thus also be seen as an attempt to +decompose this all-encompassing task into smaller interesting subtasks that can be analyzed separately. +8 +Conclusion +As of January 25, 2023, all of the runs have together invented from scratch solutions for 84587 OEIS +sequences. This is more than three times the number (27987) invented in our first experiments [6]. This +is due to several improvements. We have started to collect both the small and fast solutions, and learn +jointly from them. We have seen that this gradually leads to a large speedup of the fast programs, as the +population of programs evolves. This likely allows invention of further solutions that are within our time +limit, thanks to the use of the faster and faster invented components. The improvements (learning) on +the symbolic side thus likely complement and co-evolve with the statistical learning used for guiding +the synthesis. We have also started to use relatively fast neural translation models, their specialization +on random subsets, their combinations and continuous training, and a wide beam search instead of +MCTS. The experiments suggest that these techniques are useful, leading to a high rate of program +invention even after the 190 iterations in the longest NMT run. This is encouraging, since many of the +solved OEIS problems seem nontrivial. The trained system could thus be used as a search tool assisting +mathematicians. There is also a wealth of related mathematical tasks that can be cast in a similar synthesis +setting, and combined with other tools, such as automated theorem provers. +A crucial element of our setting is the fact that NMT is used to produce an interpretable symbolic +representation. It would be practically unusable if we relied purely on neural approximation of arbitrary +functions and the task for NMT was to directly produce the next 100 numbers in each OEIS sequence. +The interpretable symbolic representation is critical for the generalization capability of the overall system +and its capability to learn from itself. The overall alien system can also be seen as an example of a +very weakly supervised evolutionary architecture where simple high-level principles (Occam’s razor and +efficiency) govern the development and training of the lower level statistical component (NMT). Rather +than one-time training on everything that has been already invented by humans so far, as done by today’s +large language models, the system here starts from zero knowledge and progresses towards increasingly +nontrivial knowledge and skills. Such feedback loops thus seem to be a good playground for exploring +how increasingly intelligent systems emerge. Note that thanks to the Turing completeness of the language, +this particular playground is (unlike games like Chess and Go) not limited in its expressivity, and can in +principle lead to the development of arbitrary complex algorithms. +9 +Acknowledgments +We thank Chad Brown, David Cerna, Hugo Cisneros, Tom Hales, Barbora Hudcova, Jan Hula, Mikolas +Janota, Tomas Mikolov, Jelle Piepenbrock, and Martin Suda for discussions, comments and suggestions. +This work was partially supported by the CTU Global Postdoc funding scheme (TG), Czech Science +Foundation project 20-06390Y (TG), ERC-CZ project POSTMAN no. LL1902 (TG), Amazon Research +Awards (TG, JU), EU ICT-48 2020 project TAILOR no. 952215 (JU), and the European Regional +Development Fund under the Czech project AI&Reasoning no. CZ.02.1.01/0.0/0.0/15_003/0000466 +(MO, JU). +18 + +Alien Coding +Gauthier, Olšák, Urban +References +[1] Marcin Andrychowicz, Filip Wolski, Alex Ray, Jonas Schneider, Rachel Fong, Peter Welinder, Bob McGrew, +Josh Tobin, OpenAI Pieter Abbeel, and Wojciech Zaremba. Hindsight experience replay. 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CoRR, +abs/2202.11908, 2022. +[7] Richard Wesley Hamming. Mathematics on a distant planet. The American Mathematical Monthly, 105(7):640– +650, 1998. +[8] Jan Jakub˚uv and Josef Urban. Hierarchical invention of theorem proving strategies. AI Commun., 31(3):237– +250, 2018. +[9] Jan Jakub˚uv and Josef Urban. Hammering Mizar by learning clause guidance. In John Harrison, John +O’Leary, and Andrew Tolmach, editors, 10th International Conference on Interactive Theorem Proving, ITP +2019, September 9-12, 2019, Portland, OR, USA, volume 141 of LIPIcs, pages 34:1–34:8. Schloss Dagstuhl - +Leibniz-Zentrum für Informatik, 2019. +[10] Cezary Kaliszyk, Josef Urban, Henryk Michalewski, and Miroslav Olsák. Reinforcement learning of theorem +proving. In Advances in Neural Information Processing Systems 31: Annual Conference on Neural Information +Processing Systems 2018, NeurIPS 2018, 3-8 December 2018, Montréal, Canada., pages 8836–8847, 2018. +[11] Minh-Thang Luong, Eugene Brevdo, and Rui Zhao. Neural machine translation (seq2seq) tutorial. https: +//github.com/tensorflow/nmt, 2017. +[12] Donald Michie. Experiments on the Mechanization of Game-Learning Part I. Characterization of the Model +and its parameters. The Computer Journal, 6(3):232–236, 11 1963. +[13] Jelle Piepenbrock, Josef Urban, Konstantin Korovin, Miroslav Olsák, Tom Heskes, and Mikolas Janota. +Machine learning meets the Herbrand Universe. CoRR, abs/2210.03590, 2022. +[14] Bartosz Piotrowski, Josef Urban, Chad E. Brown, and Cezary Kaliszyk. Can neural networks learn symbolic +rewriting? CoRR, abs/1911.04873, 2019. +[15] Markus Norman Rabe, Dennis Lee, Kshitij Bansal, and Christian Szegedy. Mathematical reasoning via +self-supervised skip-tree training. In International Conference on Learning Representations, 2021. +[16] John Alan Robinson and Andrei Voronkov, editors. Handbook of Automated Reasoning (in 2 volumes). Elsevier +and MIT Press, 2001. +[17] David Silver, Julian Schrittwieser, Karen Simonyan, Ioannis Antonoglou, Aja Huang, Arthur Guez, Thomas +19 + +Alien Coding +Gauthier, Olšák, Urban +Hubert, Lucas Baker, Matthew Lai, Adrian Bolton, Yutian Chen, Timothy Lillicrap, Fan Hui, Laurent Sifre, +George van den Driessche, Thore Graepel, and Demis Hassabis. Mastering the game of Go without human +knowledge. Nature, 550:354–, 2017. +[18] N. J. A. Sloane. "a handbook of integer sequences" fifty years later. CoRR, abs/2301.03149, 2023. +[19] Tanel Tammet. Towards efficient subsumption. In CADE, volume 1421 of Lecture Notes in Computer Science, +pages 427–441. Springer, 1998. +[20] Josef Urban. MaLARea: a metasystem for automated reasoning in large theories. In Geoff Sutcliffe, Josef +Urban, and Stephan Schulz, editors, ESARLT, volume 257 of CEUR Workshop Proceedings. CEUR-WS.org, +2007. +[21] Josef Urban and Jan Jakubuv. First neural conjecturing datasets and experiments. In CICM, volume 12236 of +Lecture Notes in Computer Science, pages 315–323. Springer, 2020. +[22] Josef Urban, Geoff Sutcliffe, Petr Pudlák, and Jiˇrí Vyskoˇcil. MaLARea SG1 - Machine Learner for Automated +Reasoning with Semantic Guidance. In Alessandro Armando, Peter Baumgartner, and Gilles Dowek, editors, +International Joint Conference on Automated Reasoning (IJCAR), volume 5195 of LNCS, pages 441–456. +Springer, 2008. +[23] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N. Gomez, Lukasz Kaiser, +and Illia Polosukhin. Attention is all you need. In Isabelle Guyon, Ulrike von Luxburg, Samy Bengio, Hanna M. +Wallach, Rob Fergus, S. V. N. Vishwanathan, and Roman Garnett, editors, Advances in Neural Information +Processing Systems 30: Annual Conference on Neural Information Processing Systems 2017, 4-9 December +2017, Long Beach, CA, USA, pages 5998–6008, 2017. +[24] Qingxiang Wang, Cezary Kaliszyk, and Josef Urban. First experiments with neural translation of informal to +formal mathematics. In CICM, volume 11006 of Lecture Notes in Computer Science, pages 255–270. Springer, +2018. +[25] Yi Wang and Zhi-Hai Zhang. Combinatorics of generalized motzkin numbers. J. Integer Seq., 18(2):15.2.4, +2015. +20 + +Alien Coding +Gauthier, Olšák, Urban +A +Resources Used by the Experiment +The average GPU power consumption during inference is 800W: 200W per each of the four GTX 1080 +Ti cards used for inference. The average GPU power consumption during training is 350W: 175W per +each of the two GTX 1080 Ti cards used for the two kinds of training done in most of the iterations. +The training takes approximately 3 hours and the inference 10.5 hours (iteration 140 of the longest +run). This means that the average GPU consumption per hour is (3 ∗ 350 + 10.5 ∗ 800)/13.5 = 700W. +Additionally, we estimate the non-GPU (CPUs, disks, etc.) consumption to be on average 150W per hour +(the server’s CPUs are mostly idle). It took about three months to do the 190 iterations of the longest +run. This is 90 ∗ 24 = 2160 hours. The estimated power consumption for the 190 iterations is thus +2160 ∗ (700 + 150) = 1836 kWh. Assuming a cost in the range of 150-250 EUR per MWh, the main +experiment’s electricity cost is between 270 and 460 EUR. The three shorter experiments have so far +run for about half as many iterations and are also using on average less resources (fewer GPUs, shorter +inference length). The estimated power consumption for all experiments is thus likely below 4 MWh, +and the total electricity cost below 1000 EUR. +B +Triangle Coding +Our system has found a correspondence between a pair of non-negative integers (xa, xb) where xa ≥ xb, +and a single non-negative integer x by enumerating the following sequence of pairs (xa, xb) with +non-negative integers: +(0, 0), (1, 0), (1, 1), (2, 0), (2, 1), (2, 2), (3, 0), . . . . +In one direction, it computes x = xa×(xa+1) +2 ++ xb in order to “encode” the pair (xa, xb). Decoding +of a single non-negative integer x can be calculated as (xa, xb) = (f0(x) − f1(x), f0(x)), where the +functions f0, f1 can be implemented in Python for example as follows (an actual code invented by the +program generator). Note that the function f0 computes xb, and the function f1 computes xb − xa (a +non-positive integer). +def f0(X): +x = X +for y in range (1,(2 + (X // (1 + (2 + 2)))) + 1): +x = x - (y if (y - x) <= 0 else 0) +return x +def f1(X): +x = X +for y in range (1,(2 + (X // (1 + (2 + 2)))) + 1): +x = x - (0 if x <= 0 else (1 + y)) +return x +This representation was initially discovered when our system found solutions for the sequences +A2262,23 and A25581.24 Indeed, f0 is a solution for A2262 invented in the first generation and −f1 is a +solution for A25581 invented during the 9th generation. +23https://oeis.org/A002262 +24https://oeis.org/A025581 +21 + +Alien Coding +Gauthier, Olšák, Urban +Figure 12: Linear bounds for +� +2x + 1 +4 − 1 +2 invented by the system. +Figure 13: Linear bounds for +� +2x + 1 +4 − 1 +2 invented by the system. +B.1 +Number of steps +In order to calculate f0 and f1, one must perform approximately +� +2x + 1 +4 − 1 +2 steps in a subtracting loop. +Since the programming language is based on for loops, the programs are approximating the number of +subtracting steps with a function which can be easily obtained using the basic arithmetical operations. +In particular, the code above uses approximations with 2 + x/5 where 5 is written as 1 + (2 + 2) (our +basic language only supports constants 0,1,2). The program generator has experimented with many (over +80) such bounds, vast majority of such bounds are of the shape a + x/b or a + 2(x/b) where a, b are +constants. As the speed of computing with larger numbers becomes more important, the slope of the +approximation flattens, with b reaching values over 50. We have collected all such subprograms used in +the generated programs, and plotted the bounds in Figure 12 and Figure 13. The approximated function +� +2x + 1 +4 − 1 +2 is plotted with a black dashed line. All the found valid bounds are plotted green, all the +invalid bound attempts are plotted red. +B.2 +Triangle coding in action +The triangle coding turned out to be useful in many cases. The designed language for our programs only +supports a maximum of two variables but using the triangle coding, a pair of variables can be temporarily +packed into one. A simple example is sequence A27936425 – sum of 5th powers of proper divisors of n. +A human programmer could implement this sequence as: +def f(X): +res = 0 +for y in range(1,X+1): +res = res + (y**5 if (X+1) % +25https://oeis.org/A279364 +22 + +Alien Coding +Gauthier, Olšák, Urban +return res +There are three variables used in the body of the loop, in particular res, y, and X. Since the +native language supports only two variables available at every moment, this Python code cannot be +straightforwardly translated into the language of our programs. Nevertheless, the program generator has +found a workaround using the triangle coding – it packs X and y into a single natural number, so it can +perform the summing loop adding to res, and in order to decode what to add to res, it unpacks the pair, +and checks whether y divides X+1. +The code generated by NMT for A279364 +is as follows: +A279364 Sum of 5th powers of proper divisors of n. +371326 59293 555688 1 870552 1 1082401 161295 1419890 19933 2206525 1 +2476132 371537 3336950 1 4646784 1 5315740 821793 6436376 1 9301876 +16808 9868783 +size 94, time 1335684: loop2 ((loop (loop2 ((loop ((((x * x) * x) * x) +* x) 1 (1 + y)) * (if (x mod (1 + y)) <= 0 then 1 else 0)) 0 1 (1 - +(loop (x - (if x <= 0 then 0 else y)) (1 + (2 + (2 + (x div (1 + (2 * +(2 + 2))))))) (1 + x))) (loop (x - (if (y - x) <= 0 then y else 0)) (2 ++ (2 + (x div (1 + (2 * (2 + 2)))))) x)) 1 y) + x) (1 + y) x 0 (((x * +x) - x) div 2) +K K F K F K F K F B B L D J K B L D H B A I F A B B K K A L I E B C C +K B C C C D F D G D D D B K D J E K L K E L A I E C C K B C C C D F D +G D D K J N B L J K D B L D K A K K F K E C G N +def f3(X,Y): +x = 1 + Y +for y in range (1,1 + 1): +x = (((x * x) * x) * x) * x +return x +def f4(X): +x = 1 + X +for y in range (1,(1 + (2 + (2 + (X // (1 + (2 * (2 + 2))))))) + 1): +x = x - (0 if x <= 0 else y) +return x +def f5(X): +x = X +for y in range (1,(2 + (2 + (X // (1 + (2 * (2 + 2)))))) + 1): +x = x - (y if (y - x) <= 0 else 0) +return x +def f2(X): +x,y = 1 - f4(X), f5(X) +for z in range (1,1 + 1): +x,y = (f3(x,y) * (1 if (x % +return x +def f1(X,Y): +x = Y +23 + +Alien Coding +Gauthier, Olšák, Urban +for y in range (1,1 + 1): +x = f2(x) +return x +def f0(X): +x,y = 0, ((X * X) - X) // 2 +for z in range (1,X + 1): +x,y = (f1(x,y) + x), (1 + y) +return x +for x in range(32): +print (f0(x)) +Functions f4, f5 are calculating the triangle coding, f3 is the fifth power, f1 is just a dummy +function using Y instead of X for f2. Finally, f2 is returning either (b + 1)5, or 0 depending on whether +b + 1 divides a + 2 or not where (a, b) if the triangle coding pair of X, and f0 is summing over all the +pairs (a, b) going from (X-1,0) to (X-1,X-1). +C +Evolution and Proliferation +We analyze the evolution of the solutions found for the OEIS sequence of prime numbers.26 The exact +iterations of their discovery are shown in our repository, together with their size and speed.27 The 24 +invented programs are shown in Table 3. +We can observe how these 24 programs propagate through the population of all programs, as they +evolve in the iterations. This is shown in Table 4 and Table 5. We can see that, e.g., P5 (size 25, time +515990), gets quickly replaced by P6 (size 33, time 98390) and P7 (size 23, time 519654) after their +invention. P6 is more than five times faster than P5, while P7 is smaller. Therefore they are included +in the training data instead of P5 when they are invented. While there are still other programs in the +training data using P5 as a subprogram, their faster/smaller versions get eventually also reinvented as the +newly trained NMT increasingly prefers to synthesize programs that contain P6 or P7. This illustrates the +dynamics of the overall alien system. The training of the neural synthesis component (NMT) evolves, +governed by more high-level evolutionary fitness criteria, which are in our case size (Occam’s razor) and +speed (efficiency). +D +Selection of 123 Solved Sequences +Tables 6, 7, 8 present a sample of 123 sequences solved during the nmt0 and nmt1 runs. Their solutions +found by the system are presented on our web page,28 both in our language and translated to Python. +26The tables shown here for primes are also in our repository https://bit.ly/3XHZsjK, together with similar tables for the sigma +function (sum of the divisors). +27https://bit.ly/3iJ4oGd +28https://github.com/Anon52MI4/oeis-alien +24 + +Alien Coding +Gauthier, Olšák, Urban +Table 3: The 24 programs for primes. +Nr +Program +P1 +(if x <= 0 then 2 else 1) + (compr (((loop (x + x) (x mod 2) (loop (x * x) 1 (loop (x + x) (x div 2) 1))) + x) mod (1 + +x)) x) +P2 +1 + (compr ((((loop (x * x) 1 (loop (x + x) (x div 2) 1)) + x) * x) mod (1 + x)) (1 + x)) +P3 +1 + (compr (((loop (x * x) 1 (loop (x + x) (x div 2) 1)) + x) mod (1 + x)) (1 + x)) +P4 +2 + (compr ((loop2 (1 + (if (x mod (1 + y)) <= 0 then 0 else x)) (y - 1) x 1 x) mod (1 + x)) x) +P5 +1 + (compr ((loop (if (x mod (1 + y)) <= 0 then (1 + y) else x) x (1 + x)) mod (1 + x)) (1 + x)) +P6 +1 + (compr ((loop (if (x mod (1 + y)) <= 0 then (1 + y) else x) (2 + (x div (2 + (2 + 2)))) (1 + x)) mod (1 + x)) (1 + x)) +P7 +compr ((1 + (loop (if (x mod (1 + y)) <= 0 then (1 + y) else x) x x)) mod (1 + x)) (2 + x) +P8 +1 + (compr ((loop (if (x mod (1 + y)) <= 0 then (1 + y) else x) (1 + ((2 + x) div (2 + (2 + 2)))) (1 + x)) mod (1 + x)) (1 ++ x)) +P9 +compr (x - (loop (if (x mod (1 + y)) <= 0 then (1 + y) else x) x x)) (2 + x) +P10 +compr (x - (loop (if (x mod (1 + y)) <= 0 then 2 else x) (x div 2) x)) (2 + x) +P11 +1 + (compr ((loop (if (x mod (1 + y)) <= 0 then (1 + y) else x) (1 + (x div (2 + (2 + 2)))) (1 + x)) mod (1 + x)) (1 + x)) +P12 +compr ((x - (loop (if (x mod (1 + y)) <= 0 then y else x) x x)) - 2) (2 + x) +P13 +1 + (compr ((loop (if (x mod (1 + y)) <= 0 then (1 + y) else x) (2 + (x div (2 * (2 + (2 + 2))))) (1 + x)) mod (1 + x)) (1 ++ x)) +P14 +compr ((x - (loop (if (x mod (1 + y)) <= 0 then y else x) x x)) - 1) (2 + x) +P15 +1 + (compr (x - (loop (if (x mod (1 + y)) <= 0 then (1 + y) else x) (2 + (x div (2 * (2 + (2 + 2))))) (1 + x))) (1 + x)) +P16 +compr (2 - (loop (if (x mod (1 + y)) <= 0 then 0 else x) (x - 2) x)) x +P17 +1 + (compr (x - (loop (if (x mod (1 + y)) <= 0 then 2 else x) (2 + (x div (2 * (2 + (2 + 2))))) (1 + x))) (1 + x)) +P18 +1 + (compr (x - (loop (if (x mod (1 + y)) <= 0 then 2 else x) (1 + (2 + (x div (2 * (2 * (2 + 2)))))) (1 + x))) (1 + x)) +P19 +1 + (compr (x - (loop2 (loop (if (x mod (1 + y)) <= 0 then 2 else x) (2 + (y div (2 * (2 + (2 + 2))))) (1 + y)) 0 (1 - (x +mod 2)) 1 x)) (1 + x)) +P20 +1 + (compr (x - (loop2 (loop (if (x mod (1 + y)) <= 0 then 2 else x) (1 + (2 + (y div (2 * (2 * (2 + 2)))))) (1 + y)) 0 (1 - +(x mod 2)) 1 x)) (1 + x)) +P21 +1 + (compr (x - (loop2 (loop (if (x mod (2 + y)) <= 0 then 2 else x) (2 + (y div (2 * ((2 + 2) + (2 + 2))))) (1 + y)) 0 (1 - +(x mod 2)) 1 x)) (1 + x)) +P22 +1 + (compr (x - (loop2 (loop (if (x mod (2 + y)) <= 0 then 2 else x) (2 + (y div (2 * (2 * (2 + 2))))) (1 + y)) 0 (1 - (x +mod 2)) 1 x)) (1 + x)) +P23 +2 + (compr (loop (x - (if (x mod (1 + y)) <= 0 then 0 else 1)) x x) x) +P24 +loop (1 + x) (1 - x) (1 + (2 * (compr (x - (loop (if (x mod (2 + y)) <= 0 then 1 else x) (2 + (x div (2 * (2 + 2)))) (1 + (x ++ x)))) x))) +25 + +Alien Coding +Gauthier, Olšák, Urban +Table 4: Proliferation of the 24 programs for primes. +Iter +P1 +P2 +P3 +P4 +P5 +P6 +P7 +P8 +P9 +P10 +P11 +P12 +P13 +P14 +P15 +P16 +P17 +P18 +P19 +P20 +P21 +P22 +P23 +P24 +25 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +26 +6 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +27 +7 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +28 +8 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +29 +9 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +30 +10 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +31 +4 +6 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +32 +6 +6 +0 +0 +0 +0 +0 +0 +0 +0 +0 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+a(n) = U(2*n, n), where U(n, x) is the Chebyshev polynomial of the second kind. +https://oeis.org/A293339 +Greatest integer k such that k/2n < 1/e. +https://oeis.org/A1848 +Crystal ball sequence for 6-dimensional cubic lattice. +https://oeis.org/A8628 +Molien series for A5. +https://oeis.org/A259445 +Multiplicative with a(n) = n if n is odd and a(2s) = 2. +https://oeis.org/A314106 +Coordination sequence Gal.6.199.4 where G.u.t.v denotes the coordination sequence for a vertex of type v in tiling number t in +the Galebach list of u-uniform tilings +https://oeis.org/A311889 +Coordination sequence Gal.6.129.2 where G.u.t.v denotes the coordination sequence for a vertex of type v in tiling number t in +the Galebach list of u-uniform tilings. +https://oeis.org/A315334 +Coordination sequence Gal.6.623.2 where G.u.t.v denotes the coordination sequence for a vertex of type v in tiling number t in +the Galebach list of u-uniform tilings. +https://oeis.org/A315742 +Coordination sequence Gal.5.302.5 where G.u.t.v denotes the coordination sequence for a vertex of type v in tiling number t in +the Galebach list of u-uniform tilings. +https://oeis.org/A004165 +OEIS writing backwards +https://oeis.org/A83186 +Sum of first n primes whose indices are primes. +https://oeis.org/A88176 +Primes such that the previous two primes are a twin prime pair. +https://oeis.org/A96282 +Sums of successive twin primes of order 2. +https://oeis.org/A53176 +Primes p such that 2p + 1 is composite. +https://oeis.org/A267262 +Total number of OFF (white) cells after n iterations of the "Rule 111" elementary cellular automaton starting with a single ON +(black) cell. +https://oeis.org/A273385 +Number of active (ON,black) cells at stage 2n − 1 of the two-dimensional cellular automaton defined by "Rule 659", based on +the 5-celled von Neumann neighborhood. +https://oeis.org/A60431 +Number of cubefree numbers <= n. +https://oeis.org/A42731 +Denominators of continued fraction convergents to sqrt(895). +https://oeis.org/A81495 +Start with Pascal’s triangle; form a rhombus by sliding down n steps from top on both sides then sliding down inwards to +complete the rhombus and then deleting the inner numbers; a(n) = sum of entries on perimeter of rhombus. +https://oeis.org/A20027 +Nearest integer to Gamma(n + 3/8)/Gamma(3/8). +https://oeis.org/A99197 +Figurate numbers based on the 10-dimensional regular convex polytope called the 10-dimensional cross-polytope, or 10- +dimensional hyperoctahedron, which is represented by the Schlaefli symbol 3, 3, 3, 3, 3, 3, 3, 3, 4. It is the dual of the +10-dimensional hypercube. +https://oeis.org/A220469 +Fibonacci 14-step numbers, a(n) = a(n − 1) + a(n − 2) + ... + a(n − 14). +https://oeis.org/A8583 +Molien series for Weyl group E7. +https://oeis.org/A251672 +8-step Fibonacci sequence starting with 0,0,0,0,0,0,1,0. +https://oeis.org/A124615 +Poincaré series [or Poincare series] P(T3,2; x). +https://oeis.org/A79262 +Octanacci numbers: a(0) = a(1) = ... = a(6) = 0, a(7) = 1; for n >= 8, a(n) = Sumi=1..8a(n − i). +https://oeis.org/A75068 +Product of prime(n) primes starting from prime(n). +https://oeis.org/A57168 +Next larger integer with same binary weight (number of 1 bits) as n. +https://oeis.org/A1553 +a(n) = 1n + 2n + ... + 6n. +https://oeis.org/A19560 +Coordination sequence for C4 lattice. +https://oeis.org/A289834 +Number of perfect matchings on n edges which represent RNA secondary folding structures characterized by the Lyngso and +Pedersen (L&P) family and the Cao and Chen (C&C) family. +https://oeis.org/A5249 +Determinant of inverse Hilbert matrix. +https://oeis.org/A3714 +Fibbinary numbers: if n = F(i1) + F(i2) + ... + F(ik) is the Zeckendorf representation of n (i.e., write n in Fibonacci +number system) then a(n) = 2i1−2 + 2i2−2 + ... + 2ik−2. Also numbers whose binary representation contains no two +adjacent 1’s. +https://oeis.org/A4457 +Nimsum n + 16. +https://oeis.org/A92143 +Cumulative product of all divisors of 1..n. +https://oeis.org/A2119 +Bessel polynomial yn(−2). +https://oeis.org/A5913 +a(n) = [tau ∗ a(n − 1)] + [tau ∗ a(n − 2)]. +https://oeis.org/A34960 +Divide odd numbers into groups with prime(n) elements and add together. +https://oeis.org/A247395 +The smallest numbers of every class in a classification of positive numbers (see comment). +https://oeis.org/A68068 +Number of odd unitary divisors of n. d is a unitary divisor of n if d divides n and gcd(d, n/d) = 1. +28 + +Alien Coding +Gauthier, Olšák, Urban +Table 7: Samples of the solved sequences. +https://oeis.org/A30973 +[ exp(1/5) ∗ n! ]. +https://oeis.org/A54469 +A second-order recursive sequence. +https://oeis.org/A54054 +Smallest digit of n. +https://oeis.org/A36561 +Nicomachus triangle read by rows, T(n, k) = 2n−k ∗ 3k, for0 <= k <= n. +https://oeis.org/A107347 +Number of even semiprimes strictly between prime(n) and 2 * prime(n). +https://oeis.org/A123379 +Values x of the solutions (x,y) of the Diophantine equation 5 ∗ (X − Y )4 − 4XY = 0 with X >= Y. +https://oeis.org/A201204 +Half-convolution of Catalan sequence A000108 with itself. +https://oeis.org/A125494 +Composite evil numbers. +https://oeis.org/A277094 +Numbers k such that sin(k) > 0 and sin(k + 2) < 0. +https://oeis.org/A59760 +a(n) is the number of edges (one-dimensional faces) in the convex polytope of real n X n doubly stochastic matrices. +https://oeis.org/A246303 +Numbers k such that cos(k) < cos(k + 1). +https://oeis.org/A7957 +Numbers that contain an odd digit. +https://oeis.org/A7452 +Expand cosx/expx and invert nonzero coefficients. +https://oeis.org/A88896 +Length of longest integral ladder that can be moved horizontally around the right angled corner where two hallway +corridors of integral widths meet. +https://oeis.org/A131989 +Start with the symbol *| and for each iteration replace * with *| . This sequence is the number of *’s between each +dash. +https://oeis.org/A308066 +Number of triangles with perimeter n whose side lengths are even. +https://oeis.org/A11540 +Numbers that contain a digit 0. +https://oeis.org/A156660 +Characteristic function of Sophie Germain primes. +https://oeis.org/A167132 +Gaps between twin prime pairs. +https://oeis.org/A8846 +Hypotenuses of primitive Pythagorean triangles. +https://oeis.org/A332381 +a(n) is the Y-coordinate of the n-th point of the Peano curve. Sequence A332380 gives X-coordinates. +https://oeis.org/A25492 +Fixed point reached by iterating the Kempner function A002034 starting at n. +https://oeis.org/A131530 +Numbers k such that k2 − k − 1 and k2 − k + 1 are twin primes. +https://oeis.org/A143165 +Expansion of the exponential generating function arcsin(2x)/(2(1 − 2 ∗ x)3/2). +https://oeis.org/A30957 +[ exp(1/9) ∗ n! ]. +https://oeis.org/A295286 +Sum of the products of the smaller and larger parts of the partitions of n into two parts with the smaller part odd. +https://oeis.org/A86699 +Number of n X n matrices over GF(2) with rank n-1. +https://oeis.org/A2819 +Liouville’s function L(n) = partial sums of A008836. +https://oeis.org/A7318 +Pascal’s triangle read by rows: C(n, k) = binomial(n, k) = n!/(k! ∗ (n − k)!), 0 <= k <= n. +https://oeis.org/A8836 +Liouville’s function lambda(n) =(−1)k, where k is number of primes dividing n (counted with multiplicity). +https://oeis.org/A266776 +Molien series for invariants of finite Coxeter group A7. +https://oeis.org/A284115 +Hosoya triangle of Lucas type. +https://oeis.org/A45717 +For each prime p take the sum of nonprimes < p. +https://oeis.org/A307508 +Primes p for which the continued fraction expansion of sqrt(p) does not have a 1 in the second position. +https://oeis.org/A3506 +Triangle of denominators in Leibniz’s Harmonic Triangle a(n, k), n >= 1, 1 <= k <= n. +https://oeis.org/A93017 +Luhn algorithm double-and-add sum of digits of n. +https://oeis.org/A121373 +Expansion off(x) = f(x, −x2) in powers of x where f(, ) is Ramanujan’s general theta function. +https://oeis.org/A227127 +The Akiyama-Tanigawa algorithm applied to 1/(1,2,3,5,... old prime numbers). Reduced numerators of the second +row. +https://oeis.org/A39637 +Number of steps to fixed point of "n− > n/2or(n + 1)/2 until result is prime". +https://oeis.org/A548 +Squares that are not the sum of 2 nonzero squares. +https://oeis.org/A131650 +Number of symbols in Babylonian numeral representation of n. +29 + +Alien Coding +Gauthier, Olšák, Urban +Table 8: Samples of the solved sequences. +https://oeis.org/A3538 +Divisors of 230−1. +https://oeis.org/A152135 +Maximal length of rook tour on an n X n+4 board. +https://oeis.org/A8637 +Number of partitions of n into at most 8 parts. +https://oeis.org/A113953 +A Jacobsthal triangle. +https://oeis.org/A3605 +Unique monotonic sequence of nonnegative integers satisfying a(a(n)) = 3n. +https://oeis.org/A266214 +Numbers n that are not coprime to the numerator of zeta(2n)/(Pi2n). +https://oeis.org/A266778 +Molien series for invariants of finite Coxeter group A9. +https://oeis.org/A101608 +Solution to Tower of Hanoi puzzle encoded in pairs with the moves (1, 2), (2, 3), (3, 1), (2, 1), (3, 2), (1, 3). The +disks are moved from peg 1 to 2. For a tower of k disks use the first 2k − 1 number pairs. +https://oeis.org/A90971 +Sierpi´nski’s triangle, read by rows, starting from 1: T(n, k) = (T(n − 1, k) + T(n − 1, k − 1))mod2. +https://oeis.org/A83743 +a(1) = 1; if a(n − 1) + n is prime then a(n) = a(n − 1) + n, else a(n) = a(n − 1). +https://oeis.org/A79683 +Order of Burnside group B(6, n) of exponent 6 and rank n. +https://oeis.org/A34444 +a(n) is the number of unitary divisors of n (d such that d divides n, gcd(d, n/d) = 1). +https://oeis.org/A218509 +Number of partitions of n in which any two parts differ by at most 7. +https://oeis.org/A65109 +Triangle T(n, k) of coefficients relating to Bezier curve continuity. +https://oeis.org/A166555 +Triangle read by rows, Sierpinski’s gasket, A047999 * (1,2,4,8,...) diagonalized. +https://oeis.org/A119467 +A masked Pascal triangle. +https://oeis.org/A194887 +Numbers that are the sum of two powers of 12. +https://oeis.org/A1824 +Central factorial numbers. +https://oeis.org/A47780 +Number of inequivalent ways to color faces of a cube using at most n colors. +https://oeis.org/A8640 +Number of partitions of n into at most 11 parts. +https://oeis.org/A29635 +The (1,2)-Pascal triangle (or Lucas triangle) read by rows. +https://oeis.org/A45995 +Rows of Fibonacci-Pascal triangle. +https://oeis.org/A5045 +Number of restricted 3 X 3 matrices with row and column sums n. +https://oeis.org/A971 +Fermat coefficients. +https://oeis.org/A5835 +Pseudoperfect (or semiperfect) numbers n: some subset of the proper divisors of n sums to n. +https://oeis.org/A266773 +Molien series for invariants of finite Coxeter group D10 (bisected). +https://oeis.org/A69209 +Orders of non-Abelian Z-groups. +https://oeis.org/A8383 +Coordination sequence for A4 lattice. +https://oeis.org/A70896 +Determinant of the Cayley addition table of Zn. +https://oeis.org/A262 +Number of "sets of lists": number of partitions of 1,...,n into any number of lists, where a list means an ordered subset. +https://oeis.org/A23436 +Dying rabbits: a(n) = a(n − 1) + a(n − 2) − a(n − 6). +https://oeis.org/A8641 +Number of partitions of n into at most 12 parts. +https://oeis.org/A68764 +Generalized Catalan numbers. +https://oeis.org/A7856 +Subtrees in rooted plane trees on n nodes. +https://oeis.org/A271 +Sums of ménage numbers. +https://oeis.org/A199033 +Number of ways to place n non-attacking bishops on a 2 X 2n board. +https://oeis.org/A1006 +Motzkin numbers: number of ways of drawing any number of nonintersecting chords joining n (labeled) points on a +circle. +https://oeis.org/A239768 +Number of pairs of functions (f,g) from a set of n elements into itself satisfying f(x) = f(g(f(x))). +https://oeis.org/A2895 +Domb numbers: number of 2n-step polygons on diamond lattice. +https://oeis.org/A14342 +Convolution of primes with themselves. +https://oeis.org/A27847 +a(n) = Sumd|nsigma(n/d) ∗ d3. +30 + diff --git a/U9FJT4oBgHgl3EQfNyw_/content/tmp_files/load_file.txt b/U9FJT4oBgHgl3EQfNyw_/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..25e359462c1c851750b5efff264fa3ba3a39efe5 --- /dev/null +++ b/U9FJT4oBgHgl3EQfNyw_/content/tmp_files/load_file.txt @@ -0,0 +1,4282 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf,len=4281 +page_content='Alien Coding Thibault Gauthier1, Miroslav Olšák2, and Josef Urban1 1 Czech Technical University in Prague, Czech Republic 2 Institut des Hautes Etudes Scientifiques Paris, France Abstract We introduce a self-learning algorithm for synthesizing programs for OEIS sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' The algorithm starts from scratch initially generating programs at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Then it runs many iterations of a self-learning loop that interleaves (i) training neural machine translation to learn the correspondence between sequences and the programs discovered so far, and (ii) proposing many new programs for each OEIS sequence by the trained neural machine translator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' The algorithm discovers on its own programs for more than 78000 OEIS sequences, sometimes developing unusual programming methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' We analyze its behavior and the invented programs in several experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' 1 Introduction Galileo once said, "Mathematics is the language of Science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='" Hence, facing the same laws of the physical world, alien mathematics must have a good deal of similarity to ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' – R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Hamming - Mathematics on a Distant Planet [7] Most of today’s successful coding assistants, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' GitHub Copilot [2], are trained on large code repositories such as GitHub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' This makes them quite versatile and capable of coding in multiple program- ming languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' They can also transfer some of the knowledge acquired in one programming language, provided there are large training corpora for both programming languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' However, since they are trained in a supervised way to mimic existing human-written code, they may be biased towards possibly non-optimal solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' This may make them incapable of coming up with better solutions on their own.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' In order to free themselves from human bias, techniques such as reinforcement learning (RL) can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' RL techniques do not rely on training examples but on search and rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' An early example of such a system is MENACE [12] which can learn to play noughts and crosses on its own and much faster than brute-forcing all possible solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Recently, with a residual network as a machine learner, AlphaGoZero [17] has learned playing Go better than professional players using only self-play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' In this process, the system discovered many effective moves that went against 3000 years of human wisdom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' The early 3-3 invasion was considered a bad opening move but is now frequently used by professionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' The goal of this paper is to develop a self-learning system inspired by [17] capable discovering on its own interesting and possibly unusual (alien) mathematical formulas/programs from common patterns found in today’s mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' We choose those patterns to be integer sequences taken from the Online Encyclopedia of Integer Sequences (OEIS) [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Formulas for these sequences will be constructed from a small general programming language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' This makes the search space manageable, allowing our system to bootstrap itself without the need for supervised data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Our work can also be viewed as an advancement in the field of automated theorem proving (ATP) [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' There, finding existential witnesses of the form ∃f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='P(f) is a grand challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Our task is of this form: to find a program f satisfying the property “P: f generates the sequence s“.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' In today’s ATP, this often turns into brute-force enumeration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Our experience with ATP problems created from OEIS sequences is that the ATP systems have hard time deriving a witness more complicated than the doubling function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Recent efforts in automated reasoning go beyond the deduction paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' They incorporate feedback loops between machine learning and theorem proving[22, 8, 10, 9] and neural models for conjecturing and related synthesis tasks [21, 15, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='11479v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='AI] 27 Jan 2023 Alien Coding Gauthier, Olšák, Urban Search Check Learn programs examples weights Figure 1: The tree phases of the self-learning loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Overview and Contributions In order to learn how to find programs generating integer sequences, our approach relies on a self-learning loop that alternates between the three phases represented in Figure 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' During the search phase, our machine learning model synthesizes programs for integer sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' In this work we predominantly use neural machine translation (NMT) as the machine learning component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' For each OEIS sequence, the NMT trained on previous examples typically creates 240 candidate programs using beam search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' In the first iteration (generation) of this loop, programs are randomly constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Then, during the checking phase, the proposed millions of programs are checked to see if they generate their target sequence, or any other OEIS sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' The smallest and fastest programs generating an OEIS sequence are kept to produce the training examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' In the learning phase, NMT trains on these examples to translate the “solved” OEIS sequences into the best discovered program(s) generating it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' This updates the weights of the NMT network which influences the next search phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Each iteration of the self-learning loop leads to the discovery of more solutions, as well as to the optimization of the existing solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Our work builds mostly on our work in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Our contributions are the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' The tree neural network architecture is replaced by a relatively fast encoder-decoder NMT network (bidirectional LSTM with attention).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' We replace the previously used MCTS search with a relatively wide beam search during the NMT decoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Our objective function lets us collect both the smallest and fastest programs for each sequence instead of only the smallest programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' We provide many experiments comparing different parameters (embedding size, programming language, search strategy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' In particular, we experiment with local and global definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' We analyze the solutions and their evolution (getting faster and smaller) over many generations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Our longest run finds solutions for more than 78000 OEIS sequences in 190 iterations, and all our experiments have so far together produced solutions for 84587 OEIS sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' This is more than three times the number (27987) invented in our first experiments [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' 2 Components In this section, we give a technical description of the components of our system: the OEIS datasets, the programming language and it representations, and the checking phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='1 The OEIS Dataset The OEIS is a repository created and maintained by Neil Sloane where amateur and professional mathematicians can contribute integer sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' There are currently more than 350,000 sequences in this repository.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Each entry contains the terms of the sequence and a short English description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' It is referenced by a A-number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' For example, A40 is the reference for prime number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Additional information may be provided such as: alternative descriptions, links to other entries and to papers where the sequence was investigated and in about one third of the cases a program for generating the sequence is provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' These programs are written in many different languages such as PARI, Matlab, Haskell, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='. In our experiments, we ignore the human-written programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Thus, our problem set consists of all OEIS sequences (351663 as of March 2022) without any corresponding programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' 2 Alien Coding Gauthier, Olšák, Urban 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='2 The Programming Language and its Python and NMT Representations A formal description of the programming language used in this paper is given in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' It is minimalistic by design, to avoid human-informed bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Since many examples given in this paper require an understanding of the language, we briefly summarize it here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' The language contains two variables x and y, that can take as values arbitrary-precision integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' It includes the standard operators 0, 1, 2, +, ×, mod, div (integer division) and the conditional operators cond(a, b, c) := if a ≤ 0 then b else c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' These programming operators follow the standard semantics of most programming languages (including C and Python).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' In this programming language, an expression p can either be evaluated to an integer if given specific values for x and y or can be used to create a binary function f defined by f(x, y) = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' The three looping operators of this language treat their arguments in these two different manners depending on their positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Looping expressions may themselves be used as arguments of looping operators allowing for arbitrarily nesting of loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' The loop Operator This operator takes three arguments: one function and two integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' loop(f, a, b) := b if a ≤ 0 f(loop(f, a − 1, b), a) otherwise This definition is almost the same as the one used to define primitive recursion in the standard theory of primitive recursive functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' For more clarity and portability, we can translate this construction to Python.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' We capitalize the variables in a and b, which play a different role than the variables in f, to avoid undesirable variable capture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Python’s implementation F of the function that can be derived from the expression loop(f, a, b) is as follows: def F(X,Y) = x = b[x/X,y/Y] for y in range (1,a[x/X,y/Y] + 1) x = f(x,y) return x Some common uses of loop include: 2x written as loop(2 × x, x, 1) and x!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' written as loop(y × x, x, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' The loop2 Operator This operator takes five arguments: two functions and three integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' loop2(f, g, a, b, c) := b if a ≤ 0 loop2(f, g, a − 1, f(b, c), g(b, c)) otherwise This operators starts with the pair of numbers (b, c) and updates their values a times using the functions f and g before returning b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' This is a generalization of the loop operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Given g such that g(x, y) = y + 1, we have loop(f, a, b) = loop2(f, g, a, b, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Therefore, the loop operator could be removed from the language without affecting its expressiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' It is kept in the language as it is a natural and useful instantiation of loop2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Python’s implementation F of the function derived from the expression loop2(f, g, a, b, c) is: def F(X,Y) = x = b[x/X,y/Y] y = c[x/X,y/Y] for _ in range (1,a[x/X,y/Y] + 1) x = f(x,y) y = g(x,y) 3 Alien Coding Gauthier, Olšák, Urban return x The following constructions have a natural implementation using loop2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' They are however difficult to express using loop and would generally require encodings such as the Cantor pairing function: The Fibonacci function Fibonacci(x) can be implemented by the program loop2(x + y, x, x, 0, 1), and the power function xy by the program loop2(x × y, y, y, 1, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' The compr Operator The comprehension operator takes two arguments: one function and one integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' compr(f, a) := failure if a < 0 min{m | m ≥ 0 ∧ f(m, 0) ≤ 0} if a = 0 min{m | m > compr(f, a − 1) ∧ f(m, 0) ≤ 0} otherwise The comprehension expression finds the a + 1th smallest nonnegative integer m satisfying the predicate f(m, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' If the value of a is 0 then this behaves like the minimization operator µ in the theory of general recursive functions, thus making the language Turing-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' It gives a natural way of constructing in- creasing sequences of numbers (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=', sets) from a predicate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' In particular, suppose we have constructed the function fprime(x, y) = if x is prime then 0 else 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Then the expression compr(fprime, x) constructs the sequence of primes as as the value of x increases from 0 to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Note that the operator thus behaves similarly to the set comprehension operator in set theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' The Python’s implementation F of the function derived from the expression compr(f, a) is: def F(X,Y): x,i = 0,0 while i <= a[x/X,y/Y]: if f(x,0) <= 0: i = i + 1 x = x + 1 return x - 1 Linear Representation of Programs for NMT We use prefix notation (with argument order re- versed) to represent a program as a sequence of tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' The main advantage of this approach is that the notation does not require the use of parentheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' For example, the prefix notation for the program loop(x × y, x, 1) is loop 1 x × y x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' When using NMT, the 14 operators/actions/tokens [0, 1, 2, +, −, ×, div, mod, cond, loop, x, y, compr, loop2] are represented by capital letters from A to N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Thus, the program for factorial is written as J B K F L K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Definitions To allow code re-use, human programmers introduce definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' We allow NMT to produce definitions in two different settings and experiments: one using local definitions and the other using global definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' We have decided that such definitions will be arbitrary sequences of actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' This is quite a powerful setup since such definitions can represent not just subprograms, but also subprograms with holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' A sequence of actions is called a macro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' A program can be always constructed by a sequence of actions, but not every sequence of actions is a program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Local Definitions In this setting, we add to the programming language ten tokens representing ten possible local definitions (macros) that may include preceding macros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' A special action/token is used as a separator between the different macros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' The macros in the generated programs are unfolded before the 4 Alien Coding Gauthier, Olšák, Urban Figure 2: Representing local macros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' A macro version and expanded version of a program invented for A1813 (a(n) = (2n)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='/n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Note that the macro here (K D L B) is not a proper program, only a sequence of actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' checking phase takes place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Note that the naming of such macros is often inconsistent across different programs,1 possibly making them harder to learn across many examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Figure 2 shows an example of the solution found for the sequence A18132 which involved synthesis of a local macro that is then used three times in the body of the invented program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Global Definitions In this setup, we allow for arbitrarily many macros stored in a global array and shared across all programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' This makes the naming of the macros consistent in all programs, possibly making them easier to learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Programs may refer to any macro stored in the global array, by writing its index in base 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' This again requires 10 additional actions (one for each digit) and a special action to separate the references to the macros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' As in the local definition setup, the global macros may contain references to macros with lower indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Figure 3 shows an example of the solution found for the sequence A141873 which involved three global macros that are altogether used five times in the body of the invented program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Introduction and use of local macros for a particular input integer sequence is completely a “local” decision of the trained NMT that generates the particular program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' In the global case, we however need more coordination to introduce the global macros consistently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' This can be done in various ways and we now use the following method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' At every iteration of the overall loop, we add the ten most frequent sequences of actions to the array of global macros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' To force the network to learn to use the global macros, 1E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=', in program P1, a macro called m could be used to define n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=', while in P2, m could be used to define 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' 2https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A1813 - Quadruple factorial numbers: a(n) = (2n)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='/n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='. 3https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A14187 - Cubes of palindromes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' 5 Alien Coding Gauthier, Olšák, Urban Figure 3: Representing global macros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' A macro version and expanded version of a program invented for A14187 (cubes of palindromes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Note that two macros here (K C and B F F K K) are not proper programs, while the third one (D F C D C C C) is a program that evaluates to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' we greedily (starting from the macros with the lowest indices) recognize sub-sequences of actions that correspond to the macros, and replace them by the macros’ names (indices) in the programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='3 The Program Checker In its most basic form the checker takes a program and a sequence and checks if that program generates the sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Since programs may depend on two variables, we say that the program f(x, y) = p generates the finite sequence (sx)0≤x≤n if and only if ∀x ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' 0 ≤ x ≤ n ⇒ f(x, 0) = sx 6 Alien Coding Gauthier, Olšák, Urban A40 A5843 A45 0 2 2 4 6 1 1 2 3 5 7 Figure 4: Tree of OEIS sequences with branches for primes (A40), even numbers (A5843) and Fibonacci numbers (A45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' We say that a sequence s has a solution if we have found a program p generating it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' The number of OEIS sequences with at least one solution is the number reported in all our experiments under the label solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Timeout The first issue when implementing a program checker is to determine the time limit for running the (generally non-terminating) programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' In particular, to adapt the time limit for longer sequences, we compute the generated terms in the order f(0, 0), f(1, 0), f(2, 0), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' and stop the program if it has not generated the n-th term in less that n × tcall abstract time units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' This effectively means that we give a timeout of tcall time units per call with the time unused during previous calls added to the timeout of the current call.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' This abstract time unit is computed to be an approximation of the number of CPU instructions needed to perform each operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' The cost of an operation is 1 for the +, −, × operations, it is 5 for the div, mod operations, and it is the number of bits of the result if the result is larger than 64 bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Using the abstract time is also important to get accurate and repeatable measurements of the speed of the programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Hindsight Experience Replay In order to augment the training data using a limited form of hindsight experience replay [1], we check our program against all OEIS sequences at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' This can be done effectively by organizing the sequences into a tree of sequences (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' 4) and stopping the checking as soon as the generated sequence reaches a leaf in that tree or takes a non-existing branch in the tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' All sequences (typically at most one) found along the path taken by the generated sequence are said to have a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Objectives After each iterations we keep only the fastest and the smallest program (which could be the same) for each sequence s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' The speed of a program for s is the total number of abstract time units necessary to generate s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' The size of a program is the number of operators/tokens in its linear representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' As soon as the checker has found a program that is a solution for a particular OEIS sequence, we compare it with the existing solutions for that sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' We use the abstract time to select the fastest program among the ones that match the solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' The fastest and smallest programs are also used as the training examples for the next generation of the loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='4 Comprehension Limit Evaluating each term of the sequence compr(f, 0), ,compr(f, 1),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=',compr(f, n − 1) separately is most of the time too slow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' This computation can be sped up using the fact that each term can be computed from the preceding term in the sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' In general, when executing a program containing comprehension 7 Alien Coding Gauthier, Olšák, Urban operators, we precompute the results of applying compr(f, i) for each f appearing as the first argument of compr, where the number i ranges from 0 to ncompr −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' The number ncompr is a parameter called the comprehension limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' The pre-computation times out if it takes more than i × tcall time units to produce the outputs for compr(f, 0), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=', compr(f, i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' When the top program is executed, a call to a compr(f, a) subprogram times out if no precomputed value exists for the input i created by the subprogram a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Otherwise, the call returns the precomputed value for compr(f, i) to the top program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='5 Choice of the Timeout Parameters The two parameters that determine how long a program may be run for is the timer per call tcall and the comprehension limit ncompr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' A program times out if it exceeds the timeout or if one of the compr expression reaches the comprehension limit or if an integer with an absolute value larger than 10285 is produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' We may run either a fast check, a slow check or a hybrid check on the set of candidate programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' The fast check uses as parameters tcall = 1000, ncompr = 20, and the slow check uses tcall = 100000, ncompr = 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Hybrid Check The hybrid check tries to achieve the performance of the fast check while retaining most of the additional solutions found by the slow check.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' The first phase of the hybrid check is the fast check.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' After this check, we look at the programs that generated a prefix of an OEIS sequence but could not complete the full task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' At this point, if we were to perform the slow check on all those prefix-generating programs, the hybrid check would take a time equivalent to the slow check.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' To get a gain in performance, we select only the ones that are the smallest for each prefix to be tested for the longer time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Fast programs implicitly differentiate themselves from others by generating longer prefixes, therefore are also selected by the same criteria for further checking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' In most of our experiments, we use the hybrid check because it is about 15 times faster than the slow check.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' However, since it does not test all programs with the long timeout, it misses out on some solutions found by the slow check.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' In the longer NMT runs, we eventually switched from the hybrid check to the more robust slow check, to discover more solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' 3 OEIS Synthesis as an NMT Task Neural networks have in the last decade become competitive in language modeling and machine transla- tion tasks, leading to applications in many areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' In particular, recurrent neural networks (RNNs) with attention [3] and transformers [23] have been recently applied in mathematical and symbolic tasks such as rewriting [14], autoformalization [24] and synthesis of mathematical conjectures and proof steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Many of these tasks are naturally formulated as sequence-to-sequence translation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' NMT Representation The OEIS program synthesis can also be cast as such task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' In this work we therefore experiment with replacing the TNN architecture with a reasonably fast encoder-decoder neural machine translation (NMT) system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' In particular, we represent the input integer sequence as a series of digits, separated by an additional token at the integer boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Since the initial integers in a sequence are typically smaller and may be more informative for the NMT decoding phase, we reverse the input sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' The output program is also represented as a sequence of tokens in Polish notation (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Beam Search To make full use of the NMT capabilities, we also replace the original MCTS search with a wide beam search during the NMT decoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Beam search with width N is an alternative to greedy decoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Instead of a single greedily best output, NMT in beam search keeps track of the N 8 Alien Coding Gauthier, Olšák, Urban Figure 5: Representing sequences and solutions for NMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' conditionally most probable outputs, updating their ranking after each decoding step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' When the NMT decoding for a particular input OEIS sequence s is finished, the final N best outputs can be used as NMT’s N alternative suggestions of programs that solve s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' NMT Framework and Hyperparameters After several initial evaluations we have chosen for the experiments Luong’s NMT framework [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' It works efficiently on our hardware both in the training and wide-beam inference mode, and we were able to find suitable hyperparameters for it [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' In more detail, we use for most experiments a 2-layer bidirectional LSTM equipped with the “scaled Luong” attention and 512 units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' In our NMT experiments (Section 5) we start by using one NMT model for training and inference, using many default NMT hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' As the iterations progress, we gradually adjust the parameters and add more models trained differently and on differently selected data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' We also experiment with larger models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Combining NMT Models In most NMT runs we train two to four different NMT models in parallel each on its own GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' We then run the inference with two of them in parallel, thus using all 4 GPUs on the server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' The rationale behind training and inference with differently trained models is the standard portfolio argument, used routinely, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=', in automated theorem proving [19, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' A complementary portfolio of specialists typically outperforms a single general strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' In feedback loops that alternate between proof search and learning [22], this also further benefits the learning phase, since each learner can additionally use the training data accumulated by others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' In Section 5 we see that this indeed considerably improves the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Continuous training NMT models can be trained either only on the latest version of the solutions or in a continuous way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' The latter re-uses the model trained in the previous iteration and trains it on the latest data for more steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' This makes such model more stable, being eventually trained for orders of magnitude more steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' It also makes it different from the models trained from scratch only on the latest data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Even when only a few solutions arrive in the latest iteration, the network is training further on the whole latest corpus, thus becoming smarter and hopefully more competent for the next inference 9 Alien Coding Gauthier, Olšák, Urban Table 1: Solutions at generation 0,5,10,15 and 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Embedding size 16 2005 14920 19674 21163 22203 32 1972 17608 23490 25750 27351 64 2017 18156 23737 26490 28463 96 1993 16051 20127 22890 24718 96 l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' 3771 20434 25378 28361 30344 Language (embedding size 64) minimal 1157 7096 8547 9126 9965 default 2017 18156 23737 26490 28463 extra 1763 18690 23757 26905 28794 Objective (local search) both 3771 20434 25378 28361 30344 small 3725 20501 26231 29124 31520 fast 3716 21021 26270 29326 31421 phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' The models trained only on the latest corpus are on the other hand less concerned by the old (slower/longer) solutions and more focused on exploring the latest ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' 4 Experiments with a Tree Neural Network In the work of [6], a tree neural network (TNN) serves as a machine learning model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' In the following, we replicate their work and test how varying a range of parameters influence the self-learning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' To test the limit of the TNN, we run the TNN-guided learning loop for 500 generations instead of the original 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' This final experiment also provides a baseline for the NMT experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Varying Parameters We investigate the effect of three different parameters: the TNN embedding size, the choice of the programming language and the choice of the objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Unless specified otherwise, the default value for those parameters are respectively 96 for the dimension, the previously described programming language (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='2) and the selection of the smallest and fastest programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' In Table 1, we present the results of running experiments with different parameters for 20 generations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' In a first experiment (first block in the table), we observe that the number of solutions increases with the dimension until dimension 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Surprisingly, dimension 96 gave worse results, this is mainly due to the fact that larger networks are more expensive to compute and therefore produce fewer programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' As a countermeasure, we introduce a local search (l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=') that tests all programs that are one action away from being constructed in the search tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' This balances the generation time and checking time better leading to an increase in the number of solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' In a second experiment (second block in the table), we measure how changing the programming language affects the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' All the experiments presented in this block use dimension 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' The minimal row runs the self-learning loop with an even more minimalistic programming language consisting of four operators: 0, successor, predecessor, loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' This makes the learning much more challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' One of the reasons is that creating a large number such as one million in this language requires at least one million steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Therefore, it is impossible to 10 Alien Coding Gauthier, Olšák, Urban produce large numbers within the checker’s time limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Such considerations justify the inclusion in the default language of operators efficiently computed by current hardware such as + and ×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' The extra row shows what happens when we include the extra constants 3,4,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=',10 as primitive operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' This makes the computation of large numbers more efficient which increases the performance of our system slightly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' However, introducing more and more primitive operators introduces human bias that we would rather avoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' In a third experiment (third block in the table), the results of learning with different program objectives are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' All these experiments were performed with dimension 96 and local search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' The small (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' fast) row shows the effect of only collecting and learning from the smallest (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' the fastest) programs for a given OEIS sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' The results imply that focusing on one objective at a time simplifies the work of the machine learner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Yet, we expect more synergy between the two objectives to occur during longer runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Long Run The result of a long-lasting experiment, running for 500 generations with the default parameters and local search, is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' 6 (tnn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' To fit also the NMT runs, we display only the first 190 iterations, however the average increments (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' 7, tnn) between iteration 200 and 300 drops below 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' At the end of the run, the TNN seems to have reached its limit and about five new solutions are found at each generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' As we will see in Section 5, due to its larger embedding size and its more involved architecture and the introduction of continuous training, the main NMT experiment does not plateau and reaches a much higher number in an equivalent amount of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' 5 Experiments with NMT 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='1 Basic run (nmt0) In the basic NMT run4 (nmt0), we are running the loop in a way that is most similar to the TNN run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' In particular, the checking phase is interleaved with training a single NMT model0, which is then used for the search phase, implemented as NMT inference using a wide beam search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' As in the TNN experiment, we start with the initial random generation which yields 3771 solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Then we run 100 iterations of the NMT-based learn-generate-check loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Training: We use a batch size of 512 and SGD with a learning rate of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' In each iteration, we train on the latest set of solutions, initially for 12000 steps, and since iter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' 45 for 14000 steps (to adjust for the growing number of training examples).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='5 The bleu scores on small test and development sets are typically between 25 and 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' There is only one memory crash (iter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' 97).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Inference: We use two GPUs in parallel (splitting OEIS into two parts), each with a batch size of 32 and beam width 240.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' These values are determined experimentally, to load the GPUs efficiently without memory crashes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' The parallelized inference time grows from 2 to 8 hours, increasing as the iterations invent longer examples, and the trained network and the beam search become less confident about when to stop decoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Each inference phase yields 240 ∗ 351663 = 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='4M program candidates that are then checked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Checking: We use the hybrid checking mode (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='5) parallelized over 18 CPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' The checking time grows from about 2 minutes to about 8-10 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' This is negligible compared to the NMT training 4https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='com/Anon52MI4/oeis-alien/tree/master/run0 5This corresponds to 438 epochs in iter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' 3 where there are about 14k examples, 92 epochs in iter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' 44 (67k examples), 107 epochs in iter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' 45 (after switching to 14000 steps), and 76 epochs in iter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' 101 (81k examples).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' On one GPU (GTX1080 Ti, 12G RAM) the training takes on average 100m with 12000 steps and 130m with 14000 steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' 11 Alien Coding Gauthier, Olšák, Urban and inference times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' The 100 iterations of this loop took about a month of real time, reaching 46707 solutions (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' 6, nmt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' However, the increments drop below 200 and 100 after iteration 37 and 94, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='2 Long extended run (nmt1) Since the time taken by one nmt0 iteration reaches about half a day towards the end of the run, we explore more efficient approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' This leads to the longest run6 nmt1, which has at the time of submitting this paper reached 190 iterations and over 78000 solutions (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' 6, nmt1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='7 Unlike in nmt0, the number of new solutions produced in each iteration rarely drops below 200, even after many iterations (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' 7, nmt1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' This challenges the standard wisdom of “plateauing curves” appearing in the TNN and nmt0 runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Combining models: The nmt1 run bootstraps from nmt0, inheriting its first 20 iterations, thus starting with 34420 solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Then we start combining multiple NMT models (Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Since iter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' 21, we add training of model1 to model0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' It is trained only on a randomly chosen half of the training set, however for twice as many steps/epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' This yields a differently trained (and more focused) specialist in each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' The number of solution candidates produced by the inference phase thus doubles, to 168.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='8M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' After de-duplication, this yields 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='5M unique candidates in iter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' 21 compared to 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='4M in iter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' The checking (initially also using the hybrid mode) is still fast, taking 5 minutes on 18 CPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' The difference to nmt0 is remarkable: 687 new solutions in nmt1 vs 272 in nmt0 in iter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' This effect continues over the next iterations, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Further modifications: Since iter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' 70, we start training model1 in a continuous way (Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' By iter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' 190 model1 is thus trained for over 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='4M steps on the evolving data, making it quite different from other models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' As we invent longer programs, we also allow training on longer sequences, raising the default NMT values of 50 input and 50 output tokens gradually to 80 and 140, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Since iter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' 156 we also add training of model2, which takes specialization even further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' It is trained for four times as many steps as model0 on a random quarter of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' The bleu scores of model0 are at this point low, due to the larger size of the data, while model2 still achieves scores above 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' We then use model0 only as a backup when model2 diverges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Since iter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' 159 we switch from the hybrid check to the slow (full) check (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='5), raising the checking time from 45m (iter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' 158) to 6h, and the number of solutions from 178 to 860 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' 7, nmt1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' This jump is likely due to many programs using comprehension being newly allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' It includes sudden solutions for hundreds of problems that combine congruence operations and primes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='8 Bigger Network Since iter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' 170, we add continuous training of a bigger model3 which uses 1024 units instead of 512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' To keep all models and phases in sync, we train model3 for fewer steps (8000), decreasing also its batch size and inference width to 288 and 120, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Table 2 analyzes the benefits of using model3, model2 and model0 in addition to model1over 12 iterations (175-186).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' The number of unique candidates is much lower for model3 (due to the beam width) than for model2, however still higher than for model0, which is at this point likely undertrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' In general, model3 performs better than model0, but is inferior to model2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' A faster model, producing twice as many plausible candidates in the same amount of time, is here better than a slower model which is a bit more precise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='9 6https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='com/Anon52MI4/oeis-alien/tree/master/run1 7The 190 nmt1 iterations took 3 months on a 4-GPU server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' 8https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='ly/3QPkquE 9Also, model3 is trained continuously and may resemble more model1, providing fewer different candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' And model2 is trained only on a random quarter of the latest data, making it even more orthogonal to the continuous models that have seen much more data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' 12 Alien Coding Gauthier, Olšák, Urban Table 2: Influence of using models M3, M2 or M0 for inference in addition to M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' UC are unique candidates (millions), NS new solutions added, TS total solutions including all found so far, and OS own solutions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=', the sequences covered by the current iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Iter 175 176 177 178 179 180 Model M2 M3 M0 M2 M2 M2 UC 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='7 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='7 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='7 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='7 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='0 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='2 OS 74746 74916 74130 75196 75386 75313 TS 75471 75666 75795 75985 76160 76404 NS 260 195 129 190 175 244 Iter 181 182 183 184 185 186 Model M2 M2 M2 M2 M3 M3 UC 105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='8 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='1 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='7 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='1 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='5 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='7 OS 75686 75986 76271 76397 76793 77007 TS 76656 76861 77055 77244 77411 77577 NS 252 205 194 189 167 166 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='3 Runs with Local and Global Macros As the average program size grows in nmt1 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' 8), the NMT decoding times increase, taking almost 12h in the latest iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Verbatim repetition of blocks of code also feels suboptimal to human programmers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' This motivates later additional runs nmt2 and nmt3, where we experiment with using local and global macros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Apart for allowing these additional macro mechanisms (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='2), the two runs resemble run nmt1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' In each of them we train the basic model0 together with the continuous model1, and infer with both of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Since the sequences are shorter (making model0 easier to train), we have not so far experimented with model2 here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' While the two runs seem superior to nmt1 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' 7, this is mainly due to the use of both models since iter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' 2 (instead of iter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' 21 in nmt1), and also an earlier switch to the slow check (iter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' 75 in nmt2 and iter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' 67 in nmt3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' The two runs also do not significantly complement nmt1: each of them adds less than 2800 solutions to nmt1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' This may mean that the alien system nmt1 has so far less trouble than humans with expanding everything and the use of definitions is not as critical as in humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' None of the two runs has however reached iter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' 100 yet, making the comparison with nmt1 only preliminary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Especially the statistics of the use of global macros10 in nmt3 are interesting, and can be used for analyzing the evolution of its coding trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' 6 Analysis of the Results We provide a statistical analysis of the 78118 solutions found during the nmt1 run and discuss the details of some techniques developed by our system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' More information on the nmt runs is available in our anonymous repository.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='11 For some sequences, its subdirectory12 contains our analysis of the 10https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='ly/3ZNe7fm 11https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='ly/3iVIfnX 12https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='ly/3XHZsjK 13 Alien Coding Gauthier, Olšák, Urban 0 20000 40000 60000 80000 25 50 75 100 125 150 175 tnn nmt0 nmt1 nmt2 nmt3 Figure 6: All solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' 0 250 500 750 1000 25 50 75 100 125 150 175 tnn nmt0 nmt1 nmt2 nmt3 Figure 7: Increments of solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' 14 Alien Coding Gauthier, Olšák, Urban Generation Avrg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Size 0 20 40 60 25 50 75 100 125 150 175 small fast Figure 8: Avrg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' size in iters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Generation Avrg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Time 20000 40000 60000 80000 200000 400000 600000 25 50 75 100 125 150 175 fast small Figure 9: Avrg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' time in iters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' 15 Alien Coding Gauthier, Olšák, Urban Generation Avrg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Size After X Iters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' 0 10 20 30 20 40 60 80 100 Figure 10: Avrg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' size after X iters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Generation Avrg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Time After X Iters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' 0 50000 100000 150000 200000 250000 20 40 60 80 100 Figure 11: Avrg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' time after X iters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' 16 Alien Coding Gauthier, Olšák, Urban evolution13,14 and proliferation15,16 of important programs such as primes and sigma, as well as proofs that some of the alien programs match the human OEIS intention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='17,18,19 See also the appendix for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Noteworthy sequences from this repository include convolution of primes with themselves,20 showing the proficiency of our system with primes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Motzkin numbers21 [25] are an example where our synthesized programs relies on a pairing function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' This programming technique, re-invented by our system, packs two variables into one, allowing further programs (see Appendix B for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' And a solution for the unique monotonic sequence of nonnegative integers satisfying a(a(n)) = 3n is needed in solving a problem in 27th British Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Olympiad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='22 Evolution of the Programs Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' 8 shows the evolution of the average size and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' 9 the speed of the solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' We see that as new solutions are found, they become longer and typically also take more time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' However, there are interesting exceptions to the latter rule (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' 11), as more efficient code is invented and propagated by the alien system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' We thus also measure the gradual size reduction (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' 10) and time reduction (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' 11) for the short and fast solutions (respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' This is for each sequence computed for 100 iterations after its first solution was found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' We see that the iterations induce a remarkable speedup of the invented fast solutions (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Generalization of the Solutions to Larger Indices OEIS provides additional terms for some of the OEIS entries in b-files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Among the 78118 solutions, 40,577 of them have a b-file that contains 100 additional terms for their OEIS entry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' We evaluate both the small and the fast programs with the slow check parameters on the 100 additional terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Here, 14,701 small and 11,056 fast programs time out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Among the programs that do not timeout, the percentage of generalizing programs (producing matching additional terms) is 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='57% for the slow programs and 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='51% for the fast programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' A common error is reliance on an approximation for a real number, such as π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' 7 Related work The most relevant related work is summarized in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' This includes [4], where the general approach is focused on training a single model using supervised learning techniques on synthetic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' The deep reinforcement learning system DreamCoder [5] has demonstrated self-improvement from scratch in several programming tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Its main contribution is the use of definitions that compress existing solutions and facilitate building new solutions on top of the existing ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Our experiments with local and global macros are however so far inconclusive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' While humans certainly benefit from introducing new names, concepts and shortcuts, it is so far unclear if it results in a clear improvement in our current alien setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' While the previous version of our system used MCTS inspired by AlphaGoZero [17], the current NMT approach uses just straightforward beam search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' The general search-verify-learn positive feedback loop between ML-guided symbolic search and statistical learning from the verified results has been used for long time in large-theory automated theorem proving, going back at least to the MaLARea 13https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='ly/3iJ4oGd 14https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='ly/3HfwemI 15https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='ly/3ZNExO4 16https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='ly/3IVzrJz 17https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='ly/3QPB25o 18https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='ly/3WlVHPM 19https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='ly/3XJi96j 20https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='ly/3XJi96j 21https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='ly/3QPB25o 22https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='ly/3wjmqSg 17 Alien Coding Gauthier, Olšák, Urban system [20, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Its recent instances include systems such as rlCoP [10] and ENIGMA [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Such systems can be also seen as synthesis frameworks, where the mathematical object synthesized by the guided search is however the full proof of a theorem, rather than just a particular interesting witness or a conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Synthesis of full proofs of mathematical theorems is an all-encompassing task that can be extremely hard in cases like Fermat’s Last Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Our current setting may thus also be seen as an attempt to decompose this all-encompassing task into smaller interesting subtasks that can be analyzed separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' 8 Conclusion As of January 25, 2023, all of the runs have together invented from scratch solutions for 84587 OEIS sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' This is more than three times the number (27987) invented in our first experiments [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' This is due to several improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' We have started to collect both the small and fast solutions, and learn jointly from them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' We have seen that this gradually leads to a large speedup of the fast programs, as the population of programs evolves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' This likely allows invention of further solutions that are within our time limit, thanks to the use of the faster and faster invented components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' The improvements (learning) on the symbolic side thus likely complement and co-evolve with the statistical learning used for guiding the synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' We have also started to use relatively fast neural translation models, their specialization on random subsets, their combinations and continuous training, and a wide beam search instead of MCTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' The experiments suggest that these techniques are useful, leading to a high rate of program invention even after the 190 iterations in the longest NMT run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' This is encouraging, since many of the solved OEIS problems seem nontrivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' The trained system could thus be used as a search tool assisting mathematicians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' There is also a wealth of related mathematical tasks that can be cast in a similar synthesis setting, and combined with other tools, such as automated theorem provers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' A crucial element of our setting is the fact that NMT is used to produce an interpretable symbolic representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' It would be practically unusable if we relied purely on neural approximation of arbitrary functions and the task for NMT was to directly produce the next 100 numbers in each OEIS sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' The interpretable symbolic representation is critical for the generalization capability of the overall system and its capability to learn from itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' The overall alien system can also be seen as an example of a very weakly supervised evolutionary architecture where simple high-level principles (Occam’s razor and efficiency) govern the development and training of the lower level statistical component (NMT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Rather than one-time training on everything that has been already invented by humans so far, as done by today’s large language models, the system here starts from zero knowledge and progresses towards increasingly nontrivial knowledge and skills.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Such feedback loops thus seem to be a good playground for exploring how increasingly intelligent systems emerge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Note that thanks to the Turing completeness of the language, this particular playground is (unlike games like Chess and Go) not limited in its expressivity, and can in principle lead to the development of arbitrary complex algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' 9 Acknowledgments We thank Chad Brown, David Cerna, Hugo Cisneros, Tom Hales, Barbora Hudcova, Jan Hula, Mikolas Janota, Tomas Mikolov, Jelle Piepenbrock, and Martin Suda for discussions, comments and suggestions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' This work was partially supported by the CTU Global Postdoc funding scheme (TG), Czech Science Foundation project 20-06390Y (TG), ERC-CZ project POSTMAN no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' LL1902 (TG), Amazon Research Awards (TG, JU), EU ICT-48 2020 project TAILOR no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' 952215 (JU), and the European Regional Development Fund under the Czech project AI&Reasoning no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' CZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='01/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='0/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='0/15_003/0000466 (MO, JU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' 18 Alien Coding Gauthier, Olšák, Urban References [1] Marcin Andrychowicz, Filip Wolski, Alex Ray, Jonas Schneider, Rachel Fong, Peter Welinder, Bob McGrew, Josh Tobin, OpenAI Pieter Abbeel, and Wojciech Zaremba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Hindsight experience replay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Advances in neural information processing systems, 30, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' [2] Mark Chen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Jerry Tworek,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Heewoo Jun,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Qiming Yuan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Henrique Ponde de Oliveira Pinto,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Jared Kaplan,' metadata={'source': 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Reasoning (IJCAR), volume 5195 of LNCS, pages 441–456.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Springer, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' [23] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Gomez, Lukasz Kaiser, and Illia Polosukhin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Attention is all you need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' In Isabelle Guyon, Ulrike von Luxburg, Samy Bengio, Hanna M.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' First experiments with neural translation of informal to formal mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' In CICM, volume 11006 of Lecture Notes in Computer Science, pages 255–270.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Springer, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' [25] Yi Wang and Zhi-Hai Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Combinatorics of generalized motzkin numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Integer Seq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=', 18(2):15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='4, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' 20 Alien Coding Gauthier, Olšák, Urban A Resources Used by the Experiment The average GPU power consumption during inference is 800W: 200W per each of the four GTX 1080 Ti cards used for inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' The average GPU power consumption during training is 350W: 175W per each of the two GTX 1080 Ti cards used for the two kinds of training done in most of the iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' The training takes approximately 3 hours and the inference 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='5 hours (iteration 140 of the longest run).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' This means that the average GPU consumption per hour is (3 ∗ 350 + 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='5 ∗ 800)/13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='5 = 700W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Additionally, we estimate the non-GPU (CPUs, disks, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=') consumption to be on average 150W per hour (the server’s CPUs are mostly idle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' It took about three months to do the 190 iterations of the longest run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' This is 90 ∗ 24 = 2160 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' The estimated power consumption for the 190 iterations is thus 2160 ∗ (700 + 150) = 1836 kWh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Assuming a cost in the range of 150-250 EUR per MWh, the main experiment’s electricity cost is between 270 and 460 EUR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' The three shorter experiments have so far run for about half as many iterations and are also using on average less resources (fewer GPUs, shorter inference length).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' The estimated power consumption for all experiments is thus likely below 4 MWh, and the total electricity cost below 1000 EUR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' B Triangle Coding Our system has found a correspondence between a pair of non-negative integers (xa, xb) where xa ≥ xb, and a single non-negative integer x by enumerating the following sequence of pairs (xa, xb) with non-negative integers: (0, 0), (1, 0), (1, 1), (2, 0), (2, 1), (2, 2), (3, 0), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' In one direction, it computes x = xa×(xa+1) 2 + xb in order to “encode” the pair (xa, xb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Decoding of a single non-negative integer x can be calculated as (xa, xb) = (f0(x) − f1(x), f0(x)), where the functions f0, f1 can be implemented in Python for example as follows (an actual code invented by the program generator).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Note that the function f0 computes xb, and the function f1 computes xb − xa (a non-positive integer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' def f0(X): x = X for y in range (1,(2 + (X // (1 + (2 + 2)))) + 1): x = x - (y if (y - x) <= 0 else 0) return x def f1(X): x = X for y in range (1,(2 + (X // (1 + (2 + 2)))) + 1): x = x - (0 if x <= 0 else (1 + y)) return x This representation was initially discovered when our system found solutions for the sequences A2262,23 and A25581.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='24 Indeed, f0 is a solution for A2262 invented in the first generation and −f1 is a solution for A25581 invented during the 9th generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' 23https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A002262 24https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A025581 21 Alien Coding Gauthier, Olšák, Urban Figure 12: Linear bounds for � 2x + 1 4 − 1 2 invented by the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Figure 13: Linear bounds for � 2x + 1 4 − 1 2 invented by the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='1 Number of steps In order to calculate f0 and f1, one must perform approximately � 2x + 1 4 − 1 2 steps in a subtracting loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Since the programming language is based on for loops, the programs are approximating the number of subtracting steps with a function which can be easily obtained using the basic arithmetical operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' In particular, the code above uses approximations with 2 + x/5 where 5 is written as 1 + (2 + 2) (our basic language only supports constants 0,1,2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' The program generator has experimented with many (over 80) such bounds, vast majority of such bounds are of the shape a + x/b or a + 2(x/b) where a, b are constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' As the speed of computing with larger numbers becomes more important, the slope of the approximation flattens, with b reaching values over 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' We have collected all such subprograms used in the generated programs, and plotted the bounds in Figure 12 and Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' The approximated function � 2x + 1 4 − 1 2 is plotted with a black dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' All the found valid bounds are plotted green, all the invalid bound attempts are plotted red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='2 Triangle coding in action The triangle coding turned out to be useful in many cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' The designed language for our programs only supports a maximum of two variables but using the triangle coding, a pair of variables can be temporarily packed into one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' A simple example is sequence A27936425 – sum of 5th powers of proper divisors of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' A human programmer could implement this sequence as: def f(X): res = 0 for y in range(1,X+1): res = res + (y**5 if (X+1) % 25https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A279364 22 Alien Coding Gauthier, Olšák, Urban return res There are three variables used in the body of the loop, in particular res, y, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Since the native language supports only two variables available at every moment, this Python code cannot be straightforwardly translated into the language of our programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Nevertheless, the program generator has found a workaround using the triangle coding – it packs X and y into a single natural number, so it can perform the summing loop adding to res, and in order to decode what to add to res, it unpacks the pair, and checks whether y divides X+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' The code generated by NMT for A279364 is as follows: A279364 Sum of 5th powers of proper divisors of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' 371326 59293 555688 1 870552 1 1082401 161295 1419890 19933 2206525 1 2476132 371537 3336950 1 4646784 1 5315740 821793 6436376 1 9301876 16808 9868783 size 94,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' time 1335684: loop2 ((loop (loop2 ((loop ((((x * x) * x) * x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='x) 1 (1 + y)) * (if (x mod (1 + y)) <= 0 then 1 else 0)) 0 1 (1 - ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='(loop (x - (if x <= 0 then 0 else y)) (1 + (2 + (2 + (x div (1 + (2 * ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='(2 + 2))))))) (1 + x))) (loop (x - (if (y - x) <= 0 then y else 0)) (2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='+ (2 + (x div (1 + (2 * (2 + 2)))))) x)) 1 y) + x) (1 + y) x 0 (((x * ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='x) - x) div 2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='K K F K F K F K F B B L D J K B L D H B A I F A B B K K A L I E B C C ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='K B C C C D F D G D D D B K D J E K L K E L A I E C C K B C C C D F D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='G D D K J N B L J K D B L D K A K K F K E C G N ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='def f3(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='Y): x = 1 + Y for y in range (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='1 + 1): x = (((x * x) * x) * x) * x return x def f4(X): x = 1 + X for y in range (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='(1 + (2 + (2 + (X // (1 + (2 * (2 + 2))))))) + 1): x = x - (0 if x <= 0 else y) return x def f5(X): x = X for y in range (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='(2 + (2 + (X // (1 + (2 * (2 + 2)))))) + 1): x = x - (y if (y - x) <= 0 else 0) return x def f2(X): x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='y = 1 - f4(X),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' f5(X) for z in range (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='1 + 1): x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='y = (f3(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='y) * (1 if (x % return x def f1(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='Y): x = Y 23 Alien Coding Gauthier,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Olšák,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Urban for y in range (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='1 + 1): x = f2(x) return x def f0(X): x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='y = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' ((X * X) - X) // 2 for z in range (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='X + 1): x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='y = (f1(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='y) + x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' (1 + y) return x for x in range(32): print (f0(x)) Functions f4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' f5 are calculating the triangle coding,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' f3 is the fifth power,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' f1 is just a dummy function using Y instead of X for f2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Finally, f2 is returning either (b + 1)5, or 0 depending on whether b + 1 divides a + 2 or not where (a, b) if the triangle coding pair of X, and f0 is summing over all the pairs (a, b) going from (X-1,0) to (X-1,X-1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' C Evolution and Proliferation We analyze the evolution of the solutions found for the OEIS sequence of prime numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='26 The exact iterations of their discovery are shown in our repository, together with their size and speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='27 The 24 invented programs are shown in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' We can observe how these 24 programs propagate through the population of all programs, as they evolve in the iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' This is shown in Table 4 and Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' We can see that, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=', P5 (size 25, time 515990), gets quickly replaced by P6 (size 33, time 98390) and P7 (size 23, time 519654) after their invention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' P6 is more than five times faster than P5, while P7 is smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Therefore they are included in the training data instead of P5 when they are invented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' While there are still other programs in the training data using P5 as a subprogram, their faster/smaller versions get eventually also reinvented as the newly trained NMT increasingly prefers to synthesize programs that contain P6 or P7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' This illustrates the dynamics of the overall alien system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' The training of the neural synthesis component (NMT) evolves, governed by more high-level evolutionary fitness criteria, which are in our case size (Occam’s razor) and speed (efficiency).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' D Selection of 123 Solved Sequences Tables 6, 7, 8 present a sample of 123 sequences solved during the nmt0 and nmt1 runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Their solutions found by the system are presented on our web page,28 both in our language and translated to Python.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' 26The tables shown here for primes are also in our repository https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='ly/3XHZsjK, together with similar tables for the sigma function (sum of the divisors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' 27https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='ly/3iJ4oGd 28https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='com/Anon52MI4/oeis-alien 24 Alien Coding Gauthier, Olšák, Urban Table 3: The 24 programs for primes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='Nr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='Program ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='P1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='(if x <= 0 then 2 else 1) + (compr (((loop (x + x) (x mod 2) (loop (x * x) 1 (loop (x + x) (x div 2) 1))) + x) mod (1 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='x)) x) ' 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((loop2 (1 + (if (x mod (1 + y)) <= 0 then 0 else x)) (y - 1) x 1 x) mod (1 + x)) x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='P5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='1 + (compr ((loop (if (x mod (1 + y)) <= 0 then (1 + y) else x) x (1 + x)) mod (1 + x)) (1 + x)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='P6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='1 + (compr ((loop (if (x mod (1 + y)) <= 0 then (1 + y) else x) (2 + (x div (2 + (2 + 2)))) (1 + x)) mod (1 + x)) (1 + x)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='P7 ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='mod 2)) 1 x)) (1 + x)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='P23 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='2 + (compr (loop (x - (if (x mod (1 + y)) <= 0 then 0 else 1)) x x) x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='P24 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='loop (1 + x) (1 - x) (1 + (2 * (compr (x - (loop (if (x mod (2 + y)) <= 0 then 1 else x) (2 + (x div (2 * (2 + 2)))) (1 + (x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='+ x)))) x))) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='Alien Coding ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='Gauthier,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Olšák,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Urban Table 4: Proliferation of the 24 programs for primes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' ' metadata={'source': 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for 6-dimensional cubic lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A8628 Molien series for A5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A259445 Multiplicative with a(n) = n if n is odd and a(2s) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A314106 Coordination sequence Gal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='199.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='4 where G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='v denotes the coordination sequence for a vertex of type v in tiling number t in the Galebach list of u-uniform tilings https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A311889 Coordination sequence Gal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='129.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='2 where G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='v denotes the coordination sequence for a vertex of type v in tiling number t in the Galebach list of u-uniform tilings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A315334 Coordination sequence Gal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='623.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='2 where G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='v denotes the coordination sequence for a vertex of type v in tiling number t in the Galebach list of u-uniform tilings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A315742 Coordination sequence Gal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='302.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='5 where G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='v denotes the coordination sequence for a vertex of type v in tiling number t in the Galebach list of u-uniform tilings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A004165 OEIS writing backwards https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A83186 Sum of first n primes whose indices are primes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A88176 Primes such that the previous two primes are a twin prime pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A96282 Sums of successive twin primes of order 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A53176 Primes p such that 2p + 1 is composite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A267262 Total number of OFF (white) cells after n iterations of the "Rule 111" elementary cellular automaton starting with a single ON (black) cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A273385 Number of active (ON,black) cells at stage 2n − 1 of the two-dimensional cellular automaton defined by "Rule 659", based on the 5-celled von Neumann neighborhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A60431 Number of cubefree numbers <= n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A42731 Denominators of continued fraction convergents to sqrt(895).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A81495 Start with Pascal’s triangle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' form a rhombus by sliding down n steps from top on both sides then sliding down inwards to complete the rhombus and then deleting the inner numbers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' a(n) = sum of entries on perimeter of rhombus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A20027 Nearest integer to Gamma(n + 3/8)/Gamma(3/8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A99197 Figurate numbers based on the 10-dimensional regular convex polytope called the 10-dimensional cross-polytope, or 10- dimensional hyperoctahedron, which is represented by the Schlaefli symbol 3, 3, 3, 3, 3, 3, 3, 3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' It is the dual of the 10-dimensional hypercube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A220469 Fibonacci 14-step numbers, a(n) = a(n − 1) + a(n − 2) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' + a(n − 14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A8583 Molien series for Weyl group E7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A251672 8-step Fibonacci sequence starting with 0,0,0,0,0,0,1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A124615 Poincaré series [or Poincare series] P(T3,2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A79262 Octanacci numbers: a(0) = a(1) = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' = a(6) = 0, a(7) = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' for n >= 8, a(n) = Sumi=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='.8a(n − i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A75068 Product of prime(n) primes starting from prime(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A57168 Next larger integer with same binary weight (number of 1 bits) as n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A1553 a(n) = 1n + 2n + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' + 6n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A19560 Coordination sequence for C4 lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A289834 Number of perfect matchings on n edges which represent RNA secondary folding structures characterized by the Lyngso and Pedersen (L&P) family and the Cao and Chen (C&C) family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A5249 Determinant of inverse Hilbert matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A3714 Fibbinary numbers: if n = F(i1) + F(i2) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' + F(ik) is the Zeckendorf representation of n (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=', write n in Fibonacci number system) then a(n) = 2i1−2 + 2i2−2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' + 2ik−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Also numbers whose binary representation contains no two adjacent 1’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A4457 Nimsum n + 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A92143 Cumulative product of all divisors of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='.n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A2119 Bessel polynomial yn(−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A5913 a(n) = [tau ∗ a(n − 1)] + [tau ∗ a(n − 2)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A34960 Divide odd numbers into groups with prime(n) elements and add together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A247395 The smallest numbers of every class in a classification of positive numbers (see comment).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A68068 Number of odd unitary divisors of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' d is a unitary divisor of n if d divides n and gcd(d, n/d) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' 28 Alien Coding Gauthier, Olšák, Urban Table 7: Samples of the solved sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A30973 [ exp(1/5) ∗ n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A54469 A second-order recursive sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A54054 Smallest digit of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A36561 Nicomachus triangle read by rows, T(n, k) = 2n−k ∗ 3k, for0 <= k <= n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A107347 Number of even semiprimes strictly between prime(n) and 2 * prime(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A123379 Values x of the solutions (x,y) of the Diophantine equation 5 ∗ (X − Y )4 − 4XY = 0 with X >= Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A201204 Half-convolution of Catalan sequence A000108 with itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A125494 Composite evil numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A277094 Numbers k such that sin(k) > 0 and sin(k + 2) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A59760 a(n) is the number of edges (one-dimensional faces) in the convex polytope of real n X n doubly stochastic matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A246303 Numbers k such that cos(k) < cos(k + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A7957 Numbers that contain an odd digit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A7452 Expand cosx/expx and invert nonzero coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A88896 Length of longest integral ladder that can be moved horizontally around the right angled corner where two hallway corridors of integral widths meet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A131989 Start with the symbol *| and for each iteration replace * with *| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' This sequence is the number of *’s between each dash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A308066 Number of triangles with perimeter n whose side lengths are even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A11540 Numbers that contain a digit 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A156660 Characteristic function of Sophie Germain primes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A167132 Gaps between twin prime pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A8846 Hypotenuses of primitive Pythagorean triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A332381 a(n) is the Y-coordinate of the n-th point of the Peano curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Sequence A332380 gives X-coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A25492 Fixed point reached by iterating the Kempner function A002034 starting at n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A131530 Numbers k such that k2 − k − 1 and k2 − k + 1 are twin primes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A143165 Expansion of the exponential generating function arcsin(2x)/(2(1 − 2 ∗ x)3/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A30957 [ exp(1/9) ∗ n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A295286 Sum of the products of the smaller and larger parts of the partitions of n into two parts with the smaller part odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A86699 Number of n X n matrices over GF(2) with rank n-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A2819 Liouville’s function L(n) = partial sums of A008836.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A7318 Pascal’s triangle read by rows: C(n, k) = binomial(n, k) = n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='/(k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' ∗ (n − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' ), 0 <= k <= n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A8836 Liouville’s function lambda(n) =(−1)k, where k is number of primes dividing n (counted with multiplicity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A266776 Molien series for invariants of finite Coxeter group A7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A284115 Hosoya triangle of Lucas type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A45717 For each prime p take the sum of nonprimes < p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A307508 Primes p for which the continued fraction expansion of sqrt(p) does not have a 1 in the second position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A3506 Triangle of denominators in Leibniz’s Harmonic Triangle a(n, k), n >= 1, 1 <= k <= n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A93017 Luhn algorithm double-and-add sum of digits of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A121373 Expansion off(x) = f(x, −x2) in powers of x where f(, ) is Ramanujan’s general theta function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A227127 The Akiyama-Tanigawa algorithm applied to 1/(1,2,3,5,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' old prime numbers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' Reduced numerators of the second row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A39637 Number of steps to fixed point of "n− > n/2or(n + 1)/2 until result is prime".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A548 Squares that are not the sum of 2 nonzero squares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A131650 Number of symbols in Babylonian numeral representation of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' 29 Alien Coding Gauthier, Olšák, Urban Table 8: Samples of the solved sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A3538 Divisors of 230−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A152135 Maximal length of rook tour on an n X n+4 board.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A8637 Number of partitions of n into at most 8 parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A113953 A Jacobsthal triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A3605 Unique monotonic sequence of nonnegative integers satisfying a(a(n)) = 3n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A266214 Numbers n that are not coprime to the numerator of zeta(2n)/(Pi2n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A266778 Molien series for invariants of finite Coxeter group A9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A101608 Solution to Tower of Hanoi puzzle encoded in pairs with the moves (1, 2), (2, 3), (3, 1), (2, 1), (3, 2), (1, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' The disks are moved from peg 1 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' For a tower of k disks use the first 2k − 1 number pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A90971 Sierpi´nski’s triangle, read by rows, starting from 1: T(n, k) = (T(n − 1, k) + T(n − 1, k − 1))mod2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A83743 a(1) = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' if a(n − 1) + n is prime then a(n) = a(n − 1) + n, else a(n) = a(n − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A79683 Order of Burnside group B(6, n) of exponent 6 and rank n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A34444 a(n) is the number of unitary divisors of n (d such that d divides n, gcd(d, n/d) = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A218509 Number of partitions of n in which any two parts differ by at most 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A65109 Triangle T(n, k) of coefficients relating to Bezier curve continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A166555 Triangle read by rows, Sierpinski’s gasket, A047999 * (1,2,4,8,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=') diagonalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A119467 A masked Pascal triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A194887 Numbers that are the sum of two powers of 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A1824 Central factorial numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A47780 Number of inequivalent ways to color faces of a cube using at most n colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A8640 Number of partitions of n into at most 11 parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A29635 The (1,2)-Pascal triangle (or Lucas triangle) read by rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A45995 Rows of Fibonacci-Pascal triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A5045 Number of restricted 3 X 3 matrices with row and column sums n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A971 Fermat coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A5835 Pseudoperfect (or semiperfect) numbers n: some subset of the proper divisors of n sums to n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A266773 Molien series for invariants of finite Coxeter group D10 (bisected).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A69209 Orders of non-Abelian Z-groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A8383 Coordination sequence for A4 lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A70896 Determinant of the Cayley addition table of Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A262 Number of "sets of lists": number of partitions of 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=',n into any number of lists, where a list means an ordered subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A23436 Dying rabbits: a(n) = a(n − 1) + a(n − 2) − a(n − 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A8641 Number of partitions of n into at most 12 parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A68764 Generalized Catalan numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A7856 Subtrees in rooted plane trees on n nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A271 Sums of ménage numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A199033 Number of ways to place n non-attacking bishops on a 2 X 2n board.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A1006 Motzkin numbers: number of ways of drawing any number of nonintersecting chords joining n (labeled) points on a circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A239768 Number of pairs of functions (f,g) from a set of n elements into itself satisfying f(x) = f(g(f(x))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A2895 Domb numbers: number of 2n-step polygons on diamond lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A14342 Convolution of primes with themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' https://oeis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content='org/A27847 a(n) = Sumd|nsigma(n/d) ∗ d3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} +page_content=' 30' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FJT4oBgHgl3EQfNyw_/content/2301.11479v1.pdf'} diff --git a/UdAzT4oBgHgl3EQflv12/vector_store/index.faiss b/UdAzT4oBgHgl3EQflv12/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..ce619aa7351434ac9daa72cac65bfbdf6c0c10a5 --- /dev/null +++ b/UdAzT4oBgHgl3EQflv12/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:63044e55b66492fea5cb9ab0b296445fad03b644c6811143f611cd45a3e3392e +size 5767213 diff --git 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a/XtFOT4oBgHgl3EQf9DQS/content/tmp_files/2301.12968v1.pdf.txt b/XtFOT4oBgHgl3EQf9DQS/content/tmp_files/2301.12968v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a68b6932b42ea4ff2b9a74d58d32e646619f3192 --- /dev/null +++ b/XtFOT4oBgHgl3EQf9DQS/content/tmp_files/2301.12968v1.pdf.txt @@ -0,0 +1,1373 @@ +Improving Adversarial Transferability with Scheduled Step Size and Dual +Example +Zeliang Zhang +University of Rochester +Peihan Liu +Harvard University +Xiaosen Wang* +HUST +Chenliang Xu⋆ +University of Rochester +Abstract +Deep neural networks are widely known to be vulnerable +to adversarial examples, especially showing significantly +poor performance on adversarial examples generated un- +der the white-box setting. However, all white-box attack +methods rely heavily on the target model and quickly get +stuck in local optima, resulting in poor adversarial trans- +ferability. The momentum-based methods and their variants +are proposed to escape the local optima for better transfer- +ability. In this work, we notice that the transferability of +adversarial examples generated by iterative fast gradient +sign method (I-FGSM) exhibits a decreasing trend when in- +creasing the number of iterations. Motivated by this find- +ing, we argue that the information of adversarial perturba- +tions near the benign sample, especially the direction, ben- +efit more on the transferability. Thus, we propose a novel +strategy, which uses the Scheduled step size and the Dual +example (SD), to fully utilize the adversarial information +near the benign sample. Our proposed strategy can be eas- +ily integrated with existing adversarial attack methods for +better adversarial transferability. Empirical evaluations on +the standard ImageNet dataset demonstrate that our pro- +posed method can significantly enhance the transferability +of existing adversarial attacks. +1. Introduction +Adversarial examples [13,45], crafted by adding human- +imperceptible perturbations on benign samples to fool deep +neural networks (DNNs), have drawn more research inter- +est in recent years [1, 21, 66]. On the one hand, the exis- +tence of adversarial examples identifies the vulnerability of +DNNs, throwing out the severe security issues to be solved +urgently on many areas, including the autonomous driv- +ing [20], facial authentication [22], object detection [17], +et al. The adversarial examples also generalized to other +models [39,57,60], where the adversarial examples crafted +*Corresponding author +by one model can also fool other models. On the other +hand, the adversarial examples help evaluate the robustness +of DNNs and enhance the robustness of DNNs through ad- +versarial training [42,54]. Great research interest has arisen +in how to generate adversarial examples with high transfer- +ability for a better understanding of the DNNs [55]. +According to the knowledge about the victim model, the +existing adversarial attacks can be categorized into two set- +tings, namely the white-box [13, 36, 65] and the black-box +settings [19, 41, 44]. With the full knowledge of the vic- +tim model, including the architecture, the parameter, the +training loss function, etc., most of the existing adversar- +ial attacks under the white-box setting usually exhibit a +great performance but behave poorly facing the models +without knowing any information under the black-box set- +ting [28,49]. The low adversarial transferability makes ap- +plying such existing adversarial attacks inefficient in real- +world applications, where the victim models are usually un- +known to attackers [38]. +Some methods have been proposed to enhance the trans- +ferability of adversarial examples, such as gradient-based +attacks with momentum [15, 48], input transformation- +based attacks [40, 50], ensemble attacks [4, 43], etc. +Among them, the gradient-based methods mainly utilize +the momentum (MI-FGSM) [9] or pre-computation (NI- +FGSM) [29] to help escape the local optima for better ad- +versarial transferability. The input transformation [10, 59] +methods analogize the crafting of adversarial examples to +the generalization of DNNs, thus diversifying the inputs to +generate the adversarial examples with high transferability. +Different from the above adversarial attacks, we study +adversarial transferability from a different perspective. +From a simple experiment, we observe that the transfer- +ability of adversarial examples crafted using vanilla itera- +tive adversarial attacks demonstrate a trend from high to +low, which indicates that the directions of the adversarial +perturbations near the benign sample are more transferable +among different models. Motivated by this, we propose a +novel strategy with the Scheduled step size and Dual ex- +ample (SD), which fully explores and utilizes the adversar- +1 +arXiv:2301.12968v1 [cs.LG] 30 Jan 2023 + +ial perturbations with high transferability near the benign +sample. Also, it can be easily integrated on the existing +adversarial attack methods to enhance the performance on +transferability. +We apply our proposed method to several generally used +adversarial attack methods and conduct proof experiments +on the standard ImageNet dataset. The results demonstrate +that our proposed method can remarkably boost the existing +adversarial attack methods on transferability. +2. Related Work +In this section, we provide a brief overview of adversarial +attack and defense. +2.1. Adversarial Attack +Since Szegedy et al. [45] identified the vulnerability of +DNNs to adversarial examples, numerous adversarial attack +methods have been proposed, including white-box attack [8, +13, 24, 34, 36], transfer-based attack [9, 10, 48, 49, 51, 60], +score-based attack [6, 19, 26, 47], decision-based attack [3, +5,25,35,52], etc. Among the above attacks, a transfer-based +attack, as a black-box attack, does not need access to the +target model, making it popular to attack the deep models +in the real world. We provide an overview of the transfer- +based attacks as follows. +Gradient-based Attacks. Fast Gradient Sign Method +(FGSM) [13], which is the first gradient-based attack, crafts +adversarial examples by adding a large perturbation in the +gradient direction to the benign sample. +To improve its +white-box attack performance, it is further extended to an +iterative Version (I-FGSM) [24] but shows poor transfer- +ability. With the potential application of transfer-based at- +tacks in the real world, various methods have been pro- +posed to improve the transferability of adversarial exam- +ples. Dong et al. [9] introduce the momentum into I-FGSM, +dubbed Momentum I-FGSM (MI-FGSM), to stabilize the +optimization and escape the local optima. Lin et al. [29] +utilize Nesterov accelerated gradient to look ahead, denoted +as NI-FGSM, which exhibits better transferability. Wang et +al. [48] propose the definition of gradient variance, which +could be used to tune the gradient for MI-FGSM (VMI- +FGSM) and NI-FGSM (VNI-FGSM). Wang et al. [51] en- +hance the momentum with the gradient of multiple samples +in the neighborhood of the current adversarial example. +Input transformation-based attacks. Inspired by the +data augmentation in training, various input transformation- +based attacks are proposed to improve the transferability, +which can be combined with the above gradient-based at- +tacks. For instance, the diverse input method (DIM) [60] +resizes the input image into a random size, which will be +padded to a fixed size for gradient calculation. Transla- +tion invariant method (TIM) [10] adopts Gaussian smooth +on the gradient to approximate the average gradient of a set +of translated images to update the adversary. Scale-invariant +method (SIM) [29] calculates the gradient on a set of scaled +images. Admix [49] adds a small portion of images from +other categories to the input image to obtain several images +for gradient calculation. +Ensemble attacks. Liu et al. [30] first find that ensem- +ble attack, which generates adversarial examples on mul- +tiple models, can lead to better transferability. Xiong et +al. [62] reduces the gradient variance among various mod- +els to boost ensemble attack. Long et al. [33] augment the +model in the frequency domain to obtain more diverse sub- +stitute models. +Others. Gao et al. [12] reuses the cut noise introduced +by clip operation. +Wu et al. [56] argues that the attack +should maximize the difference of the attention map be- +tween benign samples and adversarial examples. +2.2. Adversarial Defense +On the other hand, to mitigate the threat of adversar- +ial attacks, a variety of defense methods have been pro- +posed, e.g. adversarial training [13, 34, 53, 64], input pre- +processing [14], certified defense [7] etc. Adversarial train- +ing has been shown as one of the most effective methods to +improve adversarial robustness [2,8]. It is first proposed by +Madry et al. [34] that including the adversarial examples in +the training data enhances the adversarial robustness. Wong +et al. [54] use the fast gradient sign method with random +initialization to generate adversarial examples for efficient +adversarial training. +3. Methods +In this section, we introduce our motivation and provide +a detailed description of the proposed strategy SD. +3.1. Motivation +Given a target deep model f and a benign sample x with +the ground-truth label y, adversarial attacks find an example +xadv similar to x (i.e., xadv ∈ Bϵ(x) = {x′ : ∥x′ − x∥p ≤ +ϵ} where ∥ · ∥p refers to the p−norm) such that the model +gives different predictions, i.e., f(x) = y ̸= f(xadv). +Gradient-based adversarial attacks (e.g., FGSM [13], I- +FGSM [24], MI-FGSM [9], NI-FGSM [29], etc.) can gen- +erate the adversarial examples xadv efficiently. Goodfellow +et al. [13] first propose the gradient sign method (FGSM) to +attack the deep learning model efficiently. However, single- +step optimization for adversarial examples may not find the +optima, which fails to fool the models under the white-box +setting, within the box constraint Bϵ(x). Iterative fast gra- +dient sign method (I-FGSM) employs the iterative process +to optimize xadv and can achieve a attack success rate of +100% or nearly 100%. +Suppose L(·, ·) is the classification loss, I-FGSM adopts +2 + +1 +2 +3 +4 +5 +Iteration +20 +40 +60 +80 +100 +Attack Success Rate(%) +ResNet-18 +ResNet-101 +ResNeXt-50 +DenseNet121 +ViT +Swin +Figure 1. Varying the maximum number of iteration, the attack +success rate of six models on the adversarial examples generated +by ResNet-18. +the iterative optimization process to craft an adversarial ex- +ample xadv which maximizes L(f(xadv), y) as follows, +xadv +t+1 = ΠBϵ(x) +� +xadv +t ++ α · sign +� +∇xL +� +f +� +xadv +t +� +, y +��� +, +(1) +where Π is the projector function, α is the step size, and +xadv +t +is the adversarial example at the t-th iteration. +To further explore how the number of iterations affects +the attack performance, we conduct I-FGSM attack on Im- +ageNet dataset with different iterations (T = 1 → 5 itera- +tions). The adversarial examples are generated on ResNet- +18 [16], which are used to attack other models (ResNet- +101 [16], ResNext-50 [61], DenseNet-121 [18], ViT [11], +Swin [31]) with the perturbation budget ϵ = 16/255 and +step size α = ϵ/T. As shown in Fig. 1, I-FGSM achieves +100% attack success rate on the white-box model (ResNet- +18) with 2 iterations. +Also, it is obvious that the trans- +ferabilty of adversarial examples generated by FGSM, i.e., +T = 1, are better than that of I-FGSM with multiple steps +(T ≥ 4). Since the difference of updates on xadv in each +iteration only falls on the direction, i.e., the sign of the gra- +dient, it naturally inspires us the following assumption, +Assumption 1. The direction of the adversarial perturba- +tion near the benign sample benefits more on adversarial +transferability than that far away. +Lin et al. [29] analogize the search for adversarial exam- +ples with the training process of models and the transfer- +ability of adversarial examples with the generalization abil- +ity. Training on the same dataset, e.g., ImageNet dataset, +different models usually have similar attentions on the fea- +tures. However, the adversarial example is one category +of the out-of-distribution (OOD) data. Evaluated on such +OOD datasets, different models may have different perfor- +mances, i.e., different attentions on the features. With an in- +creasing number of iterations, the distribution shifts become +Resnet-18 +ResNet-101 +ResNeXt-50 +DenseNet-121 +Vit +Swin +0 +20 +40 +60 +80 +100 +Attack Success Rate (%) +Identity +Ln +Linear +Power +Exp +Figure 2. With different decreasing scheduled step sizes, the attack +success rate of six model on the adversarial examples generated by +ResNet-18. +large and the performance difference, i.e., attention on the +features, will also become large. Thus, the adversarial per- +turbation added on examples computed by the white-box +model, that are far away from the benign sample, are not +generally easy to transfer to be efficient for other models. +A natural method to alleviate the above problem, caused +by distribution shifts, is to use decreasing step sizes to fully +utilize the adversarial perturbation direction near the benign +sample. Thus, we use several kinds of decreasing sequence +as the step sizes α to optimize xadv in (1) and evaluate the +attack performances on aforementioned models. +We adopt four decreasing sequences: logarithm (αi = +ln(T − i)), linear (αi = T − i), power (αi = (T − i)2), and +exponential (αi = eT −i) sequences. We normalize each +sequence by dividing the sum to satisfy Bϵ(x). As shown +in results in Fig. 2, the scheduled decreasing step size im- +proves the adversarial attack success rates under the black- +box setting, with an average of 2.66% for (Ln), 2.75% for +(Linear), 2.98% for (Power) and 2.72% for (Exp). However, +the results are still worse than that of FGSM and I-FGSM +with a few iterations on the adversarial transferability. The +reason why the proposed method does not work as expected +is that, there is a huge gap between the magnitude of the first +step and the magnitude of the last step, which leads to insuf- +ficient updates on the last several iterations of I-FGSM. The +huge gap of the schedule step sizes between the early and +last stages are also discussed in our ablation study in Sec- +tion 4.6. So, besides employing the scheduled decreasing +steps, is there any alternative way to fully utilize the infor- +mation near the benign sample to enhance the adversarial +transferability? +3.2. Method +To fully utilize the information, i.e., the direction of ad- +versarial perturbations with high transferability, except for +3 + +Figure 3. Visualization of the adversarial examples with perturbations for each iteration. The strength of the color on the mask represents +the adversarial perturbation. The sharp colors, especially red, indicate the large perturbations, and the soft colors, e.g., yellow, indicate the +small perturbations. Please view on a color screen. +using the decreasing step size, we can also use the sched- +uled increasing step size. +Scheduled step size: As aforementioned, we hope to uti- +lize the scheduled decreasing step size to rearrange larger +weights on the early updates for xadv, where the direction of +the updates, i.e., the adversarial perturbation, is more trans- +ferable. Moreover, we want the updates on the later stages +to sufficiently useful. It turns out that using the scheduled +increasing step sizes can achieve the two goals simultane- +ously. More specifically, we sample more directions of the +adversarial perturbation with high transferability and put +them on use in the later iterations to enhance the transfer- +ability of adversarial examples. +In particular, rather than using an identity step size in (1), +we made the step sizes as a increasing sequence to find xadv +as follows, +xadv +t+1 = Clipϵ +� +xadv +t ++ ˆαtsign +� +∇xadv +t +L +� +f +� +xadv +t +� +, y +��� +, +(2) +where ˆα is the scheduled step size sequence. +As shown in Fig. 2, due to the huge gap of the step size +between the early and last several iterations, the inefficient +updates on the adversarial example on the last several itera- +tions do not gain a lot for the transferability. However, with +the increasing sequence as the scheduled step size, the up- +dates on the last few iterations becomes sufficient. Also, due +to the small step size on the first few iterations, xadv stays +near around x, meaning that we can sample more directions +of adversarial perturbations with high transferability. Then, +the next question is: how can we utilize the sampled adver- +sarial perturbations with high transferability? +Dual example: In order to make full use of the transferable +directions of adversarial perturbations, we propose a novel +strategy, namely dual example, which uses two examples in +parallel but applies different updates on them. +Specifically, we initialize two examples xadv = xdual = +x, namely the decoy and dual example, respectively, in our +method. In each iteration, we use the decoy example, i.e., +xadv, to explore and sample adversarial perturbations, and +use xdual to fully utilize them. Due to the scheduled in- +creasing step size, the early sampled adversarial perturba- +tions using small step size near the benign sample are usu- +ally more transferable than the later ones. Thus, we average +all the previous adversarial perturbations as an ensemble re- +sult to compute the update on the current iteration. There +are two advantages using this way. On the one hand, even +for the last few steps, where xadv is quite far away from x +and the adversarial perturbations usually behave poorly on +the transferability, the actual updates on xdual will not be af- +fected much from the ensemble result on the early updates. +On the other hand, the large step size, for the later stage, +will help xdual jump over the local optima as well as mak- +ing each update more sufficiently useful. It addresses the +major drawback using the scheduled decreasing step size as +introduced in the previous section. +From the above analysis and design, we present our al- +gorithm in Alg. 1. We use the same scheduled step size to +update the dual examples to make the optimization of xadv +coordinated with x. +4 + +X +t=1 +t=2 +t=3 +t=4 +t=5 +t=6 +t=7 +t=8 +t=9 +t=10 +I-FGSM +std. +0 +1.75 +2.06 +2.30 +2.50 +2.69 +2.86 +3.02 +3.16 +3.28 +3.39 +Decoy +Example +std. +0 +1.75 +2.05 +2.33 +2.60 +2.84 +3.07 +3.26 +3.44 +3.60 +3.76 +Dual +Example +std. +0 +1.75 +1.83 +1.92 +2.01 +2.05 +2.11 +2.19 +2.28 +2.28 +2.29 +IterationAlgorithm 1: I-FGSM with Scheduled Adaptive +Step Size and Dual Example +Input: Classifier f(·); The benign sample x with +ground-truth label y; Loss function L(·, ·); +The number of iterations T; Decreasing +sequence ˆα; +Output: An transferable adversarial example. +1 initialize a random perturbation δ0, dual sample +xdual +0 += x0 = x, moving average update +mdual +0 += m0 = 0, step t = 0; +2 while t < T do +3 +g = ∂L(f(xt),y) +∂xt +; +▷ explore and sample +features +4 +mdual +t+1 = mdual +t +t+g +t+1 +; +▷ exploit and +smooth the features +5 +xt+1 = xt + ˆαtsign(g); +▷ update input +sample +6 +xdual +t+1 = xdual +t ++ ˆαtsign(mdual +t+1 ); +▷ update +the dual sample +7 +t ← t + 1; +8 return xdual +T +. +Further, we visualize the benign sample and adversar- +ial examples in each iterations and highlight the adversarial +perturbations in Fig. 3. The sharp colors, including the light +green, pink and red, represent large perturbations, while the +soft colors fulfilled at the background, especially the yel- +low, represent small perturbations. Besides, we also com- +pute the standard deviation (std) of the previous perturba- +tions for the current iteration. From the figure (first two +rows of images), it can be observed that the attackers’ fea- +ture attention varies a lot between different iterations. For +instance, there are large adversarial perturbations on the ta- +ble of the image in the 5, 6, 8, 9, 10-th iteration. For this, +we argue that the large variance on perturbations between +different iterations comes from the data distribution shifts, +i.e., from natural examples to adversarial examples. The +data distribution shifts lead to unstable performance on the +feature attention for models, which also causes the drop of +performance on the adversarial transferablity. On the other +hand, the proposed SD strategy, i.e., images of the last row +in Fig. 3, can alleviate this problem. It can be clearly ob- +served that the variance does not change significantly, with +stable feature attention. +3.3. Relationship to Existing Methods +The mechanism of scheduled adaptive step size and dual +example can be easily integrated with existing methods +such as other gradient-based methods, including MI-FGSM +and NI-FGSM, and transformation-based methods, includ- +ing TIM, DIM, and SIM. +Gradient-based methods: A notable difference between +I-FGSM and the others is that MI-FGSM and NI-FGSM re- +place the gradients with momentums to enhance the perfor- +mance. To integrate our method with them, it only needs to +change the line 4 in Alg. 1 with the adaptive momentum as +follows, +mt+1 = µmt + g. +(3) +Transformation-based methods: +The transformation- +based methods use T (x) instead of x to compute g, where +T (·) is an input transformation function. It only needs to +change xt of the line 3 in Alg. 1 with T (xt). +Ensemble attack methods: The ensemble attacks gener- +ate adversarial examples by attacking several models in +parallel. To integrate our method in the ensemble attack, +we can change the computation of gradient in line 3 from +the single-model setting to the ensemble-model setting, i.e. +g = +∂L( 1 +N +N +� +n=1 +fi(xt),y) +∂xt +, where N indicates the number of +ensemble models. +4. Experiments +In this section, we conduct extensive evaluations on +ImaageNet dataset to validate the effectiveness of our pro- +posed strategy, Schduled step size and Dual example (SD). +4.1. Experimental Setup +Dataset: We randomly select 1, 000 images remaining with +1, 000 categories from ILSVRC 2012 validation set [23], +which are almost classified correctly by the chosen models. +Victim Models: We evaluate the attack performance on +six popular convolution neural networks, including ResNet- +18 [16], ResNet-101 [16], ResNeXt-50 [61], DenseNet- +121 [18], Vision Transformer (ViT) [11] and Swin Trans- +former (Swin) [31]. +We also study several defense +models, including the top-3 submissions in the NIPS +2017 defense challenge, namely High-level representa- +tion guided denoiser (HGD) [27], random resizing and +padding (R&P) [58] and NIPS-r3 1, two adversarial train- +ing approaches, namely ensemble adversarially trained +model (IncRes-v2ens) [46] and Fast adversarial training +(FastAdv) [54], one certified defense, namely random- +ized smoothing (RS) [7], one deep denoiser, namely neu- +ral representation purifier (NRP) [37], and three input +transformation-based defense method, namely JPEG [14], +BitRed [63], and feature squeezing (FD) [32]. +Baselines: For gradient-based attack methods, we adopt +I-FGSM [24], MI-FGSM [9] and NI-FGSM [29] as our +baselines. For input transformation-based attacks, we treat +DIM [60], TIM [10] and SIM [29] as our baselines. +1https : / / github . com / anlthms / nips - 2017 / tree / +master/mmd +5 + +Resnet-18 +ResNet-101 +ResNeXt-50 +DenseNet-121 +ViT +Swin +0 +20 +40 +60 +80 +100 +Attack Success Rate (%) +(a) Resnet-18 +Resnet-18 +ResNet-101 +ResNeXt-50 +DenseNet-121 +ViT +Swin +0 +20 +40 +60 +80 +100 +Attack Success Rate (%) +(b) ResNet-101 +Resnet-18 +ResNet-101 +ResNeXt-50 +DenseNet-121 +ViT +Swin +0 +20 +40 +60 +80 +100 +Attack Success Rate (%) +(c) ResNeXt-50 +Resnet-18 +ResNet-101 +ResNeXt-50 +DenseNet-121 +ViT +Swin +0 +20 +40 +60 +80 +100 +Attack Success Rate (%) +(d) DenseNet-121 +Resnet-18 +ResNet-101 +ResNeXt-50 +DenseNet-121 +ViT +Swin +0 +20 +40 +60 +80 +100 +Attack Success Rate (%) +(e) ViT +Resnet-18 +ResNet-101 +ResNeXt-50 +DenseNet-121 +ViT +Swin +0 +20 +40 +60 +80 +100 +Attack Success Rate (%) +(f) Swin +I-FGSM +MI-FGSM +NI-FGSM +SD's improvement +Figure 4. The Attack success rate (%) of baseline models on the crafted adversarial examples generated by I-FGSM, MI-FGSM, NI-FGSM +and our proposed method under single model setting. We generate the adversarial examples on one model and evaluate on the other five +models. Please view on a color screen. +Resnet-18 +ResNet-101 +ResNeXt-50 +DenseNet-121 +ViT +Swin +0 +20 +40 +60 +80 +100 +Attack Success Rate (%) +(a) Resnet-18 +Resnet-18 +ResNet-101 +ResNeXt-50 +DenseNet-121 +ViT +Swin +0 +20 +40 +60 +80 +100 +Attack Success Rate (%) +(b) ResNet-101 +Resnet-18 +ResNet-101 +ResNeXt-50 +DenseNet-121 +ViT +Swin +0 +20 +40 +60 +80 +100 +Attack Success Rate (%) +(c) ResNeXt-50 +Resnet-18 +ResNet-101 +ResNeXt-50 +DenseNet-121 +ViT +Swin +0 +20 +40 +60 +80 +100 +Attack Success Rate (%) +(d) DenseNet-121 +Resnet-18 +ResNet-101 +ResNeXt-50 +DenseNet-121 +ViT +Swin +0 +20 +40 +60 +80 +100 +Attack Success Rate (%) +(e) ViT +Resnet-18 +ResNet-101 +ResNeXt-50 +DenseNet-121 +ViT +Swin +0 +20 +40 +60 +80 +100 +Attack Success Rate (%) +(f) Swin +TIM +DIM +SIM +SD's improvement +Figure 5. The Attack success rate (%) of baseline models on the crafted adversarial examples generated by DIM, TIM, SIM and our +proposed method under single model setting. We generate the adversarial examples on one model and evaluate the transferability on the +other five models. +Hyper-parameters: For the hyper-parameters, we follow +the same setting of MI-FGSM [9] with the maximum per- +turbation of ϵ = 16, the number of iterations T = 10, step +size α = 1.6, and decay factor µ = 1.0. Besides, we set +the transformation probability for DIM as 0.5 [60], the size +of Gaussian kernel for TIM as 7 × 7 [10], and the number +of scale copies for SIM as m = 5 [29]. For our method, +we set ρ = 0.9 when we integrate the proposed strategy in +MI-FGSM and NI-FGSM. +4.2. Integrated with Gradient-based Attacks +Gradient-based +adversarial +attacks +are +the +widely +adopted approaches to craft adversarial examples. Here we +first evaluate the effectiveness of SD to boost the attack +performance of existing gradient-based attacks, including +I-FGSM, MI-FGSM and NI-FGSM. The adversarial exam- +ples are generated on each model and evaluated on the other +models. Te attack success rates, which are the misclassifica- +tion rates of the victim models on the adversarial examples, +are reported in Fig. 4. +In general, we can observe that SD can effectively im- +prove the attack performance of the baselines in white-box +6 + +Resnet-18 +ResNet-101 +ResNeXt-50 +DenseNet-121 +Vit +Swin +0 +20 +40 +60 +80 +100 +Attack Success Rate (%) +I-FGSM +MI-FGSM +NI-FGSM +SD's improvement +Figure 6. The Attack success rate (%) of baseline models on the +crafted adversarial examples generated by I-FGSM, MI-FGSM, +NI-FGSM and our proposed method under an ensemble model set- +ting. We evaluate the performance on the hold-out model using the +adversarial examples generated by the other five models. +as well as black-box settings. +On average, SD helps I- +FGSM achieves the attack performance of 92.5%, which +is 19.4% higher than vanilla I-FGSM. Under the white-box +setting, MI-FGSM and NI-FGSM achieves better transfer- +ability than I-FGSM since the momentum helps stablilize +the optimization and escape local minima. Compared with +these attacks, SD can significantly boost the transferabil- +ity. In particular, SD can remarkably enhance the attack +performance of I-FGSM, which outperforms NI-FGSM and +MI-FGSM with a clear margin. SD can boost I-FGSM, NI- +FGSM and MI-FGSM at least 15.7%, 6.7%, and 5.5%, re- +spectively, showing its high effectiveness in improving the +transferability and generality to various models. +4.3. Integrated with Transformation-based Attacks +Transformation-based adversarial attack leverages mul- +tiple transformations on the inputs to improve the adver- +sarial transferability. +We are inline with the aforemen- +tioned experiments to integrate our method on three clas- +sical transformation-based attacks, namely DIM, TIM and +SIM. The reuslts can be shown in Fig. 5. +We can observe that all the transformation-based meth- +ods can achieve the attack success rate of 100.0% or near +100.0%, showing that the transformation-based methods do +not degrade and even enhance the white-box attack per- +formance. +As for the black-box performance, TIM be- +haves the poorest transferability on these normally trained +models. Surprisingly, on some models, with SD strategy, +TIM achieves even better transferability than DIM and even +SIM. For instance, on ResNeXt-50, TIM with SD strategy +achieves the attack success rate of 79.9%, which is much +high than 50.1% of DIM and 70.8% of SIM. On average +HGD +IncRes +v2ensFastAdv +RS +NRP +JPEG +BitRed +FD +0 +20 +40 +60 +80 +100 +Attack Success Rate (%) +I-FGSM +MI-FGSM +NI-FGSM +SD's improvement +Figure 7. The Attack success rate (%) of baseline models on the +crafted adversarial examples generated by I-FGSM, MI-FGSM, +NI-FGSM and our proposed method under ensemble model set- +ting. We evaluate on the advanced denfese methods. +of all experiment cases, SD can boost the transferability of +TIM, DIM and SIM with a clear margin at least 4.9%, 2.1% +and 2.8% respectively. +4.4. Integrated with Ensemble Attacks +Ensemble attacks are another effective method to gener- +ate more transferable adversarial examples by attacking sev- +eral models in parallel. We evaluate the performance of SD +when integrated with the ensemble attack method. Specif- +ically, we use the indicated single model as the evaluation +model while we generate the adversarial example using the +other five models. We present the results in Fig. 6. +It can be clearly observed that SD strategy significantly +enhances the adversarial attack performance, especially for +I-FGSM. Specifically, with SD strategy can achieve an av- +erage attack success rate of 97.5% , 98.4%,99.2% when in- +tegrated with I-FGSM, MI-FGSM and NI-FGSM, respec- +tively, showing a clear margin of 16.2%, 3.0 and 2.3% com- +pared with the vanilla ones. +4.5. Attack on Advanced Defense Methods +To further identify the effectiveness of our method, we +evaluate the performance of the crafted adversarial exam- +ples on six defense methods, including HGD, IncRes − +v2ens, FastAdv, RS, NRP, JPEG, BitRed and FD. The tar- +get model for JPEG and NRP is VGG19 and the other meth- +ods adopts the official models provided in the correspond- +ing works. We generate the adversarial examples on the en- +semble model, i.e., ResNet-18, ResNet-101, ResNeXt-50, +DenseNet-121. ViT and Swin. The results are summarized +in Fig. 7. +From the figure, SD still exhibits an excellent attack +performance towards various advanced defense methods. +7 + +ResNet-18 +ResNet-101 +ResNeXt-50 +DenseNet-121ViT +Swin +0 +20 +40 +60 +80 +100 +Attack Success Rate (%) +(a) Ablation study on the 'S' and 'D' strategy +I-FGSM +I-FGSM-D +I-FGSM-S +Ours +ResNet-18 +ResNet-101 +ResNeXt-50 +DenseNet-121ViT +Swin +0 +20 +40 +60 +80 +100 +Attack Success Rate (%) +(b) Ablation study on different increasing sequence +Exp +Power +Linear +Ln(e) +ResNet-18 +ResNet-101 +ResNeXt-50 +DenseNet-121ViT +Swin +0 +20 +40 +60 +80 +100 +Attack Success Rate (%) +(c) Ablation study on different base of log +Ln(2) +Ln(e) +Ln(10) +Figure 8. Ablation studies using the increasing sequence. +Among them, the certificate robustness defense method, +Random Smoothing (RS), is the most effective against ad- +versarial attacks, where I-FGSM, MI-FGSM and NI-FGSM +only achieves the attack success rates of 20.9%, 41.0% and +43.1% respectively. SD can remarkably enhance the attack +performance with a clear improvement of 50.8%, 32.9% +and 30.4%, respectively. On average, SD can improve the +adversarial transferability of I-FGSM, MI-FGSM and NI- +FGSM with a clear margin of 38.9%, 20.3%, and 19.8%, +respectively, showing the high effectiveness of our SD. +4.6. Ablation Studies +In this subsection, we mainly study the effectiveness of +the proposed strategy and the choice of scheduled adaptive +step sizes. We adopt ResNet-18 as the victim model to con- +duct the following experiments. +The effectiveness of the proposed strategy. There are two +major components in our proposed method, namely sched- +uled adaptive step size and dual example. As carefully dis- +cussed in section 3.2, the scheduled adaptive step size is +used to explore more on the region where there exits more +transferable adversarial examples while the dual example +is used to craft adversarial examples with high transfer- +ability. Here, we design an ablation study to identify the +effectiveness of each component as follows, 1) I-FGSM: +conventional iterative fast gradient sign method with the +identity step size; 2) I-FGSM-D: apply the dual example +to I-FGSM, where we update the dual example with per- +turbation smoothing; 3) I-FGSM-S: I-FGSM with sched- +uled adaptive step size; 4) Ours: I-FGSM with scheduled +adaptive step size and the dual example. The results can be +shown in Fig. 8 (a). We can identify the effectiveness of the +dual example from the comparison between I-FGSM and +I-FGSM-D. Although it sacrifices a little performance un- +der the white-box setting with only the scheduled step size, +the transferability of crafted adversarial examples is signif- +icantly improved. +The choice of scheduled adaptive step size. As our anal- +ysis in Section 3.1, the density of transferable adversarial +examples decreases with increasing the distance to the be- +nign example, so we adopt the increasing sequence as the +scheduled step size to fully utilize the perturbations with +good transferability. We study four increasing sequences, +exponentiation (Exp), power (Power), linear (Linear), loga- +rithm sequence (Ln(·)) respectively. The experiment results +are shown in Fig. 8 From the result in Fig. 8(b), it can be +shown that with the decreasing of second-order, the attack +performance of the increasing sequences display an increas- +ing trend, i.e. from Exp to Ln(e). We additionally adopts +different number of base in logarithm function in Fig. 8 (c). +It can be shown there exits minor difference between the +settings with different bases. +5. Conclusion +In our work, we find an interesting phenomenon, that the +FGSM and the I-FGSM with few iterations exhibit a bet- +ter adversarial transferability than I-FGSM with more iter- +ations. We hold an assumption for this phenomenon, that +the adversarial perturbations are more transferable near the +benign sample. With increasing distance of the adversarial +example relative to the benign sample, there is an increasing +difference on the feature attention between different mod- +els, caused by the different architectures and distribution +shifts. To alleviate this catastrophic performance drop, we +propose a novel strategy, which uses the scheduled step size +and dual example (SD), to fully utilize the transferable ad- +versarial perturbation near the benign sample. 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Feature squeez- +ing: Detecting adversarial examples in deep neural networks. +arXiv preprint arXiv:1704.01155, 2017. 5 +[64] Hongyang Zhang, Yaodong Yu, Jiantao Jiao, Eric P. Xing, +Laurent El Ghaoui, and Michael I. Jordan. +Theoretically +principled trade-off between robustness and accuracy. +In +Proceedings of the International Conference on Machine +Learning, 2019. 2 +[65] Yuchen Zhang and Percy Liang. Defending against white- +box adversarial attacks via randomized discretization. In The +22nd International Conference on Artificial Intelligence and +Statistics, pages 684–693. PMLR, 2019. 1 +[66] Junhua Zou, Yexin Duan, Boyu Li, Wu Zhang, Yu Pan, and +Zhisong Pan. Making adversarial examples more transfer- +able and indistinguishable. In Proceedings of the AAAI Con- +ference on Artificial Intelligence, 2022. 1 +11 + diff --git a/XtFOT4oBgHgl3EQf9DQS/content/tmp_files/load_file.txt b/XtFOT4oBgHgl3EQf9DQS/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3646ed67f67b2bf1467a8056a2ffd76075576c1f --- /dev/null +++ b/XtFOT4oBgHgl3EQf9DQS/content/tmp_files/load_file.txt @@ -0,0 +1,859 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf,len=858 +page_content='Improving Adversarial Transferability with Scheduled Step Size and Dual Example Zeliang Zhang University of Rochester Peihan Liu Harvard University Xiaosen Wang* HUST Chenliang Xu⋆ University of Rochester Abstract Deep neural networks are widely known to be vulnerable to adversarial examples, especially showing significantly poor performance on adversarial examples generated un- der the white-box setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' However, all white-box attack methods rely heavily on the target model and quickly get stuck in local optima, resulting in poor adversarial trans- ferability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' The momentum-based methods and their variants are proposed to escape the local optima for better transfer- ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' In this work, we notice that the transferability of adversarial examples generated by iterative fast gradient sign method (I-FGSM) exhibits a decreasing trend when in- creasing the number of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Motivated by this find- ing, we argue that the information of adversarial perturba- tions near the benign sample, especially the direction, ben- efit more on the transferability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Thus, we propose a novel strategy, which uses the Scheduled step size and the Dual example (SD), to fully utilize the adversarial information near the benign sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Our proposed strategy can be eas- ily integrated with existing adversarial attack methods for better adversarial transferability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Empirical evaluations on the standard ImageNet dataset demonstrate that our pro- posed method can significantly enhance the transferability of existing adversarial attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Introduction Adversarial examples [13,45], crafted by adding human- imperceptible perturbations on benign samples to fool deep neural networks (DNNs), have drawn more research inter- est in recent years [1, 21, 66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' On the one hand, the exis- tence of adversarial examples identifies the vulnerability of DNNs, throwing out the severe security issues to be solved urgently on many areas, including the autonomous driv- ing [20], facial authentication [22], object detection [17], et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' The adversarial examples also generalized to other models [39,57,60], where the adversarial examples crafted Corresponding author by one model can also fool other models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' On the other hand, the adversarial examples help evaluate the robustness of DNNs and enhance the robustness of DNNs through ad- versarial training [42,54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Great research interest has arisen in how to generate adversarial examples with high transfer- ability for a better understanding of the DNNs [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' According to the knowledge about the victim model, the existing adversarial attacks can be categorized into two set- tings, namely the white-box [13, 36, 65] and the black-box settings [19, 41, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' With the full knowledge of the vic- tim model, including the architecture, the parameter, the training loss function, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=', most of the existing adversar- ial attacks under the white-box setting usually exhibit a great performance but behave poorly facing the models without knowing any information under the black-box set- ting [28,49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' The low adversarial transferability makes ap- plying such existing adversarial attacks inefficient in real- world applications, where the victim models are usually un- known to attackers [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Some methods have been proposed to enhance the trans- ferability of adversarial examples, such as gradient-based attacks with momentum [15, 48], input transformation- based attacks [40, 50], ensemble attacks [4, 43], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Among them, the gradient-based methods mainly utilize the momentum (MI-FGSM) [9] or pre-computation (NI- FGSM) [29] to help escape the local optima for better ad- versarial transferability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' The input transformation [10, 59] methods analogize the crafting of adversarial examples to the generalization of DNNs, thus diversifying the inputs to generate the adversarial examples with high transferability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Different from the above adversarial attacks, we study adversarial transferability from a different perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' From a simple experiment, we observe that the transfer- ability of adversarial examples crafted using vanilla itera- tive adversarial attacks demonstrate a trend from high to low, which indicates that the directions of the adversarial perturbations near the benign sample are more transferable among different models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Motivated by this, we propose a novel strategy with the Scheduled step size and Dual ex- ample (SD), which fully explores and utilizes the adversar- 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='12968v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='LG] 30 Jan 2023 ial perturbations with high transferability near the benign sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Also, it can be easily integrated on the existing adversarial attack methods to enhance the performance on transferability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' We apply our proposed method to several generally used adversarial attack methods and conduct proof experiments on the standard ImageNet dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' The results demonstrate that our proposed method can remarkably boost the existing adversarial attack methods on transferability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Related Work In this section, we provide a brief overview of adversarial attack and defense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Adversarial Attack Since Szegedy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' [45] identified the vulnerability of DNNs to adversarial examples, numerous adversarial attack methods have been proposed, including white-box attack [8, 13, 24, 34, 36], transfer-based attack [9, 10, 48, 49, 51, 60], score-based attack [6, 19, 26, 47], decision-based attack [3, 5,25,35,52], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Among the above attacks, a transfer-based attack, as a black-box attack, does not need access to the target model, making it popular to attack the deep models in the real world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' We provide an overview of the transfer- based attacks as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Gradient-based Attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Fast Gradient Sign Method (FGSM) [13], which is the first gradient-based attack, crafts adversarial examples by adding a large perturbation in the gradient direction to the benign sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' To improve its white-box attack performance, it is further extended to an iterative Version (I-FGSM) [24] but shows poor transfer- ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' With the potential application of transfer-based at- tacks in the real world, various methods have been pro- posed to improve the transferability of adversarial exam- ples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' [9] introduce the momentum into I-FGSM, dubbed Momentum I-FGSM (MI-FGSM), to stabilize the optimization and escape the local optima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' [29] utilize Nesterov accelerated gradient to look ahead, denoted as NI-FGSM, which exhibits better transferability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' [48] propose the definition of gradient variance, which could be used to tune the gradient for MI-FGSM (VMI- FGSM) and NI-FGSM (VNI-FGSM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' [51] en- hance the momentum with the gradient of multiple samples in the neighborhood of the current adversarial example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Input transformation-based attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Inspired by the data augmentation in training, various input transformation- based attacks are proposed to improve the transferability, which can be combined with the above gradient-based at- tacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' For instance, the diverse input method (DIM) [60] resizes the input image into a random size, which will be padded to a fixed size for gradient calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Transla- tion invariant method (TIM) [10] adopts Gaussian smooth on the gradient to approximate the average gradient of a set of translated images to update the adversary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Scale-invariant method (SIM) [29] calculates the gradient on a set of scaled images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Admix [49] adds a small portion of images from other categories to the input image to obtain several images for gradient calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Ensemble attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' [30] first find that ensem- ble attack, which generates adversarial examples on mul- tiple models, can lead to better transferability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Xiong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' [62] reduces the gradient variance among various mod- els to boost ensemble attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Long et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' [33] augment the model in the frequency domain to obtain more diverse sub- stitute models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' [12] reuses the cut noise introduced by clip operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' [56] argues that the attack should maximize the difference of the attention map be- tween benign samples and adversarial examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Adversarial Defense On the other hand, to mitigate the threat of adversar- ial attacks, a variety of defense methods have been pro- posed, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' adversarial training [13, 34, 53, 64], input pre- processing [14], certified defense [7] etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Adversarial train- ing has been shown as one of the most effective methods to improve adversarial robustness [2,8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' It is first proposed by Madry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' [34] that including the adversarial examples in the training data enhances the adversarial robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Wong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' [54] use the fast gradient sign method with random initialization to generate adversarial examples for efficient adversarial training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Methods In this section, we introduce our motivation and provide a detailed description of the proposed strategy SD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Motivation Given a target deep model f and a benign sample x with the ground-truth label y, adversarial attacks find an example xadv similar to x (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=', xadv ∈ Bϵ(x) = {x′ : ∥x′ − x∥p ≤ ϵ} where ∥ · ∥p refers to the p−norm) such that the model gives different predictions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=', f(x) = y ̸= f(xadv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Gradient-based adversarial attacks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=', FGSM [13], I- FGSM [24], MI-FGSM [9], NI-FGSM [29], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=') can gen- erate the adversarial examples xadv efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Goodfellow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' [13] first propose the gradient sign method (FGSM) to attack the deep learning model efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' However, single- step optimization for adversarial examples may not find the optima, which fails to fool the models under the white-box setting, within the box constraint Bϵ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Iterative fast gra- dient sign method (I-FGSM) employs the iterative process to optimize xadv and can achieve a attack success rate of 100% or nearly 100%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Suppose L(·, ·) is the classification loss, I-FGSM adopts 2 1 2 3 4 5 Iteration 20 40 60 80 100 Attack Success Rate(%) ResNet-18 ResNet-101 ResNeXt-50 DenseNet121 ViT Swin Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Varying the maximum number of iteration, the attack success rate of six models on the adversarial examples generated by ResNet-18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' the iterative optimization process to craft an adversarial ex- ample xadv which maximizes L(f(xadv), y) as follows, xadv t+1 = ΠBϵ(x) � xadv t + α · sign � ∇xL � f � xadv t � , y ��� , (1) where Π is the projector function, α is the step size, and xadv t is the adversarial example at the t-th iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' To further explore how the number of iterations affects the attack performance, we conduct I-FGSM attack on Im- ageNet dataset with different iterations (T = 1 → 5 itera- tions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' The adversarial examples are generated on ResNet- 18 [16], which are used to attack other models (ResNet- 101 [16], ResNext-50 [61], DenseNet-121 [18], ViT [11], Swin [31]) with the perturbation budget ϵ = 16/255 and step size α = ϵ/T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' 1, I-FGSM achieves 100% attack success rate on the white-box model (ResNet- 18) with 2 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Also, it is obvious that the trans- ferabilty of adversarial examples generated by FGSM, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=', T = 1, are better than that of I-FGSM with multiple steps (T ≥ 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Since the difference of updates on xadv in each iteration only falls on the direction, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=', the sign of the gra- dient, it naturally inspires us the following assumption, Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' The direction of the adversarial perturba- tion near the benign sample benefits more on adversarial transferability than that far away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' [29] analogize the search for adversarial exam- ples with the training process of models and the transfer- ability of adversarial examples with the generalization abil- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Training on the same dataset, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=', ImageNet dataset, different models usually have similar attentions on the fea- tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' However, the adversarial example is one category of the out-of-distribution (OOD) data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Evaluated on such OOD datasets, different models may have different perfor- mances, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=', different attentions on the features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' With an in- creasing number of iterations, the distribution shifts become Resnet-18 ResNet-101 ResNeXt-50 DenseNet-121 Vit Swin 0 20 40 60 80 100 Attack Success Rate (%) Identity Ln Linear Power Exp Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' With different decreasing scheduled step sizes, the attack success rate of six model on the adversarial examples generated by ResNet-18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' large and the performance difference, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=', attention on the features, will also become large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Thus, the adversarial per- turbation added on examples computed by the white-box model, that are far away from the benign sample, are not generally easy to transfer to be efficient for other models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' A natural method to alleviate the above problem, caused by distribution shifts, is to use decreasing step sizes to fully utilize the adversarial perturbation direction near the benign sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Thus, we use several kinds of decreasing sequence as the step sizes α to optimize xadv in (1) and evaluate the attack performances on aforementioned models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' We adopt four decreasing sequences: logarithm (αi = ln(T − i)), linear (αi = T − i), power (αi = (T − i)2), and exponential (αi = eT −i) sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' We normalize each sequence by dividing the sum to satisfy Bϵ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' As shown in results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' 2, the scheduled decreasing step size im- proves the adversarial attack success rates under the black- box setting, with an average of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='66% for (Ln), 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='75% for (Linear), 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='98% for (Power) and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='72% for (Exp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' However, the results are still worse than that of FGSM and I-FGSM with a few iterations on the adversarial transferability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' The reason why the proposed method does not work as expected is that, there is a huge gap between the magnitude of the first step and the magnitude of the last step, which leads to insuf- ficient updates on the last several iterations of I-FGSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' The huge gap of the schedule step sizes between the early and last stages are also discussed in our ablation study in Sec- tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' So, besides employing the scheduled decreasing steps, is there any alternative way to fully utilize the infor- mation near the benign sample to enhance the adversarial transferability?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Method To fully utilize the information, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=', the direction of ad- versarial perturbations with high transferability, except for 3 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Visualization of the adversarial examples with perturbations for each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' The strength of the color on the mask represents the adversarial perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' The sharp colors, especially red, indicate the large perturbations, and the soft colors, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=', yellow, indicate the small perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Please view on a color screen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' using the decreasing step size, we can also use the sched- uled increasing step size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Scheduled step size: As aforementioned, we hope to uti- lize the scheduled decreasing step size to rearrange larger weights on the early updates for xadv, where the direction of the updates, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=', the adversarial perturbation, is more trans- ferable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Moreover, we want the updates on the later stages to sufficiently useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' It turns out that using the scheduled increasing step sizes can achieve the two goals simultane- ously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' More specifically, we sample more directions of the adversarial perturbation with high transferability and put them on use in the later iterations to enhance the transfer- ability of adversarial examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' In particular, rather than using an identity step size in (1), we made the step sizes as a increasing sequence to find xadv as follows, xadv t+1 = Clipϵ � xadv t + ˆαtsign � ∇xadv t L � f � xadv t � , y ��� , (2) where ˆα is the scheduled step size sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' 2, due to the huge gap of the step size between the early and last several iterations, the inefficient updates on the adversarial example on the last several itera- tions do not gain a lot for the transferability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' However, with the increasing sequence as the scheduled step size, the up- dates on the last few iterations becomes sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Also, due to the small step size on the first few iterations, xadv stays near around x, meaning that we can sample more directions of adversarial perturbations with high transferability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Then, the next question is: how can we utilize the sampled adver- sarial perturbations with high transferability?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Dual example: In order to make full use of the transferable directions of adversarial perturbations, we propose a novel strategy, namely dual example, which uses two examples in parallel but applies different updates on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Specifically, we initialize two examples xadv = xdual = x, namely the decoy and dual example, respectively, in our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' In each iteration, we use the decoy example, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=', xadv, to explore and sample adversarial perturbations, and use xdual to fully utilize them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Due to the scheduled in- creasing step size, the early sampled adversarial perturba- tions using small step size near the benign sample are usu- ally more transferable than the later ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Thus, we average all the previous adversarial perturbations as an ensemble re- sult to compute the update on the current iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' There are two advantages using this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' On the one hand, even for the last few steps, where xadv is quite far away from x and the adversarial perturbations usually behave poorly on the transferability, the actual updates on xdual will not be af- fected much from the ensemble result on the early updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' On the other hand, the large step size, for the later stage, will help xdual jump over the local optima as well as mak- ing each update more sufficiently useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' It addresses the major drawback using the scheduled decreasing step size as introduced in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' From the above analysis and design, we present our al- gorithm in Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' We use the same scheduled step size to update the dual examples to make the optimization of xadv coordinated with x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' 4 X t=1 t=2 t=3 t=4 t=5 t=6 t=7 t=8 t=9 t=10 I-FGSM std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='06 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='30 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='50 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='69 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='86 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='02 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='16 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='28 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='39 Decoy Example std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='33 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='60 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='84 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='07 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='26 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='44 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='60 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='76 Dual Example std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='83 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='92 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='01 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='11 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='19 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='28 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='28 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='29 IterationAlgorithm 1: I-FGSM with Scheduled Adaptive Step Size and Dual Example Input: Classifier f(·);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' The benign sample x with ground-truth label y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Loss function L(·, ·);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' The number of iterations T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Decreasing sequence ˆα;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Output: An transferable adversarial example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' 1 initialize a random perturbation δ0, dual sample xdual 0 = x0 = x, moving average update mdual 0 = m0 = 0, step t = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' 2 while t < T do 3 g = ∂L(f(xt),y) ∂xt ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' ▷ explore and sample features 4 mdual t+1 = mdual t t+g t+1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' ▷ exploit and smooth the features 5 xt+1 = xt + ˆαtsign(g);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' ▷ update input sample 6 xdual t+1 = xdual t + ˆαtsign(mdual t+1 );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' ▷ update the dual sample 7 t ← t + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' 8 return xdual T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Further, we visualize the benign sample and adversar- ial examples in each iterations and highlight the adversarial perturbations in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' The sharp colors, including the light green, pink and red, represent large perturbations, while the soft colors fulfilled at the background, especially the yel- low, represent small perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Besides, we also com- pute the standard deviation (std) of the previous perturba- tions for the current iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' From the figure (first two rows of images), it can be observed that the attackers’ fea- ture attention varies a lot between different iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' For instance, there are large adversarial perturbations on the ta- ble of the image in the 5, 6, 8, 9, 10-th iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' For this, we argue that the large variance on perturbations between different iterations comes from the data distribution shifts, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=', from natural examples to adversarial examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' The data distribution shifts lead to unstable performance on the feature attention for models, which also causes the drop of performance on the adversarial transferablity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' On the other hand, the proposed SD strategy, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=', images of the last row in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' 3, can alleviate this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' It can be clearly ob- served that the variance does not change significantly, with stable feature attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Relationship to Existing Methods The mechanism of scheduled adaptive step size and dual example can be easily integrated with existing methods such as other gradient-based methods, including MI-FGSM and NI-FGSM, and transformation-based methods, includ- ing TIM, DIM, and SIM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Gradient-based methods: A notable difference between I-FGSM and the others is that MI-FGSM and NI-FGSM re- place the gradients with momentums to enhance the perfor- mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' To integrate our method with them, it only needs to change the line 4 in Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' 1 with the adaptive momentum as follows, mt+1 = µmt + g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' (3) Transformation-based methods: The transformation- based methods use T (x) instead of x to compute g, where T (·) is an input transformation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' It only needs to change xt of the line 3 in Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' 1 with T (xt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Ensemble attack methods: The ensemble attacks gener- ate adversarial examples by attacking several models in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' To integrate our method in the ensemble attack, we can change the computation of gradient in line 3 from the single-model setting to the ensemble-model setting, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' g = ∂L( 1 N N � n=1 fi(xt),y) ∂xt , where N indicates the number of ensemble models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Experiments In this section, we conduct extensive evaluations on ImaageNet dataset to validate the effectiveness of our pro- posed strategy, Schduled step size and Dual example (SD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Experimental Setup Dataset: We randomly select 1, 000 images remaining with 1, 000 categories from ILSVRC 2012 validation set [23], which are almost classified correctly by the chosen models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Victim Models: We evaluate the attack performance on six popular convolution neural networks, including ResNet- 18 [16], ResNet-101 [16], ResNeXt-50 [61], DenseNet- 121 [18], Vision Transformer (ViT) [11] and Swin Trans- former (Swin) [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' We also study several defense models,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' including the top-3 submissions in the NIPS 2017 defense challenge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' namely High-level representa- tion guided denoiser (HGD) [27],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' random resizing and padding (R&P) [58] and NIPS-r3 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' two adversarial train- ing approaches,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' namely ensemble adversarially trained model (IncRes-v2ens) [46] and Fast adversarial training (FastAdv) [54],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' one certified defense,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' namely random- ized smoothing (RS) [7],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' one deep denoiser,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' namely neu- ral representation purifier (NRP) [37],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' and three input transformation-based defense method,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' namely JPEG [14],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' BitRed [63],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' and feature squeezing (FD) [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Baselines: For gradient-based attack methods, we adopt I-FGSM [24], MI-FGSM [9] and NI-FGSM [29] as our baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' For input transformation-based attacks, we treat DIM [60], TIM [10] and SIM [29] as our baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' 1https : / / github .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' com / anlthms / nips - 2017 / tree / ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='master/mmd ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='Resnet-18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='ResNet-101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='ResNeXt-50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='DenseNet-121 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='ViT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='Swin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='Attack Success Rate (%) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='(a) Resnet-18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='Resnet-18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='ResNet-101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='ResNeXt-50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='Attack Success Rate (%) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='(b) ResNet-101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='Resnet-18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='ResNet-101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='ResNeXt-50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='Attack Success Rate (%) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='(c) ResNeXt-50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='Resnet-18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='ResNet-101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='ResNeXt-50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='Attack Success Rate (%) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='(d) DenseNet-121 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='Resnet-18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='ResNet-101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='ResNeXt-50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='DenseNet-121 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='ViT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='Swin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='Attack Success Rate (%) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='(e) ViT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='Resnet-18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='ResNet-101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='ResNeXt-50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='DenseNet-121 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='ViT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='Swin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='Attack Success Rate (%) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='(f) Swin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='I-FGSM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='MI-FGSM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='NI-FGSM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content="SD's improvement " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' The Attack success rate (%) of baseline models on the crafted adversarial examples generated by I-FGSM, MI-FGSM, NI-FGSM and our proposed method under single model setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' We generate the adversarial examples on one model and evaluate on the other five models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Please view on a color screen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='Resnet-18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='ResNet-101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='ResNeXt-50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='DenseNet-121 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='ViT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='Swin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='Attack Success Rate (%) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='(a) Resnet-18 ' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='Attack Success Rate (%) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='(b) ResNet-101 ' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='Attack Success Rate (%) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='(c) ResNeXt-50 ' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='Attack Success Rate (%) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='(d) DenseNet-121 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='Resnet-18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='ResNet-101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='ResNeXt-50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='DenseNet-121 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='ViT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='Swin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='Attack Success Rate (%) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='(e) ViT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='Resnet-18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='ResNet-101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='ResNeXt-50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='DenseNet-121 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='ViT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='Swin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='Attack Success Rate (%) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='(f) Swin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='TIM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='DIM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='SIM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content="SD's improvement " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' The Attack success rate (%) of baseline models on the crafted adversarial examples generated by DIM, TIM, SIM and our proposed method under single model setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' We generate the adversarial examples on one model and evaluate the transferability on the other five models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Hyper-parameters: For the hyper-parameters, we follow the same setting of MI-FGSM [9] with the maximum per- turbation of ϵ = 16, the number of iterations T = 10, step size α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='6, and decay factor µ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Besides, we set the transformation probability for DIM as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='5 [60], the size of Gaussian kernel for TIM as 7 × 7 [10], and the number of scale copies for SIM as m = 5 [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' For our method, we set ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='9 when we integrate the proposed strategy in MI-FGSM and NI-FGSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Integrated with Gradient-based Attacks Gradient-based adversarial attacks are the widely adopted approaches to craft adversarial examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Here we first evaluate the effectiveness of SD to boost the attack performance of existing gradient-based attacks, including I-FGSM, MI-FGSM and NI-FGSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' The adversarial exam- ples are generated on each model and evaluated on the other models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Te attack success rates, which are the misclassifica- tion rates of the victim models on the adversarial examples, are reported in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=" In general, we can observe that SD can effectively im- prove the attack performance of the baselines in white-box 6 Resnet-18 ResNet-101 ResNeXt-50 DenseNet-121 Vit Swin 0 20 40 60 80 100 Attack Success Rate (%) I-FGSM MI-FGSM NI-FGSM SD's improvement Figure 6." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' The Attack success rate (%) of baseline models on the crafted adversarial examples generated by I-FGSM, MI-FGSM, NI-FGSM and our proposed method under an ensemble model set- ting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' We evaluate the performance on the hold-out model using the adversarial examples generated by the other five models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' as well as black-box settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' On average, SD helps I- FGSM achieves the attack performance of 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='5%, which is 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='4% higher than vanilla I-FGSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Under the white-box setting, MI-FGSM and NI-FGSM achieves better transfer- ability than I-FGSM since the momentum helps stablilize the optimization and escape local minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Compared with these attacks, SD can significantly boost the transferabil- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' In particular, SD can remarkably enhance the attack performance of I-FGSM, which outperforms NI-FGSM and MI-FGSM with a clear margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' SD can boost I-FGSM, NI- FGSM and MI-FGSM at least 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='7%, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='7%, and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='5%, re- spectively, showing its high effectiveness in improving the transferability and generality to various models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Integrated with Transformation-based Attacks Transformation-based adversarial attack leverages mul- tiple transformations on the inputs to improve the adver- sarial transferability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' We are inline with the aforemen- tioned experiments to integrate our method on three clas- sical transformation-based attacks, namely DIM, TIM and SIM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' The reuslts can be shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' We can observe that all the transformation-based meth- ods can achieve the attack success rate of 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='0% or near 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='0%, showing that the transformation-based methods do not degrade and even enhance the white-box attack per- formance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' As for the black-box performance, TIM be- haves the poorest transferability on these normally trained models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Surprisingly, on some models, with SD strategy, TIM achieves even better transferability than DIM and even SIM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' For instance, on ResNeXt-50, TIM with SD strategy achieves the attack success rate of 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='9%, which is much high than 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='1% of DIM and 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='8% of SIM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=" On average HGD IncRes v2ensFastAdv RS NRP JPEG BitRed FD 0 20 40 60 80 100 Attack Success Rate (%) I-FGSM MI-FGSM NI-FGSM SD's improvement Figure 7." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' The Attack success rate (%) of baseline models on the crafted adversarial examples generated by I-FGSM, MI-FGSM, NI-FGSM and our proposed method under ensemble model set- ting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' We evaluate on the advanced denfese methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' of all experiment cases, SD can boost the transferability of TIM, DIM and SIM with a clear margin at least 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='9%, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='1% and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='8% respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Integrated with Ensemble Attacks Ensemble attacks are another effective method to gener- ate more transferable adversarial examples by attacking sev- eral models in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' We evaluate the performance of SD when integrated with the ensemble attack method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Specif- ically, we use the indicated single model as the evaluation model while we generate the adversarial example using the other five models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' We present the results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' It can be clearly observed that SD strategy significantly enhances the adversarial attack performance, especially for I-FGSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Specifically, with SD strategy can achieve an av- erage attack success rate of 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='5% , 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='4%,99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='2% when in- tegrated with I-FGSM, MI-FGSM and NI-FGSM, respec- tively, showing a clear margin of 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='2%, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='0 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='3% com- pared with the vanilla ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Attack on Advanced Defense Methods To further identify the effectiveness of our method, we evaluate the performance of the crafted adversarial exam- ples on six defense methods, including HGD, IncRes − v2ens, FastAdv, RS, NRP, JPEG, BitRed and FD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' The tar- get model for JPEG and NRP is VGG19 and the other meth- ods adopts the official models provided in the correspond- ing works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' We generate the adversarial examples on the en- semble model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=', ResNet-18, ResNet-101, ResNeXt-50, DenseNet-121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' ViT and Swin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' The results are summarized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' From the figure, SD still exhibits an excellent attack performance towards various advanced defense methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='ResNet-18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='ResNet-101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='ResNeXt-50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='DenseNet-121ViT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='Swin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='Attack Success Rate (%) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content="(a) Ablation study on the 'S' and 'D' strategy " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='I-FGSM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='I-FGSM-D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='I-FGSM-S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='Ours ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='ResNet-18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='ResNet-101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='ResNeXt-50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='DenseNet-121ViT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='Swin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='Attack Success Rate (%) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='(b) Ablation study on different increasing sequence ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='Exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='Power ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='Linear ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='Ln(e) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='ResNet-18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='ResNet-101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='ResNeXt-50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='DenseNet-121ViT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='Swin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='Attack Success Rate (%) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='(c) Ablation study on different base of log ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='Ln(2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='Ln(e) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='Ln(10) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Ablation studies using the increasing sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Among them, the certificate robustness defense method, Random Smoothing (RS), is the most effective against ad- versarial attacks, where I-FGSM, MI-FGSM and NI-FGSM only achieves the attack success rates of 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='9%, 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='0% and 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='1% respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' SD can remarkably enhance the attack performance with a clear improvement of 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='8%, 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='9% and 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='4%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' On average, SD can improve the adversarial transferability of I-FGSM, MI-FGSM and NI- FGSM with a clear margin of 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='9%, 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='3%, and 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='8%, respectively, showing the high effectiveness of our SD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Ablation Studies In this subsection, we mainly study the effectiveness of the proposed strategy and the choice of scheduled adaptive step sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' We adopt ResNet-18 as the victim model to con- duct the following experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' The effectiveness of the proposed strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' There are two major components in our proposed method, namely sched- uled adaptive step size and dual example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' As carefully dis- cussed in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='2, the scheduled adaptive step size is used to explore more on the region where there exits more transferable adversarial examples while the dual example is used to craft adversarial examples with high transfer- ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Here, we design an ablation study to identify the effectiveness of each component as follows, 1) I-FGSM: conventional iterative fast gradient sign method with the identity step size;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' 2) I-FGSM-D: apply the dual example to I-FGSM, where we update the dual example with per- turbation smoothing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' 3) I-FGSM-S: I-FGSM with sched- uled adaptive step size;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' 4) Ours: I-FGSM with scheduled adaptive step size and the dual example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' The results can be shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' 8 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' We can identify the effectiveness of the dual example from the comparison between I-FGSM and I-FGSM-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Although it sacrifices a little performance un- der the white-box setting with only the scheduled step size, the transferability of crafted adversarial examples is signif- icantly improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' The choice of scheduled adaptive step size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' As our anal- ysis in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='1, the density of transferable adversarial examples decreases with increasing the distance to the be- nign example, so we adopt the increasing sequence as the scheduled step size to fully utilize the perturbations with good transferability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' We study four increasing sequences, exponentiation (Exp), power (Power), linear (Linear), loga- rithm sequence (Ln(·)) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' The experiment results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' 8 From the result in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' 8(b), it can be shown that with the decreasing of second-order, the attack performance of the increasing sequences display an increas- ing trend, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' from Exp to Ln(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' We additionally adopts different number of base in logarithm function in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' 8 (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' It can be shown there exits minor difference between the settings with different bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Conclusion In our work, we find an interesting phenomenon, that the FGSM and the I-FGSM with few iterations exhibit a bet- ter adversarial transferability than I-FGSM with more iter- ations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' We hold an assumption for this phenomenon, that the adversarial perturbations are more transferable near the benign sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' With increasing distance of the adversarial example relative to the benign sample, there is an increasing difference on the feature attention between different mod- els, caused by the different architectures and distribution shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' To alleviate this catastrophic performance drop, we propose a novel strategy, which uses the scheduled step size and dual example (SD), to fully utilize the transferable ad- versarial perturbation near the benign sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Extensive ex- periments shows that SD can significantly boost the attack performance of existing adversarial attack methods, includ- ing the gradient-based attacks, transformation-basd attacks and ensemble attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Also, SD can remarkably enhance the adversarial transferability of existing methods towards various advanced defensive methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' References [1] Akshay Agarwal, Nalini Ratha, Mayank Vatsa, and Richa Singh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Exploring robustness connection between artifi- 8 cial and natural adversarial examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' 1 [2] Anish Athalye, Nicholas Carlini, and David A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Wagner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' Ob- fuscated gradients give a false sense of security: Circumvent- ing defenses to adversarial examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFOT4oBgHgl3EQf9DQS/content/2301.12968v1.pdf'} +page_content=' In Proceedings of the 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Andrew L. Goodwin1 +1Inorganic Chemistry Laboratory, Department of Chemistry, University of Oxford, Oxford, UK +Disorder in crystals is rarely random, and instead involves local correlations whose presence and +nature are hidden from conventional crystallographic probes. This hidden order can sometimes +be controlled, but its importance for physical properties of materials is not well understood. Us- +ing simple models for electronic and interatomic interactions, we show how crystals with identi- +cal average structures but different types of hidden order can have very different electronic and +phononic band structures. Increasing the strength of local correlations within hidden-order states +can open band gaps and tune mode (de)localisation—both mechanisms allowing for fundamental +changes in physical properties without long-range symmetry breaking. Taken together, our results +demonstrate how control over hidden order offers a new mechanism for tuning material proper- +ties, orthogonal to the conventional principles of (ordered) structure/property relationships. +Pre-print +January 5th 2023 +1 +arXiv:2301.02035v1 [cond-mat.mtrl-sci] 5 Jan 2023 + +1 +Introduction +The delocalised electronic and vibrational states key to many physical properties of periodic solids +emerge from the collective behaviour of atoms and electrons on ordered lattices 1–3. Random disor- +der breaks this emergence and drives localisation, resulting in scattering of electronic and vibrational +states and lowering of electronic and thermal transport 4,5. For strong random disorder, transport is +completely stopped and—in the case of electronic properties—a metal-to-insulator transition can occur +through Anderson localisation 6,7. +Disordered crystals present an interesting problem that, at face value, lies between these two ex- +tremes. Disorder is rarely random, and instead many disordered crystals still obey strict local chemical +rules that do not result in long-range symmetry breaking 8. In this sense such materials support a ‘hid- +den order’ that is not evident in conventional crystallographic analysis. A well-known example is the +hydrogen-bonding network of water-ice Ih, where periodically-arranged oxygen atoms each are cova- +lently bonded to two of four nearby hydrogen atoms to give a non-periodic arrangement of H2O orienta- +tions 9. Related states have been identified in mixed-anion perovskites 10,11, Coulomb-phase pyrochlores +12–15, and metal–organic frameworks 16. An obvious and important question concerns the nature of col- +lective electronic and/or phononic states in such systems: are they similar to those in ordered crystals or +more closely related to those of amorphous solids? Or are they altogether different in character? +There are strong indications that hidden order may impact material properties. Short-range or- +der in battery materials can influence ionic conductivities and charge-storage capacities by affecting the +networks of mobile ions and vacancies 17,18. Likewise, the nature of phonon broadening in disordered +crystals has also been found to vary as a function of the type and extent of short-range order present +19–21. In the few systems known to exhibit hidden-order transitions—such as the heavy-fermion super- +conductor URu2Si2 22,23 and magnetocaloric Gd3Ga5O12 24—the emergence of hidden order couples to +thermodynamic anomalies and results in different electronic and/or magnetic behaviour. What remains +entirely unclear is the nature of this link between hidden local order and collective phenomena. +2 + +Here we address precisely this problem by exploring the consequences of hidden order on the +electronic and vibrational states of a model family of disordered crystals. The toy model we study is +chosen because there is an obvious mechanism for varying the degree and nature of hidden order it +supports. We begin by introducing this model and explaining our approach for calculating electronic +and phononic states for its various realisations. We then proceed to demonstrate a complex interplay +between hidden order and the nature of collective states. In particular, we report three key findings: (i) +that hidden order can be used to selectively broaden specific parts of the electronic or phononic band +structure, (ii) that it modulates localisation in different ways, and (iii) that it can result the opening of +band-gaps without long-range symmetry breaking. We conclude by discussing generalisations of this toy +model, and the relevance of its behaviour to a range of physical systems. +2 +Results and Discussion +Hidden-order model. A useful toy model for exploring different types of correlated disorder is the +two dimensional system A2B, where B atoms occupy a square lattice with A atoms positioned half-way +between them (Fig. 1a). By introducing a distortion such that A atoms form one stronger and one weaker +bond to neighbouring B atoms, several distinct types of disorder can be achieved. One possibility is for +random distortions of A atoms, such as illustrated in Fig. 1b. In this case the B atoms will have a varying +number of strong and weak bonds. In many real systems, however, there will be local chemical rules that +govern the types and geometries of bonds. One such example is for each B atom to have two strong and +two weak bonds, which can be satisfied by a large number of configurations, with an example shown in +Fig. 1c. These rules are similar to the two-in-two-out rule for hydrogen bonding in ice 9 and this square- +lattice representation results in the well-known ‘6-vertex’ statistical mechanical model.25 Note that there +are two types of B atom geometries, where the two strong bonds are either parallel or perpendicular to +one another. A stronger chemical rule is then to have only perpendicular strong bonds, as illustrated in +Fig. 1d, equivalent to the square-ice system 26. In this highly-constrained case, the strong A–B bonds are +ordered in one-dimensional chains, but there is no three-dimensional bond order. +These distorted systems all have identical average crystal structures and therefore identical Bragg +3 + +Figure 1: The effect of random and correlated disorder on electronic and phonon bands. a) Ordered +A2B structure with A halfway between B atoms on a square lattice. b) A random configuration of A site +distortions. c) two-in-two-out rule for correlated distortions. d) correlated disorder with perpendicular +strong bonds. e-h) The corresponding diffuse scattering patterns. The square lattice of black dots are +Bragg peaks, which are several orders of magnitude stronger than diffuse scattering. i-l) Electronic +bands for these systems. The energy scale is arbitrary and the zero point does not imply the fermi level. +m-p) Phonon bands. +4 + +diffraction intensities. In this sense the presence of additional local order is hidden from conventional +crystallographic analysis. The clearest signature of this hidden order is through weak diffuse scattering. +Fig. 1e-h shows the single-crystal scattering pattern for each system. The square grid of black dots +indicates the positions of Bragg peaks, which are several orders of magnitude stronger than the weak +diffuse scattering lying between them. In the case of the undistorted parent structure (Fig. 1e) there is +no diffuse scattering. Random distortions give broad diffuse scattering (Fig. 1f), while the two-in-two- +out locally ordered system has characteristic structured diffuse scattering with pinch-points (Fig. 1g), +reminiscent of those found in the scattering of 3D spin-ice with a similar local rule 12,27. Finally, the +system with two strong perpendicular bonds has thin lines of diffuse scattering (Fig. 1h), indicative of +long-range one-dimensional correlations. +Collective electronic behaviour. We explore the effect of varying hidden order on the electronic +properties of these models by calculating the electronic band structure using a semi-empirical tight- +binding model with nearest neighbour hopping parameters. Drawing on the conceptual analogy to H and +S arrangements in H3S 28,29, we assign to B atoms a set of s, px and py orbitals but only a single s orbital +to the A atoms. On-site energies and hopping parameters for strong and weak bonds are modelled on +the values calculated for H3S, which has 2D layers with similar distortions of H between S on a square +lattice 28,29. Using this realistic parameter set allows some general effects to be illustrated. We note that +the energy scale used is arbitrary and does not imply the Fermi energy lies at E = 0. Further details of +our calculations are given in the supporting information. +The electronic bands depend very strongly on the type and degree of hidden order. In the ordered +state the bands are well-defined in energy and disperse throughout the Brillouin zone with band crossings +at the Γ and M points (Fig.1i). Random distortions of the A sites change this picture, as shown in Fig. 1j. +While the overall features and general dispersion are very similar to the ordered case, the bands are now +much broader. Hence, as anticipated, the electronic states are no longer well-defined in energy and will +scatter as a consequence, reducing electronic transport. +By introducing the local two-in-two-out rule, significant differences to both the random and ordered +5 + +cases are found (Fig. 1k). Now some of the bands have become narrower in energy again, while the +gaps below the flat band at Γ and above the low-energy flat band at M have been filled with dilute states. +Furthermore, the crossing above the flat band at Γ has lifted and given way to a small band gap. Changing +the local order to the case of two perpendicular strong bonds per B site leads to very different effects, +as shown in Fig. 1l. The bands are now generally narrow with states well-defined in energy, meaning +electronic transport is not as hindered by scattering as in the two other disordered cases. In sharp contrast +to the random and ordered systems, the band crossing as Γ and M have now lifted and clear band-gaps +are observed. The dilute states filling some gaps in the two-in-two-out system are gone. There are also +very weak additional band-like features between the strong narrow bands. +The emergence of band gaps for the two systems with strongest hidden order is conceptually impor- +tant because, for the right filling fraction, a variation in hidden order type could lead to a metal–insulator +transition. +Collective vibrational behaviour. We observe similar effects on the phonon spectrum as a conse- +quence of correlations [Fig. 1m-p]. In our calculations, phonon energies and eigenvectors are obtained +by diagonalising the dynamical matrix using semi-empirical force constants between nearest neighbours. +An arbitrary (but sensible) set of force-constants was chosen to best illustrate the effects, as elaborated +in the supporting information. For reference, Fig. 1m shows the phonon bands for the ordered system, +where acoustic and optical phonons are well-defined in energy with crossing of four bands at the M point. +Random distortions again give phonon bands similar to the ordered system but broadened in energy, re- +sulting in increased phonon scattering (Fig. 1n). The broadening is least evident for the long-wavelength +acoustic branches as these are most insensitive to variations in local configurations. The behaviour at the +M point is now different: the four bands no longer cross as before, but change their dispersion to avoid +the crossing. +The locally-ordered two-in-two-out system has some differences to the random system in terms +of the band widths (Fig. 1o), but it is the system with strongest hidden order for which the phonon +dispersion is most different (Fig. 1p). Here, the bands are almost all narrow, and a large band gap +6 + +has opened throughout the Brillouin zone (Fig. 1p). Consequently, the type of correlations in disordered +structures can also strongly impact properties that depend on vibrations, such as thermal transport. While +some interplay between correlated disorder and phonon structure had been reported previously,19,20 a key +result of this study is the demonstration that this interplay can be sufficiently strong as to open vibrational +band gaps. +Thermodynamic stabilisation of hidden order. In Fig. 2 we show the integrated electronic and +phonon densities of states, which make clear that the band-gaps seen along high-symmetry directions do +indeed persist throughout the entire Brillouin zone. Focusing on the emergence of electronic band-gaps +we note that the energies of the corresponding valence (low-energy) edge states is reduced in the ‘per- +pendicular’ hidden order state relative to the ordered and random cases. This stabilisation implies that, +at appropriate filling fractions, the electronic energy of the system can be reduced through a concerted +distortion to the hidden order state. Such a transition is conceptually similar to a Peierls distortion, but is +fundamentally different in that it proceeds without any global symmetry lowering. Similar mechanisms +may be at play in disordered ‘orbital molecule’ states, such as in LiRh2O4 and Li2RuO3,30,31 where the +structural distortions associated with valence electron localisation are local and not long-range ordered.32 +Mode localisation. Correlations not only affect the form of the electronic and phonon band struc- +ture, but also change the delocalisation of modes. We show this in Fig. 3 by indicating the degree of +delocalisation of electronic and phonon modes weighted by the number of states (see methods for de- +tails). Taking each diagram in turn, we begin by noting that in the ordered system (Fig. 3a), electronic +bands with dispersion are generally quite delocalised, while flat bands are localised. In the randomly +distorted system (Fig. 3b), all bands have become more localised—as anticipated for disordered sys- +tems. The two-in-two-out rule gives rise to intermediate behaviour (Fig. 3c). But, most surprisingly, the +most strongly correlated state (perpendicular strong bonds), gives delocalised states (Fig. 3d). Similar +changes are observed for the phonon modes (Fig. 3e-h), for which the key difference is the resilience of +delocalisation within the long-wavelength acoustic branches. +The variance in degree of localisation is clearly exemplified by interrogating representative modes +7 + +Figure 2: Density of states plots for (a) electronic modes and (b) phonon modes. Hidden order can induce +band gaps that are not present in randomly disordered systems +in real-space. Fig. 4a-h shows two examples of electronic modes for the different systems. The orbitals +are coloured according to the wavefunction phase, while corresponding saturation is given by the wave- +function amplitude. Whenever modes are localised, atoms do not contribute equally to the wavefunction, +which fragments into small coherent regions incoherent with respect to one another (see, e.g., Fig 4b). +We provide a detailed interpretation of these images in the supporting information, but highlight for in- +terest here the unusual behaviour shown in Fig. 4h for the system with perpendicular strong bonds. This +particular mode is completely delocalised with strong coherence along chains but mixing from chain +to chain, causing chains to have phase shifts relative to each other. This is in contrast to the ordered, +8 + +Figure 3: Weighted delocalisation of electronic (a-d) and phonon modes (e-h). +random and two-in-two-out systems, where the corresponding modes are all localised to a large extent. +In a similar way, Fig. 4i-p shows the real-space representations of two types of phonon modes. Here +colours indicate displacement direction—further highlighted by arrows—while saturation gives the cor- +responding amplitudes. The two vibrational modes illustrate how correlations can have different effects +on modes, opening up the possibility of selectively (de)localising modes. These phonon modes are fur- +ther discussed in the supporting information. To summarise, we find that hidden order not only affects +the density and coherence of states, but also their degree of delocalisation—often in quite nuanced and +unexpected ways. +Extension to three dimensions. In the supporting information, we include a discussion of the +extension of our approach to three dimensions (3D). The results are qualitatively the same as for two +dimensions, albeit with some additional subtleties and avenues for control given the increased scope for +geometric isomerism in 3D. +9 + +Figure 4: Real-space view of modes. (a-h) Two types of electronic modes for the different systems. +Colour hue indicates the phase of the wave-function, while saturation indicates the amplitude. (i-p) Two +types of phonon modes for the different systems. Colour hue indicates direction of motion with saturation +indicating amplitude. Arrows inside atoms further illustrate the movements. +10 + +3 +Concluding Remarks +Perhaps our key result has been to show clearly that correlations in disordered crystals have consequences +for properties, as both electronic and vibrational modes are impacted in functionally-important ways. +Hence control over correlations offers a new handle with which to tune properties in functional materials. +Moreover, because hidden order affects electronic and vibrational states in subtly different ways, it may +prove possible to combine the effects of both to engineer functional materials with particularly desirable +properties. We offer a handful of examples to demonstrate this point. +One topical family is that of thermoelectric materials, where the design brief is to combine a low +thermal conductivity with large electrical conductivity in a gapped semiconductor, as captured by the +phonon–glass–electron–crystal paradigm 33. The conventional approach is to introduce disorder into a +subset of atoms that do not contribute to electronic conductivity 34–36. This is a design principle based +on the idea of disorder being random and creating strong scattering of modes, which is why disorder on +the substructure responsible for electronic conductivity is to be avoided. But our present study suggests +an entirely new design strategy of introducing specific kinds of hidden order that at once broaden heat- +carrying phonon modes whilst preserving narrow electronic modes in the conduction band. Additionally, +one might even use correlated disorder to tune the electronic band gap so as to optimise thermoelectric +performance 37. In this context, we note that thermoelectric half-Heusler materials can be made with +different local vacancy orderings but with identical average crystal structures and stoichiometries 38, +indicating the possibility for tuning this class of materials through the concepts presented here. +The effect of disorder on topological insulators (TI) is a problem of strong currency in the field of +functional materials design. Topological insulators are insulating in the bulk but host conducting gapless +edge or surface states. These gapless states are topologically protected and are robust against weak +disorder 39,40. For strong disorder the non-trivial topological states can break down due to localisation. +However, in some systems, strong disorder causes phase transitions from topologically-trivial to non- +trivial states, such as topological Anderson insulators (TAIs) 41,42 or disorder-induced topological Floquet +insulators 43. Quantised topological invariants are related to symmetries, but these can be broken in +11 + +strongly disordered crystals. However, it has been shown that symmetry-stabilised topological invariants +are still strictly quantised even in the presence of disorder that breaks symmetries locally yet restores +them on average 44. Since TI phases are robust to disorder, the disorder itself can be used to further +engineer their band structures 45,46. As we have shown here, the hidden order present within correlated +disordered states can be used to control band structures, underlining the importance of understanding +correlations in disordered TIs whilst also offering a new mechanism for tuning TI materials. We note that +several topological insulator materials have been found experimentally to be disordered crystals 47,48— +albeit that the nature of this disorder is not well understood, since earlier studies have only analysed +average structure. +The same principles might be used to engineer band gaps and transport properties of photovoltaics, +and—in principle—combining effects of electronic and phonon band structures could tune electron– +phonon coupling in superconductors. +In an entirely different field, we anticipate that the link we demonstrate between hidden order and +gap opening may have implications for the design of disordered photonic materials. The relatively recent +demonstration of optical transparency in hyperuniform structures has shown that subtleties of disordered +networks can have fundamentally important effects on optical band structure.49 Likewise, control over +the degree of short-range order has emerged as an unexpected design strategy for controlling visual +appearance in photonic matter.50 To the best of our knowledge, the concept of introducing hidden order +within an otherwise-crystalline photonic medium as a means of introducing transparency has not yet +explored, and may offer interesting new approaches for controlling matter–light interactions. +As a final point, we note that, because phases with different types of hidden order can have sig- +nificantly different properties, it is more important than ever to develop experimental tools for probing +hidden order in crystalline materials. The Bragg diffraction techniques used to determine crystal struc- +tures are sensitive only to long-range order, which is why it is often only the average structure of materials +that is known. By contrast, diffuse scattering is sensitive to local correlations, but is several orders of +magnitude weaker than Bragg scattering—this has limited its use historically 51. The development of +12 + +modern detectors and high-intensity x-ray, neutron and electron sources have now made it feasible to +measure diffuse scattering much more routinely, allowing for identification of distinct locally-ordered +phases 17,18,38. +4 +Acknowledgements +N.R. acknowledges the Independent Research Fund Denmark (DFF) for funding through the Interna- +tional Postdoctoral grant 1025-00016B. A.L.G. thanks A. R. Overy (Oxford), M. G. Tucker (SNS), A. +Simonov (ETH Zurich), and C. Romao (ETH Zurich) for discussions, and the European Research Coun- +cil for funding (Grant 788144). +Methods +Electronic states are calculated from supercell configurations using a semi-empirical tight binding model. +Taking φi as the ith atomic orbital in the supercell, a basis of Bloch sums for wavevector k is +Φik = +1 +√ +N +� +tm +eik(tm+vi)φi(r − tm − vi), +(1) +where tm is the position of the mth supercell origin, vi is the position of the ith atomic orbital in the +supercell, N is the number of supercells, and r is the real-space coordinate vector. In the tight-binding +approximation, the Hamiltonian then takes the form 52: +Hijk = +� +τ +eikτγijτ + δijE0i. +(2) +Here, τ are the vectors between atomic orbital i and j with nonzero matrix elements γijτ = ⟨φi(r)|H|φj(r− +τ)⟩ , δij is the Knonecker-delta and E0i the energy of orbital i on an isolated atom. Here τ is limited to +nearest-neighbours only and the matrix elements γijτ are given semi-empirical values for the different +types of orbital combinations. In the present case one type of atom is given one s orbital and the other +type one s and a set of p orbitals. The needed parameters in the present case are a set of four values +comprised of γssσ, and |γspσ| for short and long bonds, as well as parameters for E0i. Directional depen- +dence is taken into account using γspσ = lx|γspσ| for an s to px element, where lx is the x-component +13 + +of the normalised τ vector, and similarly the s to py and s to pz depend on ly and lz, respectively. p to +s orbital elements obey γpsσ = −γspσ.52 All other matrix elements are zero in this case. In other cases, +more matrix elements would be needed, such as the γppσ, γppπ +Using a custom python script the Hamiltonian is constructed and diagonalised to obtain the eigen- +vectors and eigenvalues of the system at different k. The bands are then unfolded to the Brillouin Zone +of the primitive cell by calculating the weight of each state as 53: +Wk = 1 +No +� +o∈PC +�� +i∈o +c∗ +ik +� �� +i∈o +cik +� +(3) +The sum o ∈ PC are over the different orbitals of the primitive cell, and the sum i ∈ o are those orbitals +in the supercell which are equivalent in the primitive cell. cik are the coefficients of the normalised +eigenvectors in the Bloch sum basis and No the number of orbitals in the system. The number of states +per cell for each mode is then given as 2N0∈PCWk, where 2N0∈PC is the number of orbitals in the +primitive cell and the factor of two takes into account the spin degree of freedom. The weighed degree +of delocalisation of each mode, Dk is calculated as 54: Dk = Wk/ � +i |ci|4 . +Phonon modes are calculated in a very similar way by constructing and diagonalising the mass- +adjusted dynamical matrix from the eigenvalue equation: +DU = ω2U. +(4) +Here D is the mass-adjusted dynamical matrix, U is the eigenvector of mass-adjusted elementary move- +ments and ω the energy. The method for phonon calculations follow that given in detail in Ref. 55. +Elements of D are given by +Dijk = +1 +√maimaj +� +τ +eikτKijτ, +(5) +where i and j now reference the elementary movements of all atoms in the supercell along cartesian +axes. mai is the mass of the atom to which the ith elementary movement belongs. Kijτ is the force +constant between elementary atomic movements i and j. The diagonal elements Diik need to conserve +force balance: Diik = −1/mai +� +j̸=i Kij. Again, only nearest neighbours are included. Two types of +14 + +force-constants are used: K⊥ and K∥ for perpendicular and parallel movements of nearest neighbours, +with two possibilities for short and long bonds for each. +The phonon bands are unfolded in the same way as for the electronic bands using +Wk = 1 +Nu +� +u∈PC +�� +i∈u +c∗ +ik +� �� +i∈u +cik +� +, +(6) +where u are the elementary displacements in the primitive cell, Nu the number of elementary displace- +ments in the supercell and cik the coefficients of the normalised eigenvectors of D. The weighed delo- +calisation is then calculated in the same way as for the electronic modes. +The electronic and phononic band structures were calculated on configurations with 32 by 32 atoms +and averaged over 30 different configurations. For the electronic bands values were chosen to be close +to those calculated for H3S, as to keep them realistic. This was done using the minimal tight binding +model from 28, where values for orbital energies are taken relative to the sulphur s level, with E0Ss = 0, +E0Sp = 8.16 eV and E0Hs = 6.42 eV. The hopping elements used for the 2D simulation were rounded +to nearest integer values, γssσ = −5 and −3 eV for strong and weak bonds, respectively. Similarly, +|γspσ| = 6 and 4 eV were used. For the 3D systems shown in the SI, the exact values for H3S given +in Ref. 28 were used for configurations with 16 atoms along each dimensions and averaged over 10 +configurations. These are γssσ = −4.69 and −2.98 ev and |γspσ| = 5.69 and 4.3 eV. For the phonon +calculations parameters were chosen to give clear band structures. In the 2D systems, the masses for the +two types of atom were mA = 0.8 and mB = 1. Values for the force constant were chosen as K∥ = −2 +and −1 for strong and weak bonds, respectively, as well as K⊥ = −0.6 and −0.2. 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A 75, +14–24 (2019). +20 + +Supplementary Information 1 +Powder Diffraction Patterns +Figure SI 1 shows the calculated powder diffraction patterns for the three different disorder types in +the 2D system. The upper part of the figure shows the full scale of the pattern. As these systems all +have identical average crystal structures, the intensities of the Bragg diffraction peaks are identical. The +difference in scattering comes in the diffuse signal, which is much weaker and not generally visible on +the same scale as the Bragg peaks. In the lower sub-figure, the intensities have been multiplied by 100 +to zoom in on the diffuse scattering, showing their differences.The noise seen in the diffuse scattering +is due to a limited calculation size. Calculations were made by averaging over 100 configurations with +each 60x60 atoms. To increase contrast of the diffuse scattering, Li was used as A atoms with O as B +atoms. As shown in the main paper, single crystal scattering can be a better way to identify these phases. +Supplementary Information 2 +Real space view of states +Fig. 4a-d shows a low-energy electronic mode between Γ and M composed of bonding s-orbitals from +both atom types. In the ordered case the mode is delocalised with contributions from all s-orbitals, and +the phase of the wavefunction changes slowly along the wavevector. For the randomly disordered system, +this mode has now become localised. Atoms no longer contribute equally to the wavefunction, which +consists of small coherent regions that are incoherent with respect to each other. For the two-in-two- +out correlations, the mode is more delocalised, but not fully, and there is still incoherence in the phase. +The mode becomes fully delocalised and coherent in the system with perpendicular strong bonds, and is +similar to the ordered case. The same is not true for the mode shown in Fig. e-h. This mode is on the flat +band between M and X, consisting of the B atom p orbitals and A atom s orbitals. In the ordered system +the band is localised on a single chain, and there is no mixing between chains. With random distortions +there is mixing between directions but the mode is still localised. For this mode the two-in-two-out +system is close to the random system, which was not the case for the first mode shown. However, for the +system with perpendicular strong bonds, the mode is completely delocalised with strong coherence along +chains in one dimension and mixing between chains along the other dimension, causing chains to have +21 + +Figure SI 1: Calcualted Powder X-ray Diffraction patterns for the two-dimenstional systems. Upper +figure shows the full scale, while the lower figure shows the weak diffuse scattering signal by scaling by +a factor of 100. The noise seen in the diffuse scattering is due to a limited calculation size. +phase shifts relative to each other. Thus, the electronic wavefunction can differ significantly depending +on correlations, and in some cases the wavefunction can take on unique shapes not seen in ordered or +randomly disordered systems. +22 + +Similar behaviour is seen for the phonon modes (Fig. 4i-p). Here each atom is drawn as a circle and +the colour now indicates the direction of movement while the saturation is again the amplitude. Arrows +inside the circles highlight the displacement vectors further. Fig 4i-l show a transverse acoustic shear +mode between X and Γ. In the ordered system the mode is delocalised with layers three atoms wide +moving out of phase with respect to each other, separated by a single layer of stationary A atoms. In +the random system this mode becomes more localised with coherent regions that are incoherent with +respect to each other. With the two-in-two-out correlations, the mode becomes more delocalised and +consists of almost coherent layers moving as in the ordered case, but with disorder in the amplitude and +directions of some atoms. In the system with perpendicular strong bonds, the mode becomes delocalised +with layers moving like in the ordered case, but atoms between the layers are no longer stationary and +move incoherently. This is in contrast to the optical mode shown in Fig. 4m-p, where the ordered system +and the system with perpendicular strong bonds are both delocalised and coherent, while the random and +two-in-two-out systems have incoherent localised modes. +Supplementary Information 3 +Orbital contributions to states +Figure SI 2 shows the orbital contributions weighed by the number of modes in the 2D case. +Supplementary Information 4 +Three dimensional example +The same type of effects that were observed in 2D will also apply to three-dimensional systems. As +an example, a 3D analogy to the 2D system is presented, based on a cubic A3B structure with A atoms +positioned between B atoms forming a simple cubic lattice. With A halfway between B sites, the system +is ordered (Fig. SI 3a). If A atoms distort to form one strong and one weak bond to B sites, several +disordered configurations can be obtained. With random distortions a wide variety of strong and weak +bonds around B sites are generated (Fig. SI 3b). If instead each B site has the chemical rule of 3 strong +and 3 weak bonds, a three-in three-out structure is obtained, as shown in Fig. SI 3c. In this structure there +is a mixture of B sites with the 3 A atoms coordinated meridionally (mer) and facially (fac). Stronger +chemical rules are then that the B atoms form only one of these, which in both cases leads to disordered +23 + +Figure SI 2: Weighed orbital contributions to the electronic states in the 2D systems. +systems shown in Fig SI 3d and SI 3e for mer and fac geometry. Thus, in three dimensions more different +types of short-range order can be generated due to the increased flexibility of another dimension. +Diffuse scattering patterns for these disordered systems are very different, while their Bragg diffrac- +tion intensities are identical. Fig. SI 3f-j shows the diffuse scattering down in the [111] plane of reciprocal +space. While the ordered system has no diffuse scattering and the random disorder has broad smeared out +diffuse scattering, the systems with local correlations have structured scattering. In the three-in three-out +case pinch-points are again seen, reminiscent of other systems with similar rules 12,27. In the meridional +system the diffuse scattering has condensed further to form ring-like features with several maxima, while +the facial system has strong narrow lines indicating long-range correlations of the disorder. +24 + +Figure SI 3: Effects different disorder correlations on electronic and phonon bands. a) Ordered A3B +structure with A (blue) halfway between B (grey) on a simple cubic lattice. b) A random configuration +of A site distortions. c) three-in three-out rule producing a mixture of meridional and facial geometry. +d) meridional geometry only. e) facial geometry only. f-j) Corresponding diffuse scattering patterns. +The hexagonal grid of black dots are Bragg peaks, which are several orders of magnitude stronger than +diffuse scattering. k-o) Electronic bands for these systems. The energy scale is arbitrary and the zero +point does not imply the fermi level. p-t) Phonon bands. +Similar trends to the two-dimensional system are found in the electronic (Fig. SI 3k-o) and phonon +bands (Fig. SI 3p-t). Here the electronic bands are calculated using parameters for the R-3m phase of +H3S 28. Phonon modes are calculated using parameters chosen to highlight effects of correlations. While +25 + +1random disorder tends to strongly broaden features that were present in the ordered system, correlated +disorder changes the band dispersions and broadening, opening and closing band gaps. In the electronic +bands (Fig. SI 3k-o) the low-energy band crossings at the R and Γ points can be opened to different sizes +for the different types of correlations. The meridional system introduces a new band-like feature with a +maximum at the R point in the electronic structure as well as new weak band-like features in the phonon +modes. The facial system induces new band gaps for both electronic and phonon modes. +26 + diff --git a/YdA0T4oBgHgl3EQfFv8F/content/tmp_files/load_file.txt b/YdA0T4oBgHgl3EQfFv8F/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9860c67800c5a1e2b3e8e419aa7092074c3b917e --- /dev/null +++ b/YdA0T4oBgHgl3EQfFv8F/content/tmp_files/load_file.txt @@ -0,0 +1,785 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf,len=784 +page_content='Tuning electronic and phononic states with hidden order in disordered crystals Nikolaj Roth∗1 & Andrew L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Goodwin1 1Inorganic Chemistry Laboratory, Department of Chemistry, University of Oxford, Oxford, UK Disorder in crystals is rarely random, and instead involves local correlations whose presence and nature are hidden from conventional crystallographic probes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' This hidden order can sometimes be controlled, but its importance for physical properties of materials is not well understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Us- ing simple models for electronic and interatomic interactions, we show how crystals with identi- cal average structures but different types of hidden order can have very different electronic and phononic band structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Increasing the strength of local correlations within hidden-order states can open band gaps and tune mode (de)localisation—both mechanisms allowing for fundamental changes in physical properties without long-range symmetry breaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Taken together, our results demonstrate how control over hidden order offers a new mechanism for tuning material proper- ties, orthogonal to the conventional principles of (ordered) structure/property relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Pre-print January 5th 2023 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content='02035v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content='mtrl-sci] 5 Jan 2023 1 Introduction The delocalised electronic and vibrational states key to many physical properties of periodic solids emerge from the collective behaviour of atoms and electrons on ordered lattices 1–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Random disor- der breaks this emergence and drives localisation, resulting in scattering of electronic and vibrational states and lowering of electronic and thermal transport 4,5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' For strong random disorder, transport is completely stopped and—in the case of electronic properties—a metal-to-insulator transition can occur through Anderson localisation 6,7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Disordered crystals present an interesting problem that, at face value, lies between these two ex- tremes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Disorder is rarely random, and instead many disordered crystals still obey strict local chemical rules that do not result in long-range symmetry breaking 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' In this sense such materials support a ‘hid- den order’ that is not evident in conventional crystallographic analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' A well-known example is the hydrogen-bonding network of water-ice Ih, where periodically-arranged oxygen atoms each are cova- lently bonded to two of four nearby hydrogen atoms to give a non-periodic arrangement of H2O orienta- tions 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Related states have been identified in mixed-anion perovskites 10,11, Coulomb-phase pyrochlores 12–15, and metal–organic frameworks 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' An obvious and important question concerns the nature of col- lective electronic and/or phononic states in such systems: are they similar to those in ordered crystals or more closely related to those of amorphous solids?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Or are they altogether different in character?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' There are strong indications that hidden order may impact material properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Short-range or- der in battery materials can influence ionic conductivities and charge-storage capacities by affecting the networks of mobile ions and vacancies 17,18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Likewise, the nature of phonon broadening in disordered crystals has also been found to vary as a function of the type and extent of short-range order present 19–21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' In the few systems known to exhibit hidden-order transitions—such as the heavy-fermion super- conductor URu2Si2 22,23 and magnetocaloric Gd3Ga5O12 24—the emergence of hidden order couples to thermodynamic anomalies and results in different electronic and/or magnetic behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' What remains entirely unclear is the nature of this link between hidden local order and collective phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' 2 Here we address precisely this problem by exploring the consequences of hidden order on the electronic and vibrational states of a model family of disordered crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' The toy model we study is chosen because there is an obvious mechanism for varying the degree and nature of hidden order it supports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' We begin by introducing this model and explaining our approach for calculating electronic and phononic states for its various realisations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' We then proceed to demonstrate a complex interplay between hidden order and the nature of collective states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' In particular, we report three key findings: (i) that hidden order can be used to selectively broaden specific parts of the electronic or phononic band structure, (ii) that it modulates localisation in different ways, and (iii) that it can result the opening of band-gaps without long-range symmetry breaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' We conclude by discussing generalisations of this toy model, and the relevance of its behaviour to a range of physical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' 2 Results and Discussion Hidden-order model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' A useful toy model for exploring different types of correlated disorder is the two dimensional system A2B, where B atoms occupy a square lattice with A atoms positioned half-way between them (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' 1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' By introducing a distortion such that A atoms form one stronger and one weaker bond to neighbouring B atoms, several distinct types of disorder can be achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' One possibility is for random distortions of A atoms, such as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' 1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' In this case the B atoms will have a varying number of strong and weak bonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' In many real systems, however, there will be local chemical rules that govern the types and geometries of bonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' One such example is for each B atom to have two strong and two weak bonds, which can be satisfied by a large number of configurations, with an example shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' 1c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' These rules are similar to the two-in-two-out rule for hydrogen bonding in ice 9 and this square- lattice representation results in the well-known ‘6-vertex’ statistical mechanical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content='25 Note that there are two types of B atom geometries, where the two strong bonds are either parallel or perpendicular to one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' A stronger chemical rule is then to have only perpendicular strong bonds, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' 1d, equivalent to the square-ice system 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' In this highly-constrained case, the strong A–B bonds are ordered in one-dimensional chains, but there is no three-dimensional bond order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' These distorted systems all have identical average crystal structures and therefore identical Bragg 3 Figure 1: The effect of random and correlated disorder on electronic and phonon bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' a) Ordered A2B structure with A halfway between B atoms on a square lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' b) A random configuration of A site distortions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' c) two-in-two-out rule for correlated distortions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' d) correlated disorder with perpendicular strong bonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' e-h) The corresponding diffuse scattering patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' The square lattice of black dots are Bragg peaks, which are several orders of magnitude stronger than diffuse scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' i-l) Electronic bands for these systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' The energy scale is arbitrary and the zero point does not imply the fermi level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' m-p) Phonon bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' 4 diffraction intensities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' In this sense the presence of additional local order is hidden from conventional crystallographic analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' The clearest signature of this hidden order is through weak diffuse scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' 1e-h shows the single-crystal scattering pattern for each system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' The square grid of black dots indicates the positions of Bragg peaks, which are several orders of magnitude stronger than the weak diffuse scattering lying between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' In the case of the undistorted parent structure (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' 1e) there is no diffuse scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Random distortions give broad diffuse scattering (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' 1f), while the two-in-two- out locally ordered system has characteristic structured diffuse scattering with pinch-points (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' 1g), reminiscent of those found in the scattering of 3D spin-ice with a similar local rule 12,27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Finally, the system with two strong perpendicular bonds has thin lines of diffuse scattering (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' 1h), indicative of long-range one-dimensional correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Collective electronic behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' We explore the effect of varying hidden order on the electronic properties of these models by calculating the electronic band structure using a semi-empirical tight- binding model with nearest neighbour hopping parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Drawing on the conceptual analogy to H and S arrangements in H3S 28,29, we assign to B atoms a set of s, px and py orbitals but only a single s orbital to the A atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' On-site energies and hopping parameters for strong and weak bonds are modelled on the values calculated for H3S, which has 2D layers with similar distortions of H between S on a square lattice 28,29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Using this realistic parameter set allows some general effects to be illustrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' We note that the energy scale used is arbitrary and does not imply the Fermi energy lies at E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Further details of our calculations are given in the supporting information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' The electronic bands depend very strongly on the type and degree of hidden order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' In the ordered state the bands are well-defined in energy and disperse throughout the Brillouin zone with band crossings at the Γ and M points (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content='1i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Random distortions of the A sites change this picture, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' 1j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' While the overall features and general dispersion are very similar to the ordered case, the bands are now much broader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Hence, as anticipated, the electronic states are no longer well-defined in energy and will scatter as a consequence, reducing electronic transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' By introducing the local two-in-two-out rule, significant differences to both the random and ordered 5 cases are found (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' 1k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Now some of the bands have become narrower in energy again, while the gaps below the flat band at Γ and above the low-energy flat band at M have been filled with dilute states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Furthermore, the crossing above the flat band at Γ has lifted and given way to a small band gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Changing the local order to the case of two perpendicular strong bonds per B site leads to very different effects, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' 1l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' The bands are now generally narrow with states well-defined in energy, meaning electronic transport is not as hindered by scattering as in the two other disordered cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' In sharp contrast to the random and ordered systems, the band crossing as Γ and M have now lifted and clear band-gaps are observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' The dilute states filling some gaps in the two-in-two-out system are gone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' There are also very weak additional band-like features between the strong narrow bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' The emergence of band gaps for the two systems with strongest hidden order is conceptually impor- tant because, for the right filling fraction, a variation in hidden order type could lead to a metal–insulator transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Collective vibrational behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' We observe similar effects on the phonon spectrum as a conse- quence of correlations [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' 1m-p].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' In our calculations, phonon energies and eigenvectors are obtained by diagonalising the dynamical matrix using semi-empirical force constants between nearest neighbours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' An arbitrary (but sensible) set of force-constants was chosen to best illustrate the effects, as elaborated in the supporting information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' For reference, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' 1m shows the phonon bands for the ordered system, where acoustic and optical phonons are well-defined in energy with crossing of four bands at the M point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Random distortions again give phonon bands similar to the ordered system but broadened in energy, re- sulting in increased phonon scattering (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' 1n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' The broadening is least evident for the long-wavelength acoustic branches as these are most insensitive to variations in local configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' The behaviour at the M point is now different: the four bands no longer cross as before, but change their dispersion to avoid the crossing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' The locally-ordered two-in-two-out system has some differences to the random system in terms of the band widths (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' 1o), but it is the system with strongest hidden order for which the phonon dispersion is most different (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' 1p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Here, the bands are almost all narrow, and a large band gap 6 has opened throughout the Brillouin zone (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' 1p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Consequently, the type of correlations in disordered structures can also strongly impact properties that depend on vibrations, such as thermal transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' While some interplay between correlated disorder and phonon structure had been reported previously,19,20 a key result of this study is the demonstration that this interplay can be sufficiently strong as to open vibrational band gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Thermodynamic stabilisation of hidden order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' 2 we show the integrated electronic and phonon densities of states, which make clear that the band-gaps seen along high-symmetry directions do indeed persist throughout the entire Brillouin zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Focusing on the emergence of electronic band-gaps we note that the energies of the corresponding valence (low-energy) edge states is reduced in the ‘per- pendicular’ hidden order state relative to the ordered and random cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' This stabilisation implies that, at appropriate filling fractions, the electronic energy of the system can be reduced through a concerted distortion to the hidden order state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Such a transition is conceptually similar to a Peierls distortion, but is fundamentally different in that it proceeds without any global symmetry lowering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Similar mechanisms may be at play in disordered ‘orbital molecule’ states, such as in LiRh2O4 and Li2RuO3,30,31 where the structural distortions associated with valence electron localisation are local and not long-range ordered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content='32 Mode localisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Correlations not only affect the form of the electronic and phonon band struc- ture, but also change the delocalisation of modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' We show this in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' 3 by indicating the degree of delocalisation of electronic and phonon modes weighted by the number of states (see methods for de- tails).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Taking each diagram in turn, we begin by noting that in the ordered system (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' 3a), electronic bands with dispersion are generally quite delocalised, while flat bands are localised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' In the randomly distorted system (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' 3b), all bands have become more localised—as anticipated for disordered sys- tems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' The two-in-two-out rule gives rise to intermediate behaviour (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' 3c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' But, most surprisingly, the most strongly correlated state (perpendicular strong bonds), gives delocalised states (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' 3d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Similar changes are observed for the phonon modes (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' 3e-h), for which the key difference is the resilience of delocalisation within the long-wavelength acoustic branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' The variance in degree of localisation is clearly exemplified by interrogating representative modes 7 Figure 2: Density of states plots for (a) electronic modes and (b) phonon modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Hidden order can induce band gaps that are not present in randomly disordered systems in real-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' 4a-h shows two examples of electronic modes for the different systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' The orbitals are coloured according to the wavefunction phase, while corresponding saturation is given by the wave- function amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Whenever modes are localised, atoms do not contribute equally to the wavefunction, which fragments into small coherent regions incoherent with respect to one another (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=', Fig 4b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' We provide a detailed interpretation of these images in the supporting information, but highlight for in- terest here the unusual behaviour shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' 4h for the system with perpendicular strong bonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' This particular mode is completely delocalised with strong coherence along chains but mixing from chain to chain, causing chains to have phase shifts relative to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' This is in contrast to the ordered, 8 Figure 3: Weighted delocalisation of electronic (a-d) and phonon modes (e-h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' random and two-in-two-out systems, where the corresponding modes are all localised to a large extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' In a similar way, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' 4i-p shows the real-space representations of two types of phonon modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Here colours indicate displacement direction—further highlighted by arrows—while saturation gives the cor- responding amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' The two vibrational modes illustrate how correlations can have different effects on modes, opening up the possibility of selectively (de)localising modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' These phonon modes are fur- ther discussed in the supporting information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' To summarise, we find that hidden order not only affects the density and coherence of states, but also their degree of delocalisation—often in quite nuanced and unexpected ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Extension to three dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' In the supporting information, we include a discussion of the extension of our approach to three dimensions (3D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' The results are qualitatively the same as for two dimensions, albeit with some additional subtleties and avenues for control given the increased scope for geometric isomerism in 3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' 9 Figure 4: Real-space view of modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' (a-h) Two types of electronic modes for the different systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Colour hue indicates the phase of the wave-function, while saturation indicates the amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' (i-p) Two types of phonon modes for the different systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Colour hue indicates direction of motion with saturation indicating amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Arrows inside atoms further illustrate the movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' 10 3 Concluding Remarks Perhaps our key result has been to show clearly that correlations in disordered crystals have consequences for properties, as both electronic and vibrational modes are impacted in functionally-important ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Hence control over correlations offers a new handle with which to tune properties in functional materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Moreover, because hidden order affects electronic and vibrational states in subtly different ways, it may prove possible to combine the effects of both to engineer functional materials with particularly desirable properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' We offer a handful of examples to demonstrate this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' One topical family is that of thermoelectric materials, where the design brief is to combine a low thermal conductivity with large electrical conductivity in a gapped semiconductor, as captured by the phonon–glass–electron–crystal paradigm 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' The conventional approach is to introduce disorder into a subset of atoms that do not contribute to electronic conductivity 34–36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' This is a design principle based on the idea of disorder being random and creating strong scattering of modes, which is why disorder on the substructure responsible for electronic conductivity is to be avoided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' But our present study suggests an entirely new design strategy of introducing specific kinds of hidden order that at once broaden heat- carrying phonon modes whilst preserving narrow electronic modes in the conduction band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Additionally, one might even use correlated disorder to tune the electronic band gap so as to optimise thermoelectric performance 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' In this context, we note that thermoelectric half-Heusler materials can be made with different local vacancy orderings but with identical average crystal structures and stoichiometries 38, indicating the possibility for tuning this class of materials through the concepts presented here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' The effect of disorder on topological insulators (TI) is a problem of strong currency in the field of functional materials design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Topological insulators are insulating in the bulk but host conducting gapless edge or surface states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' These gapless states are topologically protected and are robust against weak disorder 39,40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' For strong disorder the non-trivial topological states can break down due to localisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' However, in some systems, strong disorder causes phase transitions from topologically-trivial to non- trivial states, such as topological Anderson insulators (TAIs) 41,42 or disorder-induced topological Floquet insulators 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Quantised topological invariants are related to symmetries, but these can be broken in 11 strongly disordered crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' However, it has been shown that symmetry-stabilised topological invariants are still strictly quantised even in the presence of disorder that breaks symmetries locally yet restores them on average 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Since TI phases are robust to disorder, the disorder itself can be used to further engineer their band structures 45,46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' As we have shown here, the hidden order present within correlated disordered states can be used to control band structures, underlining the importance of understanding correlations in disordered TIs whilst also offering a new mechanism for tuning TI materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' We note that several topological insulator materials have been found experimentally to be disordered crystals 47,48— albeit that the nature of this disorder is not well understood, since earlier studies have only analysed average structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' The same principles might be used to engineer band gaps and transport properties of photovoltaics, and—in principle—combining effects of electronic and phonon band structures could tune electron– phonon coupling in superconductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' In an entirely different field, we anticipate that the link we demonstrate between hidden order and gap opening may have implications for the design of disordered photonic materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' The relatively recent demonstration of optical transparency in hyperuniform structures has shown that subtleties of disordered networks can have fundamentally important effects on optical band structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content='49 Likewise, control over the degree of short-range order has emerged as an unexpected design strategy for controlling visual appearance in photonic matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content='50 To the best of our knowledge, the concept of introducing hidden order within an otherwise-crystalline photonic medium as a means of introducing transparency has not yet explored, and may offer interesting new approaches for controlling matter–light interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' As a final point, we note that, because phases with different types of hidden order can have sig- nificantly different properties, it is more important than ever to develop experimental tools for probing hidden order in crystalline materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' The Bragg diffraction techniques used to determine crystal struc- tures are sensitive only to long-range order, which is why it is often only the average structure of materials that is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' By contrast, diffuse scattering is sensitive to local correlations, but is several orders of magnitude weaker than Bragg scattering—this has limited its use historically 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' The development of 12 modern detectors and high-intensity x-ray, neutron and electron sources have now made it feasible to measure diffuse scattering much more routinely, allowing for identification of distinct locally-ordered phases 17,18,38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' 4 Acknowledgements N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' acknowledges the Independent Research Fund Denmark (DFF) for funding through the Interna- tional Postdoctoral grant 1025-00016B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' thanks A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Overy (Oxford), M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Tucker (SNS), A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Simonov (ETH Zurich), and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Romao (ETH Zurich) for discussions, and the European Research Coun- cil for funding (Grant 788144).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Methods Electronic states are calculated from supercell configurations using a semi-empirical tight binding model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Taking φi as the ith atomic orbital in the supercell, a basis of Bloch sums for wavevector k is Φik = 1 √ N � tm eik(tm+vi)φi(r − tm − vi), (1) where tm is the position of the mth supercell origin, vi is the position of the ith atomic orbital in the supercell, N is the number of supercells, and r is the real-space coordinate vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' In the tight-binding approximation, the Hamiltonian then takes the form 52: Hijk = � τ eikτγijτ + δijE0i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' (2) Here, τ are the vectors between atomic orbital i and j with nonzero matrix elements γijτ = ⟨φi(r)|H|φj(r− τ)⟩ , δij is the Knonecker-delta and E0i the energy of orbital i on an isolated atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Here τ is limited to nearest-neighbours only and the matrix elements γijτ are given semi-empirical values for the different types of orbital combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' In the present case one type of atom is given one s orbital and the other type one s and a set of p orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' The needed parameters in the present case are a set of four values comprised of γssσ, and |γspσ| for short and long bonds, as well as parameters for E0i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Directional depen- dence is taken into account using γspσ = lx|γspσ| for an s to px element, where lx is the x-component 13 of the normalised τ vector, and similarly the s to py and s to pz depend on ly and lz, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' p to s orbital elements obey γpsσ = −γspσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content='52 All other matrix elements are zero in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' In other cases, more matrix elements would be needed, such as the γppσ, γppπ Using a custom python script the Hamiltonian is constructed and diagonalised to obtain the eigen- vectors and eigenvalues of the system at different k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' The bands are then unfolded to the Brillouin Zone of the primitive cell by calculating the weight of each state as 53: Wk = 1 No � o∈PC �� i∈o c∗ ik � �� i∈o cik � (3) The sum o ∈ PC are over the different orbitals of the primitive cell, and the sum i ∈ o are those orbitals in the supercell which are equivalent in the primitive cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' cik are the coefficients of the normalised eigenvectors in the Bloch sum basis and No the number of orbitals in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' The number of states per cell for each mode is then given as 2N0∈PCWk, where 2N0∈PC is the number of orbitals in the primitive cell and the factor of two takes into account the spin degree of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' The weighed degree of delocalisation of each mode, Dk is calculated as 54: Dk = Wk/ � i |ci|4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Phonon modes are calculated in a very similar way by constructing and diagonalising the mass- adjusted dynamical matrix from the eigenvalue equation: DU = ω2U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' (4) Here D is the mass-adjusted dynamical matrix, U is the eigenvector of mass-adjusted elementary move- ments and ω the energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' The method for phonon calculations follow that given in detail in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Elements of D are given by Dijk = 1 √maimaj � τ eikτKijτ, (5) where i and j now reference the elementary movements of all atoms in the supercell along cartesian axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' mai is the mass of the atom to which the ith elementary movement belongs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Kijτ is the force constant between elementary atomic movements i and j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' The diagonal elements Diik need to conserve force balance: Diik = −1/mai � j̸=i Kij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Again, only nearest neighbours are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Two types of 14 force-constants are used: K⊥ and K∥ for perpendicular and parallel movements of nearest neighbours, with two possibilities for short and long bonds for each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' The phonon bands are unfolded in the same way as for the electronic bands using Wk = 1 Nu � u∈PC �� i∈u c∗ ik � �� i∈u cik � , (6) where u are the elementary displacements in the primitive cell, Nu the number of elementary displace- ments in the supercell and cik the coefficients of the normalised eigenvectors of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' The weighed delo- calisation is then calculated in the same way as for the electronic modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' The electronic and phononic band structures were calculated on configurations with 32 by 32 atoms and averaged over 30 different configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' For the electronic bands values were chosen to be close to those calculated for H3S, as to keep them realistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' This was done using the minimal tight binding model from 28, where values for orbital energies are taken relative to the sulphur s level, with E0Ss = 0, E0Sp = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content='16 eV and E0Hs = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content='42 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' The hopping elements used for the 2D simulation were rounded to nearest integer values, γssσ = −5 and −3 eV for strong and weak bonds, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Similarly, |γspσ| = 6 and 4 eV were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' For the 3D systems shown in the SI, the exact values for H3S given in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' 28 were used for configurations with 16 atoms along each dimensions and averaged over 10 configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' These are γssσ = −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content='69 and −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content='98 ev and |γspσ| = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content='69 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content='3 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' For the phonon calculations parameters were chosen to give clear band structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' In the 2D systems, the masses for the two types of atom were mA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content='8 and mB = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Values for the force constant were chosen as K∥ = −2 and −1 for strong and weak bonds, respectively, as well as K⊥ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content='6 and −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' For the 3D systems presented in the SI, values used are mA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content='75, mB = 1, K∥ = −2 and −1 , and K⊥ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content='4 and −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' In general, the averaged values for strong and weak bonds were used for the calculation of the ordered system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' The Diffuse scattering intensity is calculated using the Scatty software 56, using configurations with 60 by 60 atoms and averaged over 100 different configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' 15 References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Bloch, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' ¨Uber die quantenmechanik der elektronen in kristallgittern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' 52, 555–600 (1929).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Debye, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Zur theorie der spezifischen w¨armen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' 344, 789–839 (1912).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Born, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' & Von Karman, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Vibrations in space gratings (molecular frequencies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' 13, 297–309 (1912).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Mott, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} 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Physics (Elsevier Science, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Deretzis, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=', Calogero, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=', Angilella, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' & La Magna, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Role of basis sets on the unfolding of supercell band structures: From tight-binding to density functional theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Europhys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' 107, 27006 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Viola, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' & Brown, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Generalized entanglement as a framework for complex quantum systems: purity versus delocalization measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' A: Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' 40, 8109 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Willis, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' & Pryor, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Thermal Vibrations in Crystallography (Cambridge University Press, 1975).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Paddison, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Ultrafast calculation of diffuse scattering from atomistic models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Acta Cryst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' A 75, 14–24 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' 20 Supplementary Information 1 Powder Diffraction Patterns Figure SI 1 shows the calculated powder diffraction patterns for the three different disorder types in the 2D system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' The upper part of the figure shows the full scale of the pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' As these systems all have identical average crystal structures, the intensities of the Bragg diffraction peaks are identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' The difference in scattering comes in the diffuse signal, which is much weaker and not generally visible on the same scale as the Bragg peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' In the lower sub-figure, the intensities have been multiplied by 100 to zoom in on the diffuse scattering, showing their differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content='The noise seen in the diffuse scattering is due to a limited calculation size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Calculations were made by averaging over 100 configurations with each 60x60 atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' To increase contrast of the diffuse scattering, Li was used as A atoms with O as B atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' As shown in the main paper, single crystal scattering can be a better way to identify these phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Supplementary Information 2 Real space view of states Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' 4a-d shows a low-energy electronic mode between Γ and M composed of bonding s-orbitals from both atom types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' In the ordered case the mode is delocalised with contributions from all s-orbitals, and the phase of the wavefunction changes slowly along the wavevector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' For the randomly disordered system, this mode has now become localised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Atoms no longer contribute equally to the wavefunction, which consists of small coherent regions that are incoherent with respect to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' For the two-in-two- out correlations, the mode is more delocalised, but not fully, and there is still incoherence in the phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' The mode becomes fully delocalised and coherent in the system with perpendicular strong bonds, and is similar to the ordered case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' The same is not true for the mode shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' e-h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' This mode is on the flat band between M and X, consisting of the B atom p orbitals and A atom s orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' In the ordered system the band is localised on a single chain, and there is no mixing between chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' With random distortions there is mixing between directions but the mode is still localised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' For this mode the two-in-two-out system is close to the random system, which was not the case for the first mode shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' However, for the system with perpendicular strong bonds, the mode is completely delocalised with strong coherence along chains in one dimension and mixing between chains along the other dimension, causing chains to have 21 Figure SI 1: Calcualted Powder X-ray Diffraction patterns for the two-dimenstional systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Upper figure shows the full scale, while the lower figure shows the weak diffuse scattering signal by scaling by a factor of 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' The noise seen in the diffuse scattering is due to a limited calculation size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' phase shifts relative to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Thus, the electronic wavefunction can differ significantly depending on correlations, and in some cases the wavefunction can take on unique shapes not seen in ordered or randomly disordered systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' 22 Similar behaviour is seen for the phonon modes (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' 4i-p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Here each atom is drawn as a circle and the colour now indicates the direction of movement while the saturation is again the amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Arrows inside the circles highlight the displacement vectors further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Fig 4i-l show a transverse acoustic shear mode between X and Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' In the ordered system the mode is delocalised with layers three atoms wide moving out of phase with respect to each other, separated by a single layer of stationary A atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' In the random system this mode becomes more localised with coherent regions that are incoherent with respect to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' With the two-in-two-out correlations, the mode becomes more delocalised and consists of almost coherent layers moving as in the ordered case, but with disorder in the amplitude and directions of some atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' In the system with perpendicular strong bonds, the mode becomes delocalised with layers moving like in the ordered case, but atoms between the layers are no longer stationary and move incoherently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' This is in contrast to the optical mode shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' 4m-p, where the ordered system and the system with perpendicular strong bonds are both delocalised and coherent, while the random and two-in-two-out systems have incoherent localised modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Supplementary Information 3 Orbital contributions to states Figure SI 2 shows the orbital contributions weighed by the number of modes in the 2D case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Supplementary Information 4 Three dimensional example The same type of effects that were observed in 2D will also apply to three-dimensional systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' As an example, a 3D analogy to the 2D system is presented, based on a cubic A3B structure with A atoms positioned between B atoms forming a simple cubic lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' With A halfway between B sites, the system is ordered (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' SI 3a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' If A atoms distort to form one strong and one weak bond to B sites, several disordered configurations can be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' With random distortions a wide variety of strong and weak bonds around B sites are generated (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' SI 3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' If instead each B site has the chemical rule of 3 strong and 3 weak bonds, a three-in three-out structure is obtained, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' SI 3c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' In this structure there is a mixture of B sites with the 3 A atoms coordinated meridionally (mer) and facially (fac).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Stronger chemical rules are then that the B atoms form only one of these, which in both cases leads to disordered 23 Figure SI 2: Weighed orbital contributions to the electronic states in the 2D systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' systems shown in Fig SI 3d and SI 3e for mer and fac geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Thus, in three dimensions more different types of short-range order can be generated due to the increased flexibility of another dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Diffuse scattering patterns for these disordered systems are very different, while their Bragg diffrac- tion intensities are identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' SI 3f-j shows the diffuse scattering down in the [111] plane of reciprocal space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' While the ordered system has no diffuse scattering and the random disorder has broad smeared out diffuse scattering, the systems with local correlations have structured scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' In the three-in three-out case pinch-points are again seen, reminiscent of other systems with similar rules 12,27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' In the meridional system the diffuse scattering has condensed further to form ring-like features with several maxima, while the facial system has strong narrow lines indicating long-range correlations of the disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' 24 Figure SI 3: Effects different disorder correlations on electronic and phonon bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' a) Ordered A3B structure with A (blue) halfway between B (grey) on a simple cubic lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' b) A random configuration of A site distortions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' c) three-in three-out rule producing a mixture of meridional and facial geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' d) meridional geometry only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' e) facial geometry only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' f-j) Corresponding diffuse scattering patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' The hexagonal grid of black dots are Bragg peaks, which are several orders of magnitude stronger than diffuse scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' k-o) Electronic bands for these systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' The energy scale is arbitrary and the zero point does not imply the fermi level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' p-t) Phonon bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Similar trends to the two-dimensional system are found in the electronic (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' SI 3k-o) and phonon bands (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' SI 3p-t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Here the electronic bands are calculated using parameters for the R-3m phase of H3S 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' Phonon modes are calculated using parameters chosen to highlight effects of correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' While 25 1random disorder tends to strongly broaden features that were present in the ordered system, correlated disorder changes the band dispersions and broadening, opening and closing band gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' In the electronic bands (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' SI 3k-o) the low-energy band crossings at the R and Γ points can be opened to different sizes for the different types of correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' The meridional system introduces a new band-like feature with a maximum at the R point in the electronic structure as well as new weak band-like features in the phonon modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' The facial system induces new band gaps for both electronic and phonon modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} +page_content=' 26' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdA0T4oBgHgl3EQfFv8F/content/2301.02035v1.pdf'} diff --git a/ZNE0T4oBgHgl3EQfnAEV/content/tmp_files/2301.02505v1.pdf.txt b/ZNE0T4oBgHgl3EQfnAEV/content/tmp_files/2301.02505v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..8ce25eb82a54c64da23072a039fcacf79b8626af --- /dev/null +++ b/ZNE0T4oBgHgl3EQfnAEV/content/tmp_files/2301.02505v1.pdf.txt @@ -0,0 +1,6113 @@ +UNSUPERVISED ATTACK PATTERN DETECTION IN HONEYPOT DATA +USING BAYESIAN TOPIC MODELLING +BY FRANCESCO SANNA PASSINO1,a +, ANASTASIA MANTZIOU2 +, +DANIYAR GHANI1 +, PHILIP THIEDE1, ROSS BEVINGTON3, +AND NICHOLAS A. HEARD1 +1Department of Mathematics, Imperial College London, London (United Kingdom), af.sannapassino@imperial.ac.uk +2The Alan Turing Institute, London (United Kingdom) +3Microsoft Threat Intelligence Center (MSTIC), Cheltenham (United Kingdom) +Cyber-systems are under near-constant threat from intrusion attempts. +Attacks types vary, but each attempt typically has a specific underlying in- +tent, and the perpetrators are typically groups of individuals with similar ob- +jectives. Clustering attacks appearing to share a common intent is very valu- +able to threat-hunting experts. This article explores topic models for cluster- +ing terminal session commands collected from honeypots, which are special +network hosts designed to entice malicious attackers. The main practical im- +plications of clustering the sessions are two-fold: finding similar groups of +attacks, and identifying outliers. A range of statistical topic models are con- +sidered, adapted to the structures of command-line syntax. In particular, con- +cepts of primary and secondary topics, and then session-level and command- +level topics, are introduced into the models to improve interpretability. The +proposed methods are further extended in a Bayesian nonparametric fashion +to allow unboundedness in the vocabulary size and the number of latent in- +tents. The methods are shown to discover an unusual MIRAI variant which +attempts to take over existing cryptocurrency coin-mining infrastructure, not +detected by traditional topic-modelling approaches. +1. Introduction. +The increasing reliance of enterprises on information technologies, +such as cloud services, gives rise to new challenges for protecting customer data and com- +puter systems from intrusions. To tackle these cyber threats, enterprises increasingly resort +to quantitative methods for the development of the next-generation intrusion detection tech- +niques. Honeypots play an important role in the detection and understanding of attacker be- +haviours. A honeypot is a host located within a computer network designed to entice ma- +licious attackers. Security teams use the commands issued by attackers during interactive +sessions with the honeypot, as well as other meta-data such as the source IP address, in order +to understand the attack and the attacker’s intent to better protect their networks from com- +promise. Honeypots therefore provide cyber analysts with session data, where each session +is comprised of multiple commands issued by the user; each command can be interpreted as a +sequence of instructions in command language (for example, shell programming languages), +similar to words in natural language. +Honeypot session data provide a rare insight into the operational techniques of cyber at- +tackers, such as their automated or interactive nature, the individual scripting styles and their +overall objectives. This makes honeypot tracking systems particularly attractive for devel- +oping robust quantitative methods for cyber-security (Highnam et al., 2021). The volume of +traffic passing through a honeypot can be surprisingly high, and so automating the under- +standing of these sessions, classifying them and detecting new emerging patterns provides a +challenging research problem which is addressed in this article. +Keywords and phrases: model-based clustering, statistical cyber-security, topic modelling. +1 +arXiv:2301.02505v1 [cs.CR] 6 Jan 2023 + +2 +Typically, attackers have one main objective after gaining access to a network host. For ex- +ample, an intruder might want to infect the machine with ransomware, build a cryptocurrency +miner, take over existing infrastructure, copy information for data leakage or sale, or collect +intel about the organisation. Therefore, each observed session could be thought to have an +underlying latent intent. Importantly, such intents evolve and change over time, creating new +threats for the security of cyber-systems. From a statistical perspective, the problem of es- +timating latent intents from a collection of attempted attacks can be framed as a clustering +task. Hence, the main objective of this work is to develop clustering models for command +line data observed in cyber-security applications. Such clustering models could then be used +for automated online classification of network intrusions, providing a valuable tool for threat +experts and enterprises to discover underlying patterns that would have not been easily de- +tectable otherwise. Automated threat detection can be viewed as complementary to deter- +ministic classification frameworks for enterprise attacks (for example, MITRE ATT&CK1), +providing a further level of sophistication to attack pattern detection. +The analysis of attacker behaviour from command logs has been mainly studied from a +machine learning perspective in the literature (Shrivastava, Bashir and Hota, 2019; Crespi +et al., 2021; Sadique and Sengupta, 2021). In the present work, ideas borrowed from the lit- +erature on topic modelling in text analysis are used to to detect attack patterns, with sessions +playing the role of documents and commands playing the role of sentences. Command line +instructions are modelled under a bag-of-words assumption, leading to a generative model for +the instructions dependent on the latent intent characterising the corresponding session. This +fundamentally differentiates our approach from mixed membership strategies to language +modelling, such as Latent Dirichlet Allocation (LDA, Blei, Ng and Jordan, 2003). The draw- +backs of LDA for the scope of modelling command lines are threefold: (i) attackers usually +have mainly one intent per session; (ii) interpreting the results of a mixed-membership model +for attack pattern detection is complex for analysts and threat experts; (iii) models based +on LDA often present unidentifiability and convergence difficulties, making reproducibility +of results problematic. Such difficulties are addressed in this work, presenting an approach +that assigns a single topic, or intent, to each session, providing a single label to threat ex- +perts which is then easy to interpret through statistical summaries of sessions assigned to the +same group. Furthermore, one class of proposed models incorporate the additional idea of +command-level intents, establishing a two-level clustering structure. This approach appears +to alleviate the convergence issues observed with LDA on session data. +Later models and inferential procedures discussed in this work admit the possibility of an +unknown and unbounded number of latent intents and an unbounded vocabulary size. These +extensions are particularly important in computer network security, since attack vectors fre- +quently evolve meaning new intents to arise, and new command line instructions will appear. +The number of topics in LDA models is usually chosen using scree-plot criteria using the +perplexities calculated from a holdout dataset (Teh et al., 2006). However, optimising for +perplexity might not yield interpretable topics (Ding, Nallapati and Xiang, 2018). In this +work, an alternative strategy based on Bayesian hierarchical nonparametric Griffiths-Engen- +McCloskey priors (GEM, Pitman, 2006) is used, admitting the possibility of previously un- +observed intents and instructions. +The rest of the article is structured as follows: in the remainder of this section, the data +sources used in this work are described along with a review of the related literature. Sec- +tion 2 describes models for session data, and Section 3 presents inferential procedures. The +methodology is then extended to the cases of unbounded numbers of topics and vocabulary +size in Section 4. Finally, the proposed methods are applied to real-world session data from +honeypots in Section 5, and the practical implications of the results are discussed. +1For more details, see https://attack.mitre.org/. + +UNSUPERVISED ATTACK PATTERN DETECTION IN HONEYPOT DATA +3 +1.1. Honeypot session data. +When a user connects to a honeypot through certain pro- +tocols, a session starts, and every action the user performs on this host is recorded until +logout, when the session ends. A user will run a sequence of commands, which are strings +of code which perform actions on the host. Each command comprises a sequence of words +drawn from the syntax of the chosen protocol. In the following example session, the in- +truder first attempts to access a convenient directory (through multiples uses of the cd com- +mand), then tries three methods of downloading a bash script from the web (wget, curl +and tftp get, representing different commands having the same underlying intention), +before attempting to execute and delete the script. The real IP address which was used in the +attack is masked using the string abc.def.ghi.jkl. +cd /tmp || cd /var/run || cd /mnt || cd /root || cd / +wget http://abc.def.ghi.jkl/Zerow.sh +curl -O http://abc.def.ghi.jkl/Zerow.sh +chmod 777 Zerow.sh +sh Zerow.sh +tftp abc.def.ghi.jkl -c get tZerow.sh +chmod 777 tZerow.sh +sh tZerow.sh +rm -rf Zerow.sh tZerow.sh +Note that transforming commands into sequences of words, known as tokenisation in +the literature, is not a trivial task in cyber-security. For example, consider the web address +http://abc.def.ghi.jkl/Zerow.sh, which appeared in the second command of +the session. One might consider the entire string as a word, or split it into different words, +such as http, abc.def.ghi.jkl and Zerow.sh. Furthermore, the entire Internet Pro- +tocol (IP) address abc.def.ghi.jkl could be considered as a word, or just its subnet +abc.def, for example. Similarly, Zerow could be considered as an individual word, ex- +cluding the file extension .sh. More details around the preprocessing of session data will be +given in Section 5.1. +1.2. Related literature. +In topic models, each document is usually considered as a bag- +of-words, and the words are assumed to be exchangeable. Under this assumption, the in- +formation carried by paragraphs and sentences in natural language is lost. In cyber-security, +documents correspond to sessions and sentences correspond to commands, which are ex- +pected to have a specific intent. For attack pattern detection, it would be informative to also +capture such latent intents at the command-level. In the literature, document and sentence +clustering have been considered as two independent problems. +The problem of clustering documents has been extensively studied in the natural language +processing, computer science and information retrieval communities (for a survey, see Ag- +garwal and Zhai, 2012, and references therein). Common approaches include matrix factori- +sation techniques (Xu, Liu and Gong, 2003) and spectral clustering (Cai, He and Han, 2011). +Furthermore, Wallach (2008) proposed a cluster-based topic model extending LDA, where +each group is assigned a cluster-specific Dirichlet prior on the document-specific topic dis- +tribution. Xie and Xing (2013) also propose a multi-grain topic model with clustering where +documents are assigned global and group-specific topics. +Sentence-level structure within topic models has been largely overlooked in the literature. +Balikas, Amini and Clausel (2016) propose to extend LDA by sampling words from sentence- +specific topic distributions. Furthermore, Jiang et al. (2019) propose to model the sentence- +specific topic distribution as a mixture between the topic distributions of adjacent sentences, +weighted by a topic association matrix. In the present article, a new framework is proposed +which permits to joint inference of the latent structure for both sessions and commands. + +4 +Usually, one of the main difficulties for LDA models is topic interpretability. Sparse topic +models (Williamson et al., 2010; Archambeau, Lakshminarayanan and Bouchard, 2015; +Zhang, 2020) alleviate this issue by enforcing sparsity in the topic-specific word distribu- +tions. Doshi-Velez, Wallace and Adams (2015) proposed Graph-Sparse LDA, that used rela- +tionships between words to improve interpretability. Also, the performance of LDA methods +heavily relies on suitable preprocessing of the data. For example, high-frequency words are +often removed, under the assumption that such words make limited contributions to the mean- +ing of the documents. In order to avoid data pruning, alternative term-weighting schemes +have been proposed in the literature (Wilson and Chew, 2010). Here a further possible so- +lution is proposed: a secondary topic, shared across all documents, can be used to capture +high-frequency words and lead to more interpretable primary topics characterising individ- +ual documents. +Another possible explanation of the issues of LDA with high-frequency terms is that, in +natural language, word counts have a power-law distribution (Sato and Nakagawa, 2010). +Therefore, Sato and Nakagawa (2010) proposed a Pitman-Yor LDA model, which admits +power-laws by construction. Another approach to the problem of modelling power-laws is +the latent IBP compound Dirichlet Allocation model (Archambeau, Lakshminarayanan and +Bouchard, 2015). It is unclear whether power-laws apply to the word counts in command +line data and cyber-security applications. Such structures could be easily accounted for in the +methodology proposed in this paper via two-parameter GEM prior distributions, correspond- +ing to stick-breaking proportions of a Pitman-Yor process (Pitman, 2006). +Cyber-security applications require the number of latent intents and vocabulary to be un- +bounded for practical deployment. Hierarchical Dirichlet Processes (Teh et al., 2006) have +been successfully used within the context of LDA models to admit an unbounded number +of topics. Furthermore, Zhai and Boyd-Graber (2013) developed an online LDA algorithm +with unbounded vocabulary size, proposing a multinomial and n-gram prior distribution for +a conventional character language. However, n-grams tend to suffer from data sparsity issues +(Allison, Guthrie and Guthrie, 2006). In this work, the words are simply interpreted as to- +kens, and therefore a GEM prior is employed instead, corresponding to a prior distribution +over the natural numbers. This strategy avoids the difficulty of specifying a prior distribution +on the command line syntax, which would be an undesirable additional task. Furthermore, +GEM priors are also assigned to the number of latent topics. It must be remarked that the +task of estimating the number of components in a finite mixture model presents issues with +consistency both under finite parametric (Cai, Campbell and Broderick, 2021) and nonpara- +metric priors (Miller and Harrison, 2013, 2014) under model misspecification, unless a prior +distribution on the concentration parameter of the Dirichlet process is appropriately specified +(Ascolani et al., 2022). Therefore, in practice, it is not expected to always recover the exact +number in the data generating process. +In cyber-security, command line data have been mainly analysed using two different ap- +proaches. The first class of studies uses machine learning techniques to understand attacker +behaviour from command logs. Notably, Sadique and Sengupta (2021) aim to predict the +next command of the attacker by using an edit distance training model on the sequence of +commands input. In a similar setup, Crespi et al. (2021) aim to identify attacker behaviours +from command logs using supervised NLP methods. Lastly, Shrivastava, Bashir and Hota +(2019) focus on classifying types of attacks from commands using a series of machine learn- +ing techniques. The second class of approaches analyses attacker behaviour from session +data using Hidden Markov Models, as seen in the studies of Rade et al. (2018) and Desh- +mukh, Rade and Kazi (2019). However, none of the aforementioned studies consider topic +modelling approaches for the analysis of sessions. + +UNSUPERVISED ATTACK PATTERN DETECTION IN HONEYPOT DATA +5 +2. Models for clustering session data. +Command line data are observed in sessions, +where each session is divided into commands, and each command is composed of different +words drawn from a vocabulary V . Following the standard LDA model (Blei, Ng and Jordan, +2003), for D observed sessions the number of commands Nd in session d and the number +Md,j of words in each command j within that session are assumed to be Poisson distributed: +Nd ∼ Poisson(ζ), d = 1,...,D, +Md,j ∼ Poisson(ω), j = 1,...,Nd, +ζ,ω ∈ R+. The i-th word in the j-th command of the d-th session is denoted wd,j,i, which +has a corresponding probability mass function ξd,j,i ∈ R|V | ++ over the vocabulary V . +The stated aim is to develop clustering algorithms for sessions, where the clusters represent +shared intents of the intruders, or groups of attackers with similar behaviour. To achieve this +aim, a range of topic model structures are now considered, establishing shared distributions +ξd,j,i across groups of sessions and commands to identify clusters. In particular, this work +focuses on two approaches: (i) Constrained: Each session has a primary topic and a global +secondary topic; (ii) Hierarchical: Each session topic is a distribution on command-level top- +ics, which introduces two layers of latent topics. The two modelling approaches are discussed +in detail in the next sections. +2.1. Constrained topic modelling with primary and secondary topics. +As a most basic +approach, each document d could have a latent assignment to one of K possible topics, +where each topic k ∈ {1,...,K} is characterised by a probability mass function φk on the +vocabulary V . Let td denote the topic assignment for document d, and let λ = (λ1,...,λK) +where λk denotes the probability that td = k. It could then be assumed ξd,j,i = φtd, such that +conditional on the session-specific topic td, all the words wd,j,i in session d are sampled from +the same distribution φtd. Assuming conjugate Dirichlet prior distributions for the probability +distributions λ and {φk} implies the following model: +λ ∼ Dirichlet(γ), +φk ∼ Dirichlet(η), k = 1,2,...,K, +td | λ ∼ Categorical(λ), d = 1,...,D, +wd,j,i | td,{φk} ∼ Categorical(φtd), i = 1,...,Md,j, j = 1,...,Nd, +(1) +where γ ∈ RK ++,η ∈ R|V | ++ . In the cyber-attack context, these topics correspond to different +intents. +The simple model above can be used as a starting point for exploring more complex clus- +tering structures. For example, to better identify differences between topic-specific distribu- +tions, it could be assumed that words in a document are sampled either from a topic-specific +probability distribution, or from a baseline probability distribution shared across all docu- +ments. The shared distribution represents words that are commonly used in all sessions, but +are not key for characterising the intent of a session. For example, in natural language, such +a baseline distribution could give probability mass to conjunctions (e.g. but, and, if), articles +(e.g. a, an, the), or pronouns (e.g. she, he, they). Similarly, for command lines in a cyber- +security context, the shared distribution might give weight to common bash commands such +as ls (list contents), ps (list the running processes), or cd (change directory). The shared +distribution will be used to make the session-specific topics more representative of the at- +tacker’s intents, excluding commonly used words. + +6 +In particular, an extended model assumes each topic has an associated probability θk ∈ +[0,1], k = 1,...,K, representing the mixing proportion between the topic-specific word dis- +tribution φk and the shared distribution φ0. Each word is then sampled with probability θtd +from φtd, or from φ0 with probability 1 − θtd, implying the following revised model: +φk ∼ Dirichlet(η), k = 0,1,2,...,K, +θk ∼ Beta(αk,α0), k = 1,...,K, +zd,j,i | td,{θk} ∼ Bernoulli(θtd),i = 1,...,Md,j, j = 1,...,Nd, +wd,j,i | zd,j,i,td,{φk} ∼ Categorical(φtdzd,j,i), i = 1,...,Md,j, j = 1,...,Nd,. +(2) +This model essentially imposes a sparsity constraint on LDA, by assuming that each docu- +ment contains only two topics: (i) a primary topic td, chosen from K primary topics, and (ii) a +secondary topic shared across all documents, denoted “topic 0” for notational convenience. +2.2. Hierarchical constrained topic modelling with session-level and command-level clus- +tering. +The models in (1) and (2) assume that words in each command within a given session +are sampled from the same topic-specific distribution, or from a distribution shared across +documents. The information about the structure of a session as a sequence of commands is +therefore ignored, which might be limiting in practical settings. Instead, it would be reason- +able to assume that session-specific intents share similar commands for specific tasks. Such +tasks could be interpreted as command-level intents, and the distribution of the tasks char- +acterises the session-level topic. Let H be the assumed number of command-level topics. It +could be assumed that each session-level topic has an associated H-dimensional probability +distribution ψk across command-level intents, with each command within a given session +being assigned a command-specific topic sd,j ∈ {1,...,H} sampled from ψtd. Conditional +on sd,j, the words in the command are then sampled independently from a |V |-dimensional +probability distribution φsd,j, specific to the command-level topic. Therefore, the model in +(1) is extended as follows: +ψk ∼ Dirichlet(τ), k = 1,2,...,K, +φh ∼ Dirichlet(η), h = 1,2,...,H, +sd,j | td,{ψk} ∼ Categorical(ψtd), j = 1,...,Nd, +(3) +wd,j,i | sd,j,{φh} ∼ Categorical(φsd,j), i = 1,...,Md,j, j = 1,...,Nd, +where τ ∈ RH ++. In this model, there are two layers of topics, and corresponding indices: +(i) Command topic indices, sd,j, used to match the words in the corresponding command +to distributions φ1,...,φH over V ; (ii) Document topic indices, td, used to match the com- +mands in the corresponding session to distributions ψ1,...,ψK over the command-level top- +ics. Letting Φ be the H × |V | matrix with j-th row φj, and letting Ψ be the K × H matrix +with k-th row ψk, then marginally ξd,j,i = λ⊤ · Ψ · Φ, whereas ξd,j,i = φsj,d conditionally. +2.3. Combining the two approaches: hierarchical constrained topic modelling with sec- +ondary topics. +In order to aid interpretability of the command-level topics, it is possible to +use the same constraint from model (2). In particular, it could be assumed that words are sam- +pled either from φsd,j, where sd,j is the command-level topic, or from a distribution φ0 shared +across all commands and sessions. As in (2), each command-level topic h ∈ {1,...,H} has +an associated mixture probability θh ∈ [0,1] for sampling words from φh. Therefore, the full +model, which combines (1), (2) and (3), takes the following form: +λ ∼ Dirichlet(γ), + +UNSUPERVISED ATTACK PATTERN DETECTION IN HONEYPOT DATA +7 +1 +2 +1 +1 +2 +2 +1 +K = 2 +H = 3 +λ +ψ1 +ψ2 +td ∼ λ +D sessions +t2 +Sessions (Documents) +1 +1 +2 +1 +3 +2 +θ1 +θ2 +θ3 +Commands (Sentences) +Nd commands +s2,4 +sd,j ∼ ψtd +1 +0 +1 +0 +1 +Md,j words, P(zd,j,i = 1) = θsd,j +Indicators +Words +tftp -g abc.def.ghi.jkl -r tftp1.sh +w2,4,2 +wd,j,i ∼ φzd,j,isd,j +FIG 1: Cartoon representation of the full Hierarchical Constrained Topic Model (HCTM). +ψk ∼ Dirichlet(τ), k = 1,2,...,K, +φh ∼ Dirichlet(η), h = 0,1,...,H, +θh ∼ Beta(αh,α0), h = 1,2,...,H, +Nd ∼ Poisson(ζ), d = 1,...,D, +Md,j ∼ Poisson(ω), d = 1,...,D, j = 1,...,Nd, +td | λ ∼ Categorical(λ), d = 1,...,D, +sd,j | td,{ψk} ∼ Categorical(ψtd), j = 1,...,Nd, +zd,j,i | sd,j,{θh} ∼ Bernoulli(θsd,j),i = 1,...,Md,j, +wd,j,i | zd,j,i,sd,j,{φh} ∼ Categorical(φsd,jzd,j,i), i = 1,...,Md,j. +(4) +A pictorial representation of model (4) is given in Figure 1. Note that it might be possible +to consider variations of (4) through making changes to the specification of the prior distri- +butions on the hyperparameters. For example, document-specific mixing topic proportions +θd ∼ Beta(α,α0) could be used. +The next section will describe inferential methods for model (4). Deriving the inferential +procedures for (1), (2) and (3) follows similar guidelines, with minor modifications to the +equations used for (4). +3. Bayesian inference via Markov Chain Monte Carlo. +This section describes infer- +ential procedures for the topic models discussed in Section 2. The full model is considered, +with primary-secondary topics and session-level and command-level clustering. The pos- +terior distribution of the parameters is only available up to a normalising constant, therefore +inference must be performed using Markov Chain Monte Carlo (MCMC) methods. The main +objective of the inferential procedure is estimating t, the session-level clusters. Hence, the re- +maining parameters could be interpreted as nuisance, and integrated out when possible. The +parameters θ,λ,{φh} and {ψk} can be analytically marginalised, resulting in the marginal + +8 +posterior density +(5) +p(z,s,t | w) ∝ p(w,z,s,t) = p(t) × p(s | t) × p(z | s) × p(w | z,s). +Each term in the right-hand side of the marginal posterior (5) can be calculated explicitly +by conjugacy of the Categorical-Dirichlet and Beta-Bernoulli distributions. The marginal +distribution for the session-level intents is: +(6) +p(t) = B(γ + T ) +B(γ) +, +where B(x) = � +i Γ(xi)/Γ(� +i xi) is the multivariate beta function, and T = (T1,...,TK), +where Tk = � +d I{k}(td) denotes the number of sessions assigned to topic k. Similar cal- +culations lead to the marginal distribution for the command-level topics, conditional on the +session-level intents: +(7) +p(s | t) = +K +� +k=1 +B(τ + Sk) +B(τ) +, +where Sk = (S1,k,...,SH,k), and Sk,h = � +d:td=k +� +j I{h}(sd,j) denotes the number of com- +mands assigned to the command-level topic h, only from the subset of sessions with session- +level topic k. Similarly, the marginal distribution of the primary-secondary topic indicators z +has a closed form expression from the Beta-Bernoulli conjugacy: +(8) +p(z | s) = +H +� +h=1 +B(Zh + αh,M∗ +h − Zh + α0) +B(αh,α0) +, +where Zh = � +(d,j):sd,j=h +�Md,j +i=1 zd,j,i denotes the number of words assigned to the primary +topic, only from commands with command-level topic h, and M∗ +h = � +(d,j):sd,j=h Md,j de- +notes the total number of words in commands with topic h, across all documents. The final +component of the marginal posterior (5) is the marginal likelihood for the observed words w, +conditional on the indicators z and topic-level allocations s: +(9) +p(w | z,s) = +H +� +h=0 +B(Wh + η) +B(η) +, +where Wh = (Wh,1,...,Wh,|V |), and Wh,v = � +i,j,d I{h}(zd,j,isd,j)I{v}(wd,j,i) denotes the +number of times word v is assigned to the command-level topic h. +The marginal distributions (6), (7), (8) and (9) are the building blocks for the collapsed +Gibbs sampler (Liu, 1994) used for inference on the model parameters. The Gibbs sampler +consists of three basic moves: resample the session-level topic allocations t, resample the +command-level topic allocations s, and resample the primary-secondary topic indicators z. +Furthermore, convergence of Gibbs sampling algorithms for clustering usually benefits from +split-merge proposals, which are evaluated using a Metropolis-Hastings acceptance ratio, +resulting in a collapsed Metropolis-within-Gibbs algorithm. In this framework, split-merge +moves can be used on the session-level topics t and command-level topics s. The next sub- +sections give a detailed description of the steps required in the Gibbs sampling algorithm. +3.1. Resampling the session-level topic allocations. +The Gibbs sampler requires to sam- +ple from the conditional distribution of a subset of the parameters, conditional on the ob- +served data and remaining parameters. Therefore, for resampling the session-level topic al- +location td for a given document, it is required to sample from p(td | t−d,w,z,s), where the + +UNSUPERVISED ATTACK PATTERN DETECTION IN HONEYPOT DATA +9 +superscript −d denotes that the calculations of the corresponding quantity exclude the d-th +document. For the r-th session-level topic, the probability can be written as: +p(td = r | t−d,w,z,s) ∝ p(td = r | t−d)p(s | td = r,t−d) ∝ +B(γ + T ) +B(γ + T −d) +K +� +k=1 +B(τ +Sk). +where the quantities T and Sk in the final part of the expression are calculated assuming +that td = r. The ratio of beta functions in the conditional distribution can be simplified us- +ing the properties of the gamma function, yielding B(γ + T )/B(γ + T −d) ∝ (γr + T −d +r +). +Similarly, the product of beta functions in the final part of the expression could be further +simplified using the fact that Sr,h = S−d +r,h + Sd +h, where S−d +k,h = � +u:tu=k,u̸=d +� +j I{h}(sj,u) and +Sd +h = � +j I{h}(sd,j). From the properties of the gamma function, the probability can be then +expressed as: +(10) +p(td = r | t−d,w,z,s) ∝ (γr + T −d +r +) +�H +h=1 +�Sd +h +ℓ=1(τh + S−d +r,h + ℓ − 1) +�Nd +ℓ=1{�H +h=1(τh + S−d +r,h) + ℓ − 1} +. +3.2. Resampling the command-level topic allocations. +The Gibbs sampler also re- +quires to sample the command-level topic allocations from the distribution p(sd,j = ℓ | +s−d,j,w,t,z), where the superscript denotes that the quantities have been calculated exclud- +ing the (d,j)-th terms. For the ℓ-th command-level topic, the probability can be factorised +as: +(11) +p(sd,j = ℓ | s−d,j,w,t,z) ∝ +∝ p(sd,j = ℓ,s−d,j | t) × p(z | sd,j = ℓ,s−d,j) × p(w | sd,j = ℓ,s−d,j,z). +The first term in the factorisation (11), corresponding to (7), can be simplified noting that +Std,ℓ = S−d,j +td,ℓ + 1: +(12) +p(sd,j = ℓ,s−d,j | t) ∝ (τℓ + S−d,j +td,ℓ ). +A similar reasoning could be used to simplify the second term in the factorisation (11). +In particular, Zℓ = Z−d,j +ℓ ++ Zd,j, where Z−d,j +h += � +(u,q):su,q=h,(u,q)̸=(d,j) +�Md,j +i=1 zu,q,i and +Zd,j = �Md,j +i=1 zd,j,i. Using the properties of the gamma function, the corresponding proba- +bility is expressed as: +(13) +p(z | sd,j = ℓ,s−d,j) ∝ +∝ +�Zd,j−1 +q=0 +(αℓ + Z−d,j +ℓ ++ q)�Md,j−Zd,j−1 +u=0 +(α0 + M∗−d,j +ℓ +− Z−d,j +ℓ ++ u) +�Md,j−1 +q=0 +(α0 + αℓ + M∗−d,j +ℓ ++ q) +. +The last term in (11) admits a similar simplification to (10), using Wℓ,v = W −d,j +ℓ,v ++ W d,j +v , +where W −d,j +h,v += � +u,q,i:zu,q,isu,q=h,(u,q)̸=(d,j) I{v}(wu,q,i), W d,j +v += � +i:zd,j,isd,j̸=0 I{v}(wd,j,i). +The resulting probability is: +(14) +p(w | sd,j = ℓ,s−d,j,z) ∝ +�|V | +v=1 +�W d,j +v +q=1 (ηv + W −d,j +ℓ,v ++ q − 1) +�� +v W d,j +v +q=1 +{�|V | +v=1(ηv + W −d,j +ℓ,v +) + q − 1} +. +The probability (11) is obtained by calculating the product of the terms (12), (13), (14), and +normalising. + +10 +3.3. Resampling the primary-secondary topic indicators. +In the models with primary- +secondary topics, the Gibbs sampler also requires to resample the binary indicators zd,j,i, +conditional on w,s,t and z−d,j,i, denoting all the indicators except zd,j,i. Each binary indi- +cator is drawn from a Bernoulli distribution with unnormalised probabilities: +(15) +p(zd,j,i = b | z−d,j,i,w,s,t) ∝ p(zd,j,i = b | z−d,j,i,s) × p(w | zd,j,i = b,z−d,j,i,s), +where b ∈ {0,1}. Noting that Zsd,j = Z−d,j,i +sd,j ++ b, where the term Z−d,j,i +sd,j +is defined as +Z−d,j,i +h += � +(u,q):su,q=h,(u,q)̸=(d,j) +�Mu,q +i=1 zu,q,i, the first term in the factorisation becomes: +p(zd,j,i = b | z−d,j,i,s) ∝ (αsd,j + Z−d,j,i +sd,j +)b(α0 + M∗ +sd,j − Z−d,j,i +sd,j +− 1)1−b. +For the marginal likelihood of observed words, the only terms affected by a change in the +binary indicator zd,j,i are W0,wd,j,i = W −d,j,i +0,wd,j,i + 1 − b and Wsd,j,wd,j,i = W −d,j,i +sd,j,wd,j,i + b, +where W −d,j,i +h,v += � +(u,q,r):zu,q,rsu,q=h,(u,q,r)̸=(d,j,i) I{v}(wu,q,r), giving: +p(w | zd,j,i = b,z−d,j,i,s) ∝ +� +W −d,j,i +0,wd,j,i + ηwd,j,i +�|V | +v=1(W −d,j,i +0,v ++ ηv) +�1−b � +W −d,j,i +sd,j,wd,j,i + ηwd,j,i +�|V | +v=1(W −d,j,i +sd,j,v + ηv) +�b +. +3.4. Split-merge topic allocations. +There are two types of split-merge moves that can be +proposed: (i) split-merge session-level topics, and (ii) split-merge command-level topics. For +the split-merge move on session-level topics, two sessions d and d′ are sampled at random +from the D observed sessions. If td = td′ = t∗, the proposal for the session-level topics splits +the sessions assigned to t∗ in two different clusters using the following iterative procedure: (i) +assign topic t∗ to document d, and topic ˜t to document d′ (where ˜t corresponds to the number +of non-empty clusters, plus one – note that if ˜t > K the split move should be immediately +rejected); (ii) documents previously assigned to topic t∗ are sequentially allocated to topics +t∗ or ˜t in random order, with probabilities proportional to the predictive distribution (10), +restricted to the session already reallocated to topics t∗ and ˜t. This allocation procedure is +adapted from common split-merge MCMC moves in related clustering problems (see, for +example, Dahl, 2003; Sanna Passino and Heard, 2020). The final proposal is denoted as t∗, +with probability q(t∗ | t), corresponding to the product of sequential probabilities obtained +in the splitting procedure. The resulting acceptance probability for the move from t to t∗ is: +(16) +min +� +1, +p(t∗)p(s | t∗) +p(t)p(s | t)q(t∗ | t) +� +. +On the other hand, if td ̸= td′, the proposal t∗ assigns topic td to all documents previously +given topic td′, corresponding to a merge move. In this case, the acceptance ratio in (16) +must be further multiplied by the proposal probability q(t | t∗), calculated by simulating +a split move from t∗ to t. Since there is only one way to merge two topics, the proposal +probability at the denominator of (16) is q(t∗ | t) = 1 for a merge move. +A similar split-merge move can be constructed for the command-level topics: two com- +mands j and j′ are randomly sampled from two random documents d and d′ respectively. +If sd,j ̸= sj′,d′, a merge move is proposed. Alternatively, if sd,j = sj′,d′ = s∗, the split move +proceeds similarly to the procedure described for the topic-level sessions, and the command +previously assigned topic s∗ are sequentially allocated to s∗ or ˜s (corresponding to the num- +ber of non-empty command-level topics, plus one) with probabilities proportional to the pre- +dictive distribution (11), limited to the commands already reassigned to s∗ and ˜s. As be- +fore, if ˜s > H, the move is rejected. In summary, the acceptance probability for a vector of +command-level topics s∗ obtained via the split-merge procedure is: +min +� +1, p(s∗ | t)p(z | s∗)p(w | z,s∗)q(s | s∗) +p(s | t)p(z | s)p(w | z,s)q(s∗ | s) +� +, + +UNSUPERVISED ATTACK PATTERN DETECTION IN HONEYPOT DATA +11 +where the probabilities q(s∗ | s) and q(s | s∗) are either 1, or the product of allocation prob- +abilities calculated from the sequential splitting procedure. +3.5. Initialisation schemes. +In MCMC, setting good initial values could be helpful to +achieve faster convergence, in particular for complex inferential tasks. In this work, two +methods for initialisation are considered, based on spectral clustering and standard LDA. +Spectral methods are commonly used for text analysis and topic modelling (Ke and Wang, +2022). In order to initialise the algorithm via spectral clustering, a (�D +d=1 Nd) × |V | word +occurrence matrix C = {Csw} is constructed, where Csw counts the number of times word +w appears in command s. Note that all commands are stacked in an individual matrix C, +initially disregarding information about the division into sessions. A truncated singular value +decomposition of C is then calculated, considering only the largest H singular values and +corresponding left singular vectors. A clustering algorithm, like k-means, is then run on the +resulting embedding, setting H clusters. For initialisation of the session-level topics, a similar +procedure is carried out, using the initial values of the command-level topics as words in a +spectral clustering algorithm, obtaining a different form of the matrix of counts C. First, the +matrix C, with dimension D × H, is constructed, where each entry Cdh counts the number +of times a command assigned to the command-level topic h appears in document d. Then, +a K-dimensional truncated spectral decomposition of C is calculated, and k-means with K +clusters is run on the resulting embedding, obtaining initial values for the session-level topics. +Alternatively, standard LDA could be used to initialise the MCMC sampler, via fast- +performing software libraries such as python’s gensim ( ˇReh˚uˇrek and Sojka, 2010). First, +LDA with H topics could be fitted, and subsequently used to predict a topic for all the words +appearing in commands and sessions. Then the most common estimated topic within each +command is selected as the initial command-level topic. If secondary topics are used, LDA +is initially fitted with H + 1 topics, and the most common estimated topic is selected as sec- +ondary topic. The command-level primary topic is then selected as the most common topic +within each command, excluding the secondary topic. After command-level topics are es- +timated, session-level topics could be initialised by running LDA with K topics, using the +estimated command-level topics as words within the algorithm. For each command-level +topic, now interpreted as a word, a topic can be estimated from the fitted LDA model, and +the session-level topics are then initialised as the most common topic within each command. +For initialisation of the primary-secondary topic indicators, zd,j,i could be initially set to +1 if the proportion of sessions or commands where the word wd,j,i appears is less than a pre- +specified threshold. This is because φ0 should represent a distribution of common words, +shared across topics. +4. Unbounded number of topics and vocabulary. +All models discussed in Section 2 +assume a fixed size |V | of the vocabulary, and a fixed number of session-level and command- +level topics, K and H respectively. Such assumptions might be problematic if the model is +used for clustering future sessions, since it would not be possible to cluster new commands, +composed of words not present in the vocabulary. Therefore, a potentially infinite vocabulary +must be considered. Also, the behaviour of attackers is expected to evolve and change over +time, and it is possible that new attack patterns or intents arise. Therefore, for real-world +attack pattern detection, it is beneficial to assume an unbounded number of session-level and +command-level topics. These allowances require a modification to the Dirichlet distributions +used in Section 2, instead assuming: +λ ∼ GEM(γ), +ψk ∼ GEM(τ), k = 0,1,2,..., +φℓ ∼ GEM(η), ℓ = 0,1,2,..., + +12 +where τ,η,γ ∈ R+. The GEM (Griffiths-Engen-McCloskey) distribution (Pitman, 2006) cor- +responds to the proportions calculated using the stick-breaking representations of the Dirich- +let process (Sethuraman, 1994). Hence, the GEM distribution also corresponds to the limit for +K → ∞, H → ∞ and |V | → ∞ of the Dirichlet distributions in Section 2 with γ = γ1K/K, +τ = τ1H/H, and η = η1|V |/|V |. To simplify the discussion on the GEM distribution, its +link to the Dirichlet process, and its representation in the posterior distribution, consider n +objects allocated to Kn non-empty groups, with labels xn = (x1,...,xn), such that xi ∈ N +and max(xn) = Kn. Under a Dirichlet process with scaling parameter β, the predictive dis- +tribution for the next label in the sequence is: +(17) +p(xn+1 | xn) = +β +β + nI{Kn+1}(xn+1) + +Kn +� +k=1 +Nkn +β + nI{k}(xn+1), +where Nkn = �n +i=1 I{k}(xi) is the number of the n objects allocated to group k. The predic- +tive equation (17) immediately provides a technique for Gibbs sampling: since the Dirichlet +process assumes exchangeability of observations, any label can be considered as the last el- +ement of the sequence, and a new value resampled using (17). This fact will be particularly +useful when implementing the sampler. Using (17), the joint distribution for the sequence is: +p(xn) = +n +� +j=1 +p(xj | xj−1) = αKnΓ(α) +Γ(α + n) +Kn +� +k=1 +Γ(Nkn). +It follows that the components of the marginalised posterior distribution (5) take the following +revised form: +p(t) = γK(t)Γ(γ) +Γ(γ + D) +K(t) +� +k=1 +Γ(Tk), +(18) +p(s | t) = +K(t) +� +k=1 +τ +�H(s) +h=1 IN+(Sk,h)Γ(τ) +Γ(τ + � +d:td=k Nd) +� +h:Sk,h>0 +Γ(Sk,h), +p(z | s) = +H(s) +� +h=1 +B(Zh + αh,M∗ +h − Zh + α0) +B(αh,α0) +, +p(w | z,s) = +H(s) +� +h=0 +η +�V (w) +v=1 IN+(Wh,v)Γ(η) +Γ(η + �V (w) +v=1 Wh,v) +� +v:Wh,v>0 +Γ(Wh,v), +where K(t) = �∞ +k=1 IN+(Tk) and H(s) = �∞ +h=1 IN+(�∞ +k=1 Sk,h) are the number of unique +session-level and command-level topics respectively, and V (w) = �∞ +v=1 IN+(�∞ +h=0 Wh,v) +is the observed number of unique words. +4.1. Bayesian inference with GEM priors. +Inference in the model with unbounded num- +ber of topics and vocabulary size can be carried out using a similar algorithm to the Gibbs +sampling described in Section 3. Only minor modifications are required, since the marginals +in (18) take a different form under the GEM priors. For example, for resampling the session- +level topics, the probabilities in (10) are modified as follows: +p(td = r | t−d,w,z,s) ∝ γI{K(t−d)+1}(r)(T −d +r +)1−I{K(t−d)+1}(r) +× +� +h:Sd +h>0 τ I{0}(S−d +r,h)(S−d +r,h)1−I{0}(S−d +r,h) ��Sd +h +ℓ=2(S−d +r,h + ℓ − 1) +�IN>1(Sd +h) +�Nd +ℓ=1{τ + (�H(s) +h=1 S−d +r,h) + ℓ − 1} +, + +UNSUPERVISED ATTACK PATTERN DETECTION IN HONEYPOT DATA +13 +where r ∈ {1,...,K(t−d) + 1}, and the convention 00 = 1 is adopted. Similarly, the proba- +bilities in (11) for resampling command-level topics become: +p(sd,j = ℓ | s−d,j,w,t,z) ∝ τ I{h:Std,h=0}(ℓ)(S−d,j +td,ℓ )1−I{h:Std,h=0}(ℓ) +× +� +v:W d,j +v +>0 ηI{0}(W −d,j +ℓ,v +)(W −d,j +ℓ,v +)1−I{0}(W −d,j +ℓ,v +) ��W d,j +v +q=2 (W −d,j +ℓ,v ++ q − 1) +�IN>1(W d,j +v +) +�� +v W d,j +v +q=1 +{η + (�V (w) +v=1 W −d,j +ℓ,v +) + q − 1} +× +�Zd,j +q=1(αℓ + Z−d,j +ℓ ++ q − 1)�Md,j−Zd,j +u=1 +(α0 + M∗−d,j +ℓ +− Z−d,j +ℓ ++ u − 1) +�Md,j +q=1 (α0 + αℓ + M∗−d,j +ℓ ++ q − 1) +, +where ℓ ∈ {1,...,H(s−d,j) + 1}. A modification is required also for the conditional proba- +bilities of resampling the indicator zd,j,i in (15), resulting in: +p(zd,j,i = b | z−d,j,i,w,s,t) ∝ (αsd,j + Z−d,j,i +sd,j +)b(α0 + M∗ +sd,j − Z−d,j,i +sd,j +− 1)1−b +× +� +� +� +ηI{0}(W −d,j,i +0,wd,j,i)(W −d,j,i +0,wd,j,i)1−I{0}(W −d,j,i +0,wd,j,i) +η + �V (w−d,j,i) +v=1 +W −d,j,i +0,v +� +� +� +1−b +× +� +ηI{0}(W −d,j,i +sd,j ,wd,j,i)(W −d,j,i +sd,j,wd,j,i)1−I{0}(W −d,j,i +sd,j ,wd,j,i) +η + �V (w−d,j,i) +v=1 +W −d,j,i +sj,d,v +�b +, +where b ∈ {0,1}. The split-merge move in Section 3.4 can be equivalently extended to the +model with GEM priors, using the same ideas presented in this section. Split-merge moves are +expected to improve convergence of Gibbs sampling algorithms in models based on Dirich- +let processes (Jain and Neal, 2004). Note that all the probabilities described in this section +are extremely similar to the equations in Section 3, with the added complexity of handling +previously unseen topics. Alternative MCMC algorithms for models based on the Dirichlet +processes are extensively discussed in the literature (for example, Neal, 2000; Ishwaran and +James, 2001; Teh et al., 2006). +5. Application to the Imperial College London honeypot data. +The models described +in Sections 2 and 4 are now applied to real data collected on a honeypot hosted within the +Imperial College London (ICL) computer network. Within a time period between 21st May, +2021, and 27th January, 2022, the ICL honeypot collected approximately 40,000 unique ses- +sions, observed over 1.3 million times. This is a large corpus of attacks for a single machine. +5.1. Data preprocessing. +As discussed in the introduction, such sessions and commands +must be tokenised to obtain the words and vocabulary. In this work, the tokenisation is per- +formed with the python package NLTK (Bird, Klein and Loper, 2009), setting the regular +expression [a-zA-Z0-9_\.\-\*]+. Also, commands observed in the ICL honeypot data +often contain combinations of strings in hexadecimal form, preceded by the letter x. An ex- +ample of such a command is +bin busybox echo -e x6b x61 x6d x69 dev dev .nippon +In these analyses, all such instances of hexadecimal strings (x6b, x61, x6d and x69 in +the above example) are replaced by the word HEX. Also, some commands display the word +GHILIMEA appended to HEX strings. These are replaced with the word GHILIMEA_word. +Similarly to standard preprocessing techniques in natural language processing, extremely + +14 +rare and extremely common words are removed from the dataset. For the ICL honeypot data, +words appearing in less than 10 commands were removed, as were words appearing in over +10% of commands. Such words are often denoted stopwords in natural language processing +and information retrieval (see, for example, Manning, Raghavan and Schütze, 2008). After +preprocessing, a vocabulary of 1,003 unique words is obtained, and 2,617 uniquely observed +sessions, for a total of 42,640 commands and 261,283 words. Each session has an average +of 16.29 commands (with median 15), whereas each command contains on average 6.12 +words (with median 2). Users who access the honeypot have malicious intent, and it is not +uncommon to observe swear words and discriminatory language in many of the sessions. +Those terms have been redacted in plots of the results. +5.2. Topic estimation. +Before describing the results, some practical details about the es- +timation of topics from MCMC chains are discussed. In general, the number of session-level +or command-level topics are unknown. The Dirichlet priors for λ and {φh} in Section 2 +assume fixed, pre-specified values of K and H. For inference with the Dirichlet prior, a max- +imum number of possible topics could be chosen, denoted Kmax and Hmax, and the underlying +number of topics could be estimated as the number of non-empty topics at each iteration of +the MCMC sampling procedure. Furthermore, estimates of topic allocations based on the +MCMC sampler described in Section 3 could be affected by the issue of label switching +(Jasra, Holmes and Stephens, 2005). Therefore, session-level topic allocations are estimated +in this work from the estimated posterior similarity between sessions i and j, calculated as +�M +s=1 1t⋆ +i,s{t⋆ +j,s}/M, where M is the total number of posterior samples and t⋆ +i,s is the s- +th sample for ti. The posterior similarity matrix is invariant to permutations of the labels +and therefore unaffected by label switching. After the posterior similarities are obtained for +all pairs of sessions, hierarchical clustering with complete linkage is applied, with distance +measure 1 − ˆπij (Medvedovic, Yeung and Bumgarner, 2004). A similar procedure could +be followed for the command-level topics, but the very large number of commands would +make the size of the similarity matrix unfeasible to calculate and store in memory on a ma- +chine. Therefore, the last sample from the MCMC chain is considered as the estimate of the +command-level topics. +5.3. Constrained topic modelling. +First, the constrained topic model in (1) is fitted on +the postprocessed ICL honeypot data. Under the Dirichlet prior for λ, the hyperparameter γ +is set to γ = 0.1 · 1Kmax, with Kmax = 30. The hyperparameter of the Dirichlet prior for the +topic-specific word distribution is set to η = 1|V |. The MCMC sampler is run for 250,000 +iterations with 50,000 burn-in, initialising the topics via spectral clustering with Kmax clus- +ters. The results are displayed in Figure 2. The session-level topics are estimated using the +procedure described in Section 5.2. Figure 2a plots the resulting barplot of topic frequen- +cies of estimated session-level topics, for a number of topics equal to the modal number of +non-empty topics, ˆK∅ = 20. Furthermore, Figure 2b displays the barplot of the estimated +distribution for the number of non-empty topics. The barplot is also compared to the dis- +tribution obtained under a GEM prior for λ, with hyperparameter γ = 3, corresponding to +Kmax × 0.1, using the same setup for the MCMC sampler. The resulting distributions show +agreement, demonstrating a similar performance of the Dirichlet and GEM priors in estimat- +ing the modal number of topics. In general, the interplay between the prior parameters η +and γ appears to have an effect on the number of small clusters that are estimated from the +data: if η increases, the clusters in the right tail of Figure 2a tend to be incorporated within +the larger clusters. On the other hand, if η decreases towards zero, it is expected to estimate +more topics. + +UNSUPERVISED ATTACK PATTERN DETECTION IN HONEYPOT DATA +15 +(A) Barplot of frequencies of estimated session-level topics +2 10 7 11 9 20 14 3 +6 12 15 13 5 +4 +8 16 1 18 17 19 +Topic label +0 +100 +200 +300 +400 +500 +600 +700 +Number of sessions +(B) Barplot of estimated K∅ +18 +19 +20 +21 +22 +Number of topics +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +Estimated frequency +Dirichlet +GEM +FIG 2: Estimated topic frequencies and estimated distribution the number of non-empty topics K∅ under the +constrained topic model in (1), fitted on the ICL honeypot data. +5.3.1. Results: the model discovers a rare and unusual MIRAI variant. +The inferred +meaning of each topic is summarised in Table 1, according to the type of sessions that are +assigned to each group. The clusters appear to mostly contain botnets and different variants +of MIRAI malware. MIRAI is a type of botnet first emerged in 2016, which was specifically +targeted towards compromising Internet of Things (IoT) devices, or launching Distributed +Denial of Service (DDoS) attacks. Recently, it has been repurposed for Bitcoin mining on +IoT devices compromised via brute-force attacks on protocols such as SSH and Telnet. Over +the years, many different variants of MIRAI have emerged, and other bots with similar struc- +ture (Lingenfelter, Vakilinia and Sengupta, 2020; Sadique and Sengupta, 2021; Zhu et al., +2022), which appear to be assigned to different topics in Table 1. +Interestingly, careful examination of the sessions assigned to topic 5 estimated via the +constrained topic model in (1) helped analysts to discover an rare and unusual variant of +MIRAI, called MinerFinder (Bevington, 2021). The objective of MinerFinder is to look for +existing coin miner configurations, and try to gain root privileges to take control of the miner +infrastructure, if found. This demonstrates that the constrained topic model with a single topic +per session, even in its simplest form, could be extremely helpful for analysts to discover +new attack patterns. Within topic 5, MinerFinder is also mixed with other more common +MIRAI variants, which share a common frequency distribution of words. Ideally, a clustering +algorithm should be able to single-out MinerFinder from other MIRAI variants, despite their +similarities. This might be possible when a hierarchical structure is added to the topics, and +the command structure is explicitly used, as demonstrated in Section 5.5. +Overall, most of the activity on the honeypot seems to be related to attempts to install +botnets or coin miners, but topics show remarkable separation between sessions and corre- +sponding intents, as demonstrated in the list of malware and objectives in Table 1. Further +intuition about the differences between topics could be provided by examining the topic- +specific word distributions, which can be displayed via wordclouds, plotted in Figure 3. The +figure shows that words within each topic are fairly heterogeneous, but a number of words +appear frequently across multiple topics, such as HEX, cd or sh. A potential solution to bet- +ter visualise and further differentiate topic-specific word distributions would be to introduce +a secondary topic, which would capture the distribution of the most common words shared +across multiple topics and sessions. This solution is explored in the next section. +5.4. Constrained topic modelling with a secondary topic. +As discussed in the previous +section, a possible solution to aid topic interpretability and further discriminate topic-specific +word distributions would be to add a secondary topic to the model. In this section, the con- +strained topic model with secondary topic in (2) is therefore fitted to the ICL honeypot data. + +16 +TABLE 1 +Estimated session-level topics and corresponding intent under the constrained topic model in (1). +Topic label +Type of malware +Objective +1 +Shellbot +Install bot +2 +(ptmx) unnamed botnet, MIRAI +Gather system information, change permissions, execute MIRAI variants +3 +MIRAI +Download and execute MIRAI variants kura and kurc, fingerprint system +4 +MIRAI +Download sora malware, write upnp and updDl malware via echoing HEX strings +5 +MIRAI, MinerFinder (new variant) +Download malware, change permissions, gather system information, fingerprint system +6 +Shellbot, SBIDIOT, coin miner +Download and execute coin miner and MIRAI malware, change SSH keys +7 +(s4y, LAYER) unnamed botnet, MIRAI +Gather system information, change permissions, execute MIRAI variants +8 +Coin miner +Download and execute coin mining malware +9 +MIRAI +Download and execute MIRAI variants PEDO, ECCHI, PEACH... +10 +MIRAI +Download and execute MIRAI variants tftp1.sh, tftp2.sh... +11 +MIRAI +Determine shell executable, check busybox is present, print error message to console +12 +(misa) unnamed botnets +Gather system information, change permissions, execute MIRAI variants +13 +Shellbot, coin miner +Scan system, look for GPUs, look for coin miners, download malware +14 +MIRAI +Download and execute MIRAI variants sora, Pemex... +15 +MIRAI +Download and execute MIRAI variant DNXFCOW via echoing single HEX strings +16 +MIRAI +Download and execute MIRAI variant DNXFCOW via echoing multiple HEX strings +17 +MikroTik bot, coin miner +Gather system information, gather MikroTik router information, look for coin miners +18 +GHILIMEA, PentaMiner coin miner script +Install coin miner, kill mining processes with high CPU usage in order to go undetected +19 +MikroTik bot +Attempt to gain access to MikroTik router +20 +Hive OS attack, coin miner +Download miner, attempt to take over configurations in Hive OS mining platform +Topic 1 +Topic 2 +Topic 3 +Topic 4 +Topic 5 +Topic 6 +Topic 7 +Topic 8 +Topic 9 +Topic 10 +Topic 11 +Topic 12 +Topic 13 +Topic 14 +Topic 15 +Topic 16 +Topic 17 +Topic 18 +Topic 19 +Topic 20 +FIG 3: Wordclouds of estimated topic-specific word distributions under the constrained topic model in (1), fitted +on the ICL honeypot data. +The setup of the MCMC sampler is chosen to be identical to the previous section, and the +additional hyperparameters are set to α0 = 0.1 and αk = 0.9 for k = 1,...,Kmax, resulting +in a prior probability of 90% for a word to be allocated to a primary topic. 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++ +ftp +7 +du +HISTFILE +uname +bash_history +S +bash +qo +norc +usr +lscpu +1sh +cpuinfo +cat +b +1og +curlr +proc +bash aliases +portainerCat +nl +ppon +proc +dvrHelper +cp +waj8a +AYKASHI +x86_64 +e +HCAEP +t +qjmiguza +IHCCE +selfrep +ping +cd +wget +shell +human +ea8u +upnp +Chmod +mika. +sh +exad +enable +ODEP +0 +tftp +x61x6d +x6dx69 +ps +PEACH +nc +bigbotreppin +rf +ECCHI +echo +mounts +STHUB12 +macHelper +telnetnotdvrHelperUNSUPERVISED ATTACK PATTERN DETECTION IN HONEYPOT DATA +17 +Topic 1 +Topic 7 +Topic 10 +Topic 14 +Secondary topic (Topic 0) +FIG 4: Wordclouds of estimated topic-specific word distributions under the constrained topic model with sec- +ondary topic in (2), fitted on the ICL honeypot data. +tialised using the topics estimated by the constrained topic model in (1), fitted in Section 5.3. +In order to suitably compare the topic-specific word distributions and avoid the label switch- +ing issue in clustering (Jasra, Holmes and Stephens, 2005), the MCMC chain for model (2) +was initialised using the output from model (1) obtained in the previous section. The indica- +tors zd,j,i are initialised from a Bernoulli distribution with probability equal to the proportion +of documents in which the word wd,j,i occurred. The resulting wordclouds for some of the +topic-specific word distributions are plotted in Figure 4, including the distribution of the sec- +ondary topic, labelled topic 0. Note that with a secondary topic, words are implicitly sampled +from the mixture distribution ˜φtd = θtdφtd + (1 − θtd)φ0. This aids interpretability of the +latent intent of the session-level topic td, since the common words shared across most topics +are filtered out and included in φ0 instead. This is confirmed when comparing Figure 4 with +Figure 3 (obtained from a model that does not include a secondary topic). Overall, Figure 4 +shows that the secondary topic captures the most common words across documents, such as +the string HEX and simple shell commands, for example cd, chmod, var, echo, shell +and tmp. When comparing with Figure 3, the word distributions in Figure 4 appear to be less +dominated by common words. For example, for topic 1, the words tmp and cd appear to be +among the most representative words in Figure 3, but they have a much less prominent role in +the wordcloud representation in Figure 4, where words such as x86_64, uname and Xorg +gain importance. These words appear to be much more representative of the actual intents of +the session, which is particularly helpful when communicating results to analysts. +Overall, the assumption of having only one topic per session might be limiting, even if an +additional secondary shared topic is added. This is mainly because most sessions could be +considered as mixtures of commands, where each command has its own intent. Therefore, +a more precise clustering of topics and commands might be provided by the Hierarchical +Constrained Topic Model (HTCM) in Section 2.2, which is considered in the next section. +5.5. Hierarchical Constrained Topic Modelling (HCTM). +As discussed in the previous +section, the HCTM in Section 2.2 could help to further elucidate the underlying group struc- +ture within the ICL honeypot data. Similarly to Section 5.3, the MCMC is run for 250,000 +iterations with 50,000 burn-in. The command-level and session-level topics are initialised +via the spectral clustering algorithm described in Section 3.5, setting Dirichlet priors of di- +mension Kmax = 50 and Hmax = 50, with hyperparameters η = 1|V |, τ = 0.1 · 1Hmax,γ = +0.1 · 1Kmax. The session-level and command-level topics are estimated following the proce- +dure described in Section 5.2, setting ˆK∅ = 36 and ˆH∅ = 38, corresponding to the modal +number of non-empty topics. Figure 5 displays the frequency distribution of the estimated +session-level (Figure 5a) and command-level topics (Figure 5c), followed by the estimated +distributions of the number of non-empty session-level (Figure 5b) and command-level topics +(Figure 5d). +The wordclouds for the resulting session-level and command-level topic-specific distri- +butions are displayed in Figure 6. Figure 6a displays the topic-specific word distributions + +tmp +history +HEX +ftp +C +chroot +bash +uname +ipsh +chmod +alive +find +pkl +mnt +X86 +64 +0 +Xorg +keep +usrAYER +bash +s1kdq1 +cowffxxna +tmp +USmnt +linuxshell +shm +Eitset +wellnope +boot +Skyline +enable +dBot1 +Akim +binary +shell +home +done +updDl +ra +ultraesgrimaprivsrc +null +N +S +KEKSEC +工 +AstroStr +s +gay +ahsoktset +DxP1 +Hazardsecurity +Switchblades1 +rf +ping +ebot1 +chmod +mioritest +webr2342e +V +6Kafck3x0a +etc +cat +read +sh +lib +cp +RpcSecurity +echo +htm3dew +ha7665cazs +Layer' +netslink +hu87VhvQPztftp1 +Gbottftp1 +bins2 +yoyotftp1 +X +Gbottftp2 +axistftpt +anable +tftp +rf +telnet1 +2g +0 +mntv +tftp +apponline957 +ftp +awoo +chmo.d +phantom2 +yoyobins +ssh2 +help +DEMONS +ftp +telnet2 +cd +root +OK4N3 +brian +Snoopy +oblivionse +ab +telnet +shell +p +tmp +curl +S +sh +anonymoushistory +bins1 +wget +Var +bins +yoyotftp2 +run +ulimit +ssh1 +Sakura +ftpget +axistftp2 +ac +skid +UwUshtZerow +t8UsA +ufonet1 +saturn mirai +Linuxshe +Chmod +ohsitsvegawellrip1 +bins1 +EkSgbins +root +binInfect1 +Pemex +Quartztf +lewdtftp2 +sensi +ohsitsvegawellrip2 +tz +76d32be02 ufonet2 +Var +Ciatftp1 +whattttttlol +bins2 +sensi1 +0x83911d24Fx +ifconfig +rf +g +tmp +0xt984767 +P +cd +C +bot +Pemex1 +ufonet +goontftp1 +V +run +0xft6426467 +EkSgtftp1 +tZerow2 +lewdbins +Beastmode +Hilix3 +wget +TelNet2Josh +lewdtftp1 +sora +sensi2 +Hilix +c0r0n4x +lewd +TelNet1 +phantom2 +t8UsA2 +phantom +8USA +inInfect2 +EkSgtftp2 +Rakitin1 +ohsitsvegawellrip +tbot +sorat +0 +Rakitin +Pemex2 +tftp +goontftp2 +Ciatftp2 +Quartztftp2 +ft +binnfect +pget +mnt +tz2 +Quartzbins +whattttttlol2 +Lerow +TelNet +76d32be01 +8USA1 +0xtf2984767 +help +creepy2 +shell +Cur +nable +Lerow +sora2( +Lucid +betaalverzoek +scorpion +76d32be0 +none +N +icy +Hilix1 +None +saturn1linuxshell +ne +tmp +777sh +echo +777enable +rfva +DNXFCOW +chmod +ftpget +.00. +shshell +echoenable +nika +S +boot +HEXcd +enable +Oh +chmodcd +chmodtmp +rfsh +iffxna +curl +etslinl +vgetenable +usrsh +echoshell +cshe11 +s4y +human +terminal +etc +basi +shellsh +n +start +shell +bin +tftp2 +nes +usrenable +GHILIMEA word +shellenable +anonymous +misa +mnt +777tmp +cdvar +icda +config +cdtmp +chnodps +cp +PEDO +varsh +echotmp +edusi +HEXshe11 +chmodmnt +warcd +tftp +catshell +usr +rf +wget +shahcd +D +ECCHI +retrieve +LAYER +fuu +g +n +e +echocd +chmodsh +rfcat +S +Va +shs +Gof +777cat +777usr +CO +usrcd +netelinhce +.cat +shtmp +tmpshel1 +shmcd +cenable +rfcd +777var +ftp1 +cdsh +cdshell +mounts +echosh +rftmp +rfenable +ptmx +HEXsh chmodenable +catsh +ngetsh +tmpsh +catenable +uny +catcd +shm +V +ootcd +rfshell +e 777cd +777shell +whilecd +chmodshell +Cdenable +ping shcat +777mnt18 +(A) Barplot of frequencies of estimated session-level topics +1 2 10 15 4 13 11 3 34 21 18 14 16 31 12 5 22 6 23 20 29 7 17 32 8 9 26 35 30 27 33 28 19 25 24 36 +Session-level topic label +0 +100 +200 +300 +400 +Number of sessions +(B) Barplot of estimated K∅ +33 +34 +35 +36 +37 +38 +39 +40 +Number of session-level topics +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +Estimated frequency +(C) Barplot of frequencies of estimated command-level topics +2 7 1 4 13 19 3 20 9 11 21 8 28 15 6 26 27 5 12 14 16 10 23 34 31 17 35 29 30 18 32 33 22 24 37 25 36 38 +Command-level topic label +0 +1000 +2000 +3000 +4000 +5000 +6000 +7000 +Number of commands +(D) Barplot of estimated H∅ +34 +35 +36 +37 +38 +39 +40 +41 +42 +43 +44 +45 +Number of command-level topics +0.00 +0.05 +0.10 +0.15 +0.20 +Estimated frequency +FIG 5: Frequency distributions of the estimated session-level and command-level topics, and estimated distribu- +tion the number of non-empty session-level topics K∅ and command-level topics H∅, under the hierarchical +constrained topic model in (3), fitted on the ICL honeypot data. +obtained from the estimated session-level topics, whereas Figure 6b plots the wordclouds cor- +responding to the estimated command-level topics. Interestingly, when compared to Figure 3, +the word distributions in Figure 6 appear more heterogeneous, especially at the command- +level. This is not surprising: the constrained topic model in (1) gives the same primary topic +to all words in a session, whereas the HCTM admits command-specific topics. In the ICL +honeypot data, and more generally in session data, individual commands tend to have a spe- +cific intent identified by specific words in the command (for example, wget for download- +ing files from a web server under the HTTP, HTTPS, and FTP protocols). Therefore, having +command-specific topics helps in identifying the intents of individual commands, making the +command-level topic-specific word distributions highly interpretable. +Similarly to the previous section, the intents for the estimated session-level and command- +level topics are summarised in Table 2 and 3. In general, the two tables show that some topics +correspond to the same intent, achieved through different words or commands. In particular, +Table 2 shows the same malware types as Table 1, with similar objectives. Similarly to the re- +sults in the previous section, different MIRAI variants are observed, and malware types such +as shellbots, coin miners, the Hive OS attack and the MikroTik bot are all allocated to sep- +arate topics. In addition to the session-level objectives in Table 1, Table 2 also shows topics +representing reconnaissance sessions, where intruders attempted to gather system informa- +tion, for example by checking directories and determining shell executables. MinerFinder +is again discovered and relevant sessions are singled-out in the topic with label 24. In par- +ticular, under the hierarchical constrained topic model, MinerFinder is explicitly split from +the similar MIRAI variants that were present in topic 5 under the constrained topic model +(cf. Table 1), which are instead allocated to topic 22 under the HCTM (cf. Table 2). It must +be remarked that MinerFinder is not detected using alternative clustering approaches based +on spectral clustering or standard LDA fitted via gensim (cf. Section 3.5). If the topics are + +UNSUPERVISED ATTACK PATTERN DETECTION IN HONEYPOT DATA +19 +(A) Session-level topic-specific word distributions +Topic 1 +Topic 2 +Topic 3 +Topic 4 +Topic 5 +Topic 6 +Topic 7 +Topic 8 +Topic 9 +Topic 10 +Topic 11 +Topic 12 +Topic 13 +Topic 14 +Topic 15 +Topic 16 +Topic 17 +Topic 18 +Topic 19 +Topic 20 +Topic 21 +Topic 22 +Topic 23 +Topic 24 +Topic 25 +Topic 26 +Topic 27 +Topic 28 +Topic 29 +Topic 30 +Topic 31 +Topic 32 +Topic 33 +Topic 34 +Topic 35 +Topic 36 +(B) Command-level topic-specific word distributions +Topic 1 +Topic 2 +Topic 3 +Topic 4 +Topic 5 +Topic 6 +Topic 7 +Topic 8 +Topic 9 +Topic 10 +Topic 11 +Topic 12 +Topic 13 +Topic 14 +Topic 15 +Topic 16 +Topic 17 +Topic 18 +Topic 19 +Topic 20 +Topic 21 +Topic 22 +Topic 23 +Topic 24 +Topic 25 +Topic 26 +Topic 27 +Topic 28 +Topic 29 +Topic 30 +Topic 31 +Topic 32 +Topic 33 +Topic 34 +Topic 35 +Topic 36 +Topic 37 +Topic 38 +FIG 6: Wordclouds of estimated session-level and command-level topic-specific word 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+8 +bigbotreppin +S +shiina +sefaexecbi +nount: +Cat +exa +dvrHelper +puinfo +echo +NsGA4 +aznuuxa +6F +notdvrHelper +passwd AbAduptin +notakame +nippon +CCAD +Proton +mika +lacHe.p +X19I239124UIU +ans +infectedByRakitin20 +TABLE 2 +Estimated session-level topics and corresponding intent under the hierarchical constrained topic model in (3). +Topic label +Type of malware +Objective +1 +MIRAI +Check shell and directories, download and execute MIRAI malware, delete files +2 +(ptmx) unnamed botnet, MIRAI +Gather system information, change permissions, execute MIRAI variants +3 +Shellbot, coin miner, MIRAI +Download and execute coin miner and MIRAI malware +4 +MIRAI +Determine shell executable, check busybox is present, print error message to console +5 +Shellbot, coin miner +SSH backdoor botnet, download malware, change SSH keys, check CPU/GPU information +6 +MIRAI +Download and execute MIRAI malware (for example, garm or gmips), delete files +7 +Shellbot, coin miner +Gather CPU/GPU information, download coin miners, kill existing coin miners +8 +Reconnaissance +Check mounted file system, gather system information, fingerprint system +9 +MIRAI +Write malware (for example dvrHelpwer and updDl) via echoing HEX strings +10 +Reconnaissance +Check shell and directories, delete files (.ptmx, Switchblades) +11 +Hive OS attack, coin miner +Download miner, attempt to take over configurations in Hive OS mining platform +12 +MIRAI +Execute MIRAI variant PEDO, fingerprint system +13 +MIRAI +Execute MIRAI variants kura and kurc, fingerprint system +14 +MIRAI +Determine shell, download and execute MIRAI variant DNXFCOW via echoing HEX strings +15 +Reconnaissance +Check shell and directories, delete files (.ptmx, .s4y) +16 +MIRAI +Download and execute MIRAI variant PEDO and mika, fingerprint system +17 +Coin miner +Download coin miner (c3pool) +18 +MIRAI +Download, execute MIRAI variant ECCHI, delete files, fingerprint system +19 +MIRAI +Download malware, write updDl malware via echoing HEX strings +20 +MIRAI +Download sora malware, write upnp and updDl malware via echoing HEX strings +21 +MIRAI +Gather system information, change permissions, execute MIRAI variants +22 +MIRAI +Download malware, change permissions, gather system information, fingerprint system +23 +Coin miner, SBIDIOT +Download and execute coin mining malware +24 +MinerFinder (new MIRAI variant) +Changes SSH keys, looks for coin miners, attempt to take over miners +25 +MIRAI +Check shell and directories, execute MIRAI malware (Skyline and Akim), delete files +26 +GHILIMEA, PentaMiner coin miner script +Install coin miner, kill mining processes with high CPU usage in order to go undetected +27 +Reconnaissance +Attempt to read and change SSH keys +28 +MIRAI +Write RONALD malware via echoing HEX strings +29 +MIRAI +Gather system information, print error message (component of DNXFCOW MIRAI variant) +30 +MIRAI +Gather system information, change permissions, execute MIRAI variant cowffxxna +31 +MikroTik bot, coin miner +Gather system information, gather MikroTik router information, kill existing coin miners +32 +MIRAI +Download and execute MIRAI variant DNXFCOW via echoing multiple HEX strings +33 +Shellbot, coin miner +Scan system, look for GPUs, look for coin miners, download malware +34 +MIRAI +Gather system information, change permissions, execute MIRAI variants +35 +Coin miner +Download coin miner (TeamTNT) +36 +MikroTik bot +Remove firewall NAT rules on MikroTik router +estimated using the procedures described in Section 3.5, MinerFinder is usually allocated to +large clusters, with a number of sessions ranging between 80 and 1,000. +Furthermore, Figure 7 displays the heatmap of Jaccard similarity scores comparing the re- +sults of the constrained model (cf. Section 5.3) and the hierarchical constrained topic model +fitted in this section, demonstrating significant agreement. The level of agreement is remark- +able, considering that the session-level topics are obtained under two different modelling +assumptions: in Section 5.3, the constrained topic model in (1) directly uses the session-level +topic to obtain the word distribution for the entire session. On the other hand, the session- +level topics in HTCMs are only estimated from sequences of command-level topics, which +are themselves unknown and therefore estimated. +More details about the commands appearing in the sessions are given in Table 3, and such +objectives can be associated with the command-level topic-specific wordclouds in Figure 6b. +Overall, analysing the results of the HTCM confirms the results obtained in the previous +section, with the added benefit of having estimated command-level topics. +6. Conclusion. +Models for clustering session data collected on honeypots were pro- +posed, with the objective of finding groups of similar attacks which could aid cyber ana- +lysts in detecting emerging intrusion attempts and vulnerabilities. The proposed models are +based on modifications of Latent Dirichlet Allocation, aimed at improving topic interpretabil- +ity and convergence properties. In particular, the concepts of primary and secondary topics +were introduced, along with session-level and command-level topics. Secondary topics are +used to represent common, high-frequency words, whereas the primary topic is used for the +words which define the latent intent of the session. Furthermore, two hierarchical layers of +topics were introduced: a command-level topic which determines the word distribution, and + +UNSUPERVISED ATTACK PATTERN DETECTION IN HONEYPOT DATA +21 +TABLE 3 +Estimated command-level topics and corresponding intent under the hierarchical constrained topic model in (3). +Topic label +Objective +1 +Attempt to write file .ptmx or LAYER to directory var, run or tmp and change directory if writeable +2 +Attempt to write file .ptmx to directory netslink, mnt or shm and change directory if writeable +3 +Attempt to copy, write and delete files +4 +Attempt to grant full permissions to all users on malware files +5 +Gather system information about CPU architectures +6 +Kill processes, gather system information, Hive OS logon attempt +7 +Check available commands, determine shell executable +8 +Check if busybox exists, print error message to console (component of PEDO MIRAI variant) +9 +Gather system information about mounted file systems, copy files +10 +Fingerprint readable and writeable directories to hidden file .nippon +11 +Download malware from internet using wget and curl +12 +Attempt to read and delete files +13 +Attempt to execute malware, delete files after execution +14 +Check if directories such as var, run or tmp exist, by attempting to change directory +15 +Download malware using tftp +16 +Download, execute and delete malware files +17 +Download malware using ftpget +18 +Download and install GHILIMEA coin mining malware, kill own processes if CPU usage is high +19 +Check if busybox exists, check available commands, determine shell executable +20 +Check if busybox exists, print error message to console (component of KURA and OWARI MIRAI variants) +21 +Check if busybox exists, print error message to console (component of ECCHI MIRAI variant) +22 +Download malware to attempt to compromise MikroTik router +23 +Look for existing coin miners, gather GPU and CPU information +24 +Attempt to gather information about MikroTik router +25 +Determine shell executable +26 +Attempt to write file .file to directory netslink, mnt or boot and change directory if writeable +27 +Copy, grant full permission, execute and delete .cowbot malware (MIRAI variant) +28 +Write malware binary to disk via echoing HEX bits +29 +Download coin miner malware from c3pool +30 +Read SSH authorised keys, attempt to delete and replace authorised keys +31 +Exit shell (part of GHILIMEA coin mining malware) +32 +Install GHILIMEA coin mining malware, kill own processes if CPU usage is high +33 +Check internet connectivity, check if GHILIMEA is already installed, kill own processes if CPU usage is high +34 +Read and delete files (for example, .none and .human) +35 +Check if busybox exists, print error message to console (component of RONALD and Akim MIRAI variant) +36 +Remove firewall NAT rules on MikroTik router +37 +Remove temporary directory p (singleton cluster) +38 +Exit shell (singleton cluster) +26 22 +4 +13 14 +5 +11 32 +1 +20 18 23 +6 +3 +2 +16 36 +7 +19 10 21 15 12 24 25 +9 +27 28 29 30 31 +8 +33 34 35 17 +Session-level topic label (hierarchical constrained topic model) +18 +5 +11 +3 +15 +13 +20 +16 +10 +4 +9 +8 +14 +6 +2 +12 +17 +1 +19 +7 +Topic label (constrained topic model) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +FIG 7: Heatmap of Jaccard similarities between the sessions assigned to session-level topics under the con- +strained topic model (cf. Section 5.3) and hierarchical constrained topic modelling (cf. Section 5.5). + +22 +a session-level topic which controls the distribution of the command-level intents. Further- +more, the proposed methodologies extend to a Bayesian nonparametric framework, admitting +an unbounded (and unknown) number of latent intents. +The proposed models could be further extended by introducing dependencies between +command-level topics. In particular, a Markov model with H states could be devised, +whereby each session-level topic corresponds to different transition probabilities between +command-level topics. Also, a possible extension of this work could consider dynamically- +evolving topics (dynamic topic modelling, Blei and Lafferty, 2006), or explicitly encode cor- +relation between topics (correlated topic models, Lafferty and Blei, 2005), or a combination +of the two approaches (see, for example, Tomasi et al., 2020). +Sections 1.1 and 5.1 briefly discussed some of the challenges related to tokenisation in +cyber-security applications. Depending on the chosen tokenisation method, important context +about the commands might be lost, which might prevent cyber-security analysts to make rapid +decisions and assessments on the content of the session. One possibility in this direction +would be to explore deep neural network methods, inspired by their use for large language +models. For example, the RAPIDS CLX library cyBERT2 uses deep neural networks and +transformers for parsing cyber-security logs, including session data. +Overall, session data collected on honeypots are a valuable resource for cyber analysts, +and principled statistical modelling is required to obtain actionable insights from these data. +The proposed methodologies have provided useful groupings of the attacks observed on a +university network, including the discovery of an unusual MIRAI variant which attempts to +take over existing coin miner infrastructure. This demonstrates the potential of topic models +to elucidate hidden structure within text data, obtaining valuable insights for cyber-defence +purposes. +Acknowledgements. +The authors thank Andy Thomas and the Information & Com- +munications Technology team at Imperial College London for their support on data pro- +cessing. FSP and NAH acknowledge funding from Microsoft Security, through the research +grant “Understanding the enterprise: Host-based event prediction for automatic defence in +cyber-security”. AM and NAH acknowledge funding from the Data-centric Engineering pro- +gramme at the Alan Turing Institute, for the Grand Challenge “Monitoring Complex Sys- +tems”. DG acknowledges funding from the Department of Mathematics at Imperial College +London. Part of this work was carried out when AM was a postdoctoral research associate +in statistical cyber-security at Imperial College London and the Alan Turing Institute. PT is +now a Quantitative Analyst at Abios, and he completed part of this work as part of a masters’ +degree at Imperial College London. +Code. +A python library that implements the methodologies proposed in this article is +available in the Github repository fraspass/lda_clust. +REFERENCES +AGGARWAL, C. C. and ZHAI, C. (2012). A Survey of Text Classification Algorithms. 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Mining Function Homology +of Bot Loaders from Honeypot Logs. arXiv e-prints. +APPENDIX A: SIMULATIONS AND RESULTS ON SYNTHETIC DATA +In this section, the proposed models are compared and contrasted on synthetic datasets, in +order to assess their performance in recovering the session-level topics, the main quantity of +inferential interest. Note that the true topics are known when data are simulated. If synthetic +data are used, the true underlying session-level topics are available, and the true allocations +could be compared to the estimated topics via the Adjusted Rand Index (ARI, Hubert and +Arabie, 1985). Each simulation is repeated for 50 datasets using different seeds, setting |V | = +50, D = 100, Nd = 10, Md,j = 10, λ = 1/K · 1K for each simulated dataset. The MCMC +sampler is run for 25,000 iterations with 10,000 burn-in, initialising the topics via spectral +clustering. Unless otherwise specified, the hyperparameter γ is set to γ = 0.1 · 1Kmax. + +UNSUPERVISED ATTACK PATTERN DETECTION IN HONEYPOT DATA +25 +(A) Boxplot of ARIs for estimated session-level topics +η = 1.0 +η = 5.0 +η = 10.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +ARI +(B) Barplot of estimated number of non-empty topics K∅ +5 +6 +K +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Estimated probability +Truth +η = 1.0 +η = 5.0 +η = 10.0 +FIG 8: Summary plots obtained from 50 simulated datasets from model (1), with |V | = 50, K = 5, D = +100, Nd = 10, Md,j = 10, λ = 1/K · 1K, using different values for η, such that η = η · 1K. The MCMC +sampler is run for 25,000 iterations with 10,000 burn-in, setting Kmax = 10. +First, datasets are simulated from model (1), setting K = 5 and using three different val- +ues of the parameter η. If η = η · 1K, the parameter η controls the spikiness of the topic- +specific word distributions: low values of η correspond to distributions which assign most of +the probability mass to a small number of words, whereas larger values of η correspond to +distributions where the probability mass is distributed more uniformly across words. In the +MCMC sampler, Kmax is set to 10, implying that the sampler should identify 5 empty topics. +The results are reported in Figure 8: Figure 8a reports the ARIs for the estimated session- +level topics, whereas Figure 8b shows the barplot of the estimated number of non-empty +topics, denoted K∅. As expected, the ARI decreases when the value of η increases, since the +topic-specific distributions become increasingly uniform and therefore similar. Increasing the +value of η also provides worse estimates of the true number of topics used in the simulation. +Ideally, K∅ should correspond to the value of K used in the simulation, provided that Kmax +in the MCMC algorithm is chosen to be larger than the true underlying K. +Next, inference is repeated on the 50 datasets simulated as in Figure 8 with η = 5 · 1K, +using different initialisation methods and priors for λ. In particular, results are compared +between the spectral and gensim initialisation schemes described in Section 3.5. Figure 9a +displays the boxplots of ARIs for the session-level topics across the different datasets, after +initialisation, before and after running the MCMC sampling scheme. The plot shows that the +spectral initialisation scheme appears to have a better performance initially, but both methods +lead to equivalent results after MCMC sampling. Furthermore, results on estimation of the +number of topics are compared between two different priors: a Kmax-dimensional Dirichlet +distribution with Kmax = 10 and γ = 0.1 · 1Kmax (also used in Figure 8b) or a GEM prior on +λ with γ = 0.1. Figure 9b reports the resulting barplot for the estimated modal number of +non-empty communities across the different datasets, suggesting that the two priors have a +similar performance. +Second, datasets are simulated from model (2), using η = 1K, K = 5 and different val- +ues of θk. The value of θk corresponds to the expected proportion of words sampled from +the primary topic. Therefore, small values of θk are expected to make inferential procedures +more difficult, since less observations are available from the primary topics, and words are +sampled instead from a secondary topic, shared across sessions and commands. Furthermore, +the secondary topic makes the primary topics more difficult to estimate, since the words are +implicitly sampled from the mixture distribution ˜φtd = θtdφtd + (1 − θtd)φ0. If θk decreases +for all k, the distributions ˜φ1,..., ˜φK become increasingly similar, and a drop in the ARI +similar to Figure 8a is expected. In the MCMC sampler, the required parameters are set to +Kmax = 10, αh = 0.9 and α0 = 0.1. The results are presented in Figure 10. As expected, + +26 +(A) Boxplot of ARIs for estimated session-level topics ob- +tained via different initialisation schemes +Spectral +gensim +Spectral +gensim +At initialisation +After MCMC +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +ARI +(B) Barplot of estimated number of non-empty topics K∅ for +different priors on λ (Kmax-dimensional Dirichlet or GEM) +5 +6 +K +0.0 +0.2 +0.4 +0.6 +0.8 +Estimated probability +Dirichlet +GEM +FIG 9: Summary plots obtained from 50 simulated datasets from model (1), with |V | = 50, K = 5, D = +100, Nd = 10, Md,j = 10, λ = 1/K · 1K, and η = 5 · 1K. The MCMC sampler is run for 25,000 iterations +with 10,000 burn-in, setting Kmax = 10 or a GEM prior for λ with γ = 0.1. +(A) Boxplot of ARIs for estimated session-level topics +θk = 0.9 +θk = 0.75 +θk = 0.5 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +ARI +(B) Barplot of estimated number of non-empty topics K∅ +2 +3 +4 +5 +6 +7 +8 +K +0.0 +0.2 +0.4 +0.6 +0.8 +Estimated probability +Truth +θk = 0.9 +θk = 0.75 +θk = 0.5 +FIG 10: Summary plots obtained from 50 simulated datasets from model (2), with |V | = 50, K = 5, D = +100, Nd = 10, Md,j = 10, λ = 1/K · 1K, η = 1K, using different values for θk. The MCMC sampler is run +for 25,000 iterations with 10,000 burn-in, setting Kmax = 10, αh = 0.9,α0 = 0.1. +(A) Boxplot of ARIs for estimated session-level topics +τ = 0.25 +τ = 0.5 +τ = 1.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +ARI +(B) Barplot of estimated number of non-empty topics K∅ +2 +3 +K +0.0 +0.2 +0.4 +0.6 +0.8 +Estimated probability +Truth +τ = 0.25 +τ = 0.5 +τ = 1.0 +FIG 11: Summary plots obtained from 50 simulated datasets from model (3), with |V | = 50, K = 3, H = 5, D = +100, Nd = 10, Md,j = 10, λ = 1/K · 1K, η = 1K, using different values for τ = τ1H. The MCMC sampler +is run for 25,000 iterations with 10,000 burn-in, setting Kmax = 10, Hmax = 10. +Figure 10a shows that the ARI for the estimated session-level topics decreases when θk de- +creases. Furthermore, Figure 10b shows that estimation of the number of non-empty primary +topics becomes increasingly imprecise when θk decreases, causing the drop in ARI observed +in Figure 10a. +Third, datasets are simulated from model (3), using η = 0.1 · 1K, K = 3, H = 5 and +τ = τ1H, for different values of τ. In this case, τ expresses how concentrated around a + +UNSUPERVISED ATTACK PATTERN DETECTION IN HONEYPOT DATA +27 +peak the command-level topic distributions are. Small values of τ imply that the probability +mass is mostly concentrated around one topic, whereas larger values correspond to more +evenly distributed probability mass functions. In the MCMC sampler, the values Kmax = 10 +and Hmax = 10 are used. Note that the task of recovering the session-level topics is much +more complex than previous simulations, since for model (3), the session-level topic only +controls the distribution of the command-level topics. Hence, the session-level topic must be +estimated from the Nd command-level topics within each document, which are themselves +estimated. This makes the inferential task substantially harder. Figure 11 displays the results: +Figure 11a shows that the ARI for the estimated session-level topics tends to decrease when +τ increases, and Figure 11b shows that estimates of the number of non-empty session-level +topics are more precise when the value of τ used in the simulation is smaller. Notice that, +since η = 0.1·1K, the topic-specific word distributions have most of their mass concentrated +around a small number of words, and the command-level topics are therefore easy to recover, +with ARIs averaging above 0.99 across the three settings for τ shown in Figure 11. + diff --git a/ZNE0T4oBgHgl3EQfnAEV/content/tmp_files/load_file.txt b/ZNE0T4oBgHgl3EQfnAEV/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b721533112dcff7ee8eb4a86f2aa215fc5a9dc48 --- /dev/null +++ b/ZNE0T4oBgHgl3EQfnAEV/content/tmp_files/load_file.txt @@ -0,0 +1,5247 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf,len=5246 +page_content='UNSUPERVISED ATTACK PATTERN DETECTION IN HONEYPOT DATA USING BAYESIAN TOPIC MODELLING BY FRANCESCO SANNA PASSINO1,a , ANASTASIA MANTZIOU2 , DANIYAR GHANI1 , PHILIP THIEDE1, ROSS BEVINGTON3, AND NICHOLAS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' HEARD1 1Department of Mathematics, Imperial College London, London (United Kingdom), af.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='sannapassino@imperial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='uk 2The Alan Turing Institute, London (United Kingdom) 3Microsoft Threat Intelligence Center (MSTIC), Cheltenham (United Kingdom) Cyber-systems are under near-constant threat from intrusion attempts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Attacks types vary, but each attempt typically has a specific underlying in- tent, and the perpetrators are typically groups of individuals with similar ob- jectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Clustering attacks appearing to share a common intent is very valu- able to threat-hunting experts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' This article explores topic models for cluster- ing terminal session commands collected from honeypots, which are special network hosts designed to entice malicious attackers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' The main practical im- plications of clustering the sessions are two-fold: finding similar groups of attacks, and identifying outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' A range of statistical topic models are con- sidered, adapted to the structures of command-line syntax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' In particular, con- cepts of primary and secondary topics, and then session-level and command- level topics, are introduced into the models to improve interpretability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' The proposed methods are further extended in a Bayesian nonparametric fashion to allow unboundedness in the vocabulary size and the number of latent in- tents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' The methods are shown to discover an unusual MIRAI variant which attempts to take over existing cryptocurrency coin-mining infrastructure, not detected by traditional topic-modelling approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' The increasing reliance of enterprises on information technologies, such as cloud services, gives rise to new challenges for protecting customer data and com- puter systems from intrusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' To tackle these cyber threats, enterprises increasingly resort to quantitative methods for the development of the next-generation intrusion detection tech- niques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Honeypots play an important role in the detection and understanding of attacker be- haviours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' A honeypot is a host located within a computer network designed to entice ma- licious attackers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Security teams use the commands issued by attackers during interactive sessions with the honeypot, as well as other meta-data such as the source IP address, in order to understand the attack and the attacker’s intent to better protect their networks from com- promise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Honeypots therefore provide cyber analysts with session data, where each session is comprised of multiple commands issued by the user;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' each command can be interpreted as a sequence of instructions in command language (for example, shell programming languages), similar to words in natural language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Honeypot session data provide a rare insight into the operational techniques of cyber at- tackers, such as their automated or interactive nature, the individual scripting styles and their overall objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' This makes honeypot tracking systems particularly attractive for devel- oping robust quantitative methods for cyber-security (Highnam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' The volume of traffic passing through a honeypot can be surprisingly high, and so automating the under- standing of these sessions, classifying them and detecting new emerging patterns provides a challenging research problem which is addressed in this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Keywords and phrases: model-based clustering, statistical cyber-security, topic modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='02505v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='CR] 6 Jan 2023 2 Typically, attackers have one main objective after gaining access to a network host.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' For ex- ample, an intruder might want to infect the machine with ransomware, build a cryptocurrency miner, take over existing infrastructure, copy information for data leakage or sale, or collect intel about the organisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Therefore, each observed session could be thought to have an underlying latent intent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Importantly, such intents evolve and change over time, creating new threats for the security of cyber-systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' From a statistical perspective, the problem of es- timating latent intents from a collection of attempted attacks can be framed as a clustering task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Hence, the main objective of this work is to develop clustering models for command line data observed in cyber-security applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Such clustering models could then be used for automated online classification of network intrusions, providing a valuable tool for threat experts and enterprises to discover underlying patterns that would have not been easily de- tectable otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Automated threat detection can be viewed as complementary to deter- ministic classification frameworks for enterprise attacks (for example, MITRE ATT&CK1), providing a further level of sophistication to attack pattern detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' The analysis of attacker behaviour from command logs has been mainly studied from a machine learning perspective in the literature (Shrivastava, Bashir and Hota, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Crespi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Sadique and Sengupta, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' In the present work, ideas borrowed from the lit- erature on topic modelling in text analysis are used to to detect attack patterns, with sessions playing the role of documents and commands playing the role of sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Command line instructions are modelled under a bag-of-words assumption, leading to a generative model for the instructions dependent on the latent intent characterising the corresponding session.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' This fundamentally differentiates our approach from mixed membership strategies to language modelling, such as Latent Dirichlet Allocation (LDA, Blei, Ng and Jordan, 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' The draw- backs of LDA for the scope of modelling command lines are threefold: (i) attackers usually have mainly one intent per session;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' (ii) interpreting the results of a mixed-membership model for attack pattern detection is complex for analysts and threat experts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' (iii) models based on LDA often present unidentifiability and convergence difficulties, making reproducibility of results problematic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Such difficulties are addressed in this work, presenting an approach that assigns a single topic, or intent, to each session, providing a single label to threat ex- perts which is then easy to interpret through statistical summaries of sessions assigned to the same group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Furthermore, one class of proposed models incorporate the additional idea of command-level intents, establishing a two-level clustering structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' This approach appears to alleviate the convergence issues observed with LDA on session data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Later models and inferential procedures discussed in this work admit the possibility of an unknown and unbounded number of latent intents and an unbounded vocabulary size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' These extensions are particularly important in computer network security, since attack vectors fre- quently evolve meaning new intents to arise, and new command line instructions will appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' The number of topics in LDA models is usually chosen using scree-plot criteria using the perplexities calculated from a holdout dataset (Teh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=', 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' However, optimising for perplexity might not yield interpretable topics (Ding, Nallapati and Xiang, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' In this work, an alternative strategy based on Bayesian hierarchical nonparametric Griffiths-Engen- McCloskey priors (GEM, Pitman, 2006) is used, admitting the possibility of previously un- observed intents and instructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' The rest of the article is structured as follows: in the remainder of this section, the data sources used in this work are described along with a review of the related literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Sec- tion 2 describes models for session data, and Section 3 presents inferential procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' The methodology is then extended to the cases of unbounded numbers of topics and vocabulary size in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Finally, the proposed methods are applied to real-world session data from honeypots in Section 5, and the practical implications of the results are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' 1For more details, see https://attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='mitre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='org/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' UNSUPERVISED ATTACK PATTERN DETECTION IN HONEYPOT DATA 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Honeypot session data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' When a user connects to a honeypot through certain pro- tocols, a session starts, and every action the user performs on this host is recorded until logout, when the session ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' A user will run a sequence of commands, which are strings of code which perform actions on the host.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Each command comprises a sequence of words drawn from the syntax of the chosen protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' In the following example session, the in- truder first attempts to access a convenient directory (through multiples uses of the cd com- mand), then tries three methods of downloading a bash script from the web (wget, curl and tftp get, representing different commands having the same underlying intention), before attempting to execute and delete the script.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' The real IP address which was used in the attack is masked using the string abc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='ghi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='jkl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' cd /tmp || cd /var/run || cd /mnt || cd /root || cd / wget http://abc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='ghi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='jkl/Zerow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='sh curl -O http://abc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='ghi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='jkl/Zerow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='sh chmod 777 Zerow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='sh sh Zerow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='sh tftp abc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='ghi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='jkl -c get tZerow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='sh chmod 777 tZerow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='sh sh tZerow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='sh rm -rf Zerow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='sh tZerow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='sh Note that transforming commands into sequences of words, known as tokenisation in the literature, is not a trivial task in cyber-security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' For example, consider the web address http://abc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='ghi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='jkl/Zerow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='sh, which appeared in the second command of the session.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' One might consider the entire string as a word, or split it into different words, such as http, abc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='ghi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='jkl and Zerow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='sh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Furthermore, the entire Internet Pro- tocol (IP) address abc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='ghi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='jkl could be considered as a word, or just its subnet abc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='def, for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Similarly, Zerow could be considered as an individual word, ex- cluding the file extension .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='sh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' More details around the preprocessing of session data will be given in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Related literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' In topic models, each document is usually considered as a bag- of-words, and the words are assumed to be exchangeable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Under this assumption, the in- formation carried by paragraphs and sentences in natural language is lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' In cyber-security, documents correspond to sessions and sentences correspond to commands, which are ex- pected to have a specific intent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' For attack pattern detection, it would be informative to also capture such latent intents at the command-level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' In the literature, document and sentence clustering have been considered as two independent problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' The problem of clustering documents has been extensively studied in the natural language processing, computer science and information retrieval communities (for a survey, see Ag- garwal and Zhai, 2012, and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Common approaches include matrix factori- sation techniques (Xu, Liu and Gong, 2003) and spectral clustering (Cai, He and Han, 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Furthermore, Wallach (2008) proposed a cluster-based topic model extending LDA, where each group is assigned a cluster-specific Dirichlet prior on the document-specific topic dis- tribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Xie and Xing (2013) also propose a multi-grain topic model with clustering where documents are assigned global and group-specific topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Sentence-level structure within topic models has been largely overlooked in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Balikas, Amini and Clausel (2016) propose to extend LDA by sampling words from sentence- specific topic distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Furthermore, Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' (2019) propose to model the sentence- specific topic distribution as a mixture between the topic distributions of adjacent sentences, weighted by a topic association matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' In the present article, a new framework is proposed which permits to joint inference of the latent structure for both sessions and commands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' 4 Usually, one of the main difficulties for LDA models is topic interpretability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Sparse topic models (Williamson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=', 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Archambeau, Lakshminarayanan and Bouchard, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Zhang, 2020) alleviate this issue by enforcing sparsity in the topic-specific word distribu- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Doshi-Velez, Wallace and Adams (2015) proposed Graph-Sparse LDA, that used rela- tionships between words to improve interpretability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Also, the performance of LDA methods heavily relies on suitable preprocessing of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' For example, high-frequency words are often removed, under the assumption that such words make limited contributions to the mean- ing of the documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' In order to avoid data pruning, alternative term-weighting schemes have been proposed in the literature (Wilson and Chew, 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Here a further possible so- lution is proposed: a secondary topic, shared across all documents, can be used to capture high-frequency words and lead to more interpretable primary topics characterising individ- ual documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Another possible explanation of the issues of LDA with high-frequency terms is that, in natural language, word counts have a power-law distribution (Sato and Nakagawa, 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Therefore, Sato and Nakagawa (2010) proposed a Pitman-Yor LDA model, which admits power-laws by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Another approach to the problem of modelling power-laws is the latent IBP compound Dirichlet Allocation model (Archambeau, Lakshminarayanan and Bouchard, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' It is unclear whether power-laws apply to the word counts in command line data and cyber-security applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Such structures could be easily accounted for in the methodology proposed in this paper via two-parameter GEM prior distributions, correspond- ing to stick-breaking proportions of a Pitman-Yor process (Pitman, 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Cyber-security applications require the number of latent intents and vocabulary to be un- bounded for practical deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Hierarchical Dirichlet Processes (Teh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=', 2006) have been successfully used within the context of LDA models to admit an unbounded number of topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Furthermore, Zhai and Boyd-Graber (2013) developed an online LDA algorithm with unbounded vocabulary size, proposing a multinomial and n-gram prior distribution for a conventional character language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' However, n-grams tend to suffer from data sparsity issues (Allison, Guthrie and Guthrie, 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' In this work, the words are simply interpreted as to- kens, and therefore a GEM prior is employed instead, corresponding to a prior distribution over the natural numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' This strategy avoids the difficulty of specifying a prior distribution on the command line syntax, which would be an undesirable additional task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Furthermore, GEM priors are also assigned to the number of latent topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' It must be remarked that the task of estimating the number of components in a finite mixture model presents issues with consistency both under finite parametric (Cai, Campbell and Broderick, 2021) and nonpara- metric priors (Miller and Harrison, 2013, 2014) under model misspecification, unless a prior distribution on the concentration parameter of the Dirichlet process is appropriately specified (Ascolani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Therefore, in practice, it is not expected to always recover the exact number in the data generating process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' In cyber-security, command line data have been mainly analysed using two different ap- proaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' The first class of studies uses machine learning techniques to understand attacker behaviour from command logs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Notably, Sadique and Sengupta (2021) aim to predict the next command of the attacker by using an edit distance training model on the sequence of commands input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' In a similar setup, Crespi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' (2021) aim to identify attacker behaviours from command logs using supervised NLP methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Lastly, Shrivastava, Bashir and Hota (2019) focus on classifying types of attacks from commands using a series of machine learn- ing techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' The second class of approaches analyses attacker behaviour from session data using Hidden Markov Models, as seen in the studies of Rade et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' (2018) and Desh- mukh, Rade and Kazi (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' However, none of the aforementioned studies consider topic modelling approaches for the analysis of sessions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' UNSUPERVISED ATTACK PATTERN DETECTION IN HONEYPOT DATA 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Models for clustering session data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Command line data are observed in sessions, where each session is divided into commands, and each command is composed of different words drawn from a vocabulary V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Following the standard LDA model (Blei, Ng and Jordan, 2003), for D observed sessions the number of commands Nd in session d and the number Md,j of words in each command j within that session are assumed to be Poisson distributed: Nd ∼ Poisson(ζ), d = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=',D, Md,j ∼ Poisson(ω), j = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=',Nd, ζ,ω ∈ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' The i-th word in the j-th command of the d-th session is denoted wd,j,i, which has a corresponding probability mass function ξd,j,i ∈ R|V | + over the vocabulary V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' The stated aim is to develop clustering algorithms for sessions, where the clusters represent shared intents of the intruders, or groups of attackers with similar behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' To achieve this aim, a range of topic model structures are now considered, establishing shared distributions ξd,j,i across groups of sessions and commands to identify clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' In particular, this work focuses on two approaches: (i) Constrained: Each session has a primary topic and a global secondary topic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' (ii) Hierarchical: Each session topic is a distribution on command-level top- ics, which introduces two layers of latent topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' The two modelling approaches are discussed in detail in the next sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Constrained topic modelling with primary and secondary topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' As a most basic approach, each document d could have a latent assignment to one of K possible topics, where each topic k ∈ {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=',K} is characterised by a probability mass function φk on the vocabulary V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Let td denote the topic assignment for document d, and let λ = (λ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=',λK) where λk denotes the probability that td = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' It could then be assumed ξd,j,i = φtd, such that conditional on the session-specific topic td, all the words wd,j,i in session d are sampled from the same distribution φtd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Assuming conjugate Dirichlet prior distributions for the probability distributions λ and {φk} implies the following model: λ ∼ Dirichlet(γ), φk ∼ Dirichlet(η), k = 1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=',K, td | λ ∼ Categorical(λ), d = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=',D, wd,j,i | td,{φk} ∼ Categorical(φtd), i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=',Md,j, j = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=',Nd, (1) where γ ∈ RK +,η ∈ R|V | + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' In the cyber-attack context, these topics correspond to different intents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' The simple model above can be used as a starting point for exploring more complex clus- tering structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' For example, to better identify differences between topic-specific distribu- tions, it could be assumed that words in a document are sampled either from a topic-specific probability distribution, or from a baseline probability distribution shared across all docu- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' The shared distribution represents words that are commonly used in all sessions, but are not key for characterising the intent of a session.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' For example, in natural language, such a baseline distribution could give probability mass to conjunctions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' but, and, if), articles (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' a, an, the), or pronouns (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' she, he, they).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Similarly, for command lines in a cyber- security context, the shared distribution might give weight to common bash commands such as ls (list contents), ps (list the running processes), or cd (change directory).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' The shared distribution will be used to make the session-specific topics more representative of the at- tacker’s intents, excluding commonly used words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' 6 In particular, an extended model assumes each topic has an associated probability θk ∈ [0,1], k = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=',K, representing the mixing proportion between the topic-specific word dis- tribution φk and the shared distribution φ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Each word is then sampled with probability θtd from φtd, or from φ0 with probability 1 − θtd, implying the following revised model: φk ∼ Dirichlet(η), k = 0,1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=',K, θk ∼ Beta(αk,α0), k = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=',K, zd,j,i | td,{θk} ∼ Bernoulli(θtd),i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=',Md,j, j = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=',Nd, wd,j,i | zd,j,i,td,{φk} ∼ Categorical(φtdzd,j,i), i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=',Md,j, j = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=',Nd,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' (2) This model essentially imposes a sparsity constraint on LDA, by assuming that each docu- ment contains only two topics: (i) a primary topic td, chosen from K primary topics, and (ii) a secondary topic shared across all documents, denoted “topic 0” for notational convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Hierarchical constrained topic modelling with session-level and command-level clus- tering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' The models in (1) and (2) assume that words in each command within a given session are sampled from the same topic-specific distribution, or from a distribution shared across documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' The information about the structure of a session as a sequence of commands is therefore ignored, which might be limiting in practical settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Instead, it would be reason- able to assume that session-specific intents share similar commands for specific tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Such tasks could be interpreted as command-level intents, and the distribution of the tasks char- acterises the session-level topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Let H be the assumed number of command-level topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' It could be assumed that each session-level topic has an associated H-dimensional probability distribution ψk across command-level intents, with each command within a given session being assigned a command-specific topic sd,j ∈ {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=',H} sampled from ψtd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Conditional on sd,j, the words in the command are then sampled independently from a |V |-dimensional probability distribution φsd,j, specific to the command-level topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Therefore, the model in (1) is extended as follows: ψk ∼ Dirichlet(τ), k = 1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=',K, φh ∼ Dirichlet(η), h = 1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=',H, sd,j | td,{ψk} ∼ Categorical(ψtd), j = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=',Nd, (3) wd,j,i | sd,j,{φh} ∼ Categorical(φsd,j), i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=',Md,j, j = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=',Nd, where τ ∈ RH +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' In this model, there are two layers of topics, and corresponding indices: (i) Command topic indices, sd,j, used to match the words in the corresponding command to distributions φ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=',φH over V ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' (ii) Document topic indices, td, used to match the com- mands in the corresponding session to distributions ψ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=',ψK over the command-level top- ics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Letting Φ be the H × |V | matrix with j-th row φj, and letting Ψ be the K × H matrix with k-th row ψk, then marginally ξd,j,i = λ⊤ · Ψ · Φ, whereas ξd,j,i = φsj,d conditionally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Combining the two approaches: hierarchical constrained topic modelling with sec- ondary topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' In order to aid interpretability of the command-level topics, it is possible to use the same constraint from model (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' In particular, it could be assumed that words are sam- pled either from φsd,j, where sd,j is the command-level topic, or from a distribution φ0 shared across all commands and sessions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' As in (2), each command-level topic h ∈ {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=',H} has an associated mixture probability θh ∈ [0,1] for sampling words from φh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Therefore, the full model, which combines (1), (2) and (3), takes the following form: λ ∼ Dirichlet(γ), UNSUPERVISED ATTACK PATTERN DETECTION IN HONEYPOT DATA 7 1 2 1 1 2 2 1 K = 2 H = 3 λ ψ1 ψ2 td ∼ λ D sessions t2 Sessions (Documents) 1 1 2 1 3 2 θ1 θ2 θ3 Commands (Sentences) Nd commands s2,4 sd,j ∼ ψtd 1 0 1 0 1 Md,j words, P(zd,j,i = 1) = θsd,j Indicators Words tftp -g abc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='ghi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='jkl -r tftp1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='sh w2,4,2 wd,j,i ∼ φzd,j,isd,j FIG 1: Cartoon representation of the full Hierarchical Constrained Topic Model (HCTM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' ψk ∼ Dirichlet(τ), k = 1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=',K, φh ∼ Dirichlet(η), h = 0,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=',H, θh ∼ Beta(αh,α0), h = 1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=',H, Nd ∼ Poisson(ζ), d = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=',D, Md,j ∼ Poisson(ω), d = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=',D, j = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=',Nd, td | λ ∼ Categorical(λ), d = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=',D, sd,j | td,{ψk} ∼ Categorical(ψtd), j = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=',Nd, zd,j,i | sd,j,{θh} ∼ Bernoulli(θsd,j),i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=',Md,j, wd,j,i | zd,j,i,sd,j,{φh} ∼ Categorical(φsd,jzd,j,i), i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=',Md,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' (4) A pictorial representation of model (4) is given in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Note that it might be possible to consider variations of (4) through making changes to the specification of the prior distri- butions on the hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' For example, document-specific mixing topic proportions θd ∼ Beta(α,α0) could be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' The next section will describe inferential methods for model (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Deriving the inferential procedures for (1), (2) and (3) follows similar guidelines, with minor modifications to the equations used for (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Bayesian inference via Markov Chain Monte Carlo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' This section describes infer- ential procedures for the topic models discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' The full model is considered, with primary-secondary topics and session-level and command-level clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' The pos- terior distribution of the parameters is only available up to a normalising constant, therefore inference must be performed using Markov Chain Monte Carlo (MCMC) methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' The main objective of the inferential procedure is estimating t, the session-level clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Hence, the re- maining parameters could be interpreted as nuisance, and integrated out when possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' The parameters θ,λ,{φh} and {ψk} can be analytically marginalised, resulting in the marginal 8 posterior density (5) p(z,s,t | w) ∝ p(w,z,s,t) = p(t) × p(s | t) × p(z | s) × p(w | z,s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Each term in the right-hand side of the marginal posterior (5) can be calculated explicitly by conjugacy of the Categorical-Dirichlet and Beta-Bernoulli distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' The marginal distribution for the session-level intents is: (6) p(t) = B(γ + T ) B(γ) , where B(x) = � i Γ(xi)/Γ(� i xi) is the multivariate beta function, and T = (T1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=',TK), where Tk = � d I{k}(td) denotes the number of sessions assigned to topic k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Similar cal- culations lead to the marginal distribution for the command-level topics, conditional on the session-level intents: (7) p(s | t) = K � k=1 B(τ + Sk) B(τ) , where Sk = (S1,k,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=',SH,k), and Sk,h = � d:td=k � j I{h}(sd,j) denotes the number of com- mands assigned to the command-level topic h, only from the subset of sessions with session- level topic k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Similarly, the marginal distribution of the primary-secondary topic indicators z has a closed form expression from the Beta-Bernoulli conjugacy: (8) p(z | s) = H � h=1 B(Zh + αh,M∗ h − Zh + α0) B(αh,α0) , where Zh = � (d,j):sd,j=h �Md,j i=1 zd,j,i denotes the number of words assigned to the primary topic, only from commands with command-level topic h, and M∗ h = � (d,j):sd,j=h Md,j de- notes the total number of words in commands with topic h, across all documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' The final component of the marginal posterior (5) is the marginal likelihood for the observed words w, conditional on the indicators z and topic-level allocations s: (9) p(w | z,s) = H � h=0 B(Wh + η) B(η) , where Wh = (Wh,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=',Wh,|V |), and Wh,v = � i,j,d I{h}(zd,j,isd,j)I{v}(wd,j,i) denotes the number of times word v is assigned to the command-level topic h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' The marginal distributions (6), (7), (8) and (9) are the building blocks for the collapsed Gibbs sampler (Liu, 1994) used for inference on the model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' The Gibbs sampler consists of three basic moves: resample the session-level topic allocations t, resample the command-level topic allocations s, and resample the primary-secondary topic indicators z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Furthermore, convergence of Gibbs sampling algorithms for clustering usually benefits from split-merge proposals, which are evaluated using a Metropolis-Hastings acceptance ratio, resulting in a collapsed Metropolis-within-Gibbs algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' In this framework, split-merge moves can be used on the session-level topics t and command-level topics s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' The next sub- sections give a detailed description of the steps required in the Gibbs sampling algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Resampling the session-level topic allocations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' The Gibbs sampler requires to sam- ple from the conditional distribution of a subset of the parameters, conditional on the ob- served data and remaining parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Therefore, for resampling the session-level topic al- location td for a given document, it is required to sample from p(td | t−d,w,z,s), where the UNSUPERVISED ATTACK PATTERN DETECTION IN HONEYPOT DATA 9 superscript −d denotes that the calculations of the corresponding quantity exclude the d-th document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' For the r-th session-level topic, the probability can be written as: p(td = r | t−d,w,z,s) ∝ p(td = r | t−d)p(s | td = r,t−d) ∝ B(γ + T ) B(γ + T −d) K � k=1 B(τ +Sk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' where the quantities T and Sk in the final part of the expression are calculated assuming that td = r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' The ratio of beta functions in the conditional distribution can be simplified us- ing the properties of the gamma function, yielding B(γ + T )/B(γ + T −d) ∝ (γr + T −d r ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Similarly, the product of beta functions in the final part of the expression could be further simplified using the fact that Sr,h = S−d r,h + Sd h, where S−d k,h = � u:tu=k,u̸=d � j I{h}(sj,u) and Sd h = � j I{h}(sd,j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' From the properties of the gamma function, the probability can be then expressed as: (10) p(td = r | t−d,w,z,s) ∝ (γr + T −d r ) �H h=1 �Sd h ℓ=1(τh + S−d r,h + ℓ − 1) �Nd ℓ=1{�H h=1(τh + S−d r,h) + ℓ − 1} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Resampling the command-level topic allocations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' The Gibbs sampler also re- quires to sample the command-level topic allocations from the distribution p(sd,j = ℓ | s−d,j,w,t,z), where the superscript denotes that the quantities have been calculated exclud- ing the (d,j)-th terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' For the ℓ-th command-level topic, the probability can be factorised as: (11) p(sd,j = ℓ | s−d,j,w,t,z) ∝ ∝ p(sd,j = ℓ,s−d,j | t) × p(z | sd,j = ℓ,s−d,j) × p(w | sd,j = ℓ,s−d,j,z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' The first term in the factorisation (11), corresponding to (7), can be simplified noting that Std,ℓ = S−d,j td,ℓ + 1: (12) p(sd,j = ℓ,s−d,j | t) ∝ (τℓ + S−d,j td,ℓ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' A similar reasoning could be used to simplify the second term in the factorisation (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' In particular, Zℓ = Z−d,j ℓ + Zd,j, where Z−d,j h = � (u,q):su,q=h,(u,q)̸=(d,j) �Md,j i=1 zu,q,i and Zd,j = �Md,j i=1 zd,j,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Using the properties of the gamma function, the corresponding proba- bility is expressed as: (13) p(z | sd,j = ℓ,s−d,j) ∝ ∝ �Zd,j−1 q=0 (αℓ + Z−d,j ℓ + q)�Md,j−Zd,j−1 u=0 (α0 + M∗−d,j ℓ − Z−d,j ℓ + u) �Md,j−1 q=0 (α0 + αℓ + M∗−d,j ℓ + q) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' The last term in (11) admits a similar simplification to (10), using Wℓ,v = W −d,j ℓ,v + W d,j v , where W −d,j h,v = � u,q,i:zu,q,isu,q=h,(u,q)̸=(d,j) I{v}(wu,q,i), W d,j v = � i:zd,j,isd,j̸=0 I{v}(wd,j,i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' The resulting probability is: (14) p(w | sd,j = ℓ,s−d,j,z) ∝ �|V | v=1 �W d,j v q=1 (ηv + W −d,j ℓ,v + q − 1) �� v W d,j v q=1 {�|V | v=1(ηv + W −d,j ℓ,v ) + q − 1} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' The probability (11) is obtained by calculating the product of the terms (12), (13), (14), and normalising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' 10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Resampling the primary-secondary topic indicators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' In the models with primary- secondary topics, the Gibbs sampler also requires to resample the binary indicators zd,j,i, conditional on w,s,t and z−d,j,i, denoting all the indicators except zd,j,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Each binary indi- cator is drawn from a Bernoulli distribution with unnormalised probabilities: (15) p(zd,j,i = b | z−d,j,i,w,s,t) ∝ p(zd,j,i = b | z−d,j,i,s) × p(w | zd,j,i = b,z−d,j,i,s), where b ∈ {0,1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Noting that Zsd,j = Z−d,j,i sd,j + b, where the term Z−d,j,i sd,j is defined as Z−d,j,i h = � (u,q):su,q=h,(u,q)̸=(d,j) �Mu,q i=1 zu,q,i, the first term in the factorisation becomes: p(zd,j,i = b | z−d,j,i,s) ∝ (αsd,j + Z−d,j,i sd,j )b(α0 + M∗ sd,j − Z−d,j,i sd,j − 1)1−b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' For the marginal likelihood of observed words, the only terms affected by a change in the binary indicator zd,j,i are W0,wd,j,i = W −d,j,i 0,wd,j,i + 1 − b and Wsd,j,wd,j,i = W −d,j,i sd,j,wd,j,i + b, where W −d,j,i h,v = � (u,q,r):zu,q,rsu,q=h,(u,q,r)̸=(d,j,i) I{v}(wu,q,r), giving: p(w | zd,j,i = b,z−d,j,i,s) ∝ � W −d,j,i 0,wd,j,i + ηwd,j,i �|V | v=1(W −d,j,i 0,v + ηv) �1−b � W −d,j,i sd,j,wd,j,i + ηwd,j,i �|V | v=1(W −d,j,i sd,j,v + ηv) �b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Split-merge topic allocations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' There are two types of split-merge moves that can be proposed: (i) split-merge session-level topics, and (ii) split-merge command-level topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' For the split-merge move on session-level topics, two sessions d and d′ are sampled at random from the D observed sessions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' If td = td′ = t∗, the proposal for the session-level topics splits the sessions assigned to t∗ in two different clusters using the following iterative procedure: (i) assign topic t∗ to document d, and topic ˜t to document d′ (where ˜t corresponds to the number of non-empty clusters, plus one – note that if ˜t > K the split move should be immediately rejected);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' (ii) documents previously assigned to topic t∗ are sequentially allocated to topics t∗ or ˜t in random order, with probabilities proportional to the predictive distribution (10), restricted to the session already reallocated to topics t∗ and ˜t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' This allocation procedure is adapted from common split-merge MCMC moves in related clustering problems (see, for example, Dahl, 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Sanna Passino and Heard, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' The final proposal is denoted as t∗, with probability q(t∗ | t), corresponding to the product of sequential probabilities obtained in the splitting procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' The resulting acceptance probability for the move from t to t∗ is: (16) min � 1, p(t∗)p(s | t∗) p(t)p(s | t)q(t∗ | t) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' On the other hand, if td ̸= td′, the proposal t∗ assigns topic td to all documents previously given topic td′, corresponding to a merge move.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' In this case, the acceptance ratio in (16) must be further multiplied by the proposal probability q(t | t∗), calculated by simulating a split move from t∗ to t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Since there is only one way to merge two topics, the proposal probability at the denominator of (16) is q(t∗ | t) = 1 for a merge move.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' A similar split-merge move can be constructed for the command-level topics: two com- mands j and j′ are randomly sampled from two random documents d and d′ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' If sd,j ̸= sj′,d′, a merge move is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Alternatively, if sd,j = sj′,d′ = s∗, the split move proceeds similarly to the procedure described for the topic-level sessions, and the command previously assigned topic s∗ are sequentially allocated to s∗ or ˜s (corresponding to the num- ber of non-empty command-level topics, plus one) with probabilities proportional to the pre- dictive distribution (11), limited to the commands already reassigned to s∗ and ˜s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' As be- fore, if ˜s > H, the move is rejected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' In summary, the acceptance probability for a vector of command-level topics s∗ obtained via the split-merge procedure is: min � 1, p(s∗ | t)p(z | s∗)p(w | z,s∗)q(s | s∗) p(s | t)p(z | s)p(w | z,s)q(s∗ | s) � , UNSUPERVISED ATTACK PATTERN DETECTION IN HONEYPOT DATA 11 where the probabilities q(s∗ | s) and q(s | s∗) are either 1, or the product of allocation prob- abilities calculated from the sequential splitting procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Initialisation schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' In MCMC, setting good initial values could be helpful to achieve faster convergence, in particular for complex inferential tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' In this work, two methods for initialisation are considered, based on spectral clustering and standard LDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Spectral methods are commonly used for text analysis and topic modelling (Ke and Wang, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' In order to initialise the algorithm via spectral clustering, a (�D d=1 Nd) × |V | word occurrence matrix C = {Csw} is constructed, where Csw counts the number of times word w appears in command s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Note that all commands are stacked in an individual matrix C, initially disregarding information about the division into sessions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' A truncated singular value decomposition of C is then calculated, considering only the largest H singular values and corresponding left singular vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' A clustering algorithm, like k-means, is then run on the resulting embedding, setting H clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' For initialisation of the session-level topics, a similar procedure is carried out, using the initial values of the command-level topics as words in a spectral clustering algorithm, obtaining a different form of the matrix of counts C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' First, the matrix C, with dimension D × H, is constructed, where each entry Cdh counts the number of times a command assigned to the command-level topic h appears in document d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Then, a K-dimensional truncated spectral decomposition of C is calculated, and k-means with K clusters is run on the resulting embedding, obtaining initial values for the session-level topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Alternatively, standard LDA could be used to initialise the MCMC sampler, via fast- performing software libraries such as python’s gensim ( ˇReh˚uˇrek and Sojka, 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' First, LDA with H topics could be fitted, and subsequently used to predict a topic for all the words appearing in commands and sessions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Then the most common estimated topic within each command is selected as the initial command-level topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' If secondary topics are used, LDA is initially fitted with H + 1 topics, and the most common estimated topic is selected as sec- ondary topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' The command-level primary topic is then selected as the most common topic within each command, excluding the secondary topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' After command-level topics are es- timated, session-level topics could be initialised by running LDA with K topics, using the estimated command-level topics as words within the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' For each command-level topic, now interpreted as a word, a topic can be estimated from the fitted LDA model, and the session-level topics are then initialised as the most common topic within each command.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' For initialisation of the primary-secondary topic indicators, zd,j,i could be initially set to 1 if the proportion of sessions or commands where the word wd,j,i appears is less than a pre- specified threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' This is because φ0 should represent a distribution of common words, shared across topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Unbounded number of topics and vocabulary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' All models discussed in Section 2 assume a fixed size |V | of the vocabulary, and a fixed number of session-level and command- level topics, K and H respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Such assumptions might be problematic if the model is used for clustering future sessions, since it would not be possible to cluster new commands, composed of words not present in the vocabulary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Therefore, a potentially infinite vocabulary must be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Also, the behaviour of attackers is expected to evolve and change over time, and it is possible that new attack patterns or intents arise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Therefore, for real-world attack pattern detection, it is beneficial to assume an unbounded number of session-level and command-level topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' These allowances require a modification to the Dirichlet distributions used in Section 2, instead assuming: λ ∼ GEM(γ), ψk ∼ GEM(τ), k = 0,1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=', φℓ ∼ GEM(η), ℓ = 0,1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=', 12 where τ,η,γ ∈ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' The GEM (Griffiths-Engen-McCloskey) distribution (Pitman, 2006) cor- responds to the proportions calculated using the stick-breaking representations of the Dirich- let process (Sethuraman, 1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Hence, the GEM distribution also corresponds to the limit for K → ∞, H → ∞ and |V | → ∞ of the Dirichlet distributions in Section 2 with γ = γ1K/K, τ = τ1H/H, and η = η1|V |/|V |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' To simplify the discussion on the GEM distribution, its link to the Dirichlet process, and its representation in the posterior distribution, consider n objects allocated to Kn non-empty groups, with labels xn = (x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=',xn), such that xi ∈ N and max(xn) = Kn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Under a Dirichlet process with scaling parameter β, the predictive dis- tribution for the next label in the sequence is: (17) p(xn+1 | xn) = β β + nI{Kn+1}(xn+1) + Kn � k=1 Nkn β + nI{k}(xn+1), where Nkn = �n i=1 I{k}(xi) is the number of the n objects allocated to group k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' The predic- tive equation (17) immediately provides a technique for Gibbs sampling: since the Dirichlet process assumes exchangeability of observations, any label can be considered as the last el- ement of the sequence, and a new value resampled using (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' This fact will be particularly useful when implementing the sampler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Using (17), the joint distribution for the sequence is: p(xn) = n � j=1 p(xj | xj−1) = αKnΓ(α) Γ(α + n) Kn � k=1 Γ(Nkn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' It follows that the components of the marginalised posterior distribution (5) take the following revised form: p(t) = γK(t)Γ(γ) Γ(γ + D) K(t) � k=1 Γ(Tk),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' (18) p(s | t) = K(t) � k=1 τ �H(s) h=1 IN+(Sk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='h)Γ(τ) Γ(τ + � d:td=k Nd) � h:Sk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='h>0 Γ(Sk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='h),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' p(z | s) = H(s) � h=1 B(Zh + αh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='M∗ h − Zh + α0) B(αh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='α0) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' p(w | z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='s) = H(s) � h=0 η �V (w) v=1 IN+(Wh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='v)Γ(η) Γ(η + �V (w) v=1 Wh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='v) � v:Wh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='v>0 Γ(Wh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='v),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' where K(t) = �∞ k=1 IN+(Tk) and H(s) = �∞ h=1 IN+(�∞ k=1 Sk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='h) are the number of unique session-level and command-level topics respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' and V (w) = �∞ v=1 IN+(�∞ h=0 Wh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='v) is the observed number of unique words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Bayesian inference with GEM priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Inference in the model with unbounded num- ber of topics and vocabulary size can be carried out using a similar algorithm to the Gibbs sampling described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Only minor modifications are required, since the marginals in (18) take a different form under the GEM priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' For example, for resampling the session- level topics, the probabilities in (10) are modified as follows: p(td = r | t−d,w,z,s) ∝ γI{K(t−d)+1}(r)(T −d r )1−I{K(t−d)+1}(r) × � h:Sd h>0 τ I{0}(S−d r,h)(S−d r,h)1−I{0}(S−d r,h) ��Sd h ℓ=2(S−d r,h + ℓ − 1) �IN>1(Sd h) �Nd ℓ=1{τ + (�H(s) h=1 S−d r,h) + ℓ − 1} , UNSUPERVISED ATTACK PATTERN DETECTION IN HONEYPOT DATA 13 where r ∈ {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=',K(t−d) + 1}, and the convention 00 = 1 is adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Similarly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' the proba- bilities in (11) for resampling command-level topics become: p(sd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='j = ℓ | s−d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='z) ∝ τ I{h:Std,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='h=0}(ℓ)(S−d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='j td,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='ℓ )1−I{h:Std,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='h=0}(ℓ) × � v:W d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='j v >0 ηI{0}(W −d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='j ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='v )(W −d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='j ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='v )1−I{0}(W −d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='j ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='v ) ��W d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='j v q=2 (W −d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='j ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='v + q − 1) �IN>1(W d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='j v ) �� v W d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='j v q=1 {η + (�V (w) v=1 W −d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='j ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='v ) + q − 1} × �Zd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='j q=1(αℓ + Z−d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='j ℓ + q − 1)�Md,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='j−Zd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='j u=1 (α0 + M∗−d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='j ℓ − Z−d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='j ℓ + u − 1) �Md,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='j q=1 (α0 + αℓ + M∗−d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='j ℓ + q − 1) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' where ℓ ∈ {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=',H(s−d,j) + 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' A modification is required also for the conditional proba- bilities of resampling the indicator zd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='i in (15),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' resulting in: p(zd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='i = b | z−d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='t) ∝ (αsd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='j + Z−d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='i sd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='j )b(α0 + M∗ sd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='j − Z−d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='i sd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='j − 1)1−b × � � � ηI{0}(W −d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='i 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='wd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='i)(W −d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='i 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='wd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='i)1−I{0}(W −d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='i 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='wd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='i) η + �V (w−d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='i) v=1 W −d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='i 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='v � � � 1−b × � ηI{0}(W −d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='i sd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='j ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='wd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='i)(W −d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='i sd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='wd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='i)1−I{0}(W −d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='i sd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='j ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='wd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='i) η + �V (w−d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='i) v=1 W −d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='i sj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='v �b ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' where b ∈ {0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' The split-merge move in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='4 can be equivalently extended to the model with GEM priors, using the same ideas presented in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Split-merge moves are expected to improve convergence of Gibbs sampling algorithms in models based on Dirich- let processes (Jain and Neal, 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Note that all the probabilities described in this section are extremely similar to the equations in Section 3, with the added complexity of handling previously unseen topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Alternative MCMC algorithms for models based on the Dirichlet processes are extensively discussed in the literature (for example, Neal, 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Ishwaran and James, 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Teh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=', 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Application to the Imperial College London honeypot data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' The models described in Sections 2 and 4 are now applied to real data collected on a honeypot hosted within the Imperial College London (ICL) computer network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Within a time period between 21st May, 2021, and 27th January, 2022, the ICL honeypot collected approximately 40,000 unique ses- sions, observed over 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='3 million times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' This is a large corpus of attacks for a single machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Data preprocessing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' As discussed in the introduction, such sessions and commands must be tokenised to obtain the words and vocabulary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' In this work, the tokenisation is per- formed with the python package NLTK (Bird, Klein and Loper, 2009), setting the regular expression [a-zA-Z0-9_\\.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='\\-\\*]+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Also, commands observed in the ICL honeypot data often contain combinations of strings in hexadecimal form, preceded by the letter x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' An ex- ample of such a command is bin busybox echo -e x6b x61 x6d x69 dev dev .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='nippon In these analyses, all such instances of hexadecimal strings (x6b, x61, x6d and x69 in the above example) are replaced by the word HEX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Also, some commands display the word GHILIMEA appended to HEX strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' These are replaced with the word GHILIMEA_word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Similarly to standard preprocessing techniques in natural language processing, extremely 14 rare and extremely common words are removed from the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' For the ICL honeypot data, words appearing in less than 10 commands were removed, as were words appearing in over 10% of commands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Such words are often denoted stopwords in natural language processing and information retrieval (see, for example, Manning, Raghavan and Schütze, 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' After preprocessing, a vocabulary of 1,003 unique words is obtained, and 2,617 uniquely observed sessions, for a total of 42,640 commands and 261,283 words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Each session has an average of 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='29 commands (with median 15), whereas each command contains on average 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='12 words (with median 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Users who access the honeypot have malicious intent, and it is not uncommon to observe swear words and discriminatory language in many of the sessions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Those terms have been redacted in plots of the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Topic estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Before describing the results, some practical details about the es- timation of topics from MCMC chains are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' In general, the number of session-level or command-level topics are unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' The Dirichlet priors for λ and {φh} in Section 2 assume fixed, pre-specified values of K and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' For inference with the Dirichlet prior, a max- imum number of possible topics could be chosen, denoted Kmax and Hmax, and the underlying number of topics could be estimated as the number of non-empty topics at each iteration of the MCMC sampling procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Furthermore, estimates of topic allocations based on the MCMC sampler described in Section 3 could be affected by the issue of label switching (Jasra, Holmes and Stephens, 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Therefore, session-level topic allocations are estimated in this work from the estimated posterior similarity between sessions i and j, calculated as �M s=1 1t⋆ i,s{t⋆ j,s}/M, where M is the total number of posterior samples and t⋆ i,s is the s- th sample for ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' The posterior similarity matrix is invariant to permutations of the labels and therefore unaffected by label switching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' After the posterior similarities are obtained for all pairs of sessions, hierarchical clustering with complete linkage is applied, with distance measure 1 − ˆπij (Medvedovic, Yeung and Bumgarner, 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' A similar procedure could be followed for the command-level topics, but the very large number of commands would make the size of the similarity matrix unfeasible to calculate and store in memory on a ma- chine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Therefore, the last sample from the MCMC chain is considered as the estimate of the command-level topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Constrained topic modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' First, the constrained topic model in (1) is fitted on the postprocessed ICL honeypot data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Under the Dirichlet prior for λ, the hyperparameter γ is set to γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='1 · 1Kmax, with Kmax = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' The hyperparameter of the Dirichlet prior for the topic-specific word distribution is set to η = 1|V |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' The MCMC sampler is run for 250,000 iterations with 50,000 burn-in, initialising the topics via spectral clustering with Kmax clus- ters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' The results are displayed in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' The session-level topics are estimated using the procedure described in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Figure 2a plots the resulting barplot of topic frequen- cies of estimated session-level topics, for a number of topics equal to the modal number of non-empty topics, ˆK∅ = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Furthermore, Figure 2b displays the barplot of the estimated distribution for the number of non-empty topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' The barplot is also compared to the dis- tribution obtained under a GEM prior for λ, with hyperparameter γ = 3, corresponding to Kmax × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='1, using the same setup for the MCMC sampler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' The resulting distributions show agreement, demonstrating a similar performance of the Dirichlet and GEM priors in estimat- ing the modal number of topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' In general, the interplay between the prior parameters η and γ appears to have an effect on the number of small clusters that are estimated from the data: if η increases, the clusters in the right tail of Figure 2a tend to be incorporated within the larger clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' On the other hand, if η decreases towards zero, it is expected to estimate more topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' UNSUPERVISED ATTACK PATTERN DETECTION IN HONEYPOT DATA 15 (A) Barplot of frequencies of estimated session-level topics 2 10 7 11 9 20 14 3 6 12 15 13 5 4 8 16 1 18 17 19 Topic label 0 100 200 300 400 500 600 700 Number of sessions (B) Barplot of estimated K∅ 18 19 20 21 22 Number of topics 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='6 Estimated frequency Dirichlet GEM FIG 2: Estimated topic frequencies and estimated distribution the number of non-empty topics K∅ under the constrained topic model in (1), fitted on the ICL honeypot data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Results: the model discovers a rare and unusual MIRAI variant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' The inferred meaning of each topic is summarised in Table 1, according to the type of sessions that are assigned to each group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' The clusters appear to mostly contain botnets and different variants of MIRAI malware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' MIRAI is a type of botnet first emerged in 2016, which was specifically targeted towards compromising Internet of Things (IoT) devices, or launching Distributed Denial of Service (DDoS) attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Recently, it has been repurposed for Bitcoin mining on IoT devices compromised via brute-force attacks on protocols such as SSH and Telnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Over the years, many different variants of MIRAI have emerged, and other bots with similar struc- ture (Lingenfelter, Vakilinia and Sengupta, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Sadique and Sengupta, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=', 2022), which appear to be assigned to different topics in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Interestingly, careful examination of the sessions assigned to topic 5 estimated via the constrained topic model in (1) helped analysts to discover an rare and unusual variant of MIRAI, called MinerFinder (Bevington, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' The objective of MinerFinder is to look for existing coin miner configurations, and try to gain root privileges to take control of the miner infrastructure, if found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' This demonstrates that the constrained topic model with a single topic per session, even in its simplest form, could be extremely helpful for analysts to discover new attack patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Within topic 5, MinerFinder is also mixed with other more common MIRAI variants, which share a common frequency distribution of words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Ideally, a clustering algorithm should be able to single-out MinerFinder from other MIRAI variants, despite their similarities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' This might be possible when a hierarchical structure is added to the topics, and the command structure is explicitly used, as demonstrated in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Overall, most of the activity on the honeypot seems to be related to attempts to install botnets or coin miners, but topics show remarkable separation between sessions and corre- sponding intents, as demonstrated in the list of malware and objectives in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Further intuition about the differences between topics could be provided by examining the topic- specific word distributions, which can be displayed via wordclouds, plotted in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' The figure shows that words within each topic are fairly heterogeneous, but a number of words appear frequently across multiple topics, such as HEX, cd or sh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' A potential solution to bet- ter visualise and further differentiate topic-specific word distributions would be to introduce a secondary topic, which would capture the distribution of the most common words shared across multiple topics and sessions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' This solution is explored in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Constrained topic modelling with a secondary topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' As discussed in the previous section, a possible solution to aid topic interpretability and further discriminate topic-specific word distributions would be to add a secondary topic to the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' In this section, the con- strained topic model with secondary topic in (2) is therefore fitted to the ICL honeypot data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' 16 TABLE 1 Estimated session-level topics and corresponding intent under the constrained topic model in (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Topic label Type of malware Objective 1 Shellbot Install bot 2 (ptmx) unnamed botnet,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' MIRAI Gather system information,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' change permissions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' execute MIRAI variants 3 MIRAI Download and execute MIRAI variants kura and kurc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' fingerprint system 4 MIRAI Download sora malware,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' write upnp and updDl malware via echoing HEX strings 5 MIRAI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' MinerFinder (new variant) Download malware,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' change permissions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' gather system information,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' fingerprint system 6 Shellbot,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' SBIDIOT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' coin miner Download and execute coin miner and MIRAI malware,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' change SSH keys 7 (s4y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' LAYER) unnamed botnet,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' MIRAI Gather system information,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' change permissions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' execute MIRAI variants 8 Coin miner Download and execute coin mining malware 9 MIRAI Download and execute MIRAI variants PEDO,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' ECCHI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' PEACH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' 10 MIRAI Download and execute MIRAI variants tftp1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='sh, tftp2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='sh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' 11 MIRAI Determine shell executable, check busybox is present, print error message to console 12 (misa) unnamed botnets Gather system information, change permissions, execute MIRAI variants 13 Shellbot, coin miner Scan system, look for GPUs, look for coin miners, download malware 14 MIRAI Download and execute MIRAI variants sora, Pemex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' 15 MIRAI Download and execute MIRAI variant DNXFCOW via echoing single HEX strings 16 MIRAI Download and execute MIRAI variant DNXFCOW via echoing multiple HEX strings 17 MikroTik bot,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' coin miner Gather system information,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' gather MikroTik router information,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' look for coin miners 18 GHILIMEA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' PentaMiner coin miner script Install coin miner,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' kill mining processes with high CPU usage in order to go undetected 19 MikroTik bot Attempt to gain access to MikroTik router 20 Hive OS attack,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' coin miner Download miner,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' attempt to take over configurations in Hive OS mining platform Topic 1 Topic 2 Topic 3 Topic 4 Topic 5 Topic 6 Topic 7 Topic 8 Topic 9 Topic 10 Topic 11 Topic 12 Topic 13 Topic 14 Topic 15 Topic 16 Topic 17 Topic 18 Topic 19 Topic 20 FIG 3: Wordclouds of estimated topic-specific word distributions under the constrained topic model in (1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' fitted on the ICL honeypot data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' The setup of the MCMC sampler is chosen to be identical to the previous section, and the additional hyperparameters are set to α0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='1 and αk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='9 for k = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='..' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='Chmod ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='mika.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' sh exad enable ODEP 0 tftp x61x6d x6dx69 ps PEACH nc bigbotreppin rf ECCHI echo mounts STHUB12 macHelper telnetnotdvrHelperUNSUPERVISED ATTACK PATTERN DETECTION IN HONEYPOT DATA 17 Topic 1 Topic 7 Topic 10 Topic 14 Secondary topic (Topic 0) FIG 4: Wordclouds of estimated topic-specific word distributions under the constrained topic model with sec- ondary topic in (2), fitted on the ICL honeypot data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' tialised using the topics estimated by the constrained topic model in (1), fitted in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' In order to suitably compare the topic-specific word distributions and avoid the label switch- ing issue in clustering (Jasra, Holmes and Stephens, 2005), the MCMC chain for model (2) was initialised using the output from model (1) obtained in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' The indica- tors zd,j,i are initialised from a Bernoulli distribution with probability equal to the proportion of documents in which the word wd,j,i occurred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' The resulting wordclouds for some of the topic-specific word distributions are plotted in Figure 4, including the distribution of the sec- ondary topic, labelled topic 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Note that with a secondary topic, words are implicitly sampled from the mixture distribution ˜φtd = θtdφtd + (1 − θtd)φ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' This aids interpretability of the latent intent of the session-level topic td, since the common words shared across most topics are filtered out and included in φ0 instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' This is confirmed when comparing Figure 4 with Figure 3 (obtained from a model that does not include a secondary topic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Overall, Figure 4 shows that the secondary topic captures the most common words across documents, such as the string HEX and simple shell commands, for example cd, chmod, var, echo, shell and tmp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' When comparing with Figure 3, the word distributions in Figure 4 appear to be less dominated by common words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' For example, for topic 1, the words tmp and cd appear to be among the most representative words in Figure 3, but they have a much less prominent role in the wordcloud representation in Figure 4, where words such as x86_64, uname and Xorg gain importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' These words appear to be much more representative of the actual intents of the session, which is particularly helpful when communicating results to analysts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Overall, the assumption of having only one topic per session might be limiting, even if an additional secondary shared topic is added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' This is mainly because most sessions could be considered as mixtures of commands, where each command has its own intent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Therefore, a more precise clustering of topics and commands might be provided by the Hierarchical Constrained Topic Model (HTCM) in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='2, which is considered in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Hierarchical Constrained Topic Modelling (HCTM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' As discussed in the previous section, the HCTM in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='2 could help to further elucidate the underlying group struc- ture within the ICL honeypot data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Similarly to Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='3, the MCMC is run for 250,000 iterations with 50,000 burn-in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' The command-level and session-level topics are initialised via the spectral clustering algorithm described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='5, setting Dirichlet priors of di- mension Kmax = 50 and Hmax = 50, with hyperparameters η = 1|V |, τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='1 · 1Hmax,γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='1 · 1Kmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' The session-level and command-level topics are estimated following the proce- dure described in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='2, setting ˆK∅ = 36 and ˆH∅ = 38, corresponding to the modal number of non-empty topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Figure 5 displays the frequency distribution of the estimated session-level (Figure 5a) and command-level topics (Figure 5c), followed by the estimated distributions of the number of non-empty session-level (Figure 5b) and command-level topics (Figure 5d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' The wordclouds for the resulting session-level and command-level topic-specific distri- butions are displayed in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Figure 6a displays the topic-specific word distributions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='tmp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='history ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='chmodshell ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='Cdenable ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='ping shcat ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='777mnt18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='(A) Barplot of frequencies of estimated session-level topics ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='1 2 10 15 4 13 11 3 34 21 18 14 16 31 12 5 22 6 23 20 29 7 17 32 8 9 26 35 30 27 33 28 19 25 24 36 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='Session-level topic label ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='Number of sessions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='(B) Barplot of estimated K∅ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='33 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='34 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='35 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='36 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='37 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='38 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='39 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='Number of session-level topics ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='5 Estimated frequency (C) Barplot of frequencies of estimated command-level topics 2 7 1 4 13 19 3 20 9 11 21 8 28 15 6 26 27 5 12 14 16 10 23 34 31 17 35 29 30 18 32 33 22 24 37 25 36 38 Command-level topic label 0 1000 2000 3000 4000 5000 6000 7000 Number of commands (D) Barplot of estimated H∅ 34 35 36 37 38 39 40 41 42 43 44 45 Number of command-level topics 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='20 Estimated frequency FIG 5: Frequency distributions of the estimated session-level and command-level topics, and estimated distribu- tion the number of non-empty session-level topics K∅ and command-level topics H∅, under the hierarchical constrained topic model in (3), fitted on the ICL honeypot data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' obtained from the estimated session-level topics, whereas Figure 6b plots the wordclouds cor- responding to the estimated command-level topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Interestingly, when compared to Figure 3, the word distributions in Figure 6 appear more heterogeneous, especially at the command- level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' This is not surprising: the constrained topic model in (1) gives the same primary topic to all words in a session, whereas the HCTM admits command-specific topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' In the ICL honeypot data, and more generally in session data, individual commands tend to have a spe- cific intent identified by specific words in the command (for example, wget for download- ing files from a web server under the HTTP, HTTPS, and FTP protocols).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Therefore, having command-specific topics helps in identifying the intents of individual commands, making the command-level topic-specific word distributions highly interpretable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Similarly to the previous section, the intents for the estimated session-level and command- level topics are summarised in Table 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' In general, the two tables show that some topics correspond to the same intent, achieved through different words or commands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' In particular, Table 2 shows the same malware types as Table 1, with similar objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Similarly to the re- sults in the previous section, different MIRAI variants are observed, and malware types such as shellbots, coin miners, the Hive OS attack and the MikroTik bot are all allocated to sep- arate topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' In addition to the session-level objectives in Table 1, Table 2 also shows topics representing reconnaissance sessions, where intruders attempted to gather system informa- tion, for example by checking directories and determining shell executables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' MinerFinder is again discovered and relevant sessions are singled-out in the topic with label 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' In par- ticular, under the hierarchical constrained topic model, MinerFinder is explicitly split from the similar MIRAI variants that were present in topic 5 under the constrained topic model (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Table 1), which are instead allocated to topic 22 under the HCTM (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' It must be remarked that MinerFinder is not detected using alternative clustering approaches based on spectral clustering or standard LDA fitted via gensim (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' If the topics are ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='UNSUPERVISED ATTACK PATTERN DETECTION IN HONEYPOT DATA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='19 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='(A) Session-level topic-specific word distributions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='Topic 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='Topic 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='Topic 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='Topic 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='Topic 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='Topic 6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='Topic 7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='Topic 8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='Topic 9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='Topic 10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='Topic 11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='Topic 12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='Topic 13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='Topic 14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='Topic 15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='Topic 16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='Topic 17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='Topic 18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='nippon ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='CCAD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='Proton ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='mika ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='lacHe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='p X19I239124UIU ans infectedByRakitin20 TABLE 2 Estimated session-level topics and corresponding intent under the hierarchical constrained topic model in (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Topic label Type of malware Objective 1 MIRAI Check shell and directories,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' download and execute MIRAI malware,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' delete files 2 (ptmx) unnamed botnet,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' MIRAI Gather system information,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' change permissions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' execute MIRAI variants 3 Shellbot,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' coin miner,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' MIRAI Download and execute coin miner and MIRAI malware 4 MIRAI Determine shell executable,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' check busybox is present,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' print error message to console 5 Shellbot,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' coin miner SSH backdoor botnet,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' download malware,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' change SSH keys,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' check CPU/GPU information 6 MIRAI Download and execute MIRAI malware (for example,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' garm or gmips),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' delete files 7 Shellbot,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' coin miner Gather CPU/GPU information,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' download coin miners,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' kill existing coin miners 8 Reconnaissance Check mounted file system,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' gather system information,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' fingerprint system 9 MIRAI Write malware (for example dvrHelpwer and updDl) via echoing HEX strings 10 Reconnaissance Check shell and directories,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' delete files (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='ptmx, Switchblades) 11 Hive OS attack, coin miner Download miner, attempt to take over configurations in Hive OS mining platform 12 MIRAI Execute MIRAI variant PEDO, fingerprint system 13 MIRAI Execute MIRAI variants kura and kurc, fingerprint system 14 MIRAI Determine shell, download and execute MIRAI variant DNXFCOW via echoing HEX strings 15 Reconnaissance Check shell and directories, delete files (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='ptmx, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='s4y) 16 MIRAI Download and execute MIRAI variant PEDO and mika,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' fingerprint system 17 Coin miner Download coin miner (c3pool) 18 MIRAI Download,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' execute MIRAI variant ECCHI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' delete files,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' fingerprint system 19 MIRAI Download malware,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' write updDl malware via echoing HEX strings 20 MIRAI Download sora malware,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' write upnp and updDl malware via echoing HEX strings 21 MIRAI Gather system information,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' change permissions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' execute MIRAI variants 22 MIRAI Download malware,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' change permissions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' gather system information,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' fingerprint system 23 Coin miner,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' SBIDIOT Download and execute coin mining malware 24 MinerFinder (new MIRAI variant) Changes SSH keys,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' looks for coin miners,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' attempt to take over miners 25 MIRAI Check shell and directories,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' execute MIRAI malware (Skyline and Akim),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' delete files 26 GHILIMEA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' PentaMiner coin miner script Install coin miner,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' kill mining processes with high CPU usage in order to go undetected 27 Reconnaissance Attempt to read and change SSH keys 28 MIRAI Write RONALD malware via echoing HEX strings 29 MIRAI Gather system information,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' print error message (component of DNXFCOW MIRAI variant) 30 MIRAI Gather system information,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' change permissions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' execute MIRAI variant cowffxxna 31 MikroTik bot,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' coin miner Gather system information,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' gather MikroTik router information,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' kill existing coin miners 32 MIRAI Download and execute MIRAI variant DNXFCOW via echoing multiple HEX strings 33 Shellbot,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' coin miner Scan system,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' look for GPUs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' look for coin miners,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' download malware 34 MIRAI Gather system information,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' change permissions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' execute MIRAI variants 35 Coin miner Download coin miner (TeamTNT) 36 MikroTik bot Remove firewall NAT rules on MikroTik router estimated using the procedures described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='5, MinerFinder is usually allocated to large clusters, with a number of sessions ranging between 80 and 1,000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Furthermore, Figure 7 displays the heatmap of Jaccard similarity scores comparing the re- sults of the constrained model (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='3) and the hierarchical constrained topic model fitted in this section, demonstrating significant agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' The level of agreement is remark- able, considering that the session-level topics are obtained under two different modelling assumptions: in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='3, the constrained topic model in (1) directly uses the session-level topic to obtain the word distribution for the entire session.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' On the other hand, the session- level topics in HTCMs are only estimated from sequences of command-level topics, which are themselves unknown and therefore estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' More details about the commands appearing in the sessions are given in Table 3, and such objectives can be associated with the command-level topic-specific wordclouds in Figure 6b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Overall, analysing the results of the HTCM confirms the results obtained in the previous section, with the added benefit of having estimated command-level topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Models for clustering session data collected on honeypots were pro- posed, with the objective of finding groups of similar attacks which could aid cyber ana- lysts in detecting emerging intrusion attempts and vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' The proposed models are based on modifications of Latent Dirichlet Allocation, aimed at improving topic interpretabil- ity and convergence properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' In particular, the concepts of primary and secondary topics were introduced, along with session-level and command-level topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Secondary topics are used to represent common, high-frequency words, whereas the primary topic is used for the words which define the latent intent of the session.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Furthermore, two hierarchical layers of topics were introduced: a command-level topic which determines the word distribution, and UNSUPERVISED ATTACK PATTERN DETECTION IN HONEYPOT DATA 21 TABLE 3 Estimated command-level topics and corresponding intent under the hierarchical constrained topic model in (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Topic label Objective 1 Attempt to write file .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='ptmx or LAYER to directory var, run or tmp and change directory if writeable 2 Attempt to write file .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='ptmx to directory netslink,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' mnt or shm and change directory if writeable 3 Attempt to copy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' write and delete files 4 Attempt to grant full permissions to all users on malware files 5 Gather system information about CPU architectures 6 Kill processes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' gather system information,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Hive OS logon attempt 7 Check available commands,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' determine shell executable 8 Check if busybox exists,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' print error message to console (component of PEDO MIRAI variant) 9 Gather system information about mounted file systems,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' copy files 10 Fingerprint readable and writeable directories to hidden file .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='nippon 11 Download malware from internet using wget and curl 12 Attempt to read and delete files 13 Attempt to execute malware,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' delete files after execution 14 Check if directories such as var,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' run or tmp exist,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' by attempting to change directory 15 Download malware using tftp 16 Download,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' execute and delete malware files 17 Download malware using ftpget 18 Download and install GHILIMEA coin mining malware,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' kill own processes if CPU usage is high 19 Check if busybox exists,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' check available commands,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' determine shell executable 20 Check if busybox exists,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' print error message to console (component of KURA and OWARI MIRAI variants) 21 Check if busybox exists,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' print error message to console (component of ECCHI MIRAI variant) 22 Download malware to attempt to compromise MikroTik router 23 Look for existing coin miners,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' gather GPU and CPU information 24 Attempt to gather information about MikroTik router 25 Determine shell executable 26 Attempt to write file .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='file to directory netslink, mnt or boot and change directory if writeable 27 Copy, grant full permission, execute and delete .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='cowbot malware (MIRAI variant) 28 Write malware binary to disk via echoing HEX bits 29 Download coin miner malware from c3pool 30 Read SSH authorised keys,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' attempt to delete and replace authorised keys 31 Exit shell (part of GHILIMEA coin mining malware) 32 Install GHILIMEA coin mining malware,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' kill own processes if CPU usage is high 33 Check internet connectivity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' check if GHILIMEA is already installed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' kill own processes if CPU usage is high 34 Read and delete files (for example,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='none and .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='human) 35 Check if busybox exists,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' print error message to console (component of RONALD and Akim MIRAI variant) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='36 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='Remove firewall NAT rules on MikroTik router ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='37 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='Remove temporary directory p (singleton cluster) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='38 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='Exit shell (singleton cluster) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='26 22 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='13 14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='11 32 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='20 18 23 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='16 36 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='19 10 21 15 12 24 25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='27 28 29 30 31 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='33 34 35 17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='Session-level topic label (hierarchical constrained topic model) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='19 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='Topic label (constrained topic model) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='0 FIG 7: Heatmap of Jaccard similarities between the sessions assigned to session-level topics under the con- strained topic model (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='3) and hierarchical constrained topic modelling (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' 22 a session-level topic which controls the distribution of the command-level intents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Further- more, the proposed methodologies extend to a Bayesian nonparametric framework, admitting an unbounded (and unknown) number of latent intents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' The proposed models could be further extended by introducing dependencies between command-level topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' In particular, a Markov model with H states could be devised, whereby each session-level topic corresponds to different transition probabilities between command-level topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Also, a possible extension of this work could consider dynamically- evolving topics (dynamic topic modelling, Blei and Lafferty, 2006), or explicitly encode cor- relation between topics (correlated topic models, Lafferty and Blei, 2005), or a combination of the two approaches (see, for example, Tomasi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Sections 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='1 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='1 briefly discussed some of the challenges related to tokenisation in cyber-security applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Depending on the chosen tokenisation method, important context about the commands might be lost, which might prevent cyber-security analysts to make rapid decisions and assessments on the content of the session.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' One possibility in this direction would be to explore deep neural network methods, inspired by their use for large language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' For example, the RAPIDS CLX library cyBERT2 uses deep neural networks and transformers for parsing cyber-security logs, including session data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Overall, session data collected on honeypots are a valuable resource for cyber analysts, and principled statistical modelling is required to obtain actionable insights from these data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' The proposed methodologies have provided useful groupings of the attacks observed on a university network, including the discovery of an unusual MIRAI variant which attempts to take over existing coin miner infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' This demonstrates the potential of topic models to elucidate hidden structure within text data, obtaining valuable insights for cyber-defence purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' The authors thank Andy Thomas and the Information & Com- munications Technology team at Imperial College London for their support on data pro- cessing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' FSP and NAH acknowledge funding from Microsoft Security, through the research grant “Understanding the enterprise: Host-based event prediction for automatic defence in cyber-security”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' AM and NAH acknowledge funding from the Data-centric Engineering pro- gramme at the Alan Turing Institute, for the Grand Challenge “Monitoring Complex Sys- tems”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' DG acknowledges funding from the Department of Mathematics at Imperial College London.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Part of this work was carried out when AM was a postdoctoral research associate in statistical cyber-security at Imperial College London and the Alan Turing Institute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' PT is now a 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' and BOYD-GRABER, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Online Latent Dirichlet Allocation with Infinite Vocabulary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' In Pro- ceedings of the 30th International Conference on Machine Learning (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' DASGUPTA and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' MCALLESTER, eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Proceedings of Machine Learning Research 28 561–569.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' PMLR, Atlanta, Georgia, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' ZHANG, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Bayesian Latent Feature Modelling for Unstructured Data, PhD thesis, Imperial College London.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' ZHU, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=', CHEN, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=', YAN, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=', WANG, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=', LI, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=', PENG, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' and ZHAO, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Mining Function Homology of Bot Loaders from Honeypot Logs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' arXiv e-prints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' APPENDIX A: SIMULATIONS AND RESULTS ON SYNTHETIC DATA In this section, the proposed models are compared and contrasted on synthetic datasets, in order to assess their performance in recovering the session-level topics, the main quantity of inferential interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Note that the true topics are known when data are simulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' If synthetic data are used, the true underlying session-level topics are available, and the true allocations could be compared to the estimated topics via the Adjusted Rand Index (ARI, Hubert and Arabie, 1985).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Each simulation is repeated for 50 datasets using different seeds, setting |V | = 50, D = 100, Nd = 10, Md,j = 10, λ = 1/K · 1K for each simulated dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' The MCMC sampler is run for 25,000 iterations with 10,000 burn-in, initialising the topics via spectral clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Unless otherwise specified, the hyperparameter γ is set to γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='1 · 1Kmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' UNSUPERVISED ATTACK PATTERN DETECTION IN HONEYPOT DATA 25 (A) Boxplot of ARIs for estimated session-level topics η = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='0 η = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='0 η = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='0 ARI (B) Barplot of estimated number of non-empty topics K∅ 5 6 K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='0 Estimated probability Truth η = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='0 η = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='0 η = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='0 FIG 8: Summary plots obtained from 50 simulated datasets from model (1), with |V | = 50, K = 5, D = 100, Nd = 10, Md,j = 10, λ = 1/K · 1K, using different values for η, such that η = η · 1K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' The MCMC sampler is run for 25,000 iterations with 10,000 burn-in, setting Kmax = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' First, datasets are simulated from model (1), setting K = 5 and using three different val- ues of the parameter η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' If η = η · 1K, the parameter η controls the spikiness of the topic- specific word distributions: low values of η correspond to distributions which assign most of the probability mass to a small number of words, whereas larger values of η correspond to distributions where the probability mass is distributed more uniformly across words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' In the MCMC sampler, Kmax is set to 10, implying that the sampler should identify 5 empty topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' The results are reported in Figure 8: Figure 8a reports the ARIs for the estimated session- level topics, whereas Figure 8b shows the barplot of the estimated number of non-empty topics, denoted K∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' As expected, the ARI decreases when the value of η increases, since the topic-specific distributions become increasingly uniform and therefore similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Increasing the value of η also provides worse estimates of the true number of topics used in the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Ideally, K∅ should correspond to the value of K used in the simulation, provided that Kmax in the MCMC algorithm is chosen to be larger than the true underlying K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Next, inference is repeated on the 50 datasets simulated as in Figure 8 with η = 5 · 1K, using different initialisation methods and priors for λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' In particular, results are compared between the spectral and gensim initialisation schemes described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Figure 9a displays the boxplots of ARIs for the session-level topics across the different datasets, after initialisation, before and after running the MCMC sampling scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' The plot shows that the spectral initialisation scheme appears to have a better performance initially, but both methods lead to equivalent results after MCMC sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Furthermore, results on estimation of the number of topics are compared between two different priors: a Kmax-dimensional Dirichlet distribution with Kmax = 10 and γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='1 · 1Kmax (also used in Figure 8b) or a GEM prior on λ with γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Figure 9b reports the resulting barplot for the estimated modal number of non-empty communities across the different datasets, suggesting that the two priors have a similar performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Second, datasets are simulated from model (2), using η = 1K, K = 5 and different val- ues of θk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' The value of θk corresponds to the expected proportion of words sampled from the primary topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Therefore, small values of θk are expected to make inferential procedures more difficult, since less observations are available from the primary topics, and words are sampled instead from a secondary topic, shared across sessions and commands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Furthermore, the secondary topic makes the primary topics more difficult to estimate, since the words are implicitly sampled from the mixture distribution ˜φtd = θtdφtd + (1 − θtd)φ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' If θk decreases for all k, the distributions ˜φ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=', ˜φK become increasingly similar, and a drop in the ARI similar to Figure 8a is expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' In the MCMC sampler, the required parameters are set to Kmax = 10, αh = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='9 and α0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' The results are presented in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' As expected, 26 (A) Boxplot of ARIs for estimated session-level topics ob- tained via different initialisation schemes Spectral gensim Spectral gensim At initialisation After MCMC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='0 ARI (B) Barplot of estimated number of non-empty topics K∅ for different priors on λ (Kmax-dimensional Dirichlet or GEM) 5 6 K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='8 Estimated probability Dirichlet GEM FIG 9: Summary plots obtained from 50 simulated datasets from model (1), with |V | = 50, K = 5, D = 100, Nd = 10, Md,j = 10, λ = 1/K · 1K, and η = 5 · 1K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' The MCMC sampler is run for 25,000 iterations with 10,000 burn-in, setting Kmax = 10 or a GEM prior for λ with γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' (A) Boxplot of ARIs for estimated session-level topics θk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='9 θk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='75 θk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='0 ARI (B) Barplot of estimated number of non-empty topics K∅ 2 3 4 5 6 7 8 K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='8 Estimated probability Truth θk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='9 θk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='75 θk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='5 FIG 10: Summary plots obtained from 50 simulated datasets from model (2), with |V | = 50, K = 5, D = 100, Nd = 10, Md,j = 10, λ = 1/K · 1K, η = 1K, using different values for θk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' The MCMC sampler is run for 25,000 iterations with 10,000 burn-in, setting Kmax = 10, αh = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='9,α0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' (A) Boxplot of ARIs for estimated session-level topics τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='25 τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='5 τ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='0 ARI (B) Barplot of estimated number of non-empty topics K∅ 2 3 K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='8 Estimated probability Truth τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='25 τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='5 τ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='0 FIG 11: Summary plots obtained from 50 simulated datasets from model (3), with |V | = 50, K = 3, H = 5, D = 100, Nd = 10, Md,j = 10, λ = 1/K · 1K, η = 1K, using different values for τ = τ1H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' The MCMC sampler is run for 25,000 iterations with 10,000 burn-in, setting Kmax = 10, Hmax = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Figure 10a shows that the ARI for the estimated session-level topics decreases when θk de- creases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Furthermore, Figure 10b shows that estimation of the number of non-empty primary topics becomes increasingly imprecise when θk decreases, causing the drop in ARI observed in Figure 10a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Third, datasets are simulated from model (3), using η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='1 · 1K, K = 3, H = 5 and τ = τ1H, for different values of τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' In this case, τ expresses how concentrated around a UNSUPERVISED ATTACK PATTERN DETECTION IN HONEYPOT DATA 27 peak the command-level topic distributions are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Small values of τ imply that the probability mass is mostly concentrated around one topic, whereas larger values correspond to more evenly distributed probability mass functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' In the MCMC sampler, the values Kmax = 10 and Hmax = 10 are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Note that the task of recovering the session-level topics is much more complex than previous simulations, since for model (3), the session-level topic only controls the distribution of the command-level topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Hence, the session-level topic must be estimated from the Nd command-level topics within each document, which are themselves estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' This makes the inferential task substantially harder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Figure 11 displays the results: Figure 11a shows that the ARI for the estimated session-level topics tends to decrease when τ increases, and Figure 11b shows that estimates of the number of non-empty session-level topics are more precise when the value of τ used in the simulation is smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content=' Notice that, since η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='1·1K, the topic-specific word distributions have most of their mass concentrated around a small number of words, and the command-level topics are therefore easy to recover, with ARIs averaging above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} +page_content='99 across the three settings for τ shown in Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf'} diff --git a/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf b/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..4910029476ac8a1939f34da6d7c6cd00e39ad978 --- /dev/null +++ b/_NAzT4oBgHgl3EQfS_tm/content/2301.01241v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid 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a/aNE2T4oBgHgl3EQfvgjw/content/tmp_files/2301.04093v1.pdf.txt b/aNE2T4oBgHgl3EQfvgjw/content/tmp_files/2301.04093v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..33a66363183c2708495d806cc3966b190b9efa3a --- /dev/null +++ b/aNE2T4oBgHgl3EQfvgjw/content/tmp_files/2301.04093v1.pdf.txt @@ -0,0 +1,7802 @@ +On the Robustness of AlphaFold: A COVID-19 Case Study +Ismail Alkhouri1, Sumit Jha2, Andre Beckus3, George Atia1, Alvaro Velasquez4, Rickard Ewetz1, +Arvind Ramanathan5, Susmit Jha6 +1 Electrical & Computer Engineering Department, University of Central Florida, Orlando, FL 32816 +2 Computer Science Department, University of Texas at San Antonio, TX 78249 +3 Information Directorate, Air Force Research Laboratory, Rome, NY 13441 +4 Defense Advanced Research Projects Agency, Arlington, VA 22203 +5 Data Science and Learning, Argonne National Laboratory, Lemont, IL, 60439 +6 Computer Science Laboratory, SRI International, Menlo Park, CA, 94709 +Abstract +Protein folding neural networks (PFNNs) such as AlphaFold +predict remarkably accurate structures of proteins compared to +other approaches. However, the robustness of such networks +has heretofore not been explored. This is particularly relevant +given the broad social implications of such technologies and +the fact that biologically small perturbations in the protein +sequence do not generally lead to drastic changes in the pro- +tein structure. In this paper, we demonstrate that AlphaFold +does not exhibit such robustness despite its high accuracy. +This raises the challenge of detecting and quantifying the ex- +tent to which these predicted protein structures can be trusted. +To measure the robustness of the predicted structures, we +utilize (i) the root-mean-square deviation (RMSD) and (ii) +the Global Distance Test (GDT) similarity measure between +the predicted structure of the original sequence and the struc- +ture of its adversarially perturbed version. We prove that the +problem of minimally perturbing protein sequences to fool +protein folding neural networks is NP-complete. Based on +the well-established BLOSUM62 sequence alignment scoring +matrix, we generate adversarial protein sequences and show +that the RMSD between the predicted protein structure and +the structure of the original sequence are very large when the +adversarial changes are bounded by (i) 20 units in the BLO- +SUM62 distance, and (ii) five residues (out of hundreds or +thousands of residues) in the given protein sequence. In our +experimental evaluation, we consider 111 COVID-19 proteins +in the Universal Protein resource (UniProt), a central resource +for protein data managed by the European Bioinformatics +Institute, Swiss Institute of Bioinformatics, and the US Pro- +tein Information Resource. These result in an overall GDT +similarity test score average of around 34%, demonstrating a +substantial drop in the performance of AlphaFold. +Introduction +Proteins form the building blocks of life as they enable a vari- +ety of vital functions essential to life and reproduction. Natu- +rally occurring proteins are bio-polymers typically composed +of 20 amino acids and this primary sequence of amino acids +is well known for many proteins, thanks to high-throughput +sequencing techniques. However, in order to understand the +functions of different protein molecules and complexes, it is +essential to comprehend their three-dimensional (3D) struc- +tures. Until recently, one of the grand challenges in structural +biology has been the accurate determination of the 3D struc- +ture of the protein from its primary sequence. Such accurate +ID: O43765 𝑛 = 313 +RMSD = 12.051Å +Figure 1: The structure of the original (black) and adversarial +(red) sequences predicted using AlphaFold for the Small glutamine- +rich tetratricopeptide repeat-containing protein alpha sequence. The +length of the protein sequence is denoted by n. For structures, after +their alignment using PyMol (Schrödinger and DeLano), the Root +Mean Square Deviation (RMSD) is given in Angstroms (equal to +10−10 meters and denoted by Å). +predictive protein folding promises to have a profound impact +on the design of therapeutics for diseases and drug discovery +(Chan et al. 2019). +AlphaFold (Jumper et al. 2021a) achieved unparalleled suc- +cess in predicting protein structures using neural networks +and remains first at the Critical Assessment of protein Struc- +ture Prediction (CASP14), which corresponds to year 2020, +competition. While this has been touted as a breakthrough +for structural biology (Bagdonas et al. 2021), the robustness +of its predictions has not yet been explored. The main con- +tribution of this paper is to demonstrate the susceptibility +of AlphaFold to adversarial sequences by generating sev- +eral examples where protein sequences that vary only in +five residues out of hundreds or thousands of residues re- +sult in very different 3D protein structures. We present the +problem of adversarial attacks on Protein Folding Neural Net- +work (PFNN) and prove that the problem is NP-complete. +We use sequence alignment scores (Henikoff and Henikoff +1992) such as those derived from Block Substitution Matrices +(BLOSUM62) to identify a space of similar protein sequences +used in constructing adversarial perturbations. For the output +arXiv:2301.04093v1 [cs.LG] 10 Jan 2023 + +structures, we leverage the standard metrics commonly used +in CASP, namely (i) the root-mean-square deviation (RMSD) +and (ii) the Global Distance Test (GDT) similarity measure +between the predicted structure and the structure of its adver- +sarially perturbed sequence. See Figure 1 and its caption for +an example. +Moreover, we conduct two experiments investigating the +choice of the BLOSUM threshold and the use of the pre- +diction, per-residue, confidence information obtained from +AlphaFold. Our experiments show that different input pro- +tein sequences have very different adversarial robustness as +determined by the RMSD (GDT-TS) in the protein structure +predicted by AlphaFold. These values range from 1.011Å +(0.43%) to 49.531Å (98.8%) when the BLOSUM62 distance +between the original and adversarial sequences is bounded +by a threshold of 20 units with a hamming distance of 5 +residues only. Hence, our proposed approach is a first step in +the direction of identifying protein sequences on which the +predicted 3D structure cannot be trusted. +Summary and Related work +PFNNs (Jumper et al. 2021b; Baek et al. 2021) should be +expected to obey the natural observation that biologically +small changes in the sequence of a protein usually do not +lead to drastic changes in the protein structure. Almost four +decades ago, it was noted that two structures with 50% se- +quence identity align within approximately 1Å RMSD from +each other (Chothia and Lesk 1986). Two proteins with even +40% sequence identity and at least 35 aligned residues align +within approximately 2.5Å (Sander and Schneider 1991). The +phenomenon of sequence-similar proteins producing similar +structures have also been observed in larger studies (Rost +1999). As with almost any rule in biology, a small number +of counterexamples to the conventional wisdom of similar +sequences leading to similar structures do exist, wherein even +small perturbations can potentially alter the entire fold of a +protein. However, such exceptions are not frequent and often +lead to exciting investigations (Cordes et al. 2000; Tuinstra +et al. 2008). +Manipulating the multiple sequence alignment step of Al- +phaFold has been studied in (Stein and Mchaourab 2021) +using in silico mutagenesis. However, there, the goal is not +to study the robustness of the protein folding neural net- +works, but rather to enhance the prediction capability of +AlphaFold in terms of the intrinsic conformational hetero- +geneity of proteins. The authors in (Del Alamo et al. 2022), +present a method that manipulates inputs to obtain diverse +distinct structures that are absent from the AlphaFold training +data. Using membrane proteins, the authors show that their +method enhances the multiple sequence alignment step while +generating more accurate structures. +The work in (Jha et al. 2021) is aimed at generating adver- +sarial sequences in order to cause significant damage to the +output predicted structure of RosettaFold (Baek et al. 2021), +which, according to CASP, is the second best protein folding +neural network. However, the authors only show results for a +few proteins and do not consider all the standard metrics for +measuring the output structures. In contrast, in this paper, we +present results for more than 100 sequences, derive a com- +plexity proof for the problem of finding adversarial protein +sequences, and, based on the CASP competition, utilize all +the standard metrics for measuring the output structures. +Robustness Metric using Adversarial Attacks +The similar-sequence implies similar-structure paradigm dic- +tates that PFNNs should make robust predictions. Given a +protein sequence of n residues S = s1s2 . . . sn with a three- +dimensional structure A(S) = (x1, y1, z1), . . . , (xn, yn, zn), +we define a notion of biologically similar sequences V us- +ing Block Substitution Matrices (BLOSUM) (Henikoff and +Henikoff 1992), and then employ formulations of adversar- +ial attacks (Goodfellow, McDaniel, and Papernot 2018) on +PFNNs within this space of similar sequences to identify +a sequence Sadv ∈ V that produces a maximally different +three-dimensional structure A(Sadv). We then compute the +RMSD and GDT between the structures for the original and +adversarial inputs (A(S) and A(Sadv)), and use these met- +rics as the robustness measure. If the RMSD (GDT) is small +(high), the response of the PFNN is deemed robust; a large +(small) RMSD (GDT) indicates that the predicted structure +is not robust. +BLOSUM Similarity Measures +Given two sequences of n residues S = s1s2 . . . sn and +S′ = s′ +1s′ +2 . . . s′ +n, in which every residue si (or s′ +i) is from +the set X = {A, R, N, D, C, Q, E, G, H, I, L, K, M, F, P, +S, T, W, Y, V } of amino acids, a natural question is how to +compute the sequence similarity Dseq between these proteins. +A naive approach would be to count the number of residues +that are different, i.e., the Hamming distance. However, an +analysis of naturally occurring proteins shows that not all +changes in residues have the same impact on protein struc- +tures. Changes to one type of residue are more likely to cause +structural variations than changes to another type of residue. +Early work in bioinformatics focused on properties of +amino acids and reliance on genetic codes. However, more +modern methods have relied on the creation of amino acid +scoring matrices that are derived from empirical observations +of frequencies of amino acid replacements in homologous +sequences (Dayhoff, Schwartz, and Orcutt 1978; Jones, Tay- +lor, and Thornton 1992). The original scoring matrix, called +the PAM250 matrix, was based on empirical analysis of 1572 +mutations observed in 71 families of closely-related proteins +that are 85% or more identical after they have been aligned. +The PAM1 model-based scoring matrix was obtained by nor- +malizing the frequency of mutations to achieve a 99% identity +between homologous proteins. These results were then extrap- +olated to create the PAM10, PAM30, PAM70 and PAM120 +matrices with 90%, 75%, 55%, and 37% identity between +homologous proteins. +Another interesting approach (Henikoff and Henikoff +1992) to understanding protein similarity is the direct count- +ing of replacement frequencies using the so-called Block Sub- +stitution Matrices (BLOSUM). Instead of relying solely on +sequences of homologous proteins that are relatively harder +to find, the BLOSUM approach focuses on identifying con- +served blocks or conserved sub-sequences in a larger variety + +of proteins potentially unrelated by evolutionary pathways +and counts the frequency of replacements within these con- +served sub-sequences. BLOSUM62 (Figure 2), BLOSUM80 +and BLOSUM90 denote block substitution matrices that are +obtained from blocks or subsequences with at least 62%, +80%, and 90% similarity, respectively. The BLOSUM matrix +[Bij] is a matrix of integers where each entry denotes the +similarity between residue of type bi ∈ X and type bj ∈ X. +We identify the space of biologically similar sequences +V for a given protein sequence S with respect to the BLO- +SUM distance. We expect the predicted structures for the +similar sequences to be similar. If there is a large RMSD (or +small GDT) between the predicted structure A(S) and the +structure of the adversarial sequence A(Sadv), it would re- +flect a lack of robustness in the prediction of the network. We +adopt a sequence similarity measure that counts replacement +frequencies in conserved blocks across different proteins. +Approach +Our approach to evaluating the robustness of PFNNs is based +on two main ideas: (i) the existence of adversarial exam- +ples in PFNNs that produce adversarial structures possibly +very different from the original structure, and (ii) the use of +BLOSUM matrices for identifying a neighborhood of a given +sequence that are biologically similar and hence expected to +have similar 3D structures. We utilize the RMSD and GDT +between the structure of an original protein sequence and the +structure of the adversarial sequence as a measure of robust- +ness of a protein folding network on the given input. In this +work, we focus on the state-of-the-art AlphaFold model, the +winner of the 1st place in CASP2020. +Sequence Similarity Measures +Given two sequences S = s1s2 . . . sn and S′ = s′ +1s′ +2 . . . s′ +n, +the BLOSUM distance between the two sequences is given +by Equation (1) below. For an illustrative example of Dseq, +see Figure 3. +Dseq(S, S′) = +� +i∈[n] +� +Bsisi − Bsis′ +i +� +. +(1) +A +R +N +D +C +Q +E +G +H +I +L +K M +F +P +S +T +W Y +V +A +4 +-1 -2 -2 +0 +-1 -1 +0 +-2 -1 -1 -1 -1 -2 -1 +1 +0 +-3 -2 +0 +R +-1 +5 +0 +-2 -3 +1 +0 +-2 +0 +-3 -2 +2 +-1 -3 -2 -1 -1 -3 -2 -3 +N -2 +0 +6 +1 +-3 +0 +0 +0 +1 +-3 -3 +0 +-2 -3 -2 +1 +0 +-4 -2 -3 +D -2 -2 +1 +6 +-3 +0 +2 +-1 -1 -3 -4 -1 -3 -3 -1 +0 +-1 -4 -3 -3 +C +0 +-3 -3 -3 +9 +-3 -4 -3 -3 -1 -1 -3 -1 -2 -3 -1 -1 -2 -2 -1 +Q -1 +1 +0 +0 +-3 +5 +2 +-2 +0 +-3 -2 +1 +0 +-3 -1 +0 +-1 -2 -1 -2 +E +-1 +0 +0 +2 +-4 +2 +5 +-2 +0 +-3 -3 +1 +-2 -3 -1 +0 +-1 -3 -2 -2 +G +0 +-2 +0 +-1 -3 -2 -2 +6 +-2 -4 -4 -2 -3 -3 -2 +0 +-2 -2 -3 -3 +H -2 +0 +1 +-1 -3 +0 +0 +-2 +8 +-3 -3 -1 -2 -1 -2 -1 -2 -2 +2 +-3 +I +-1 -3 -3 -3 -1 -3 -3 -4 -3 +4 +2 +-3 +1 +0 +-3 -2 -1 -3 -1 +3 +L +-1 -2 -3 -4 -1 -2 -3 -4 -3 +2 +4 +-2 +2 +0 +-3 -2 -1 -2 -1 +1 +K +-1 +2 +0 +-1 -3 +1 +1 +-2 -1 -3 -2 +5 +-1 -3 -1 +0 +-1 -3 -2 -2 +M -1 -1 -2 -3 -1 +0 +-2 -3 -2 +1 +2 +-1 +5 +0 +-2 -1 -1 -1 -1 +1 +F +-2 -3 -3 -3 -2 -3 -3 -3 -1 +0 +0 +-3 +0 +6 +-4 -2 -2 +1 +3 +-1 +P +-1 -2 -2 -1 -3 -1 -1 -2 -2 -3 -3 -1 -2 -4 +7 +-1 -1 -4 -3 -2 +S +1 +-1 +1 +0 +-1 +0 +0 +0 +-1 -2 -2 +0 +-1 -2 -1 +4 +1 +-3 -2 -2 +T +0 +-1 +0 +-1 -1 -1 -1 -2 -2 -1 -1 -1 -1 -2 -1 +1 +5 +-2 -2 +0 +W -3 -3 -4 -4 -2 -2 -3 -2 -2 -3 -2 -3 -1 +1 +-4 -3 -2 11 2 +-3 +Y +-2 -2 -2 -3 -2 -1 -2 -3 +2 +-1 -1 -2 -1 +3 +-3 -2 -2 +2 +7 +-1 +V +0 +-3 -3 -3 -1 -2 -2 -3 -3 +3 +1 +-2 +1 +-1 -2 -2 +0 +-3 -1 +4 +Figure 2: The BLOSUM62 matrix. +Output Structural Measure +Given a sequence of n residues S = s1s2 . . . sn, its three +dimensional structure A(S) is an ordered n-tuple of three- +dimensional co-ordinates (x1, y1, z1), . . . (xn, yn, zn). Our +goal is to utilize a structural distance measure that captures +the variations in the two structures A(S) and A(S′) and is +invariant to rigid-body motion. Therefore, in this work, we +use standard structural distances, namely the RMSD, mea- +sured in Å, and the GDT with its two variants: (i) the Total +Score (TS) and (ii) the High Accuracy (HA) (Zemla 2003). +Given the output structure of the adversarial sequence +A(S′), an alignment algorithm is employed before comput- +ing the RMSD and GDT measures between the two structures +of interest. We use the alignment procedure implemented in +PyMOL (Schrödinger and DeLano) to align A(S′) with re- +gard to the target structure A(S). Let the aligned structure +be denoted by ˆ +A(S′) = (ˆx′ +1, ˆy′ +1, ˆz′ +1), . . . , (ˆx′ +n, ˆy′ +n, ˆz′ +n). Then, +the RMSD, measured in Å, is obtained as +RMSD(A(S), ˆ +A(S′)) = +� +� +� +� 1 +n +� +i∈[n] +d(A(S)i, ˆ +A(S′)i) , +(2) +where d(A(S)i, ˆ +A(S′)i) = (xi−ˆx′ +i)2+(yi−ˆy′ +i)2+(zi−ˆz′ +i)2 +and A(S)i represents the 3D carbon-alpha coordinates of the +ith residue. Using the carbon-alpha coordinates is the standard +approach in CASP (Zemla 2003). +Another standard metric for gauging the similarity of pro- +tein structures is the GDT similarity measure, introduced by +(Zemla 2003) and commonly used in the CASP competition +along with the RMSD. In some cases, the latter is known to +be sensitive to outliers (Zemla 2003). The GDT score returns +a value in [0, 1] where 1 indicates identical structures, and is +computed with respect to four thresholds, δj, as +GDT(A(S), ˆ +A(S′)) = +1 +4n +� +j∈[4] +� +i∈[n] +1 +� +d(A(S)i, ˆ +A(S′)i) < δj +� +, +(3) +where the thresholds δ1, δ2, δ3, and δ4 for TS (HA) are given +by 1(0.5), 2(1), 4(2), and 8(4) for j equals to 1, 2, 3, and 4 +respectively, and 1(·) is the indicator function. In (3), each +j ∈ [4] reflects the number of residues in the structures for +which the distance is less than δj. +Adversarial Attacks on PFNNs +Small carefully crafted changes in a few pixels of input +images cause well-trained neural networks with otherwise +high accuracy to consistently produce incorrect responses +in domains such as computer vision (Croce et al. 2020; An- +driushchenko et al. 2020; Bai et al. 2020; Croce and Hein +2021). Given a neural network A mapping a sequence S of +residues to a three-dimensional geometry A(S) describing +the structure of the protein, we seek to obtain a sequence +S′ such that the sequence similarity measure Dseq(S, S′) be- +tween S and S′ is small and some structural distance measure +Dstr(A(S), A(S′)) is maximized. This can be achieved by + +SFP1_1 +MDLFMRFFTLGSITAQPVKIDNASPASTVHATATIPLQASLPFGWLVIGVAFLAVFQSATKIIALNKRWQLALYKGFQFICNLLLLFVTIYSHLLLVAAG +𝑆 +𝑆1 +′ +𝑆2 +′ +𝑆3 +′ +𝑆4 +′ +𝑆5 +′ +𝑆6 +′ +𝑆7 +′ +𝑆8 +′ +𝑆9 +′ +𝑆10 +′ +DVPSMRFFTLGSITAQPVKIDNASPASTVHATATIPLQASLPFGWLVIGVAFLAVFQSATKIIALNKRWQLALYKGFQFICNLLLLFVTIYSHLLLVAAG +FGCYMRFFTLGSITAQPVKIDNASPASTVHATATIPLQASLPFGWLVIGVAFLAVFQSATKIIALNKRWQLALYKGFQFICNLLLLFVTIYSHLLLVAAG +MDLFMRFFTLGSITAQPIRVPNASPASTVHATATIPLQASLPFGWLVIGVAFLAVFQSATKIIALNKRWQLALYKGFQFICNLLLLFVTIYSHLLLVAAG +MDLFMRFFTLGSITAQPVKIDNASPASTVHATATIPLQASLWAYKLVIGVAFLAVFQSATKIIALNKRWQLALYKGFQFICNLLLLFVTIYSHLLLVAAG +MDLFMRFFVIAAVTAQPVKIDNASPASTVHATATIPLQASLPFGWLVIGVAFLAVFQSATKIIALNKRWQLALYKGFQFICNLLLLFVTIYSHLLLVAAG +MDLFMRFFTLGSITAQPVKIDNASPASTVHATATIPLQASDIERRLVIGVAFLAVFQSATKIIALNKRWQLALYKGFQFICNLLLLFVTIYSHLLLVAAG +MDLFMRFFTLGSITAQPVKIDNASPASTVHATATIPLQASLPFGWLVIGVAFLAVFQSATRVLTMKKRWQLALYKGFQFICNLLLLFVTIYSHLLLVAAG +MDLFMRFFTLGSITAQPVKIDNASPASTVHATATIPLQAVEFQLVLVIGVAFLAVFQSATKIIALNKRWQLALYKGFQFICNLLLLFVTIYSHLLLVAAG +MDLFMRFFTLGSITAQPVKIDNASPASTVHATATIPLQASLPFGWLVIGVAFLAVFQSATKIIALNKRWQLALYKGFQFICNLACMYISMYSHLLLVAAG +MDLFMRFFTLGSITAQPVKIDNASPASTVHATATIPLQASDIGIINGIGVAFLAVFQSATKIIALNKRWQLALYKGFQFICNLLLLFVTIYSHLLLVAAG +𝐷seq +32 +20 +12 +42 +17 +49 +18 +57 +23 +65 +𝐷ham +4 +4 +4 +4 +5 +5 +6 +6 +7 +7 +Original and Adversarial Sequences +Figure 3: The original sequence S is followed by 10 sequences generated by changing 4, 5, 6, and 7 residues. The sequences are samples from +the space in (5) with different values of L and H. The distance Dseq is calculated using (1). +solving the following optimization problem +max +S′ Dstr (A(S), A(S′)) s.t. Dseq(S, S′) ≤ L . +(4) +In our experiments, we set L = 20 and Dstr as the RMSD +measure. Given the discrete nature of the input sequences, +well-known methods for generating adversarial examples (e.g. +gradient-based methods) fail to produce valid and accurate +results. As such, we propose a solution based on a brute-force +exploration in the space of biologically similar sequences that, +given a sequence of interest S with n residues, can be defined +as +VL,H(S) = {S′ ∈ X n | Dseq(S, S′) ≤ L and +Dham(S, S′) ≤ H} , +(5) +where X n is the set of all possible sequences over X of +length n, Dham is the hamming distance, and H is a prede- +fined threshold. For long sequences, the search space can be +extensively large. Therefore, we select random samples from +VL,H(S) and choose the sequence that returns the maximum +value based on the RMSD measure. Our approach to generat- +ing adversarial sequences falls under the class of black-box +attacks. This means that we only have access to the output of +the network (Papernot et al. 2017). +It is worth noting that the inference time of complex pro- +tein folding systems, which apply multiple processing and +alignment steps prior to the use of any neural network, such +as AlphaFold is extremely high compared to NN-based im- +age classifiers. The forward pass of such systems involves +a large number of computations. This fact, along with the +discrete nature of the input space, are the bottleneck of de- +veloping more complex black-box attacks (Mahmood et al. +2021), which in general require a high number of queries. +Complexity +In this section, we formalize the problem of generating an +adversarial attack for PFNNs and establish its complexity. +Definition 1 (PFNN Adversarial Attack (PAA) Problem). +Given a learning model A(. ; θ) : X n → (R × R × R)n +mapping residues to 3-dimensional coordinates and parame- +terized by θ, a sequence S ∈ X n, and a sequence alignment +scoring matrix B, find an input sequence S′ ∈ X n such that +Dseq(S, S′) ≤ L and Dstr(A(S), A(S′)) ≥ U, where the +bounds L and U and distance functions d and D are given. +We prove that the PAA problem is NP-complete. This es- +tablishes that, in general, there is no polynomial-time solution +to the PAA problem unless P = NP. Due to this complexity +and for ease of presentation, we adopt simple perturbation +attacks for our experiments in the next section. We begin +by defining the NP-complete problem to be reduced to an +instance of the PAA problem. +Definition 2 (CLIQUE Problem). Given an undirected graph +G = (V, E) and an integer k, find a fully connected sub- +graph induced by V ′ ⊆ V such that |V ′| = k. +Theorem 1. The PFNN Adversarial Attack (PAA) problem +in Definition 1 is NP-complete. +Proof. It is easy to verify that the PAA problem is in NP +since, given a solution sequence S′, one can check whether +the constraints Dseq(S, S′) ≤ L and Dstr(A(S), A(S′)) ≥ +U are satisfied in polynomial time. It remains to be shown +whether the PAA problem is NP-hard. We establish this +result via a reduction from the CLIQUE problem in Defi- +nition 2. Given a CLIQUE instance ⟨G = (V, E), k⟩ with +|V | = n and |E| = m, we construct its corresponding PAA +instance ⟨A(. ; θ), S, B, L, U⟩ as follows. Without loss of +generality, let us consider a restricted version of the PAA +problem where there are only two residue types {N, K} with +the corresponding BLOSUM62 sub-matrix B′ = 6 · I, where +I denotes the identity matrix. Following the one-hot repre- +sentation of residues adopted in (Jumper et al. 2021b), any +input tensor over {N, K} is represented as a one-hot encod- +ing Sin ∈ (B × B)n to be used as an input tensor to A, where +sin +i0 = 1 (sin +i1 = 1) denotes that residue sin +i is of type N (K). +Let S = (N, N, . . . , N) denote the all-N sequence. We set +L = 6k and U = k(k−1) +2 +� +3 +n. The connectivity structure of +A is derived from the edges E in the CLIQUE instance as +follows. The first column of the input tensor corresponding +to sin +i0 for all i ≤ n is disconnected from the network and +the second column corresponding to sin +i1 is connected to A +such that, for each edge (vi, vj) ∈ E, we have a connection +from sin +i1 and sin +j1 to each of the three outputs in the first three- +dimensional coordinate of A(Sin)1. All connections have a +weight of unity and this defines the parameters θ of the model +A. Therefore, without loss of generality, we are only consid- +ering the first of the n output three-dimensional coordinates + +A(Sin)1. In particular, these values keep track of the number +of edges induced by the vertices in G corresponding to the +non-zero entries in sin +11, . . . , sin +1n. We now prove that there is a +clique of size k in G if and only if there is a feasible solution +Sin = S′ to the reduced PAA instance. +( =⇒ ) Assume there is a clique of size k in G. We can +derive a feasible solution S′ to the reduced PAA instance as +follows. For every vertex vi ∈ V (not) in the clique, let (s′ +i0 = +1) s′ +i1 = 1. Since S is the all-N sequence, its corresponding +one-hot encoding consists of si0 = 1 for all 1 ≤ i ≤ n. Thus, +the corresponding BLOSUM62 distance is +Dseq(S, S′) = +� +1≤i≤n +(6 − 6 · 1(si ̸= s′ +i)) = 6k . +(6) +This satisfies the sequence alignment constraint defined +by Dseq(S, S′) ≤ L = 6k. Furthermore, the solution S′ +induces outputs of x′ +1 = y′ +1 = z′ +1 = k(k − 1)/2, leading +to an RMSD of U. Without loss of generality, we omit the +alignment step in computing the RMSD and therefore assume +that A(S′) = +ˆ +A(S′). The corresponding RMSD distance +Dstr(A(S), ˆ +A(S′)) in output predictions is presented below. +Recall that x1 = y1 = z1 = 0 for the the all-N sequence S +because its corresponding column in the one-hot encoding is +disconnected from the network. +Dstr(A(S), A(S′)) = +� +� +� +� 1 +n +� +i∈[n] +d(A(S)i, ˆ +A(S′)i) += +� +� +� +� 1 +n +� +3 +� +0 − k(k − 1) +2 +�2� += k(k − 1) +2 +� +3 +n . +(7) +Thus, the constraint Dstr(S, S′) ≥ U = +k(k−1) +2 +� +3 +n is +satisfied. +( ⇐= ) We prove the contrapositive. That is, if there is +no clique of size k in G, then the reduced PAA instance +is infeasible. We proceed by showing that there must be +exactly k non-zero entries in the column vector {s′ +i1|i ≤ +n} in order to satisfy constraints Dseq(S, S′) ≤ L = 6k +and Dstr(A(S), A(S′)) ≥ U and that, if there is no clique +of size k, then there is no choice of k non-zero entries in +{s′ +i1|i ≤ n} that will satisfy these constraints. Let k′ denote +the number of non-zero entries in {s′ +i1|i ≤ n}. To satisfy +Dseq(S, S′) ≤ L = 6k, it follows that k′ ≤ k. If k′ < k, then +the maximum value of Dstr(A(S), A(S′)) is k′(k′−1) +2 +� +3 +n < +k(k−1) +2 +� +3 +n and denotes to the case where the k′ non-zero +entries correspond to a clique of size k′ in G. The strict +inequality is due to the monotonically increasing nature of +this equation. Therefore, it must be that k = k′ and we have +outputs x′ +1 = y′ +1 = z′ +1 = k(k − 1)/2 as before. Suppose that +the k′ non-zero entries in {s′ +i1|i ≤ n} do not correspond to +a clique in G. Then the values x′ +1, y′ +1, and z′ +1 output by A +and corresponding to the number of edges induced by the +chosen non-zero entries would be strictly less than k(k−1)/2. +Therefore, we would have Dstr(A(S), A(S′)) < U. This +proves that the reduced PAA is infeasible. +Table 1: RMSD results when L ∈ {20, 30, 40}. +Seq. ID +n +L +RMSD +µall +µdiff +µ′ +all +µ′ +diff +Q14653 +427 +20 +18.87 +79.76 +92.92 +79.46 +86.29 +Q14653 +427 +30 +22.42 +79.76 +93.15 +77.45 +64.12 +Q14653 +427 +40 +28.28 +79.76 +90.49 +79.42 +69.026 +Q5BJD5 +291 +20 +14.311 +82.23 +89.77 +80.6 +80.64 +Q5BJD5 +291 +30 +15.708 +82.23 +59.26 +83.13 +43.53 +Q5BJD5 +291 +40 +17.132 +82.23 +62.02 +83.21 +62.83 +P59595 +422 +20 +24.321 +68.25 +91.44 +67.05 +89.51 +P59595 +422 +30 +30.139 +68.25 +93.142 +67.44 +89.29 +P59595 +422 +40 +30.675 +68.25 +46.87 +66.4 +29.33 +P0DTC9 +419 +20 +26.51 +68.39 +80.32 +68.09 +80.316 +P0DTC9 +419 +30 +26.27 +68.39 +68.05 +68.61 +65.18 +P0DTC9 +419 +40 +31.33 +68.39 +40.52 +67.76 +35.56 +P07711 +333 +20 +7.09 +93.68 +92.4 +93.2 +81.12 +P07711 +333 +30 +8.52 +93.68 +95.91 +92.95 +92.69 +P07711 +333 +40 +9.246 +93.68 +95.91 +92.85 +95.76 +Q9Y397 +364 +20 +11.184 +84.24 +97.35 +83.85 +95.81 +Q9Y397 +364 +30 +11.828 +84.24 +95.91 +83.51 +85.416 +Q9Y397 +364 +40 +14.222 +84.24 +95.91 +83.71 +89.79 +Table 2: RMSD results for the three considered categories. +Seq. ID +n +Category +RMSD +µall +µdiff +µ′ +all +µ′ +diff +Q01629 +132 +MIN. +6.02 +64.63 +32.44 +60.99 +36.99 +Q01629 +132 +AVG. +19.92 +64.63 +64.75 +63.77 +69.57 +Q01629 +132 +MAX. +19.906 +64.63 +66.99 +90.21 +90.19 +Q5BJD5 +291 +MIN. +14.023 +82.23 +38.86 +81.22 +37.79 +Q5BJD5 +291 +AVG. +14.232 +82.23 +82.24 +81.17 +77.23 +Q5BJD5 +291 +MAX. +13.567 +82.23 +98.17 +82.42 +98.1 +P59595 +422 +MIN. +24.74 +68.25 +29.13 +67.57 +31.5 +P59595 +422 +AVG. +28.164 +68.25 +69.44 +68.69 +69.04 +P59595 +422 +MAX. +24.62 +68.25 +96.14 +67.51 +96.44 +P59633 +154 +MIN. +21.67 +44.82 +27.14 +44.15 +38.75 +P59633 +154 +AVG. +21.52 +44.82 +45.1 +43.8 +42.26 +P59633 +154 +MAX. +23.13 +44.82 +61.26 +46.13 +54.84 +P0DTC9 +419 +MIN. +25.593 +68.39 +28.46 +67.9 +28.83 +P0DTC9 +419 +AVG. +21.767 +68.39 +68.37 +68.5 +70.83 +P0DTC9 +419 +MAX. +23.685 +68.39 +97.1 +68.64 +96.94 +Experimental Results +For our experimental setup, we use the default settings of +the latest version of AlphaFold 1. This includes the initial +multi-sequence alignment (MSA) step, the five-model ensem- +bles predictions, recycling, output confidence ranking, and +amber relaxation. For further details about each step, we refer +the reader to (Jumper et al. 2021a) and its supplementary +information. We include results from using the high-accuracy +full database configuration of the initial AlphaFold MSA step +along with the less accurate (and faster) reduced database +option. In order to compute the RMSD and GDT, we need +to employ an alignment algorithm. In this paper, we use the +built-in alignment PyMOL procedure without outlier rejec- +tions (Schrödinger and DeLano). The parameters of PyMOL +alignment are selected using the default settings, which in- +clude an outlier rejection cutoff of 2, a maximum number +of outlier rejection cycles of 5, and the use of the structural +superposition step . We note that these outliers only impact +the calculations of the RMSD. +Our adversarial sequences are generated by randomly sam- +pling 20 sequences from the set VL,H in (5) with H = 5 +and L = 20. Then, we pick the sequence that returns the +maximum value in RMSD structural distance. We use an +1https://github.com/deepmind/alphafold + +GDT = 17.1% +ID: P0DTC6 +𝑛=61 +Sim = 91.8% +RMSD = 5.4Å +GDT = 54.1% +ID: P0DTC8 +𝑛=121 +Sim = 95.9% +RMSD = 17.7Å +GDT = 11.1% +ID: P13164 +𝑛=125 +Sim = 96% +RMSD = 30.6Å +GDT = 4.4% +ID: O43765 𝑛 = 313 +RMSD = 12.051Å +ID: P04439 𝑛 = 365 +RMSD = 9.016Å +ID: P56962 𝑛 = 302 +RMSD = 25.013Å +ID: P59632 𝑛 = 274 +RMSD = 18.052Å +Figure 4: The structures of the original (black) and adversarial (red) sequences from AlphaFold. The 3D plots, aligned using PyMol +(Schrödinger and DeLano), are for proteins O43765 (first), P04439 (second), P56962 (third), and P59632 (fourth). For structure differences, +the RMSD values are reported. The structures of the complete list of sequences are given in the supplementary material. +Table 3: RMSD, GDT-TS, and GDT-HA results using the full database AlphaFold configuration with L = 20 and H = 5. The average +columns correspond to 20 adversarial samples for each protein ID. The complete table is placed in the supplementary material. +Seq. ID +n +Similarity (%) +RMSD +Avg. RMSD +GDT-TS (%) +Avg. GDT-TS (%) +GDT-HA (%) +Avg. GDT-HA (%) +run-time (days) +O43765 +313 +98.4026 +14.438 +9.1741 +13.9776 +35.4832 +2.8754 +17.6358 +1.6068 +P56962 +302 +98.3444 +22.301 +15.8695 +12.3344 +18.6921 +3.4768 +5.803 +0.5959 +P04439 +365 +98.6301 +6.162 +3.7942 +47.7397 +68.2705 +25.0 +45.774 +0.6429 +Q99836 +296 +98.3108 +8.761 +5.2907 +24.1554 +46.6723 +7.6858 +26.2584 +0.6246 +P59632 +274 +98.1752 +13.018 +8.4704 +24.8175 +41.0401 +9.0328 +21.6834 +0.5214 +AMD EPYC 7702 64-Core Processor with 1 TiB of RAM +and NVIDIA A100 GPU. We generate adversarial sequences +against the COVID-19 protein sequences from the UniProt +database considered by AlphaFold in (Jumper et al. 2020). +The original fasta (file extension for protein sequences) se- +quence files are available online 2. Additionally, we generate +adversarial sequences against most of the the UniProt (Uni- +versal Protein resource, a central repository of protein data +created by combining the Swiss-Prot, TrEMBL and PIR-PSD +databases (uni 2021)). Our code is provided as supplementary +material. +BLOSUM Threshold Experiment +In this subsection, we want to investigate how a change in +the bound on biological similarity changes the adversarial +sequence. In other words, we show the impact of using differ- +ent BLOSUM thresholds in set VL,H. As such, we randomly +select 6 sequences and generate adversarial sequences by +configuring the BLOSUM threshold, L, to be 20, 30, and +40 (we use strict equalities to ensure the exact BLOSUM +distance) and set H = 5. For each case, we obtain the RMSD +after alignment as reported in the fourth column of Table 1. +Furthermore, we present the average confidence percentage +level of the prediction of the original (adversarial) sequence +as reported by AlphaFold and denoted by µall (µ′ +all). Addi- +tionally, in the 6th and 8th columns, we report the average +confidence values for the residues that are different between +the original and adversarial sequences. These are denoted by +µdiff and µ′ +diff, respectively. We observe that, in general, when +the BLOSUM threshold distance increases, the RMSD also +increases. This means that biologically increased distance +in the input space, in general, causes higher changes in the +output predictions of AlphaFold. In terms of the confidence +scores, we observe that the change in the overall average con- +fidence between the original and perturbed sequence is not +2https://ftp.uniprot.org/pub/databases/uniprot/pre_release/ +covid-19.fasta +significant. However, in almost all the considered cases, we +notice that the prediction confidence of the altered residues +has reduced for the adversarial sequence when compared to +the ones reported for the original sequence. +Confidence Experiment +Given a sequence S, per residue, AlphaFold generates an +estimate of its prediction confidence in the form of a value in +[0, 100]. This value is called the predicted Local Distance Test +(pLDDT) and represents the predicted value on the lDDT-Cα +metric (Mariani et al. 2013). +In this subsection, we answer the following question. Does +selecting the residues to be changed based on their low (or +high) confidence scores impact the resulting RMSD between +the original and adversarial structure prediction? Phrased dif- +ferently, in terms of the RMSD, we illustrate the impact of +using the prediction confidence scores of every residue of the +predicted structure of the original sequence in determining +the location of the residues to be altered in the adversarial se- +quence generation method presented in the previous section. +As such, five, not cherry picked, randomly selected sequences +are used. Then, the locations of the 5 residues to be altered +are taken based on three categories as follows. Residues are +selected with confidence values near the (i) minimum con- +fidence score (MIN. category), (ii) the average score (AVG. +category), and (iii) the maximum confidence score (MAX. +category). Results are presented in Table 2. We observe that, +in general, selecting residues with low or high confidence +scores is not related to the amount of the induced RMSD +at the output. As such, in our method, the locations of the +flipped residues are selected independent of the confidence +scores. +COVID-19 Case Studies +We apply our adversarial approach to 111 publicly available +COVID-19 protein sequences as of the time of this writing +per the UniProt database using AlphaFold full database con- +figuration. Additionally, in the supplementary material, we + +Table 4: Prediction confidence results using the full database AlphaFold configuration with L = 20. +Seq. ID +n +RMSD +µall +σall +µdiff +σdiff +µ′ +all +σ′ +all +µ′ +diff +σ′ +diff +O43765 +313 +14.438 +80.221 +19.634 +94.786 +1.027 +80.554 +19.423 +93.71 +1.392 +P56962 +302 +22.301 +69.172 +23.753 +96.516 +0.409 +69.342 +23.759 +96.54 +0.463 +P04439 +365 +6.162 +86.845 +18.995 +44.23 +2.704 +86.921 +19.068 +44.678 +3.968 +Q99836 +296 +8.761 +81.213 +13.817 +78.914 +7.454 +80.918 +13.971 +72.198 +6.253 +P59632 +274 +13.018 +58.367 +18.783 +66.136 +2.029 +57.364 +18.794 +60.926 +4.362 +Table 5: Overall Prediction and attack results for the reduced and full database configurations of AlphaFold. +Configuration. +Avg. n +Std. n +Avg. µall +Std. µall +Avg. RMSD +Std. RMSD +Avg. GDT +Std. GDT +Avg. run-time +Std. run-time +reduced database +480.53 +416.66 +78.22 +10.96 +15.31 +11.24 +34.08 +28.39 +0.68 +0.59 +full database +410.73 +336.63 +78.25 +10.23 +14.78 +11.18 +34.95 +28.16 +0.86 +0.63 +provide complete results using the reduced AlphaFold con- +figuration. The BLOSUM62 distance between the original +and adversarial sequences is at most 20, thus they are biolog- +ically close to each other (Chothia and Lesk 1986; Sander +and Schneider 1991). Given the long list of the considered +sequences, we describe only the following. SGTA_HUMAN +Small glutamine-rich tetratricopeptide repeat-containing pro- +tein alpha (O43765), HLAA_HUMAN HLA class I histocom- +patibility antigen, A alpha chain (P04439), STX17_HUMAN +Syntaxin-17 (P56962), AP3A_SARS ORF3a (P59632), and +MYD88_HUMAN Myeloid differentiation primary response +protein MyD88 (Q99836). The cases covered include homo +sapiens and severe acute respiratory syndrome coronavirus +2 (2019-nCoV) (SARS-CoV-2) organisms which provide a +wide variety of proteins. The considered sequences vary in +length as they range from n = 22 to n = 2511. +Figures 1 and 4 show the aligned predicted structures of +the proteins described earlier where the original sequence +is given in black and the adversarial sequence is given in +red. We observe that, independent of the predicted struc- +ture of the original sequence, a small change in the input +sequence results in significant changes in the output struc- +tures. The resulting structural distances (similarities) mea- +sured in Å (percentage) are given in terms of the RMSD +(GDT-TS) in the fourth (sixth) column of Table 9 for the full +database configuration. Furthermore, we report the results +using GDT-HA in the eighth column. The high similarity +between the original and adversarial sequences is observed +from the third column. The similarity percentage is calculated +as 100(n − Dham(S, S′))/n, where Dham(S, S′) ≤ H = 5. +The complete results of all the considered proteins, includ- +ing reduced AlphaFold configuration, are provided in the +supplementary material. +As observed from the RMSD and GDT results in Table 9, +small changes in the input sequence corresponding to only +five residues cause AlphaFold to predict structures that are +highly divergent from the predicted structure of the original +sequence. The last column in Table 9 reports the total execu- +tion time (in days) of running the 20 adversarial sequences +that were randomly selected from the set VL,H, which is +shown to scale with the sequence length. We only select 20 +samples given the long time incurred by AlphaFold to predict +the output structure. +Additionally, in Table 13, we report the average (devi- +ation) prediction confidence results as for all the residues +(designated with subscript ‘all’) and for the 5 altered residues +(subscript ‘diff’). The standard deviation is denoted σ. We ob- +serve that, independent of the average prediction confidence, +the RMSD between the original and adversarial predicted +structures is always high. This is noted for both the full and +reduced database configurations of AlphaFold. Moreover, we +observe that AlphaFold predicts the adversarial structure with +similar confidence values to the original sequence (e.g., see +the 4th and 8th columns in both tables). The same observa- +tion holds for the entire sequence and for the altered residues +(columns 6 and 10). +In Tables 14 and 15 of the supplementary, we break down +GDT scores between the structures of the original and per- +turbed sequences based on the prediction confidence scores +of the original sequence. We use the regions (1 to 4) defined +by AlphaFold. As observed w.r.t. all regions, GDT scores are, +in general, low. +For the considered dataset, the values presented in Table 5 +gauge the overall robustness of AlphaFold to adversarial se- +quences. As indicated in the documentation of AlphaFold, for +better accuracy, the full database configuration incurs a higher +execution time compared to the reduced database configura- +tion. The reported average values of the RMSD and GDT-TS +measures are 14.78Å and 37.95%, respectively. In CASP14 +(year 2020), AlphaFold achieved a median GDT-TS score +of 92.4%, and 88% of their predictions fall under RMSD = +4Å 3. These results are obtained by comparing the predicted +and ground truth structures. The CASP14 AlphaFold results +underscore the significance of the values reported in Tables 9 +and 5, as they show how small changes in the input sequences +could damage the predictions (See columns 6 to 9 in Table 5). +The key takeaway is that AlphaFold is generally not robust +even when a basic approach is used to generate perturbations +of the input protein sequence. +Conclusion +The groundbreaking progress made in recent years on the +prediction of protein folding structures promises to enable +profound advances in the understanding of diseases, the map- +ping of the human proteome, and the design of drugs and +therapeutics. However, until these predictions are shown to +be robust, we argue that the grand challenge of predictive +protein folding persists. In this paper, we have presented +the first work in this direction by demonstrating that Pro- +tein Folding Neural Networks (PFNNs) are often susceptible +to adversarial attacks in the form of minor perturbations to +the input protein sequence. These perturbations can induce +3https://predictioncenter.org/casp14/index.cgi + +great changes in the predicted protein structure and the re- +sulting lack of robustness precludes the adoption of such +PFNNs in safety-critical applications. We have employed +standard protein structural distance and similarity to measure +the robustness of AlphaFold. While the perturbation methods +employed in this paper were basic for the purposes of illus- +trating the lack of robustness of PFNNs, the results presented +herein can be readily used as a baseline for future work on +adversarial attacks on PFNNs and their robustness. +References +2021. 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GDT-HA (%) +run-time (days) +A0A663 +38 +86.8421 +3.129 +1.655 +78.9474 +95.0 +57.2368 +84.5395 +0.4173 +MPROTEI +193 +97.4093 +2.06 +1.2348 +83.1606 +97.0531 +60.6218 +89.6308 +0.7081 +NSP2NNN +546 +99.0842 +19.561 +11.4867 +4.2125 +19.9657 +0.3663 +7.0696 +1.6596 +NSP4NNN +494 +98.9879 +11.389 +2.9463 +6.5789 +72.7277 +0.253 +51.1943 +1.6327 +NSP6NNN +290 +98.2759 +8.921 +3.5994 +29.4828 +79.7716 +9.3966 +62.7629 +0.9444 +O00327 +626 +99.2013 +10.445 +5.7079 +26.3578 +52.6637 +9.6645 +31.4177 +0.5849 +O14745 +358 +98.6034 +12.34 +8.3659 +7.6117 +24.7835 +0.5587 +8.6627 +0.4956 +O14786 +923 +99.4583 +33.465 +11.134 +0.5688 +45.6744 +0.0271 +27.1682 +0.7755 +O15393 +492 +98.9837 +24.847 +13.9194 +6.7073 +32.9014 +1.1179 +15.3938 +0.5225 +O15455 +904 +99.4469 +24.886 +11.3427 +5.3927 +33.4748 +0.9403 +17.1308 +0.7941 +O43765 +313 +98.4026 +12.051 +9.9661 +17.0927 +25.1917 +2.7157 +8.746 +0.4918 +O94826 +608 +99.1776 +11.824 +6.2432 +50.6579 +65.514 +29.4819 +44.4881 +0.5685 +O95721 +258 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20 and H = 5. The average +results correspond to 20 adversarial samples for each protein ID. Part II of II. +Protein ID +n +Similarity (%) +RMSD +Avg. RMSD +GDT-TS (%) +Avg. GDT-TS (%) +GDT-HA (%) +Avg. GDT-HA (%) +run-time (days) +P52292 +529 +99.0548 +9.065 +4.3164 +44.9433 +75.0071 +23.3459 +54.5723 +0.531 +P52948 +1817 +99.7248 +56.34 +39.782 +1.9125 +8.2327 +0.0413 +2.4869 +2.0914 +P56962 +302 +98.3444 +25.013 +19.3739 +10.3477 +18.0505 +0.4139 +5.1283 +0.5233 +P59594 +1255 +99.6016 +4.432 +2.9699 +81.1355 +88.4303 +59.6016 +72.8357 +0.9841 +P59595 +422 +98.8152 +26.286 +18.0807 +4.0284 +15.0474 +0.7109 +3.2524 +0.4821 +P59596 +221 +97.7376 +9.173 +4.2628 +30.3167 +71.1369 +10.0679 +55.6618 +0.4292 +P59632 +274 +98.1752 +18.052 +14.8883 +9.854 +21.3777 +1.2774 +7.6505 +0.452 +P59633 +154 +96.7532 +20.183 +12.9835 +6.3312 +22.8247 +0.487 +8.4253 +0.4349 +P59634 +63 +92.0635 +2.5 +1.8276 +74.2063 +86.2103 +49.2063 +64.8214 +0.3953 +P59635 +122 +95.9016 +5.955 +3.2856 +56.3525 +76.2807 +32.582 +54.0984 +0.4216 +P59636 +98 +94.898 +3.095 +1.6404 +74.7449 +92.8061 +50.0 +82.3469 +0.4042 +P59637 +76 +93.4211 +1.743 +1.4686 +88.8158 +94.7039 +69.0789 +78.5362 +0.398 +P62937 +165 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+34.4523 +0.4998 +Q99836 +296 +98.3108 +6.525 +4.4038 +31.8412 +49.6833 +11.7399 +26.7314 +0.4677 +Q9BV40 +100 +95.0 +3.427 +2.0524 +60.75 +82.1125 +37.25 +68.025 +0.4523 +Q9BYF1 +480 +98.9583 +5.509 +1.9196 +65.6771 +92.0182 +41.875 +76.8854 +0.4854 +Q9BYX4 +1025 +99.5122 +4.818 +3.6154 +42.7317 +57.2427 +19.2439 +33.9329 +1.1546 +Q9C000 +1473 +99.6606 +35.861 +17.441 +2.6986 +25.5431 +0.3904 +10.5821 +1.398 +Q9H074 +479 +98.9562 +30.3 +19.6118 +6.524 +19.6425 +0.8351 +7.8314 +0.5462 +Q9NR97 +1041 +99.5197 +21.869 +8.5147 +5.3074 +41.9681 +1.2248 +24.9412 +1.0298 +Q9NRS4 +437 +98.8558 +6.233 +3.0814 +53.833 +79.7368 +29.7483 +61.4045 +0.5264 +Q9NVJ2 +186 +97.3118 +1.945 +1.2634 +88.4409 +97.0027 +68.9516 +90.4368 +0.4276 +Q9NYK1 +1049 +99.5234 +12.803 +7.1807 +25.9771 +46.2345 +6.9828 +25.8115 +1.0763 +Q9UHD2 +729 +99.3141 +5.063 +2.4523 +73.5597 +92.3525 +50.5487 +83.5391 +0.7376 +Q9ULC8 +765 +99.3464 +34.128 +26.1615 +5.1961 +15.9297 +1.3725 +6.2614 +0.8275 +Q9Y2I7 +2098 +99.7617 +34.987 +27.8546 +8.9967 +22.4285 +0.9771 +9.4948 +4.8776 +Q9Y397 +364 +98.6264 +11.551 +6.3126 +40.4533 +65.783 +19.7802 +48.2521 +0.4675 + +Table 8: RMSD, GDT-TS, and GDT-HA results using the full database AlphaFold configuration with L = 20 and H = 5. The average results +correspond to 20 adversarial samples for each protein ID. Part I of II. +Seq. ID +n +Similarity (%) +RMSD +Avg. RMSD +GDT-TS (%) +Avg. GDT-TS (%) +GDT-HA (%) +Avg. GDT-HA (%) +run-time (days) +A0A663 +38 +86.8421 +2.886 +1.7076 +82.8947 +95.0 +63.8158 +85.7566 +0.4058 +MPROTEI +193 +97.4093 +5.562 +4.8324 +37.5648 +45.2785 +13.4715 +23.1995 +0.5288 +NSP2NNN +546 +99.0842 +22.333 +8.7108 +8.3791 +33.6332 +1.3736 +16.0119 +0.885 +NSP4NNN +494 +98.9879 +16.995 +14.4184 +7.2368 +13.4261 +1.0121 +2.9681 +0.6312 +NSP6NNN +290 +98.2759 +5.41 +3.2315 +66.4655 +85.2543 +42.931 +67.944 +0.5433 +O00327 +626 +99.2013 +18.69 +5.133 +10.1438 +62.7456 +0.7188 +41.1621 +1.8352 +O14745 +358 +98.6034 +7.395 +3.6869 +23.3939 +57.2521 +4.8184 +33.7919 +0.8124 +O14786 +923 +99.4583 +36.546 +17.6305 +1.0022 +31.0455 +0.0271 +16.2649 +1.4629 +O15393 +492 +98.9837 +19.978 +13.4431 +9.4512 +33.5823 +2.6423 +15.0737 +0.8496 +O15455 +904 +99.4469 +20.539 +8.6482 +8.4071 +39.6391 +1.3551 +20.0733 +2.2111 +O43765 +313 +98.4026 +14.438 +9.1741 +13.9776 +35.4832 +2.8754 +17.6358 +1.6068 +O94826 +608 +99.1776 +8.407 +5.513 +48.7253 +70.4996 +27.426 +52.021 +2.4019 +O95721 +258 +98.062 +11.099 +4.2622 +32.2674 +69.6754 +11.8217 +49.4234 +0.5642 +O95992 +272 +98.1618 +5.988 +3.0947 +54.3199 +79.5864 +31.1581 +64.3244 +0.5627 +P00973 +400 +98.75 +14.825 +7.3132 +24.0625 +52.4594 +6.9375 +30.7406 +0.6362 +P01185 +164 +96.9512 +7.641 +3.0531 +32.4695 +67.4085 +10.8232 +43.628 +0.4879 +P01889 +362 +98.6188 +9.441 +5.624 +40.5387 +58.7017 +19.4061 +36.5608 +0.638 +P02649 +317 +98.4227 +22.675 +13.6351 +2.3659 +24.1916 +0.0789 +11.455 +0.6008 +P04233 +296 +98.3108 +18.279 +7.6704 +15.625 +39.9535 +2.2804 +18.6529 +0.5748 +P04439 +365 +98.6301 +6.162 +3.7942 +47.7397 +68.2705 +25.0 +45.774 +0.6429 +P05109 +93 +94.6237 +1.19 +0.9509 +96.2366 +98.9247 +92.4731 +96.707 +0.4963 +P05161 +165 +96.9697 +2.569 +1.2816 +80.6061 +96.4621 +57.8788 +90.3485 +0.5225 +P05231 +212 +97.6415 +9.402 +3.6258 +28.8915 +64.7642 +9.7877 +42.8715 +0.5111 +P07711 +333 +98.4985 +7.479 +1.6907 +44.0691 +93.0856 +21.1712 +82.9467 +0.6467 +P08887 +468 +98.9316 +10.171 +6.0542 +36.3782 +50.86 +15.812 +29.3109 +0.8998 +P09429 +215 +97.6744 +20.391 +12.9117 +7.6744 +26.4767 +1.2791 +10.6628 +0.5901 +P09958 +794 +99.3703 +32.057 +17.4892 +4.8489 +22.9455 +0.3778 +9.3923 +1.5547 +P0DTC2 +1273 +99.6072 +4.917 +3.6252 +81.1076 +88.0027 +60.546 +71.8274 +1.5762 +P0DTC3 +275 +98.1818 +11.391 +7.6561 +28.9091 +47.5182 +11.6364 +27.0682 +0.5612 +P0DTC4 +75 +93.3333 +2.496 +1.4206 +73.6667 +92.8667 +51.0 +77.7667 +0.4491 +P0DTC5 +222 +97.7477 +6.436 +1.9211 +67.2297 +91.9032 +46.7342 +81.6667 +0.5269 +P0DTC6 +61 +91.8033 +4.88 +1.547 +56.9672 +95.4508 +33.1967 +85.6352 +0.4218 +P0DTC7 +121 +95.8678 +4.492 +2.8855 +53.719 +77.0558 +30.1653 +55.0103 +0.4462 +P0DTC8 +121 +95.8678 +17.601 +12.2174 +11.5702 +25.9194 +2.4793 +9.6798 +0.4668 +P0DTC9 +419 +98.8067 +29.977 +18.2825 +3.58 +17.148 +0.358 +6.8377 +0.604 +P0DTD2 +97 +94.8454 +4.156 +2.0488 +65.9794 +89.884 +42.268 +74.9485 +0.439 +P0DTD3 +73 +93.1507 +17.276 +12.5488 +3.7671 +16.2671 +0.0 +3.8699 +0.4229 +P0DTD8 +43 +88.3721 +2.394 +1.2436 +90.1163 +99.4477 +73.8372 +97.936 +0.4274 +P0DTF1 +22 +77.2727 +2.033 +1.5082 +86.3636 +97.6136 +64.7727 +90.3409 +0.4111 +P0DTG0 +57 +91.2281 +13.398 +7.9086 +17.1053 +43.9693 +3.5088 +23.8596 +0.4064 +P0DTG1 +41 +87.8049 +17.037 +8.2412 +7.3171 +45.8537 +0.6098 +28.811 +0.4122 +P11226 +248 +97.9839 +21.819 +10.5974 +8.871 +31.744 +0.6048 +14.7782 +0.6851 +P13164 +125 +96.0 +8.675 +3.759 +30.2 +69.17 +10.8 +47.44 +0.4743 +P13747 +358 +98.6034 +12.178 +9.1998 +27.8631 +42.6013 +8.8687 +22.8631 +0.6393 +P17181 +557 +99.1023 +21.61 +15.1002 +16.3375 +30.7989 +5.0718 +13.8353 +1.1425 +P17405 +631 +99.2076 +12.066 +4.2296 +52.6149 +75.9311 +29.8336 +56.6442 +0.9839 +P20701 +1170 +99.5726 +8.323 +4.1428 +34.9786 +62.5865 +12.3504 +39.1603 +2.4989 +P26022 +381 +98.6877 +14.411 +9.0528 +18.2415 +41.9029 +4.3963 +23.5138 +0.949 +P26715 +233 +97.8541 +22.384 +10.099 +12.2318 +35.0215 +2.1459 +16.0193 +0.5796 +P29597 +1187 +99.5788 +32.804 +22.2887 +3.4751 +31.9482 +0.5897 +20.8888 +3.6416 +P30556 +359 +98.6072 +10.111 +5.7957 +23.6769 +61.3962 +8.2173 +40.3343 +0.6767 +P33076 +1130 +99.5575 +26.719 +20.1579 +13.6504 +26.344 +0.4646 +10.1427 +2.2339 +P35232 +272 +98.1618 +3.447 +1.6141 +63.2353 +91.5074 +39.4301 +78.4559 +0.5927 +P35613 +385 +98.7013 +8.896 +3.4822 +27.2078 +68.9123 +7.5325 +46.8344 +1.032 +P40189 +918 +99.4553 +24.632 +10.274 +27.9139 +50.4112 +11.7647 +30.6727 +2.0398 +P47901 +424 +98.8208 +5.702 +3.889 +49.8821 +71.8013 +28.3019 +51.418 +0.7334 +P48551 +515 +99.0291 +28.717 +14.5889 +4.9029 +19.2087 +0.0485 +6.4102 +0.7453 +P51149 +207 +97.5845 +6.188 +2.8336 +37.8019 +80.8092 +15.2174 +61.3285 +0.5977 + +Table 9: RMSD, GDT-TS, and GDT-HA results using the full database AlphaFold configuration with L = 20 and H = 5. The average results +correspond to 20 adversarial samples for each protein ID. This is part II of II. +Seq. ID +n +Similarity (%) +RMSD +Avg. RMSD +GDT-TS (%) +Avg. GDT-TS (%) +GDT-HA (%) +Avg. GDT-HA (%) +run-time (days) +P56962 +302 +98.3444 +22.301 +15.8695 +12.3344 +18.6921 +3.4768 +5.803 +0.5959 +P59596 +221 +97.7376 +7.506 +2.1014 +33.9367 +90.1923 +11.7647 +79.2251 +0.5245 +P59632 +274 +98.1752 +13.018 +8.4704 +24.8175 +41.0401 +9.0328 +21.6834 +0.5214 +P59633 +154 +96.7532 +20.993 +15.7533 +5.3571 +15.7468 +0.3247 +4.3912 +0.4625 +P59634 +63 +92.0635 +2.514 +1.5476 +82.1429 +94.3849 +58.7302 +77.619 +0.4214 +P59635 +122 +95.9016 +6.527 +4.9216 +37.7049 +58.4016 +14.3443 +34.7746 +0.4458 +P59636 +98 +94.898 +5.647 +1.8721 +47.7041 +89.3622 +24.4898 +78.3673 +0.4361 +P59637 +76 +93.4211 +3.488 +1.6534 +64.1447 +88.6513 +40.4605 +70.5592 +0.4352 +P62937 +165 +96.9697 +0.963 +0.8934 +99.5455 +99.947 +97.1212 +98.8333 +0.5403 +P68104 +462 +98.9177 +1.547 +1.3632 +93.1818 +96.1607 +75.974 +85.1677 +0.8504 +P84022 +425 +98.8235 +16.633 +7.2522 +15.4706 +52.5882 +3.4706 +31.4382 +0.67 +PLRPOCT +355 +98.5915 +3.824 +1.4276 +75.7746 +93.4648 +53.662 +78.5493 +0.5569 +Q01628 +133 +96.2406 +26.947 +15.7051 +3.0075 +21.8797 +0.3759 +9.2011 +0.498 +Q01629 +132 +96.2121 +18.508 +10.094 +16.0985 +32.3201 +4.1667 +15.322 +0.4848 +Q10589 +180 +97.2222 +11.884 +3.1079 +11.8056 +73.3403 +0.9722 +53.7847 +0.5251 +Q13241 +179 +97.2067 +11.456 +5.3773 +23.4637 +50.4958 +6.8436 +28.9874 +0.5468 +Q13568 +498 +98.996 +29.991 +17.9838 +1.2048 +13.637 +0.1004 +5.1657 +0.7167 +Q14653 +427 +98.829 +16.703 +9.6337 +15.0468 +28.0884 +3.9227 +11.1797 +0.6629 +Q16236 +605 +99.1736 +38.828 +18.9672 +4.5455 +22.1591 +0.0826 +9.7376 +0.8244 +Q16552 +155 +96.7742 +13.575 +8.3168 +17.4194 +30.0242 +3.5484 +12.7823 +0.4973 +Q16553 +131 +96.1832 +10.901 +5.9358 +46.7557 +64.9523 +28.4351 +46.3836 +0.4789 +Q16665 +826 +99.3947 +41.354 +31.4424 +5.9322 +14.3644 +0.4843 +4.3614 +2.0399 +Q4KMQ2 +910 +99.4505 +4.59 +2.1627 +83.022 +94.5618 +62.6374 +87.2802 +1.1123 +Q5BJD5 +291 +98.2818 +13.792 +9.6387 +27.6632 +44.4674 +7.5601 +24.317 +0.6327 +Q5W0Z9 +365 +98.6301 +15.887 +6.2479 +22.3288 +63.8938 +7.1918 +42.9863 +0.6469 +Q7TFA0 +39 +87.1795 +2.505 +1.6814 +87.8205 +94.7115 +66.6667 +82.8205 +0.4365 +Q7TFA1 +44 +88.6364 +2.106 +1.552 +94.8864 +97.358 +80.1136 +90.5114 +0.4189 +Q7TLC7 +70 +92.8571 +24.995 +6.955 +2.1429 +54.0893 +0.0 +35.9464 +0.4291 +Q7Z434 +540 +99.0741 +50.815 +34.506 +3.4722 +8.9236 +0.5093 +2.8449 +0.7655 +Q80H93 +84 +94.0476 +14.707 +2.8159 +17.8571 +82.2173 +3.2738 +63.6012 +0.446 +Q86U44 +580 +99.1379 +17.742 +12.9209 +22.2414 +40.4203 +4.8707 +19.8448 +0.7916 +Q86WV6 +379 +98.6807 +20.858 +8.6755 +11.9393 +65.4617 +3.628 +53.5026 +0.6223 +Q8IUC6 +712 +99.2978 +49.827 +26.3508 +0.6671 +11.1464 +0.0351 +2.8002 +0.813 +Q8N3R9 +675 +99.2593 +35.139 +15.0821 +4.4815 +29.0204 +0.7037 +13.45 +0.9995 +Q8N884 +522 +99.0421 +29.73 +20.0914 +7.6149 +16.2883 +1.5805 +4.9449 +0.7615 +Q8NAC3 +791 +99.3679 +11.202 +7.0906 +29.3616 +49.9621 +10.9039 +27.9772 +0.98 +Q8NHX9 +752 +99.3351 +6.888 +5.0382 +56.9149 +72.9887 +32.879 +52.387 +1.1902 +Q92499 +740 +99.3243 +12.723 +5.6603 +20.8784 +55.125 +5.9459 +34.5507 +1.7048 +Q96F46 +866 +99.4226 +36.383 +22.73 +2.5115 +21.1345 +0.3176 +8.177 +1.0736 +Q96JC1 +886 +99.4357 +3.565 +2.164 +66.0835 +82.9628 +41.4221 +63.5468 +1.1447 +Q96P20 +1036 +99.5174 +14.659 +6.1123 +19.6911 +55.3933 +4.778 +33.8694 +2.4871 +Q96PD4 +163 +96.9325 +8.703 +4.1108 +43.5583 +74.6242 +22.8528 +54.3942 +0.506 +Q99836 +296 +98.3108 +8.761 +5.2907 +24.1554 +46.6723 +7.6858 +26.2584 +0.6246 +Q9BV40 +100 +95.0 +3.375 +2.3022 +62.5 +77.55 +38.0 +59.6125 +0.4611 +Q9BYF1 +480 +98.9583 +5.231 +1.4446 +67.0312 +94.7708 +43.125 +80.4167 +0.7056 +Q9H074 +479 +98.9562 +36.353 +28.2003 +1.4614 +8.63 +0.0522 +2.44 +0.738 +Q9NR97 +1041 +99.5197 +21.572 +11.2926 +6.4121 +23.5387 +1.3929 +8.2841 +2.3966 +Q9NRS4 +437 +98.8558 +6.429 +2.8379 +51.2586 +85.5807 +28.8902 +70.7466 +0.7937 +Q9NVJ2 +186 +97.3118 +1.744 +1.2656 +91.129 +96.8683 +72.8495 +89.3952 +0.6283 +Q9ULC8 +765 +99.3464 +38.808 +32.3238 +3.4314 +7.8088 +0.3922 +1.8399 +1.0204 +Q9Y2I7 +2098 +99.7617 +33.211 +23.7625 +5.8508 +16.2095 +0.9175 +5.4469 +3.7743 +Q9Y397 +364 +98.6264 +9.301 +2.9895 +44.9176 +82.3867 +23.3516 +66.1504 +0.6409 + +Table 10: Prediction confidence results using the reduced database AlphaFold configuration with L = 20. This is part I of II. +Protein ID +n +RMSD +µall +σall +µdiff +σdiff +µ′ +all +σ′ +all +µ′ +diff +σ′ +diff +A0A663 +38 +3.129 +75.048 +11.991 +71.45 +1.508 +74.755 +11.327 +74.862 +4.587 +MPROTEI +193 +2.06 +89.041 +7.416 +87.138 +2.009 +87.461 +8.837 +73.19 +3.874 +NSP2NNN +546 +19.561 +74.536 +17.778 +92.656 +1.044 +73.385 +17.646 +91.094 +1.21 +NSP4NNN +494 +11.389 +83.936 +8.896 +89.444 +1.356 +83.913 +8.92 +87.668 +1.288 +NSP6NNN +290 +8.921 +77.331 +11.445 +73.286 +2.769 +76.104 +12.64 +65.228 +0.762 +O00327 +626 +10.445 +65.242 +25.756 +32.584 +3.532 +65.128 +25.804 +32.296 +1.221 +O14745 +358 +12.34 +72.042 +17.526 +88.666 +0.565 +72.051 +17.223 +70.686 +7.533 +O14786 +923 +33.465 +79.024 +21.94 +36.21 +1.406 +78.925 +21.993 +35.662 +2.1 +O15393 +492 +24.847 +80.455 +20.693 +41.292 +2.675 +80.765 +20.643 +41.088 +3.315 +O15455 +904 +24.886 +90.367 +14.664 +96.286 +1.564 +90.261 +14.604 +91.518 +0.356 +O43765 +313 +12.051 +79.03 +20.468 +88.548 +3.314 +79.191 +20.501 +88.86 +2.758 +O94826 +608 +11.824 +79.953 +25.515 +25.302 +1.419 +80.241 +26.177 +24.912 +2.005 +O95721 +258 +8.478 +75.906 +22.281 +89.91 +0.914 +75.923 +21.394 +89.766 +1.254 +O95786 +925 +33.92 +83.194 +16.732 +88.724 +1.844 +83.717 +16.674 +86.648 +2.241 +O95992 +272 +6.531 +91.633 +13.904 +96.86 +0.666 +91.815 +14.142 +96.806 +0.787 +P00973 +400 +7.988 +83.749 +21.993 +89.328 +1.047 +83.832 +21.867 +90.164 +0.926 +P01185 +164 +5.552 +79.569 +16.773 +86.778 +1.34 +78.21 +16.506 +79.71 +1.177 +P01889 +362 +9.904 +87.936 +18.463 +47.662 +2.89 +87.711 +18.817 +42.058 +1.925 +P02649 +317 +16.897 +75.319 +18.538 +45.584 +0.691 +74.833 +19.063 +43.904 +3.122 +P04233 +296 +17.217 +70.433 +20.278 +90.33 +2.496 +69.938 +20.327 +87.284 +3.829 +P04439 +365 +9.016 +86.959 +19.05 +43.686 +2.588 +87.121 +19.044 +44.268 +3.233 +P05109 +93 +1.953 +92.956 +10.483 +98.21 +0.266 +93.971 +10.54 +97.618 +0.329 +P05161 +165 +2.557 +86.236 +11.269 +88.324 +3.105 +87.591 +10.99 +92.202 +0.684 +P05231 +212 +7.214 +84.805 +16.694 +58.22 +1.7 +84.561 +17.821 +57.118 +5.322 +P07711 +333 +8.263 +93.679 +11.289 +96.096 +0.739 +93.987 +9.796 +92.25 +2.42 +P08887 +468 +11.083 +78.202 +23.606 +48.592 +1.57 +78.237 +23.467 +40.974 +4.313 +P09429 +215 +24.633 +76.632 +23.082 +87.266 +1.572 +76.506 +22.79 +85.194 +1.507 +P09958 +794 +24.603 +85.088 +20.744 +55.2 +2.152 +85.213 +20.625 +48.182 +2.726 +P0DTC2 +1273 +10.44 +78.887 +16.851 +43.964 +11.301 +78.793 +17.001 +41.546 +12.094 +P0DTC3 +275 +16.932 +49.31 +20.799 +68.958 +1.12 +49.737 +18.922 +66.964 +2.356 +P0DTC4 +75 +3.403 +74.108 +15.66 +86.564 +2.455 +70.975 +16.037 +78.342 +3.36 +P0DTC5 +222 +9.978 +84.285 +13.438 +77.74 +1.38 +84.175 +13.413 +83.272 +0.819 +P0DTC6 +61 +5.379 +86.46 +13.406 +97.194 +0.606 +84.249 +13.294 +88.176 +2.458 +P0DTC7 +121 +8.067 +76.98 +15.559 +68.908 +1.929 +77.021 +13.534 +78.996 +2.271 +P0DTC8 +121 +17.706 +80.17 +13.321 +50.0 +2.248 +60.707 +22.596 +35.768 +1.302 +P0DTC9 +419 +27.007 +68.399 +23.597 +36.932 +3.69 +68.28 +23.457 +36.95 +3.116 +P0DTD2 +97 +4.388 +79.607 +16.769 +92.54 +0.706 +79.076 +16.655 +91.874 +0.678 +P0DTD3 +73 +24.441 +68.544 +14.985 +57.288 +2.791 +69.92 +14.985 +63.542 +5.297 +P0DTD8 +43 +2.458 +89.464 +10.491 +85.754 +6.336 +89.428 +10.44 +84.788 +2.948 +P0DTF1 +22 +2.589 +95.196 +6.768 +98.532 +0.162 +88.977 +9.041 +86.57 +0.939 +P0DTG0 +57 +12.352 +47.494 +6.446 +46.364 +2.274 +51.696 +8.039 +54.49 +2.471 +P0DTG1 +41 +15.533 +74.977 +7.67 +79.804 +2.456 +73.7 +9.233 +78.844 +1.75 +P11226 +248 +16.516 +80.221 +20.491 +45.93 +3.496 +80.601 +19.994 +53.552 +2.939 +P13164 +125 +30.581 +65.86 +17.167 +57.746 +1.217 +66.137 +17.299 +68.906 +1.428 +P13747 +358 +14.9 +87.122 +17.953 +65.358 +3.687 +86.866 +18.362 +45.174 +1.508 +P17181 +557 +12.674 +78.051 +21.024 +41.674 +5.38 +78.097 +20.97 +43.752 +3.981 +P17405 +631 +12.182 +87.596 +22.476 +48.034 +3.069 +87.558 +22.861 +33.232 +6.77 +P20701 +1170 +10.129 +82.3 +16.239 +31.852 +2.669 +82.095 +16.892 +24.556 +1.216 +P26022 +381 +15.865 +79.175 +19.768 +54.22 +5.098 +77.667 +21.36 +42.768 +2.566 +P26715 +233 +30.415 +74.997 +23.568 +36.586 +0.839 +73.933 +24.603 +38.216 +4.465 +P29597 +1187 +5.571 +81.737 +20.108 +93.04 +4.918 +81.602 +20.193 +84.85 +6.157 +P30556 +359 +12.313 +81.815 +19.517 +95.35 +0.857 +81.98 +20.04 +96.014 +0.413 +P33076 +1130 +30.469 +68.876 +27.612 +25.838 +1.988 +69.022 +27.382 +27.406 +2.7 +P35232 +272 +2.652 +88.354 +9.295 +88.634 +0.647 +87.474 +10.562 +65.122 +2.361 +P35613 +385 +9.773 +86.321 +16.975 +41.688 +2.507 +86.056 +17.439 +41.862 +2.688 +P40189 +918 +22.039 +75.005 +25.376 +73.17 +2.298 +74.838 +25.38 +67.82 +2.657 +P47901 +424 +17.192 +73.874 +23.532 +86.392 +2.649 +73.475 +24.844 +84.59 +2.816 +P48551 +515 +36.573 +62.979 +23.882 +52.708 +2.159 +63.525 +23.313 +54.156 +2.675 +P49327 +2511 +33.768 +85.931 +12.562 +95.704 +3.157 +85.719 +12.081 +88.756 +4.571 +P49754 +854 +6.766 +87.457 +12.041 +36.328 +3.905 +87.275 +12.067 +33.306 +4.098 +P51149 +207 +7.341 +88.012 +16.049 +95.476 +2.282 +86.78 +17.15 +91.898 +4.241 + +Table 11: Prediction confidence results using the reduced database AlphaFold configuration with L = 20. This is part II of II. +Protein ID +n +RMSD +µall +σall +µdiff +σdiff +µ′ +all +σ′ +all +µ′ +diff +σ′ +diff +P52292 +529 +9.065 +87.35 +19.082 +64.18 +15.207 +87.229 +19.347 +49.878 +11.916 +P52948 +1817 +56.34 +55.901 +24.723 +34.444 +2.948 +55.807 +24.315 +35.338 +1.097 +P56962 +302 +25.013 +69.898 +22.843 +92.422 +1.212 +69.777 +22.207 +77.418 +5.058 +P59594 +1255 +4.432 +80.532 +15.908 +91.996 +0.605 +80.352 +16.139 +90.166 +0.838 +P59595 +422 +26.286 +68.249 +23.318 +34.632 +2.009 +67.744 +23.489 +32.506 +1.776 +P59596 +221 +9.173 +82.821 +13.832 +78.104 +1.736 +81.492 +14.351 +73.706 +2.18 +P59632 +274 +18.052 +48.927 +18.922 +42.85 +3.674 +47.395 +18.279 +42.27 +4.011 +P59633 +154 +20.183 +44.815 +10.281 +29.56 +1.883 +44.543 +8.393 +53.31 +1.047 +P59634 +63 +2.5 +81.663 +14.713 +93.518 +1.588 +81.059 +14.849 +94.688 +0.757 +P59635 +122 +5.955 +76.497 +14.537 +68.356 +1.203 +75.91 +15.363 +65.254 +1.946 +P59636 +98 +3.095 +81.38 +15.126 +71.648 +10.347 +81.841 +14.977 +76.852 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+19.538 +84.994 +1.66 +Q96PD4 +163 +14.017 +87.556 +15.56 +80.29 +2.432 +85.833 +18.033 +49.242 +4.759 +Q99623 +299 +8.752 +84.62 +11.565 +84.124 +2.261 +85.089 +12.409 +83.036 +4.335 +Q99836 +296 +6.525 +81.472 +14.828 +69.176 +10.703 +81.182 +14.261 +68.55 +9.133 +Q9BV40 +100 +3.427 +89.359 +13.065 +93.94 +0.823 +88.458 +12.435 +92.318 +0.979 +Q9BYF1 +480 +5.509 +92.351 +10.146 +51.158 +1.636 +92.898 +9.614 +52.218 +3.92 +Q9BYX4 +1025 +4.818 +79.75 +21.358 +81.918 +2.421 +79.805 +21.328 +72.982 +3.576 +Q9C000 +1473 +35.861 +69.449 +25.846 +86.358 +4.214 +69.717 +25.962 +72.072 +4.221 +Q9H074 +479 +30.3 +72.33 +24.787 +57.078 +4.864 +71.602 +26.038 +42.676 +2.225 +Q9NR97 +1041 +21.869 +87.848 +14.307 +54.966 +4.4 +87.815 +14.379 +56.082 +2.013 +Q9NRS4 +437 +6.233 +87.694 +13.102 +90.044 +2.06 +87.436 +13.579 +89.578 +1.338 +Q9NVJ2 +186 +1.945 +91.479 +11.254 +80.562 +3.887 +91.257 +11.377 +67.22 +5.577 +Q9NYK1 +1049 +12.803 +87.369 +15.844 +59.046 +4.871 +87.454 +15.725 +48.7 +3.93 +Q9UHD2 +729 +5.063 +89.556 +13.939 +97.104 +1.418 +89.359 +14.059 +95.536 +1.129 +Q9ULC8 +765 +34.128 +57.602 +26.325 +97.412 +0.375 +57.685 +26.027 +96.774 +0.453 +Q9Y2I7 +2098 +34.987 +64.043 +28.71 +28.668 +1.65 +64.029 +29.133 +21.178 +0.918 +Q9Y397 +364 +11.551 +84.236 +21.308 +55.18 +1.752 +84.093 +21.323 +50.83 +4.146 + +Table 12: Prediction confidence results using the full database AlphaFold configuration. This is part I of II. +Seq. ID +n +RMSD +µall +σall +µdiff +σdiff +µ′ +all +σ′ +all +µ′ +diff +σ′ +diff +A0A663 +38 +2.886 +75.736 +10.537 +88.266 +1.015 +70.263 +11.001 +75.346 +1.262 +MPROTEI +193 +5.562 +87.9 +7.153 +92.318 +1.308 +87.857 +6.955 +88.774 +1.525 +NSP2NNN +546 +22.333 +83.737 +12.059 +94.264 +1.16 +83.833 +11.752 +91.19 +0.457 +NSP4NNN +494 +16.995 +81.696 +9.823 +89.742 +1.546 +81.481 +9.997 +88.32 +1.986 +NSP6NNN +290 +5.41 +78.09 +12.781 +79.708 +2.607 +77.337 +11.29 +81.604 +1.085 +O00327 +626 +18.69 +65.332 +25.637 +34.454 +4.124 +65.298 +26.183 +29.684 +3.145 +O14745 +358 +7.395 +71.834 +17.666 +87.546 +0.83 +71.453 +17.52 +71.164 +6.795 +O14786 +923 +36.546 +79.032 +21.688 +36.114 +1.665 +79.167 +21.792 +37.798 +2.439 +O15393 +492 +19.978 +80.703 +21.253 +35.588 +3.047 +80.475 +21.314 +38.482 +2.24 +O15455 +904 +20.539 +89.822 +14.659 +93.75 +0.576 +89.897 +14.681 +88.368 +1.734 +O43765 +313 +14.438 +80.221 +19.634 +94.786 +1.027 +80.554 +19.423 +93.71 +1.392 +O94826 +608 +8.407 +81.232 +25.994 +24.44 +1.076 +81.309 +25.374 +24.338 +1.394 +O95721 +258 +11.099 +76.044 +20.028 +94.472 +1.042 +75.623 +20.744 +89.75 +0.788 +O95992 +272 +5.988 +91.775 +13.799 +96.072 +0.75 +91.048 +13.976 +90.196 +1.92 +P00973 +400 +14.825 +83.661 +22.121 +87.986 +2.211 +83.888 +22.197 +88.732 +1.671 +P01185 +164 +7.641 +79.565 +16.968 +54.8 +2.856 +79.506 +16.961 +55.752 +3.383 +P01889 +362 +9.441 +87.63 +18.772 +44.886 +1.572 +87.651 +18.719 +46.628 +2.721 +P02649 +317 +22.675 +74.197 +18.551 +44.938 +1.656 +74.255 +17.582 +44.798 +2.96 +P04233 +296 +18.279 +68.498 +20.332 +90.584 +2.164 +68.742 +20.641 +89.472 +1.721 +P04439 +365 +6.162 +86.845 +18.995 +44.23 +2.704 +86.921 +19.068 +44.678 +3.968 +P05109 +93 +1.19 +94.633 +10.47 +96.582 +2.721 +94.62 +10.248 +90.894 +6.768 +P05161 +165 +2.569 +85.182 +10.732 +87.384 +3.029 +85.738 +10.568 +90.89 +0.545 +P05231 +212 +9.402 +85.294 +16.248 +57.324 +4.406 +84.919 +17.756 +55.27 +3.901 +P07711 +333 +7.479 +93.624 +11.188 +96.6 +0.67 +93.941 +9.784 +92.306 +2.45 +P08887 +468 +10.171 +79.042 +23.058 +50.664 +1.41 +79.053 +23.11 +44.186 +3.148 +P09429 +215 +20.391 +75.711 +23.003 +94.868 +0.426 +76.624 +22.933 +93.982 +0.536 +P09958 +794 +32.057 +85.332 +20.268 +55.18 +2.153 +85.024 +20.889 +36.474 +0.395 +P0DTC2 +1273 +4.917 +78.96 +16.515 +44.794 +11.075 +78.972 +16.515 +41.002 +13.017 +P0DTC3 +275 +11.391 +61.089 +21.166 +50.25 +1.085 +61.363 +20.931 +46.506 +3.1 +P0DTC4 +75 +2.496 +75.297 +15.67 +91.574 +1.843 +74.004 +15.183 +82.986 +3.486 +P0DTC5 +222 +6.436 +83.012 +14.062 +89.206 +2.336 +83.312 +13.759 +89.546 +1.351 +P0DTC6 +61 +4.88 +86.241 +12.854 +96.814 +0.676 +84.169 +13.171 +89.606 +2.674 +P0DTC7 +121 +4.492 +76.821 +15.179 +66.336 +1.601 +77.019 +15.334 +68.648 +1.564 +P0DTC8 +121 +17.601 +56.598 +10.34 +63.222 +3.508 +48.462 +12.791 +53.566 +2.139 +P0DTC9 +419 +29.977 +67.367 +23.173 +88.718 +2.222 +67.564 +23.335 +85.494 +3.082 +P0DTD2 +97 +4.156 +79.424 +16.645 +92.134 +0.784 +78.68 +16.368 +90.232 +0.666 +P0DTD3 +73 +17.276 +69.06 +15.256 +47.152 +3.291 +69.138 +15.24 +44.438 +3.966 +P0DTD8 +43 +2.394 +89.893 +10.325 +85.21 +6.189 +89.745 +10.28 +83.602 +3.102 +P0DTF1 +22 +2.033 +95.286 +6.738 +98.574 +0.134 +93.265 +8.046 +96.042 +0.735 +P0DTG0 +57 +13.398 +46.508 +6.478 +44.294 +2.44 +47.424 +5.993 +47.484 +1.588 +P0DTG1 +41 +17.037 +79.198 +8.769 +85.9 +2.624 +77.292 +10.147 +78.726 +2.596 +P11226 +248 +21.819 +80.394 +20.402 +47.02 +1.473 +80.227 +20.867 +48.33 +2.905 +P13164 +125 +8.675 +65.077 +17.669 +56.108 +2.975 +63.893 +16.589 +54.514 +2.112 +P13747 +358 +12.178 +86.76 +18.681 +49.512 +2.503 +86.547 +18.838 +57.836 +2.876 +P17181 +557 +21.61 +79.093 +20.317 +43.486 +1.523 +78.859 +20.886 +48.594 +2.501 +P17405 +631 +12.066 +87.796 +22.376 +47.99 +2.672 +87.682 +22.809 +29.628 +3.587 +P20701 +1170 +8.323 +82.167 +16.158 +32.766 +0.935 +82.255 +16.678 +24.696 +1.215 +P26022 +381 +14.411 +80.061 +19.401 +50.256 +4.399 +79.295 +19.914 +37.094 +2.276 +P26715 +233 +22.384 +74.56 +23.912 +36.474 +1.217 +74.551 +24.108 +34.286 +1.773 +P29597 +1187 +32.804 +82.201 +20.483 +95.13 +2.856 +82.191 +20.266 +84.898 +8.554 +P30556 +359 +10.111 +80.902 +19.02 +39.53 +2.981 +81.31 +18.965 +38.666 +3.949 +P33076 +1130 +26.719 +69.118 +27.69 +52.374 +5.287 +69.073 +27.682 +69.802 +1.087 +P35232 +272 +3.447 +88.919 +9.067 +88.67 +1.189 +88.485 +8.855 +87.73 +1.575 +P35613 +385 +8.896 +85.87 +17.193 +43.738 +2.005 +86.146 +17.188 +46.344 +2.166 +P40189 +918 +24.632 +74.808 +25.439 +75.236 +2.187 +74.814 +25.51 +68.942 +3.959 +P47901 +424 +5.702 +74.001 +23.301 +88.66 +2.375 +74.045 +23.157 +85.132 +2.005 +P48551 +515 +28.717 +64.011 +23.755 +70.534 +3.521 +63.891 +23.481 +61.75 +2.599 +P51149 +207 +6.188 +88.239 +16.587 +95.512 +2.591 +87.067 +17.216 +91.842 +4.365 + +Table 13: Prediction confidence results using the full database AlphaFold configuration. This is part II of II. +Seq. ID +n +RMSD +µall +σall +µdiff +σdiff +µ′ +all +σ′ +all +µ′ +diff +σ′ +diff +P56962 +302 +22.301 +69.172 +23.753 +96.516 +0.409 +69.342 +23.759 +96.54 +0.463 +P59596 +221 +7.506 +82.061 +12.733 +76.868 +1.775 +81.175 +13.742 +72.064 +2.043 +P59632 +274 +13.018 +58.367 +18.783 +66.136 +2.029 +57.364 +18.794 +60.926 +4.362 +P59633 +154 +20.993 +47.859 +10.608 +38.736 +4.455 +46.228 +8.664 +48.764 +0.907 +P59634 +63 +2.514 +81.124 +14.464 +91.134 +2.227 +79.701 +14.365 +84.632 +4.937 +P59635 +122 +6.527 +76.232 +15.74 +69.514 +0.897 +76.129 +16.043 +54.806 +3.045 +P59636 +98 +5.647 +80.765 +16.225 +91.182 +5.274 +77.654 +15.556 +79.584 +6.025 +P59637 +76 +3.488 +75.038 +15.62 +66.71 +6.263 +75.677 +15.389 +65.872 +5.204 +P62937 +165 +0.963 +98.037 +2.658 +98.78 +0.11 +97.592 +2.813 +96.842 +1.058 +P68104 +462 +1.547 +87.439 +10.419 +90.584 +1.789 +88.26 +9.399 +93.154 +1.275 +P84022 +425 +16.633 +83.759 +20.532 +45.254 +6.648 +83.581 +20.67 +41.162 +5.792 +PLRPOCT +355 +3.824 +90.168 +6.528 +92.868 +1.836 +89.47 +7.723 +84.62 +3.832 +Q01628 +133 +26.947 +59.485 +12.738 +55.092 +3.368 +60.204 +15.275 +49.082 +4.223 +Q01629 +132 +18.508 +62.633 +17.46 +77.596 +2.819 +62.329 +17.48 +70.92 +1.877 +Q10589 +180 +11.884 +84.566 +17.867 +93.704 +0.746 +83.726 +18.104 +90.02 +1.893 +Q13241 +179 +11.456 +87.467 +17.008 +93.192 +0.895 +87.165 +16.87 +88.192 +1.946 +Q13568 +498 +29.991 +72.656 +23.921 +92.118 +0.769 +72.434 +23.764 +79.794 +5.125 +Q14653 +427 +16.703 +79.881 +21.726 +88.028 +4.931 +79.761 +22.05 +89.036 +3.51 +Q16236 +605 +38.828 +60.894 +24.052 +70.194 +0.603 +60.396 +24.197 +51.342 +4.096 +Q16552 +155 +13.575 +83.831 +14.869 +87.152 +2.516 +83.098 +15.725 +56.958 +4.194 +Q16553 +131 +10.901 +78.582 +14.393 +78.83 +0.746 +78.854 +14.611 +69.338 +1.745 +Q16665 +826 +41.354 +59.753 +27.088 +84.802 +2.557 +59.815 +26.839 +83.638 +2.931 +Q4KMQ2 +910 +4.59 +81.331 +15.895 +29.766 +1.54 +81.46 +15.823 +28.168 +3.134 +Q5BJD5 +291 +13.792 +82.742 +17.059 +43.9 +1.158 +82.653 +19.347 +27.804 +1.888 +Q5W0Z9 +365 +15.887 +85.224 +23.712 +98.5 +0.087 +85.247 +23.511 +97.632 +0.322 +Q7TFA0 +39 +2.505 +78.834 +15.988 +87.908 +9.073 +79.225 +15.951 +89.984 +5.317 +Q7TFA1 +44 +2.106 +85.691 +14.274 +84.248 +5.805 +85.609 +14.322 +82.498 +5.857 +Q7TLC7 +70 +24.995 +70.072 +14.57 +52.722 +2.59 +70.101 +14.933 +45.152 +2.867 +Q7Z434 +540 +50.815 +55.255 +20.386 +95.978 +0.55 +55.05 +19.831 +85.694 +3.344 +Q80H93 +84 +14.707 +63.83 +6.81 +71.14 +1.769 +52.502 +12.869 +40.918 +1.147 +Q86U44 +580 +17.742 +74.735 +24.442 +35.918 +1.898 +74.552 +24.301 +31.652 +0.349 +Q86WV6 +379 +20.858 +83.949 +16.754 +96.908 +0.31 +83.971 +16.467 +89.178 +2.384 +Q8IUC6 +712 +49.827 +62.905 +24.297 +93.142 +0.412 +62.708 +23.747 +84.022 +1.321 +Q8N3R9 +675 +35.139 +78.299 +22.289 +55.756 +7.659 +78.394 +22.344 +56.592 +5.767 +Q8N884 +522 +29.73 +77.606 +25.97 +38.534 +3.736 +77.404 +26.054 +36.74 +1.55 +Q8NAC3 +791 +11.202 +74.645 +21.198 +41.694 +2.118 +74.661 +21.317 +35.41 +3.997 +Q8NHX9 +752 +6.888 +80.556 +17.278 +91.308 +0.553 +80.289 +17.483 +87.504 +0.721 +Q92499 +740 +12.723 +86.385 +14.576 +89.098 +1.926 +86.362 +14.876 +88.688 +1.804 +Q96F46 +866 +36.383 +68.955 +26.798 +49.514 +2.823 +68.911 +26.894 +36.258 +3.614 +Q96JC1 +886 +3.565 +88.113 +8.225 +93.376 +0.356 +88.239 +8.657 +91.282 +0.696 +Q96P20 +1036 +14.659 +80.375 +19.634 +23.954 +2.496 +80.21 +19.724 +24.76 +1.336 +Q96PD4 +163 +8.703 +87.758 +14.681 +82.896 +1.929 +86.532 +16.772 +56.214 +6.728 +Q99836 +296 +8.761 +81.213 +13.817 +78.914 +7.454 +80.918 +13.971 +72.198 +6.253 +Q9BV40 +100 +3.375 +89.26 +11.772 +96.458 +0.289 +84.495 +13.064 +85.25 +2.311 +Q9BYF1 +480 +5.231 +92.48 +10.124 +48.952 +1.565 +92.981 +9.549 +53.718 +4.395 +Q9H074 +479 +36.353 +72.107 +25.823 +56.444 +4.022 +71.939 +25.903 +41.594 +2.434 +Q9NR97 +1041 +21.572 +87.692 +14.66 +60.438 +3.784 +86.732 +15.638 +35.11 +2.076 +Q9NRS4 +437 +6.429 +87.475 +13.523 +57.542 +2.717 +87.623 +13.176 +59.396 +2.244 +Q9NVJ2 +186 +1.744 +92.241 +10.762 +78.798 +6.333 +91.723 +11.576 +55.84 +2.358 +Q9ULC8 +765 +38.808 +57.841 +25.532 +97.696 +0.34 +57.592 +25.77 +97.14 +0.448 +Q9Y2I7 +2098 +33.211 +64.167 +27.302 +27.08 +0.878 +64.111 +27.934 +23.858 +0.46 +Q9Y397 +364 +9.301 +83.666 +21.359 +55.914 +2.097 +83.822 +21.283 +52.324 +3.18 + +Table 14: GDT-TS results based on AlphaFold predefined confidence regions using the full database AlphaFold configuration. Regions R1 to +R4 correspond to confidence scores of above 90%, 70% to 90%, 50% to 70%, and below 50%, respectively. This is part I of II. +Seq. ID +n +R1 GDT (%) +R1 (%) +R2 GDT (%) +R2 (%) +R3 GDT (%) +R3 (%) +R4 GDT (%) +R4 (%) +A0A663 +38 +N/A +0.0 +89.815 +71.053 +65.909 +28.947 +N/A +0.0 +MPROTEI +193 +38.918 +50.259 +36.648 +45.596 +31.25 +4.145 +N/A +0.0 +NSP2NNN +546 +8.447 +40.11 +9.449 +46.52 +3.448 +10.623 +8.333 +2.747 +NSP4NNN +494 +5.67 +19.636 +9.91 +67.409 +2.917 +12.146 +6.25 +0.81 +NSP6NNN +290 +85.227 +7.586 +76.279 +74.138 +42.143 +12.069 +11.111 +6.207 +O00327 +626 +21.014 +33.067 +8.43 +13.738 +7.188 +12.78 +2.767 +40.415 +O14745 +358 +32.927 +11.453 +26.117 +50.0 +22.619 +23.464 +8.333 +15.084 +O14786 +923 +1.244 +43.554 +0.949 +34.236 +0.0 +4.659 +1.235 +17.551 +O15393 +492 +11.643 +56.301 +7.692 +21.138 +16.176 +6.911 +0.974 +15.65 +O15455 +904 +10.731 +72.677 +0.872 +19.027 +2.083 +3.982 +8.333 +4.314 +O43765 +313 +17.296 +50.799 +11.194 +21.406 +12.5 +14.696 +7.317 +13.099 +O94826 +608 +57.007 +69.243 +67.727 +9.046 +40.385 +2.138 +11.555 +19.572 +O95721 +258 +44.196 +43.411 +34.694 +18.992 +22.273 +21.318 +10.714 +16.279 +O95992 +272 +58.798 +85.662 +41.25 +7.353 +25.0 +1.471 +10.0 +5.515 +P00973 +400 +27.099 +65.5 +28.929 +17.5 +23.438 +4.0 +2.404 +13.0 +P01185 +164 +35.959 +44.512 +27.703 +22.561 +32.292 +29.268 +20.833 +3.659 +P01889 +362 +69.069 +75.691 +24.074 +7.459 +14.286 +5.801 +25.0 +11.05 +P02649 +317 +3.883 +32.492 +2.151 +29.338 +1.761 +22.397 +0.5 +15.773 +P04233 +296 +25.725 +23.311 +26.765 +28.716 +27.869 +20.608 +8.642 +27.365 +P04439 +365 +58.086 +73.699 +36.719 +8.767 +16.304 +6.301 +18.902 +11.233 +P05109 +93 +97.126 +93.548 +100.0 +1.075 +91.667 +3.226 +62.5 +2.151 +P05161 +165 +90.678 +35.758 +79.62 +55.758 +41.667 +5.455 +50.0 +3.03 +P05231 +212 +37.316 +64.151 +25.781 +15.094 +5.263 +17.925 +4.167 +2.83 +P07711 +333 +47.518 +84.685 +38.71 +9.309 +5.769 +3.904 +0.0 +2.102 +P08887 +468 +47.94 +57.051 +41.667 +12.179 +17.188 +6.838 +11.607 +23.932 +P09429 +215 +12.371 +45.116 +7.273 +25.581 +1.667 +6.977 +0.521 +22.326 +P09958 +794 +6.733 +66.877 +2.009 +14.106 +0.0 +6.549 +0.505 +12.469 +P0DTC2 +1273 +93.421 +28.358 +86.278 +49.804 +69.966 +11.705 +34.109 +10.134 +P0DTC3 +275 +50.0 +1.091 +40.551 +46.182 +25.568 +16.0 +15.099 +36.727 +P0DTC4 +75 +63.095 +28.0 +85.87 +30.667 +73.333 +40.0 +25.0 +1.333 +P0DTC5 +222 +72.917 +37.838 +69.393 +48.198 +53.125 +7.207 +43.333 +6.757 +P0DTC6 +61 +55.0 +57.377 +60.0 +24.59 +59.091 +18.033 +N/A +0.0 +P0DTC7 +121 +78.571 +17.355 +63.492 +52.066 +47.917 +19.835 +44.231 +10.744 +P0DTC8 +121 +N/A +0.0 +37.5 +1.653 +13.202 +73.554 +10.833 +24.793 +P0DTC9 +419 +4.762 +15.036 +7.902 +41.527 +4.082 +11.695 +1.88 +31.742 +P0DTD2 +97 +76.163 +44.33 +69.643 +28.866 +56.667 +15.464 +29.545 +11.34 +P0DTD3 +73 +0.0 +12.329 +4.63 +36.986 +2.679 +38.356 +13.889 +12.329 +P0DTD8 +43 +93.333 +69.767 +85.0 +23.256 +75.0 +6.977 +N/A +0.0 +P0DTF1 +22 +96.053 +86.364 +66.667 +13.636 +N/A +0.0 +N/A +0.0 +P0DTG0 +57 +N/A +0.0 +N/A +0.0 +28.333 +26.316 +15.476 +73.684 +P0DTG1 +41 +N/A +0.0 +8.088 +82.927 +3.571 +17.073 +N/A +0.0 +P11226 +248 +15.741 +54.435 +1.515 +13.306 +0.0 +15.726 +0.61 +16.532 +P13164 +125 +36.765 +13.6 +43.966 +23.2 +33.5 +40.0 +6.897 +23.2 +P13747 +358 +34.457 +74.581 +17.857 +5.866 +8.594 +8.939 +3.289 +10.615 +P17181 +557 +25.722 +49.731 +11.822 +23.16 +8.696 +8.259 +0.476 +18.851 +P17405 +631 +72.793 +82.567 +50.0 +0.634 +33.333 +2.377 +9.066 +14.422 +P20701 +1170 +50.441 +38.803 +52.667 +44.872 +41.912 +8.718 +9.831 +7.607 +P26022 +381 +34.637 +46.982 +10.476 +27.559 +0.521 +12.598 +3.571 +12.861 +P26715 +233 +22.458 +50.644 +3.261 +9.871 +2.0 +10.73 +1.119 +28.755 +P29597 +1187 +4.882 +53.496 +2.594 +29.233 +1.364 +4.634 +4.0 +12.637 +P30556 +359 +30.307 +49.861 +23.969 +27.019 +15.278 +10.028 +4.255 +13.092 +P33076 +1130 +23.591 +36.106 +13.542 +25.487 +9.551 +7.876 +3.043 +30.531 +P35232 +272 +69.591 +62.868 +58.523 +32.353 +8.333 +3.309 +18.75 +1.471 +P35613 +385 +32.722 +67.273 +28.175 +16.364 +7.292 +6.234 +1.282 +10.13 +P40189 +918 +44.341 +49.564 +22.619 +18.301 +12.162 +4.031 +4.651 +28.105 +P47901 +424 +61.08 +41.509 +61.48 +23.113 +56.618 +8.019 +21.121 +27.358 +P48551 +515 +17.383 +24.854 +17.391 +17.864 +6.553 +20.0 +1.302 +37.282 +P51149 +207 +42.007 +71.014 +38.889 +13.043 +27.381 +10.145 +2.083 +5.797 + +Table 15: GDT-TS results based on AlphaFold predefined confidence regions using the full database AlphaFold configuration. Regions R1 to +R4 correspond to confidence scores of above 90%, 70% to 90%, 50% to 70%, and below 50%, respectively. This is part II of II. +Seq. ID +n +R1 GDT (%) +R1 (%) +R2 GDT (%) +R2 (%) +R3 GDT (%) +R3 (%) +R4 GDT (%) +R4 (%) +P56962 +302 +28.958 +39.735 +5.952 +6.954 +4.583 +19.868 +2.97 +33.444 +P59596 +221 +37.5 +26.244 +36.94 +60.633 +20.0 +6.787 +5.357 +6.335 +P59632 +274 +N/A +0.0 +34.184 +35.766 +35.87 +25.182 +14.72 +39.051 +P59633 +154 +N/A +0.0 +N/A +0.0 +11.486 +48.052 +4.688 +51.948 +P59634 +63 +91.071 +44.444 +76.471 +26.984 +84.722 +28.571 +N/A +0.0 +P59635 +122 +54.762 +17.213 +39.286 +51.639 +29.167 +19.672 +19.643 +11.475 +P59636 +98 +61.735 +50.0 +46.875 +24.49 +22.368 +19.388 +16.667 +6.122 +P59637 +76 +88.889 +23.684 +61.607 +36.842 +53.571 +36.842 +25.0 +2.632 +P62937 +165 +100.0 +98.788 +100.0 +0.606 +100.0 +0.606 +N/A +0.0 +P68104 +462 +98.235 +55.195 +90.123 +35.065 +80.682 +9.524 +50.0 +0.216 +P84022 +425 +40.67 +64.941 +27.692 +15.294 +6.25 +4.706 +2.734 +15.059 +PLRPOCT +355 +79.418 +65.352 +67.308 +32.958 +100.0 +1.69 +N/A +0.0 +Q01628 +133 +N/A +0.0 +12.069 +21.805 +3.788 +49.624 +1.974 +28.571 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Rickard Ewetz1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Arvind Ramanathan5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Susmit Jha6 1 Electrical & Computer Engineering Department,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' University of Central Florida,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Orlando,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' FL 32816 2 Computer Science Department,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' University of Texas at San Antonio,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' TX 78249 3 Information Directorate,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Air Force Research Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Rome,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' NY 13441 4 Defense Advanced Research Projects Agency,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Arlington,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' VA 22203 5 Data Science and Learning,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Argonne National Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Lemont,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' IL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' 60439 6 Computer Science Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' SRI International,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Menlo Park,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' CA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' 94709 Abstract Protein folding neural networks (PFNNs) such as AlphaFold predict remarkably accurate structures of proteins compared to other approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' However, the robustness of such networks has heretofore not been explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' This is particularly relevant given the broad social implications of such technologies and the fact that biologically small perturbations in the protein sequence do not generally lead to drastic changes in the pro- tein structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' In this paper, we demonstrate that AlphaFold does not exhibit such robustness despite its high accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' This raises the challenge of detecting and quantifying the ex- tent to which these predicted protein structures can be trusted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' To measure the robustness of the predicted structures, we utilize (i) the root-mean-square deviation (RMSD) and (ii) the Global Distance Test (GDT) similarity measure between the predicted structure of the original sequence and the struc- ture of its adversarially perturbed version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' We prove that the problem of minimally perturbing protein sequences to fool protein folding neural networks is NP-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Based on the well-established BLOSUM62 sequence alignment scoring matrix, we generate adversarial protein sequences and show that the RMSD between the predicted protein structure and the structure of the original sequence are very large when the adversarial changes are bounded by (i) 20 units in the BLO- SUM62 distance, and (ii) five residues (out of hundreds or thousands of residues) in the given protein sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' In our experimental evaluation, we consider 111 COVID-19 proteins in the Universal Protein resource (UniProt), a central resource for protein data managed by the European Bioinformatics Institute, Swiss Institute of Bioinformatics, and the US Pro- tein Information Resource.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' These result in an overall GDT similarity test score average of around 34%, demonstrating a substantial drop in the performance of AlphaFold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Introduction Proteins form the building blocks of life as they enable a vari- ety of vital functions essential to life and reproduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Natu- rally occurring proteins are bio-polymers typically composed of 20 amino acids and this primary sequence of amino acids is well known for many proteins, thanks to high-throughput sequencing techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' However, in order to understand the functions of different protein molecules and complexes, it is essential to comprehend their three-dimensional (3D) struc- tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Until recently, one of the grand challenges in structural biology has been the accurate determination of the 3D struc- ture of the protein from its primary sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Such accurate ID: O43765 𝑛 = 313 RMSD = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='051Å Figure 1: The structure of the original (black) and adversarial (red) sequences predicted using AlphaFold for the Small glutamine- rich tetratricopeptide repeat-containing protein alpha sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' The length of the protein sequence is denoted by n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' For structures, after their alignment using PyMol (Schrödinger and DeLano), the Root Mean Square Deviation (RMSD) is given in Angstroms (equal to 10−10 meters and denoted by Å).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' predictive protein folding promises to have a profound impact on the design of therapeutics for diseases and drug discovery (Chan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' AlphaFold (Jumper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' 2021a) achieved unparalleled suc- cess in predicting protein structures using neural networks and remains first at the Critical Assessment of protein Struc- ture Prediction (CASP14), which corresponds to year 2020, competition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' While this has been touted as a breakthrough for structural biology (Bagdonas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' 2021), the robustness of its predictions has not yet been explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' The main con- tribution of this paper is to demonstrate the susceptibility of AlphaFold to adversarial sequences by generating sev- eral examples where protein sequences that vary only in five residues out of hundreds or thousands of residues re- sult in very different 3D protein structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' We present the problem of adversarial attacks on Protein Folding Neural Net- work (PFNN) and prove that the problem is NP-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' We use sequence alignment scores (Henikoff and Henikoff 1992) such as those derived from Block Substitution Matrices (BLOSUM62) to identify a space of similar protein sequences used in constructing adversarial perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' For the output arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='04093v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='LG] 10 Jan 2023 structures, we leverage the standard metrics commonly used in CASP, namely (i) the root-mean-square deviation (RMSD) and (ii) the Global Distance Test (GDT) similarity measure between the predicted structure and the structure of its adver- sarially perturbed sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' See Figure 1 and its caption for an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Moreover, we conduct two experiments investigating the choice of the BLOSUM threshold and the use of the pre- diction, per-residue, confidence information obtained from AlphaFold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Our experiments show that different input pro- tein sequences have very different adversarial robustness as determined by the RMSD (GDT-TS) in the protein structure predicted by AlphaFold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' These values range from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='011Å (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='43%) to 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='531Å (98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='8%) when the BLOSUM62 distance between the original and adversarial sequences is bounded by a threshold of 20 units with a hamming distance of 5 residues only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Hence, our proposed approach is a first step in the direction of identifying protein sequences on which the predicted 3D structure cannot be trusted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Summary and Related work PFNNs (Jumper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' 2021b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Baek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' 2021) should be expected to obey the natural observation that biologically small changes in the sequence of a protein usually do not lead to drastic changes in the protein structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Almost four decades ago, it was noted that two structures with 50% se- quence identity align within approximately 1Å RMSD from each other (Chothia and Lesk 1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Two proteins with even 40% sequence identity and at least 35 aligned residues align within approximately 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='5Å (Sander and Schneider 1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' The phenomenon of sequence-similar proteins producing similar structures have also been observed in larger studies (Rost 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' As with almost any rule in biology, a small number of counterexamples to the conventional wisdom of similar sequences leading to similar structures do exist, wherein even small perturbations can potentially alter the entire fold of a protein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' However, such exceptions are not frequent and often lead to exciting investigations (Cordes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Tuinstra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Manipulating the multiple sequence alignment step of Al- phaFold has been studied in (Stein and Mchaourab 2021) using in silico mutagenesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' However, there, the goal is not to study the robustness of the protein folding neural net- works, but rather to enhance the prediction capability of AlphaFold in terms of the intrinsic conformational hetero- geneity of proteins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' The authors in (Del Alamo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' 2022), present a method that manipulates inputs to obtain diverse distinct structures that are absent from the AlphaFold training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Using membrane proteins, the authors show that their method enhances the multiple sequence alignment step while generating more accurate structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' The work in (Jha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' 2021) is aimed at generating adver- sarial sequences in order to cause significant damage to the output predicted structure of RosettaFold (Baek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' 2021), which, according to CASP, is the second best protein folding neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' However, the authors only show results for a few proteins and do not consider all the standard metrics for measuring the output structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' In contrast, in this paper, we present results for more than 100 sequences, derive a com- plexity proof for the problem of finding adversarial protein sequences, and, based on the CASP competition, utilize all the standard metrics for measuring the output structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Robustness Metric using Adversarial Attacks The similar-sequence implies similar-structure paradigm dic- tates that PFNNs should make robust predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Given a protein sequence of n residues S = s1s2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' sn with a three- dimensional structure A(S) = (x1, y1, z1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' , (xn, yn, zn), we define a notion of biologically similar sequences V us- ing Block Substitution Matrices (BLOSUM) (Henikoff and Henikoff 1992), and then employ formulations of adversar- ial attacks (Goodfellow, McDaniel, and Papernot 2018) on PFNNs within this space of similar sequences to identify a sequence Sadv ∈ V that produces a maximally different three-dimensional structure A(Sadv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' We then compute the RMSD and GDT between the structures for the original and adversarial inputs (A(S) and A(Sadv)), and use these met- rics as the robustness measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' If the RMSD (GDT) is small (high), the response of the PFNN is deemed robust;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' a large (small) RMSD (GDT) indicates that the predicted structure is not robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' BLOSUM Similarity Measures Given two sequences of n residues S = s1s2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' sn and S′ = s′ 1s′ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' s′ n, in which every residue si (or s′ i) is from the set X = {A, R, N, D, C, Q, E, G, H, I, L, K, M, F, P, S, T, W, Y, V } of amino acids, a natural question is how to compute the sequence similarity Dseq between these proteins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' A naive approach would be to count the number of residues that are different, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=', the Hamming distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' However, an analysis of naturally occurring proteins shows that not all changes in residues have the same impact on protein struc- tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Changes to one type of residue are more likely to cause structural variations than changes to another type of residue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Early work in bioinformatics focused on properties of amino acids and reliance on genetic codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' However, more modern methods have relied on the creation of amino acid scoring matrices that are derived from empirical observations of frequencies of amino acid replacements in homologous sequences (Dayhoff, Schwartz, and Orcutt 1978;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Jones, Tay- lor, and Thornton 1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' The original scoring matrix, called the PAM250 matrix, was based on empirical analysis of 1572 mutations observed in 71 families of closely-related proteins that are 85% or more identical after they have been aligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' The PAM1 model-based scoring matrix was obtained by nor- malizing the frequency of mutations to achieve a 99% identity between homologous proteins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' These results were then extrap- olated to create the PAM10, PAM30, PAM70 and PAM120 matrices with 90%, 75%, 55%, and 37% identity between homologous proteins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Another interesting approach (Henikoff and Henikoff 1992) to understanding protein similarity is the direct count- ing of replacement frequencies using the so-called Block Sub- stitution Matrices (BLOSUM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Instead of relying solely on sequences of homologous proteins that are relatively harder to find, the BLOSUM approach focuses on identifying con- served blocks or conserved sub-sequences in a larger variety of proteins potentially unrelated by evolutionary pathways and counts the frequency of replacements within these con- served sub-sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' BLOSUM62 (Figure 2), BLOSUM80 and BLOSUM90 denote block substitution matrices that are obtained from blocks or subsequences with at least 62%, 80%, and 90% similarity, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' The BLOSUM matrix [Bij] is a matrix of integers where each entry denotes the similarity between residue of type bi ∈ X and type bj ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' We identify the space of biologically similar sequences V for a given protein sequence S with respect to the BLO- SUM distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' We expect the predicted structures for the similar sequences to be similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' If there is a large RMSD (or small GDT) between the predicted structure A(S) and the structure of the adversarial sequence A(Sadv), it would re- flect a lack of robustness in the prediction of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' We adopt a sequence similarity measure that counts replacement frequencies in conserved blocks across different proteins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Approach Our approach to evaluating the robustness of PFNNs is based on two main ideas: (i) the existence of adversarial exam- ples in PFNNs that produce adversarial structures possibly very different from the original structure, and (ii) the use of BLOSUM matrices for identifying a neighborhood of a given sequence that are biologically similar and hence expected to have similar 3D structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' We utilize the RMSD and GDT between the structure of an original protein sequence and the structure of the adversarial sequence as a measure of robust- ness of a protein folding network on the given input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' In this work, we focus on the state-of-the-art AlphaFold model, the winner of the 1st place in CASP2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Sequence Similarity Measures Given two sequences S = s1s2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' sn and S′ = s′ 1s′ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' s′ n, the BLOSUM distance between the two sequences is given by Equation (1) below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' For an illustrative example of Dseq, see Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Dseq(S, S′) = � i∈[n] � Bsisi − Bsis′ i � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='R ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='1 -2 -2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='3 -1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='Figure 2: The BLOSUM62 matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Output Structural Measure Given a sequence of n residues S = s1s2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' sn, its three dimensional structure A(S) is an ordered n-tuple of three- dimensional co-ordinates (x1, y1, z1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' (xn, yn, zn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Our goal is to utilize a structural distance measure that captures the variations in the two structures A(S) and A(S′) and is invariant to rigid-body motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Therefore, in this work, we use standard structural distances, namely the RMSD, mea- sured in Å, and the GDT with its two variants: (i) the Total Score (TS) and (ii) the High Accuracy (HA) (Zemla 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Given the output structure of the adversarial sequence A(S′), an alignment algorithm is employed before comput- ing the RMSD and GDT measures between the two structures of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' We use the alignment procedure implemented in PyMOL (Schrödinger and DeLano) to align A(S′) with re- gard to the target structure A(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Let the aligned structure be denoted by ˆ A(S′) = (ˆx′ 1, ˆy′ 1, ˆz′ 1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' , (ˆx′ n, ˆy′ n, ˆz′ n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Then, the RMSD, measured in Å, is obtained as RMSD(A(S), ˆ A(S′)) = � � � � 1 n � i∈[n] d(A(S)i, ˆ A(S′)i) , (2) where d(A(S)i, ˆ A(S′)i) = (xi−ˆx′ i)2+(yi−ˆy′ i)2+(zi−ˆz′ i)2 and A(S)i represents the 3D carbon-alpha coordinates of the ith residue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Using the carbon-alpha coordinates is the standard approach in CASP (Zemla 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Another standard metric for gauging the similarity of pro- tein structures is the GDT similarity measure, introduced by (Zemla 2003) and commonly used in the CASP competition along with the RMSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' In some cases, the latter is known to be sensitive to outliers (Zemla 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' The GDT score returns a value in [0, 1] where 1 indicates identical structures, and is computed with respect to four thresholds, δj, as GDT(A(S), ˆ A(S′)) = 1 4n � j∈[4] � i∈[n] 1 � d(A(S)i, ˆ A(S′)i) < δj � , (3) where the thresholds δ1, δ2, δ3, and δ4 for TS (HA) are given by 1(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='5), 2(1), 4(2), and 8(4) for j equals to 1, 2, 3, and 4 respectively, and 1(·) is the indicator function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' In (3), each j ∈ [4] reflects the number of residues in the structures for which the distance is less than δj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Adversarial Attacks on PFNNs Small carefully crafted changes in a few pixels of input images cause well-trained neural networks with otherwise high accuracy to consistently produce incorrect responses in domains such as computer vision (Croce et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' An- driushchenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Bai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Croce and Hein 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Given a neural network A mapping a sequence S of residues to a three-dimensional geometry A(S) describing the structure of the protein, we seek to obtain a sequence S′ such that the sequence similarity measure Dseq(S, S′) be- tween S and S′ is small and some structural distance measure Dstr(A(S), A(S′)) is maximized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' This can be achieved by ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='SFP1_1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='MDLFMRFFTLGSITAQPVKIDNASPASTVHATATIPLQASLPFGWLVIGVAFLAVFQSATKIIALNKRWQLALYKGFQFICNLLLLFVTIYSHLLLVAAG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='𝑆 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='𝑆1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='𝑆2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='𝑆3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='𝑆4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='𝑆5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='𝑆6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='𝑆7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='𝑆8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='𝑆9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='𝑆10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='DVPSMRFFTLGSITAQPVKIDNASPASTVHATATIPLQASLPFGWLVIGVAFLAVFQSATKIIALNKRWQLALYKGFQFICNLLLLFVTIYSHLLLVAAG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='FGCYMRFFTLGSITAQPVKIDNASPASTVHATATIPLQASLPFGWLVIGVAFLAVFQSATKIIALNKRWQLALYKGFQFICNLLLLFVTIYSHLLLVAAG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='MDLFMRFFTLGSITAQPIRVPNASPASTVHATATIPLQASLPFGWLVIGVAFLAVFQSATKIIALNKRWQLALYKGFQFICNLLLLFVTIYSHLLLVAAG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='MDLFMRFFTLGSITAQPVKIDNASPASTVHATATIPLQASLWAYKLVIGVAFLAVFQSATKIIALNKRWQLALYKGFQFICNLLLLFVTIYSHLLLVAAG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='MDLFMRFFVIAAVTAQPVKIDNASPASTVHATATIPLQASLPFGWLVIGVAFLAVFQSATKIIALNKRWQLALYKGFQFICNLLLLFVTIYSHLLLVAAG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='MDLFMRFFTLGSITAQPVKIDNASPASTVHATATIPLQASDIERRLVIGVAFLAVFQSATKIIALNKRWQLALYKGFQFICNLLLLFVTIYSHLLLVAAG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='MDLFMRFFTLGSITAQPVKIDNASPASTVHATATIPLQASLPFGWLVIGVAFLAVFQSATRVLTMKKRWQLALYKGFQFICNLLLLFVTIYSHLLLVAAG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='MDLFMRFFTLGSITAQPVKIDNASPASTVHATATIPLQAVEFQLVLVIGVAFLAVFQSATKIIALNKRWQLALYKGFQFICNLLLLFVTIYSHLLLVAAG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='MDLFMRFFTLGSITAQPVKIDNASPASTVHATATIPLQASLPFGWLVIGVAFLAVFQSATKIIALNKRWQLALYKGFQFICNLACMYISMYSHLLLVAAG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='MDLFMRFFTLGSITAQPVKIDNASPASTVHATATIPLQASDIGIINGIGVAFLAVFQSATKIIALNKRWQLALYKGFQFICNLLLLFVTIYSHLLLVAAG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='𝐷seq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='32 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='42 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='49 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='57 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='23 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='65 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='𝐷ham ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='Original and Adversarial Sequences ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='Figure 3: The original sequence S is followed by 10 sequences generated by changing 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' and 7 residues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' The sequences are samples from the space in (5) with different values of L and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' The distance Dseq is calculated using (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' solving the following optimization problem max S′ Dstr (A(S), A(S′)) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Dseq(S, S′) ≤ L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' (4) In our experiments, we set L = 20 and Dstr as the RMSD measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Given the discrete nature of the input sequences, well-known methods for generating adversarial examples (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' gradient-based methods) fail to produce valid and accurate results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' As such, we propose a solution based on a brute-force exploration in the space of biologically similar sequences that, given a sequence of interest S with n residues, can be defined as VL,H(S) = {S′ ∈ X n | Dseq(S, S′) ≤ L and Dham(S, S′) ≤ H} , (5) where X n is the set of all possible sequences over X of length n, Dham is the hamming distance, and H is a prede- fined threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' For long sequences, the search space can be extensively large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Therefore, we select random samples from VL,H(S) and choose the sequence that returns the maximum value based on the RMSD measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Our approach to generat- ing adversarial sequences falls under the class of black-box attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' This means that we only have access to the output of the network (Papernot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' It is worth noting that the inference time of complex pro- tein folding systems, which apply multiple processing and alignment steps prior to the use of any neural network, such as AlphaFold is extremely high compared to NN-based im- age classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' The forward pass of such systems involves a large number of computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' This fact, along with the discrete nature of the input space, are the bottleneck of de- veloping more complex black-box attacks (Mahmood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' 2021), which in general require a high number of queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Complexity In this section, we formalize the problem of generating an adversarial attack for PFNNs and establish its complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Definition 1 (PFNN Adversarial Attack (PAA) Problem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Given a learning model A(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' θ) : X n → (R × R × R)n mapping residues to 3-dimensional coordinates and parame- terized by θ, a sequence S ∈ X n, and a sequence alignment scoring matrix B, find an input sequence S′ ∈ X n such that Dseq(S, S′) ≤ L and Dstr(A(S), A(S′)) ≥ U, where the bounds L and U and distance functions d and D are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' We prove that the PAA problem is NP-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' This es- tablishes that, in general, there is no polynomial-time solution to the PAA problem unless P = NP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Due to this complexity and for ease of presentation, we adopt simple perturbation attacks for our experiments in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' We begin by defining the NP-complete problem to be reduced to an instance of the PAA problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Definition 2 (CLIQUE Problem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Given an undirected graph G = (V, E) and an integer k, find a fully connected sub- graph induced by V ′ ⊆ V such that |V ′| = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' The PFNN Adversarial Attack (PAA) problem in Definition 1 is NP-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' It is easy to verify that the PAA problem is in NP since, given a solution sequence S′, one can check whether the constraints Dseq(S, S′) ≤ L and Dstr(A(S), A(S′)) ≥ U are satisfied in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' It remains to be shown whether the PAA problem is NP-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' We establish this result via a reduction from the CLIQUE problem in Defi- nition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Given a CLIQUE instance ⟨G = (V, E), k⟩ with |V | = n and |E| = m, we construct its corresponding PAA instance ⟨A(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' θ), S, B, L, U⟩ as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Without loss of generality, let us consider a restricted version of the PAA problem where there are only two residue types {N, K} with the corresponding BLOSUM62 sub-matrix B′ = 6 · I, where I denotes the identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Following the one-hot repre- sentation of residues adopted in (Jumper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' 2021b), any input tensor over {N, K} is represented as a one-hot encod- ing Sin ∈ (B × B)n to be used as an input tensor to A, where sin i0 = 1 (sin i1 = 1) denotes that residue sin i is of type N (K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Let S = (N, N, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' , N) denote the all-N sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' We set L = 6k and U = k(k−1) 2 � 3 n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' The connectivity structure of A is derived from the edges E in the CLIQUE instance as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' The first column of the input tensor corresponding to sin i0 for all i ≤ n is disconnected from the network and the second column corresponding to sin i1 is connected to A such that, for each edge (vi, vj) ∈ E, we have a connection from sin i1 and sin j1 to each of the three outputs in the first three- dimensional coordinate of A(Sin)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' All connections have a weight of unity and this defines the parameters θ of the model A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Therefore, without loss of generality, we are only consid- ering the first of the n output three-dimensional coordinates A(Sin)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' In particular, these values keep track of the number of edges induced by the vertices in G corresponding to the non-zero entries in sin 11, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' , sin 1n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' We now prove that there is a clique of size k in G if and only if there is a feasible solution Sin = S′ to the reduced PAA instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' ( =⇒ ) Assume there is a clique of size k in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' We can derive a feasible solution S′ to the reduced PAA instance as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' For every vertex vi ∈ V (not) in the clique, let (s′ i0 = 1) s′ i1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Since S is the all-N sequence, its corresponding one-hot encoding consists of si0 = 1 for all 1 ≤ i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Thus, the corresponding BLOSUM62 distance is Dseq(S, S′) = � 1≤i≤n (6 − 6 · 1(si ̸= s′ i)) = 6k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' (6) This satisfies the sequence alignment constraint defined by Dseq(S, S′) ≤ L = 6k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Furthermore, the solution S′ induces outputs of x′ 1 = y′ 1 = z′ 1 = k(k − 1)/2, leading to an RMSD of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Without loss of generality, we omit the alignment step in computing the RMSD and therefore assume that A(S′) = ˆ A(S′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' The corresponding RMSD distance Dstr(A(S), ˆ A(S′)) in output predictions is presented below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Recall that x1 = y1 = z1 = 0 for the the all-N sequence S because its corresponding column in the one-hot encoding is disconnected from the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Dstr(A(S), A(S′)) = � � � � 1 n � i∈[n] d(A(S)i, ˆ A(S′)i) = � � � � 1 n � 3 � 0 − k(k − 1) 2 �2� = k(k − 1) 2 � 3 n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' (7) Thus, the constraint Dstr(S, S′) ≥ U = k(k−1) 2 � 3 n is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' ( ⇐= ) We prove the contrapositive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' That is, if there is no clique of size k in G, then the reduced PAA instance is infeasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' We proceed by showing that there must be exactly k non-zero entries in the column vector {s′ i1|i ≤ n} in order to satisfy constraints Dseq(S, S′) ≤ L = 6k and Dstr(A(S), A(S′)) ≥ U and that, if there is no clique of size k, then there is no choice of k non-zero entries in {s′ i1|i ≤ n} that will satisfy these constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Let k′ denote the number of non-zero entries in {s′ i1|i ≤ n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' To satisfy Dseq(S, S′) ≤ L = 6k, it follows that k′ ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' If k′ < k, then the maximum value of Dstr(A(S), A(S′)) is k′(k′−1) 2 � 3 n < k(k−1) 2 � 3 n and denotes to the case where the k′ non-zero entries correspond to a clique of size k′ in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' The strict inequality is due to the monotonically increasing nature of this equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Therefore, it must be that k = k′ and we have outputs x′ 1 = y′ 1 = z′ 1 = k(k − 1)/2 as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Suppose that the k′ non-zero entries in {s′ i1|i ≤ n} do not correspond to a clique in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Then the values x′ 1, y′ 1, and z′ 1 output by A and corresponding to the number of edges induced by the chosen non-zero entries would be strictly less than k(k−1)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Therefore, we would have Dstr(A(S), A(S′)) < U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' This proves that the reduced PAA is infeasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Table 1: RMSD results when L ∈ {20, 30, 40}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Seq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' ID n L RMSD µall µdiff µ′ all µ′ diff Q14653 427 20 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='87 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='76 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='92 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='46 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='29 Q14653 427 30 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='42 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='76 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='15 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='45 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='12 Q14653 427 40 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='28 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='76 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='49 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='42 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='026 Q5BJD5 291 20 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='311 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='23 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='77 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='6 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='64 Q5BJD5 291 30 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='708 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='23 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='52 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='82 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='1 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='8 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='26 P59633 154 MAX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='13 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='82 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='26 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='13 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='84 P0DTC9 419 MIN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='593 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='39 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='46 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='9 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='83 P0DTC9 419 AVG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='767 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='39 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='37 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='5 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='83 P0DTC9 419 MAX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='685 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='39 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='1 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='64 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='94 Experimental Results For our experimental setup, we use the default settings of the latest version of AlphaFold 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' This includes the initial multi-sequence alignment (MSA) step, the five-model ensem- bles predictions, recycling, output confidence ranking, and amber relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' For further details about each step, we refer the reader to (Jumper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' 2021a) and its supplementary information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' We include results from using the high-accuracy full database configuration of the initial AlphaFold MSA step along with the less accurate (and faster) reduced database option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' In order to compute the RMSD and GDT, we need to employ an alignment algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' In this paper, we use the built-in alignment PyMOL procedure without outlier rejec- tions (Schrödinger and DeLano).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' The parameters of PyMOL alignment are selected using the default settings, which in- clude an outlier rejection cutoff of 2, a maximum number of outlier rejection cycles of 5, and the use of the structural superposition step .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' We note that these outliers only impact the calculations of the RMSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Our adversarial sequences are generated by randomly sam- pling 20 sequences from the set VL,H in (5) with H = 5 and L = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Then, we pick the sequence that returns the maximum value in RMSD structural distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' We use an 1https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='com/deepmind/alphafold GDT = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='1% ID: P0DTC6 𝑛=61 Sim = 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='8% RMSD = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='4Å GDT = 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='1% ID: P0DTC8 𝑛=121 Sim = 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='9% RMSD = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='7Å GDT = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='1% ID: P13164 𝑛=125 Sim = 96% RMSD = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='6Å GDT = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='4% ID: O43765 𝑛 = 313 RMSD = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='051Å ID: P04439 𝑛 = 365 RMSD = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='016Å ID: P56962 𝑛 = 302 RMSD = 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='013Å ID: P59632 𝑛 = 274 RMSD = 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='052Å Figure 4: The structures of the original (black) and adversarial (red) sequences from AlphaFold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' The 3D plots, aligned using PyMol (Schrödinger and DeLano), are for proteins O43765 (first), P04439 (second), P56962 (third), and P59632 (fourth).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' For structure differences, the RMSD values are reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' The structures of the complete list of sequences are given in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Table 3: RMSD, GDT-TS, and GDT-HA results using the full database AlphaFold configuration with L = 20 and H = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' The average columns correspond to 20 adversarial samples for each protein ID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' The complete table is placed in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Seq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' ID n Similarity (%) RMSD Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' RMSD GDT-TS (%) Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' GDT-TS (%) GDT-HA (%) Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' GDT-HA (%) run-time (days) O43765 313 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='4026 14.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='6858 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='2584 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='6246 P59632 274 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='1752 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='018 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='4704 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='8175 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='0401 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='0328 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='6834 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='5214 AMD EPYC 7702 64-Core Processor with 1 TiB of RAM and NVIDIA A100 GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' We generate adversarial sequences against the COVID-19 protein sequences from the UniProt database considered by AlphaFold in (Jumper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' The original fasta (file extension for protein sequences) se- quence files are available online 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Additionally, we generate adversarial sequences against most of the the UniProt (Uni- versal Protein resource, a central repository of protein data created by combining the Swiss-Prot, TrEMBL and PIR-PSD databases (uni 2021)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Our code is provided as supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' BLOSUM Threshold Experiment In this subsection, we want to investigate how a change in the bound on biological similarity changes the adversarial sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' In other words, we show the impact of using differ- ent BLOSUM thresholds in set VL,H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' As such, we randomly select 6 sequences and generate adversarial sequences by configuring the BLOSUM threshold, L, to be 20, 30, and 40 (we use strict equalities to ensure the exact BLOSUM distance) and set H = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' For each case, we obtain the RMSD after alignment as reported in the fourth column of Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Furthermore, we present the average confidence percentage level of the prediction of the original (adversarial) sequence as reported by AlphaFold and denoted by µall (µ′ all).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Addi- tionally, in the 6th and 8th columns, we report the average confidence values for the residues that are different between the original and adversarial sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' These are denoted by µdiff and µ′ diff, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' We observe that, in general, when the BLOSUM threshold distance increases, the RMSD also increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' This means that biologically increased distance in the input space, in general, causes higher changes in the output predictions of AlphaFold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' In terms of the confidence scores, we observe that the change in the overall average con- fidence between the original and perturbed sequence is not 2https://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='uniprot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='org/pub/databases/uniprot/pre_release/ covid-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='fasta significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' However, in almost all the considered cases, we notice that the prediction confidence of the altered residues has reduced for the adversarial sequence when compared to the ones reported for the original sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Confidence Experiment Given a sequence S, per residue, AlphaFold generates an estimate of its prediction confidence in the form of a value in [0, 100].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' This value is called the predicted Local Distance Test (pLDDT) and represents the predicted value on the lDDT-Cα metric (Mariani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' In this subsection, we answer the following question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Does selecting the residues to be changed based on their low (or high) confidence scores impact the resulting RMSD between the original and adversarial structure prediction?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Phrased dif- ferently, in terms of the RMSD, we illustrate the impact of using the prediction confidence scores of every residue of the predicted structure of the original sequence in determining the location of the residues to be altered in the adversarial se- quence generation method presented in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' As such, five, not cherry picked, randomly selected sequences are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Then, the locations of the 5 residues to be altered are taken based on three categories as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Residues are selected with confidence values near the (i) minimum con- fidence score (MIN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' category), (ii) the average score (AVG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' category), and (iii) the maximum confidence score (MAX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' category).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Results are presented in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' We observe that, in general, selecting residues with low or high confidence scores is not related to the amount of the induced RMSD at the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' As such, in our method, the locations of the flipped residues are selected independent of the confidence scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' COVID-19 Case Studies We apply our adversarial approach to 111 publicly available COVID-19 protein sequences as of the time of this writing per the UniProt database using AlphaFold full database con- figuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Additionally, in the supplementary material, we Table 4: Prediction confidence results using the full database AlphaFold configuration with L = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Seq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' ID n RMSD µall σall µdiff σdiff µ′ all σ′ all µ′ diff σ′ diff O43765 313 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='438 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='221 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='634 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='786 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='027 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='554 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='423 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='342 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='759 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='463 P04439 365 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='162 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='845 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='995 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='23 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='704 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='921 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='068 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='678 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='968 Q99836 296 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='761 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='213 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='817 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='914 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='454 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='918 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='971 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='198 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='253 P59632 274 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='018 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='367 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='783 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='136 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='029 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='364 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='794 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='926 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='362 Table 5: Overall Prediction and attack results for the reduced and full database configurations of AlphaFold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' n Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' n Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' µall Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' µall Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' RMSD Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' RMSD Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' GDT Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' GDT Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' run-time Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' run-time reduced database 480.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='53 416.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='66 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='22 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='96 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='31 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='24 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='08 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='59 full database 410.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='73 336.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='63 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='25 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='23 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='78 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='18 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='95 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='86 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='63 provide complete results using the reduced AlphaFold con- figuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' The BLOSUM62 distance between the original and adversarial sequences is at most 20, thus they are biolog- ically close to each other (Chothia and Lesk 1986;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Sander and Schneider 1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Given the long list of the considered sequences, we describe only the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' SGTA_HUMAN Small glutamine-rich tetratricopeptide repeat-containing pro- tein alpha (O43765), HLAA_HUMAN HLA class I histocom- patibility antigen, A alpha chain (P04439), STX17_HUMAN Syntaxin-17 (P56962), AP3A_SARS ORF3a (P59632), and MYD88_HUMAN Myeloid differentiation primary response protein MyD88 (Q99836).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' The cases covered include homo sapiens and severe acute respiratory syndrome coronavirus 2 (2019-nCoV) (SARS-CoV-2) organisms which provide a wide variety of proteins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' The considered sequences vary in length as they range from n = 22 to n = 2511.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Figures 1 and 4 show the aligned predicted structures of the proteins described earlier where the original sequence is given in black and the adversarial sequence is given in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' We observe that, independent of the predicted struc- ture of the original sequence, a small change in the input sequence results in significant changes in the output struc- tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' The resulting structural distances (similarities) mea- sured in Å (percentage) are given in terms of the RMSD (GDT-TS) in the fourth (sixth) column of Table 9 for the full database configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Furthermore, we report the results using GDT-HA in the eighth column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' The high similarity between the original and adversarial sequences is observed from the third column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' The similarity percentage is calculated as 100(n − Dham(S, S′))/n, where Dham(S, S′) ≤ H = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' The complete results of all the considered proteins, includ- ing reduced AlphaFold configuration, are provided in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' As observed from the RMSD and GDT results in Table 9, small changes in the input sequence corresponding to only five residues cause AlphaFold to predict structures that are highly divergent from the predicted structure of the original sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' The last column in Table 9 reports the total execu- tion time (in days) of running the 20 adversarial sequences that were randomly selected from the set VL,H, which is shown to scale with the sequence length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' We only select 20 samples given the long time incurred by AlphaFold to predict the output structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Additionally, in Table 13, we report the average (devi- ation) prediction confidence results as for all the residues (designated with subscript ‘all’) and for the 5 altered residues (subscript ‘diff’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' The standard deviation is denoted σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' We ob- serve that, independent of the average prediction confidence, the RMSD between the original and adversarial predicted structures is always high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' This is noted for both the full and reduced database configurations of AlphaFold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Moreover, we observe that AlphaFold predicts the adversarial structure with similar confidence values to the original sequence (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=', see the 4th and 8th columns in both tables).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' The same observa- tion holds for the entire sequence and for the altered residues (columns 6 and 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' In Tables 14 and 15 of the supplementary, we break down GDT scores between the structures of the original and per- turbed sequences based on the prediction confidence scores of the original sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' We use the regions (1 to 4) defined by AlphaFold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' As observed w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' all regions, GDT scores are, in general, low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' For the considered dataset, the values presented in Table 5 gauge the overall robustness of AlphaFold to adversarial se- quences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' As indicated in the documentation of AlphaFold, for better accuracy, the full database configuration incurs a higher execution time compared to the reduced database configura- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' The reported average values of the RMSD and GDT-TS measures are 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='78Å and 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='95%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' In CASP14 (year 2020), AlphaFold achieved a median GDT-TS score of 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='4%, and 88% of their predictions fall under RMSD = 4Å 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' These results are obtained by comparing the predicted and ground truth structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' The CASP14 AlphaFold results underscore the significance of the values reported in Tables 9 and 5, as they show how small changes in the input sequences could damage the predictions (See columns 6 to 9 in Table 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' The key takeaway is that AlphaFold is generally not robust even when a basic approach is used to generate perturbations of the input protein sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Conclusion The groundbreaking progress made in recent years on the prediction of protein folding structures promises to enable profound advances in the understanding of diseases, the map- ping of the human proteome, and the design of drugs and therapeutics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' However, until these predictions are shown to be robust, we argue that the grand challenge of predictive protein folding persists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' In this paper, we have presented the first work in this direction by demonstrating that Pro- tein Folding Neural Networks (PFNNs) are often susceptible to adversarial attacks in the form of minor perturbations to the input protein sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' These perturbations can induce 3https://predictioncenter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='org/casp14/index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='cgi great changes in the predicted protein structure and the re- sulting lack of robustness precludes the adoption of such PFNNs in safety-critical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' We have employed standard protein structural distance and similarity to measure the robustness of AlphaFold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' While the perturbation methods employed in this paper were basic for the purposes of illus- trating the lack of robustness of PFNNs, the results presented herein can be readily used as a baseline for future work on adversarial attacks on PFNNs and their robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' References 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' UniProt: the universal protein knowledgebase in 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Nucleic acids research, 49(D1): D480–D489.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' and Volkman, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Interconversion between two unrelated protein folds in the lymphotactin native state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Proceedings of the National Academy of Sciences, 105(13): 5057–5062.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Zemla, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' LGA: a method for finding 3D similarities in protein structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Nucleic acids research, 31(13): 3370– 3374.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Table 6: RMSD, GDT-TS, and GDT-HA results using the reduced database AlphaFold configuration with L = 20 and H = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' The average results correspond to 20 adversarial samples for each protein ID.' metadata={'source': 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+page_content=' ID n Similarity (%) RMSD Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' RMSD GDT-TS (%) Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' GDT-TS (%) GDT-HA (%) Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' GDT-HA (%) run-time (days) P56962 302 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='3444 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='301 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='8695 12.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='822 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='283 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='324 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='18 Table 14: GDT-TS results based on AlphaFold predefined confidence regions using the full database AlphaFold configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Regions R1 to R4 correspond to confidence scores of above 90%, 70% to 90%, 50% to 70%, and below 50%, respectively.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='007 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='014 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='889 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='043 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='381 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='145 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='083 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='797 Table 15: GDT-TS results based on AlphaFold predefined confidence regions using the full database AlphaFold configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Regions R1 to R4 correspond to confidence scores of above 90%, 70% to 90%, 50% to 70%, and below 50%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' This is part II of II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' Seq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content=' ID n R1 GDT (%) R1 (%) R2 GDT (%) R2 (%) R3 GDT (%) R3 (%) R4 GDT (%) R4 (%) P56962 302 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='958 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='735 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='952 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='954 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='583 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} +page_content='868 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE2T4oBgHgl3EQfvgjw/content/2301.04093v1.pdf'} 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--git a/bNE2T4oBgHgl3EQfwAiL/content/tmp_files/2301.04097v1.pdf.txt b/bNE2T4oBgHgl3EQfwAiL/content/tmp_files/2301.04097v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..cc5aaa107ef21325a0f93dcca5b7e4c9e53cd779 --- /dev/null +++ b/bNE2T4oBgHgl3EQfwAiL/content/tmp_files/2301.04097v1.pdf.txt @@ -0,0 +1,1575 @@ +arXiv:2301.04097v1 [math.AP] 10 Jan 2023 +STABILITY OF HARDY-LITTLEWOOD-SOBOLEV +INEQUALITIES WITH EXPLICIT LOWER BOUNDS +LU CHEN, GUOZHEN LU, AND HANLI TANG +Abstract. In this paper, we establish the stability for the Hardy-Littlewood- +Sobolev (HLS) inequalities with explicit lower bounds. +By establishing +the relation between the stability of HLS inequalities and the stability of +fractional Sobolev inequalities, we also give the stability of the fractional +Sobolev inequalities with the lower bounds. +This extends the stability +of Sobolev inequalities with the explicit lower bounds established by Dol- +beault, Esteban, Figalli, Frank and Loss in [14] to the fractional order case. +Our proofs are based on the competing symmetries, the continuous Steiner +symmetrization inequality for the HLS integral and the dual stability the- +ory. +1. Introduction +The main purpose of this paper is to give the lower bound estimate of +the sharp constants of the stability of the Hardy-Littlewood-Sobolev (HLS) +inequality and fractional Sobolev inequality. +The classical Hardy-Littlewood-Sobolev (HLS) inequality ([22, 28]) states +� +Rn +� +Rn |x − y|−(n−2s)f(x)g(y)dxdy ≤ Cn,p,q′∥f∥Lq′(Rn)∥g∥Lp(Rn), +(1.1) +with 1 < q′, p < ∞, 0 < s < n +2 and 1 +q′ + 1 +p − 2s +n = 1. Lieb and Loss [25] applied +the layer cake representation formula to give an explicit upper bound for the +sharp constant Cn,p,q′. +More precisely, they showed that the best constant +Cn,p,q′ satisfies the following estimate +Cn,p,q′ ≤ +n +n − λ +� +π +λ +2 +Γ(1 + n +2) +� λ +n 1 +q′p +� +( +λq′ +n(q′ − 1)) +λ +n + ( +λp +n(p − 1)) +λ +n +� +. +In the special diagonal case q′ = p = +2n +n+2 (s = 1), Aubin [1] and Talenti +[30] derived the sharp constants of HLS inequality by classifying the extremal +of classical Sobolev inequality which is a dual form of HLS inequality. For +the HLS inequality of general diagonal case, Lieb [24] classified the extremal +function of HLS inequality and obtained the best constant +Cn,p,q′ = π +λ +2 Γ( n +2 − λ +2) +Γ(n − λ +2) +�Γ(n) +Γ( n +2) +�1− λ +n. +Recently, the authors of [10, 17, 18, 19] developed a rearrangement-free ar- +gument to obtain the sharp HLS inequality. Weighted HLS inequalities, also +1 + +2 +LU CHEN, GUOZHEN LU, AND HANLI TANG +known as the Stein-Weiss inequalities [29], and their inequalities, sharp con- +stants and existence of extremal functions have also been studied in Euclidean +spaces and the Heisenberg group, see e.g., [4, 13, 20, 21]. +The stability of the fractional Sobolev inequality states that there exists +some positive constant Cn,s such that for any U ∈ ˙Hs(Rn), there holds +��(−∆)s/2U +��2 +2 − Ss,n∥U∥2 +2n +n−2s ≥ Cn,s inf +h∈MS ∥(−∆)s/2(U − h)∥2 +2, +where ˙Hs(Rn) denotes the s-order homogenous Sobolev space in Rn: the com- +pletion of C∞ +c (Rn) under the norm +� � +Rn |(−∆) +s +2U|2dx +� 1 +2, +MS := +� +c( +2b +b2 + |x − a|2) +n−2s +2 , a ∈ Rn, b > 0, c ∈ R +� +is the extremal set of the fractional Sobolev inequality and +Ss,n = (4π)s Γ( n+2s +2 ) +Γ( n−2s +2 ) +�Γ( n +2) +Γ(n) +�2s/n += Γ( n+2s +2 ) +Γ( n−2s +2 ) |Sn|2s/n +(1.2) +is the sharp constant of the fractional Sobolev inequality. This kind of stability +inequality was proposed in Brezis and Lieb’s work in [5]. Bianchi and Egnell +in [3] first obtained the stability of the first order Sobolev inequality. Stability +of Sobolev inequality for s = 2 and even integers s < n +2 were established by the +second author and Wei [26], and by Bartsch, Weth and Willem [2] respectively. +Chen, Frank and Weth [11] established the stability of Sobolev inequality for +all 0 < s < n/2. It should be noted that although the fractional Sobolev +inequality is equivalent to the HLS inequality of diagonal case, their stabilities +are not equivalent. Carlen [8] developed a duality stability theory based on +the quantitative convexity and deduce the following stability of HLS inequality +from the stability of fractional Sobolev inequality: There exists some constant +� +Cn,s such that for any g ∈ L +2n +n+2s(Rn), there holds +∥g∥2 +2n +n+2s − S−1 +n,s∥(−∆)−s/2g∥2 +2 ≥ � +Cn,s +inf +h∈MHLS ∥g − h∥2 +2n +n+2s, +where +MHLS := +� +c( +2b +b2 + |x − a|2) +n+2s +2 , a ∈ Rn, b > 0, c ∈ R +� +. +Stability of Sobolev inequality is proved by establishing the local stabil- +ity of Sobolev inequality based on the spectrum analysis of elliptic or high- +order elliptic operator and using the Lions concentration compactness tech- +nique to obtain global stability of Sobolev inequality. However, this method +does not tell us any lower bound information about the sharp constant of +stability of Sobolev inequality. Recently, Dolbeault, Esteban, Figalli, Frank + +STABILITY OF HLS INEQUALITY WITH LOWER BOUNDS +3 +and Loss in [14] first obtained the stability for first order Sobolev inequal- +ity with the explicit lower bound by competing symmetries, the continu- +ous Steiner symmetrization inequality (Poly´a-Szeg¨o inequality) for L2 inte- +gral of gradient u and the fact ∥∇u∥2 +L2 = ∥∇u+∥2 +L2 + ∥∇u−∥2 +L2. +However +this method is not applicable to the stability of fractional Sobolev inequality +(s < n +2) because of the absence of continuous Steiner symmetrization inequal- +ity for L2 integral of high-order derivatives and the identity ∥(∆)s/2u∥2 +L2 = +∥(∆)s/2(u+)∥2 +L2 + ∥(∆)s/2(u−)∥2 +L2. In this paper, we will overcome these dif- +ficulties and derive the stability of fractional Sobolev inequality with explicit +lower bounds. Instead of directly considering the stability of Sobolev inequal- +ity, we will first establish the stability for the HLS inequality by competing +symmetries, the continuous Steiner symmetrization inequality for HLS integral +and local dual stability theory. We also note that some relationship between +the global geometric and functional inequalities and their local version of sta- +bility inequalities in a certain sense has been explored in [12]. Our first result +states: +Theorem 1.1. Let SHLS(g) denote the HLS stability functional given by +SHLS(g) = +∥g∥2 +2n +n+2s − S−1 +n,s∥(−∆)−s/2g∥2 +2 +inf +h∈MHLS ∥g − h∥2 +2n +n+2s +. +(1.3) +Denote by Kn,s = sup +0<δ<1 +δ +2 +n−2s +n+2s min{ m(2δ) +Sn,s +n−2s +n+2s, 1}, where +m(δ) = +4s +n + 2s + 2 − 2 +2∗s +2∗ +s +� +k=3 +2∗ +s(2∗ +s − 1) · · ·(2∗ +s − k + 1) +k! +l(δ)k−2 +if 2∗ +s = +2n +n−2s is an integer and otherwise +m(δ) = +4s +n + 2s + 2 − 2 +2∗s +[2∗ +s] +� +k=3 +2∗ +s(2∗ +s − 1) · · ·(2∗ +s − k + 1) +k! +l(δ)k−2 − 2 +2∗s +l(δ)2∗ +s−2, +and l(δ) = +� +δ +1−δ. Then there holds +inf +g∈L +2n +n+2s (Rn)\MHLS +SHLS(g) ≥ 1 +2 min{Kn,s, min{2 +n+2s +n +− 2, 1}}. +By the dual stability theory, we can deduce the following stability of frac- +tional Sobolev inequality with explicit lower bounds from the stability of HLS +inequality with explicit lower bounds, i.e., Theorem 1.1: +Theorem 1.2. For f ∈ +˙Hs(Rn) \ MS, denote SS(f) the Sobolev stability +functional given by +SS(f) = Sn,s∥(−∆)s/2f∥2 +2 − ∥f∥2 +2∗ +inf +h∈MS ∥(−∆)s/2(f − h)∥2 +2 +. +(1.4) + +4 +LU CHEN, GUOZHEN LU, AND HANLI TANG +Then +inf +f∈ ˙Hs(Rn)\MS +SS(f) ≥ Sn,s min{Kn,s, min{2 +n+2s +n +− 2, 1}} +4 +. +Remark 1. We also note that K¨onig in [23] proved that +inf +f∈ ˙Hs(Rn)\MS +SS(f) < +4s +n + 2 + 2s +by doing the third-order Taylor expansion to fractional Sobolev functional. +This paper is organized as follows. Section 2 is devoted to some preliminar- +ies. In Section 3, we will give the stability of HLS inequality with explicit lower +bounds using competing symmetries, the continuous Steiner symmetrization +inequality for the HLS integral and the local dual stability theory. In Section +4, we will establish the stability of fractional Sobolev inequality with explicit +lower bounds. +2. Preliminaries +In this section, we will state some tools, including competing symmetries +theorem, a continuous rearrangement flow and expansions with remainder term +and so on, which will be used in our proof. +Let f ∈ Lp(Rn), 1 < p < ∞ and fk = (RU)kf, k ∈ N, where Rf = f ∗ is +the decreasing rearrangement of f and +(Uf)(x) = +� +2 +|x − en|2 +� n +p +f +� +x1 +|x − en|2, · · · , +xn−1 +|x − en|2, |x|2 − 1 +|x − en|2 +� +, +en = (0, · · · , 0, 1) ∈ Rn. Carlen and Loss [9] proved the following competing +symmetry theorem. +Theorem 2.1. Let 0 ≤ f ∈ Lp (1 < p < ∞), fk = (RU)kf and h(x) = +∥f∥p|Sn|−1/p � +2 +1+|x|2 +�n/p +. Then +lim +k→∞ ∥fk(x) − h(x)∥p = 0. +A continuous rearrangement flow which interpolates between a function and +its symmetric decreasing rearrangement introduced in [14] based on Brock’s +flow ( [6], [7] plays an important part in our proof. More specifically, there +exists a flow fτ, τ ∈ [0, ∞], such that +f0 = f, f∞ = f ∗. +And if 0 ≤ f ∈ Lp(Rn) for some 1 ≤ p < ∞, then τ → fτ is continuous in +Lp(Rn). In our setting we need to prove τ → +inf +h∈MHLS ∥fτ −h∥ +2n +n+2s is continuous. +Lemma 2.1. Let 0 ≤ f ∈ L +2n +n+2s. Then the function τ → +inf +h∈MHLS ∥fτ − h∥ +2n +n+2s +is continuous. + +STABILITY OF HLS INEQUALITY WITH LOWER BOUNDS +5 +Proof. Let us prove +inf +h∈MHLS ∥fτ − h∥ +2n +n+2s → +inf +h∈MHLS ∥fτ0 − h∥ +2n +n+2s as τ → τ0. +For any ε > 0, there exists a g1 ∈ MHLS such that +inf +h∈MHLS ∥fτ − h∥ +2n +n+2s ≥ ∥fτ − g1∥ +2n +n+2s − ε. +Then +inf +h∈MHLS ∥fτ − h∥ +2n +n+2s − +inf +h∈MHLS ∥fτ0 − h∥ +2n +n+2s ≥ ∥fτ − g1∥ +2n +n+2s − ∥fτ0 − g1∥ +2n +n+2s − ε. +(2.1) +On the other hand, there exist a g0 ∈ MHLS such that +inf +h∈MHLS ∥fτ0 − h∥ +2n +n+2s ≥ ∥fτ0 − g0∥ +2n +n+2s − ε. +Thus +inf +h∈MHLS ∥fτ − h∥ +2n +n+2s − +inf +h∈MHLS ∥fτ0 − h∥ +2n +n+2s ≤ ∥fτ − g0∥ +2n +n+2s − ∥fτ0 − g0∥ +2n +n+2s + ε. +(2.2) +By the triangle inequality and the fact that τ → fτ is continuous in L +2n +n+2s(Rn), +we have +|∥fτ − gi∥ +2n +n+2s − ∥fτ0 − gi∥ +2n +n+2s| ≤ ∥fτ − fτ0∥ +2n +n+2s → 0, τ → τ0 i = 0, 1. (2.3) +Combining (2.1), (2.2) and (2.3), we complete the proof. +□ +The following expansions of ∥u + r∥2 +2∗s are also needed in our proof. +Lemma 2.2. Let u, r ∈ L2∗ +s(Rn), u + r ≥ 0 and u ≥ 0. If 2∗ +s is not an integer, +then +∥u + r∥2 +2∗s ≤ ∥u∥2 +2∗s + 2 +2∗s +[2∗ +s] +� +k=1 +2∗ +s(2∗ +s − 1) · · ·(2∗ +s − k + 1) +k! +∥u∥2−2∗ +s +2∗s +� +Rn u2∗ +s−krkdx ++ 2 +2∗ +s +∥u∥2−2∗ +s +2∗s +∥r∥2∗ +s +2∗s, +where [2∗ +s] is the integer part of 2∗ +s. If 2∗ +s is an integer, then +∥u + r∥2 +2∗s ≤ ∥u∥2 +2∗s + 2 +2∗ +s +[2∗ +s] +� +k=1 +2∗ +s(2∗ +s − 1) · · ·(2∗ +s − k + 1) +k! +∥u∥2−2∗ +s +2∗s +� +Rn u2∗ +s−krkdx. +In order to prove Lemma 2.2, we need the following technical lemma. +Lemma 2.3. (1) For all x ≥ −1, q ≥ 1, q is not an integer, +(1 + x)q ≤ 1 + +[q] +� +k=1 +q(q − 1) · · ·(q − k + 1) +k! +xk + |x|q, +where [q] is the integer part of q. + +6 +LU CHEN, GUOZHEN LU, AND HANLI TANG +(2) For all x ≥ −1, q ≥ 1, q is an integer, +(1 + x)q = 1 + +q +� +k=1 +q(q − 1) · · ·(q − k + 1) +k! +xk. +Proof. We only need to consider the case when q is not an integer. Let +f(x) = (1 + x)q − 1 − +[q] +� +k=1 +q(q − 1) · · ·(q − k + 1) +k! +xk − |x|q. +We first prove that f(x) ≤ 0 when x ≥ 0. Since +f ′(x) = q(1 + x)q−1 − +[q] +� +k=1 +q(q − 1) · · ·(q − k + 1) +(k − 1)! +xk−1 − qxq−1, +· · ·, +f ([q])(x) = q(q − 1) · · ·(q − [q] + 1) +� +(1 + x)q−[q] − 1 − xq−[q]� +, +then +f(0) = f ′(0) = · · · = f ([q])(0) = 0. +Using the fundamental inequality (1 + x)α ≤ 1 + xα for 0 < α < 1 and x ≥ 0, +we obtain f ([q])(x) ≤ 0 (x > 0). Then f ([q]−1)(x) is decreasing on (0, +∞), +which implies f ([q]−1)(x) ≤ f ([q]−1)(0) = 0. Repeat the deduction and we can +obtain f(x) ≤ 0 when x ≥ 0. +When −1 ≤ x ≤ 0, we have +f ′(x) = q(1 + x)q−1 − +[q] +� +k=1 +q(q − 1) · · ·(q − k + 1) +k! +xk−1 + q(−x)q−1, +· · · , +f ([q])(x) = q(q − 1) · · · (q − [q] + 1) +� +(1 + x)q−[q] − 1 − (−1)[q](−x)q−[q]� +, +and there still holds +f(0) = f ′(0) = · · · = f ([q])(0) = 0. +If [q] is even, using the fundamental inequality +(1 + x)α ≤ 1 + (−x)α, 0 < α < 1, −1 ≤ x ≤ 0 +we know f ([q])(x) ≤ 0. Then f ([q]−1)(x) is decreasing on [−1, 0], which implies +f ([q]−1)(x) ≥ f ([q]−1)(0) = 0. Thus f ([q]−2)(x) is increasing on [−1, 0], which +means f ([q]−2)(x) ≤ f ([q]−2)(0) = 0. Repeat the deduction and we can obtain +f (2k)(x) ≤ 0, f (2k+1)(x) ≥ 0 when k = 0, 1, · · · , [q] +2 . +Now let us deal with the case when [q] is odd. Since the function +h(x) = (1 + x)α − 1 + (−x)α (0 < α < 0) +satisfy that h′(x) = α[(1 + x)α−1 − (−x)α−1] is nonnegative on [−1, −1/2) and +nonpositive on (−1/2, 0], then f ([q])(x) is increasing on [−1, −1/2) and de- +creasing on (−1/2, 0], which implies f ([q])(x) ≥ f ([q])(0) = f ([q])(−1) = 0. Thus + +STABILITY OF HLS INEQUALITY WITH LOWER BOUNDS +7 +f ([q]−1)(x) is increasing on [−1, 0] which means f ([q]−1)(x) ≤ f ([q]−1)(0) = 0, x ∈ +[−1, 0]. Then f ([q]−2)(x) is decreasing on [−1, 0], which means f ([q]−2)(x) ≥ +f [q]−2(0) = 0. Repeat the deduction and we can obtain f (2k)(x) ≤ 0, f (2k+1)(x) ≥ +0 when k = 0, 1, · · · , [q]−1 +2 , which complete the proof. +□ +Now let us prove Lemma 2.2. Here we only state the proof when 2∗ +s is not +an integer. By Proposition 1, we derive that +(u + r)2∗ +s ≤ u2∗ +s + +[2∗ +s] +� +k=1 +2∗ +s(2∗ +s − 1) · · · (2∗ +s − k + 1) +k! +u2∗ +s−krk + |r|2∗ +s, +thus +� +Rn(u+r)2∗ +sdx ≤ +� +Rn u2∗ +sdx+ +[2∗ +s] +� +k=1 +2∗ +s(2∗ +s − 1) · · · (2∗ +s − k + 1) +k! +� +Rn u2∗ +s−krkdx +� +Rn |r|2∗ +sdx. +Using the inequality (1+x)α ≤ 1+αx for x ≥ −1 and 0 < α < 1, we complete +the proof of Lemma 2.2. +3. Stability of the Hardy-Littlewood-Sobolev inequality +In this section, we will set up the stability of the Hardy-Littlewood-Sobolev +inequality with explicit constant. First we will establish the local version of the +stability of the fractional Sobolev inequality for nonnegative functions. Then, +using the dual method from [8] by Carlen, we can deduce the local stability for +the HLS inequality for nonnegative function (where the aim function is close +to the manifold of the HLS optimizers). When the aim function is positive +and away from the manifold of HLS optimizers, we will handle them by the +competing symmetries and Block’s flow. At last, we will deal with the stability +of HLS inequality when the aim function is not necessarily non-negative. +3.1. Local stability of fractional Sobolev inequality. In the subsection +we will set up the local stability of fractional Sobolev inequality with explicit +lower bounds (Lemma 3.2) for 0 < s < n +2, while the case s = 1 was proved in +[14]. Without loss of generality, we only give the proof when 2∗ +s is an integer. +Let 0 ≤ f ∈ ˙Hs(Rn). It is well known that there exists an 0 ≤ u ∈ MS such +that ∥(−∆)s/2(f − u)∥2 +2 = inf +h∈MS ∥(−∆)s/2(f − h)∥2 +2. Then r = f − u satisfies +� +Rn[(−∆)s/2u][(−∆)s/2r]dx = 0, +and +� +Rn u2∗ +s−1rdx = 0, +since (−∆)su = cuu2∗ +s−1 with cu = ∥(−∆)s/2u∥2 +2 +∥u∥2 +2∗s +. Then +∥(−∆)s/2f∥2 +2 −Sn,s∥f∥2 +2∗s = ∥(−∆)s/2u∥2 +2 +∥(−∆)s/2r∥2 +2 −Sn,s∥u+r∥2 +2∗s. (3.1) + +8 +LU CHEN, GUOZHEN LU, AND HANLI TANG +Next we will estimate the Sobolev deficit ∥(−∆)s/2f∥2 +2 − Sn,s∥f∥2 +2∗ by the +expansion from Lemma 2.2 and the following spectral gap inequalities which +are due to Rey [27] (Eq. (D.1)) and Esposito [16] (Lemma 2.1) for s = 1, to +De Nitti and K¨onig [15] (Prop. 3.4) for 0 < s < n/2. +Lemma 3.1. Given f ∈ ˙Hs(Rn). Let u ∈ MS such that +∥(−∆)s/2(f − u)∥2 +2 = inf +h∈MS ∥(−∆)s/2(f − h)∥2 +2. +Then r = f − u satisfies +∥(−∆)s/2r∥2 +2 − (2∗ +s − 1)Sn,s∥u∥2−2∗ +s +2∗s +� +Rn u2∗ +s−2r2dx ≥ +4s +n + 2s + 2∥(−∆)s/2r∥2 +2. +Now by (3.1) and Lemma 2.2, we have +∥(−∆)s/2f∥2 +2 − Sn,s∥f∥2 +2∗s ≥ ∥(−∆)s/2r∥2 +2 − (2∗ +s − 1)Sn,s∥u∥2−2∗ +s +2∗s +� +Rn u2∗ +s−2r2dx +− Sn,s + + + +2 +2∗s +[2∗ +s] +� +k=3 +2∗ +s(2∗ +s − 1) · · · (2∗ +s − k + 1) +k! +∥u∥2−2∗ +s +2∗s +� +Rn u2∗ +s−krkdx + 2 +2∗s +∥u∥2−2∗ +s +2∗s +∥r∥2∗ +s +2∗s + + + . +(3.2) +By H¨older’s inequality and sharp Sobolev inequality, there holds +∥u∥2−2∗ +s +2∗s +� +Rn u2∗ +s−krkdx ≤ ∥u∥2−k +2∗s ∥r∥k +2∗s ≤ +1 +Sn,s +�∥r∥2∗s +∥u∥2∗s +�k−2 +∥(−∆)s/2r∥2 +2, +and +∥u∥2−2∗ +s +2∗s +∥r∥2∗ +s +2∗s ≤ +1 +Sn,s +�∥r∥2∗s +∥u∥2∗s +�2∗−2 +∥(−∆)s/2r∥2 +2. +Therefore combining Lemma 3.1, (3.2) and the above estimates, we get +∥(−∆)s/2f∥2 +2 − Sn,s∥f∥2 +2∗s +∥(−∆)s/2r∥2 +2 +≥ + + + +4s +n + 2s + 2 − 2 +2∗s +[2∗ +s] +� +k=3 +2∗ +s(2∗ +s − 1) · · · (2∗ +s − k + 1) +k! +�∥r∥2∗s +∥u∥2∗s +�k−2 +− 2 +2∗s +�∥r∥2∗s +∥u∥2∗s +�2∗ +s−2 + + + . +(3.3) +Now we are in the position to prove the local stability of Sobolev inequality +for nonnegative functions. Let l(δ) = +� +δ +1−δ and define +ν(δ) = inf +� +SSob(f) : 0 ≤ f ∈ ˙Hs(Rn) \ MS, inf +g∈MS ∥(−∆)s/2(f − g)∥2 +2 ≤ δ∥(−∆)s/2f∥2 +2 +� +, +and +m(δ) = +4s +n + 2s + 2− 2 +2∗ +s +2∗ +s +� +k=3 +2∗ +s(2∗ +s − 1) · · ·(2∗ +s − k + 1) +k! +l(δ)k−2, when 2∗ +s is an integer, + +STABILITY OF HLS INEQUALITY WITH LOWER BOUNDS +9 +otherwise +m(δ) = +4s +n + 2s + 2 − 2 +2∗ +[2∗ +s] +� +k=3 +2∗ +s(2∗ +s − 1) · · ·(2∗ − k + 1) +k! +l(δ)k−2 − 2 +2∗s +l(δ)2∗ +s−2. +Lemma 3.2. With the above notations, we have ν(δ) ≥ m(δ). +Proof. Since ∥(−∆)s/2f∥2 +2 = ∥(−∆)s/2r∥2 +2 + ∥(−∆)s/2u∥2 +2, then +∥r∥2∗s +∥u∥2∗s +≤ ∥(−∆)s/2r∥2 +∥(−∆)s/2u∥2 += ∥(−∆)s/2r∥2 +∥(−∆)s/2f∥2 +1 +� +1 − ∥(−∆)s/2r∥2 +2 +∥(−∆)s/2f∥2 +2 +≤ +� +δ +1 − δ = l(δ). +(3.4) +Therefor by (3.3) and (3.4) +ν(δ) ≥ + + + +4s +n + 2s + 2 − 2 +2∗ +s +[2∗ +s] +� +k=3 +2∗ +s(2∗ +s − 1) · · ·(2∗ +s − k + 1) +k! +l(δ)k−2 − 2 +2∗ +s +l(δ)2∗ +s−2 + + + , +which completes the proof of local stability of fractional Sobolev inequality +with explicit lower bounds. +□ +3.2. Stability of HLS inequality for positive functions. In this subsec- +tion, we will consider the stability of HLS inequality for positive functions. +Choose δ small enough, and define +µ(δ) = inf{SHLS(g) : 0 ≤ g ∈ L +2n +n+2s(Rn)\MHLS, +inf +h∈MHLS ∥g−h∥2 +2n +n+2s ≤ δ∥g∥2 +2n +n+2s}. +First we obtain the local stability of HLS inequality by the following dual +lemma from Carlen [8] and the local stability of Sobolev inequality for non- +negative functions. +Lemma 3.3. If SS(f) ≥ m(δ) for all nonnegative function f with +inf +h∈MS ∥(−∆)s/2(f − h)∥2 +2 ≤ δ∥(−∆)s/2f∥2 +2, +then +SHLS(g) ≥ 1 +2 +n − 2s +n + 2s min{m(δ) +Sn,s +n − 2s +n + 2s, 1}, +for all 0 < g ∈ L +2n +n+2s(Rn) \ MHLS satisfying +inf +h∈MHLS ∥g − h∥2 +2n +n+2s ≤ δ +2∥g∥2 +2n +n+2s. +Proof. If ∥g∥2 +2n +n+2s ≥ 2S−1 +n,s∥(−∆)−s/2g∥2 +2, then +∥g∥2 +2n +n+2s − S−1 +n,s∥(−∆)−s/2g∥2 +2 ≥ 1 +2∥g∥2 +2n +n+2s ≥ 1 +2 +inf +h∈MHLS ∥g − h∥2 +2n +n+2s, +thus SHLS(g) ≥ 1 +2. When ∥g∥2 +2n +n+2s ≤ 2S−1 +n,s∥(−∆)−s/2g∥2 +2, using Lemma 3.4 in +[8], we can obtain SHLS(g) ≥ 1 +2 +n−2s +n+2s min{ m(δ) +Sn,s +n−2s +n+2s, 1}. Since 1 +2 +n−2s +n+2s min{ m(δ) +Sn,s +n−2s +n+2s, 1} ≤ + +10 +LU CHEN, GUOZHEN LU, AND HANLI TANG +1 +2, then +SHLS(g) ≥ 1 +2 +n − 2s +n + 2s min{m(δ) +Sn,s +n − 2s +n + 2s, 1} +if +inf +h∈MHLS ∥g − h∥2 +2n +n+2s ≤ δ +2∥g∥2 +2n +n+2s. This accomplishes the proof of Lemma 3.3. +□ +Next let us handle the stability of HLS inequality when the nonnegative +function g satisfies +inf +h∈MHLS ∥g − h∥2 +2n +n+2s > δ∥g∥2 +2n +n+2s. +Lemma 3.4. For any 0 ≤ g ∈ L +2n +n+2s(Rn) satisfying +inf +h∈MHLS ∥g − h∥2 +2n +n+2s > +δ∥g∥2 +2n +n+2s, there holds +SHLS(g) ≥ δµ(δ). +Proof. Assume that 0 ≤ g ∈ L +2n +n+2s(Rn) satisfying inf +h∈M ∥g − h∥2 +2n +n+2s > δ∥g∥2 +2n +n+2s. +Let gk = (RU)kg (see the definition of RU in section 2). By Theorem 2.1 we +know +∥gk − hg∥ +2n +n+2s → 0, k → ∞, +where hg = ∥g∥ +2n +n+2s|Sn|− n−2s +2n ( +2 +1+|x|2) +n+2s +2 +∈ MHLS. It is well known that +k → ∥(−∆)− s +2gk∥2 +2 +is increasing by Riesz rearrangement inequality and ∥gk∥ +2n +n+2s = ∥g∥ +2n +n+2s. Thus +SHLS(g) ≥ +∥g∥2 +2n +n+2s − S−1 +n,s∥(−∆)−s/2g∥2 +2 +∥g∥2 +2n +n+2s +≥ 1 − S−1 +n,s∥(−∆)−s/2g∥2 +2 +∥g∥2 +2n +n+2s +≥ +∥gk∥2 +2n +n+2s − S−1 +n,s∥(−∆)−s/2gk∥2 +2 +∥gk∥2 +2n +n+2s +. +(3.5) +Since ∥gk − hg∥ +2n +n+2s → 0 as k → ∞ and hg ∈ MHLS, then there exist a k0 ∈ N +such that +inf +h∈MHLS ∥gk0 − h∥2 +2n +n+2s > δ∥gk0∥2 +2n +n+2s +and +inf +h∈MHLS ∥gk0+1 − h∥2 +2n +n+2s ≤ δ∥gk0+1∥2 +2n +n+2s. +Denote g0 = Ugk0, g∞ = gk0+1, then +inf +h∈MHLS ∥g0 − h∥2 +2n +n+2s = +inf +h∈MHLS ∥gk0 − h∥2 +2n +n+2s > δ∥gk0∥2 +2n +n+2s = δ∥g0∥2 +2n +n+2s. +Now using the continuous rearrangement flow gτ (0 ≤ τ ≤ ∞) introduced in +section 2, we conclude that gτ satisfies +∥(−∆)− s +2gτ∥2 +2 ≥ ∥(−∆)− s +2g0∥2 +2 = ∥(−∆)− s +2gk0∥2 +2, and ∥gτ∥ +2n +n+2s = ∥g0∥ +2n +n+2s = ∥g∥ +2n +n+2s. + +STABILITY OF HLS INEQUALITY WITH LOWER BOUNDS +11 +Since τ → +inf +h∈MHLS ∥gτ − h∥2 +2n +n+2s is continuous by lemma 2.1, then there exists +τ0 ∈ (0, ∞) such that +inf +h∈MHLS ∥gτ0 − h∥2 +2n +n+2s = δ∥gτ0∥2 +2n +n+2s. +(3.6) +Therefore by (3.5) and (3.6), +SHLS(g) ≥ +∥g0∥2 +2n +n+2s − S−1 +n,s∥(−∆)−s/2g0∥2 +2 +∥g0∥2 +2n +n+2s +≥ +∥gτ0∥2 +2n +n+2s − S−1 +n,s∥(−∆)−s/2gτ0∥2 +2 +∥gτ0∥2 +2n +n+2s += δ +∥gτ0∥2 +2n +n+2s − S−1 +n,s∥(−∆)−s/2gτ0∥2 +2 +inf +h∈MHLS ∥gτ0 − h∥2 +2n +n+2s +≥ δµ(δ), +where in the second inequality we use the following Riesz’s inequality for con- +tinuous convex rearrangement (see Appendix A in [14]): For nonnegative func- +tions f, g, there holds +�� +Rn×Rn +fτ(x)gτ(y) +|x − y|n−2sdxdy ≥ +�� +Rn×Rn +f(x)g(y) +|x − y|n−2sdxdy. +This proves Lemma 3.4. +□ +According to the definition of µ(δ), through Lemma 3.3, we deduce that +µ(δ) ≥ 1 +2 +n − 2s +n + 2s min{m(2δ) +Sn,s +n − 2s +n + 2s, 1}. +Combining this and Lemma 3.4, we derive that for any 0 ≤ f ∈ L +2n +n+2s(Rn)\ +MHLS, there holds +SHLS(g) ≥ δ +2 +n − 2s +n + 2s min{m(2δ) +Sn,s +n − 2s +n + 2s, 1}. +Therefor we have established the stability of HLS inequality with explicit lower +bounds for nonnegative functions. +3.3. Stability of HLS inequality with explicit lower bounds. In this +subsection, we will prove the stability of HLS inequality with explicit lower +bounds for general function f ∈ L +2n +n+2s(Rn) \ MHLS, namely we shall give +the proof of Theorem 1.1. Let us denote by CHLS the optimal constant for +stability of HLS inequality and denote by Cpos +HLS the optimal constant in (1.3) +when restricted to nonnegative functions. We first state a relationship between +these two optimal constants. +Lemma 3.5. +CHLS ≥ 1 +2 min{Cpos +HLS, min{2 +n+2s +n +− 2, 1}}. + +12 +LU CHEN, GUOZHEN LU, AND HANLI TANG +Proof. Let D(g) = ∥g∥2 +2n +n+2s −S−1 +n,s∥(−∆)−s/2f∥2 +2 and f± denote the positive and +negative parts of f. Then +D(f) ≥ ∥g∥2 +2n +n+2s − S−1 +n,s∥(−∆)−s/2g+∥2 +2 − S−1 +n,s∥(−∆)−s/2g−∥2 +2 += D(g+) + D(g−) + ∥g∥2 +2n +n+2s − ∥g+∥2 +2n +n+2s − ∥g−∥2 +2n +n+2s. +(3.7) +Without loss of generality, we may assume +∥g∥2 +2n +n+2s = 1, and m = ∥g−∥ +2n +n+2s +2n +n+2s ∈ [0, 1/2]. +Let h(m) = 1 − m +n+2s +n +− (1 − m) +n+2s +n +and t(m) = 1 − (1 − m) +n+2s +n +− 2(1 − +(1/2) +n+2s +n )m. It is easy to check t(0) = t(1/2) = 0 and t′′(m) ≤ 0 on [0, 1/2], +then t(m) ≥ 0 on [0, 1/2], which means 1 − (1 − m) +n+2s +n +≥ 2(1 − (1/2) +n+2s +n )m. +Thus +h(m) ≥ 2(1 − (1/2) +n+2s +n )m − m +n+2s +n , m ∈ [0, 1/2]. +Since the function 2(1 − (1/2) +n+2s +n )m− 2s +n − 1 is decreasing on [0, 1/2], then +h(m) ≥ (2 +n+2s +n +− 2)m +n+2s +n . +(3.8) +By (3.8) and sharp HLS inequality we have +∥g∥2 +2n +n+2s−∥g+∥2 +2n +n+2s−∥g−∥2 +2n +n+2s ≥ (2 +n+2s +n −2)∥g−∥2 +2n +n+2s ≥ (2 +n+2s +n −2)S−1 +n,s∥(−∆)s/2g−∥2 +2. +Along with (3.7), +D(g) ≥ D(g+) + D(g−) + (2 +n+2s +n +− 2)S−1 +n,s∥(−∆)−s/2g−∥2 +2. +Let h+ ∈ MHLS such that ∥g+ − h+∥ +2n +n+2s = +inf +h∈MHLS ∥g+ − h∥ +2n +n+2s. Since +D(g−) + S−1 +n,s∥(−∆)−s/2(g−)∥2 +2 = ∥g−∥2 +2n +n+2s, +thus +D(g) ≥ D(g+) + min{2 +n+2s +n +− 2, 1}∥g−∥2 +2n +n+2s +≥ Cpos +HLS∥g+ − h+∥2 +2n +n+2s + min{2 +n+2s +n +− 2, 1}∥g−∥2 +2n +n+2s +≥ min{Cpos +HLS, min{2 +n+2s +n +− 2, 1}} +� +∥g+ − h+∥ +2n +n+2s + ∥g−∥ +2n +n+2s +�2 +2 +≥ 1 +2 min{Cpos +HLS, min{2 +n+2s +n +− 2, 1}}∥g − h+∥2 +2n +n+2s +≥ 1 +2 min{Cpos +HLS, min{2 +n+2s +n +− 2, 1}} +inf +h∈MHLS ∥g − h∥2 +2n +n+2s +which completes the proof of Lemma 3.5. +□ + +STABILITY OF HLS INEQUALITY WITH LOWER BOUNDS +13 +Now we are in position to prove Theorem 1.1. Denote by +Kn,s = sup +0<δ<1 +δ +2 +n − 2s +n + 2s min{m(2δ) +Sn,s +n − 2s +n + 2s, 1}. +We have already established SHLS(g) ≥ Kn,s for all 0 ≤ g ∈ L +2n +n+2s(Rn) \ +MHLS in subsection 3.2. This together with Lemma 3.5 yields that for all +g ∈ L +2n +n+2s(Rn) \ MHLS, there holds +SHLS(g) ≥ 1 +2 min{Kn,s, min{2 +n+2s +n +− 2, 1}}. +Then we finish the proof of Theorem 1.1. +4. HLS stability implies Sobolev stability +In this section, we will prove the stability of fractional Sobolev inequality +with explicit lower bounds for all 0 < s < n/2 from the stability of Hardy- +Littlewood-Sobolev inequality with explicit lower bounds. +Let f ∈ ˙Hs(Rn). Define +F(f) = Sn,s∥(−∆)s/2f∥2 +2, E(f) = ∥f∥2 +2∗s. +Then the Legendre transform F ∗ of a convex functional F : ˙Hs(Rn) → [0, +∞) +defined on ˙H−s(Rn) is given by +F ∗(g) = +sup +f∈Hs(Rn) +{2 +� +Rn f(x)g(x)dx − F(f)}. +A simple calculation gives F ∗(g) = S−1 +n,s∥(−∆)−s/2g∥2 +2. +Similarly, the Le- +gendre transform E∗ of a convex functional E : L2∗ +s(Rn) → [0, +∞) defined +on L +2n +n+2s(Rn) is given by +E∗(g) = +sup +f∈L2∗s (Rn) +{2 +� +Rn f(x)g(x)dx − E(f)}. +Obviously, E∗(g) = ∥g∥2 +2n +n+2s. +Choose g = ∥f∥2−2∗ +s +2∗ +|f|2∗ +s−1sgn(f) and f1 = +S−1 +n,s(−∆)−sg, we can check that +E(f) + E∗(g) = 2 +� +Rn fgdx +(4.1) +and +F(f) = Sn,s∥(−∆)s/2f∥2 +2 += Sn,s∥(−∆)s/2f1∥2 +2 + Sn,s∥(−∆)s/2(f − f1)∥2 +2 + 2 +� +Rn(f − f1)(Sn,s(−∆)sf1)dx += S−1 +n,s∥(−∆)−s/2g∥2 +2 + 2 +� +Rn(f − S−1 +n,s(−∆)−sg)gdx + Sn,s∥(−∆)s/2(f − f1)∥2 +2 += 2 +� +Rn fgdx − F ∗(g) + Sn,s∥(−∆)s/2(f − f1)∥2 +2. +(4.2) + +14 +LU CHEN, GUOZHEN LU, AND HANLI TANG +Combining (4.1) with (4.2), we have +F(f) − E(f) = E∗(g) − F ∗(g) + Sn,s∥(−∆)s/2(f − f1)∥2 +2. +(4.3) +Since we have already proved the stability of HLS inequality in Theorem 1.1 +E∗(g) − F ∗(g) ≥ 1 +2 min{Kn,s, min{2 +n+2s +n +− 2, 1}} +inf +h∈MHLS ∥g − h∥2 +2n +n+2s, +(4.4) +then for any ǫ > 0, there exists a g0 ∈ MHLS such that +E∗(g)−F ∗(g) ≥ 1 +2 min{Kn,s, min{2 +n+2s +n −2, 1}}∥g−g0∥2 +2n +n+2s−ε+Sn,s∥(−∆)s/2(f−f1)∥2 +2. +Denote by kn,s = 1 +2 min{Kn,s, min{2 +n+2s +n +− 2, 1}}, by (4.3), (4.4), sharp HLS +inequality, (−∆)−sg0 ∈ MS, we derive +F(f) − E(f) ≥ kn,sS−1 +n,s∥(−∆)−s/2(g − g0)∥2 +2 − ε + Sn,s∥(−∆)s/2(f − f1)∥2 +2 += kn,s∥S−1/2 +n,s (−∆)−s/2(g − g0)∥2 +2 − ε + ∥S1/2 +n,s (−∆)s/2f − S−1/2 +n,s (−∆)−s/2g∥2 +2 +≥ kn,s +2 ∥S1/2 +n,s (−∆)s/2f − S−1/2 +n,s (−∆)−s/2g0∥2 +2 − ε +≥ Sn,skn,s +2 +inf +h∈MS ∥(−∆)s/2(f − h)∥2 +2 − ε, +which completes the proof of Theorem 1.2. +References +[1] T. 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Pura Appl. 110 (1976), +353–372. +(Lu Chen) School of Mathematics and Statistics, Beijing Institute of Tech- +nology, Beijing 100081, PR China +Email address: chenlu5818804@163.com +(Guozhen Lu) Department of Mathematics, University of Connecticut, Storrs, +CT 06269, USA +Email address: guozhen.lu@uconn.edu +(Hanli Tang) Laboratory of Mathematics and Complex Systems (Ministry of +Education), School of Mathematical Sciences, Beijing Normal University, +Beijing, 100875, China +Email address: hltang@bnu.edu.cn + diff --git a/bNE2T4oBgHgl3EQfwAiL/content/tmp_files/load_file.txt b/bNE2T4oBgHgl3EQfwAiL/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9825972e6ab0d38c9b7692070c93defa26695034 --- /dev/null +++ b/bNE2T4oBgHgl3EQfwAiL/content/tmp_files/load_file.txt @@ -0,0 +1,507 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf,len=506 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='04097v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='AP] 10 Jan 2023 STABILITY OF HARDY-LITTLEWOOD-SOBOLEV INEQUALITIES WITH EXPLICIT LOWER BOUNDS LU CHEN, GUOZHEN LU, AND HANLI TANG Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' In this paper, we establish the stability for the Hardy-Littlewood- Sobolev (HLS) inequalities with explicit lower bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' By establishing the relation between the stability of HLS inequalities and the stability of fractional Sobolev inequalities, we also give the stability of the fractional Sobolev inequalities with the lower bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' This extends the stability of Sobolev inequalities with the explicit lower bounds established by Dol- beault, Esteban, Figalli, Frank and Loss in [14] to the fractional order case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Our proofs are based on the competing symmetries, the continuous Steiner symmetrization inequality for the HLS integral and the dual stability the- ory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Introduction The main purpose of this paper is to give the lower bound estimate of the sharp constants of the stability of the Hardy-Littlewood-Sobolev (HLS) inequality and fractional Sobolev inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' The classical Hardy-Littlewood-Sobolev (HLS) inequality ([22, 28]) states � Rn � Rn |x − y|−(n−2s)f(x)g(y)dxdy ≤ Cn,p,q′∥f∥Lq′(Rn)∥g∥Lp(Rn), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='1) with 1 < q′, p < ∞, 0 < s < n 2 and 1 q′ + 1 p − 2s n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Lieb and Loss [25] applied the layer cake representation formula to give an explicit upper bound for the sharp constant Cn,p,q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' More precisely, they showed that the best constant Cn,p,q′ satisfies the following estimate Cn,p,q′ ≤ n n − λ � π λ 2 Γ(1 + n 2) � λ n 1 q′p � ( λq′ n(q′ − 1)) λ n + ( λp n(p − 1)) λ n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' In the special diagonal case q′ = p = 2n n+2 (s = 1), Aubin [1] and Talenti [30] derived the sharp constants of HLS inequality by classifying the extremal of classical Sobolev inequality which is a dual form of HLS inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' For the HLS inequality of general diagonal case, Lieb [24] classified the extremal function of HLS inequality and obtained the best constant Cn,p,q′ = π λ 2 Γ( n 2 − λ 2) Γ(n − λ 2) �Γ(n) Γ( n 2) �1− λ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Recently, the authors of [10, 17, 18, 19] developed a rearrangement-free ar- gument to obtain the sharp HLS inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Weighted HLS inequalities, also 1 2 LU CHEN, GUOZHEN LU, AND HANLI TANG known as the Stein-Weiss inequalities [29], and their inequalities, sharp con- stants and existence of extremal functions have also been studied in Euclidean spaces and the Heisenberg group, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=', [4, 13, 20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' The stability of the fractional Sobolev inequality states that there exists some positive constant Cn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='s such that for any U ∈ ˙Hs(Rn),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' there holds ��(−∆)s/2U ��2 2 − Ss,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='n∥U∥2 2n n−2s ≥ Cn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='s inf h∈MS ∥(−∆)s/2(U − h)∥2 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' where ˙Hs(Rn) denotes the s-order homogenous Sobolev space in Rn: the com- pletion of C∞ c (Rn) under the norm � � Rn |(−∆) s 2U|2dx � 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' MS := � c( 2b b2 + |x − a|2) n−2s 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' a ∈ Rn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' b > 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' c ∈ R � is the extremal set of the fractional Sobolev inequality and Ss,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='n = (4π)s Γ( n+2s 2 ) Γ( n−2s 2 ) �Γ( n 2) Γ(n) �2s/n = Γ( n+2s 2 ) Γ( n−2s 2 ) |Sn|2s/n (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='2) is the sharp constant of the fractional Sobolev inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' This kind of stability inequality was proposed in Brezis and Lieb’s work in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Bianchi and Egnell in [3] first obtained the stability of the first order Sobolev inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Stability of Sobolev inequality for s = 2 and even integers s < n 2 were established by the second author and Wei [26], and by Bartsch, Weth and Willem [2] respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Chen, Frank and Weth [11] established the stability of Sobolev inequality for all 0 < s < n/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' It should be noted that although the fractional Sobolev inequality is equivalent to the HLS inequality of diagonal case, their stabilities are not equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Carlen [8] developed a duality stability theory based on the quantitative convexity and deduce the following stability of HLS inequality from the stability of fractional Sobolev inequality: There exists some constant � Cn,s such that for any g ∈ L 2n n+2s(Rn), there holds ∥g∥2 2n n+2s − S−1 n,s∥(−∆)−s/2g∥2 2 ≥ � Cn,s inf h∈MHLS ∥g − h∥2 2n n+2s, where MHLS := � c( 2b b2 + |x − a|2) n+2s 2 , a ∈ Rn, b > 0, c ∈ R � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Stability of Sobolev inequality is proved by establishing the local stabil- ity of Sobolev inequality based on the spectrum analysis of elliptic or high- order elliptic operator and using the Lions concentration compactness tech- nique to obtain global stability of Sobolev inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' However, this method does not tell us any lower bound information about the sharp constant of stability of Sobolev inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Recently, Dolbeault, Esteban, Figalli, Frank STABILITY OF HLS INEQUALITY WITH LOWER BOUNDS 3 and Loss in [14] first obtained the stability for first order Sobolev inequal- ity with the explicit lower bound by competing symmetries, the continu- ous Steiner symmetrization inequality (Poly´a-Szeg¨o inequality) for L2 inte- gral of gradient u and the fact ∥∇u∥2 L2 = ∥∇u+∥2 L2 + ∥∇u−∥2 L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' However this method is not applicable to the stability of fractional Sobolev inequality (s < n 2) because of the absence of continuous Steiner symmetrization inequal- ity for L2 integral of high-order derivatives and the identity ∥(∆)s/2u∥2 L2 = ∥(∆)s/2(u+)∥2 L2 + ∥(∆)s/2(u−)∥2 L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' In this paper, we will overcome these dif- ficulties and derive the stability of fractional Sobolev inequality with explicit lower bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Instead of directly considering the stability of Sobolev inequal- ity, we will first establish the stability for the HLS inequality by competing symmetries, the continuous Steiner symmetrization inequality for HLS integral and local dual stability theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' We also note that some relationship between the global geometric and functional inequalities and their local version of sta- bility inequalities in a certain sense has been explored in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Our first result states: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Let SHLS(g) denote the HLS stability functional given by SHLS(g) = ∥g∥2 2n n+2s − S−1 n,s∥(−∆)−s/2g∥2 2 inf h∈MHLS ∥g − h∥2 2n n+2s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='3) Denote by Kn,s = sup 0<δ<1 δ 2 n−2s n+2s min{ m(2δ) Sn,s n−2s n+2s, 1}, where m(δ) = 4s n + 2s + 2 − 2 2∗s 2∗ s � k=3 2∗ s(2∗ s − 1) · · ·(2∗ s − k + 1) k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' l(δ)k−2 if 2∗ s = 2n n−2s is an integer and otherwise m(δ) = 4s n + 2s + 2 − 2 2∗s [2∗ s] � k=3 2∗ s(2∗ s − 1) · · ·(2∗ s − k + 1) k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' l(δ)k−2 − 2 2∗s l(δ)2∗ s−2, and l(δ) = � δ 1−δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Then there holds inf g∈L 2n n+2s (Rn)\\MHLS SHLS(g) ≥ 1 2 min{Kn,s, min{2 n+2s n − 2, 1}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' By the dual stability theory, we can deduce the following stability of frac- tional Sobolev inequality with explicit lower bounds from the stability of HLS inequality with explicit lower bounds, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=', Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='1: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' For f ∈ ˙Hs(Rn) \\ MS, denote SS(f) the Sobolev stability functional given by SS(f) = Sn,s∥(−∆)s/2f∥2 2 − ∥f∥2 2∗ inf h∈MS ∥(−∆)s/2(f − h)∥2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='4) 4 LU CHEN, GUOZHEN LU, AND HANLI TANG Then inf f∈ ˙Hs(Rn)\\MS SS(f) ≥ Sn,s min{Kn,s, min{2 n+2s n − 2, 1}} 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' We also note that K¨onig in [23] proved that inf f∈ ˙Hs(Rn)\\MS SS(f) < 4s n + 2 + 2s by doing the third-order Taylor expansion to fractional Sobolev functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Section 2 is devoted to some preliminar- ies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' In Section 3, we will give the stability of HLS inequality with explicit lower bounds using competing symmetries, the continuous Steiner symmetrization inequality for the HLS integral and the local dual stability theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' In Section 4, we will establish the stability of fractional Sobolev inequality with explicit lower bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Preliminaries In this section, we will state some tools, including competing symmetries theorem, a continuous rearrangement flow and expansions with remainder term and so on, which will be used in our proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Let f ∈ Lp(Rn), 1 < p < ∞ and fk = (RU)kf, k ∈ N, where Rf = f ∗ is the decreasing rearrangement of f and (Uf)(x) = � 2 |x − en|2 � n p f � x1 |x − en|2, · · · , xn−1 |x − en|2, |x|2 − 1 |x − en|2 � , en = (0, · · · , 0, 1) ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Carlen and Loss [9] proved the following competing symmetry theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Let 0 ≤ f ∈ Lp (1 < p < ∞), fk = (RU)kf and h(x) = ∥f∥p|Sn|−1/p � 2 1+|x|2 �n/p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Then lim k→∞ ∥fk(x) − h(x)∥p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' A continuous rearrangement flow which interpolates between a function and its symmetric decreasing rearrangement introduced in [14] based on Brock’s flow ( [6], [7] plays an important part in our proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' More specifically, there exists a flow fτ, τ ∈ [0, ∞], such that f0 = f, f∞ = f ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' And if 0 ≤ f ∈ Lp(Rn) for some 1 ≤ p < ∞, then τ → fτ is continuous in Lp(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' In our setting we need to prove τ → inf h∈MHLS ∥fτ −h∥ 2n n+2s is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Let 0 ≤ f ∈ L 2n n+2s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Then the function τ → inf h∈MHLS ∥fτ − h∥ 2n n+2s is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' STABILITY OF HLS INEQUALITY WITH LOWER BOUNDS 5 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Let us prove inf h∈MHLS ∥fτ − h∥ 2n n+2s → inf h∈MHLS ∥fτ0 − h∥ 2n n+2s as τ → τ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' For any ε > 0, there exists a g1 ∈ MHLS such that inf h∈MHLS ∥fτ − h∥ 2n n+2s ≥ ∥fτ − g1∥ 2n n+2s − ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Then inf h∈MHLS ∥fτ − h∥ 2n n+2s − inf h∈MHLS ∥fτ0 − h∥ 2n n+2s ≥ ∥fτ − g1∥ 2n n+2s − ∥fτ0 − g1∥ 2n n+2s − ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='1) On the other hand, there exist a g0 ∈ MHLS such that inf h∈MHLS ∥fτ0 − h∥ 2n n+2s ≥ ∥fτ0 − g0∥ 2n n+2s − ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Thus inf h∈MHLS ∥fτ − h∥ 2n n+2s − inf h∈MHLS ∥fτ0 − h∥ 2n n+2s ≤ ∥fτ − g0∥ 2n n+2s − ∥fτ0 − g0∥ 2n n+2s + ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='2) By the triangle inequality and the fact that τ → fτ is continuous in L 2n n+2s(Rn), we have |∥fτ − gi∥ 2n n+2s − ∥fτ0 − gi∥ 2n n+2s| ≤ ∥fτ − fτ0∥ 2n n+2s → 0, τ → τ0 i = 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='3) Combining (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='1), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='2) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='3), we complete the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' □ The following expansions of ∥u + r∥2 2∗s are also needed in our proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Let u, r ∈ L2∗ s(Rn), u + r ≥ 0 and u ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' If 2∗ s is not an integer, then ∥u + r∥2 2∗s ≤ ∥u∥2 2∗s + 2 2∗s [2∗ s] � k=1 2∗ s(2∗ s − 1) · · ·(2∗ s − k + 1) k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' ∥u∥2−2∗ s 2∗s � Rn u2∗ s−krkdx + 2 2∗ s ∥u∥2−2∗ s 2∗s ∥r∥2∗ s 2∗s, where [2∗ s] is the integer part of 2∗ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' If 2∗ s is an integer, then ∥u + r∥2 2∗s ≤ ∥u∥2 2∗s + 2 2∗ s [2∗ s] � k=1 2∗ s(2∗ s − 1) · · ·(2∗ s − k + 1) k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' ∥u∥2−2∗ s 2∗s � Rn u2∗ s−krkdx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' In order to prove Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='2, we need the following technical lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' (1) For all x ≥ −1, q ≥ 1, q is not an integer, (1 + x)q ≤ 1 + [q] � k=1 q(q − 1) · · ·(q − k + 1) k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' xk + |x|q, where [q] is the integer part of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' 6 LU CHEN, GUOZHEN LU, AND HANLI TANG (2) For all x ≥ −1, q ≥ 1, q is an integer, (1 + x)q = 1 + q � k=1 q(q − 1) · · ·(q − k + 1) k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' xk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' We only need to consider the case when q is not an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Let f(x) = (1 + x)q − 1 − [q] � k=1 q(q − 1) · · ·(q − k + 1) k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' xk − |x|q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' We first prove that f(x) ≤ 0 when x ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Since f ′(x) = q(1 + x)q−1 − [q] � k=1 q(q − 1) · · ·(q − k + 1) (k − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' xk−1 − qxq−1, · ·, f ([q])(x) = q(q − 1) · · ·(q − [q] + 1) � (1 + x)q−[q] − 1 − xq−[q]� , then f(0) = f ′(0) = · · · = f ([q])(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Using the fundamental inequality (1 + x)α ≤ 1 + xα for 0 < α < 1 and x ≥ 0, we obtain f ([q])(x) ≤ 0 (x > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Then f ([q]−1)(x) is decreasing on (0, +∞), which implies f ([q]−1)(x) ≤ f ([q]−1)(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Repeat the deduction and we can obtain f(x) ≤ 0 when x ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' When −1 ≤ x ≤ 0, we have f ′(x) = q(1 + x)q−1 − [q] � k=1 q(q − 1) · · ·(q − k + 1) k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' xk−1 + q(−x)q−1, · · , f ([q])(x) = q(q − 1) · · · (q − [q] + 1) � (1 + x)q−[q] − 1 − (−1)[q](−x)q−[q]� , and there still holds f(0) = f ′(0) = · · · = f ([q])(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' If [q] is even, using the fundamental inequality (1 + x)α ≤ 1 + (−x)α, 0 < α < 1, −1 ≤ x ≤ 0 we know f ([q])(x) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Then f ([q]−1)(x) is decreasing on [−1, 0], which implies f ([q]−1)(x) ≥ f ([q]−1)(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Thus f ([q]−2)(x) is increasing on [−1, 0], which means f ([q]−2)(x) ≤ f ([q]−2)(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Repeat the deduction and we can obtain f (2k)(x) ≤ 0, f (2k+1)(x) ≥ 0 when k = 0, 1, · · · , [q] 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Now let us deal with the case when [q] is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Since the function h(x) = (1 + x)α − 1 + (−x)α (0 < α < 0) satisfy that h′(x) = α[(1 + x)α−1 − (−x)α−1] is nonnegative on [−1, −1/2) and nonpositive on (−1/2, 0], then f ([q])(x) is increasing on [−1, −1/2) and de- creasing on (−1/2, 0], which implies f ([q])(x) ≥ f ([q])(0) = f ([q])(−1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Thus STABILITY OF HLS INEQUALITY WITH LOWER BOUNDS 7 f ([q]−1)(x) is increasing on [−1, 0] which means f ([q]−1)(x) ≤ f ([q]−1)(0) = 0, x ∈ [−1, 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Then f ([q]−2)(x) is decreasing on [−1, 0], which means f ([q]−2)(x) ≥ f [q]−2(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Repeat the deduction and we can obtain f (2k)(x) ≤ 0, f (2k+1)(x) ≥ 0 when k = 0, 1, · · · , [q]−1 2 , which complete the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' □ Now let us prove Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Here we only state the proof when 2∗ s is not an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' By Proposition 1, we derive that (u + r)2∗ s ≤ u2∗ s + [2∗ s] � k=1 2∗ s(2∗ s − 1) · · · (2∗ s − k + 1) k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' u2∗ s−krk + |r|2∗ s, thus � Rn(u+r)2∗ sdx ≤ � Rn u2∗ sdx+ [2∗ s] � k=1 2∗ s(2∗ s − 1) · · · (2∗ s − k + 1) k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' � Rn u2∗ s−krkdx � Rn |r|2∗ sdx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Using the inequality (1+x)α ≤ 1+αx for x ≥ −1 and 0 < α < 1, we complete the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Stability of the Hardy-Littlewood-Sobolev inequality In this section, we will set up the stability of the Hardy-Littlewood-Sobolev inequality with explicit constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' First we will establish the local version of the stability of the fractional Sobolev inequality for nonnegative functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Then, using the dual method from [8] by Carlen, we can deduce the local stability for the HLS inequality for nonnegative function (where the aim function is close to the manifold of the HLS optimizers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' When the aim function is positive and away from the manifold of HLS optimizers, we will handle them by the competing symmetries and Block’s flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' At last, we will deal with the stability of HLS inequality when the aim function is not necessarily non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Local stability of fractional Sobolev inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' In the subsection we will set up the local stability of fractional Sobolev inequality with explicit lower bounds (Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='2) for 0 < s < n 2, while the case s = 1 was proved in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Without loss of generality, we only give the proof when 2∗ s is an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Let 0 ≤ f ∈ ˙Hs(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' It is well known that there exists an 0 ≤ u ∈ MS such that ∥(−∆)s/2(f − u)∥2 2 = inf h∈MS ∥(−∆)s/2(f − h)∥2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Then r = f − u satisfies � Rn[(−∆)s/2u][(−∆)s/2r]dx = 0, and � Rn u2∗ s−1rdx = 0, since (−∆)su = cuu2∗ s−1 with cu = ∥(−∆)s/2u∥2 2 ∥u∥2 2∗s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Then ∥(−∆)s/2f∥2 2 −Sn,s∥f∥2 2∗s = ∥(−∆)s/2u∥2 2 +∥(−∆)s/2r∥2 2 −Sn,s∥u+r∥2 2∗s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='1) 8 LU CHEN, GUOZHEN LU, AND HANLI TANG Next we will estimate the Sobolev deficit ∥(−∆)s/2f∥2 2 − Sn,s∥f∥2 2∗ by the expansion from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='2 and the following spectral gap inequalities which are due to Rey [27] (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='1)) and Esposito [16] (Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='1) for s = 1, to De Nitti and K¨onig [15] (Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='4) for 0 < s < n/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Given f ∈ ˙Hs(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Let u ∈ MS such that ∥(−∆)s/2(f − u)∥2 2 = inf h∈MS ∥(−∆)s/2(f − h)∥2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Then r = f − u satisfies ∥(−∆)s/2r∥2 2 − (2∗ s − 1)Sn,s∥u∥2−2∗ s 2∗s � Rn u2∗ s−2r2dx ≥ 4s n + 2s + 2∥(−∆)s/2r∥2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Now by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='1) and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='2, we have ∥(−∆)s/2f∥2 2 − Sn,s∥f∥2 2∗s ≥ ∥(−∆)s/2r∥2 2 − (2∗ s − 1)Sn,s∥u∥2−2∗ s 2∗s � Rn u2∗ s−2r2dx − Sn,s \uf8f1 \uf8f2 \uf8f3 2 2∗s [2∗ s] � k=3 2∗ s(2∗ s − 1) · · · (2∗ s − k + 1) k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' ∥u∥2−2∗ s 2∗s � Rn u2∗ s−krkdx + 2 2∗s ∥u∥2−2∗ s 2∗s ∥r∥2∗ s 2∗s \uf8fc \uf8fd \uf8fe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='2) By H¨older’s inequality and sharp Sobolev inequality, there holds ∥u∥2−2∗ s 2∗s � Rn u2∗ s−krkdx ≤ ∥u∥2−k 2∗s ∥r∥k 2∗s ≤ 1 Sn,s �∥r∥2∗s ∥u∥2∗s �k−2 ∥(−∆)s/2r∥2 2, and ∥u∥2−2∗ s 2∗s ∥r∥2∗ s 2∗s ≤ 1 Sn,s �∥r∥2∗s ∥u∥2∗s �2∗−2 ∥(−∆)s/2r∥2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Therefore combining Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='1, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='2) and the above estimates, we get ∥(−∆)s/2f∥2 2 − Sn,s∥f∥2 2∗s ∥(−∆)s/2r∥2 2 ≥ \uf8f1 \uf8f2 \uf8f3 4s n + 2s + 2 − 2 2∗s [2∗ s] � k=3 2∗ s(2∗ s − 1) · · · (2∗ s − k + 1) k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' �∥r∥2∗s ∥u∥2∗s �k−2 − 2 2∗s �∥r∥2∗s ∥u∥2∗s �2∗ s−2 \uf8fc \uf8fd \uf8fe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='3) Now we are in the position to prove the local stability of Sobolev inequality for nonnegative functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Let l(δ) = � δ 1−δ and define ν(δ) = inf � SSob(f) : 0 ≤ f ∈ ˙Hs(Rn) \\ MS, inf g∈MS ∥(−∆)s/2(f − g)∥2 2 ≤ δ∥(−∆)s/2f∥2 2 � , and m(δ) = 4s n + 2s + 2− 2 2∗ s 2∗ s � k=3 2∗ s(2∗ s − 1) · · ·(2∗ s − k + 1) k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' l(δ)k−2, when 2∗ s is an integer, STABILITY OF HLS INEQUALITY WITH LOWER BOUNDS 9 otherwise m(δ) = 4s n + 2s + 2 − 2 2∗ [2∗ s] � k=3 2∗ s(2∗ s − 1) · · ·(2∗ − k + 1) k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' l(δ)k−2 − 2 2∗s l(δ)2∗ s−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' With the above notations, we have ν(δ) ≥ m(δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Since ∥(−∆)s/2f∥2 2 = ∥(−∆)s/2r∥2 2 + ∥(−∆)s/2u∥2 2, then ∥r∥2∗s ∥u∥2∗s ≤ ∥(−∆)s/2r∥2 ∥(−∆)s/2u∥2 = ∥(−∆)s/2r∥2 ∥(−∆)s/2f∥2 1 � 1 − ∥(−∆)s/2r∥2 2 ∥(−∆)s/2f∥2 2 ≤ � δ 1 − δ = l(δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='4) Therefor by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='3) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='4) ν(δ) ≥ \uf8f1 \uf8f2 \uf8f3 4s n + 2s + 2 − 2 2∗ s [2∗ s] � k=3 2∗ s(2∗ s − 1) · · ·(2∗ s − k + 1) k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' l(δ)k−2 − 2 2∗ s l(δ)2∗ s−2 \uf8fc \uf8fd \uf8fe , which completes the proof of local stability of fractional Sobolev inequality with explicit lower bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Stability of HLS inequality for positive functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' In this subsec- tion, we will consider the stability of HLS inequality for positive functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Choose δ small enough, and define µ(δ) = inf{SHLS(g) : 0 ≤ g ∈ L 2n n+2s(Rn)\\MHLS, inf h∈MHLS ∥g−h∥2 2n n+2s ≤ δ∥g∥2 2n n+2s}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' First we obtain the local stability of HLS inequality by the following dual lemma from Carlen [8] and the local stability of Sobolev inequality for non- negative functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' If SS(f) ≥ m(δ) for all nonnegative function f with inf h∈MS ∥(−∆)s/2(f − h)∥2 2 ≤ δ∥(−∆)s/2f∥2 2, then SHLS(g) ≥ 1 2 n − 2s n + 2s min{m(δ) Sn,s n − 2s n + 2s, 1}, for all 0 < g ∈ L 2n n+2s(Rn) \\ MHLS satisfying inf h∈MHLS ∥g − h∥2 2n n+2s ≤ δ 2∥g∥2 2n n+2s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' If ∥g∥2 2n n+2s ≥ 2S−1 n,s∥(−∆)−s/2g∥2 2, then ∥g∥2 2n n+2s − S−1 n,s∥(−∆)−s/2g∥2 2 ≥ 1 2∥g∥2 2n n+2s ≥ 1 2 inf h∈MHLS ∥g − h∥2 2n n+2s, thus SHLS(g) ≥ 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' When ∥g∥2 2n n+2s ≤ 2S−1 n,s∥(−∆)−s/2g∥2 2, using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='4 in [8], we can obtain SHLS(g) ≥ 1 2 n−2s n+2s min{ m(δ) Sn,s n−2s n+2s, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Since 1 2 n−2s n+2s min{ m(δ) Sn,s n−2s n+2s, 1} ≤ 10 LU CHEN, GUOZHEN LU, AND HANLI TANG 1 2, then SHLS(g) ≥ 1 2 n − 2s n + 2s min{m(δ) Sn,s n − 2s n + 2s, 1} if inf h∈MHLS ∥g − h∥2 2n n+2s ≤ δ 2∥g∥2 2n n+2s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' This accomplishes the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' □ Next let us handle the stability of HLS inequality when the nonnegative function g satisfies inf h∈MHLS ∥g − h∥2 2n n+2s > δ∥g∥2 2n n+2s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' For any 0 ≤ g ∈ L 2n n+2s(Rn) satisfying inf h∈MHLS ∥g − h∥2 2n n+2s > δ∥g∥2 2n n+2s, there holds SHLS(g) ≥ δµ(δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Assume that 0 ≤ g ∈ L 2n n+2s(Rn) satisfying inf h∈M ∥g − h∥2 2n n+2s > δ∥g∥2 2n n+2s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Let gk = (RU)kg (see the definition of RU in section 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='1 we know ∥gk − hg∥ 2n n+2s → 0, k → ∞, where hg = ∥g∥ 2n n+2s|Sn|− n−2s 2n ( 2 1+|x|2) n+2s 2 ∈ MHLS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' It is well known that k → ∥(−∆)− s 2gk∥2 2 is increasing by Riesz rearrangement inequality and ∥gk∥ 2n n+2s = ∥g∥ 2n n+2s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Thus SHLS(g) ≥ ∥g∥2 2n n+2s − S−1 n,s∥(−∆)−s/2g∥2 2 ∥g∥2 2n n+2s ≥ 1 − S−1 n,s∥(−∆)−s/2g∥2 2 ∥g∥2 2n n+2s ≥ ∥gk∥2 2n n+2s − S−1 n,s∥(−∆)−s/2gk∥2 2 ∥gk∥2 2n n+2s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='5) Since ∥gk − hg∥ 2n n+2s → 0 as k → ∞ and hg ∈ MHLS, then there exist a k0 ∈ N such that inf h∈MHLS ∥gk0 − h∥2 2n n+2s > δ∥gk0∥2 2n n+2s and inf h∈MHLS ∥gk0+1 − h∥2 2n n+2s ≤ δ∥gk0+1∥2 2n n+2s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Denote g0 = Ugk0, g∞ = gk0+1, then inf h∈MHLS ∥g0 − h∥2 2n n+2s = inf h∈MHLS ∥gk0 − h∥2 2n n+2s > δ∥gk0∥2 2n n+2s = δ∥g0∥2 2n n+2s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Now using the continuous rearrangement flow gτ (0 ≤ τ ≤ ∞) introduced in section 2, we conclude that gτ satisfies ∥(−∆)− s 2gτ∥2 2 ≥ ∥(−∆)− s 2g0∥2 2 = ∥(−∆)− s 2gk0∥2 2, and ∥gτ∥ 2n n+2s = ∥g0∥ 2n n+2s = ∥g∥ 2n n+2s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' STABILITY OF HLS INEQUALITY WITH LOWER BOUNDS 11 Since τ → inf h∈MHLS ∥gτ − h∥2 2n n+2s is continuous by lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='1, then there exists τ0 ∈ (0, ∞) such that inf h∈MHLS ∥gτ0 − h∥2 2n n+2s = δ∥gτ0∥2 2n n+2s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='6) Therefore by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='5) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='6),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' SHLS(g) ≥ ∥g0∥2 2n n+2s − S−1 n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='s∥(−∆)−s/2g0∥2 2 ∥g0∥2 2n n+2s ≥ ∥gτ0∥2 2n n+2s − S−1 n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='s∥(−∆)−s/2gτ0∥2 2 ∥gτ0∥2 2n n+2s = δ ∥gτ0∥2 2n n+2s − S−1 n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='s∥(−∆)−s/2gτ0∥2 2 inf h∈MHLS ∥gτ0 − h∥2 2n n+2s ≥ δµ(δ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' where in the second inequality we use the following Riesz’s inequality for con- tinuous convex rearrangement (see Appendix A in [14]): For nonnegative func- tions f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' there holds �� Rn×Rn fτ(x)gτ(y) |x − y|n−2sdxdy ≥ �� Rn×Rn f(x)g(y) |x − y|n−2sdxdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' This proves Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' □ According to the definition of µ(δ), through Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='3, we deduce that µ(δ) ≥ 1 2 n − 2s n + 2s min{m(2δ) Sn,s n − 2s n + 2s, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Combining this and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='4, we derive that for any 0 ≤ f ∈ L 2n n+2s(Rn)\\ MHLS, there holds SHLS(g) ≥ δ 2 n − 2s n + 2s min{m(2δ) Sn,s n − 2s n + 2s, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Therefor we have established the stability of HLS inequality with explicit lower bounds for nonnegative functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Stability of HLS inequality with explicit lower bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' In this subsection, we will prove the stability of HLS inequality with explicit lower bounds for general function f ∈ L 2n n+2s(Rn) \\ MHLS, namely we shall give the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Let us denote by CHLS the optimal constant for stability of HLS inequality and denote by Cpos HLS the optimal constant in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='3) when restricted to nonnegative functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' We first state a relationship between these two optimal constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' CHLS ≥ 1 2 min{Cpos HLS, min{2 n+2s n − 2, 1}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' 12 LU CHEN, GUOZHEN LU, AND HANLI TANG Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Let D(g) = ∥g∥2 2n n+2s −S−1 n,s∥(−∆)−s/2f∥2 2 and f± denote the positive and negative parts of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Then D(f) ≥ ∥g∥2 2n n+2s − S−1 n,s∥(−∆)−s/2g+∥2 2 − S−1 n,s∥(−∆)−s/2g−∥2 2 = D(g+) + D(g−) + ∥g∥2 2n n+2s − ∥g+∥2 2n n+2s − ∥g−∥2 2n n+2s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='7) Without loss of generality, we may assume ∥g∥2 2n n+2s = 1, and m = ∥g−∥ 2n n+2s 2n n+2s ∈ [0, 1/2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Let h(m) = 1 − m n+2s n − (1 − m) n+2s n and t(m) = 1 − (1 − m) n+2s n − 2(1 − (1/2) n+2s n )m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' It is easy to check t(0) = t(1/2) = 0 and t′′(m) ≤ 0 on [0, 1/2], then t(m) ≥ 0 on [0, 1/2], which means 1 − (1 − m) n+2s n ≥ 2(1 − (1/2) n+2s n )m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Thus h(m) ≥ 2(1 − (1/2) n+2s n )m − m n+2s n , m ∈ [0, 1/2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Since the function 2(1 − (1/2) n+2s n )m− 2s n − 1 is decreasing on [0, 1/2], then h(m) ≥ (2 n+2s n − 2)m n+2s n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='8) By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='8) and sharp HLS inequality we have ∥g∥2 2n n+2s−∥g+∥2 2n n+2s−∥g−∥2 2n n+2s ≥ (2 n+2s n −2)∥g−∥2 2n n+2s ≥ (2 n+2s n −2)S−1 n,s∥(−∆)s/2g−∥2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Along with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='7), D(g) ≥ D(g+) + D(g−) + (2 n+2s n − 2)S−1 n,s∥(−∆)−s/2g−∥2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Let h+ ∈ MHLS such that ∥g+ − h+∥ 2n n+2s = inf h∈MHLS ∥g+ − h∥ 2n n+2s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Since D(g−) + S−1 n,s∥(−∆)−s/2(g−)∥2 2 = ∥g−∥2 2n n+2s, thus D(g) ≥ D(g+) + min{2 n+2s n − 2, 1}∥g−∥2 2n n+2s ≥ Cpos HLS∥g+ − h+∥2 2n n+2s + min{2 n+2s n − 2, 1}∥g−∥2 2n n+2s ≥ min{Cpos HLS, min{2 n+2s n − 2, 1}} � ∥g+ − h+∥ 2n n+2s + ∥g−∥ 2n n+2s �2 2 ≥ 1 2 min{Cpos HLS, min{2 n+2s n − 2, 1}}∥g − h+∥2 2n n+2s ≥ 1 2 min{Cpos HLS, min{2 n+2s n − 2, 1}} inf h∈MHLS ∥g − h∥2 2n n+2s which completes the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' □ STABILITY OF HLS INEQUALITY WITH LOWER BOUNDS 13 Now we are in position to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Denote by Kn,s = sup 0<δ<1 δ 2 n − 2s n + 2s min{m(2δ) Sn,s n − 2s n + 2s, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' We have already established SHLS(g) ≥ Kn,s for all 0 ≤ g ∈ L 2n n+2s(Rn) \\ MHLS in subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' This together with Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='5 yields that for all g ∈ L 2n n+2s(Rn) \\ MHLS, there holds SHLS(g) ≥ 1 2 min{Kn,s, min{2 n+2s n − 2, 1}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Then we finish the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' HLS stability implies Sobolev stability In this section, we will prove the stability of fractional Sobolev inequality with explicit lower bounds for all 0 < s < n/2 from the stability of Hardy- Littlewood-Sobolev inequality with explicit lower bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Let f ∈ ˙Hs(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Define F(f) = Sn,s∥(−∆)s/2f∥2 2, E(f) = ∥f∥2 2∗s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Then the Legendre transform F ∗ of a convex functional F : ˙Hs(Rn) → [0, +∞) defined on ˙H−s(Rn) is given by F ∗(g) = sup f∈Hs(Rn) {2 � Rn f(x)g(x)dx − F(f)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' A simple calculation gives F ∗(g) = S−1 n,s∥(−∆)−s/2g∥2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Similarly, the Le- gendre transform E∗ of a convex functional E : L2∗ s(Rn) → [0, +∞) defined on L 2n n+2s(Rn) is given by E∗(g) = sup f∈L2∗s (Rn) {2 � Rn f(x)g(x)dx − E(f)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Obviously, E∗(g) = ∥g∥2 2n n+2s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Choose g = ∥f∥2−2∗ s 2∗ |f|2∗ s−1sgn(f) and f1 = S−1 n,s(−∆)−sg, we can check that E(f) + E∗(g) = 2 � Rn fgdx (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='1) and F(f) = Sn,s∥(−∆)s/2f∥2 2 = Sn,s∥(−∆)s/2f1∥2 2 + Sn,s∥(−∆)s/2(f − f1)∥2 2 + 2 � Rn(f − f1)(Sn,s(−∆)sf1)dx = S−1 n,s∥(−∆)−s/2g∥2 2 + 2 � Rn(f − S−1 n,s(−∆)−sg)gdx + Sn,s∥(−∆)s/2(f − f1)∥2 2 = 2 � Rn fgdx − F ∗(g) + Sn,s∥(−∆)s/2(f − f1)∥2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='2) 14 LU CHEN, GUOZHEN LU, AND HANLI TANG Combining (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='1) with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='2), we have F(f) − E(f) = E∗(g) − F ∗(g) + Sn,s∥(−∆)s/2(f − f1)∥2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='3) Since we have already proved the stability of HLS inequality in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='1 E∗(g) − F ∗(g) ≥ 1 2 min{Kn,s, min{2 n+2s n − 2, 1}} inf h∈MHLS ∥g − h∥2 2n n+2s, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='4) then for any ǫ > 0, there exists a g0 ∈ MHLS such that E∗(g)−F ∗(g) ≥ 1 2 min{Kn,s, min{2 n+2s n −2, 1}}∥g−g0∥2 2n n+2s−ε+Sn,s∥(−∆)s/2(f−f1)∥2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content=' Denote by kn,s = 1 2 min{Kn,s, min{2 n+2s n − 2, 1}}, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='3), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='4), sharp HLS inequality, (−∆)−sg0 ∈ MS, we derive F(f) − E(f) ≥ kn,sS−1 n,s∥(−∆)−s/2(g − g0)∥2 2 − ε + Sn,s∥(−∆)s/2(f − f1)∥2 2 = kn,s∥S−1/2 n,s (−∆)−s/2(g − g0)∥2 2 − ε + ∥S1/2 n,s (−∆)s/2f − S−1/2 n,s (−∆)−s/2g∥2 2 ≥ kn,s 2 ∥S1/2 n,s (−∆)s/2f − S−1/2 n,s (−∆)−s/2g0∥2 2 − ε ≥ Sn,skn,s 2 inf h∈MS ∥(−∆)s/2(f − h)∥2 2 − ε, which completes the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} 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Laboratory of Mathematics and Complex Systems (Ministry of Education), School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, China Email address: hltang@bnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} +page_content='cn' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNE2T4oBgHgl3EQfwAiL/content/2301.04097v1.pdf'} diff --git a/btE1T4oBgHgl3EQfKwNx/content/tmp_files/2301.02968v1.pdf.txt b/btE1T4oBgHgl3EQfKwNx/content/tmp_files/2301.02968v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..b041fd1bb21700d4261490fd49d5b214afc5f0f7 --- /dev/null +++ b/btE1T4oBgHgl3EQfKwNx/content/tmp_files/2301.02968v1.pdf.txt @@ -0,0 +1,397 @@ +Resistive Read-out in Thin Silicon Sensors with +Internal Gain +N. Cartiglia1, F. Moscatelli4,6, R. Arcidiacono1,2, P. Asenov5,6, M. +Costa1,3, T. Croci6, M. Ferrero1, A. Fondacci5,6, L. Lanteri1,3, L. +Menzio1,3, A. Morozzi6, R. Mulargia1,3, D.Passeri5,6, F. Siviero1, V. +Sola1,3, M. Tornago1,3 +1 INFN, Torino, Italy, 2 Universita del Piemonte orientale, Italy 3 Universita di Torino, +Italy, 4 CNR-IOM, Perugia, Italy, 5 Universita di Perugia, Italy. 6 INFN Perugia, Italy. +E-mail: cartiglia@to.infn.it +Two design innovations, low-gain avalanche (Low-Gain Avalance Diode, LGAD) and resistive +read-out (Resistive Silicon Detector, RSD), have brought strong performance improvements +to silicon sensors. Large signals, due to the added gain mechanism, lead to improved temporal +precision, while charge sharing, introduced by resistive read-out, allows for achieving excellent +spatial resolution even with large pixels. LGAD- and RSD- based silicon sensors are now +adopted, or considered, in several future experiments and are the basis for almost every next +4D-trackers. New results obtained with sensors belonging to the second FBK production of +RSD (RSD2) demonstrate how a combined resolution of 30 ps and 30 µm can be obtained +with pixels as large as 1 × 1 mm2. +KEYWORDS: resistive read-out, silicon sensors, low-gain avalanche diode +1. +Introduction +In the past few years, the performance capabilities of silicon sensors in terms of combined +spatial and temporal resolutions have improved significantly. This evolution is due to the +introduction of two innovations in the design of silicon sensors: (i) Low-Gain Avalanche +Diode and (ii) Resistive Read-out. +The Low-Gain Avalanche Diode [1,2] design has been first introduced to compensate for +the loss of signal due to charge trapping in irradiated sensors. However, this design found its +main application in the field of precision timing, with the introduction of Ultra-Fast Silicon +Detector (UFSD) [3]. This R&D has spurred a strong evolution in the field of accurate timing +using silicon detectors. +Resistive read-out was first proposed to achieve an LGAD design with 100% fill factor [4] +(the so-called AC-LGAD). Then, it was subsequently recognized to lead to excellent spatial +precision with a concurrent reduction of the number of read-out electrodes [5, 6]. Sensors +based on resistive read-out are called RSDs (Resistive Silicon Detectors). In the productions +completed so far +[7–9], RSDs have an AC-coupled design; recently, the design has been +extended to DC-coupled read-out [10] (DC-RSD). The key feature of RSDs is that the signal +is shared among the electrodes near the hit point analogously to a current divider: each pad i +sees a fraction Ii of the total signal Io that depends on the impedance Zj between the impact +point and the pads, Ii = Io(1/Zi)/ �n +1(1/Zj). +Figure 1 shows the sketch of a silicon sensor that incorporates both the LGAD and RSD +innovations. The design is based on an n-in-p sensor, has a continuous gain implant just +1 +arXiv:2301.02968v1 [physics.ins-det] 8 Jan 2023 + +underneath the cathode, and the cathode is resistive to ensure electrode isolation and signal +sharing. The presence of the gain implant, the signature feature of the LGAD design, creates +a high electric field in the volume underneath the n+ resistive layer and leads to signal +multiplication. In this sketch, the metal electrodes are directly implanted in the resistive n+ +sheet to ensure a DC coupling between the sensor and the electronics (DC-RSD). The RSD +design has uniform electric and weighting fields over the whole sensor volume, a requisite for +good temporal resolution. +Fig. 1. +Sketch of a resistive silicon detector with DC read-out (DC-RSD). +. +2. +Future Silicon Trackers +According to the ECFA roadmap [11], the next generation of large silicon trackers will be +deployed at either circular (FCC) or linear (ILC, CLIC) lepton colliders. The physics aims +of these future lepton colliders, revolving around very precise flavor physics, can only be +achieved with silicon trackers with very small impact parameter resolutions. Table I reports +a few important parameters of future silicon vertex trackers at e+e− colliders. +Table I. +Parameters of future silicon vertex trackers at e+e− colliders. +Facility: +FCC-ee +ILC +CLIC +σhit pos [µm] +∼ 5 +< 3 +< 3 +Thickness [µm of Si] +∼ 100 +∼ 100 +∼ 100 +Hit rate [106/s/cm2] +∼ 20 +∼ 0.2 +∼ 1 +Power dissipation [W/cm2] +0.1 -0.2 +0.1 +0.1 +Pixel size [µm2] +25 × 25 +25 × 25 +25 × 25 +The trackers need to have a superb position resolution (less than 5 µm), be very thin, +sustain a high hit particle rate, and use very little power. Two quantities need to be minimized +to achieve these goals: (i) the single hit resolution σhit pos, and (ii) the multiple scattering +resolution σMS. The two terms σhit pos and σMS are linked to each other and to the type of +read-out architecture. The pixel size determines the spatial resolution if a tracker employs +single-pixel read-out. Only very small pixels (25x25 µm2) achieve the required precision of +5 µm, and it is practically impossible to reach better resolutions. On the other hand, if a +2 + +Gain implant +Cathode +Anode +E Fieldtracker uses the traditional design for charge sharing to improve σhit pos, the sensor needs to be +quite thick (at least 200 µm), leading to a large σMS contribution. The very mechanism that +optimizes σhit pos is detrimental to σMS: thick sensors, necessary for signal sharing, cause +significant multiple scattering and deteriorate the overall accuracy of the tracker system. +Multiple scattering is further increased by the cooling infrastructures: for this reason, the +levels of power consumption reported in Table I are such that air cooling can be employed. +This request is particularly difficult to achieve when using single pixel read-out as the power +used by millions of pixels is large. Although research and development in silicon detectors +is very active in many fields, currently, no design can achieve the performance listed in +Table I [12]. +3. +A Tracker Based on Resistive Read-out in Thin Silicon Sensors with +Internal Gain +According to our present R&D studies, the key to meeting the demand of the next +generation of lepton colliders is to use thin silicon sensors that combine resistive read-out +and internal gain. +In the present silicon tracker paradigm, the targeted position resolution determines the +pixel size. These systems reach an excellent spatial resolution using millions of pixels and +amplifiers that, at any given time, are mostly empty; in many situations, less than 0.1 % of +pixels see a signal. In a much more efficient design, the density of particles determines the +pixel size. For example, the pixel size should be such that less than a few per mille of pixels +are hit by 2 particles at the same time. Resistive read-out allows reaching excellent position +resolution while using large pixels, so the pixel size is determined by occupancy and not by +the needed position resolution. The presence of internal gain boosts the signal and allows +using thin sensors. +Figure 2 illustrates why the combination of resistive read-out and low-gain amplification is +so powerful. With this design, the signals are shared among a few pads, have large amplitudes, +are uniform over the sensor surface, and are short. +Fig. 2. +The combination of resistive read-out and low-gain amplification leads to the experimental +features required by the next generation of experiments. +The experimental features arising from the design innovations can be summarised as: +(1) Sharing allows using large pixels, i.e., having enough space for the read-out electronics. +(2) Sharing combined with internal amplification achieves excellent spatial precision. +(3) Sharing (fewer pixels) combined with internal amplification (less need for amplification +in the electronics) reduces power consumption. +3 + +Sensor design +Signal property +nnovation +Experimental feature +1) Enough space for electronics +Resistive cathode +Shared +2)Excellent spatialprecision +3)Reducedpower consumption +Resistiyeread-out +Continuousaainlayer +Large +4)Excellenttemporalprecision +Uniform +5) 100% fillfactor +Continuous cathode +Low-gain amplification +6)100%efficiency +Short +Z) High rate +Thin +8) Low material budget(4) Large and short signals combined with a uniform response yield excellent temporal +precision. +(5) Large signals and a uniform response yield 100% fill factor. +(6) Large signals and a uniform response yield 100% efficiency. +(7) Short signals make it possible to work at a high rate. +(8) Thin sensors (and thin front-end electronics) enable a low material budget. +3.1 +Sensor Simulation +The simulation of RSD presents several unique challenges linked to the complex nature of +its design and to the large pixel size. The defining feature of RSD, built-in charge sharing over +distances that can be as large as a millimeter, represents a formidable challenge for TCAD, the +standard simulation tool. A single 3D TCAD simulation of an RSD pixel 100×100 µm2 takes +about 12 hours. Since the simulation time is an increasing function of the simulated volume, +the time needed to perform a 3D simulation of a 1 × 1 mm2 pixel is too long to be used in +an R&D phase, where many different options need to be tested. It is, therefore, impossible +to approach the simulation of RSDs using the standard TCAD method. To circumvent this +problem, a mixed-mode approach to simulation was developed [13], shown graphically in +Figure 3. +Fig. 3. +A graphical representation of the mixed-mode approach for the simulation of RSD. Left +pane: the building block of the SPICE-based RSD model. The simulation package Weightfield2 (WF2) +is used to generate the input signals. Right pane: the 3D TCAD volume, with 4 pads. +In the first phase, the properties of RSD are simulated using a SPICE-based analog +electronic circuit software. In this step, the key elements of an RSD are represented by +electrical components (left side of Figure 3). The resistive plane is modeled by a network of +resistors, the sensor bulk by capacitors, the metal pads by areas with zero-ohm resistance, +and the front-end electronics by resistors that approximate the read-out input impedance. In +this framework, signal sharing is studied by injecting a current stimulus in pre-determined +positions on the resistive plane and measuring the current collected in each pad. With this +approach, each simulation takes about 1 minute. A very important aspect of this simulation +4 + +R2 +(R) +C1 +R4 +R3 +(R +(R) +{C_backplane} +>R1 +(R) +Resistive sheet +Bac +MIP +MIP +MIPapproach is the shape of the current stimulus. Since the propagation of a signal on an RC +network depends on its frequency components, the signals used in this study need to have the +appropriate shape. To this end, the Weightfield2 (WF2) simulation package [14], has been +used. WF2 is a software program specifically developed to simulate signal formation in silicon +sensors with or without gain, and it can provide accurate predictions of the induced current +signal in RSD. +The outcome of this first phase is used in 3D TCAD simulations to determine the actual +sensor design. The most promising values used in SPICE are converted into actual design +parameters. For example, the n+ sheet resistance is obtained with appropriate n+ electrode +doping and profile combinations. Once the construction parameters of RSD are finalized, a +set of 3D TCAD simulations of small-sized pixels are performed to cross-check the SPICE- +based approach predictions. The right side of Figure 3 shows the volume of an RSD device, +with an area of 100 × 100 µm2, in a 3D TCAD simulation with four read-out pads. +3.2 +The design of the electronics +The signals generated by particles in the RSD sensor define how the first amplification +stage should be designed. The electronic design differs considerably depending on the tracker +goals. +• Excellent spatial resolution, stringent requirements on material budget. For +this configuration, the most important aspect is to precisely measure the charge on each +pad and keep the power consumption very low. Therefore, the most promising front-end +configuration is a charge integrator followed by an ADC. This design needs very low +power, and it can match the requirement of less than 80-100 mW/cm2, the maximum +power budget where air-cooling can be used. +• Combined good spatial and temporal resolutions, moderate requirements on +material budget.For this configuration, a time tagging circuit is necessary, which leads +to higher power consumption. Depending on the spatial precision required, the signal +can be sampled once using the standard Time-over-Threshold (ToT) approach, twice +with a dual ToT system, or multiple times using a waveform sampler (WFS). The choice +among ToT, dual ToT, and WFS depends upon the power available and the required +precision. +Figure 4 schematically shows the options for the two cases. +Fig. 4. +First and second stage architectures for the two options considered in the text. +In RSDs, each pad sees a modified version of the original signal. During the propagation +on the n+ resistive surface, the signal becomes smaller, wider, with slower leading and trailing +5 + +(i) Excellent spatial +Charge integrator +ADC +resolution, very low +material budget. +Time over Threshold +Increasing +(I) Combined good spatial +power +Dual +and temporal resolutions +Trans-Impedance +Time over Threshold +moderate material budget. +Wafeform sampleredges, and is delayed. Each of these aspects is a valuable piece of information that should be +used in the reconstruction of the hit position and time. For this reason, the finer the signal +sampling, the better the hit location can be determined. The ToT solution provides only a +limited subset of information to the reconstruction stage and might lead to a degradation of +performance. Double ToT considerably increases the information available to the reconstruc- +tion as it provides the signal derivative of the signal leading and trailing edges. A waveform +sampler (WFS) is the option that provides the most information, however, at the price of a +steep increase in power consumption. +3.3 +The Design of a New Silicon Tracker +Figure 5 shows the transformation brought about by the combination of resistive read-out +and internal gain. In present systems (left pane), sensors are made of many independent p-n +diodes, each with its own electronics. A minimum thickness of about 150 µm is needed to +ensure good efficiency. In the RSD design (right pane), there is a single p-n diode with a +p-doped bulk and an n-doped resistive cathode. The signal is boosted by built-in amplifica- +tion, generated by the high field created by an extra p-doped layer, and it is shared among +contiguous pixels. The sensor thickness is 30-50 µm. The large sizes of the cathode and anode +are also instrumental in providing a very uniform weighting field, and uniform charge carriers +drift velocities, two key features to achieving 100% signal uniformity, 100% efficiency, and +excellent temporal resolution. +Fig. 5. +Sketch of a present (left) and RSD-based (right) silicon detector. The two detector designs +yield the same spatial precision. However, to cover an area of 600 x 600 µm2 the standard detector +uses about 575 pixels while the RSD uses 4 pixels. +4. +The Second FBK RSD production +The second RSD production at FBK [9] (RSD2) was designed to test several optimizations +of the first FBK RSD production. An in-depth study of the RSD2 spatial and temporal +resolutions can be found in [15]. An important improvement has been the introduction of +cross-shaped electrodes, which significantly increase the response uniformity. Figure 6 shows +on the left pane the layout of a sensor composed of an array of 450 µm pixels with cross- +shaped electrodes while on the right, a photograph of the sensors. Wirebonders are visible +for a group of 16 electrodes. +The combined spatial and temporal resolutions for several sensors with cross-shaped +electrodes are presented in Figure 7. The presented results are obtained at a gain of about +30. +6 + +Pixel size +~ 25 x 25 +Pixel size +250 x 250 um +～50 +μm +Resistive implantFig. 6. +Layout and a photograph of a sensor from the RSD2 FBK production with cross-shaped +electrodes. +Fig. 7. +Summary of the spatial and temporal resolutions as a function of the pixel size for RSD2 +sensors with cross-shaped electrodes [15]. +The two key points are: (i) The spatial resolution is about 3% of the pixel size, and it +scales linearly with the pixel size, (i) the temporal resolution is fairly constant at about 35-40 +ps as a function of the pixel size. +5. +Conclusions +The development of resistive read-out in thin sensors with internal multiplication is driven +by the needs of future experiments and offers a viable solution to meet the demanding re- +quirements of future charged particle trackers. More generally, it will be relevant for all +applications requiring accurate photons and charged particle localization. Our preliminary +results show that a concurrent spatial resolution of about 3% of the pixel size and temporal +resolution of about 35 - 40 ps is achievable. The development of the read-out electronics +should be tailored to the need of the specific application: at low power consumption, a simple +ToT system is a good choice, while for higher power consumption, a double ToT read-out or +a waveform sampler can be considered. +7 + +FBK-IHOFN-UMIY +RSD2台RsD2 crosses: spatial and temporal resolutions +when total AC amplitude = 60 mV (gain = 30) +50 +50 +40 +40 +[sd] +Spatial resolution [um] +I resolution +30 +30 +20 +20 +Spatial resolution +10 +o Temporal resolution +0 +0 +1000 +1400 +0 +200 +400 +600 +800 +1200 +Pitch [um]6. +Acknowledgments +We kindly acknowledge the following funding agencies and collaborations: INFN – FBK +agreement on sensor production; Dipartimenti di Eccellenza, Univ. of Torino (ex L. 232/2016, +art. 1, cc. 314-337); Ministero della Ricerca, Italia, PRIN 2017, Grant 2017L2XKTJ – 4Din- +SiDe; Ministero della Ricerca, Italia, FARE, Grant R165xr8frt fare, Compagnia di San Paolo, +Italia, Grant TRAPEZIO 2021. +References +[1] P. Fern´andez-Mart´ınez, et al., Simulation of new p-type strip detectors with trench to enhance the +charge multiplication effect in the n-type electrodes, Nucl. Inst. Meth. A 658 (1) (2011) 98–102, +rESMDD 2010. +[2] G. Pellegrini, et al., Technology developments and first measurements of Low Gain Avalanche +Detectors (LGAD) for high energy physics applications, Nucl. Inst. Meth. A 765 (2014) 12. +[3] N. Cartiglia, et al., Design optimization of ultra-fast silicon detectors, Nucl. Inst. Meth. A 796 +(2015) 141. +[4] N. Cartiglia, et al., Issues in the design of Ultra-Fast Silicon Detectors, in: TREDI2015: 10th +“Trento” Workshop on Advanced Silicon Radiation Detectors, 2015. +URL https://indico.cern.ch/event/351695/contributions/828366 +[5] F. Siviero, et al., First application of machine learning algorithms to the position reconstruction +in Resistive Silicon Detectors, Journal of Instrumentation 16 (03) (2021) P03019. +[6] M. Tornago, et al., Resistive AC-Coupled Silicon Detectors: principles of operation and first +results from a combined analysis of beam test and laser data, Nucl. Inst. Meth. A 1003 (2021) +165319, arXiv: 2007.09528. +[7] G. Giacomini, et al., Fabrication and performance of AC-coupled LGADs, J. Instrum. 14 (2019) +P09004. +[8] R. Heller, et al., Characterization of bnl and hpk ac-lgad sensors with a 120 gev proton beam, +Journal of Instrumentation 17 (05) (2022) P05001. +[9] M. Mandurrino, et al., The second production of RSD (AC-LGAD) at FBK, JINST 17 (08) (2022) +C08001. +[10] L. Menzio, et al., DC-coupled resistive silicon detectors for 4D tracking, in: VCI2022 conference, +2022. +URL https://arxiv.org/abs/2204.07226 +[11] E. D. R. R. P. Group, The 2021 ECFA detector research and development roadmap, Tech. rep., +Geneva (2020). +[12] N. Cartiglia, et al., 4d tracking: present status and perspectives, Nucl. Inst. Meth. A 1040 (2022) +167228. doi:https://doi.org/10.1016/j.nima.2022.167228. +URL https://www.sciencedirect.com/science/article/pii/S0168900222005824 +[13] T. Croci, et al, A two-prong approach to the simulation of DC-RSD: TCAD and Spice, in: 2022 +IEEE Nuclear Science Symposium and Medical Imaging Conference (NSS/MIC). +[14] F. Cenna, et al., Weightfield2: A fast simulator for silicon and diamond solid state detector, Nucl. +Inst. Meth. A 796 (2015) 149. doi:https://doi.org/10.1016/j.nima.2015.04.015. +[15] R. Arcidiacono, et al., High-precision 4d tracking with large pixels using thin resistive silicon +detectors (2022). +URL https://arxiv.org/abs/2211.13809 +8 + diff --git a/btE1T4oBgHgl3EQfKwNx/content/tmp_files/load_file.txt b/btE1T4oBgHgl3EQfKwNx/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8f76be9eb0a6b90683b981341a322a5bff80721e --- /dev/null +++ b/btE1T4oBgHgl3EQfKwNx/content/tmp_files/load_file.txt @@ -0,0 +1,303 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf,len=302 +page_content='Resistive Read-out in Thin Silicon Sensors with Internal Gain N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' Cartiglia1, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' Moscatelli4,6, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' Arcidiacono1,2, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' Asenov5,6, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' Costa1,3, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' Croci6, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' Ferrero1, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' Fondacci5,6, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' Lanteri1,3, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' Menzio1,3, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' Morozzi6, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' Mulargia1,3, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content='Passeri5,6, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' Siviero1, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' Sola1,3, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' Tornago1,3 1 INFN, Torino, Italy, 2 Universita del Piemonte orientale, Italy 3 Universita di Torino, Italy, 4 CNR-IOM, Perugia, Italy, 5 Universita di Perugia, Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' 6 INFN Perugia, Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' E-mail: cartiglia@to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content='infn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content='it Two design innovations, low-gain avalanche (Low-Gain Avalance Diode, LGAD) and resistive read-out (Resistive Silicon Detector, RSD), have brought strong performance improvements to silicon sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' Large signals, due to the added gain mechanism, lead to improved temporal precision, while charge sharing, introduced by resistive read-out, allows for achieving excellent spatial resolution even with large pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' LGAD- and RSD- based silicon sensors are now adopted, or considered, in several future experiments and are the basis for almost every next 4D-trackers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' New results obtained with sensors belonging to the second FBK production of RSD (RSD2) demonstrate how a combined resolution of 30 ps and 30 µm can be obtained with pixels as large as 1 × 1 mm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' KEYWORDS: resistive read-out, silicon sensors, low-gain avalanche diode 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' Introduction In the past few years, the performance capabilities of silicon sensors in terms of combined spatial and temporal resolutions have improved significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' This evolution is due to the introduction of two innovations in the design of silicon sensors: (i) Low-Gain Avalanche Diode and (ii) Resistive Read-out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' The Low-Gain Avalanche Diode [1,2] design has been first introduced to compensate for the loss of signal due to charge trapping in irradiated sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' However, this design found its main application in the field of precision timing, with the introduction of Ultra-Fast Silicon Detector (UFSD) [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' This R&D has spurred a strong evolution in the field of accurate timing using silicon detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' Resistive read-out was first proposed to achieve an LGAD design with 100% fill factor [4] (the so-called AC-LGAD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' Then, it was subsequently recognized to lead to excellent spatial precision with a concurrent reduction of the number of read-out electrodes [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' Sensors based on resistive read-out are called RSDs (Resistive Silicon Detectors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' In the productions completed so far [7–9], RSDs have an AC-coupled design;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' recently, the design has been extended to DC-coupled read-out [10] (DC-RSD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' The key feature of RSDs is that the signal is shared among the electrodes near the hit point analogously to a current divider: each pad i sees a fraction Ii of the total signal Io that depends on the impedance Zj between the impact point and the pads, Ii = Io(1/Zi)/ �n 1(1/Zj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' Figure 1 shows the sketch of a silicon sensor that incorporates both the LGAD and RSD innovations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' The design is based on an n-in-p sensor, has a continuous gain implant just 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content='02968v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content='ins-det] 8 Jan 2023 underneath the cathode, and the cathode is resistive to ensure electrode isolation and signal sharing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' The presence of the gain implant, the signature feature of the LGAD design, creates a high electric field in the volume underneath the n+ resistive layer and leads to signal multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' In this sketch, the metal electrodes are directly implanted in the resistive n+ sheet to ensure a DC coupling between the sensor and the electronics (DC-RSD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' The RSD design has uniform electric and weighting fields over the whole sensor volume, a requisite for good temporal resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' Sketch of a resistive silicon detector with DC read-out (DC-RSD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' Future Silicon Trackers According to the ECFA roadmap [11], the next generation of large silicon trackers will be deployed at either circular (FCC) or linear (ILC, CLIC) lepton colliders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' The physics aims of these future lepton colliders, revolving around very precise flavor physics, can only be achieved with silicon trackers with very small impact parameter resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' Table I reports a few important parameters of future silicon vertex trackers at e+e− colliders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' Parameters of future silicon vertex trackers at e+e− colliders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' Facility: FCC-ee ILC CLIC σhit pos [µm] ∼ 5 < 3 < 3 Thickness [µm of Si] ∼ 100 ∼ 100 ∼ 100 Hit rate [106/s/cm2] ∼ 20 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content='2 ∼ 1 Power dissipation [W/cm2] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content='1 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content='1 Pixel size [µm2] 25 × 25 25 × 25 25 × 25 The trackers need to have a superb position resolution (less than 5 µm), be very thin, sustain a high hit particle rate, and use very little power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' Two quantities need to be minimized to achieve these goals: (i) the single hit resolution σhit pos, and (ii) the multiple scattering resolution σMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' The two terms σhit pos and σMS are linked to each other and to the type of read-out architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' The pixel size determines the spatial resolution if a tracker employs single-pixel read-out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' Only very small pixels (25x25 µm2) achieve the required precision of 5 µm, and it is practically impossible to reach better resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' On the other hand, if a 2 Gain implant Cathode Anode E Fieldtracker uses the traditional design for charge sharing to improve σhit pos, the sensor needs to be quite thick (at least 200 µm), leading to a large σMS contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' The very mechanism that optimizes σhit pos is detrimental to σMS: thick sensors, necessary for signal sharing, cause significant multiple scattering and deteriorate the overall accuracy of the tracker system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' Multiple scattering is further increased by the cooling infrastructures: for this reason, the levels of power consumption reported in Table I are such that air cooling can be employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' This request is particularly difficult to achieve when using single pixel read-out as the power used by millions of pixels is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' Although research and development in silicon detectors is very active in many fields, currently, no design can achieve the performance listed in Table I [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' A Tracker Based on Resistive Read-out in Thin Silicon Sensors with Internal Gain According to our present R&D studies, the key to meeting the demand of the next generation of lepton colliders is to use thin silicon sensors that combine resistive read-out and internal gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' In the present silicon tracker paradigm, the targeted position resolution determines the pixel size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' These systems reach an excellent spatial resolution using millions of pixels and amplifiers that, at any given time, are mostly empty;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' in many situations, less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content='1 % of pixels see a signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' In a much more efficient design, the density of particles determines the pixel size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' For example, the pixel size should be such that less than a few per mille of pixels are hit by 2 particles at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' Resistive read-out allows reaching excellent position resolution while using large pixels, so the pixel size is determined by occupancy and not by the needed position resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' The presence of internal gain boosts the signal and allows using thin sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' Figure 2 illustrates why the combination of resistive read-out and low-gain amplification is so powerful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' With this design, the signals are shared among a few pads, have large amplitudes, are uniform over the sensor surface, and are short.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' The combination of resistive read-out and low-gain amplification leads to the experimental features required by the next generation of experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' The experimental features arising from the design innovations can be summarised as: (1) Sharing allows using large pixels, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=', having enough space for the read-out electronics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' (2) Sharing combined with internal amplification achieves excellent spatial precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' (3) Sharing (fewer pixels) combined with internal amplification (less need for amplification in the electronics) reduces power consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content='Sensor design ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content='Signal property ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content='nnovation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content='Experimental feature ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content='1) Enough space for electronics ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content='Resistive cathode ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content='Shared ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content='2)Excellent spatialprecision ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content='3)Reducedpower consumption ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content='Resistiyeread-out ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content='Continuousaainlayer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content='Large ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content='4)Excellenttemporalprecision ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content='Uniform ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content='5) 100% fillfactor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content='Continuous cathode ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content='Low-gain amplification ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content='6)100%efficiency ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content='Short ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content='Z) High rate ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content='Thin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content='8) Low material budget(4) Large and short signals combined with a uniform response yield excellent temporal ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content='precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' (5) Large signals and a uniform response yield 100% fill factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' (6) Large signals and a uniform response yield 100% efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' (7) Short signals make it possible to work at a high rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' (8) Thin sensors (and thin front-end electronics) enable a low material budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content='1 Sensor Simulation The simulation of RSD presents several unique challenges linked to the complex nature of its design and to the large pixel size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' The defining feature of RSD, built-in charge sharing over distances that can be as large as a millimeter, represents a formidable challenge for TCAD, the standard simulation tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' A single 3D TCAD simulation of an RSD pixel 100×100 µm2 takes about 12 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' Since the simulation time is an increasing function of the simulated volume, the time needed to perform a 3D simulation of a 1 × 1 mm2 pixel is too long to be used in an R&D phase, where many different options need to be tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' It is, therefore, impossible to approach the simulation of RSDs using the standard TCAD method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' To circumvent this problem, a mixed-mode approach to simulation was developed [13], shown graphically in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' A graphical representation of the mixed-mode approach for the simulation of RSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' Left pane: the building block of the SPICE-based RSD model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' The simulation package Weightfield2 (WF2) is used to generate the input signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' Right pane: the 3D TCAD volume, with 4 pads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' In the first phase, the properties of RSD are simulated using a SPICE-based analog electronic circuit software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' In this step, the key elements of an RSD are represented by electrical components (left side of Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' The resistive plane is modeled by a network of resistors, the sensor bulk by capacitors, the metal pads by areas with zero-ohm resistance, and the front-end electronics by resistors that approximate the read-out input impedance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' In this framework, signal sharing is studied by injecting a current stimulus in pre-determined positions on the resistive plane and measuring the current collected in each pad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' With this approach, each simulation takes about 1 minute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' A very important aspect of this simulation 4 R2 (R) C1 R4 R3 (R (R) {C_backplane} >R1 (R) Resistive sheet Bac MIP MIP MIPapproach is the shape of the current stimulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' Since the propagation of a signal on an RC network depends on its frequency components, the signals used in this study need to have the appropriate shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' To this end, the Weightfield2 (WF2) simulation package [14], has been used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' WF2 is a software program specifically developed to simulate signal formation in silicon sensors with or without gain, and it can provide accurate predictions of the induced current signal in RSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' The outcome of this first phase is used in 3D TCAD simulations to determine the actual sensor design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' The most promising values used in SPICE are converted into actual design parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' For example, the n+ sheet resistance is obtained with appropriate n+ electrode doping and profile combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' Once the construction parameters of RSD are finalized, a set of 3D TCAD simulations of small-sized pixels are performed to cross-check the SPICE- based approach predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' The right side of Figure 3 shows the volume of an RSD device, with an area of 100 × 100 µm2, in a 3D TCAD simulation with four read-out pads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content='2 The design of the electronics The signals generated by particles in the RSD sensor define how the first amplification stage should be designed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' The electronic design differs considerably depending on the tracker goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' Excellent spatial resolution, stringent requirements on material budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' For this configuration, the most important aspect is to precisely measure the charge on each pad and keep the power consumption very low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' Therefore, the most promising front-end configuration is a charge integrator followed by an ADC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' This design needs very low power, and it can match the requirement of less than 80-100 mW/cm2, the maximum power budget where air-cooling can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' Combined good spatial and temporal resolutions, moderate requirements on material budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content='For this configuration, a time tagging circuit is necessary, which leads to higher power consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' Depending on the spatial precision required, the signal can be sampled once using the standard Time-over-Threshold (ToT) approach, twice with a dual ToT system, or multiple times using a waveform sampler (WFS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' The choice among ToT, dual ToT, and WFS depends upon the power available and the required precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' Figure 4 schematically shows the options for the two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' First and second stage architectures for the two options considered in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' In RSDs, each pad sees a modified version of the original signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' During the propagation on the n+ resistive surface, the signal becomes smaller, wider, with slower leading and trailing 5 (i) Excellent spatial Charge integrator ADC resolution, very low material budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' Time over Threshold Increasing (I) Combined good spatial power Dual and temporal resolutions Trans-Impedance Time over Threshold moderate material budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' Wafeform sampleredges, and is delayed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' Each of these aspects is a valuable piece of information that should be used in the reconstruction of the hit position and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' For this reason, the finer the signal sampling, the better the hit location can be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' The ToT solution provides only a limited subset of information to the reconstruction stage and might lead to a degradation of performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' Double ToT considerably increases the information available to the reconstruc- tion as it provides the signal derivative of the signal leading and trailing edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' A waveform sampler (WFS) is the option that provides the most information, however, at the price of a steep increase in power consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content='3 The Design of a New Silicon Tracker Figure 5 shows the transformation brought about by the combination of resistive read-out and internal gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' In present systems (left pane), sensors are made of many independent p-n diodes, each with its own electronics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' A minimum thickness of about 150 µm is needed to ensure good efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' In the RSD design (right pane), there is a single p-n diode with a p-doped bulk and an n-doped resistive cathode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' The signal is boosted by built-in amplifica- tion, generated by the high field created by an extra p-doped layer, and it is shared among contiguous pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' The sensor thickness is 30-50 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' The large sizes of the cathode and anode are also instrumental in providing a very uniform weighting field, and uniform charge carriers drift velocities, two key features to achieving 100% signal uniformity, 100% efficiency, and excellent temporal resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' Sketch of a present (left) and RSD-based (right) silicon detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' The two detector designs yield the same spatial precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' However, to cover an area of 600 x 600 µm2 the standard detector uses about 575 pixels while the RSD uses 4 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' The Second FBK RSD production The second RSD production at FBK [9] (RSD2) was designed to test several optimizations of the first FBK RSD production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' An in-depth study of the RSD2 spatial and temporal resolutions can be found in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' An important improvement has been the introduction of cross-shaped electrodes, which significantly increase the response uniformity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' Figure 6 shows on the left pane the layout of a sensor composed of an array of 450 µm pixels with cross- shaped electrodes while on the right, a photograph of the sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' Wirebonders are visible for a group of 16 electrodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' The combined spatial and temporal resolutions for several sensors with cross-shaped electrodes are presented in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' The presented results are obtained at a gain of about 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' 6 Pixel size ~ 25 x 25 Pixel size 250 x 250 um ～50 μm Resistive implantFig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' Layout and a photograph of a sensor from the RSD2 FBK production with cross-shaped electrodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' Summary of the spatial and temporal resolutions as a function of the pixel size for RSD2 sensors with cross-shaped electrodes [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' The two key points are: (i) The spatial resolution is about 3% of the pixel size, and it scales linearly with the pixel size, (i) the temporal resolution is fairly constant at about 35-40 ps as a function of the pixel size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' Conclusions The development of resistive read-out in thin sensors with internal multiplication is driven by the needs of future experiments and offers a viable solution to meet the demanding re- quirements of future charged particle trackers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' More generally, it will be relevant for all applications requiring accurate photons and charged particle localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' Our preliminary results show that a concurrent spatial resolution of about 3% of the pixel size and temporal resolution of about 35 - 40 ps is achievable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' The development of the read-out electronics should be tailored to the need of the specific application: at low power consumption, a simple ToT system is a good choice, while for higher power consumption, a double ToT read-out or a waveform sampler can be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' 7 FBK-IHOFN-UMIY RSD2台RsD2 crosses: spatial and temporal resolutions when total AC amplitude = 60 mV (gain = 30) 50 50 40 40 [sd] Spatial resolution [um] I resolution 30 30 20 20 Spatial resolution 10 o Temporal resolution 0 0 1000 1400 0 200 400 600 800 1200 Pitch [um]6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' Acknowledgments We kindly acknowledge the following funding agencies and collaborations: INFN – FBK agreement on sensor production;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' Dipartimenti di Eccellenza, Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' of Torino (ex L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' 232/2016, art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' 1, cc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' 314-337);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' Ministero della Ricerca, Italia, PRIN 2017, Grant 2017L2XKTJ – 4Din- SiDe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' Ministero della Ricerca, Italia, FARE, Grant R165xr8frt fare, Compagnia di San Paolo, Italia, Grant TRAPEZIO 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} +page_content=' References [1] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE1T4oBgHgl3EQfKwNx/content/2301.02968v1.pdf'} diff --git a/c9FPT4oBgHgl3EQfyDV1/content/tmp_files/2301.13170v1.pdf.txt b/c9FPT4oBgHgl3EQfyDV1/content/tmp_files/2301.13170v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..3d75e23ca5321690ac6e5a49141774cd9dba46e0 --- /dev/null +++ b/c9FPT4oBgHgl3EQfyDV1/content/tmp_files/2301.13170v1.pdf.txt @@ -0,0 +1,780 @@ +Hamiltonian-Oriented Homotopy QAOA +Akash Kundu1,2∗, Ludmila Botelho1,2∗†, Adam Glos1,3 +1Institute of Theoretical and Applied Informatics, Polish Academy of Sciences, +Ba�ltycka 5, Gliwice, Poland +2Joint Doctoral School, Silesian University of Technology, +Akademicka 2A, Gliwice, Poland +3Algorithmiq Ltd, Kanavakatu 3C 00160 Helsinki, Finland +Abstract +The classical homotopy optimization approach has the potential to deal with highly nonlinear +landscape, such as the energy landscape of QAOA problems. Following this motivation, we introduce +Hamiltonian-Oriented Homotopy QAOA (HOHo-QAOA), that is a heuristic method for combinato- +rial optimization using QAOA, based on classical homotopy optimization. The method consists of +a homotopy map that produces an optimization problem for each value of interpolating parameter. +Therefore, HOHo-QAOA decomposes the optimization of QAOA into several loops, each using a +mixture of the mixer and the objective Hamiltonian for cost function evaluation. Furthermore, we +conclude that the HOHo-QAOA improves the search for low energy states in the nonlinear energy +landscape and outperforms other variants of QAOA. +1 +Introduction +Speedup of practical applications is yet to be realized for quantum devices as they are small and noise +prone. The limitations of available hardware initiated the Noisy Intermediate Scale Quantum (NISQ) +era [1]. The NISQ algorithms [2] can operate on limited amount of resources., in particular by distributing +tasks between quantum and classical devices. Many of those algorithms are represented by a broad class +of variational quantum algorithms (VQAs) [3]. Their generic structure consists of two subroutines: a +parametric quantum circuit (PQC) is implemented on quantum hardware that generates a quantum +state, and classical hardware calculates the cost function and optimizes the parameters of PQC. One of +the advantages of VQAs is that they can be easily adapted to various computational problems as long +as the Hamiltonian can be designed whose ground state corresponds to the solution of the problem. To +mention a few, VQAs has potential applications in finding the ground state of a molecule [4], solving +linear [5] and nonlinear [6] system of equations, quantum state-diagonalization [7], and quantum device +certification [8]. A detailed review can be found in [3]. +Quantum approximate optimization algorithm (QAOA) [9] is a variational quantum algorithm dedi- +cated to combinatorial optimization problems. The PQC in QAOA is a trotterized adiabatic evolution i.e. +the circuit consist of interchangeably applied so-called mixer and problem Hamiltonians. It has potential +application in solving problems like graph coloring [10–12], MaxE3Lin2 [13], Max-K-Vertex Cover [14], +or traveling salesman problem [12, 15]. To improve the performance of QAOA, multiple optimization +strategies have been introduced [16–24]. This is becuase given the limited resources of quantum com- +puters it is essential to effectively explore the landscape of cost function for PQC. On the othe hand, the +landscape of energy function in QAOA is highly nonlinear and to deal with such complicated landscapes, +sophisticated methods are necessary. +This motivate us to formulate a heuristic optimization strategy that uses classical homotopy optimiza- +tion for QAOA. The homotopy optimization has potential application in dealing with highly nonlinear +functions [25]. The homotopy method comprises a homotopy map, which for each value of interpolating +∗A. Kundu and L. Botelho contributed equally to this work +†corresponding author, lbotelho@iitis.pl +1 +arXiv:2301.13170v1 [quant-ph] 30 Jan 2023 + +parameter α ∈ [0, 1] outputs an optimization problem. In particular, for α = 0, the problem is easy- +to-solve, and for α = 1 the homotopy map returns the problem of interest. During the interpolation +process, which changes the value of α from 0 to 1, the solution continuously changes and is expected to +be optimal, or close to optimal for the intermediate problems. If the intermediate optimization succeed, +in the end we obtain the optimum of the target problem. One can see quantum annealing as a particular +type of homotopy optimization. A homotopy optimization for VQE was already proposed in [26] and +improved in [27,28]. However, its applicability for QAOA was only briefly mentioned in [29]. +The introduced Hamiltonian-Oriented Homotopy QAOA (HOHo-QAOA) decomposes the optimiza- +tion into several loops. The homotopy map smoothens between the mixer Hamiltonian and the problem +Hamiltonian during the optimization and each loop uses the mixture of these two Hamiltonians for cost +function evaluation. In each loop, the quantum state is optimized with respect to such intermediate +cost functions. This strategy simplifies the search for good QAOA parameters while keeping the PQC +unchanged. To show this, first we empirically analyze the impact of the choice of the homotopy pa- +rameters: the initial αinit value and the step parameter αstep which defines the difference between two +consecutive α values. Although theoretically, a choice of αinit and αstep very close to zero provides a +better approximation to the optimal solution, empirically we show that one can still get a good ap- +proximation to the optimal solution even if αinit and αstep are detached from zero. This hugely reduces +the computational cost of HOHo-QAOA. Finally, we compare HOHo-QAOA with other commonly used +QAOA optimization strategies [9,22]. +The rest of the paper is organized in the following way. In Section 2, we provide a brief overview of +the adiabatic quantum computing, variants of QAOA and the homotopy method. Throughout Section 3, +we numerically investigate the efficient settings of the homotopy parameters. Furthermore, we compare +HOHo-QAOA with the other variants of QAOA considered in the literature. Finally, we conclude the +article in Section 4. +2 +Preliminaries +2.1 +QAOA +The core concept of Adiabatic Quantum Computing (AQC) lies in the adiabatic theorem. Let us consider +H(s) = H(t/T), a time-dependent smoothly varying Hamiltonian for all t ∈ [0, T] i.e. s ∈ [0, 1], where +T is the total time of evolution. Let us denote by |Ei(s)⟩ an eigenvector of H(s) with corresponding +eigenvalue Ei(s), where we assume E0(s) ≤ E1(s) ≤ . . .. The adiabatic theorem roughly states that a +system that is initially prepared in |E0(0)⟩ of H(t = 0), after time-evolution that is piloted by Schr¨odinger +equation with the given Hamiltonian H(s), will approximately keep the state of the system in the |E0(1)⟩ +at t = T, provided that the change in H(s) is “sufficiently slow”. Traditionally the sufficiently slow change +is given by the condition [30,31] +T ≫ ∆−2 max +s∈[0,1] +����� +�∂H(s) +∂t +�2�����, +(1) +where ∆ = mins (E1(s) − E0(s)), is the spectral gap. A class of independent conditions on T has been +discussed in [32–35]. AQC has the potential to take a initial Hamiltonian say Hmix whose ground state is +easy-to-prepare to the ground state of a computationally hard problem Hamiltonian Hobj. A particular +time-dependent Hamiltonian interpolates between the Hmix and Hobj as +H(s) = (1 − s) Hmix + sHobj, +(2) +AQC in the form of quantum annealing has been used for a variety of applications including real-world +problems [36–41], and in quantum chemistry [42]. For a rigorous review of AQC check [31,43]. +The Quantum Approximate Optimization Algorithm (QAOA), uses the first order Suzuki-Trotter +transformation of exp(−iH(s)) as the variational ansatz to solve combinatorial optimization problems. +The trotterization gives rise to the operators exp(−iγjHobj) and exp(−iβjHmix), where γj is the param- +eter corresponding to objective Hamiltonian and βj corresponds to mixer Hamiltonian for j-th step. The +mixer Hamiltonian is traditionally expressed as Hmix = − � +i Xi, where Xi is Pauli X operator acting on +i-th qubit and Hobj is the objective Ising Hamiltonian whose ground state encodes the optimal solution +of the problem. This results in state +|⃗γ, ⃗β⟩ = +L +� +j=1 +exp (−iβjHmix) exp (−iγjHobj) |+⟩⊗N, +(3) +2 + +where N is the number of qubits, L is the number layers that defines the number of repeated application +of mixer and objective Hamiltonian, and |+⟩⊗N is the ground state of − � +i Xi. The algorithm utilizes +quantum hardware to evaluate the energy expectation value E(⃗γ, ⃗β) = ⟨⃗γ, ⃗β|Hobj|⃗γ, ⃗β⟩. Then the pa- +rameters ⃗γ and ⃗β are optimized using classical optimization methods so that the energy is minimized. +This energy evaluation along with classical optimization QAOA is well defined for any combinatorial +optimization problems as long as Hobj can be implemented efficiently. While the proposed X-mixer +combined with 2-local Ising model is frequently used in the literature, different choices were also consid- +ered [12,15,44–46]. +Heuristic learning of QAOA has been explored in trajectories QAOA (T-QAOA) [22]. +T-QAOA +is a heuristic strategy that utilizes interpolation-based prediction of good QAOA parameters. +With +the random initialization, the cost for optimization of QAOA is exponential in the number of layers of +QAOA [22]. On the other hand, with increased number of layers, Hmix may gradually turn off while the +Hobj turns on, which is reminiscent of AQC. However, QAOA could learn via following a diabatic path +to achieve higher success probability [47–49], which is beyond the adiabatic process natural for AQC. +This fact was used in T-QAOA by reusing the optimal angles found for L-layers in the (L + 1)-layers +PQC. +The T-QAOA variant considered in this paper runs as follow. It starts with a number of layer L0, +and finds the locally optimal parameters (⃗γL0, ⃗βL0). Then it uses the optimal parameters of layer L0 +to construct the initial parameters for the layer L0 + 1 by sampling the last entries of ⃗γL0+1 from a +uniform random distribution and setting ⃗βL0+1 = 0. With such initialization, the (L0 + 1)-th layer PQC +is optimized, and the procedure is repeated until a final number of layer L is reached. Note that different +interpolation method can be used [22]. +Note that for QAOA, the energy landscape with respect to a single parameter θ is related to the +following process. +First, an initial quantum state is prepared. +Then, if applicable, all the unitary +operations which precedes the θ-dependent operation are applied, which transforms the initial state into +0 +π +2 +π +γj +0.34 +0.36 +0.38 +0.40 +0.42 +0.44 +Enorm +0 +π +2 +π +βj +1st layer +2nd layer +3rd layer +Figure 1: Illustration of highly nonlinear energy landscape of QAOA for Max-Cut for 10 nodes with +weighted Barab´asi-Albert graph for objective Hamiltonian (left) and mixer Hamiltonian (right). Enorm +is a standarized energy of the objective Hamiltonian, so that the eigenvalues lies in [0, 1] +a different state (possibly a mixed state for noisy evolution). Afterwards, under an assumption of pure +evolution, a unitary exp(−iθH) for mixer or objective Hamiltonian H is applied. Finally, the remaining +operations are applied and the energy estimation with respect to observable is conducted. As shown in +the Appendix B, the energy function with respect to θ takes the form +C + +� +i>j +Ai,j cos(θ(Ei − Ej) + Bi,j), +(4) +in which {Ei} is the set of all eigenvalues of the operator H, and real parameters C, Ai,j, Bi,j depend +on the initial state, observable, and θ-independent quantum operations. Note that Eq. (4) is highly +3 + +nonlinear, therefore its optimization may be difficult in practice. This is in contrast to typically used +VQE approaches, in which the parameter-dependent unitary can be reduced to a single-qubit gate, which +in turn may result in a simple, yet powerful gradient-free optimization technique [50,51]. +Unfortunately, the number of cosines in Eq. (4) may grow quadratically with number of distinct +eigenvalues of the considered Hamiltonian. In the case of the objective Hamiltonian the number may be +particularly high. While for many simple problems like unweighted Max-Cut or Max-SAT the number +of different eigenvalues usually grows polynomially with the size of the data, for weighted Max-Cut each +partition may result in a different objective value, which may give O(2n) different energies in general. A +complicated energy landscape can be seen already even for a small and simple instance, see Fig. 1. For +problems generating such a complicated landscapes, more sophisticated methods may be at hand. +2.2 +Homotopy optimization method +One of the well-known methods to solve a system of highly nonlinear problems is homotopy optimization, +where a homotopy map is constructed between two systems. The solution corresponding to one of the +systems is transformed into the solution of the other system. For example, consider the function ftarg(x) +which encodes a computationally hard problem and finit(x) which is a problem with an easy-to-find +solution. Then the particular homotopy map between the systems can be given as +F(α, x) = g1(α)ftarg(x) + g2(α)finit(x), +0 ≤ α ≤ 1, +(5) +where +g1(0) = 0, +g2(0) = 1, +g1(1) = 1, +g2(1) = 0. +(6) +Here, we get a family of problems corresponding to minx F(α, x) = 0 for each α value from 0 to 1. We +track the optimized solutions starting from (α, x) = (0, x0), as α moves from 0 to 1, which for a successful +homotopy map leads to (α, x) = (1, x1), where x1 is ideally the optimal solution of ftarg. +The state-of-the-art approach is to start from (αinit, xinit) with xinit minimizing F(0, x) = finit(x). +Then the problem minx F(α + αstep, x) = 0 is iteratively solved using the solution of minx F(α, x) as a +starting point, for sufficiently small αstep > 0 [25]. +3 +Hamiltonian-Oriented Homotopy QAOA +3.1 +Proposed method +The Hamiltonian-oriented homotopy QAOA decomposes the optimization process of the objective Hamil- +tonian into several optimization loops. Each loop optimizes the energy +Eα(⃗γ, ⃗β) = ⟨⃗γ, ⃗β|H(α)|⃗γ, ⃗β⟩, +(7) +where H(α) encodes the homotopy map +H(α) = g1(α)Hmix + g2(α)Hobj, +0 ≤ α ≤ 1. +(8) +For α = 1 the expectation value in Eq. (7) is the energy corresponding to the Hobj. While there is a +freedom in the choice of g1 and g2, throughout the paper we a simple case +g1(α) = 1 − α, +g2(α) = α. +(9) +During the optimization process, we choose an initialization of mixer and objective parameters (at α = 0) +in such a way that the parameters corresponding to the mixer are sampled from the uniform random +distribution U(a, b) in an interval [a = 0, b = 2π] and the objective parameters are all set to 0. With +this initialization we make sure that the homotopy starts from the exact ground state of the mixer on +a noise-free setting, as application of mixer on its eigenstate does not change the state. For α′ > α ≥ 0 +the initial parameters are chosen as +(⃗γ, ⃗β)init +α′ = (⃗γ, ⃗β)∗ +α, +(10) +here ∗ denotes the optimal parameters for α. +4 + +0.0 +0.5 +1.0 +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +E∗norm(α) +RR +0.0 +0.5 +1.0 +α +NZR +0.0 +0.5 +1.0 +ZR +Figure 2: The impact of different methods of initialization of γj, βj on HOHo-QAOA. The left, the +middle and the right figures are representing the convergence for RR (Random Random), NZR (Near- +Zero Random) with parameter v = 0.05, and ZR (Zero Random) initialization respectively, see Sec. 3.2 +for details. It is visible that the ZR is outperforming the other two initializations. Although for αinit ≤ 0.2 +the performance of NZR and ZR are comparable but as we tune αinit > 0.2, the minima for NZR scatters +in region 0.10 < Enorm < 0.15 whereas the minima for ZR clusters in a very narrow Enorm-width. +It should be noted that each run of HOHo-QAOA follows the generic structure of homotopy process +as in Eq. (8) where the “run-time” of HOHo-QAOA is characterized by the αstep, for a fixed αinit. The +parameter αinit fixes the initial α value. Generally, it can be inferred that better approximation to the +optimal solution can be achieved if we choose a sufficiently small value of αstep and αinit. They can be +described in a more elaborated way as follows. Small value of αstep helps us realizing the homotopy of +Eq. (8) and at the same time if we initiate with αinit → 0, it becomes easier to find the ground state for +the first step. To show this, throughout the paper, we investigate the normalized energy +Enorm(Eα(⃗γ, ⃗β), α) = Eα(⃗γ, ⃗β) − minH(α) +maxH(α) − minH(α), +(11) +with respect to parameters of HOHo-QAOA, where Enorm(α) = 0, is the normalized ground energy for +any α ∈ [0, 1], and min H(α) (max H(α)) denotes minimum (maximum) of H(α). +3.2 +Initialization strategy +In the following, first we numerically discuss proposed settings for initial QAOA parameters (⃗γ, ⃗β)init. +With this setting we show that the homotopy parameters i.e. αinit, αstep can be chosen detached from +zero without compromising the efficiency of the method. We consider and optimized energy E∗ +norm, or in +the case of HOHo-QAOA also an intermediate optimized step energy E∗ +norm(α). In the numerical results +the E∗ +norm is averaged over 100 experiments. Details of the experiment can be found in Appendix A. +For the numerical investigation of optimal QAOA parameters, which is illustrated in Figure 2, we +consider three possible initialization choices of the mixer and objective parameters at α = αinit: +1. RR (Random Random): When the parameters corresponding to mixer and objective Hamiltonians +are chosen from a uniform random distribution U(0, 2π) i.e. γinit +j +∼ U(0, 2π), βinit +j +∼ U(0, 2π). +2. NZR (Near-Zero Random): The parameters corresponding to mixer Hamiltonian are chosen from +U(0, 2π) but objective parameters are sampled from the values very close zero i.e. γinit +j +∼ U(0, v), βinit +j +∼ +U(0, 2π), where v is 0.05. +3. ZR (Zero Random): Mixer parameters are sampled from U(0, 2π) chosen and objective is all zeros +i.e. γinit +j += 0, βinit +j +∼ U(0, 2π) as proposed before. +From Figure 2 we conclude that ZR gives the best approximation to the ground state. This is because, +under the ZR setting the initial parameters of QAOA always starts corresponding to the exact ground +5 + +0.00 +0.25 +0.50 +0.75 +1.00 +αinit +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +E∗norm(α = 1) +(a) +10−4 +10−2 +100 +αstep +10−2 +10−1 +(b) +6 qubits +10 qubits +16 qubits +Figure 3: We illustrate the dependency of E∗ +norm with αinit and αstep. In (a) the variation of E∗ +norm with +αinit for 3 layers of HOHo-QAOA is presented, with γinit +j += 0, βinit +j +∼ U(0, 2π). In the figure, we see +a region of stability of HOHo-QAOA in respect with αinit in the range 0.0 to 0.50. In (b) we present +E∗ +norm vs αstep using 10 layers of HOHo-QAOA. Just like in the case of αinit, for αstep a same region of +stability can be observed. This gives us the preference on the choice of step parameter while utilizing +HOHo-QAOA. It should be noted that the y-axis in (a) is in linear scale and whereas in (b) it is in +log scale. The lines in both the plots are taken αinit and αstep-wise and is the mean of 100 experiments. +The area under the plots are standard deviation of energies. +state of Hmix while the Hobj is turned off. This is within the spirit of the homotopy optimization, in +which starting in the optimal solution of the initial system is critical. Hence this good approximation to +the initial parameters lead us to the better solution to the ground state of Hobj. Keeping in mind that +if we sample αinit in the range 0 ≤ αinit ≤ 0.2, we see that NZR shows comparable performance to ZR +and the choice of initialization of γj, βj can be either one of them, relaxing the conditions on the choice +of ⃗γ and ⃗β. In the remaining of this paper all the numerical results are initialized with the ZR setting. +Now we move to the analysis of the choice of suitable αinit. In the Figure [3] we investigate the +αinit dependency of the E∗ +norm, where the energy is averaged over 100 experiments. From Figure [3](a) +we take 3 layers of HOHo-QAOA and observe that the mean optimal energy and the corresponding +standard deviation remain unchanged (which we term as region of stability) with respect to αinit in the +range 0 ≤ αinit ≤ 0.5. With an increase in the number of nodes from 6 to 16, the region of stability shifts +upwards but remains in the range 0 ≤ αinit ≤ 0.5. This observation lead us to conclude that αinit can +be chosen detached from zero without degrading the performance of HOHo-QAOA, or that at least that +the region of stability does not shrink rapidly with the increased size of the problem. So setting αinit +in the region of stability along with γinit +j += 0, βinit +j +∼ U(0, 2π) yields a solution with particularly small +energy value. +In Figure [3](b) we investigate how the efficiency of the optimization depends on the αstep. During this +investigation, we take 10 layers of HOHo-QAOA. We observe that in the range 10−4 ≤ αstep < 0.5 the +approximation to the ground energy and the corresponding standard deviation with increasing αstep → 0 +remains almost unchanged, giving rise to a region of stability with respect to αstep. This behavior of +E∗ +norm with αstep is similar to what we can observe for αinit. This lead us to a conclusion that one can +choose αstep detached from zero for HOHo-QAOA. It should be noted that due to high simulation cost +the experiment for 16 qubits is halted at the αstep = 10−2, whereas the investigation for 6, 16 qubits is +extended to 10−4. +The discussion and numerical results from the previous paragraphs give us the following initialization +rules of HOHo-QAOA, which leads to a high efficiency of the method: +1. The parameters of mixer and objective should be initialized with ZR setting i.e. γinit +j += 0, βinit +j +∼ +U(0, 2π), +2. Although one can infer that αinit → 0 along with αstep → 0 gives the best result, our investigations +6 + +25 +50 +75 +100 +0.1 +0.2 +0.3 +E∗ +norm +QAOA +rand-rand +zero-rand +25 +50 +75 +100 +10−2 +10−1 +T-QAOA +rand-rand +zero-rand +Number of layers +Figure 4: Comparison of different initialization for QAOA and T-QAOA. In the left (right) figure we +illustrate how the E∗ +norm changes with increasing number of layers in QAOA (T-QAOA) under the RR +and ZR settings. The solid line is the median energy over 100 experiments, meanwhile the dashed line +represents the best sample, taken layer-wise and node-wise by choosing the minimum energy among all +the experiments. The areas are delimited by the first and third quartile. +25 +50 +75 +100 +Number of layers +10−3 +10−2 +10−1 +E∗ +norm +10 +15 +Number of nodes +10−1 +QAOA +T-QAOA +HOHo-QAOA +Figure 5: Performance of HOHo-QAOA compared to QAOA and T-QAOA. On both figures, for all the +QAOA methods, we applied the ZR settings. The areas are delimited by the first and third quartile. +The solid line presents E∗ +norm median over 100 experiments for the left figure and 50 experiments for the +right figure, and the dashed line represents the best sample, taken layer-wise and node-wise by choosing +the minimum energy among all the experiments. On the left figure, the number of nodes is fixed to 10. +On the right, the number of layers is fixed to 5 and the energy is sampled within 6 to 18 nodes. The +homotopy parameters are set as αinit = 0 and αstep = 0.01. One can see that in both cases the averaged +energy as well as the best sample of HOHo-QAOA outperforms the other variants of QAOA. +show that one can choose the homotopy parameters detached from zero. This greatly reduces the +cost of simulating HOHo-QAOA +7 + +3.3 +Performance analysis +In this section we analyze the performance of the introduced algorithm with respect to other optimization +strategies introduced above. While it is natural for HOHo-QAOA to initialize using ZR strategy, it is +unclear whether this choice will improve or worsen the results for QAOA or T-QAOA. Therefore before +comparing state-of-the-art methods to the introduced one, we verify whether there is any difference in the +performance for QAOA and T-QAOA with respect to the initialization of the optimized angles. In Fig. 4 +we investigate state-of-the art methods for parameters (γj, βj)init initialized with RR and ZR strategy. +We observe that the performance of QAOA and T-QAOA is not influenced by the chosen strategies. +This justifies using ZR strategy when comparing QAOA, T-QAOA and HOHo-QAOA. +Note that for QAOA we are observing undesired non-monotonic behavior with respect to the number +of layers. We claim that this is caused because of a complicated landscape of the energy function, which +makes difficult to optimize it if no information about the problem instance is used during the initialization +from large number of nodes. This argument is complies with good performance of T-QAOA where the +initial parameters of (L + 1)-layer step is evaluated based on local optimal solutions of the L-layers step. +In Fig. 5 we compare the performance of HOHo-QAOA with the other variants when (γj, βj)init are +initialized using ZR setting. In the first experiment we run the algorithms with a fixed number of nodes +while increasing number layers. In the second experiment the number of layers is fix while we vary +the number of nodes. The plots present optimized energy values, averaged respectively over 100 and +50 instances. The data shows that the introduced HOHo-QAOA gives us significantly smaller energy +in both experiment setups. Good improvements remains as more layers of the HOHo-QAOA are used +and also outperforms the other varients of QAOA for higher number of nodes. This conclusions remain +valid also for the best sample solution chosen (dashed line). It should be noted that the HOHo-QAOA +outperforms QAOA and the T-QAOA in each and every layer starting from initial layer 5 to final layer +100. +4 +Conclusion +In the article we present a novel algorithm for combinatorial optimization. The method is a combination +of homotopy optimization with an application in QAOA. In our method the observable used for computing +the energy is changed during the optimization process. The process starts with observable being a mixer, +for which the initial state of QAOA is a grounds state, and is slowly moved into the objective Hamiltonian. +In addition we verify that, although traditionally in the homotopy method the initial value of transition +parameter α should be 0 and the step should be as small as possible, for QAOA for the value of considered +parameters can be detached from 0. +A homotopy optimization is an algorithm dedicated for nonlinear optimized functions, and since even +simple QAOA landscape is a linear combination of many – for some problems exponentially many – +sinusoidal functions, our approach is well motivated for such energy function. This is in contrast to +typical VQE optimization process, in which the function landscape with respect to a single parameter +is just a sine. By comparing our approach and QAOA algorithm with the typical choice of optimization +strategies we numerically confirmed that our method outperforms state-of-the-art approaches. +While our algorithm was only presented for QUBO and X-mixer,it is not restricted to it. In particular, +if the transition function is of the form H(α) = g1(α)Hmix + g2(α)Hobj, we only require energy of the +Hmixer to be efficiently computable. This includes XY-mixer [46] and Grover mixer [45] for which the +initial state can be efficiently prepared. Moreover, our approach remains also valid for higher-order binary +problems [11,15] and more advanced pseudo-code based QAOA Hamiltonian implementation [12]. +Acknowledgment +A.K., A.G. and L .B. has been partially supported by Polish National Science +Center under grant agreements 2019/33/B/ST6/02011. A.G. acknowledges support from National Sci- +ence Center under grant agreement 2020/37/N/ST6/02220. 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Muthukrishnan, T. Albash, and D. A. Lidar, “Tunneling and speedup in quantum optimization +for permutation-symmetric problems,” Physical Review X, vol. 6, p. 031010, Jul 2016. +[49] T. Albash and D. A. Lidar, “Adiabatic quantum computation,” Review of Modern Physics, vol. 90, +p. 015002, Jan 2018. +[50] M. Ostaszewski, E. Grant, and M. Benedetti, “Structure optimization for parameterized quantum +circuits,” Quantum, vol. 5, p. 391, 2021. +[51] K. M. Nakanishi, K. Fujii, and S. Todo, “Sequential minimal optimization for quantum-classical +hybrid algorithms,” Physical Review Research, vol. 2, no. 4, p. 043158, 2020. +A +Experiment details +In order to enable the simple reproduction of our results, we publish our code on ... The algorithms for +generating data and plotting were implemented in Julia and Python programming languages. Versions +of the software and additional packages are listed in... +Experiments +Each experiment of HOHo-QAOA is uniquely characterized by the random graph G = +(V, E), that is chosen from Barab´asi-Albert distribution with 6, 8, . . . 18 nodes and with m = 2, where m +defines the number of edges to be attached from a new node to existing ones. The weights corresponding +to the edges are picked up from a uniform set of integer weights wjj′ ∈ {1, . . . , 10} for each edge {j, j′}. +Data Sampling +For sampling the objective Hamiltonian, we started by generating graph objects an +them converting to Pauli operators objects and Hamiltonian matrices with Qiskit. We generated 100 +samples for each graph setup. We sampled the initial optimization parameters in a random distribution +for RR and ZR approaches. We emulated the quantum evolution and take an exact expectation energy +and gradient of the state during the optimization. We choose the L-BFGS algorithm implemented in +Julia’s Optim package as a subroutine. The optimization has no periodic or bounds conditions. We setup +Optim with absolute tolerance, relative tolerance and absolute tolerance in gradient equal to 1−9. We +allowed steps that increase the objective value and maximum number of iterations is 10000. +11 + +T-QAOA +For T-QAOA implementation, we initialize with a minimum number of levels L0 = 4 and +run the optimization similarly to the state of art QAOA with the a given parameters initiation strategy. +The method proceeds checking the convergence of the solution and moving to the next layer L0 + 1, +using the previous optimized parameters with the addition of a zero for the mixer Hamiltonian and a +value sampled from a uniform random distribution U(0, 2π). +B +Proof of nonlinear landscape for QAOA +Theorem 1. Let ϱ be an arbitrary quantum state, H be an arbitrary Hamiltonian with spectrum set +{E1, . . . , Ek} and O be an arbitrary observable. Then +tr(exp(−iθH)ϱ exp(iθH)O) = C + +� +i>j +Ai,j cos(θ(Ei − Ej) + Bi,j), +(12) +for some real values C, Ai,j, Bi,j. +Proof. Let U be a unitary that diagonalizes the Hamiltonian H. Then we have +tr(exp(−iθH)ϱ exp(iθH)O) = tr +� +� +k +� +i=1 +(Ue−iθEi |i⟩⟨i| U †)ϱ +k +� +j=1 +(UeiθEj |j⟩⟨j| U †)O +� +� += +k +� +i=1 +k +� +j=1 +eiθ(Ej−Ei) tr +� +U |i⟩⟨i| U †ϱU |j⟩⟨j| U †O +� += +k +� +i=1 +k +� +j=1 +eiθ(Ej−Ei) tr (|i⟩⟨i| ϱ′ |j⟩⟨j| O′) += +k +� +i=1 +k +� +j=1 +eiθ(Ej−Ei) ⟨i| ϱ′ |j⟩ ⟨j| O′ |i⟩ , +(13) +where ϱ′ = U †ϱU and O′ = U †OU. Since ϱ′ is a hermitian operator and therefore ⟨i| ϱ |j⟩ = ⟨j| ϱ |i⟩, and +similarly for O′, therefore for any i, j the term for i > j is a conjugate of the term i < j. Hence +k +� +i=1 +k +� +j=1 +eiθ(Ej−Ei) ⟨i| ϱ′ |j⟩ ⟨j| O′ |i⟩ = +k +� +i=1 +⟨i| ϱ′ |i⟩ ⟨i| O′ |i⟩ + 2 +� +i>j +Re eiθ(Ej−Ei) ⟨i| ϱ′ |j⟩ ⟨j| O′ |i⟩ . +(14) +Note that the left hand side sum in the above above is a free term and is a real number. Starting from +now we will assume that the Hamiltonian H is non-degenerate – otherwise the corresponding element +of the right sum will contribute to the free term. Taking xi,j + iyi,j := ⟨i| ϱ′ |j⟩ ⟨j| O′ |i⟩ for some real +xi,j, yi,j we have +Re eiθ(Ej−Ei) ⟨i| ϱ′ |j⟩ ⟨j| O′ |i⟩ = Re(cos(θ(Ej − Ei)) + i sin(θ(Ej − Ei)))(xi,j + iyi,j) += xi,j cos(θ(Ej − Ei)) − yi,j sin(θ(Ej − Ei)) += +� +x2 +i,j + y2 +i,j +� +� +xi,j +� +x2 +i,j + y2 +i,j +cos(θ(Ej − Ei)) − +yi,j +� +x2 +i,j + y2 +i,j +sin(θ(Ej − Ei)) +� +� += +� +x2 +i,j + y2 +i,j (cos(αi,j) cos(θ(Ej − Ei)) − sin(αi,j) sin(θ(Ej − Ei))) , +(15) +where αi,j is such a real number for which the above transformation holds. Note that such a number α +can always be found as the replaced fraction squared sum to 1 and one can use Pythagorean trigonometric +identity. Finally we have +� +x2 +i,j + y2 +i,j (cos(αi,j) cos(θ(Ej − Ei)) − sin(αi,j) sin(θ(Ej − Ei))) += +� +x2 +i,j + y2 +i,j cos(θ(Ej − Ei) + αi,j), +(16) +which proves the statement of the theorem. +Note that the case of Hamiltonian with two different eigenvalues was already presented in [50,51]. +12 + diff --git a/c9FPT4oBgHgl3EQfyDV1/content/tmp_files/load_file.txt b/c9FPT4oBgHgl3EQfyDV1/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7c6addcb6b61e9753cdcade01ec33e6d4a183919 --- /dev/null +++ b/c9FPT4oBgHgl3EQfyDV1/content/tmp_files/load_file.txt @@ -0,0 +1,806 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf,len=805 +page_content='Hamiltonian-Oriented Homotopy QAOA Akash Kundu1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='2∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' Ludmila Botelho1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='2∗†,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' Adam Glos1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='3 1Institute of Theoretical and Applied Informatics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' Polish Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' Ba�ltycka 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' Gliwice,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' Poland 2Joint Doctoral School,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' Silesian University of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' Akademicka 2A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' Gliwice,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' Poland 3Algorithmiq Ltd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' Kanavakatu 3C 00160 Helsinki,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' Finland Abstract The classical homotopy optimization approach has the potential to deal with highly nonlinear landscape,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' such as the energy landscape of QAOA problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' Following this motivation, we introduce Hamiltonian-Oriented Homotopy QAOA (HOHo-QAOA), that is a heuristic method for combinato- rial optimization using QAOA, based on classical homotopy optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' The method consists of a homotopy map that produces an optimization problem for each value of interpolating parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' Therefore, HOHo-QAOA decomposes the optimization of QAOA into several loops, each using a mixture of the mixer and the objective Hamiltonian for cost function evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' Furthermore, we conclude that the HOHo-QAOA improves the search for low energy states in the nonlinear energy landscape and outperforms other variants of QAOA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' 1 Introduction Speedup of practical applications is yet to be realized for quantum devices as they are small and noise prone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' The limitations of available hardware initiated the Noisy Intermediate Scale Quantum (NISQ) era [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' The NISQ algorithms [2] can operate on limited amount of resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=', in particular by distributing tasks between quantum and classical devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' Many of those algorithms are represented by a broad class of variational quantum algorithms (VQAs) [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' Their generic structure consists of two subroutines: a parametric quantum circuit (PQC) is implemented on quantum hardware that generates a quantum state, and classical hardware calculates the cost function and optimizes the parameters of PQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' One of the advantages of VQAs is that they can be easily adapted to various computational problems as long as the Hamiltonian can be designed whose ground state corresponds to the solution of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' To mention a few, VQAs has potential applications in finding the ground state of a molecule [4], solving linear [5] and nonlinear [6] system of equations, quantum state-diagonalization [7], and quantum device certification [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' A detailed review can be found in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' Quantum approximate optimization algorithm (QAOA) [9] is a variational quantum algorithm dedi- cated to combinatorial optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' The PQC in QAOA is a trotterized adiabatic evolution i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' the circuit consist of interchangeably applied so-called mixer and problem Hamiltonians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' It has potential application in solving problems like graph coloring [10–12], MaxE3Lin2 [13], Max-K-Vertex Cover [14], or traveling salesman problem [12, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' To improve the performance of QAOA, multiple optimization strategies have been introduced [16–24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' This is becuase given the limited resources of quantum com- puters it is essential to effectively explore the landscape of cost function for PQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' On the othe hand, the landscape of energy function in QAOA is highly nonlinear and to deal with such complicated landscapes, sophisticated methods are necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' This motivate us to formulate a heuristic optimization strategy that uses classical homotopy optimiza- tion for QAOA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' The homotopy optimization has potential application in dealing with highly nonlinear functions [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' The homotopy method comprises a homotopy map, which for each value of interpolating ∗A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' Kundu and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' Botelho contributed equally to this work †corresponding author, lbotelho@iitis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='pl 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='13170v1 [quant-ph] 30 Jan 2023 parameter α ∈ [0, 1] outputs an optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' In particular, for α = 0, the problem is easy- to-solve, and for α = 1 the homotopy map returns the problem of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' During the interpolation process, which changes the value of α from 0 to 1, the solution continuously changes and is expected to be optimal, or close to optimal for the intermediate problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' If the intermediate optimization succeed, in the end we obtain the optimum of the target problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' One can see quantum annealing as a particular type of homotopy optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' A homotopy optimization for VQE was already proposed in [26] and improved in [27,28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' However, its applicability for QAOA was only briefly mentioned in [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' The introduced Hamiltonian-Oriented Homotopy QAOA (HOHo-QAOA) decomposes the optimiza- tion into several loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' The homotopy map smoothens between the mixer Hamiltonian and the problem Hamiltonian during the optimization and each loop uses the mixture of these two Hamiltonians for cost function evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' In each loop, the quantum state is optimized with respect to such intermediate cost functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' This strategy simplifies the search for good QAOA parameters while keeping the PQC unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' To show this, first we empirically analyze the impact of the choice of the homotopy pa- rameters: the initial αinit value and the step parameter αstep which defines the difference between two consecutive α values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' Although theoretically, a choice of αinit and αstep very close to zero provides a better approximation to the optimal solution, empirically we show that one can still get a good ap- proximation to the optimal solution even if αinit and αstep are detached from zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' This hugely reduces the computational cost of HOHo-QAOA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' Finally, we compare HOHo-QAOA with other commonly used QAOA optimization strategies [9,22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' The rest of the paper is organized in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' In Section 2, we provide a brief overview of the adiabatic quantum computing, variants of QAOA and the homotopy method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' Throughout Section 3, we numerically investigate the efficient settings of the homotopy parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' Furthermore, we compare HOHo-QAOA with the other variants of QAOA considered in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' Finally, we conclude the article in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' 2 Preliminaries 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='1 QAOA The core concept of Adiabatic Quantum Computing (AQC) lies in the adiabatic theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' Let us consider H(s) = H(t/T), a time-dependent smoothly varying Hamiltonian for all t ∈ [0, T] i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' s ∈ [0, 1], where T is the total time of evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' Let us denote by |Ei(s)⟩ an eigenvector of H(s) with corresponding eigenvalue Ei(s), where we assume E0(s) ≤ E1(s) ≤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='. The adiabatic theorem roughly states that a system that is initially prepared in |E0(0)⟩ of H(t = 0), after time-evolution that is piloted by Schr¨odinger equation with the given Hamiltonian H(s), will approximately keep the state of the system in the |E0(1)⟩ at t = T, provided that the change in H(s) is “sufficiently slow”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' Traditionally the sufficiently slow change is given by the condition [30,31] T ≫ ∆−2 max s∈[0,1] ����� �∂H(s) ∂t �2�����, (1) where ∆ = mins (E1(s) − E0(s)), is the spectral gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' A class of independent conditions on T has been discussed in [32–35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' AQC has the potential to take a initial Hamiltonian say Hmix whose ground state is easy-to-prepare to the ground state of a computationally hard problem Hamiltonian Hobj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' A particular time-dependent Hamiltonian interpolates between the Hmix and Hobj as H(s) = (1 − s) Hmix + sHobj, (2) AQC in the form of quantum annealing has been used for a variety of applications including real-world problems [36–41], and in quantum chemistry [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' For a rigorous review of AQC check [31,43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' The Quantum Approximate Optimization Algorithm (QAOA), uses the first order Suzuki-Trotter transformation of exp(−iH(s)) as the variational ansatz to solve combinatorial optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' The trotterization gives rise to the operators exp(−iγjHobj) and exp(−iβjHmix), where γj is the param- eter corresponding to objective Hamiltonian and βj corresponds to mixer Hamiltonian for j-th step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' The mixer Hamiltonian is traditionally expressed as Hmix = − � i Xi, where Xi is Pauli X operator acting on i-th qubit and Hobj is the objective Ising Hamiltonian whose ground state encodes the optimal solution of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' This results in state |⃗γ, ⃗β⟩ = L � j=1 exp (−iβjHmix) exp (−iγjHobj) |+⟩⊗N, (3) 2 where N is the number of qubits, L is the number layers that defines the number of repeated application of mixer and objective Hamiltonian, and |+⟩⊗N is the ground state of − � i Xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' The algorithm utilizes quantum hardware to evaluate the energy expectation value E(⃗γ, ⃗β) = ⟨⃗γ, ⃗β|Hobj|⃗γ, ⃗β⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' Then the pa- rameters ⃗γ and ⃗β are optimized using classical optimization methods so that the energy is minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' This energy evaluation along with classical optimization QAOA is well defined for any combinatorial optimization problems as long as Hobj can be implemented efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' While the proposed X-mixer combined with 2-local Ising model is frequently used in the literature, different choices were also consid- ered [12,15,44–46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' Heuristic learning of QAOA has been explored in trajectories QAOA (T-QAOA) [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' T-QAOA is a heuristic strategy that utilizes interpolation-based prediction of good QAOA parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' With the random initialization, the cost for optimization of QAOA is exponential in the number of layers of QAOA [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' On the other hand, with increased number of layers, Hmix may gradually turn off while the Hobj turns on, which is reminiscent of AQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' However, QAOA could learn via following a diabatic path to achieve higher success probability [47–49], which is beyond the adiabatic process natural for AQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' This fact was used in T-QAOA by reusing the optimal angles found for L-layers in the (L + 1)-layers PQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' The T-QAOA variant considered in this paper runs as follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' It starts with a number of layer L0, and finds the locally optimal parameters (⃗γL0, ⃗βL0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' Then it uses the optimal parameters of layer L0 to construct the initial parameters for the layer L0 + 1 by sampling the last entries of ⃗γL0+1 from a uniform random distribution and setting ⃗βL0+1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' With such initialization, the (L0 + 1)-th layer PQC is optimized, and the procedure is repeated until a final number of layer L is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' Note that different interpolation method can be used [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' Note that for QAOA, the energy landscape with respect to a single parameter θ is related to the following process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' First, an initial quantum state is prepared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' Then, if applicable, all the unitary operations which precedes the θ-dependent operation are applied, which transforms the initial state into 0 π 2 π γj 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='44 Enorm 0 π 2 π βj 1st layer 2nd layer 3rd layer Figure 1: Illustration of highly nonlinear energy landscape of QAOA for Max-Cut for 10 nodes with weighted Barab´asi-Albert graph for objective Hamiltonian (left) and mixer Hamiltonian (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' Enorm is a standarized energy of the objective Hamiltonian, so that the eigenvalues lies in [0, 1] a different state (possibly a mixed state for noisy evolution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' Afterwards, under an assumption of pure evolution, a unitary exp(−iθH) for mixer or objective Hamiltonian H is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' Finally, the remaining operations are applied and the energy estimation with respect to observable is conducted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' As shown in the Appendix B, the energy function with respect to θ takes the form C + � i>j Ai,j cos(θ(Ei − Ej) + Bi,j), (4) in which {Ei} is the set of all eigenvalues of the operator H, and real parameters C, Ai,j, Bi,j depend on the initial state, observable, and θ-independent quantum operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' Note that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' (4) is highly 3 nonlinear, therefore its optimization may be difficult in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' This is in contrast to typically used VQE approaches, in which the parameter-dependent unitary can be reduced to a single-qubit gate, which in turn may result in a simple, yet powerful gradient-free optimization technique [50,51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' Unfortunately, the number of cosines in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' (4) may grow quadratically with number of distinct eigenvalues of the considered Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' In the case of the objective Hamiltonian the number may be particularly high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' While for many simple problems like unweighted Max-Cut or Max-SAT the number of different eigenvalues usually grows polynomially with the size of the data, for weighted Max-Cut each partition may result in a different objective value, which may give O(2n) different energies in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' A complicated energy landscape can be seen already even for a small and simple instance, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' For problems generating such a complicated landscapes, more sophisticated methods may be at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='2 Homotopy optimization method One of the well-known methods to solve a system of highly nonlinear problems is homotopy optimization, where a homotopy map is constructed between two systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' The solution corresponding to one of the systems is transformed into the solution of the other system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' For example, consider the function ftarg(x) which encodes a computationally hard problem and finit(x) which is a problem with an easy-to-find solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' Then the particular homotopy map between the systems can be given as F(α, x) = g1(α)ftarg(x) + g2(α)finit(x), 0 ≤ α ≤ 1, (5) where g1(0) = 0, g2(0) = 1, g1(1) = 1, g2(1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' (6) Here, we get a family of problems corresponding to minx F(α, x) = 0 for each α value from 0 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' We track the optimized solutions starting from (α, x) = (0, x0), as α moves from 0 to 1, which for a successful homotopy map leads to (α, x) = (1, x1), where x1 is ideally the optimal solution of ftarg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' The state-of-the-art approach is to start from (αinit, xinit) with xinit minimizing F(0, x) = finit(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' Then the problem minx F(α + αstep, x) = 0 is iteratively solved using the solution of minx F(α, x) as a starting point, for sufficiently small αstep > 0 [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' 3 Hamiltonian-Oriented Homotopy QAOA 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='1 Proposed method The Hamiltonian-oriented homotopy QAOA decomposes the optimization process of the objective Hamil- tonian into several optimization loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' Each loop optimizes the energy Eα(⃗γ, ⃗β) = ⟨⃗γ, ⃗β|H(α)|⃗γ, ⃗β⟩, (7) where H(α) encodes the homotopy map H(α) = g1(α)Hmix + g2(α)Hobj, 0 ≤ α ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' (8) For α = 1 the expectation value in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' (7) is the energy corresponding to the Hobj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' While there is a freedom in the choice of g1 and g2, throughout the paper we a simple case g1(α) = 1 − α, g2(α) = α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' (9) During the optimization process, we choose an initialization of mixer and objective parameters (at α = 0) in such a way that the parameters corresponding to the mixer are sampled from the uniform random distribution U(a, b) in an interval [a = 0, b = 2π] and the objective parameters are all set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' With this initialization we make sure that the homotopy starts from the exact ground state of the mixer on a noise-free setting, as application of mixer on its eigenstate does not change the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' For α′ > α ≥ 0 the initial parameters are chosen as (⃗γ, ⃗β)init α′ = (⃗γ, ⃗β)∗ α, (10) here ∗ denotes the optimal parameters for α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='30 E∗norm(α) RR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='0 α NZR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='0 ZR Figure 2: The impact of different methods of initialization of γj, βj on HOHo-QAOA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' The left, the middle and the right figures are representing the convergence for RR (Random Random), NZR (Near- Zero Random) with parameter v = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='05, and ZR (Zero Random) initialization respectively, see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='2 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' It is visible that the ZR is outperforming the other two initializations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' Although for αinit ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='2 the performance of NZR and ZR are comparable but as we tune αinit > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='2, the minima for NZR scatters in region 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='10 < Enorm < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='15 whereas the minima for ZR clusters in a very narrow Enorm-width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' It should be noted that each run of HOHo-QAOA follows the generic structure of homotopy process as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' (8) where the “run-time” of HOHo-QAOA is characterized by the αstep, for a fixed αinit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' The parameter αinit fixes the initial α value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' Generally, it can be inferred that better approximation to the optimal solution can be achieved if we choose a sufficiently small value of αstep and αinit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' They can be described in a more elaborated way as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' Small value of αstep helps us realizing the homotopy of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' (8) and at the same time if we initiate with αinit → 0, it becomes easier to find the ground state for the first step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' To show this, throughout the paper, we investigate the normalized energy Enorm(Eα(⃗γ, ⃗β), α) = Eα(⃗γ, ⃗β) − minH(α) maxH(α) − minH(α), (11) with respect to parameters of HOHo-QAOA, where Enorm(α) = 0, is the normalized ground energy for any α ∈ [0, 1], and min H(α) (max H(α)) denotes minimum (maximum) of H(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='2 Initialization strategy In the following, first we numerically discuss proposed settings for initial QAOA parameters (⃗γ, ⃗β)init.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' With this setting we show that the homotopy parameters i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' αinit, αstep can be chosen detached from zero without compromising the efficiency of the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' We consider and optimized energy E∗ norm, or in the case of HOHo-QAOA also an intermediate optimized step energy E∗ norm(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' In the numerical results the E∗ norm is averaged over 100 experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' Details of the experiment can be found in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' For the numerical investigation of optimal QAOA parameters, which is illustrated in Figure 2, we consider three possible initialization choices of the mixer and objective parameters at α = αinit: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' RR (Random Random): When the parameters corresponding to mixer and objective Hamiltonians are chosen from a uniform random distribution U(0, 2π) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' γinit j ∼ U(0, 2π), βinit j ∼ U(0, 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' NZR (Near-Zero Random): The parameters corresponding to mixer Hamiltonian are chosen from U(0, 2π) but objective parameters are sampled from the values very close zero i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' γinit j ∼ U(0, v), βinit j ∼ U(0, 2π), where v is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' ZR (Zero Random): Mixer parameters are sampled from U(0, 2π) chosen and objective is all zeros i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' γinit j = 0, βinit j ∼ U(0, 2π) as proposed before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' From Figure 2 we conclude that ZR gives the best approximation to the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' This is because, under the ZR setting the initial parameters of QAOA always starts corresponding to the exact ground 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='00 αinit 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='35 E∗norm(α = 1) (a) 10−4 10−2 100 αstep 10−2 10−1 (b) 6 qubits 10 qubits 16 qubits Figure 3: We illustrate the dependency of E∗ norm with αinit and αstep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' In (a) the variation of E∗ norm with αinit for 3 layers of HOHo-QAOA is presented, with γinit j = 0, βinit j ∼ U(0, 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' In the figure, we see a region of stability of HOHo-QAOA in respect with αinit in the range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='0 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' In (b) we present E∗ norm vs αstep using 10 layers of HOHo-QAOA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' Just like in the case of αinit, for αstep a same region of stability can be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' This gives us the preference on the choice of step parameter while utilizing HOHo-QAOA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' It should be noted that the y-axis in (a) is in linear scale and whereas in (b) it is in log scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' The lines in both the plots are taken αinit and αstep-wise and is the mean of 100 experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' The area under the plots are standard deviation of energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' state of Hmix while the Hobj is turned off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' This is within the spirit of the homotopy optimization, in which starting in the optimal solution of the initial system is critical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' Hence this good approximation to the initial parameters lead us to the better solution to the ground state of Hobj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' Keeping in mind that if we sample αinit in the range 0 ≤ αinit ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='2, we see that NZR shows comparable performance to ZR and the choice of initialization of γj, βj can be either one of them, relaxing the conditions on the choice of ⃗γ and ⃗β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' In the remaining of this paper all the numerical results are initialized with the ZR setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' Now we move to the analysis of the choice of suitable αinit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' In the Figure [3] we investigate the αinit dependency of the E∗ norm, where the energy is averaged over 100 experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' From Figure [3](a) we take 3 layers of HOHo-QAOA and observe that the mean optimal energy and the corresponding standard deviation remain unchanged (which we term as region of stability) with respect to αinit in the range 0 ≤ αinit ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' With an increase in the number of nodes from 6 to 16, the region of stability shifts upwards but remains in the range 0 ≤ αinit ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' This observation lead us to conclude that αinit can be chosen detached from zero without degrading the performance of HOHo-QAOA, or that at least that the region of stability does not shrink rapidly with the increased size of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' So setting αinit in the region of stability along with γinit j = 0, βinit j ∼ U(0, 2π) yields a solution with particularly small energy value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' In Figure [3](b) we investigate how the efficiency of the optimization depends on the αstep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' During this investigation, we take 10 layers of HOHo-QAOA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' We observe that in the range 10−4 ≤ αstep < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='5 the approximation to the ground energy and the corresponding standard deviation with increasing αstep → 0 remains almost unchanged, giving rise to a region of stability with respect to αstep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' This behavior of E∗ norm with αstep is similar to what we can observe for αinit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' This lead us to a conclusion that one can choose αstep detached from zero for HOHo-QAOA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' It should be noted that due to high simulation cost the experiment for 16 qubits is halted at the αstep = 10−2, whereas the investigation for 6, 16 qubits is extended to 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' The discussion and numerical results from the previous paragraphs give us the following initialization rules of HOHo-QAOA, which leads to a high efficiency of the method: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' The parameters of mixer and objective should be initialized with ZR setting i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' γinit j = 0, βinit j ∼ U(0, 2π), 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' Although one can infer that αinit → 0 along with αstep → 0 gives the best result, our investigations 6 25 50 75 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='3 E∗ norm QAOA rand-rand zero-rand 25 50 75 100 10−2 10−1 T-QAOA rand-rand zero-rand Number of layers Figure 4: Comparison of different initialization for QAOA and T-QAOA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' In the left (right) figure we illustrate how the E∗ norm changes with increasing number of layers in QAOA (T-QAOA) under the RR and ZR settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' The solid line is the median energy over 100 experiments, meanwhile the dashed line represents the best sample, taken layer-wise and node-wise by choosing the minimum energy among all the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' The areas are delimited by the first and third quartile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' 25 50 75 100 Number of layers 10−3 10−2 10−1 E∗ norm 10 15 Number of nodes 10−1 QAOA T-QAOA HOHo-QAOA Figure 5: Performance of HOHo-QAOA compared to QAOA and T-QAOA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' On both figures, for all the QAOA methods, we applied the ZR settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' The areas are delimited by the first and third quartile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' The solid line presents E∗ norm median over 100 experiments for the left figure and 50 experiments for the right figure, and the dashed line represents the best sample, taken layer-wise and node-wise by choosing the minimum energy among all the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' On the left figure, the number of nodes is fixed to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' On the right, the number of layers is fixed to 5 and the energy is sampled within 6 to 18 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' The homotopy parameters are set as αinit = 0 and αstep = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' One can see that in both cases the averaged energy as well as the best sample of HOHo-QAOA outperforms the other variants of QAOA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' show that one can choose the homotopy parameters detached from zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' This greatly reduces the cost of simulating HOHo-QAOA 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='3 Performance analysis In this section we analyze the performance of the introduced algorithm with respect to other optimization strategies introduced above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' While it is natural for HOHo-QAOA to initialize using ZR strategy, it is unclear whether this choice will improve or worsen the results for QAOA or T-QAOA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' Therefore before comparing state-of-the-art methods to the introduced one, we verify whether there is any difference in the performance for QAOA and T-QAOA with respect to the initialization of the optimized angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' 4 we investigate state-of-the art methods for parameters (γj, βj)init initialized with RR and ZR strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' We observe that the performance of QAOA and T-QAOA is not influenced by the chosen strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' This justifies using ZR strategy when comparing QAOA, T-QAOA and HOHo-QAOA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' Note that for QAOA we are observing undesired non-monotonic behavior with respect to the number of layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' We claim that this is caused because of a complicated landscape of the energy function, which makes difficult to optimize it if no information about the problem instance is used during the initialization from large number of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' This argument is complies with good performance of T-QAOA where the initial parameters of (L + 1)-layer step is evaluated based on local optimal solutions of the L-layers step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' 5 we compare the performance of HOHo-QAOA with the other variants when (γj, βj)init are initialized using ZR setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' In the first experiment we run the algorithms with a fixed number of nodes while increasing number layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' In the second experiment the number of layers is fix while we vary the number of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' The plots present optimized energy values, averaged respectively over 100 and 50 instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' The data shows that the introduced HOHo-QAOA gives us significantly smaller energy in both experiment setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' Good improvements remains as more layers of the HOHo-QAOA are used and also outperforms the other varients of QAOA for higher number of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' This conclusions remain valid also for the best sample solution chosen (dashed line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' It should be noted that the HOHo-QAOA outperforms QAOA and the T-QAOA in each and every layer starting from initial layer 5 to final layer 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' 4 Conclusion In the article we present a novel algorithm for combinatorial optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' The method is a combination of homotopy optimization with an application in QAOA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' In our method the observable used for computing the energy is changed during the optimization process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' The process starts with observable being a mixer, for which the initial state of QAOA is a grounds state, and is slowly moved into the objective Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' In addition we verify that, although traditionally in the homotopy method the initial value of transition parameter α should be 0 and the step should be as small as possible, for QAOA for the value of considered parameters can be detached from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' A homotopy optimization is an algorithm dedicated for nonlinear optimized functions, and since even simple QAOA landscape is a linear combination of many – for some problems exponentially many – sinusoidal functions, our approach is well motivated for such energy function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' This is in contrast to typical VQE optimization process, in which the function landscape with respect to a single parameter is just a sine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' By comparing our approach and QAOA algorithm with the typical choice of optimization strategies we numerically confirmed that our method outperforms state-of-the-art approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' While our algorithm was only presented for QUBO and X-mixer,it is not restricted to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' In particular, if the transition function is of the form H(α) = g1(α)Hmix + g2(α)Hobj, we only require energy of the Hmixer to be efficiently computable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' This includes XY-mixer [46] and Grover mixer [45] for which the initial state can be efficiently prepared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' Moreover, our approach remains also valid for higher-order binary problems [11,15] and more advanced pseudo-code based QAOA Hamiltonian implementation [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' Acknowledgment A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=', A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' and L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' has been partially supported by Polish National Science Center under grant agreements 2019/33/B/ST6/02011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' acknowledges support from National Sci- ence Center under grant agreement 2020/37/N/ST6/02220.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' The authors would like to thank Zolt´an Zim- bor´as, ¨Ozlem Salehi and Jaros�law A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' Miszczak for valuable discussions and comments on the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' Data and code availability Data and code available in https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='5281/zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='7585691 References [1] J.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' 5, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' 391, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' [51] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' Nakanishi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' Fujii, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' Todo, “Sequential minimal optimization for quantum-classical hybrid algorithms,” Physical Review Research, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' 2, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' 4, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' 043158, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' A Experiment details In order to enable the simple reproduction of our results, we publish our code on .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' The algorithms for generating data and plotting were implemented in Julia and Python programming languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' Versions of the software and additional packages are listed in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' Experiments Each experiment of HOHo-QAOA is uniquely characterized by the random graph G = (V, E), that is chosen from Barab´asi-Albert distribution with 6, 8, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' 18 nodes and with m = 2, where m defines the number of edges to be attached from a new node to existing ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' The weights corresponding to the edges are picked up from a uniform set of integer weights wjj′ ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' , 10} for each edge {j, j′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' Data Sampling For sampling the objective Hamiltonian, we started by generating graph objects an them converting to Pauli operators objects and Hamiltonian matrices with Qiskit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' We generated 100 samples for each graph setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' We sampled the initial optimization parameters in a random distribution for RR and ZR approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' We emulated the quantum evolution and take an exact expectation energy and gradient of the state during the optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' We choose the L-BFGS algorithm implemented in Julia’s Optim package as a subroutine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' The optimization has no periodic or bounds conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' We setup Optim with absolute tolerance, relative tolerance and absolute tolerance in gradient equal to 1−9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' We allowed steps that increase the objective value and maximum number of iterations is 10000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' 11 T-QAOA For T-QAOA implementation, we initialize with a minimum number of levels L0 = 4 and run the optimization similarly to the state of art QAOA with the a given parameters initiation strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' The method proceeds checking the convergence of the solution and moving to the next layer L0 + 1, using the previous optimized parameters with the addition of a zero for the mixer Hamiltonian and a value sampled from a uniform random distribution U(0, 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' B Proof of nonlinear landscape for QAOA Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' Let ϱ be an arbitrary quantum state, H be an arbitrary Hamiltonian with spectrum set {E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' , Ek} and O be an arbitrary observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' Then tr(exp(−iθH)ϱ exp(iθH)O) = C + � i>j Ai,j cos(θ(Ei − Ej) + Bi,j), (12) for some real values C, Ai,j, Bi,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' Let U be a unitary that diagonalizes the Hamiltonian H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' Then we have tr(exp(−iθH)ϱ exp(iθH)O) = tr � � k � i=1 (Ue−iθEi |i⟩⟨i| U †)ϱ k � j=1 (UeiθEj |j⟩⟨j| U †)O � � = k � i=1 k � j=1 eiθ(Ej−Ei) tr � U |i⟩⟨i| U †ϱU |j⟩⟨j| U †O � = k � i=1 k � j=1 eiθ(Ej−Ei) tr (|i⟩⟨i| ϱ′ |j⟩⟨j| O′) = k � i=1 k � j=1 eiθ(Ej−Ei) ⟨i| ϱ′ |j⟩ ⟨j| O′ |i⟩ , (13) where ϱ′ = U †ϱU and O′ = U †OU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' Since ϱ′ is a hermitian operator and therefore ⟨i| ϱ |j⟩ = ⟨j| ϱ |i⟩, and similarly for O′, therefore for any i, j the term for i > j is a conjugate of the term i < j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' Hence k � i=1 k � j=1 eiθ(Ej−Ei) ⟨i| ϱ′ |j⟩ ⟨j| O′ |i⟩ = k � i=1 ⟨i| ϱ′ |i⟩ ⟨i| O′ |i⟩ + 2 � i>j Re eiθ(Ej−Ei) ⟨i| ϱ′ |j⟩ ⟨j| O′ |i⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' (14) Note that the left hand side sum in the above above is a free term and is a real number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' Starting from now we will assume that the Hamiltonian H is non-degenerate – otherwise the corresponding element of the right sum will contribute to the free term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' Taking xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='j + iyi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='j := ⟨i| ϱ′ |j⟩ ⟨j| O′ |i⟩ for some real xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' yi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='j we have Re eiθ(Ej−Ei) ⟨i| ϱ′ |j⟩ ⟨j| O′ |i⟩ = Re(cos(θ(Ej − Ei)) + i sin(θ(Ej − Ei)))(xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='j + iyi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='j) = xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='j cos(θ(Ej − Ei)) − yi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='j sin(θ(Ej − Ei)) = � x2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='j + y2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='j � � xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='j � x2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='j + y2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='j cos(θ(Ej − Ei)) − yi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='j � x2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='j + y2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='j sin(θ(Ej − Ei)) � � = � x2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='j + y2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='j (cos(αi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='j) cos(θ(Ej − Ei)) − sin(αi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='j) sin(θ(Ej − Ei))) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' (15) where αi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content='j is such a real number for which the above transformation holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' Note that such a number α can always be found as the replaced fraction squared sum to 1 and one can use Pythagorean trigonometric identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' Finally we have � x2 i,j + y2 i,j (cos(αi,j) cos(θ(Ej − Ei)) − sin(αi,j) sin(θ(Ej − Ei))) = � x2 i,j + y2 i,j cos(θ(Ej − Ei) + αi,j), (16) which proves the statement of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' Note that the case of Hamiltonian with two different eigenvalues was already presented in [50,51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} +page_content=' 12' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FPT4oBgHgl3EQfyDV1/content/2301.13170v1.pdf'} diff --git a/cdE5T4oBgHgl3EQfEw5P/vector_store/index.faiss b/cdE5T4oBgHgl3EQfEw5P/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..4b93f989170f5f49fbf4d658789c3ccfb4d1dca9 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sha256:5ac79dd6f69dfc28ef288d0e4b47641ba2aafe12d68f1eba5faef026b845c56a +size 90591 diff --git a/gtE0T4oBgHgl3EQf6QKw/content/tmp_files/2301.02762v1.pdf.txt b/gtE0T4oBgHgl3EQf6QKw/content/tmp_files/2301.02762v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..c88be1a0e093e5792200625db8e92ad17c9c4481 --- /dev/null +++ b/gtE0T4oBgHgl3EQf6QKw/content/tmp_files/2301.02762v1.pdf.txt @@ -0,0 +1,1171 @@ +Active control of higher-order topological corner states +in a piezoelectric elastic plate +Ze Ma, Yang Liu, Yu-Xin Xie,∗ and Yue-Sheng Wang +School of Mechanical Engineering, Tianjin University, Tianjin 300350, China +(Dated: January 10, 2023) +Different from the traditional bulk-edge correspondence principle, the discovery of higher-order +topological states has generated widespread interest. +In a second-order, two-dimensional elastic +wave topological insulator, the fluctuation information can be confined to the corners, with the state +being topologically protected. In order to better apply topological corner states, this paper designs +a two-dimensional elastic plate with adjustable topological corner states by means of piezoelectric +control capability. By selectively connecting negative capacitance circuits to piezoelectric sheets +on the honeycomb elastic plate, the energy band can be flipped. The topological corner states at +the 2π/3 corner were observed at the boundary of two different topological phase structures in the +finite lattice with finite element software. The strong robustness of the topological corner states +was verified by setting up defective control groups at the corner. In addition, the topological corner +states of this piezoelectric elastic plate are discussed accordingly in terms of their tunability in +frequency and position. The piezoelectric elastic plate is expected to provide a reference for the +design of elastic wave local control and energy harvesting devices due to its adjustable topological +corner states, which facilitate the application of topological corner states in practice. +Keywords: High-order topological insulators, Adjustable corner state, Piezoelectric elastic plate +I. +INTRODUCTION +Topological Insulators (TIs) were originally studied in +condensed matter physics[1–3]. +Their properties such +as excellent suppression of fluctuation backscattering +and strong robustness have attracted many researchers +to explore them. +Depending on the characterization +of their topological invariants, they can be classified +as quantum Hall insulators, quantum spin Hall insula- +tors, and quantum valley Hall insulators[4–7]. In recent +years, topological devices have been extensively studied +in the field of electromagnetic[8–10], acoustic[11–15] and +elastic[16–22] waves. Their construction replies primarily +on breaking the temporal and spatial inverse symmetry +of the structure. Recently, higher-order TIs(HOTIs) have +been theoretically predicted based on the extended bulk- +edge correspondence principle[23, 24]. +Unlike conven- +tional first-order TIs, in second-order two-dimensional +TIs, the (2-1)1D boundary state lacks topological pro- +tection, but maintains good robustness in the (2-2)0D +corner state[25, 26]. +Experimental studies of HOTIs +have also developed rapidly, as higher-order topological +states have been better verified in topological metamate- +rials with quantized quadrupole[27–29]. By constructing +a magnetic field to generate π flux in the electromag- +netic tetragonal lattice, the corner modes of the two- +dimensional material can be observed accordingly[26]. +Based on an extension of the one-dimensional SSH model, +the corner states of two-dimensional HOTIs are realized +in lattices such as kagome, tetragonal and hexagonal +lattices[25–28, 30–41]. +Active modulation of band gap frequencies and waves +∗ xyx@tju.edu.cn +can be achieved by introducing some external factors into +the phononic crystal or metamaterial[42]. +In previous +first-order elastic wave TIs, Gao et al. broke the mir- +ror symmetry by rotating Y-shaped steel prisms on a +substrate to achieve wide bandgap reconfigurable paths +for topological valley transmission[43]. Zhang et al. de- +signed a programmable lifting magnetic cavity to con- +trol the filling of the magnetic fluid to break the spa- +tial inversion symmetry, and observed tunable topologi- +cal valley transport in the experiment[44]. By simulating +the quantum valley Hall effect, Liu et al. achieved re- +configured topological waveguides via switching the sub- +stable structure in the unit cell[45]. Taking advantage +of the electrodynamic deformation properties of the di- +electric elastomer, Chen et al. have investigated the ac- +tive control of pseudospin boundary states in the soft- +film type metamaterial, widening the frequency operat- +ing range[46]. +By using the piezoelectric control tech- +nique, Li et al. controlled the appearance and disappear- +ance of the double Dirac cone by adjusting negative ca- +pacitance circuits connected to piezoelectric sheets, and +verified the strong robustness of the topological waveg- +uide in the experiments[47]. +Piezoelectric shunt tech- +nology was first proposed by Forward, as well as being +well developed for active control in the field of sound +and vibration[48–54]. By connecting a negative capaci- +tance circuit to the piezoelectric sheet of the metamate- +rial plate, it can be regarded as an element of adjustable +stiffness. The local resonant frequency of the resonator +can be controlled by adjusting the negative capacitance +circuit[50, 54]. +Following the study of HOTIs for electromagnetic and +acoustic waves, the corner states of elastic wave HOTIs +have recently begun to be explored as well. Fan et al. de- +signed elastic honeycomb phonon crystal plates with dif- +ferent intercellular and intracellular coupling strengths +arXiv:2301.02762v1 [physics.app-ph] 7 Jan 2023 + +2 +and experimentally observed the presence of topologi- +cal corner states[40]. +Based on the study of acoustic +Wannier-type HOTIs, Wu et al. +have experimentally +verified two-dimensional second-order corner states in an +elastic Kagome lattice plate[55]. Chen et al. observed +mechanical topological corner states at the interface be- +tween two structures, topologically trivial and topolog- +ically non-trivial, by cyclically mounting steel bolts on +the aluminum plate[41]. However, there lacks research on +the tunable corner states of elastic-wave two-dimensional +HOTIs. +Thus, inspired by the structure of the one- +dimensional local resonance tunable topological states by +Liu et al.[53] we attached negative capacitance circuits +to piezoelectric sheets on the inner or outer supports of +the hexagonal lattice. In this way, the topological cor- +ner states of the elastic plate can be actively tuned. By +adjusting the size of the capacitor parameter, the operat- +ing frequency range of its topological corner states can be +widened. And by selectively connecting negative capac- +itance circuits to the piezoelectric sheets of the holder, +active control of the topological corner states position +can be achieved. +II. +STRUCTURAL DESIGN AND +TOPOLOGICAL ENERGY BAND INVERSION +OF THE PIEZOELECTRIC ELASTIC PLATE +As shown in Fig. 1a, the piezoelectric elastic plate de- +signed in this paper is composed of the substrate, the +cylindrical oscillators and the piezoelectric sheets. The +material of the substrate is epoxy resin in the shape of +a continuous hexagonal lattice with thickness d = 2 mm, +side length L = 15 mm and width w = 5 mm. At the +top and bottom of the substrate are pasted P-4 piezoelec- +tric sheets of the same thickness as the substrate. The +six corners of the hexagonal lattice have cylindrical mag- +net oscillators glued above and below them, where the +oscillators have radius r = 2.5 mm and height h = 5d. +The material parameters of the magnet, the epoxy and +the P-4 piezoelectric sheet are shown in Tab. +I. The +multi-cell structure of the piezoelectric elastic plate can +be seen in Fig. 1b. The part of the diagram enclosed +by the yellow wire frame is taken as the unit cell, with +the length of the cell a = 45 mm. In order to distin- +guish between the piezoelectric pieces, we marked the +inner and outer hexagonal piezoelectric pieces as purple +and blue respectively. The negative capacitance circuit +in the active control system is illustrated in Fig. 1(c). +It includes capacitor Cp, compensation resistor R0, oper- +ational amplifier LM324N, fixed resistor R1 and sliding +resistor R2. Tab. II provides the parameters for each de- +vice of the negative capacitance circuit. The equivalent +bending stiffness of the piezoelectric sheet can be con- +trolled using a negative capacitance circuit connected to +the P-4 piezoelectric sheet. A topological phase change +of the elastic plate is thus achieved. +In the case when the piezoelectric sheet is connected +with a negative capacitance circuit, we derive the equiv- +alent elastic module of the piezoelectric sheet as follows. +The x, y and z directions are denoted as 1, 2 and 3 re- +spectively. Since the piezoelectric sheet is subjected to +an electric field along the z-axis only, the intrinsic equa- +tion of the piezoelectric material in the plane stress state +can be expressed as[47, 53, 54] +S1 = sE +11T1 + d31E3, +D3 = d31T1 + εT +33E3, +(1) +where S1 and T1 are the strain and stress along the x- +direction, sE +11 represents the coefficient of flexibility under +a constant electric field, d31 and εT +33 are the piezoelectric +and dielectric constants, respectively, and E3 and D3 de- +scribe the electric field strength and potential shift along +the z-direction. +At a constant strain, the intrinsic capacitance of a +piezoelectric sheet can be denoted as +Cp = εT +33Ash−1 +p , +(2) +where As and hp indicate the area and thickness of the +piezoelectric sheet respectively. +Then, introducing the +complex impedance Z, the relationship between strain +and stress can be derived as +S1 = +� +sE +11 − +sZd2 +31As +(1 + sZCp)hp +� +T1, +(3) +where s is the Laplace parameter. +The equivalent modulus of a piezoelectric sheet con- +nected to a circuit with negative capacitance can be de- +rived from equation (3) as follows: +Ep = +hp(1 + sZCp) +hpsE +11(1 + sZCp) − sZd2 +31As +. +(4) +Its complex impedance Z is expressed as +Z = − (αsCp)−1 , +(5) +where α = R2C(R1Cp)−1. +Therefore, we can make the elastic modulus of the +piezoelectric sheet varied by adjusting the parameter α, +i.e., regulating the joint stiffness of the piezoelectric elas- +tic sheet. +Next, the band structures of the unit cell were analyzed +using COMSOL Multiphysics software. The stiffness fac- +tor is the same for each holder when neither the inner +nor outer piezoelectric sheet is connected to the negative +capacitance circuit. In Fig. 2(a), the four bands degen- +erate at point Γ, forming a double Dirac cone. In the +band structures we are only concerned with the out-of- +plane modes (marked in red), and the in-plane modes +(marked in grey) are not considered. The stiffness factor +of the corresponding elastic support is enhanced when +an internal or external piezoelectric sheet is connected +to a negative capacitance circuit. As a result, the dou- +ble Dirac cone disappears and a band gap forms at the + +3 +y +x +z +𝑤 +𝑙 +Negative +capacitance +circuit ++ +− +𝑅0 +𝐶0 +𝐶𝑝 +𝑅1 +𝑅2 +LM324N +(a) +(b) +(c) +Epoxy shelf +P-4 piezoelectric sheet +Lead cylinder +𝑎 +h +𝑟 +𝑑 +FIG. 1. +(a) Schematic of the unit +cell structure of a piezoelectric elastic +plate. The yellow honeycomb support +serves as the flexible plate substrate. +P-4 piezoelectric sheets are attached +to the top and bottom of the holder, +with the piezoelectric sheets angled at +120◦ at both ends. The parts marked +in purple are the internal hexagonal +piezoelectric sheets, and the external +ones are marked in blue. Cylindrical +magnet oscillators are glued to the top +and bottom of the six corners of the +honeycomb elastic plate. Negative ca- +pacitance circuitry is connected to the +surface of the piezoelectric sheets. (b) +Composite unit structure of piezoelec- +tric elastic plates. The part enclosed +by the yellow line frame is taken as the +unit cell. (c) Schematic of the negative +capacitance circuit. +TABLE I. Material parameters for piezoelectric elastic plate. +Material +Modulus +E(Pa) +Density +ρ(kg/m3) +Compliance +coefficient +sE +11 (m2/N) +Piezoelectric strain +coefficient +d31(C/N) +Permittivity +εT +33(F/m) +Magnet +41×109 +7400 +... +... +... +Epoxy +3×109 +1180 +... +... +... +P-4 +piezoelectric sheet +8.83×1010 +7450 +1.2×10−11 +-1×10−10 +1.2×10−8 +TABLE II. The parameters of the negative capacitance +circuit. +C(pF) Cp(pF) R2(kΩ) R1(kΩ) R0(kΩ) Operational amplifier +9.684 +8.155 +51.54 +68 +2000 +LM324N +corresponding location nearby. As can be seen in Fig. +2(b), the forbidden band width widens as the parame- +ter α in the negative capacitance circuit increases. The +d modes in the band structure are above the band gap +and the p modes are below when the external piezoelec- +tric sheets are connected to negative capacitance circuits. +Instead, as the internal piezoelectric sheets are connected +to negative capacitance circuits, the d and p modes are +interchanged in their distribution above and below the +band gap. +A transition from a trivial state to a non- +trivial state occurs in the topological phase during this +period. Specifically, external powering produces the triv- +ial bandgap (Fig. +2(c)), while internal powering cor- +responds to the non-trivial bandgap (Fig. +2(d)). +The +eigenmode diagrams are plotted for the high symmetry +point Γ in the upper and lower bands forming the band +gap. They are shown accordingly at the top and bot- +tom of the band structures diagram. In this case, the d +modes represent quantized four polarization. The mode +distribution of dx2−y2 is symmetric about the x and y +axes, while the mode of dxy is antisymmetric. +The p +modes represent quantized couple polarization. px and +py modes are antisymmetrically distributed about the y +and x axes respectively[19]. +III. +DISCUSSION OF TOPOLOGICAL CORNER +STATES +Further, the supercell dispersion relation of the piezo- +electric elastic plate was analyzed. This supercell con- +sists of six internally powered unit cells on the left and +six externally powered unit cells on the right. The left +and right ends of the supercell are the free boundary +condition, and the Floquet-period boundary condition +along the y-axis is imposed at its upper and lower ends. +The dispersion relation obtained from the computational +analysis is shown in Fig. 3(a), with the grey and light +blue marked points being the in-plane and out-of-plane +modes respectively. We focus only on the out-of-plane +modes, with two clear topological boundary states (green +and purple) in the middle part, corresponding to the chi- +ral propagation model generated by the pseudo-spin Hall +effect. +The topological boundary vibration mode dia- +grams corresponding to points A1 and A2 are shown in +Fig. 3(b), where it can be observed that the displace- +ment is greatest at the intermediate interface. To bet- +ter describe the two topological pseudo-spin states, we +show the energy flow arrows (yellow) at the interface + +Ky +K +M +K" +r +/K +K +K +K'4 +(c) +(d) +Non-trivial +p +d +𝑑𝑥2−𝑦2 +𝑑𝑥𝑦 +min +max +𝑝𝑦 +𝑝𝑥 +Trivial +d +p +𝑝𝑦 +𝑝𝑥 +min +max +(a) +(b) +𝑑𝑥2−𝑦2 +𝑑𝑥𝑦 +FIG. +2. +(a) +The +unit +cell +dispersion +curve +for +the piezoelectric sheet with- +out +the +negative +capaci- +tance circuit connected. (b) +Curves for the variation of +band gap width with pa- +rameter α when the honey- +comb elastic plate is inter- +nally or externally charged. +(c), (d) Band structures and +their +intrinsic +mode +pat- +terns when the external and +internal piezoelectric sheets +are charged for α= 0.9. +External powering +Internal powering +𝐴2 +max +min +(a) +(c) +𝐴1 +0 +𝜋 +−𝜋 +𝐴1 +(b) +y +x +z +𝐴2 +Arg(w) +Fig 3. +(a) Dispersion re- +lation diagram of the su- +percell. +(b) Schematic of +the structure of the super- +cell and the corresponding +vibrational modes at points +A1 and A2. +The color bar +represents the size of the +absolute value of the dis- +placement in the z-direction +of the model. +(c) Cloud +field plot of the phase dis- +tribution and energy flow +arrows for the z-direction +eigenstates at the modal in- +terface of points A1 and A2. +as in Fig. +3(c). +In the boundary mode at points A1 +and A2, the energy flow appears in clockwise and coun- +terclockwise directions respectively. Meanwhile, we plot +the phase field of the eigenstates at the boundary as the +cloud field. It can be observed that the phases of the +two boundary modes at the same position are opposite +to each other. +This suggests that the two pseudospin +boundary states have propagation modes in opposite di- +rections. But such boundary states differ from the gap- +less edge states of fermion pseudospin protected by time- +reversal symmetry[56]. +In the middle of the two edge +states of this model band structure, there is a grey for- +bidden band. The band gap is created by the breaking of +C6 crystal symmetry at the boundary of different topo- +logical phase structures[41]. We try to find the topologi- +cal corner state by referring to the frequency distribution + +6000 +5000 +Frequency(Hz) +4000 +3000 +- +2000 +1000 +-- +1 +-- +--. +0 +M +K +M +Wave vector- +5000 +--- +4000 +1 +1 +3000 +2000 +DoubleDirac +Cone +1 +1000 +-- +- +o +0 +M +K +M +Wavevector4400 +4400 +dimodes +4200 +p;modes +4200 +4000 +4000 +3800 +External powering +3800 +3600 +3600 +Gap +3400 +3400 +Gap +3200 +Internal powering +3200 +3000 +3000 +06'0 88'0 98'0 80 8'0 08*006'0 88'0 98*0 8'0 80 08*0 +aout +ain6000 +5000 +Frequency(Hz) +4000 +3000 +- +1 +2000 +1000 +-- +- +-- +- +0 +M +T +K +M +Wavevector4400 +4200 +4000 +3800 +3600 +3400 +3200 +3000 +0 +K.(π/a)9M5 +of this band gap. +In order to analyze the topological corner states of +the piezoelectric elastic plate, we constructed a 13×13 +diamond-shaped finite lattice model as shown in Fig. +4(a). The parameter α for the negative capacitance cir- +cuit connected to the piezoelectric sheet is set to 0.9. The +external bracket of the 7×7 unit cell inside the model is +charged, corresponding to the trivial phase structure an- +alyzed above. The outer unit cell is then a non-trivial +phase structure. The two different phase structures thus +form a diamond-shaped boundary. The eigenfrequencies +of the model were calculated in COMSOL Multiphysics +software and the resulting eigenfrequency spectrum is +shown in Fig. 4(b). The presence of bulk states (grey +circles), edge states (blue square triangles), topological +corner states (red rhombuses) and trivial corner state +(green inverted triangle) can be observed. +The corre- +sponding vibration mode diagrams for the edge and bulk +states are plotted in Figs. 4(c, d). We are primarily con- +cerned with topological corner states that occur in the +bandgap range, which are topologically protected. The +topological corner state mode diagram corresponding to +the eigenfrequency of 3670.9 Hz is shown in Fig. 4(e). It +can be observed in the diagram that the larger absolute +values of out-of-plane displacement |w| are concentrated +at the 2π/3 corners of the bend at the rhombic boundary. +The corner state corresponding to 3649.3 Hz below the +band gap is the trivial corner state. In the corresponding +mode vibration diagram, the larger displacements occur +at the π/3 corners of the boundary. In analogy to the +theoretical explanation in the higher-order topology of +electromagnetic waves, there is a zero mode (topologi- +cal corner mode) at each uncoupled waveguide[57]. Each +zero mode holds one topological charge (+ or −). N+ and +N− are used to indicate the numbers of eigenstates of the +chiral symmetric operator Π with topological charges + +and −, respectively[40]. There are three zero modes at +the 2π/3 corner, two of which contain the same topo- +logical charge and one of which contains the opposite +topological charge. +As a result, the topological index +N = |N+ − N−| = 1 ̸= 0, representing the topological +corner state. The details can be found in Appendix B. +While the topological charge is positive at the two zero +modes at π/3, that at the other two zero modes is neg- +ative, so the topological index N=0. These theories can +help to explain the phenomenon that the topological cor- +ner states of the present honeycomb piezoelectric elastic +plate appear only at the 2π/3 corners. +Next, we verified the strong robustness of the piezo- +electric elastic plate topological corner states. The 2π/3 +corner of the interface between the different phase struc- +tures in the finite lattice is displayed magnified. Fig. 5(a) +shows the defect-free state, while Fig. 5(b, c) represent +the disturbed and cavity models, respectively. A point +at the 2π/3 corner of the boundary is chosen to plot the +normalized frequency energy spectrum as shown in Fig. +5(d). There is a clear high-energy peak in the grey band +gap of the defect-free structure, which corresponds to +Trival +Non-trival +(a) +min +max +7 × 7 +(c) +3741.8Hz +(e) +3670.9Hz +3869.2Hz +(f) +3649.3Hz +(d) +(b) +Fig 4. +Calculated eigenmodes of the piezoelectric elastic +plate.(a) Schematic diagram of a diamond-shaped finite lat- +tice. The interior is the externally charged trivial phase struc- +ture (cyan) and the periphery is the internally charged non- +trivial phase structure (yellow). (b) Numerical calculation of +the eigenfrequencies obtained. The red rhombus represents +the topological corner state, while the grey circle, blue square +triangle and green inverted triangle represent the bulk, edge +and trivial corner states respectively. (c) Edge state at 3741.8 +Hz. (d) Bulk state at 3869.2 Hz. (e) Topological corner state +at 3670.9 Hz. (f) Trivial corner state at 3649.3 Hz. The color +bar indicates the absolute magnitude of the displacement in +the z-direction. +the presence of the topological corner states. In the pres- +ence of disturbance and vacancy defects, the frequency +interval with the higher energy peak remains confined to +the band gap. +And the average elastic energy density +peaks at the corner of the defective structures are close +to the peak in the case of the defect-free structure, with +less dissipation. The resulting analysis indicates that the +topological corner state of the piezoelectric elastic plate +is insensitive to small defects with strong robustness. +To emphasize the frequency tunability of the topolog- +ical corner states of this piezoelectric elastic plate, we +investigated the topological corner state spectrum vari- +ation of its finite lattice as shown in Fig. 6, with the +piezoelectric parameter α as a variable. As α increases, +the stiffness of the holders connected to negative capac- +itance circuits rises and the topological corner states of +the piezoelectric elastic plate appear with an accordingly +higher frequency. Also, as the parameter increases, the +two topological corner states become closer and closer. +It means that the piezoelectric elastic plate can achieve + +4200 +Bulk +Edge +4000 +Topological corner (120°) +Trivial corner (60°) +3800 +3600 +3400 +3200 +0 +20 +40 +60 +80 +100 +120 +Solution number6 +No +defect +(a) +Disorder +(b) +Cavity +(c) +(d) +Fig 5. +(a) Partial diagram of +the finite lattice at the 2π/3 +corner. The purple holders in- +dicate that a negative capaci- +tance circuit is connected. Non- +trivial phase structures outside +the corner and trivial phase +structures inside the corner. (b) +Disturbed finite lattice model. +A holder at the corner is not +connected to the negative ca- +pacitance circuit, creating a dis- +turbance to the topology model. +(c) Cavity finite lattice model. +The absence of two oscillators +above and below the support at +the corner forms a cavity de- +fect topology model. +(d) Nor- +malized energy spectrum curves +for the non-defective and de- +fective +models. +The +grey +band represents the boundary +state band gap of the supercell. +The black wire represents the +non-defective model, with the +red and blue wires representing +the disturbed and cavity defect +models respectively. +Fig 6. Variation of the finite lattice frequency spectrum with +the piezoelectric parameter α. The grey dots indicate the bulk +and edge states and the red triangles indicate the topological +corner states. +topological corner states in multiple frequency ranges, +and the way to go is to tune the piezoelectric parameter. +Besides, compared to the finite lattice model in Fig. 4(a), +the total number of cells is kept constant by reducing and +increasing the number of trivial phase unit cells by 5×5 +and 9×9 respectively as in Fig. 7(a, b). We subjected +the two finite lattices to spectral analysis and could ob- +serve an inward shift (Fig. 7(c)) and an outward shift +Trival +Non-trival +5 × 5 +Non-trival +Trival +9 × 9 +(a) +(b) +(d) +Corner state outward shift +Corner state inward shift +min +max +(c) +Fig 7. (a)-(b) Schematic of the corner state inward and out- +ward shift model for the finite lattice (13*13 cells). (c)-(d) +Inward and outward shifted vibration mode graphs for corner +states. +(Fig. 7(d)) in the topological corner state. In practice, +the topological corner state position can be adjusted by +simply connecting the negative capacitance circuit to the +corresponding piezoelectric sheet. + +1.0 +No defect +Disorder +0.8 +Cavity +0.6 +0. 4 +0. 2 +0. 0 +3300 +3400 +3500 +3600 +3700 +3800 +3900 +4000 +4100 +Frequency(Hz)4000 +Frequency(Hz) +3800 +3600 +3400 +3200 +0.80 +0.82 +0.84 +0.86 +0.88 +0.90 +αEdge and Bulk +Topological corner7 +IV. +CONCLUSION +In this paper, a honeycomb-shaped piezoelectric elas- +tic plate is designed to achieve tunable topological corner +states of the elastic wave HOTI. By connecting negative +capacitance circuits to each of the inner or outer piezo- +electric sheets in the unit cell, the topological phase is +transformed accordingly. The band gap in the boundary +state is present in supercell containing both trivial and +non-trivial phase structures. The spectral analysis of its +finite lattice reveals the presence of topologically pro- +tected corner states. Echoing the theory, the topological +corner states in the hexagonal lattice were found to ex- +ist only at the 2π/3 corner. We designed both disturbed +and vacant defect models and plotted the frequency en- +ergy spectrum curves. The strong robustness of the topo- +logical corner states is verified by comparison with the +defect-free model. In addition, to highlight the tunabil- +ity of the topological corner states of the piezoelectric +elastic plate, we adjusted the piezoelectric parameter α +to broaden the operating frequency range of the topo- +logical corner states. Switching the position of the neg- +ative capacitance circuit connection enables inward and +outward shifts of the topological corner states, meaning +that the topological corner states can appear at any posi- +tion in the plane. The tunable topological corner state of +this piezoelectric elastic plate enriches the study of topo- +logical physical phenomena in mechanical metamateri- +als. This study is of reference value in applications such +as elastic wave energy localization, information transfer, +and energy harvesting. +ACKNOWLEDGEMENTS +This research is sponsored by the National Natural +Science Foundation of China (Grant nos. +12021002, +12072225, and 11991031). We thank Professor Yibin Fu +at Keele University for helpful discussions and valuable +advice. +Appendix A: Characterization of topological corner +states +By understanding the theory and methods in these +articles[46, 57, 58], we verify the topological corner prop- +erties of the piezoelectric elastic plate. Firstly the bulk +polarisation here is expressed as +χ(6) = ([M], [K]) +(A1) +where [M] and [K] are C2 and C3 invariants respec- +tively. They are integers that can be expressed as +[M] = #M1 − #Γ(2) +1 +[K] = #K1 − #Γ(3) +1 +(A2) +where M1 (Γ(2) +1 ) is the number of energy bands with C2 +(π) rotation eigenvalue +1 below the band gap at point +M (Γ) in the Brillouin zone. K1 (Γ(3) +1 ) is the number of +energy bands with C3 (π/3) rotation eigenvalue +1 be- +low the band gap at point K (Γ) in the Brillouin zone. +From Fig.??, we observe that the eigenmodes of the three +bands below the band gap are [M] = 0 for a trivial struc- +ture M1 = 1 and Γ(2) +1 += 1. For non-trivial structure M1 += 1 and Γ(2) +1 += 3, so [M] = 2 (ignoring the negative +sign). The C3 rotation operator and the chiral operator +are swapped in this piezoelectric elastic plate model. It +can be obtained that for both structures the invariant +[K] = 0. We summarise the topological invariants in the +two structures as follows +([M], [K]) = +� +(0, 0) +external +powering +(2, 0) +internal +powering. +(A3) +From the corner charge Q(6) +corner=[M]/4+[K]/6 in the +literature, it follows that +Q(6) +corner = +� +0 +external +powering +1/2 +internal +powering. +(A4) +To account for the difference in position between +the emergence of topological and trivial corner states, +the topological index N has been introduced to the +characterization[40, 57]. The topological index N cap- +tures the interaction between the topology of the bulk +Hamiltonian and the topology of the defect, representing +the stable mode at the corners of the boundary. From +Fig.8, it can be seen that N+=1 and N−=1 at the edges +not containing the corners of the rhombic finite lattice. +At the π/3 corners, N+= 2 and N−= 2. Therefore the +topological index at the edges and π/3 corners of the +piezoelectric elastic plate is 0. 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Hughes, Quantization of +fractional corner charge in cn-symmetric higher-order +topological crystalline insulators, Physical Review B 99, +245151 (2019). + diff --git a/gtE0T4oBgHgl3EQf6QKw/content/tmp_files/load_file.txt b/gtE0T4oBgHgl3EQf6QKw/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5a40445df7f8f89b0a89f8b2cd8682f56283ee8d --- /dev/null +++ b/gtE0T4oBgHgl3EQf6QKw/content/tmp_files/load_file.txt @@ -0,0 +1,643 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf,len=642 +page_content='Active control of higher-order topological corner states in a piezoelectric elastic plate Ze Ma, Yang Liu, Yu-Xin Xie,∗ and Yue-Sheng Wang School of Mechanical Engineering, Tianjin University, Tianjin 300350, China (Dated: January 10, 2023) Different from the traditional bulk-edge correspondence principle, the discovery of higher-order topological states has generated widespread interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' In a second-order, two-dimensional elastic wave topological insulator, the fluctuation information can be confined to the corners, with the state being topologically protected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' In order to better apply topological corner states, this paper designs a two-dimensional elastic plate with adjustable topological corner states by means of piezoelectric control capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' By selectively connecting negative capacitance circuits to piezoelectric sheets on the honeycomb elastic plate, the energy band can be flipped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' The topological corner states at the 2π/3 corner were observed at the boundary of two different topological phase structures in the finite lattice with finite element software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' The strong robustness of the topological corner states was verified by setting up defective control groups at the corner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' In addition, the topological corner states of this piezoelectric elastic plate are discussed accordingly in terms of their tunability in frequency and position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' The piezoelectric elastic plate is expected to provide a reference for the design of elastic wave local control and energy harvesting devices due to its adjustable topological corner states, which facilitate the application of topological corner states in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' Keywords: High-order topological insulators, Adjustable corner state, Piezoelectric elastic plate I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' INTRODUCTION Topological Insulators (TIs) were originally studied in condensed matter physics[1–3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' Their properties such as excellent suppression of fluctuation backscattering and strong robustness have attracted many researchers to explore them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' Depending on the characterization of their topological invariants, they can be classified as quantum Hall insulators, quantum spin Hall insula- tors, and quantum valley Hall insulators[4–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' In recent years, topological devices have been extensively studied in the field of electromagnetic[8–10], acoustic[11–15] and elastic[16–22] waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' Their construction replies primarily on breaking the temporal and spatial inverse symmetry of the structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' Recently, higher-order TIs(HOTIs) have been theoretically predicted based on the extended bulk- edge correspondence principle[23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' Unlike conven- tional first-order TIs, in second-order two-dimensional TIs, the (2-1)1D boundary state lacks topological pro- tection, but maintains good robustness in the (2-2)0D corner state[25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' Experimental studies of HOTIs have also developed rapidly, as higher-order topological states have been better verified in topological metamate- rials with quantized quadrupole[27–29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' By constructing a magnetic field to generate π flux in the electromag- netic tetragonal lattice, the corner modes of the two- dimensional material can be observed accordingly[26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' Based on an extension of the one-dimensional SSH model, the corner states of two-dimensional HOTIs are realized in lattices such as kagome, tetragonal and hexagonal lattices[25–28, 30–41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' Active modulation of band gap frequencies and waves ∗ xyx@tju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content='cn can be achieved by introducing some external factors into the phononic crystal or metamaterial[42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' In previous first-order elastic wave TIs, Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' broke the mir- ror symmetry by rotating Y-shaped steel prisms on a substrate to achieve wide bandgap reconfigurable paths for topological valley transmission[43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' de- signed a programmable lifting magnetic cavity to con- trol the filling of the magnetic fluid to break the spa- tial inversion symmetry, and observed tunable topologi- cal valley transport in the experiment[44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' By simulating the quantum valley Hall effect, Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' achieved re- configured topological waveguides via switching the sub- stable structure in the unit cell[45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' Taking advantage of the electrodynamic deformation properties of the di- electric elastomer, Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' have investigated the ac- tive control of pseudospin boundary states in the soft- film type metamaterial, widening the frequency operat- ing range[46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' By using the piezoelectric control tech- nique, Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' controlled the appearance and disappear- ance of the double Dirac cone by adjusting negative ca- pacitance circuits connected to piezoelectric sheets, and verified the strong robustness of the topological waveg- uide in the experiments[47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' Piezoelectric shunt tech- nology was first proposed by Forward, as well as being well developed for active control in the field of sound and vibration[48–54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' By connecting a negative capaci- tance circuit to the piezoelectric sheet of the metamate- rial plate, it can be regarded as an element of adjustable stiffness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' The local resonant frequency of the resonator can be controlled by adjusting the negative capacitance circuit[50, 54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' Following the study of HOTIs for electromagnetic and acoustic waves, the corner states of elastic wave HOTIs have recently begun to be explored as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' de- signed elastic honeycomb phonon crystal plates with dif- ferent intercellular and intracellular coupling strengths arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content='02762v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content='app-ph] 7 Jan 2023 2 and experimentally observed the presence of topologi- cal corner states[40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' Based on the study of acoustic Wannier-type HOTIs, Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' have experimentally verified two-dimensional second-order corner states in an elastic Kagome lattice plate[55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' observed mechanical topological corner states at the interface be- tween two structures, topologically trivial and topolog- ically non-trivial, by cyclically mounting steel bolts on the aluminum plate[41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' However, there lacks research on the tunable corner states of elastic-wave two-dimensional HOTIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' Thus, inspired by the structure of the one- dimensional local resonance tunable topological states by Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' [53] we attached negative capacitance circuits to piezoelectric sheets on the inner or outer supports of the hexagonal lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' In this way, the topological cor- ner states of the elastic plate can be actively tuned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' By adjusting the size of the capacitor parameter, the operat- ing frequency range of its topological corner states can be widened.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' And by selectively connecting negative capac- itance circuits to the piezoelectric sheets of the holder, active control of the topological corner states position can be achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' STRUCTURAL DESIGN AND TOPOLOGICAL ENERGY BAND INVERSION OF THE PIEZOELECTRIC ELASTIC PLATE As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' 1a, the piezoelectric elastic plate de- signed in this paper is composed of the substrate, the cylindrical oscillators and the piezoelectric sheets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' The material of the substrate is epoxy resin in the shape of a continuous hexagonal lattice with thickness d = 2 mm, side length L = 15 mm and width w = 5 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' At the top and bottom of the substrate are pasted P-4 piezoelec- tric sheets of the same thickness as the substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' The six corners of the hexagonal lattice have cylindrical mag- net oscillators glued above and below them, where the oscillators have radius r = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content='5 mm and height h = 5d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' The material parameters of the magnet, the epoxy and the P-4 piezoelectric sheet are shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' The multi-cell structure of the piezoelectric elastic plate can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' 1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' The part of the diagram enclosed by the yellow wire frame is taken as the unit cell, with the length of the cell a = 45 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' In order to distin- guish between the piezoelectric pieces, we marked the inner and outer hexagonal piezoelectric pieces as purple and blue respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' The negative capacitance circuit in the active control system is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' 1(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' It includes capacitor Cp, compensation resistor R0, oper- ational amplifier LM324N, fixed resistor R1 and sliding resistor R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' II provides the parameters for each de- vice of the negative capacitance circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' The equivalent bending stiffness of the piezoelectric sheet can be con- trolled using a negative capacitance circuit connected to the P-4 piezoelectric sheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' A topological phase change of the elastic plate is thus achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' In the case when the piezoelectric sheet is connected with a negative capacitance circuit, we derive the equiv- alent elastic module of the piezoelectric sheet as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' The x, y and z directions are denoted as 1, 2 and 3 re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' Since the piezoelectric sheet is subjected to an electric field along the z-axis only,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' the intrinsic equa- tion of the piezoelectric material in the plane stress state can be expressed as[47,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' 53,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' 54] S1 = sE 11T1 + d31E3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' D3 = d31T1 + εT 33E3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' (1) where S1 and T1 are the strain and stress along the x- direction,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' sE 11 represents the coefficient of flexibility under a constant electric field,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' d31 and εT 33 are the piezoelectric and dielectric constants,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' and E3 and D3 de- scribe the electric field strength and potential shift along the z-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' At a constant strain, the intrinsic capacitance of a piezoelectric sheet can be denoted as Cp = εT 33Ash−1 p , (2) where As and hp indicate the area and thickness of the piezoelectric sheet respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' Then, introducing the complex impedance Z, the relationship between strain and stress can be derived as S1 = � sE 11 − sZd2 31As (1 + sZCp)hp � T1, (3) where s is the Laplace parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' The equivalent modulus of a piezoelectric sheet con- nected to a circuit with negative capacitance can be de- rived from equation (3) as follows: Ep = hp(1 + sZCp) hpsE 11(1 + sZCp) − sZd2 31As .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' (4) Its complex impedance Z is expressed as Z = − (αsCp)−1 , (5) where α = R2C(R1Cp)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' Therefore, we can make the elastic modulus of the piezoelectric sheet varied by adjusting the parameter α, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=', regulating the joint stiffness of the piezoelectric elas- tic sheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' Next, the band structures of the unit cell were analyzed using COMSOL Multiphysics software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' The stiffness fac- tor is the same for each holder when neither the inner nor outer piezoelectric sheet is connected to the negative capacitance circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' 2(a), the four bands degen- erate at point Γ, forming a double Dirac cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' In the band structures we are only concerned with the out-of- plane modes (marked in red), and the in-plane modes (marked in grey) are not considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' The stiffness factor of the corresponding elastic support is enhanced when an internal or external piezoelectric sheet is connected to a negative capacitance circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' As a result, the dou- ble Dirac cone disappears and a band gap forms at the 3 y x z 𝑤 𝑙 Negative capacitance circuit + − 𝑅0 𝐶0 𝐶𝑝 𝑅1 𝑅2 LM324N (a) (b) (c) Epoxy shelf P-4 piezoelectric sheet Lead cylinder 𝑎 h 𝑟 𝑑 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' (a) Schematic of the unit cell structure of a piezoelectric elastic plate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' The yellow honeycomb support serves as the flexible plate substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' P-4 piezoelectric sheets are attached to the top and bottom of the holder, with the piezoelectric sheets angled at 120◦ at both ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' The parts marked in purple are the internal hexagonal piezoelectric sheets, and the external ones are marked in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' Cylindrical magnet oscillators are glued to the top and bottom of the six corners of the honeycomb elastic plate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' Negative ca- pacitance circuitry is connected to the surface of the piezoelectric sheets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' (b) Composite unit structure of piezoelec- tric elastic plates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' The part enclosed by the yellow line frame is taken as the unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' (c) Schematic of the negative capacitance circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' Material parameters for piezoelectric elastic plate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' Material Modulus E(Pa) Density ρ(kg/m3) Compliance coefficient sE 11 (m2/N) Piezoelectric strain coefficient d31(C/N) Permittivity εT 33(F/m) Magnet 41×109 7400 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' Epoxy 3×109 1180 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' P-4 piezoelectric sheet 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content='83×1010 7450 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content='2×10−11 1×10−10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content='2×10−8 TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' The parameters of the negative capacitance circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' C(pF) Cp(pF) R2(kΩ) R1(kΩ) R0(kΩ) Operational amplifier 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content='684 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content='155 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content='54 68 2000 LM324N corresponding location nearby.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' As can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' 2(b), the forbidden band width widens as the parame- ter α in the negative capacitance circuit increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' The d modes in the band structure are above the band gap and the p modes are below when the external piezoelec- tric sheets are connected to negative capacitance circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' Instead, as the internal piezoelectric sheets are connected to negative capacitance circuits, the d and p modes are interchanged in their distribution above and below the band gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' A transition from a trivial state to a non- trivial state occurs in the topological phase during this period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' Specifically, external powering produces the triv- ial bandgap (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' 2(c)), while internal powering cor- responds to the non-trivial bandgap (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' 2(d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' The eigenmode diagrams are plotted for the high symmetry point Γ in the upper and lower bands forming the band gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' They are shown accordingly at the top and bot- tom of the band structures diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' In this case, the d modes represent quantized four polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' The mode distribution of dx2−y2 is symmetric about the x and y axes, while the mode of dxy is antisymmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' The p modes represent quantized couple polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' px and py modes are antisymmetrically distributed about the y and x axes respectively[19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' DISCUSSION OF TOPOLOGICAL CORNER STATES Further, the supercell dispersion relation of the piezo- electric elastic plate was analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' This supercell con- sists of six internally powered unit cells on the left and six externally powered unit cells on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' The left and right ends of the supercell are the free boundary condition, and the Floquet-period boundary condition along the y-axis is imposed at its upper and lower ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' The dispersion relation obtained from the computational analysis is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' 3(a), with the grey and light blue marked points being the in-plane and out-of-plane modes respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' We focus only on the out-of-plane modes, with two clear topological boundary states (green and purple) in the middle part, corresponding to the chi- ral propagation model generated by the pseudo-spin Hall effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' The topological boundary vibration mode dia- grams corresponding to points A1 and A2 are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' 3(b), where it can be observed that the displace- ment is greatest at the intermediate interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' To bet- ter describe the two topological pseudo-spin states, we show the energy flow arrows (yellow) at the interface Ky K M K" r /K K K K\'4 (c) (d) Non-trivial p d 𝑑𝑥2−𝑦2 𝑑𝑥𝑦 min max 𝑝𝑦 𝑝𝑥 Trivial d p 𝑝𝑦 𝑝𝑥 min max (a) (b) 𝑑𝑥2−𝑦2 𝑑𝑥𝑦 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' (a) The unit cell dispersion curve for the piezoelectric sheet with- out the negative capaci- tance circuit connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' (b) Curves for the variation of band gap width with pa- rameter α when the honey- comb elastic plate is inter- nally or externally charged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' (c), (d) Band structures and their intrinsic mode pat- terns when the external and internal piezoelectric sheets are charged for α= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' External powering Internal powering 𝐴2 max min (a) (c) 𝐴1 0 𝜋 −𝜋 𝐴1 (b) y x z 𝐴2 Arg(w) Fig 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' (a) Dispersion re- lation diagram of the su- percell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' (b) Schematic of the structure of the super- cell and the corresponding vibrational modes at points A1 and A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' The color bar represents the size of the absolute value of the dis- placement in the z-direction of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' (c) Cloud field plot of the phase dis- tribution and energy flow arrows for the z-direction eigenstates at the modal in- terface of points A1 and A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' 3(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' In the boundary mode at points A1 and A2, the energy flow appears in clockwise and coun- terclockwise directions respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' Meanwhile, we plot the phase field of the eigenstates at the boundary as the cloud field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' It can be observed that the phases of the two boundary modes at the same position are opposite to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' This suggests that the two pseudospin boundary states have propagation modes in opposite di- rections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' But such boundary states differ from the gap- less edge states of fermion pseudospin protected by time- reversal symmetry[56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' In the middle of the two edge states of this model band structure, there is a grey for- bidden band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' The band gap is created by the breaking of C6 crystal symmetry at the boundary of different topo- logical phase structures[41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' We try to find the topologi- cal corner state by referring to the frequency distribution 6000 5000 Frequency(Hz) 4000 3000 2000 1000 -- 1 -- --.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' 0 M K M Wave vector- 5000 --- 4000 1 1 3000 2000 DoubleDirac Cone 1 1000 -- o 0 M K M Wavevector4400 4400 dimodes 4200 p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content="modes 4200 4000 4000 3800 External powering 3800 3600 3600 Gap 3400 3400 Gap 3200 Internal powering 3200 3000 3000 06'0 88'0 98'0 80 8'0 08*006'0 88'0 98*0 8'0 80 08*0 aout ain6000 5000 Frequency(Hz) 4000 3000 1 2000 1000 -- -- 0 M T K M Wavevector4400 4200 4000 3800 3600 3400 3200 3000 0 K." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content='(π/a)9M5 of this band gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' In order to analyze the topological corner states of the piezoelectric elastic plate, we constructed a 13×13 diamond-shaped finite lattice model as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' The parameter α for the negative capacitance cir- cuit connected to the piezoelectric sheet is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' The external bracket of the 7×7 unit cell inside the model is charged, corresponding to the trivial phase structure an- alyzed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' The outer unit cell is then a non-trivial phase structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' The two different phase structures thus form a diamond-shaped boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' The eigenfrequencies of the model were calculated in COMSOL Multiphysics software and the resulting eigenfrequency spectrum is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' 4(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' The presence of bulk states (grey circles), edge states (blue square triangles), topological corner states (red rhombuses) and trivial corner state (green inverted triangle) can be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' The corre- sponding vibration mode diagrams for the edge and bulk states are plotted in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' 4(c, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' We are primarily con- cerned with topological corner states that occur in the bandgap range, which are topologically protected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' The topological corner state mode diagram corresponding to the eigenfrequency of 3670.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content='9 Hz is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' 4(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' It can be observed in the diagram that the larger absolute values of out-of-plane displacement |w| are concentrated at the 2π/3 corners of the bend at the rhombic boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' The corner state corresponding to 3649.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content='3 Hz below the band gap is the trivial corner state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' In the corresponding mode vibration diagram, the larger displacements occur at the π/3 corners of the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' In analogy to the theoretical explanation in the higher-order topology of electromagnetic waves, there is a zero mode (topologi- cal corner mode) at each uncoupled waveguide[57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' Each zero mode holds one topological charge (+ or −).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' N+ and N− are used to indicate the numbers of eigenstates of the chiral symmetric operator Π with topological charges + and −, respectively[40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' There are three zero modes at the 2π/3 corner, two of which contain the same topo- logical charge and one of which contains the opposite topological charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' As a result, the topological index N = |N+ − N−| = 1 ̸= 0, representing the topological corner state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' The details can be found in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' While the topological charge is positive at the two zero modes at π/3, that at the other two zero modes is neg- ative, so the topological index N=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' These theories can help to explain the phenomenon that the topological cor- ner states of the present honeycomb piezoelectric elastic plate appear only at the 2π/3 corners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' Next, we verified the strong robustness of the piezo- electric elastic plate topological corner states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' The 2π/3 corner of the interface between the different phase struc- tures in the finite lattice is displayed magnified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' 5(a) shows the defect-free state, while Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' 5(b, c) represent the disturbed and cavity models, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' A point at the 2π/3 corner of the boundary is chosen to plot the normalized frequency energy spectrum as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' 5(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' There is a clear high-energy peak in the grey band gap of the defect-free structure, which corresponds to Trival Non-trival (a) min max 7 × 7 (c) 3741.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content='8Hz (e) 3670.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content='9Hz 3869.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content='2Hz (f) 3649.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content='3Hz (d) (b) Fig 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' Calculated eigenmodes of the piezoelectric elastic plate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' (a) Schematic diagram of a diamond-shaped finite lat- tice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' The interior is the externally charged trivial phase struc- ture (cyan) and the periphery is the internally charged non- trivial phase structure (yellow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' (b) Numerical calculation of the eigenfrequencies obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' The red rhombus represents the topological corner state, while the grey circle, blue square triangle and green inverted triangle represent the bulk, edge and trivial corner states respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' (c) Edge state at 3741.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content='8 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' (d) Bulk state at 3869.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content='2 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' (e) Topological corner state at 3670.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content='9 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' (f) Trivial corner state at 3649.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content='3 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' The color bar indicates the absolute magnitude of the displacement in the z-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' the presence of the topological corner states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' In the pres- ence of disturbance and vacancy defects, the frequency interval with the higher energy peak remains confined to the band gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' And the average elastic energy density peaks at the corner of the defective structures are close to the peak in the case of the defect-free structure, with less dissipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' The resulting analysis indicates that the topological corner state of the piezoelectric elastic plate is insensitive to small defects with strong robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' To emphasize the frequency tunability of the topolog- ical corner states of this piezoelectric elastic plate, we investigated the topological corner state spectrum vari- ation of its finite lattice as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' 6, with the piezoelectric parameter α as a variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' As α increases, the stiffness of the holders connected to negative capac- itance circuits rises and the topological corner states of the piezoelectric elastic plate appear with an accordingly higher frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' Also, as the parameter increases, the two topological corner states become closer and closer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' It means that the piezoelectric elastic plate can achieve 4200 Bulk Edge 4000 Topological corner (120°) Trivial corner (60°) 3800 3600 3400 3200 0 20 40 60 80 100 120 Solution number6 No defect (a) Disorder (b) Cavity (c) (d) Fig 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' (a) Partial diagram of the finite lattice at the 2π/3 corner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' The purple holders in- dicate that a negative capaci- tance circuit is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' Non- trivial phase structures outside the corner and trivial phase structures inside the corner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' (b) Disturbed finite lattice model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' A holder at the corner is not connected to the negative ca- pacitance circuit, creating a dis- turbance to the topology model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' (c) Cavity finite lattice model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' The absence of two oscillators above and below the support at the corner forms a cavity de- fect topology model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' (d) Nor- malized energy spectrum curves for the non-defective and de- fective models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' The grey band represents the boundary state band gap of the supercell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' The black wire represents the non-defective model, with the red and blue wires representing the disturbed and cavity defect models respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' Fig 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' Variation of the finite lattice frequency spectrum with the piezoelectric parameter α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' The grey dots indicate the bulk and edge states and the red triangles indicate the topological corner states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' topological corner states in multiple frequency ranges, and the way to go is to tune the piezoelectric parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' Besides, compared to the finite lattice model in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' 4(a), the total number of cells is kept constant by reducing and increasing the number of trivial phase unit cells by 5×5 and 9×9 respectively as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' 7(a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' We subjected the two finite lattices to spectral analysis and could ob- serve an inward shift (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' 7(c)) and an outward shift Trival Non-trival 5 × 5 Non-trival Trival 9 × 9 (a) (b) (d) Corner state outward shift Corner state inward shift min max (c) Fig 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' (a)-(b) Schematic of the corner state inward and out- ward shift model for the finite lattice (13*13 cells).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' (c)-(d) Inward and outward shifted vibration mode graphs for corner states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' 7(d)) in the topological corner state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' In practice, the topological corner state position can be adjusted by simply connecting the negative capacitance circuit to the corresponding piezoelectric sheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content='0 No defect Disorder 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content='8 Cavity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' 0 3300 3400 3500 3600 3700 3800 3900 4000 4100 Frequency(Hz)4000 Frequency(Hz) 3800 3600 3400 3200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content='84 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content='86 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content='90 αEdge and Bulk Topological corner7 IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' CONCLUSION In this paper, a honeycomb-shaped piezoelectric elas- tic plate is designed to achieve tunable topological corner states of the elastic wave HOTI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' By connecting negative capacitance circuits to each of the inner or outer piezo- electric sheets in the unit cell, the topological phase is transformed accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' The band gap in the boundary state is present in supercell containing both trivial and non-trivial phase structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' The spectral analysis of its finite lattice reveals the presence of topologically pro- tected corner states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' Echoing the theory, the topological corner states in the hexagonal lattice were found to ex- ist only at the 2π/3 corner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' We designed both disturbed and vacant defect models and plotted the frequency en- ergy spectrum curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' The strong robustness of the topo- logical corner states is verified by comparison with the defect-free model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' In addition, to highlight the tunabil- ity of the topological corner states of the piezoelectric elastic plate, we adjusted the piezoelectric parameter α to broaden the operating frequency range of the topo- logical corner states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' Switching the position of the neg- ative capacitance circuit connection enables inward and outward shifts of the topological corner states, meaning that the topological corner states can appear at any posi- tion in the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' The tunable topological corner state of this piezoelectric elastic plate enriches the study of topo- logical physical phenomena in mechanical metamateri- als.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' This study is of reference value in applications such as elastic wave energy localization, information transfer, and energy harvesting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' ACKNOWLEDGEMENTS This research is sponsored by the National Natural Science Foundation of China (Grant nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' 12021002, 12072225, and 11991031).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' We thank Professor Yibin Fu at Keele University for helpful discussions and valuable advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' Appendix A: Characterization of topological corner states By understanding the theory and methods in these articles[46, 57, 58], we verify the topological corner prop- erties of the piezoelectric elastic plate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' Firstly the bulk polarisation here is expressed as χ(6) = ([M], [K]) (A1) where [M] and [K] are C2 and C3 invariants respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' They are integers that can be expressed as [M] = #M1 − #Γ(2) 1 [K] = #K1 − #Γ(3) 1 (A2) where M1 (Γ(2) 1 ) is the number of energy bands with C2 (π) rotation eigenvalue +1 below the band gap at point M (Γ) in the Brillouin zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' K1 (Γ(3) 1 ) is the number of energy bands with C3 (π/3) rotation eigenvalue +1 be- low the band gap at point K (Γ) in the Brillouin zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=', we observe that the eigenmodes of the three bands below the band gap are [M] = 0 for a trivial struc- ture M1 = 1 and Γ(2) 1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' For non-trivial structure M1 = 1 and Γ(2) 1 = 3, so [M] = 2 (ignoring the negative sign).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' The C3 rotation operator and the chiral operator are swapped in this piezoelectric elastic plate model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' It can be obtained that for both structures the invariant [K] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' We summarise the topological invariants in the two structures as follows ([M], [K]) = � (0, 0) external powering (2, 0) internal powering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' (A3) From the corner charge Q(6) corner=[M]/4+[K]/6 in the literature, it follows that Q(6) corner = � 0 external powering 1/2 internal powering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' (A4) To account for the difference in position between the emergence of topological and trivial corner states, the topological index N has been introduced to the characterization[40, 57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' The topological index N cap- tures the interaction between the topology of the bulk Hamiltonian and the topology of the defect, representing the stable mode at the corners of the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content='8, it can be seen that N+=1 and N−=1 at the edges not containing the corners of the rhombic finite lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' At the π/3 corners, N+= 2 and N−= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' Therefore the topological index at the edges and π/3 corners of the piezoelectric elastic plate is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE0T4oBgHgl3EQf6QKw/content/2301.02762v1.pdf'} +page_content=' While at the 2π/3 cor- ners N+=1, N−=2, the topological 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Distributed GPU-accelerated Library +towards Scalable Evolutionary Computation +Beichen Huang, Ran Cheng,Yaochu Jin, Fellow, IEEE, and Kay Chen Tan, Fellow, IEEE +Abstract—During the past decades, evolutionary computation +(EC) has demonstrated promising potential in solving various +complex optimization problems of relatively small scales. Nowa- +days, however, ongoing developments in modern science and +engineering are bringing increasingly grave challenges to the con- +ventional EC paradigm in terms of scalability. As problem scales +increase, on the one hand, the encoding spaces (i.e., dimensions +of the decision vectors) are intrinsically larger; on the other +hand, EC algorithms often require growing numbers of function +evaluations (and probably larger population sizes as well) to +work properly. To meet such emerging challenges, not only does +it require delicate algorithm designs, but more importantly, a +high-performance computing framework is indispensable. Hence, +we develop a distributed GPU-accelerated algorithm library — +EvoX. First, we propose a generalized workflow for imple- +menting general EC algorithms. Second, we design a scalable +computing framework for running EC algorithms on distributed +GPU devices. Third, we provide user-friendly interfaces to both +researchers and practitioners for benchmark studies as well +as extended real-world applications. Empirically, we assess the +promising scalability of EvoX via a series of benchmark experi- +ments with problem dimensions/population sizes up to millions. +Moreover, we demonstrate the easy usability of EvoX by applying +it to solving reinforcement learning tasks on OpenAI Gym. To +the best of our knowledge, this is the first library supporting +distributed GPU computing in the EC literature. The code of +EvoX is available at https://github.com/EMI-Group/EvoX. +Index Terms—Evolutionary multi-objective optimization, neu- +ral architecture search, deep learning. +I. INTRODUCTION +W +ITH the development of modern science and engi- +neering, various emerging optimization problems are +posing stiff challenges to the optimization algorithms. Despite +that evolutionary computation (EC) has been shown to be +a promising tool for solving complex optimization problems +of relatively small scales, it has come to a consensus that +the conventional EC paradigm suffers from the curse of +dimensionality [1], [2] - the phenomenon that the search +space of a problem grows exponentially with the number +of dimensions. Since EC algorithms often rely on random +search/sampling to find solutions, the very large number of +possible solutions can make it difficult for EC algorithms +to explore the space effectively in high-dimensional complex +search spaces. Additionally, the computational complexities of +EC algorithms may also grow with the number of dimensions, +making them slow and impractical for large-scale optimization +problems [3]. +To improve the scalability of EC algorithms, researchers +have made persistent efforts during the past decade [4]–[6]. +In the early days, most efforts were mainly dedicated to +making improvements from the methodology point of view, +by proposing tailored algorithm frameworks/operators. For +example, the cooperative coevolution (CC) [7]–[9] is among +the popular frameworks tailored for improving the scalability +of EC algorithms. In the CC framework, a problem is divided +into a collection of lower-dimensional subproblems. Each +subproblem is optimized individually and its population of +candidate solutions is coevolved with the other subproblems +in a round-robin fashion; then, representative solutions from +each subproblem are combined to form a context vector, which +is used to evaluate the overall solution; finally, the context +vector is updated iteratively and serves as the context for +cooperation between the subproblems. Some other researchers +proposed various tailored operators with better scalability, +including variants of the differential evolution (DE) [10]–[12], +the particle swarm optimization (PSO) [13]–[16], as well as +the estimation of distribution algorithms (EDAs) [17]–[20], +among many others. +Undoubtedly, algorithmic innovations can improve the scal- +ability of EC algorithms. Nowadays, however, when consider- +ing the performance of an algorithm, it is also important to take +into consideration the essential roles of modern computing +architectures/devices [21]. As the most indicative example, the +rapid development of modern deep learning algorithms can be +attributed, in part, to the advancement of GPUs for training +and running deep neural networks more efficiently. Inherently, +the population-based nature of EC algorithms also makes it +possible to potentially parallelize the computing process. There +have been some attempts that aim to improve the scalability +of EC algorithms through GPU computing or distributed +computing: a parallelized version of DE was proposed to use +GPU computing and can handle continuous problems with up +to 1000 dimensions [22]; a parallel version of the compact +genetic algorithm for CPU/GPU architectures was proposed +and tested on OneMax and noisy OneMax with up to one +billion variables [23]; an improved GPU-based model for MA- +SW-Chain was proposed and tested on a scaled version of +the CEC’2013 large-scale benchmarking suite on functions +with up to 100 million decision variables [24]. Despite a few +individual works along this line, there has not yet been a +systematic research effort in the EC field similar to what has +been seen in the deep learning area. +The literature has already demonstrated the potential for EC +algorithms to improve scalability, whether through algorithmic +innovations or hardware acceleration. However, there is still +much room for improvement and further research, especially +in terms of developing more efficient and effective algorithms +by utilizing modern computing architectures and devices to +arXiv:2301.12457v1 [cs.NE] 29 Jan 2023 + +2 +their full potential. Additionally, it would be beneficial to +investigate the potential applications of EC in various domains +and industries. Nonetheless, there are three main issues that are +currently hindering the further development of scalable EC. +‚ Workflow: In a typical EC workflow, the main components +(i.e., crossover/mutation, fitness evaluation, selection) are ex- +ecuted sequentially in a main loop. While this structure is +organized and cohesive, it is not compatible with asynchronous +computing methods such as GPU computing or distributed +computing, making it challenging to implement or modify +algorithms for improved flexibility and concurrency. +ƒ Computational Cost: The main source of computational +cost in EC algorithms is the fitness evaluations needed for +population-based stochastic search. From a statistical perspec- +tive, it is often necessary to increase the number of samples +(i.e., fitness evaluations) exponentially as the dimension of the +search space grows linearly in order to accurately approximate +a target distribution. +„ Running Environment: While most EC algorithms were +initially developed for solving pure numerical optimization +problems and do not have specific requirements for the run- +ning environment, the running environments for large-scale +optimization tasks are often specific to the task at hand. For +example, neuroevolution tasks are often closely tied to deep +learning scenarios, which require running environments with +software/hardware. +To push the boundary of EC towards better scalability and +wider applicability by addressing the aforementioned issues, +we develop a distributed GPU-accelerated algorithm library – +EvoX. In summary, the main contributions are: +– We propose a generalized EC workflow. On the one hand, it +fully decouples the implementation of algorithms, problems, +user-friendly monitors, and possible population decoders or +fitness transformers, bringing more flexibility and generality to +EC than before. On the other hand, complete modularization +makes it possible for both researchers and practitioners to eas- +ily parallelize EC algorithms via high-performance computing +frameworks. +– We develop a scalable computing framework for running +EC algorithms on distributed GPUs. Based on the proposed +generalized EC workflow, a powerful distributed GPU accel- +eration library EvoX is developed. First, EvoX supports ready- +to-use GPU computing acceleration, such that users can easily +run EC algorithms on GPU(s) without any additional engi- +neering work. Second, EvoX supports ready-to-use distributed +computing framework, such that users can easily deploy EC +algorithms on distributed machines at hand. +– We provide a user-friendly interface for both numerical +benchmark tests and other challenging problems in real-world +applications. First, EvoX provides a generalized Problem +module to fully support running EC algorithms for challenging +data-related tasks (e.g. neuroevolution) via GPU computing. +Second, EvoX provides a tailored interface to provide seam- +less connections to complex environments (e.g. those in rein- +forcement learning) on top of a high-performance distributed +computing framework. +The remainder of this paper will be organized as follows. +Section II provides the necessary background information. +Section III presents a generalized EC workflow. Section IV +and Section V describe the design and contents of EvoX, +respectively. Section VI contains the experiments conducted on +EvoX. Finally, Section VII concludes the paper and discusses +our future work. +II. BACKGROUND AND RELATED WORK +In this section, we first briefly overview some representa- +tive EC libraries; then we provide background knowledge of +neuroevolution, evolutionary reinforcement learning; finally, +we introduce the related techniques of GPU computing and +distributed computing. +A. EC libraries +In the EC field, the Python programming language has +emerged as a popular choice for implementing EC algorithms. +This is due in part to the availability of powerful and easy- +to-use EC libraries, such as DEAP (Distributed Evolutionary +Algorithms in Python) [25], PyGAD (Python Genetic Algo- +rithms and Differential Evolution Framework) [26], Pymoo +(Python Multi-Objective Optimization) [27], and Pagmo (Par- +allel Global Multiobjective Optimizer) [28]. In this subsection, +we will review the features and capabilities of these libraries. +DEAP is a long-standing and feature-rich framework for +implementing evolutionary algorithms in Python. It offers +support for a wide range of EC algorithms, including both +single and multiple objective algorithms. DEAP also includes +a broad range of built-in benchmark problems, making it easy +to evaluate the performance of EC algorithms. DEAP is well- +suited for rapid prototyping and testing of ideas and is a +popular choice among researchers in the field of EC. Another +prominent feature of DEAP is its support for parallelization. +DEAP allows evaluations to be easily parallelized, making it +possible to run them on multiple cores or even across multiple +machines through scoop [29]. +PyGAD is a library for implementing genetic algorithms +in Python. It offers different types of crossover, mutation, +and parent selection operators for implementing genetic algo- +rithms. What makes PyGAD unique is its focus on machine +learning tasks. PyGAD includes features and tools specifically +designed for training artificial neural networks, making it a +good choice for applying evolutionary computation (EC) in +machine learning. +Pymoo is a library that focuses on multi-objective optimiza- +tion algorithms in Python. Its main strength lies in its com- +prehensive support for multi-objective optimization. Pymoo +includes a wide range of benchmark problems and state-of- +the-art multi-objective algorithms. Pymoo also includes many +operators suitable for multi-objective algorithms, allowing +users to easily customize the algorithms. Furthermore, Pymoo +has a set of powerful and flexible features related to multi- +objective optimization, such as visualization and decision- +making. All these features building towards multi-objective +optimization make Pymoo a suitable tool for the task. +Pagmo is a C++ library for massively parallel optimization, +with a Python binding called Pygmo. It is built around the + +3 +TABLE I: Summarized main features of EC libraries +Name +System +Algorithms +Problems +GPU +Computing +Distributed +Computing +Single-objective +Algorithms +Multi-objective +Algorithms +Numerical +Benchmarks +Neuroevolution +Tasks +Reinforcement Learning +Tasks +DEAP +✓ +✓ +✓ +✓ +PyGAD +✓ +✓ +✓ +pymoo +✓ +✓ +✓ +pagmo +✓ +✓ +✓ +EvoX +✓ +✓ +✓ +✓ +✓ +✓ +✓ +generalized island model [30], which allows coarse-grained +parallelization. It offers a variety of algorithms, benchmark +problems, and migration policies, making it easy for users +to implement parallelized algorithms. Additionally, it supports +batch fitness evaluation, enabling users to perform parallel +fitness evaluations using their own methods. +Despite the attractive features introduced above, these exist- +ing libraries have a common deficiency: the lack of scalability. +Potentially, both GPU computing and distributed computing +are powerful tools for improving the scalability of EC, partic- +ularly in scenarios involving large amounts of data or complex +calculations. However, none of these existing supports GPU +computing, while the support for distributed computing is +either missing or inefficient. +As a result, the lack of support for GPU computing or dis- +tributed computing in these libraries has largely limited their +performance and applicability for certain types of optimization +problems. Besides, the extendability of these libraries towards +more complex problems is also very limited, making it difficult +for EC practitioners to get involved. In detail, the key features +of these libraries in comparison with EvoX are summarized +in Table I. +B. Neuroevolution +Neuroevolution is a field focusing on using evolutionary +computation (EC) algorithms to optimize artificial neural net- +works (ANNs). It has a long history of development and is +facing some emerging challenges and opportunities [31]. +In the field of neuroevolution, the use of EC algorithms for +optimizing ANNs has gained significant attention due to its +potential advantages over traditional gradient-based methods. +Since EC algorithms can explore a much larger search space +than gradient-based methods, they have better potential for +discovering more diverse and novel solutions. Broadly speak- +ing, neuroevolution is able to evolve various aspects of neural +networks, such as the building blocks, architectures, weights, +and even the training rules. +Although neuroevolution was initially considered an alter- +native to backpropagation for optimizing the weights of small +and fixed-topology ANNs, some attempts quickly turned to +evolving network architectures as well [32]–[34]. With the +booming development of deep learning, researchers are now +paying increasing attention to the automatic design of deep +neural networks (DNNs) via neuroevolution - the evolutionary +neural architecture search (ENAS) [35] - which is particularly +useful when facing complex scenarios involving multiple ob- +jectives to be optimized simultaneously (e.g. hardware-aware +deployment of DNNs [36]). +Despite the biologically plausible and technically attractive +characteristics of neuroevolution, one of the main limitations +is the computational complexity, which is particularly chal- +lenging when dealing with DNNs [37]. Intuitively, one way +to improve the scalability of neuroevolution algorithms is to +make full use of the computing power of GPUs and distributed +computing systems, such that different candidate solutions (i.e. +networks) in the population can be evaluated (i.e. trained) +simultaneously. This would allow for more efficient and faster +training of ANNs. However, to the best of our knowledge, none +of the existing EC libraries currently supports distributed GPU +acceleration for neuroevolution. +C. Evolutionary Reinforcement Learning +Reinforcement learning (RL) is a powerful and widely- +studied framework for learning and decision-making in com- +plex, dynamic environments. At its core, RL is concerned with +how an agent should act to maximize a reward signal over +time. This requires the agent to learn a policy, or a mapping +from states to actions, that will allow it to take actions that lead +to the most reward from the environment. The environment +is typically modeled as a Markov decision process (MDP), +consisting of: +• set of states S with an initial state distribution P(s0), +• set of possible actions A, +• reward function R : S × A → R, +• transition function T : S × A → P(S), +• discount factor γ ∈ [0, 1]. +At each time step t, the agent can choose an action at ∈ A +based on the current state st and a policy π : S → P(A). The +objective is to find the optimal policy π∗ that maximizes the +expected reward: +π∗ = argmax +π +EP (s0),π,T [ +T −1 +� +t=0 +γtrt], +(1) +where T is length of the episode, at ∼ π(st), st+1 ∼ +T (·|st, at), and rt = R(st, at). +Evolutionary Reinforcement Learning (EvoRL) [38] specif- +ically focuses on using EC algorithms to deal with various +challenging optimization problems in RL, such as hyperparam- +eter optimization [39], policy search [40], reward shaping [41], +exploration [42], among many others [43]. One of the key +advantages of EvoRL is that it can explore a much larger + +4 +search space than gradient-based RL methods such as Q- +learning. This allows for the discovery of a wider range of +potential policies, which can lead to better performance in +terms of reward and other metrics. +In recent years, as tasks and environments have become +increasingly complex, scalability has become a major bot- +tleneck for EvoRL. On the one hand, due to its population- +based nature, EvoRL can be computationally intensive, making +it difficult to apply in real-time or real-world situations. On +the other hand, the large number of parameters in the policy +models can cause a severe curse of dimensionality for the +EC algorithm when applied to RL tasks. To address these +challenges, researchers have proposed a number of methods +to improve the scalability of EvoRL. These methods include +using parallel computing to distribute the computational work- +load across multiple machines or processors [38], [44], as well +as using more efficient algorithms and strategies for evolving +the population [45], [46]. However, the development of EvoRL +is still in its infancy, and there is a lack of a high-performance +EC library with a user-friendly interface for solving RL tasks. +D. JAX for GPU Computing +GPU computing is a technology that utilizes GPUs to +conduct general-purpose computing instead of CPUs. A GPU +usually consists of tens or hundreds of thousands of cores, +which is thousands of times more than a typical CPU. To +match the computational capabilities of GPUs, their memories +are often way faster in terms of bandwidth compared to CPU +memories. Moreover, to provide fast synchronization between +different cores, GPUs usually have much larger and faster +shared caches compared to CPUs. These properties make it +possible to process data on GPUs with higher parallelization +than CPUs. +In the past decade, GPU computing has been one of the driv- +ing forces of deep learning. Libraries like PyTorch [47] have +introduced the ability to utilize GPUs. Nonetheless, few works +have been dedicated to accelerating EC algorithms via GPU +computing. Since EC algorithms usually involve a chain of +computationally cheap operations, they are often constrained +by memory bandwidth more than computing power. Hence, +the decrease in the number of memory accesses can drastically +improve their performance when parallelized on GPUs. +Recently, the JAX has been released as a library offering +a NumPy-like API for GPU-accelerated numeral calcula- +tions [48]. With just-in-time compilation features, JAX can run +on multiple hardware backends including both CPU and GPU +by optimizing the Python functions. The optimization in the +compilation provides automatic fusing of small operators, thus +substantially helping to save memory bandwidth Such features +of JAX are particularly beneficial for the parallelization of EC +algorithms. +E. Ray for Distributed Computing +Distributed computing allows for the collective power of +multiple computers to be harnessed in order to solve complex +problems that would be difficult or impossible to solve using a +single computer. Since computers coordinate with each other +by passing messages through the network, the communication +cost in a distributed system usually has a huge impact on the +overall scalability. In a distributed system, each computer (or +node) has its own local memory that cannot be accessed by +other computers. To solve a problem collaboratively, comput- +ers need to communicate with each other by message passing +through the network. +Ray is a popular framework for distributed computing in +Python and has been shown to be well-suited for applications +in machine learning and other scientific computing tasks [49]. +As a user-friendly framework, Ray provides both actor-parallel +and task-parallel programming abstraction, and the communi- +cation between actors and tasks will be handled automatically. +One of the key features of Ray is its distributed scheduler, +which is composed of both a global scheduler and per-node +local schedulers. This design allows Ray to efficiently schedule +tasks to run on the appropriate node, both locally and across +the distributed system, providing improved scalability. With its +scheduler, users can specify resource requirements for actors +and tasks, and Ray will automatically place them on nodes +with adequate resources. Such features will allow us to easily +scale the EC algorithms across multiple machines. +F. Discussions +EC has been shown to have promising potential in tack- +ling complex tasks such as those found in neuroevolution +and EvoRL. However, the lack of support for computing +acceleration in existing EC libraries presents a challenge to +further development in more advanced EC algorithms. To +address this limitation, we have initiated the development +of EvoX, a new library that improves the scalability of EC +algorithms by leveraging the strengths of recently-developed +high-performance computing tools: JAX and Ray. +On the one hand, JAX is well-suited for GPU computing +of EC algorithms due to its use of just-in-time compilation +and support of CPU/GPU backends, which fuses operations +and minimizes memory accesses during acceleration. On the +other hand, Ray is a distributed framework that allows for +the scheduling of computations across multiple machines and +CPU/GPU resources. By combining the capabilities of both, +EvoX is able to improve the scalability of EC and extend the +applications towards larger and more complex problems. +III. GENERALIZED EC WORKFLOW +Generally, distributed GPU acceleration of EC workflow +may face two main issues: on the one hand, each component +in an EC workflow may have its own way of parallelization; +on the other hand, different components in an EC workflow +must be synchronized in the distributed system. To address +such issues, we propose a generalized EC workflow on top of +an ask-and-tell interface, which considers an EC workflow as +an agent (i.e. an EC algorithm) which iteratively transitions +through the states by performing ask and tell actions +for problem-solving. +Specifically, given θ and D are the hyperparameters for +defining the problem and the algorithm respectively, the al- +gorithm can be characterized by Aθ = ⟨θ, gask, gtell⟩, problem + +5 +TABLE II: Summary of notations +Notation +Description +t +The generation counter +Aθ +The algorithm parameterized by θ +PD +The problem parameterized by D +SAθ +t +The state of Aθ at generation t +SPD +t +The state of PD at generation t +f +The fitness function +h +The decoder +gask +The ask method of the algorithm, +used to give out candidate solutions. +gtell +The tell method of the algorithm, +used to update the state based on the fitness. +Xt +The candidate solutions at generation t +yt +The fitness values at generation t +PD = ⟨D, f⟩, where gask and gtell are ask and tell actions +for generating a new population and updating the algorithm +state. A simple iteration on generation t is: +Xt, SAθ +t+1 = gask +θ (SAθ +t +), +(2) +yt, SPD +t+1 = fD(SPD +t +, h(Xt)), +(3) +SAθ +t+1 = gtell +θ (SAθ +t+1, yt), +(4) +where Xt, yt denote the population of candidate solutions and +the corresponding fitness values at generation respectively; +SAθ, SPD are the state of the algorithm and problem re- +spectively; h is the optional decoder function. A summary of +notations is given in II. +Based on the formulation above, EvoX can fully decouple +algorithms and problems and leaves more flexibility to the +workflow. On one hand, neither gask nor gtell calls f internally. +On the other hand, f is ignorant of the implementation of +algorithm functions gask and gtell, such that the user can easily +change the problem to a validation/test phase by simple set- +tings. The generalized workflow also allows each component +to work with its own way of parallelization. Moreover, since +SAθ and SPD can capture the randomness by explicitly storing +the pseudo-random number generator key inside, the states +can be easily synchronized when running EC algorithms in a +distributed systems. +Most importantly, the proposed generalized EC workflow +strictly follows the paradigm of functional programming, +which is intrinsically compatible with JAX-based implemen- +tations. +IV. ENGINEERING DESIGNS +On the basis of the generalized EC workflow as formulated +above, this section will further introduce the detailed engineer- +ing designs of EvoX. +A. Main Pipeline +In GPU computing, tensor is the essential data structure for +GPU acceleration. Hence, in EvoX, we view the pipeline of +running an EC algorithm for problem-solving as an iterative +procedure of processing the tensor of a population. +As shown in Fig. 1, generally, the Pipeline passes the +tensor of population Xt (as well as the corresponding fitness +Monitor +Fig. 1: One iteration of the standard pipeline. It starts with gask, +which gives a tensor Xt, representing the candidate population. +Then Xt is optionally sent to the decoder h and then passed +to f to evaluate its fitness yt. Finally, yt is passed to gtell, +where the algorithm can utilize this information and update +its state. Throughout this iteration, one can optionally choose +to monitor certain values such as Xt and yt. +Weights +Fig. 2: An illustration on how decoder can be used in +neuroevolution tasks. Here decoder decodes Xt into Xt +′, +which represents a set of weights for neural networks. +values yt) through the Algorithm module and Problem +module, with the support of optional modules of Monitor +and Decoder. +The Decoder module is designed to transform the popu- +lation Xt encoded by an EC algorithm into decision vectors +in the original problem space. For example, when training an +ANN, the weights are typically represented by a set of tensors, +one for each layer. However, EC algorithms often output a +single, tightly packed tensor as the population. In this case, +the Decoder can be used to decode the tightly packed tensor +Xt into a set of tensors representing the weights of an ANN, +as shown in Figure 2. +It is important to note that the Decoder is an optional +module, as it is not necessary for plain numerical optimization +where Xt is already defined in the original problem space. +The Monitor is designed for observing intermediate re- +sults in each iteration via visualization tools. For example, +the users may observe the population Xt to analyze how +the algorithm behaves in a certain fitness landscape. Be- +sides, the users may also observe the fitness yt to check +how the optimization process goes on. Additionally, another +functionality of Monitor is to check whether the termination +criterion (e.g. maximum number of iterations/evaluations) of +the optimization process is reached. +In the following subsections, we will elaborate another two +modules Algorithm and Problem in more detail. +B. Algorithm Module +As is described in section III, Algorithm in EvoX is +basically a class initialized with a set of hyperparameters θ, +consisting of two methods – ask and tell, maintaining a + +6 +1 import jax +2 import jax.numpy as jnp +3 from evox import State, Algorithm +4 +5 +6 class SimpleES(Algorithm): +7 +def __init__(self, dim, pop_size, topk): +8 +self.dim = dim +9 +self.pop_size = pop_size +10 +self.topk = topk +11 +12 +def setup(self, key): +13 +mean = jnp.zeros((dim,)) +14 +stdev = jnp.ones((self.dim,)) +15 +return State( +16 +mean=mean, +17 +stdev=stdev, +18 +key=key +19 +) +20 +21 +def ask(self, state): +22 +key, subkey = jax.random.split(state.key) +23 +noise = jax.random.normal( +24 +subkey, +25 +(self.pop_size, self.dim) +26 +) +27 +pop = state.mean + state.stdev * noise +28 +new_state = state.update( +29 +key=key, +30 +pop=pop +31 +) +32 +return sample, new_state +33 +34 +def tell(self, state, fitness): +35 +_topk_value, topk_index = jax.lax.top_k( +36 +fitness, +37 +self.topk +38 +) +39 +elite = state.pop[topk_index] +40 +new_mean = jnp.mean(elite, axis=0) +41 +new_stdev = jnp.std(elite, axis=0) +42 +new_state = state.update( +43 +mean=new_mean, +44 +stdev=new_stdev +45 +) +46 +return new_state +Listing 1: Vanilla implementation of evolution strategies in +EvoX. The implementation has four main parts: __init__, +setup, ask and tell. __init__ and setup are used +to initialize the algorithm. ask and tell contain the main +operations of the algorithm. +global state SAθ – an independent object in EvoX. Addition- +ally, Algorithm also has a setup method for generating +the initial state SAθ +0 . As an illustrative example, we present +the implementation of vanilla evolution strategy in EvoX, as +listed in Lst. 1. +To start with, in lines 7 to 10, the __init__ method +initializes three hyperparameters in the constructor of this +class. dim is the problem dimension, pop_size is the +population size, and topk indicates the top k individuals of +the population are considered as elite to be selected. +In lines 12 to 19, the setup method initializes its internal +state SAθ +0 . In this vanilla evolution strategy, we keep an inde- +pendent normal distribution for each decision variable, such +that the mean and standard deviation vectors are initialized +independently. In addition, we record key in the state as the +  +Take a batch according to +Calculate +Loss +Fig. 3: A demonstration of how one problem can handle +different evaluation modes. In this case, SP +t indicates that the +second batch from the training dataset should be used. +seed for the pseudo-random number generator. +In lines 21 to 32, the ask method is defined in correspon- +dence to gask +θ . In this method, the algorithm splits the key and +samples a new candidate population according to the mean and +standard deviation. Then a new state is generated by updating +the key and adding the candidate population to it. +In lines 34 to 45, the tell method of the algorithm is +defined in correspondence to gtell +θ . In this method, the algo- +rithm picks the top k individuals in the candidate population +according to their fitness ranking. Then, the mean and the +standard deviation are adapted to the mean and the standard +deviation of the elite population. +Finally, in line 46, the newly generated candidate population +is returned via the new_state. +C. Problem Module +In contrast to the past when EC algorithms were mostly +tested on pre-configured numerical benchmark problems, op- +timization problems of today are becoming increasingly com- +plex – usually involving data-related configurations. Therefore, +in our design, Problem is parameterized by dataset D with +internal state SPD +t +. +As illustrated in Fig. 3, in the case of ANN training, the +dataset D = {Dtrn, Dvld, Dtst, ...} can consist of training +data, validation data, and test data; correspondingly, the prob- +lem state SPD +t +will record the choice of dataset together with +other essential parameters such as batch index. With such a +design principal, Problem is well extensible to support a +wide spectrum of problems ranging from numerical optimiza- +tion (leaving D and SPD +t +empty) to other data-related tasks. +In the following, we will elaborate three typical scenarios +for defining problems using the tailored Problem Module +in EvoX. +1) Numerical Optimization: Since the plain numerical op- +timization problems are often well-formulated functions with +basic maths operations, output yt is only determined by the +input Xt. Thus in EvoX, such problems can be implemented +by simply leaving D and SPD +t +empty. +2) Neuroevolution: As introduced in Section II-B, a neu- +roevolution task usually involves the training of an ANN to +fit a certain dataset. Since a forward pass of an ANN is +expensive, in EvoX, we adopt the common workflow as in +mini-batch gradient descent. First, the whole training dataset +Dtrn is split into small batches of data B1, B1, .., Bn. Then, at + +7 +Controller +Environment +Worker +Worker +Worker +... +Fig. 4: An illustration of running RL tasks in EvoX. Xt +′ is +a population of ANN weights given by the decoder. These +weights will be evenly distributed to a set of workers. Each +worker will run a complete episode to obtain the rewards from +the RL environment. +each iteration, only one batch of data is used to calculate the +fitness of each individual as y(i) +t += 1 +n +� +x∈Bk L(x, Xt +′), where +L(·) is the loss function, i denotes the ith row in the column +vector yt, k denotes the batch index as stored in SPD +t +, and n +denotes the batch size. +In practice, since large batch sizes may be intractable to +calculate in a single forward pass, we also introduced a +parameter called num_passes, to allow the loss of a batch +to be calculated by multiple passes. +3) Reinforcement Learning: As introduced in Section II-C, +a typical RL task aims to train an agent for maximizing the +total reward in a certain environment. To obtain the reward, +it usually takes multiple steps for an agent to complete an +episode by interacting with the environment. Our EvoX helps +bridge the gap between EC and RL tasks via the tailored +Problem module. Specifically, EvoX has the candidate pop- +ulation Xt encode a set of parameters for the policy model, +where for each candidate solution, EvoX runs a complete +episode using this policy and return the sum of all rewards +as the fitness value. It is worth noting that, since there can +be multiple rewards in some RL environments, one may treat +these multiple rewards just as in the case of multi-objective +optimization. +In practice, it can be computationally expensive to complete +an episode to get the final reward, while some popular RL +environment (e.g. Gym) merely provides a single-thread inter- +face. This can be a painful bottleneck for the population-based +EC algorithms which require batches of fitness evaluations in +each generation, substantially hindering the development of +EvoRL. Hence, to speed up the fitness evaluation process for +RL tasks, we design a Ray-based interface for running RL +environments with parallel computing. As illustrated in Fig. 4, +with our interface, the function evaluation process for RL tasks +can be deployed in a parallel computing environment with +multiple workers, where each worker is created on CPU/GPU +core for running the RL environment(s) in parallel. +D. Distributed Acceleration +In a typical EC workflow, the main computational cost +usually comes from a large number of fitness evaluations of the +Node +Fitness Exchange +... +... +... +... +CPU +GPU +Node +CPU +GPU +Node +CPU +GPU +Fig. 5: The task-parallel workflow using distributed +pipeline with n nodes. Each node will evaluate only +1 +n +of the population and then exchange the fitness with others. +candidate solutions. Since the fitness evaluation of each candi- +date solution is intrinsically isolated, theoretically, we can have +an EC algorithm deployed on a distributed computing system +to improve the concurrency of fitness evaluations. However, +when scaling beyond the boundary of physical machines, +the communication cost between different machines becomes +another obstacle. Intuitively, we may attempt to achieve dis- +tributed fitness evaluations by running the algorithm in one +node and sending the candidate solutions to the other nodes +for evaluation. However, when it comes to complex tasks +such as neuroevolution, each candidate solution may consist +of millions (or even larger number) of decision variables, and +thus the communication overhead for sending the candidate +solutions to each node will be prohibitively huge. Hence, as the +problem dimension increases, such an intuitive method suffers +from the sharply increasing communication costs between the +nodes, thus leading to poor scalability. +In order to accelerate the EC workflow by distributed +computing in a scalable manner, we design a distributed +pipeline on top of Ray, partially inspired by the work +in [46]. As illustrated in Fig. 5, given a number of n +nodes (i.e. machines) in distributed computing systems, +the distributed pipeline creates a copy of standard +pipeline (refer to Section IV-A) on each node; hhen the +pipeline on each node only takes care of 1 +n of the candidate +solutions for fitness evaluations, running in an isolated and +concurrent manner as if the other nodes do not exist; finally, +the fitness values yt +′ as obtained by each node are exchanged +to generate the entire fitness vector yt to be passed to the next +generation. +Another important issue to be considered is the synchro- +nization – to have the algorithm running on each pipeline +image performing synchronous behaviors, such that the fitness +information is always exchangeable. To this end, we have the +pipeline copies on each node initialized with the same +random seed, such that the subsequent behaviors (despite +the randomness as generated via the same random seed) of +the algorithm will be naturally synchronous per iteration t. +This synchronization method also gurantees that the same +experiments will always end up with the same result regardless +of the number of nodes in use, thus further strengthening the +reproducibility of the experiments conducted with EvoX. + +8 +EvoX +Operators +Algorithms +Problems +PSO +DE +CMA-ES +NSGA-II +MOEA/D +IBEA +Single-objective +Multi-objective +Selection +Reproduction +Numerical +Extended +Tournament +Best +BitFlip +SBX +Sphere +Ackley +Rosenbrock +TorchVision +Gym +... +... +... +... +... +... +Gaussian +Roulette +Fig. 6: Main library content of EvoX. The Algorithms component contains two sub-components: Single-objective and Multi- +objective. The Operators component contains two subcomponents: Selection and Reproduction. The Problems component +currently contains two subcomponents: Numerical Benchmarks, and Extended Applications which include tasks like neuroevo- +lution and RL. +V. LIBRARY CONTENT +As summarized in Fig. 6, the content of EvoX mainly +consists of three main components: the Algorithms component, +the Operators component, and the Problems component. In the +following, we will introduce each component in detail. +The Algorithms component includes all EC algorithms +available in EvoX, consisting of two subcomponents: Single- +objective and Multi-objective. The Single-objective subcom- +ponent includes algorithms for single-objective optimiza- +tion(PSO [50], [51], DE [52], CSO [13], etc.), while the +Multi-objective subcomponent includes EC algorithms for +multi-objective optimization (NSGA-II [53], MOEA/D [54], +RVEA [55], etc.). +The Problems component includes all the pre-defined prob- +lems in EvoX, consisting of two subcomponents: Numerical +and Extended. The Numerical subcomponent includes all pos- +sible numerical benchmark problems, including not only basic +ones for single-objective optimization (Sphere, Ackley [56], +etc.), but also composite ones for multi-objective optimization +(ZDT [57], DTLZ [58], etc.). The Extended subcomponent +includes potential extended application problems, currently +mainly related to neuroevolution and RL tasks. +The Operators component includes commonly used oper- +ators in EC algorithms, consisting of two subcomponents: +Selection and Reproduction. The Selection subcomponent in- +cludes all possible selection operators that can be adopted for +selecting candidate solutions (Tournament Selection, Random +Selection, etc). The Reproduction subcomponent includes all +possible operators for reproducing new candidate solutions +(BitFlip Mutation, SBX Crossover, etc.). +Apart from the above components, EvoX also allows the +user to implement their own components. Thanks to the well +decoupled modular engineering designs, the users may replace +any of the existing components with their own tailored one(s) +while reusing all other contents provided by EvoX. +VI. EXPERIMENTAL STUDY +To demonstrate the performance of EvoX, we will conduct +a series of experiments in this section. First, Section VI-A in- +troduces the general experimental settings. Then, Section VI-B +demonstrates the scalability performance brought about by +GPU computing. Moreover, Section VI-C demonstrates the +acceleration performance brought about by distributed com- +puting Finally, Section VI-D demonstrates the extendability +of EvoX towards complex RL tasks. +A. Experiment Settings +The experiment in Section VI-B is carried out on a physical +machine equipped with an Intel Core i9-10900X CPU @ +3.70GHz and a single NVIDIA RTX 3090 GPU. All the other +experiments are conducted on a cluster with 8 nodes, where +each node has 16 cores and 32 threads from an Intel Xeon +Gold 6226R CPU @ 2.90GHz CPU and a single NVIDIA +RTX 3090 GPU. Specifically, the experiment in Section VI-C +uses up to 6 nodes while the experiment in Section VI-D only +use a single node. +All experiments were repeated for 11 times using different +random seeds. +B. Scalability Test via GPU Computing +To demonstrate the scalability performance of EvoX, we +conduct benchmark experiments by running both single- +objective and multi-objective algorithms on a single GPU +device, in comparison with conventional CPU computing. The +benchmark program is first run with GPU enabled and disabled +respectively. In the single-objective benchmarks, we have the +classic PSO run on the Ackley problem. In multi-objective +benchmarks, we have the NSGA-II run on the ZDT1. In +both single-objective and multi-objective benchmarks, we first +fix the population size to 128 and scale up the number of +dimensions of the problem, and then we fix the number of +dimensions to 128 and scale up the population size. +As shown in Fig. 7a, when scaling up the number of +problem dimensions with PSO, CPU computing and GPU +computing have similar performance when the number of +problem dimensions is very smaller, but as the number +of problem dimensions becomes larger, the performance of + +9 +101 102 103 104 105 106 +Problem dimensions +101 +102 +103 +104 +Time per iteration (ms) +CPU +GPU +(a) Scaling the number of di- +mensions with single-objective +algorithm PSO. +101 102 103 104 105 106 +Population size +101 +102 +103 +Time per iteration (ms) +CPU +GPU +(b) +Scaling +population +size +with single-objective algorithm +PSO. +101 102 103 104 105 106 +Problem dimensions +101 +102 +103 +104 +Time per iteration (ms) +CPU +GPU +(c) Scaling the number of di- +mensions with multi-objective +algorithm NSGA-II. +101 102 103 104 105 +Population size +101 +102 +103 +104 +105 +Time per iteration (ms) +CPU +GPU +(d) +Scaling +population +size +with multi-objective algorithm +NSGA-II. +Fig. 7: Results of scalability test. The dark lines represent +the mean values runs and shaded regions are bounded by +the standard deviations. Both the x-axis and the y-axis are +in logarithmic scale. +GPU computing quickly surpasses CPU computing. When the +number of problem dimensions is larger than 100,000, GPU +computing is almost 100x more efficient in terms of time per +iteration. As shown in Fig. 7b, scaling up population size in +PSO shows similar observations. +As shown in Fig. 7c, when scaling up the number of +problem dimensions with NSGA-II, we can also observe +that GPU computing constantly performs better than CPU +computing. As shown in Fig. 7d, when it comes to scaling +up population size with NSGA-II, GPU computing also scales +much better than CPU computing. In comparison with PSO, +however, the general computing cost of NSGA-II increases +sharper in terms of population size, mainly due to the higher +algorithmic computational complexity. +C. Acceleration Test via Distributed Computing +To assess the acceleration performance of EvoX, we con- +duct an experiment on the task of neuroevolution for image +classification on multiple GPU devices. Specifically, we train +a convolution neural network on the CIFAR-10 dataset [59] +using 1 to 8 GPUs and measure the total runtime. CIFAR-10 +is an image classification dataset containing a total of 60,000 +32 × 32 color images. The architecture of the convolution +neural network is given in Tab. III, and ReLU [60] is used as +the activation function in between layers. For the algorithm +module, we adopt PGPE [61] with ClipUp [62], and the +population size is set to 300. On the problem module, we set +TABLE III: Architecture of the ANN adopted in the neuroevo- +lution experiment. +Input Shape +Layer +Filter Shape +Strides +32 × 32 × 3 +Conv +3 × 3 × 3 × 32 +1 +30 × 30 × 32 +Max Pooling +2 × 2 +2 +15 × 15 × 32 +Conv +3 × 3 × 32 × 32 +1 +13 × 13 × 32 +Max Pooling +2 × 2 +2 +6 × 6 × 32 +Conv +3 × 3 × 32 × 32 +1 +512 +Fully Connected +512 × 64 +— +64 +Fully Connected +64 × 10 +— +1 +2 +3 +4 +5 +6 +Number of nodes +200 +400 +600 +Total runtime (s) +(a) Runtime +1 +2 +3 +4 +5 +6 +Number of nodes +0.002 +0.004 +0.006 +0.008 +Performance (s +1) +(b) Performance +Fig. 8: Results of acceleration test. The number of GPU nodes +are ranged from 1 to 6. The same data is presented in two +ways, in (a) the y-axis is the total runtime, and in (b) the y- +axis is the performance which is measured by the inverse of +the runtime. +the batch size to 3000. When conducting the experiment with +n nodes, since each node only evaluates +1 +n of the candidate +population, we also tune the num_passes parameter to 30 +n +in order to fully utilize the each GPU device. The experiment +runs 1000 training iterations in total, with a validation phase +for every 100 iterations. +Fig. 8a and Fig. 8b present the runtime and the perfor- +mance with different numbers of GPU nodes respectively. The +runtime includes both the training phase and the validation +phase. It can be observed that the runtime rapidly decreases +as the number of GPU nodes grows, achieving near-linear +acceleration rate. This observation is consistent with our +expectation as the distributed pipeline only requires +exchanges of cheap fitness values among the workers, being +communication-efficient. +D. Extensibility to RL Tasks +To assess the usability of EvoX, when extended to solving +complex application problems, we conduct an experiment on +two representative RL tasks as shown in Fig. 9. We will +demonstrate that EvoX can handle both tasks fluently and +efficiently despite their challenging features. +• Bipedal Walker: In this RL task, the agent controls a 4- +joint walker robot. The goal is to move forward without +falling and apply as little torque as possible. At any given +moment, the agent can observe 24 real values given by +the sensors and control the torque applied to the four +motors on the robot, ranging from −1 to 1. Specifically, +the agent adopts a policy model of an MLP consisting +of two hidden layers, each containing 64 neurons. To + +10 +(a) Bipedal Walker +(b) ATARI Pong +Fig. 9: In-game images of two RL tasks. (a) The Bipedal +Walker: in this RL task, the agent receives the observation +as a vector and controls the robot to move forward. (b) The +ATARI game Pong: in this RL task, the agent can observe the +raw image of the game and control the right paddle. +TABLE IV: Parameter settings of CMA-ES +Parameter +Value +Population Size +128 +Initial step size +0.1 +TABLE V: Parameter settings of PGPE +Parameter +Value +Population Size +128 +Learning rate for standard deviation +0.01 +Adaptive Optimizer +Adam [63] +Fitness Shaping +Yes +Standard deviation max change +0.2 +learn the policy model, there are a total number of 6020 +parameters to be optimized. Here, we apply the CMA-ES +algorithm as the optimizer, where the detailed algorithm +settings can be found at Tab. IV. +• ATART Pong: In this RL task, the agent controls a paddle +on the right side of the screen and tries to use it to hit +a ball away from its own goal and into the opponent’s +goal. For decision makings, the agent observes the raw +frame from the game, which is a 210 × 160 colored +image. To reduce the complexity, we pre-process the +image first by converting the full-size colored image into +gray-scale one with a smaller size (80 × 80). The agent +employs a policy model of an MLP with 2 hidden layers, +where the first and second hidden layers have 64 and 32 +neurons respectively. The model takes the pre-processed +image as input and outputs the probability of all available +actions. To simplify the decision-making process, we +always choose the action with the highest probability. In +order to learn this policy model, there are a total number +of 411,942 parameters to be optimized. Here, we use +PGPE as the training algorithm to optimize the policy +of the agent, where the detailed algorithm settings can be +found in Tab. V. +As shown in Fig. 10, in both RL tasks, utilizing multiple +processes results in a significant performance improvement +in terms of computational time per iteration. It is worth +noting that although both tasks are mainly executed via CPU +computing, the entire algorithm workflow still runs on GPU. +Multi- +processing +Single- +processing +0 +10 +20 +30 +40 +Time per iteration (sec) +(a) Bipedal Walker +Multi- +processing +Single- +processing +0 +25 +50 +75 +100 +Time per iteration (sec) +(b) ATARI Pong +Fig. 10: Performance comparison of two RL tasks with mul- +tiprocessing enabled and disabled, in terms of computational +time per iteration. (a) In the Bipedal Walker task, enabling +multiprocessing resulted in almost 6× performance improve- +ment. (b) In ATARI Pong, enabling multiprocessing resulted +in almost 12× performance improvement. +0 +40000 +80000 +120000 +Episodes +50 +0 +50 +100 +150 +200 +250 +300 +Reward +(a) Bipedal Walker +0 +40000 +80000 +120000 +Episodes +25 +20 +15 +10 +5 +0 +5 +10 +15 +Reward +(b) ATARI Pong +Fig. 11: The best rewards achieved on each RL task per +episode across different seeds. At each iteration, a policy is +evaluated by running the policy in the environment once and +the highest reward in the population is recorded. The dark line +indicates the average highest reward across different runs, and +the shaded region is bounded by its standard deviation. +This behavior is made possible by our design as described +in Section IV, where the algorithm module and the problem +module are decoupled to allow for tasks that are CPU-intensive +and not native to EvoX to achieve high parallelism. +In addition to performance improvement in terms of com- +putational time, the experimental results also demonstrate the +potential of EC algorithms in tackling challenging RL tasks. +The Bipedal Walker task poses a significant challenge in +evolving a valid walking strategy. As shown in Fig. 11a, there + +11 +is a large variation during the later part of the optimization +process. The reason for this is that the point when an agent +learns to walk marks the edge of a plateau in the optimization +process. Before this point, the agent receives little to no +reward, but once it starts moving forward, it quickly earns +high rewards. So the high variance reflects the fact that the +algorithm sometimes discovers the walking strategy early on, +leading to significantly better performance. However, in some +attempts, the algorithm fails to discover a decent strategy, +leading to poor performance. +In the ATARI Pong task, despite having a large number +of trainable parameters (400K), the algorithm, as shown in +Fig. 11b, steadily improves the policy towards a promising +solution, with little variance. +In summary, EvoX provides comprehensive support for +extending EC algorithms to RL tasks, as well as other com- +plex application problems. With little additional engineering +work, practitioners can instantly enjoy the benefits of EvoX +by connecting their problems via the seamless and friendly +interface in the problem module. +VII. CONCLUSION AND FUTURE WORK +In this paper, we presented EvoX, a distributed GPU- +accelerated EC library that significantly improves the scala- +bility of the EC workflow. When using a single machine, we +observed a two-orders-of-magnitude speed-up in certain set- +tings. By running on multiple machines, we were able to scale +the workflow even further. This advancement allows existing +EC algorithms to solve problems with high-dimensional search +spaces more efficiently. Our work also paves the way for future +EC development, making it easy to develop more complex +EC algorithms that can leverage increasing computing power. +Furthermore, meta-learning algorithms could also be greatly +improved by the increased computing power, as the process of +meta-learning is known to be computationally demanding. Ad- +ditionally, we simplify the process of applying EC algorithms +to extended applications. With EvoX, EC algorithms can +work with neuroevolution or RL tasks seamlessly, substantially +reducing the barrier for researchers to tackle these problems. +However, during the development of EvoX, we also iden- +tified certain limitations. First, many algorithms were not +designed with parallelism in mind, thus making it difficult +to accelerate them via advanced GPU computing techniques. +Moreover, many extended applications are CPU-intensive and +tend to be the most time-consuming part of the workflow, thus +somehow diminishing the impact of acceleration in EC algo- +rithms. In the future, we plan to incorporate more parallelism +within existing algorithms and integrate external tasks directly +into our library to enable GPU computing in both algorithms +and problems. +ACKNOWLEDGEMENT +We wish to thank Zhenyu Liang for his contributions to the +multiple-objective algorithms and Kebin Sun for helping with +the single-objective algorithms. Additionally, we would like +to express our appreciation to Lishuang Wang and Jiachun +Li for their efforts in testing out the library, and for their +miscellaneous contributions to the library. +REFERENCES +[1] N. Gould, D. Orban, and P. 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Available: http://arxiv.org/abs/1412.6980 + diff --git a/jdFMT4oBgHgl3EQf5zEl/content/tmp_files/load_file.txt b/jdFMT4oBgHgl3EQf5zEl/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..911b3af970fd0dfb1393383d03d1a0e0edd1e4fa --- /dev/null +++ b/jdFMT4oBgHgl3EQf5zEl/content/tmp_files/load_file.txt @@ -0,0 +1,1028 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf,len=1027 +page_content='1 EvoX: A Distributed GPU-accelerated Library towards Scalable Evolutionary Computation Beichen Huang, Ran Cheng,Yaochu Jin, Fellow, IEEE, and Kay Chen Tan, Fellow, IEEE Abstract—During the past decades, evolutionary computation (EC) has demonstrated promising potential in solving various complex optimization problems of relatively small scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Nowa- days, however, ongoing developments in modern science and engineering are bringing increasingly grave challenges to the con- ventional EC paradigm in terms of scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' As problem scales increase, on the one hand, the encoding spaces (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=', dimensions of the decision vectors) are intrinsically larger;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' on the other hand, EC algorithms often require growing numbers of function evaluations (and probably larger population sizes as well) to work properly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' To meet such emerging challenges, not only does it require delicate algorithm designs, but more importantly, a high-performance computing framework is indispensable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Hence, we develop a distributed GPU-accelerated algorithm library — EvoX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' First, we propose a generalized workflow for imple- menting general EC algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Second, we design a scalable computing framework for running EC algorithms on distributed GPU devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Third, we provide user-friendly interfaces to both researchers and practitioners for benchmark studies as well as extended real-world applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Empirically, we assess the promising scalability of EvoX via a series of benchmark experi- ments with problem dimensions/population sizes up to millions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Moreover, we demonstrate the easy usability of EvoX by applying it to solving reinforcement learning tasks on OpenAI Gym.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' To the best of our knowledge, this is the first library supporting distributed GPU computing in the EC literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' The code of EvoX is available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content='com/EMI-Group/EvoX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Index Terms—Evolutionary multi-objective optimization, neu- ral architecture search, deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' INTRODUCTION W ITH the development of modern science and engi- neering, various emerging optimization problems are posing stiff challenges to the optimization algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Despite that evolutionary computation (EC) has been shown to be a promising tool for solving complex optimization problems of relatively small scales, it has come to a consensus that the conventional EC paradigm suffers from the curse of dimensionality [1], [2] - the phenomenon that the search space of a problem grows exponentially with the number of dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Since EC algorithms often rely on random search/sampling to find solutions, the very large number of possible solutions can make it difficult for EC algorithms to explore the space effectively in high-dimensional complex search spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Additionally, the computational complexities of EC algorithms may also grow with the number of dimensions, making them slow and impractical for large-scale optimization problems [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' To improve the scalability of EC algorithms, researchers have made persistent efforts during the past decade [4]–[6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' In the early days, most efforts were mainly dedicated to making improvements from the methodology point of view, by proposing tailored algorithm frameworks/operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' For example, the cooperative coevolution (CC) [7]–[9] is among the popular frameworks tailored for improving the scalability of EC algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' In the CC framework, a problem is divided into a collection of lower-dimensional subproblems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Each subproblem is optimized individually and its population of candidate solutions is coevolved with the other subproblems in a round-robin fashion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' then, representative solutions from each subproblem are combined to form a context vector, which is used to evaluate the overall solution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' finally, the context vector is updated iteratively and serves as the context for cooperation between the subproblems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Some other researchers proposed various tailored operators with better scalability, including variants of the differential evolution (DE) [10]–[12], the particle swarm optimization (PSO) [13]–[16], as well as the estimation of distribution algorithms (EDAs) [17]–[20], among many others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Undoubtedly, algorithmic innovations can improve the scal- ability of EC algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Nowadays, however, when consider- ing the performance of an algorithm, it is also important to take into consideration the essential roles of modern computing architectures/devices [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' As the most indicative example, the rapid development of modern deep learning algorithms can be attributed, in part, to the advancement of GPUs for training and running deep neural networks more efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Inherently, the population-based nature of EC algorithms also makes it possible to potentially parallelize the computing process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' There have been some attempts that aim to improve the scalability of EC algorithms through GPU computing or distributed computing: a parallelized version of DE was proposed to use GPU computing and can handle continuous problems with up to 1000 dimensions [22];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' a parallel version of the compact genetic algorithm for CPU/GPU architectures was proposed and tested on OneMax and noisy OneMax with up to one billion variables [23];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' an improved GPU-based model for MA- SW-Chain was proposed and tested on a scaled version of the CEC’2013 large-scale benchmarking suite on functions with up to 100 million decision variables [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Despite a few individual works along this line, there has not yet been a systematic research effort in the EC field similar to what has been seen in the deep learning area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' The literature has already demonstrated the potential for EC algorithms to improve scalability, whether through algorithmic innovations or hardware acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' However, there is still much room for improvement and further research, especially in terms of developing more efficient and effective algorithms by utilizing modern computing architectures and devices to arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content='12457v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content='NE] 29 Jan 2023 2 their full potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Additionally, it would be beneficial to investigate the potential applications of EC in various domains and industries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Nonetheless, there are three main issues that are currently hindering the further development of scalable EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' \x82 Workflow: In a typical EC workflow, the main components (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=', crossover/mutation, fitness evaluation, selection) are ex- ecuted sequentially in a main loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' While this structure is organized and cohesive, it is not compatible with asynchronous computing methods such as GPU computing or distributed computing, making it challenging to implement or modify algorithms for improved flexibility and concurrency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' \x83 Computational Cost: The main source of computational cost in EC algorithms is the fitness evaluations needed for population-based stochastic search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' From a statistical perspec- tive, it is often necessary to increase the number of samples (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=', fitness evaluations) exponentially as the dimension of the search space grows linearly in order to accurately approximate a target distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' \x84 Running Environment: While most EC algorithms were initially developed for solving pure numerical optimization problems and do not have specific requirements for the run- ning environment, the running environments for large-scale optimization tasks are often specific to the task at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' For example, neuroevolution tasks are often closely tied to deep learning scenarios, which require running environments with software/hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' To push the boundary of EC towards better scalability and wider applicability by addressing the aforementioned issues, we develop a distributed GPU-accelerated algorithm library – EvoX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' In summary, the main contributions are: – We propose a generalized EC workflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' On the one hand, it fully decouples the implementation of algorithms, problems, user-friendly monitors, and possible population decoders or fitness transformers, bringing more flexibility and generality to EC than before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' On the other hand, complete modularization makes it possible for both researchers and practitioners to eas- ily parallelize EC algorithms via high-performance computing frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' – We develop a scalable computing framework for running EC algorithms on distributed GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Based on the proposed generalized EC workflow, a powerful distributed GPU accel- eration library EvoX is developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' First, EvoX supports ready- to-use GPU computing acceleration, such that users can easily run EC algorithms on GPU(s) without any additional engi- neering work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Second, EvoX supports ready-to-use distributed computing framework, such that users can easily deploy EC algorithms on distributed machines at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' – We provide a user-friendly interface for both numerical benchmark tests and other challenging problems in real-world applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' First, EvoX provides a generalized Problem module to fully support running EC algorithms for challenging data-related tasks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' neuroevolution) via GPU computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Second, EvoX provides a tailored interface to provide seam- less connections to complex environments (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' those in rein- forcement learning) on top of a high-performance distributed computing framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' The remainder of this paper will be organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Section II provides the necessary background information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Section III presents a generalized EC workflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Section IV and Section V describe the design and contents of EvoX, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Section VI contains the experiments conducted on EvoX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Finally, Section VII concludes the paper and discusses our future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' BACKGROUND AND RELATED WORK In this section, we first briefly overview some representa- tive EC libraries;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' then we provide background knowledge of neuroevolution, evolutionary reinforcement learning;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' finally, we introduce the related techniques of GPU computing and distributed computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' EC libraries In the EC field, the Python programming language has emerged as a popular choice for implementing EC algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' This is due in part to the availability of powerful and easy- to-use EC libraries, such as DEAP (Distributed Evolutionary Algorithms in Python) [25], PyGAD (Python Genetic Algo- rithms and Differential Evolution Framework) [26], Pymoo (Python Multi-Objective Optimization) [27], and Pagmo (Par- allel Global Multiobjective Optimizer) [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' In this subsection, we will review the features and capabilities of these libraries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' DEAP is a long-standing and feature-rich framework for implementing evolutionary algorithms in Python.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' It offers support for a wide range of EC algorithms, including both single and multiple objective algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' DEAP also includes a broad range of built-in benchmark problems, making it easy to evaluate the performance of EC algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' DEAP is well- suited for rapid prototyping and testing of ideas and is a popular choice among researchers in the field of EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Another prominent feature of DEAP is its support for parallelization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' DEAP allows evaluations to be easily parallelized, making it possible to run them on multiple cores or even across multiple machines through scoop [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' PyGAD is a library for implementing genetic algorithms in Python.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' It offers different types of crossover, mutation, and parent selection operators for implementing genetic algo- rithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' What makes PyGAD unique is its focus on machine learning tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' PyGAD includes features and tools specifically designed for training artificial neural networks, making it a good choice for applying evolutionary computation (EC) in machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Pymoo is a library that focuses on multi-objective optimiza- tion algorithms in Python.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Its main strength lies in its com- prehensive support for multi-objective optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Pymoo includes a wide range of benchmark problems and state-of- the-art multi-objective algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Pymoo also includes many operators suitable for multi-objective algorithms, allowing users to easily customize the algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Furthermore, Pymoo has a set of powerful and flexible features related to multi- objective optimization, such as visualization and decision- making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' All these features building towards multi-objective optimization make Pymoo a suitable tool for the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Pagmo is a C++ library for massively parallel optimization, with a Python binding called Pygmo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' It is built around the 3 TABLE I: Summarized main features of EC libraries Name System Algorithms Problems GPU Computing Distributed Computing Single-objective Algorithms Multi-objective Algorithms Numerical Benchmarks Neuroevolution Tasks Reinforcement Learning Tasks DEAP ✓ ✓ ✓ ✓ PyGAD ✓ ✓ ✓ pymoo ✓ ✓ ✓ pagmo ✓ ✓ ✓ EvoX ✓ ✓ ✓ ✓ ✓ ✓ ✓ generalized island model [30], which allows coarse-grained parallelization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' It offers a variety of algorithms, benchmark problems, and migration policies, making it easy for users to implement parallelized algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Additionally, it supports batch fitness evaluation, enabling users to perform parallel fitness evaluations using their own methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Despite the attractive features introduced above, these exist- ing libraries have a common deficiency: the lack of scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Potentially, both GPU computing and distributed computing are powerful tools for improving the scalability of EC, partic- ularly in scenarios involving large amounts of data or complex calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' However, none of these existing supports GPU computing, while the support for distributed computing is either missing or inefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' As a result, the lack of support for GPU computing or dis- tributed computing in these libraries has largely limited their performance and applicability for certain types of optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Besides, the extendability of these libraries towards more complex problems is also very limited, making it difficult for EC practitioners to get involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' In detail, the key features of these libraries in comparison with EvoX are summarized in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Neuroevolution Neuroevolution is a field focusing on using evolutionary computation (EC) algorithms to optimize artificial neural net- works (ANNs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' It has a long history of development and is facing some emerging challenges and opportunities [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' In the field of neuroevolution, the use of EC algorithms for optimizing ANNs has gained significant attention due to its potential advantages over traditional gradient-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Since EC algorithms can explore a much larger search space than gradient-based methods, they have better potential for discovering more diverse and novel solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Broadly speak- ing, neuroevolution is able to evolve various aspects of neural networks, such as the building blocks, architectures, weights, and even the training rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Although neuroevolution was initially considered an alter- native to backpropagation for optimizing the weights of small and fixed-topology ANNs, some attempts quickly turned to evolving network architectures as well [32]–[34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' With the booming development of deep learning, researchers are now paying increasing attention to the automatic design of deep neural networks (DNNs) via neuroevolution - the evolutionary neural architecture search (ENAS) [35] - which is particularly useful when facing complex scenarios involving multiple ob- jectives to be optimized simultaneously (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' hardware-aware deployment of DNNs [36]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Despite the biologically plausible and technically attractive characteristics of neuroevolution, one of the main limitations is the computational complexity, which is particularly chal- lenging when dealing with DNNs [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Intuitively, one way to improve the scalability of neuroevolution algorithms is to make full use of the computing power of GPUs and distributed computing systems, such that different candidate solutions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' networks) in the population can be evaluated (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' trained) simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' This would allow for more efficient and faster training of ANNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' However, to the best of our knowledge, none of the existing EC libraries currently supports distributed GPU acceleration for neuroevolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Evolutionary Reinforcement Learning Reinforcement learning (RL) is a powerful and widely- studied framework for learning and decision-making in com- plex, dynamic environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' At its core, RL is concerned with how an agent should act to maximize a reward signal over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' This requires the agent to learn a policy, or a mapping from states to actions, that will allow it to take actions that lead to the most reward from the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' The environment is typically modeled as a Markov decision process (MDP), consisting of: set of states S with an initial state distribution P(s0), set of possible actions A, reward function R : S × A → R, transition function T : S × A → P(S), discount factor γ ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' At each time step t, the agent can choose an action at ∈ A based on the current state st and a policy π : S → P(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' The objective is to find the optimal policy π∗ that maximizes the expected reward: π∗ = argmax π EP (s0),π,T [ T −1 � t=0 γtrt], (1) where T is length of the episode, at ∼ π(st), st+1 ∼ T (·|st, at), and rt = R(st, at).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Evolutionary Reinforcement Learning (EvoRL) [38] specif- ically focuses on using EC algorithms to deal with various challenging optimization problems in RL, such as hyperparam- eter optimization [39], policy search [40], reward shaping [41], exploration [42], among many others [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' One of the key advantages of EvoRL is that it can explore a much larger 4 search space than gradient-based RL methods such as Q- learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' This allows for the discovery of a wider range of potential policies, which can lead to better performance in terms of reward and other metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' In recent years, as tasks and environments have become increasingly complex, scalability has become a major bot- tleneck for EvoRL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' On the one hand, due to its population- based nature, EvoRL can be computationally intensive, making it difficult to apply in real-time or real-world situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' On the other hand, the large number of parameters in the policy models can cause a severe curse of dimensionality for the EC algorithm when applied to RL tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' To address these challenges, researchers have proposed a number of methods to improve the scalability of EvoRL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' These methods include using parallel computing to distribute the computational work- load across multiple machines or processors [38], [44], as well as using more efficient algorithms and strategies for evolving the population [45], [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' However, the development of EvoRL is still in its infancy, and there is a lack of a high-performance EC library with a user-friendly interface for solving RL tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' JAX for GPU Computing GPU computing is a technology that utilizes GPUs to conduct general-purpose computing instead of CPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' A GPU usually consists of tens or hundreds of thousands of cores, which is thousands of times more than a typical CPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' To match the computational capabilities of GPUs, their memories are often way faster in terms of bandwidth compared to CPU memories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Moreover, to provide fast synchronization between different cores, GPUs usually have much larger and faster shared caches compared to CPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' These properties make it possible to process data on GPUs with higher parallelization than CPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' In the past decade, GPU computing has been one of the driv- ing forces of deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Libraries like PyTorch [47] have introduced the ability to utilize GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Nonetheless, few works have been dedicated to accelerating EC algorithms via GPU computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Since EC algorithms usually involve a chain of computationally cheap operations, they are often constrained by memory bandwidth more than computing power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Hence, the decrease in the number of memory accesses can drastically improve their performance when parallelized on GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Recently, the JAX has been released as a library offering a NumPy-like API for GPU-accelerated numeral calcula- tions [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' With just-in-time compilation features, JAX can run on multiple hardware backends including both CPU and GPU by optimizing the Python functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' The optimization in the compilation provides automatic fusing of small operators, thus substantially helping to save memory bandwidth Such features of JAX are particularly beneficial for the parallelization of EC algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Ray for Distributed Computing Distributed computing allows for the collective power of multiple computers to be harnessed in order to solve complex problems that would be difficult or impossible to solve using a single computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Since computers coordinate with each other by passing messages through the network, the communication cost in a distributed system usually has a huge impact on the overall scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' In a distributed system, each computer (or node) has its own local memory that cannot be accessed by other computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' To solve a problem collaboratively, comput- ers need to communicate with each other by message passing through the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Ray is a popular framework for distributed computing in Python and has been shown to be well-suited for applications in machine learning and other scientific computing tasks [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' As a user-friendly framework, Ray provides both actor-parallel and task-parallel programming abstraction, and the communi- cation between actors and tasks will be handled automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' One of the key features of Ray is its distributed scheduler, which is composed of both a global scheduler and per-node local schedulers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' This design allows Ray to efficiently schedule tasks to run on the appropriate node, both locally and across the distributed system, providing improved scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' With its scheduler, users can specify resource requirements for actors and tasks, and Ray will automatically place them on nodes with adequate resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Such features will allow us to easily scale the EC algorithms across multiple machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Discussions EC has been shown to have promising potential in tack- ling complex tasks such as those found in neuroevolution and EvoRL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' However, the lack of support for computing acceleration in existing EC libraries presents a challenge to further development in more advanced EC algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' To address this limitation, we have initiated the development of EvoX, a new library that improves the scalability of EC algorithms by leveraging the strengths of recently-developed high-performance computing tools: JAX and Ray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' On the one hand, JAX is well-suited for GPU computing of EC algorithms due to its use of just-in-time compilation and support of CPU/GPU backends, which fuses operations and minimizes memory accesses during acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' On the other hand, Ray is a distributed framework that allows for the scheduling of computations across multiple machines and CPU/GPU resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' By combining the capabilities of both, EvoX is able to improve the scalability of EC and extend the applications towards larger and more complex problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' GENERALIZED EC WORKFLOW Generally, distributed GPU acceleration of EC workflow may face two main issues: on the one hand, each component in an EC workflow may have its own way of parallelization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' on the other hand, different components in an EC workflow must be synchronized in the distributed system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' To address such issues, we propose a generalized EC workflow on top of an ask-and-tell interface, which considers an EC workflow as an agent (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' an EC algorithm) which iteratively transitions through the states by performing ask and tell actions for problem-solving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Specifically,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' given θ and D are the hyperparameters for defining the problem and the algorithm respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' the al- gorithm can be characterized by Aθ = ⟨θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' gask,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' gtell⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' problem 5 TABLE II: Summary of notations Notation Description t The generation counter Aθ The algorithm parameterized by θ PD The problem parameterized by D SAθ t The state of Aθ at generation t SPD t The state of PD at generation t f The fitness function h The decoder gask The ask method of the algorithm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' used to give out candidate solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' gtell The tell method of the algorithm, used to update the state based on the fitness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Xt The candidate solutions at generation t yt The fitness values at generation t PD = ⟨D, f⟩, where gask and gtell are ask and tell actions for generating a new population and updating the algorithm state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' A simple iteration on generation t is: Xt, SAθ t+1 = gask θ (SAθ t ), (2) yt, SPD t+1 = fD(SPD t , h(Xt)), (3) SAθ t+1 = gtell θ (SAθ t+1, yt), (4) where Xt, yt denote the population of candidate solutions and the corresponding fitness values at generation respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' SAθ, SPD are the state of the algorithm and problem re- spectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' h is the optional decoder function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' A summary of notations is given in II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Based on the formulation above, EvoX can fully decouple algorithms and problems and leaves more flexibility to the workflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' On one hand, neither gask nor gtell calls f internally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' On the other hand, f is ignorant of the implementation of algorithm functions gask and gtell, such that the user can easily change the problem to a validation/test phase by simple set- tings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' The generalized workflow also allows each component to work with its own way of parallelization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Moreover, since SAθ and SPD can capture the randomness by explicitly storing the pseudo-random number generator key inside, the states can be easily synchronized when running EC algorithms in a distributed systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Most importantly, the proposed generalized EC workflow strictly follows the paradigm of functional programming, which is intrinsically compatible with JAX-based implemen- tations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' ENGINEERING DESIGNS On the basis of the generalized EC workflow as formulated above, this section will further introduce the detailed engineer- ing designs of EvoX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Main Pipeline In GPU computing, tensor is the essential data structure for GPU acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Hence, in EvoX, we view the pipeline of running an EC algorithm for problem-solving as an iterative procedure of processing the tensor of a population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' 1, generally, the Pipeline passes the tensor of population Xt (as well as the corresponding fitness Monitor Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' 1: One iteration of the standard pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' It starts with gask, which gives a tensor Xt, representing the candidate population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Then Xt is optionally sent to the decoder h and then passed to f to evaluate its fitness yt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Finally, yt is passed to gtell, where the algorithm can utilize this information and update its state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Throughout this iteration, one can optionally choose to monitor certain values such as Xt and yt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Weights Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' 2: An illustration on how decoder can be used in neuroevolution tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Here decoder decodes Xt into Xt ′, which represents a set of weights for neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' values yt) through the Algorithm module and Problem module, with the support of optional modules of Monitor and Decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' The Decoder module is designed to transform the popu- lation Xt encoded by an EC algorithm into decision vectors in the original problem space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' For example, when training an ANN, the weights are typically represented by a set of tensors, one for each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' However, EC algorithms often output a single, tightly packed tensor as the population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' In this case, the Decoder can be used to decode the tightly packed tensor Xt into a set of tensors representing the weights of an ANN, as shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' It is important to note that the Decoder is an optional module, as it is not necessary for plain numerical optimization where Xt is already defined in the original problem space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' The Monitor is designed for observing intermediate re- sults in each iteration via visualization tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' For example, the users may observe the population Xt to analyze how the algorithm behaves in a certain fitness landscape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Be- sides, the users may also observe the fitness yt to check how the optimization process goes on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Additionally, another functionality of Monitor is to check whether the termination criterion (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' maximum number of iterations/evaluations) of the optimization process is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' In the following subsections, we will elaborate another two modules Algorithm and Problem in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Algorithm Module As is described in section III, Algorithm in EvoX is basically a class initialized with a set of hyperparameters θ, consisting of two methods – ask and tell, maintaining a 6 1 import jax 2 import jax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content='numpy as jnp 3 from evox import State, Algorithm 4 5 6 class SimpleES(Algorithm): 7 def __init__(self, dim, pop_size, topk): 8 self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content='dim = dim 9 self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content='pop_size = pop_size 10 self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content='topk = topk 11 12 def setup(self, key): 13 mean = jnp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content='zeros((dim,)) 14 stdev = jnp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content='ones((self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content='dim,)) 15 return State( 16 mean=mean, 17 stdev=stdev, 18 key=key 19 ) 20 21 def ask(self, state): 22 key, subkey = jax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content='random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content='split(state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content='key) 23 noise = jax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content='random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content='normal( 24 subkey, 25 (self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content='pop_size, self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content='dim) 26 ) 27 pop = state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content='mean + state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content='stdev * noise 28 new_state = state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content='update( 29 key=key, 30 pop=pop 31 ) 32 return sample, new_state 33 34 def tell(self, state, fitness): 35 _topk_value, topk_index = jax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content='lax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content='top_k( 36 fitness, 37 self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content='topk 38 ) 39 elite = state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content='pop[topk_index] 40 new_mean = jnp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content='mean(elite, axis=0) 41 new_stdev = jnp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content='std(elite, axis=0) 42 new_state = state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content='update( 43 mean=new_mean, 44 stdev=new_stdev 45 ) 46 return new_state Listing 1: Vanilla implementation of evolution strategies in EvoX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' The implementation has four main parts: __init__, setup, ask and tell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' __init__ and setup are used to initialize the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' ask and tell contain the main operations of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' global state SAθ – an independent object in EvoX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Addition- ally, Algorithm also has a setup method for generating the initial state SAθ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' As an illustrative example, we present the implementation of vanilla evolution strategy in EvoX, as listed in Lst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' To start with, in lines 7 to 10, the __init__ method initializes three hyperparameters in the constructor of this class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' dim is the problem dimension, pop_size is the population size, and topk indicates the top k individuals of the population are considered as elite to be selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' In lines 12 to 19, the setup method initializes its internal state SAθ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' In this vanilla evolution strategy, we keep an inde- pendent normal distribution for each decision variable, such that the mean and standard deviation vectors are initialized independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' In addition, we record key in the state as the Take a batch according to Calculate Loss Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' 3: A demonstration of how one problem can handle different evaluation modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' In this case, SP t indicates that the second batch from the training dataset should be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' seed for the pseudo-random number generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' In lines 21 to 32, the ask method is defined in correspon- dence to gask θ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' In this method, the algorithm splits the key and samples a new candidate population according to the mean and standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Then a new state is generated by updating the key and adding the candidate population to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' In lines 34 to 45, the tell method of the algorithm is defined in correspondence to gtell θ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' In this method, the algo- rithm picks the top k individuals in the candidate population according to their fitness ranking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Then, the mean and the standard deviation are adapted to the mean and the standard deviation of the elite population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Finally, in line 46, the newly generated candidate population is returned via the new_state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Problem Module In contrast to the past when EC algorithms were mostly tested on pre-configured numerical benchmark problems, op- timization problems of today are becoming increasingly com- plex – usually involving data-related configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Therefore, in our design, Problem is parameterized by dataset D with internal state SPD t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' 3, in the case of ANN training, the dataset D = {Dtrn, Dvld, Dtst, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content='} can consist of training data, validation data, and test data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' correspondingly, the prob- lem state SPD t will record the choice of dataset together with other essential parameters such as batch index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' With such a design principal, Problem is well extensible to support a wide spectrum of problems ranging from numerical optimiza- tion (leaving D and SPD t empty) to other data-related tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' In the following, we will elaborate three typical scenarios for defining problems using the tailored Problem Module in EvoX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' 1) Numerical Optimization: Since the plain numerical op- timization problems are often well-formulated functions with basic maths operations, output yt is only determined by the input Xt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Thus in EvoX, such problems can be implemented by simply leaving D and SPD t empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' 2) Neuroevolution: As introduced in Section II-B, a neu- roevolution task usually involves the training of an ANN to fit a certain dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Since a forward pass of an ANN is expensive, in EvoX, we adopt the common workflow as in mini-batch gradient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' First, the whole training dataset Dtrn is split into small batches of data B1, B1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content='., Bn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Then, at 7 Controller Environment Worker Worker Worker .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' 4: An illustration of running RL tasks in EvoX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Xt ′ is a population of ANN weights given by the decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' These weights will be evenly distributed to a set of workers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Each worker will run a complete episode to obtain the rewards from the RL environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' each iteration, only one batch of data is used to calculate the fitness of each individual as y(i) t = 1 n � x∈Bk L(x, Xt ′), where L(·) is the loss function, i denotes the ith row in the column vector yt, k denotes the batch index as stored in SPD t , and n denotes the batch size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' In practice, since large batch sizes may be intractable to calculate in a single forward pass, we also introduced a parameter called num_passes, to allow the loss of a batch to be calculated by multiple passes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' 3) Reinforcement Learning: As introduced in Section II-C, a typical RL task aims to train an agent for maximizing the total reward in a certain environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' To obtain the reward, it usually takes multiple steps for an agent to complete an episode by interacting with the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Our EvoX helps bridge the gap between EC and RL tasks via the tailored Problem module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Specifically, EvoX has the candidate pop- ulation Xt encode a set of parameters for the policy model, where for each candidate solution, EvoX runs a complete episode using this policy and return the sum of all rewards as the fitness value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' It is worth noting that, since there can be multiple rewards in some RL environments, one may treat these multiple rewards just as in the case of multi-objective optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' In practice, it can be computationally expensive to complete an episode to get the final reward, while some popular RL environment (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Gym) merely provides a single-thread inter- face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' This can be a painful bottleneck for the population-based EC algorithms which require batches of fitness evaluations in each generation, substantially hindering the development of EvoRL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Hence, to speed up the fitness evaluation process for RL tasks, we design a Ray-based interface for running RL environments with parallel computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' 4, with our interface, the function evaluation process for RL tasks can be deployed in a parallel computing environment with multiple workers, where each worker is created on CPU/GPU core for running the RL environment(s) in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Distributed Acceleration In a typical EC workflow, the main computational cost usually comes from a large number of fitness evaluations of the Node Fitness Exchange .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' CPU GPU Node CPU GPU Node CPU GPU Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' 5: The task-parallel workflow using distributed pipeline with n nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Each node will evaluate only 1 n of the population and then exchange the fitness with others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' candidate solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Since the fitness evaluation of each candi- date solution is intrinsically isolated, theoretically, we can have an EC algorithm deployed on a distributed computing system to improve the concurrency of fitness evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' However, when scaling beyond the boundary of physical machines, the communication cost between different machines becomes another obstacle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Intuitively, we may attempt to achieve dis- tributed fitness evaluations by running the algorithm in one node and sending the candidate solutions to the other nodes for evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' However, when it comes to complex tasks such as neuroevolution, each candidate solution may consist of millions (or even larger number) of decision variables, and thus the communication overhead for sending the candidate solutions to each node will be prohibitively huge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Hence, as the problem dimension increases, such an intuitive method suffers from the sharply increasing communication costs between the nodes, thus leading to poor scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' In order to accelerate the EC workflow by distributed computing in a scalable manner, we design a distributed pipeline on top of Ray, partially inspired by the work in [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' 5, given a number of n nodes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' machines) in distributed computing systems, the distributed pipeline creates a copy of standard pipeline (refer to Section IV-A) on each node;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' hhen the pipeline on each node only takes care of 1 n of the candidate solutions for fitness evaluations, running in an isolated and concurrent manner as if the other nodes do not exist;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' finally, the fitness values yt ′ as obtained by each node are exchanged to generate the entire fitness vector yt to be passed to the next generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Another important issue to be considered is the synchro- nization – to have the algorithm running on each pipeline image performing synchronous behaviors, such that the fitness information is always exchangeable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' To this end, we have the pipeline copies on each node initialized with the same random seed, such that the subsequent behaviors (despite the randomness as generated via the same random seed) of the algorithm will be naturally synchronous per iteration t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' This synchronization method also gurantees that the same experiments will always end up with the same result regardless of the number of nodes in use, thus further strengthening the reproducibility of the experiments conducted with EvoX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' 8 EvoX Operators Algorithms Problems PSO DE CMA-ES NSGA-II MOEA/D IBEA Single-objective Multi-objective Selection Reproduction Numerical Extended Tournament Best BitFlip SBX Sphere Ackley Rosenbrock TorchVision Gym .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Gaussian Roulette Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' 6: Main library content of EvoX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' The Algorithms component contains two sub-components: Single-objective and Multi- objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' The Operators component contains two subcomponents: Selection and Reproduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' The Problems component currently contains two subcomponents: Numerical Benchmarks, and Extended Applications which include tasks like neuroevo- lution and RL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' LIBRARY CONTENT As summarized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' 6, the content of EvoX mainly consists of three main components: the Algorithms component, the Operators component, and the Problems component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' In the following, we will introduce each component in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' The Algorithms component includes all EC algorithms available in EvoX, consisting of two subcomponents: Single- objective and Multi-objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' The Single-objective subcom- ponent includes algorithms for single-objective optimiza- tion(PSO [50], [51], DE [52], CSO [13], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' ), while the Multi-objective subcomponent includes EC algorithms for multi-objective optimization (NSGA-II [53], MOEA/D [54], RVEA [55], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' The Problems component includes all the pre-defined prob- lems in EvoX, consisting of two subcomponents: Numerical and Extended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' The Numerical subcomponent includes all pos- sible numerical benchmark problems, including not only basic ones for single-objective optimization (Sphere, Ackley [56], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' ), but also composite ones for multi-objective optimization (ZDT [57], DTLZ [58], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' The Extended subcomponent includes potential extended application problems, currently mainly related to neuroevolution and RL tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' The Operators component includes commonly used oper- ators in EC algorithms, consisting of two subcomponents: Selection and Reproduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' The Selection subcomponent in- cludes all possible selection operators that can be adopted for selecting candidate solutions (Tournament Selection, Random Selection, etc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' The Reproduction subcomponent includes all possible operators for reproducing new candidate solutions (BitFlip Mutation, SBX Crossover, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Apart from the above components, EvoX also allows the user to implement their own components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Thanks to the well decoupled modular engineering designs, the users may replace any of the existing components with their own tailored one(s) while reusing all other contents provided by EvoX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' EXPERIMENTAL STUDY To demonstrate the performance of EvoX, we will conduct a series of experiments in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' First, Section VI-A in- troduces the general experimental settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Then, Section VI-B demonstrates the scalability performance brought about by GPU computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Moreover, Section VI-C demonstrates the acceleration performance brought about by distributed com- puting Finally, Section VI-D demonstrates the extendability of EvoX towards complex RL tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Experiment Settings The experiment in Section VI-B is carried out on a physical machine equipped with an Intel Core i9-10900X CPU @ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content='70GHz and a single NVIDIA RTX 3090 GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' All the other experiments are conducted on a cluster with 8 nodes, where each node has 16 cores and 32 threads from an Intel Xeon Gold 6226R CPU @ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content='90GHz CPU and a single NVIDIA RTX 3090 GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Specifically, the experiment in Section VI-C uses up to 6 nodes while the experiment in Section VI-D only use a single node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' All experiments were repeated for 11 times using different random seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Scalability Test via GPU Computing To demonstrate the scalability performance of EvoX, we conduct benchmark experiments by running both single- objective and multi-objective algorithms on a single GPU device, in comparison with conventional CPU computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' The benchmark program is first run with GPU enabled and disabled respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' In the single-objective benchmarks, we have the classic PSO run on the Ackley problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' In multi-objective benchmarks, we have the NSGA-II run on the ZDT1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' In both single-objective and multi-objective benchmarks, we first fix the population size to 128 and scale up the number of dimensions of the problem, and then we fix the number of dimensions to 128 and scale up the population size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' 7a, when scaling up the number of problem dimensions with PSO, CPU computing and GPU computing have similar performance when the number of problem dimensions is very smaller, but as the number of problem dimensions becomes larger, the performance of 9 101 102 103 104 105 106 Problem dimensions 101 102 103 104 Time per iteration (ms) CPU GPU (a) Scaling the number of di- mensions with single-objective algorithm PSO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' 101 102 103 104 105 106 Population size 101 102 103 Time per iteration (ms) CPU GPU (b) Scaling population size with single-objective algorithm PSO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' 101 102 103 104 105 106 Problem dimensions 101 102 103 104 Time per iteration (ms) CPU GPU (c) Scaling the number of di- mensions with multi-objective algorithm NSGA-II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' 101 102 103 104 105 Population size 101 102 103 104 105 Time per iteration (ms) CPU GPU (d) Scaling population size with multi-objective algorithm NSGA-II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' 7: Results of scalability test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' The dark lines represent the mean values runs and shaded regions are bounded by the standard deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Both the x-axis and the y-axis are in logarithmic scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' GPU computing quickly surpasses CPU computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' When the number of problem dimensions is larger than 100,000, GPU computing is almost 100x more efficient in terms of time per iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' 7b, scaling up population size in PSO shows similar observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' 7c, when scaling up the number of problem dimensions with NSGA-II, we can also observe that GPU computing constantly performs better than CPU computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' 7d, when it comes to scaling up population size with NSGA-II, GPU computing also scales much better than CPU computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' In comparison with PSO, however, the general computing cost of NSGA-II increases sharper in terms of population size, mainly due to the higher algorithmic computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Acceleration Test via Distributed Computing To assess the acceleration performance of EvoX, we con- duct an experiment on the task of neuroevolution for image classification on multiple GPU devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Specifically, we train a convolution neural network on the CIFAR-10 dataset [59] using 1 to 8 GPUs and measure the total runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' CIFAR-10 is an image classification dataset containing a total of 60,000 32 × 32 color images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' The architecture of the convolution neural network is given in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' III, and ReLU [60] is used as the activation function in between layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' For the algorithm module, we adopt PGPE [61] with ClipUp [62], and the population size is set to 300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' On the problem module, we set TABLE III: Architecture of the ANN adopted in the neuroevo- lution experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Input Shape Layer Filter Shape Strides 32 × 32 × 3 Conv 3 × 3 × 3 × 32 1 30 × 30 × 32 Max Pooling 2 × 2 2 15 × 15 × 32 Conv 3 × 3 × 32 × 32 1 13 × 13 × 32 Max Pooling 2 × 2 2 6 × 6 × 32 Conv 3 × 3 × 32 × 32 1 512 Fully Connected 512 × 64 — 64 Fully Connected 64 × 10 — 1 2 3 4 5 6 Number of nodes 200 400 600 Total runtime (s) (a) Runtime 1 2 3 4 5 6 Number of nodes 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content='008 Performance (s 1) (b) Performance Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' 8: Results of acceleration test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' The number of GPU nodes are ranged from 1 to 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' The same data is presented in two ways, in (a) the y-axis is the total runtime, and in (b) the y- axis is the performance which is measured by the inverse of the runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' the batch size to 3000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' When conducting the experiment with n nodes, since each node only evaluates 1 n of the candidate population, we also tune the num_passes parameter to 30 n in order to fully utilize the each GPU device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' The experiment runs 1000 training iterations in total, with a validation phase for every 100 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' 8a and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' 8b present the runtime and the perfor- mance with different numbers of GPU nodes respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' The runtime includes both the training phase and the validation phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' It can be observed that the runtime rapidly decreases as the number of GPU nodes grows, achieving near-linear acceleration rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' This observation is consistent with our expectation as the distributed pipeline only requires exchanges of cheap fitness values among the workers, being communication-efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Extensibility to RL Tasks To assess the usability of EvoX, when extended to solving complex application problems, we conduct an experiment on two representative RL tasks as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' We will demonstrate that EvoX can handle both tasks fluently and efficiently despite their challenging features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Bipedal Walker: In this RL task, the agent controls a 4- joint walker robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' The goal is to move forward without falling and apply as little torque as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' At any given moment, the agent can observe 24 real values given by the sensors and control the torque applied to the four motors on the robot, ranging from −1 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Specifically, the agent adopts a policy model of an MLP consisting of two hidden layers, each containing 64 neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' To 10 (a) Bipedal Walker (b) ATARI Pong Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' 9: In-game images of two RL tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' (a) The Bipedal Walker: in this RL task, the agent receives the observation as a vector and controls the robot to move forward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' (b) The ATARI game Pong: in this RL task, the agent can observe the raw image of the game and control the right paddle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' TABLE IV: Parameter settings of CMA-ES Parameter Value Population Size 128 Initial step size 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content='1 TABLE V: Parameter settings of PGPE Parameter Value Population Size 128 Learning rate for standard deviation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content='01 Adaptive Optimizer Adam [63] Fitness Shaping Yes Standard deviation max change 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content='2 learn the policy model, there are a total number of 6020 parameters to be optimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Here, we apply the CMA-ES algorithm as the optimizer, where the detailed algorithm settings can be found at Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' ATART Pong: In this RL task, the agent controls a paddle on the right side of the screen and tries to use it to hit a ball away from its own goal and into the opponent’s goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' For decision makings, the agent observes the raw frame from the game, which is a 210 × 160 colored image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' To reduce the complexity, we pre-process the image first by converting the full-size colored image into gray-scale one with a smaller size (80 × 80).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' The agent employs a policy model of an MLP with 2 hidden layers, where the first and second hidden layers have 64 and 32 neurons respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' The model takes the pre-processed image as input and outputs the probability of all available actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' To simplify the decision-making process, we always choose the action with the highest probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' In order to learn this policy model, there are a total number of 411,942 parameters to be optimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Here, we use PGPE as the training algorithm to optimize the policy of the agent, where the detailed algorithm settings can be found in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' 10, in both RL tasks, utilizing multiple processes results in a significant performance improvement in terms of computational time per iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' It is worth noting that although both tasks are mainly executed via CPU computing, the entire algorithm workflow still runs on GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Multi- processing Single- processing 0 10 20 30 40 Time per iteration (sec) (a) Bipedal Walker Multi- processing Single- processing 0 25 50 75 100 Time per iteration (sec) (b) ATARI Pong Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' 10: Performance comparison of two RL tasks with mul- tiprocessing enabled and disabled, in terms of computational time per iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' (a) In the Bipedal Walker task, enabling multiprocessing resulted in almost 6× performance improve- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' (b) In ATARI Pong, enabling multiprocessing resulted in almost 12× performance improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' 0 40000 80000 120000 Episodes 50 0 50 100 150 200 250 300 Reward (a) Bipedal Walker 0 40000 80000 120000 Episodes 25 20 15 10 5 0 5 10 15 Reward (b) ATARI Pong Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' 11: The best rewards achieved on each RL task per episode across different seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' At each iteration, a policy is evaluated by running the policy in the environment once and the highest reward in the population is recorded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' The dark line indicates the average highest reward across different runs, and the shaded region is bounded by its standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' This behavior is made possible by our design as described in Section IV, where the algorithm module and the problem module are decoupled to allow for tasks that are CPU-intensive and not native to EvoX to achieve high parallelism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' In addition to performance improvement in terms of com- putational time, the experimental results also demonstrate the potential of EC algorithms in tackling challenging RL tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' The Bipedal Walker task poses a significant challenge in evolving a valid walking strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' 11a, there 11 is a large variation during the later part of the optimization process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' The reason for this is that the point when an agent learns to walk marks the edge of a plateau in the optimization process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Before this point, the agent receives little to no reward, but once it starts moving forward, it quickly earns high rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' So the high variance reflects the fact that the algorithm sometimes discovers the walking strategy early on, leading to significantly better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' However, in some attempts, the algorithm fails to discover a decent strategy, leading to poor performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' In the ATARI Pong task, despite having a large number of trainable parameters (400K), the algorithm, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' 11b, steadily improves the policy towards a promising solution, with little variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' In summary, EvoX provides comprehensive support for extending EC algorithms to RL tasks, as well as other com- plex application problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' With little additional engineering work, practitioners can instantly enjoy the benefits of EvoX by connecting their problems via the seamless and friendly interface in the problem module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' CONCLUSION AND FUTURE WORK In this paper, we presented EvoX, a distributed GPU- accelerated EC library that significantly improves the scala- bility of the EC workflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' When using a single machine, we observed a two-orders-of-magnitude speed-up in certain set- tings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' By running on multiple machines, we were able to scale the workflow even further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' This advancement allows existing EC algorithms to solve problems with high-dimensional search spaces more efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Our work also paves the way for future EC development, making it easy to develop more complex EC algorithms that can leverage increasing computing power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Furthermore, meta-learning algorithms could also be greatly improved by the increased computing power, as the process of meta-learning is known to be computationally demanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Ad- ditionally, we simplify the process of applying EC algorithms to extended applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' With EvoX, EC algorithms can work with neuroevolution or RL tasks seamlessly, substantially reducing the barrier for researchers to tackle these problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' However, during the development of EvoX, we also iden- tified certain limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' First, many algorithms were not designed with parallelism in mind, thus making it difficult to accelerate them via advanced GPU computing techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Moreover, many extended applications are CPU-intensive and tend to be the most time-consuming part of the workflow, thus somehow diminishing the impact of acceleration in EC algo- rithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' In the future, we plan to incorporate more parallelism within existing algorithms and integrate external tasks directly into our library to enable GPU computing in both algorithms and problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' ACKNOWLEDGEMENT We wish to thank Zhenyu Liang for his contributions to the multiple-objective algorithms and Kebin Sun for helping with the single-objective algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Additionally, we would like to express our appreciation to Lishuang Wang and Jiachun Li for their efforts in testing out the library, and for their miscellaneous contributions to the library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' REFERENCES [1] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Gould, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Orban, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content=' Toint, “Numerical methods for large-scale nonlinear optimization,” Acta Numerica, vol.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content='org/abs/1412.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} +page_content='6980' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFMT4oBgHgl3EQf5zEl/content/2301.12457v1.pdf'} diff --git a/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf b/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..59afe65ebf0da621e166d27f12cdce85f2289207 --- /dev/null +++ b/jtE4T4oBgHgl3EQfsg2s/content/2301.05217v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:87f22c856229292d364cecdfc67ad91fb99fccc631a72d35ef4051cefda27c5b +size 3033480 diff --git a/jtE4T4oBgHgl3EQfsg2s/vector_store/index.faiss b/jtE4T4oBgHgl3EQfsg2s/vector_store/index.faiss new file mode 100644 index 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9 Jan 2023 +ON ABELIAN-BY-CYCLIC MOUFANG LOOPS +ALEˇS DR´APAL AND PETR VOJTˇECHOVSK´Y +Abstract. We study abelian-by-cyclic Moufang loops. We construct all split 3-divisible +abelian-by-cyclic Moufang loops from so-called Moufang permutations on abelian groups +(X, +), which are permutations that deviate from an automorphism of (X, +) by an al- +ternating biadditive mapping (satisfying certain properties). +More generally, we obtain +additional abelian-by-cyclic Moufang loops from so-called construction pairs. As an aside, +we show that in the Moufang loops Q obtained from a construction pair on (X, +) the +abelian normal subgroup (X, +) induces an abelian congruence of Q if and only if Q is a +group. +1. Introduction +Moufang loops were introduced in 1935 [32] and studied ever since. Yet there remain +serious gaps in our understanding of basic structural concepts for Moufang loops. One of +the reasons for the state of affairs is the lack of constructions. +In this paper we start a systematic approach to constructions of abelian-by-cyclic Moufang +loops, that is, Moufang loops Q possessing an abelian normal subgroup X such that Q/X is +cyclic. The main results are Theorem 4.7 (a construction of many abelian-by-cyclic Moufang +loops from so-called construction pairs), Theorem 10.1 (every conjugation by a in a Moufang +loop restricts to a so-called Moufang permutation on X and the multiplication on the subloop +⟨a3⟩X is just like the abstract multiplication formula of Theorem 4.7), Theorem 10.4 (all +3-divisible abelian-by-cyclic Moufang loops are homomorphic images of the loops obtained +from Theorem 4.7 with construction pairs induced by Moufang permutations), and Theorem +10.6 (a characterization of split 3-divisible abelian-by-cyclic Moufang loops). +1.1. A brief overview of constructions of Moufang loops. The motivating and most +important examples of Moufang loops come from the multiplicative loops of nonzero oc- +tonions [2, 11, 31, 41]. These octonionic Moufang loops have additional strong structural +properties not shared by all Moufang loops. For instance, in the 16-element loop O16 of basic +units of real octonions, every square associates with all elements (making O16 an extra loop +[10, 16]) and there is in fact a unique nonidentity square (making O16 a code loop [9, 24]). +All nonassociative finite simple Moufang loops are obtained as central factors of the loop of +unit elements in split octonion algebras over finite fields [30, 36, 44]. +Another interesting class of Moufang loops are Moufang p-loops. Moufang p-loops are +centrally nilpotent (see [20] for p odd, [21] for p = 2, and [14] for a recent elementary +2020 Mathematics Subject Classification. 20N05. +Key words and phrases. Abelian by cyclic Moufang loop, Moufang loop, conjugation in Moufang loops, +Moufang permutation, solvability, congruence solvability. +A. Dr´apal supported by the INTER-EXCELLENCE project LTAUSA19070 of MˇSMT Czech Republic. +P. Vojtˇechovsk´y supported by the Simons Foundation Mathematics and Physical Sciences Collaboration +Grant for Mathematicians no. 855097. +1 + +proof of both cases). Using geometric considerations, Bol [4] constructed a nonassociative +commutative Moufang loop of order 34 (also see [25] for all commutative Moufang loops of +order less than 36). Later, Bruck [5] constructed a nonassociative Moufang loop of order +p5 for every prime p, and Nagy and Valsecchi [27] classified Moufang loops of order p5 for +all primes p > 3. Using a computational approach to central extensions, all Moufang loops +of orders 26 and 34 (resp. 35) were classified in [34] (resp. [40]). The aforementioned code +loops are equivalently described as Moufang 2-loops Q possessing a central subloop Z of +order 2 such that Q/Z is an elementary abelian 2-group, cf. [1, 24]. Code loops of order ≤ 29 +were enumerated in [35]. The constructions of Moufang p-loops are often of combinatorial +character and they say little about Moufang loops that are not of prime power order. +Yet another source of constructions is connected with the (still open) question for which +integers n there exists a nonassociative Moufang loop of order n. Chein described all Moufang +loops of order less than 32 in [7] and all Moufang loops of order less than 64 in [8] (see +also [22, 33] for a catalog of all nonassociative Moufang loops of order less than 64). The +constructions that appear in Chein’s classification of small Moufang loops include the well- +known Chein double M(G, 2) of a group G as well as several detailed variations on the +Chein double. Leong and Rajah [28] proved that any Moufang loop of order pαqα1 +1 · · · qαk +k +with p < q1 < · · · < qk odd primes and with α ≤ 3 and αi ≤ 2 is a group, and similarly +for the case p > 3 and α = 4. The intermediate order factorizations between this lower +bound (on exponents) and the upper bound furnished by nonassociative Moufang p-loops +was investigated extensively by Rajah and various coauthors, cf., for instance, [6, 39]. By +the very nature of the question, the constructions in this area of research are mostly ad +hoc since a single nonassociative example is required to settle the existence question for any +given order. +As far as general constructions of Moufang loops are concerned, there are substantial +results of Kinyon and Kunen on semidirect products in the context of extra loops [26]. For +(not necessarily extra) Moufang loops, the first step was a formula discovered by Gagola [19], +cf. our (2.12), that gives a necessary condition for the existence of a semidirect product of a +Moufang loop (which is normal in the product) and a cyclic group of order coprime to three. +Sufficient conditions for the existence of such a semidirect product were then given by Dr´apal +[13]. The conditions of [13] are in the form of certain requirements on semiautomorphisms +of the normal subloop and they are difficult to work with. The present paper was inspired +by [13] but is independent of it. +Finally, we wish to mention the papers [18, 23, 29] that deal with semidirect products +and/or split extensions of Moufang loops. +1.2. A summary of main results. The following definition is key to obtaining many +abelian-by-cyclic Moufang loops: +Definition 1.1. Let (X, +) be an abelian group, g a permutation of X and γ : X ×X → X +a mapping. Then (g, γ) is a construction pair on (X, +) if γ is a symmetric alternating +biadditive mapping, +g−1(g(x) + g(y)) = x + y + γ(x, y) + g−1(γ(x, y)) + g−2(γ(x, y)) +(C1) +holds for all x, y ∈ X, +γ(γ(x, y), z) = 0 +(C2) +2 + +holds for all x, y, z ∈ X, and +g−1(γ(x, y)) = γ(g(x), y) +(C3) +holds for all x, y ∈ X. +Note that the condition (C2) may be equivalently expressed by stating that the image of +γ is contained in the radical Rad(γ) = {x ∈ X : γ(x, y) = 0 for all y ∈ X} of γ. +For i, j ∈ Z, define the interval I(i, j) of Z by +I(i, j) = + + + +∅, +if i = j, +{i, i + 1, . . . , j − 1}, +if i < j, +{j, j + 1, . . . , i − 1}, +if j < i. +(1.1) +Given an abelian group (X, +) and a cyclic group C = ⟨b⟩, we show that the formula +(bi, x) · (bj, y) = +� +bi+j, g−j(x) + y + +� +k∈I(i+j,−j) +g−k(γ(x, y)) +� +(1.2) +correctly defines a multiplication on C × X if and only if either C is infinite, or C is finite, +|g| divides |C| and � +0≤k<|C| gk(x) ∈ Rad(γ) for all x ∈ X. (The condition � +0≤k<|C| gk(x) ∈ +Rad(γ) is satisfied whenever |C| is finite, |g| divides |C| and Rad(γ) contains no elements of +order 3.) In that case the resulting groupoid is in fact a Moufang loop Q = C ⋉(g,γ) X that +contains a normal subloop 1 × X such that Q/(1 × X) is isomorphic to C, cf. Theorem 4.7. +In addition, (C × 0) ∩ (1 × X) = 1, so Q is a split extension of X by C. +Several key properties of the loops C ⋉(g,γ) X are established in Section 5, such as the +formula for an elementwise commutator, elementwise associator, the associator subloop, the +commutator subloop, the derived subloop, the nucleus and the center. +In the brief Section 6 we show that in the Moufang loop Q = C ⋉(g,γ) X the normal +subloop 1×X induces an abelian congruence of Q if and only if Q is a group. This is related +to another current line of investigation of ours concerned with the two notions of solvability +in loops, cf. [15, 42, 43]. The results of Section 6 are not used elsewhere in the paper. +The rest of the paper is concerned with a partial converse of Theorem 4.7. +It is easy to see that Theorem 4.7 does not yield all abelian-by-cyclic Moufang loops. For +instance, the quaternion group Q8 is abelian-by-cyclic but it is not split and hence it cannot +be obtained from the construction of Theorem 4.7. (A smallest nonassociative abelian-by- +cyclic Moufang loop that is not split is of order 32.) In fact, Theorem 4.7 does not yield all +split abelian-by-cyclic Moufang loops either. For instance, the nonassociative commutative +Moufang loop of order 81 and exponent 3 is a split extension of X = Z3 × Z3 × Z3 by +C = Z3, but Theorem 4.7 does not yield any nonassociative commutative Moufang loops, +cf. Corollary 5.4. +On the other hand, Theorem 4.7 yields all abelian-by-cyclic Moufang loops that are split +and 3-divisible, cf. Theorem 10.6. To prove this fact, we will need the following notion similar +to construction pairs: +Definition 1.2. Let (X, +) be an abelian group. A permutation f on X is said to be a +Moufang permutation on (X, +) if the mapping β : X × X → X defined by +β(x, y) = f −1(f(x) + f(y)) − x − y +(P1) +3 + +is (symmetric) alternating and biadditive, +β(β(x, y), z) = 0 +(P2) +holds for all x, y, z ∈ X, and +β(f(x), f(y)) = f(β(f 3(x), y)) +(P3) +holds for all x, y ∈ X. When f is a Moufang permutation and β is defined by (P1), we call +β the biadditive mapping associated withf and the tuple (f, β) a Moufang pair. +The relation between construction pairs and Moufang pairs is that if (f, β) is a Moufang +pair then (f 3, β) is a construction pair, cf. Proposition 9.8. In particular, when the remaining +assumptions of Theorem 4.7 are satisfied, we can use the multiplication formula (1.2) with +(g, γ) = (f 3, β). If also b = a3 for some a ∈ C (which certainly holds when C is 3-divisible), +the formula (1.2) becomes +(a3i, x) · (a3j, y) = +� +a3(i+j), f −3j(x) + y + +� +k∈I(i+j,−j) +f −3k(β(x, y)) +� +. +(1.3) +After deriving some rather general results on pseudoautomorphisms and semiautomor- +phisms of Moufang loops induced by inner mappings in Sections 7 and 8, we prove that +for every abelian normal subgroup X of a Moufang loop Q and every element a ∈ Q the +restriction of the “conjugation” Ta to X is a Moufang permutation on X, cf. Proposition +8.4. +In Section 9 we study abstract properties of Moufang permutations f and their powers. +Among other results, we derive a formula for the expression f −i(f i(x)+f i(y)), cf. Proposition +9.5. +We then prove in Theorem 10.1 that if (X, +) is an abelian normal subgroup of a Moufang +loop Q and if a ∈ Q then +a3ix · a3jy = a3(i+j)� +f −3j(x) + y + +� +k∈I(i+j,−j) +f −3k(β(x, y)) +� +(1.4) +holds for all i, j ∈ Z and x, y ∈ X, where f is the restriction of Ta to X, which we know from +Proposition 8.4 is a Moufang permutation on (X, +). Here and throughout the paper, if +(X, +) is a normal subloop of (Q, ·), we write either x+ y or x· y for the product of x, y ∈ X +in Q. +Note that (1.4) does not necessarily describe the multiplication for all pairs of elements +of Q. Nevertheless, it easily follows, cf. Corollary 10.2, that if (X, +) ⊴ Q and Q = ⟨a3⟩X +for some a ∈ Q, then (1.4) does fully describe the multiplication of Q. +It is then not difficult to show that all 3-divisible abelian-by-cyclic Moufang loops Q = +CX are homomorphic images of the loops C ⋉(f3,β) X, where (f, β) is a Moufang pair on +(X, +). Here, |f 3| must divide |C|, else C ⋉(f3,β) X is not even defined. The kernels of +the homomorphisms are described in Proposition 10.5. Finally, 3-divisible abelian-by-cyclic +Moufang loops Q = CX that are split are precisely the the loops C ⋉(f3,β) X with (f, β) a +Moufang pair on (X, +), cf. Theorem 10.6. +Let us conclude the summary of main results with a few comments. +4 + +The multiplication formula (1.4) might appear rather complicated and unnatural. A point +of departure in deriving the formula is the important identity +a3ix · ya3j = a3(i+j)T −i−2j +a +(T i−j +a +(x)T i−j +a +(y)) +valid in all Moufang loops. We record this identity and all its variants in Proposition 2.6, +including the already mentioned identity (2.12) of Gagola [19]. +It turns out that in the finite case each construction pair (g, γ) on (X, +) induces an integer +m and a mapping U ×U → V that is alternating and bilinear over R, where U = X/Rad(γ), +V = ⟨Img(γ)⟩ and R = F2[x]/(xm − 1). To describe construction pairs for a given abelian +group (X, +) thus means to start from a classification of the relevant alternating bilinear +mappings and then include several steps of lifting. +We intend to study the problem of +obtaining construction pairs and of classifying the resulting Moufang loops up to isomorphism +in a future paper. +2. Background +2.1. Mappings. Let (X, +) be an abelian group. Let us call a mapping γ : X × X → X +symmetric if γ(x, y) = γ(y, x) for all x, y ∈ X, antisymmetric if γ(x, y) = −γ(y, x) for +all x, y ∈ X, alternating if γ(x, x) = 0 for all x ∈ X, and biadditive if γ(x + y, z) = +γ(x, z) + γ(y, z) and γ(x, y + z) = γ(x, y) + γ(x, z) for all x, y, z ∈ X. +Note that a biadditive mapping satisfies γ(0, x) = 0 = γ(x, 0), γ(−x, y) = −γ(x, y) = +γ(x, −y), γ(x−y, z) = γ(x, z) −γ(y, z) and γ(x, y −z) = γ(x, y) −γ(x, z) for all x, y, z ∈ X. +Indeed, γ(0, x) = γ(0 + 0, x) = 2γ(0, x) yields γ(0, x) = 0, then 0 = γ(x + (−x), y) = +γ(x, y)+γ(−x, y) implies γ(−x, y) = −γ(x, y), and finally γ(x−y, z) = γ(x, z)+γ(−y, z) = +γ(x, z) − γ(y, z) +An alternating biadditive mapping is antisymmetric since 0 = γ(x + y, x + y) = γ(x, x) + +γ(x, y) + γ(y, x) + γ(y, y) = γ(x, y) + γ(y, x). A symmetric alternating biadditive mapping +then satisfies 2γ(x, y) = γ(2x, y) = 0 because γ(x, y) = γ(y, x) = −γ(x, y). +All these +properties will be used without reference throughout the paper. +The image of any mapping g will be denoted by Img(g). As in the introduction, for a +symmetric mapping γ : X × X → X, the radical of γ is defined by +Rad(γ) = {x ∈ X : γ(x, y) = 0 for all y ∈ X}. +If γ is symmetric and biadditive, Rad(γ) is a subgroup of (X, +). +Notation. In order to improve legibility, we will often omit parentheses around arguments +of unary mappings. For instance, if g : X → X and γ : X ×X → X, we might write gγ(x, y) +instead of g(γ(x, y)), and γ(gx, y) instead of γ(g(x), y). +2.2. Loops. See [5, 37] for an introduction to loop theory. +A set Q with a binary operation · and an element 1 ∈ Q is a loop if for every x ∈ Q we +have x · 1 = 1 · x = x and the translations +Lx : Q → Q, Lx(y) = x · y +and +Rx : Q → Q, Rx(y) = y · x +are permutations of Q. The induced division operations will be denoted by x\y = L−1 +x (y) +and x/y = R−1 +y (x). +5 + +Associative subloops will be referred to as subgroups. A loop Q is power associative if +every element of Q generates a subgroup. A loop Q is diassociative if every two elements of +Q generate a subgroup. +Notation. We will often write xy instead of x · y and use · to indicate priority of multipli- +cation, e.g., x · yz stands for x · (y · z). +The multiplication group of a loop Q is the permutation group Mlt(Q) = ⟨Lx, Rx : x ∈ Q⟩, +and the inner mapping group Inn(Q) of Q is the stabilizer of 1 in Mlt(Q). It is well known +that Inn(Q) = ⟨Tx, Lx,y, Rx,y : x, y ∈ Q⟩, where +Tx = R−1 +x Lx, +Lx,y = L−1 +xy LxLy +and +Rx,y = R−1 +xy RyRx. +We refer to the inner mappings Tx as conjugations. +The nucleus Nuc(Q) of Q consists of all x ∈ Q such that x(yz) = (xy)z, y(xz) = (yx)z +and y(zx) = (yz)x for all y, z ∈ Q. The center Z(Q) of Q consists of all x ∈ Nuc(Q) such +that xy = yx for all y ∈ Q. The commutator [x, y] of x, y ∈ Q is defined by (yx)[x, y] = xy. +The associator [x, y, z] of x, y, z ∈ Q is defined by (x · yz)[x, y, z] = xy · z. +2.3. Morphisms and topisms. Denote by Sym(Q) the symmetric group on Q. A tuple +(c, f) ∈ Q × Sym(Q) is a (left) pseudoautomorphism of Q if +cf(x) · f(y) = cf(xy) +(2.1) +holds for every x, y ∈ Q. The element c is called a companion of f. The set of all pseudoau- +tomorphisms of Q forms a group Psaℓ(Q) under the operations +(c, f)(d, g) = (cf(d), fc) +and +(c, f)−1 = (f −1(c\1), f −1). +(2.2) +A permutation f ∈ Sym(Q) is a semiautomorphism of Q if f(1) = 1 and +f(x · yx) = f(x) · f(y)f(x) +(2.3) +holds for all x, y ∈ Q. +If f is a semiautomorphism of a power associative loop Q then +f(xi) = f(x)i for every i ∈ Z. +The condition (2.3) can be written as fLxRx = Lf(x)Rf(x)f and therefore also as +L−1 +f(x)fLx = Rf(x)fR−1 +x . +(2.4) +Thus any permutation f of a loop Q that satisfies f(1) = 1 and (2.4) is a semiautomorphism. +An autotopism of a loop Q is a triple (α, β, γ) of permutations of Q such that +α(x)β(y) = γ(xy) +holds for all x, y ∈ Q. Autotopisms of Q may be composed and inverted componentwise, +forming the autotopism group Atp(Q). +Note that (c, f) is a pseudoautomorphism of Q if and only if (Lcf, f, Lcf) is an autotopism +of Q. We use this fact in the proof of the following observation about autotopisms in which +the middle component fixes the identity element. +Lemma 2.1. Let (Q, ·, 1) be a loop. Suppose that (α, β, γ) ∈ Atp(Q) and β(1) = 1. Then +(α(1), β) ∈ Psaℓ(Q). +Proof. Let c = α(1). By our assumption, α(x) = α(x)β(1) = γ(x · 1) = γ(x) and cβ(x) = +α(1)β(x) = γ(1 · x) = γ(x) for every x ∈ Q. Hence (α, β, γ) = (Lcβ, β, Lcβ). +□ +6 + +2.4. Moufang loops. A loop Q is Moufang if it satisfies any one of the equivalent Moufang +identities +xy · zx = (x · yz)x, +(M1) +xy · zx = x(yz · x), +(M2) +x(y · zy) = (xy · z)y, +(M3) +x(y · xz) = (xy · x)z. +(M4) +Let us summarize a few facts about Moufang loops, consisting of easy observations and +standard results on Moufang loops [5]. +The identity (M1) can be stated as (Lx, Rx, RxLx) ∈ Atp(Q). This is a useful characteri- +zation of Moufang loops in terms of autotopisms. +By Moufang Theorem [12, 32], if three elements x, y and z of a Moufang loop associate, +that is, x(yz) = (xy)z, then the subloop ⟨x, y, z⟩ is a group. Consequently, Moufang loops are +diassociative, power associative, satisfy the flexible law x(yx) = (xy)x, the inverse properties +x−1(xy) = y = (yx)x−1, and so on. We take advantage of diassociativity in Moufang loops +and write unambiguously xyx, xy = y−1xy, [x, y] = x−1y−1xy, etc. +Lemma 2.2. [15] Let Q be a Moufang loop. Then +x−1(xy · z) = yx−1 · xz, +(2.5) +(z · yx)x−1 = zx · x−1y +(2.6) +for every x, y, z ∈ Q. +All inner mappings of a Moufang loop can be seen as pseudoautomorphisms, with suitable +companions. In particular, +(x−3, Tx), +([y, x], Rx,y) +and +([x−1, y−1], Lx,y) +(2.7) +are pseudoautomorphisms in a Moufang loop. Every pseudoautomorphism of a Moufang +loop is a semiautomorphism. +Lemma 2.3. Let Q be a Moufang loop, c ∈ Q and f ∈ Sym(Q). Then (c, f) ∈ Psaℓ(Q) if +and only if +xc−1 · cy = f(f −1(x)f −1(y)) +for all x, y ∈ Q. +Proof. Substituting f −1(x) for x and f −1(y) for y into cf(x) · f(y) = cf(xy) yields cx · y = +cf(f −1(x)f −1(y)). We are done upon multiplying by c−1 on the left and applying (2.5). +□ +Corollary 2.4. Let Q be a Moufang loop. Then +xa−3 · a3y = T −1 +a (Ta(x)Ta(y)) +for all a, x, y ∈ Q. +The equation (2.8) below is of crucial importance for this paper. +Proposition 2.5. Let Q be a Moufang loop. Then +a3ix · ya3j = a3i · T j−i +a +(T i−j +a +(x)T i−j +a +(y)) · a3j +(2.8) +for all a, x, y ∈ Q and all i, j ∈ Z. +7 + +Proof. Let b = a3i and δ = i − j. By diassociativity and (M1), bix · ybj = bj(bδx) · ybj = +bj(bδx · y)bj. By (2.5) and Corollary 2.4, +bδx · y = bδ(xb−δ · bδy) = bδTa−δ(Taδ(x)Taδ(y)) = bδT −δ +a (T δ +a(x)T δ +a(y)), +where we have used Tak = T k +a in the last step. Combining, we get +bix · ybj = bj(bδx · y)bj = bj · bδT −δ +a (T δ +a(x)T δ +a(y)) · bj = bi · T −δ +a (T δ +a(x)T δ +a(y)) · bj. +□ +Here are some variations on the identity (2.8): +Proposition 2.6. Let Q be a Moufang loop. Then +a3ix · a3jy = a3(i+j)T −i−2j +a +(T i−j +a +(x)T i+2j +a +(y)) += T 2i+j +a +(T i−j +a +(x)T i+2j +a +(y))a3(i+j), +(2.9) +a3ix · ya3j = a3(i+j)T −i−2j +a +(T i−j +a +(x)T i−j +a +(y)) += T 2i+j +a +(T i−j +a +(x)T i−j +a +(y))a3(i+j), +(2.10) +xa3i · a3jy = a3(i+j)T −i−2j +a +(T −2i−j +a +(x)T i+2j +a +(y)) = T 2i+j +a +(T −2i−j +a +(x)T i+2j +a +(y))a3(i+j), +(2.11) +xa3i · ya3j = a3(i+j)T −i−2j +a +(T −2i−j +a +(x)T i−j +a +(y)) = T 2i+j +a +(T −2i−j +a +(x)T i−j +a +(y))a3(i+j) +(2.12) +for all a, x, y ∈ Q and all i, j ∈ Z. +Proof. Note that akz = akza−kak = T k +a (z)ak, so it suffices to establish the first equality in +each (2.9)–(2.12). With u = T j−i +a +(T i−j +a +(x)T i−j +a +(y)), equation (2.8) yields +a3ix · ya3j = a3iua3j = T 3i +a (u)a3ia3j = T 2i+j +a +(T i−j +a +(x)T i−j +a +(y))a3(i+j), +which is (2.10). +The remaining identities then follow from (2.10) as a3ix · a3jy = a3ix · +T 3j +a (y)a3j, xa3i · a3jy = a3iT −3i +a +(x) · T 3j +a (y)a3j and xa3i · ya3j = a3iT −3i +a +(x) · ya3j. +□ +3. Construction pairs and their properties +Construction pairs were defined in the Introduction, cf. Definition 1.1. In this section we +will establish basic properties of construction pairs (g, γ) on (X, +). The condition (C1) will +be often used in the form +g(x) + g(y) = g(x + y + γ(x, y) + g−1(γ(x, y)) + g−2(γ(x, y))). +Lemma 3.1. Let (g, γ) be a construction pair on (X, +) and let i be an integer. Then: +(i) g(0) = 0, g(2x) = 2g(x) and g(−x) = −g(x) for all x ∈ X, +(ii) Img(γ) ⊆ Rad(γ), 2X ⊆ Rad(γ) and 2Img(γ) = 0, +(iii) gi permutes both Img(γ) and Rad(γ), +(iv) gi(x + y) = gi(x) + gi(y) whenever {x, y} ∩ Rad(γ) ̸= ∅, +(v) gi restricts to an automorphism of Rad(γ). +Proof. (i) We have γ(x, 0) = 0 thanks to biadditivity. +Hence g−1(0) = g−1γ(0, 0) = +γ(g(0), 0) = 0 by (C3) and g(0) = 0 follows. +Since γ is alternating and g(0) = 0, we +then have 0 = γ(x, x)+g−1γ(x, x)+g−2γ(x, x) = g−1(g(x)+g(x))−x−x = g−1(2g(x))−2x +by (C1), which yields 2g(x) = g(2x). Finally, γ(x, −x) = −γ(x, x) = 0, so 0 = γ(x, −x) + +g−1γ(x, −x) + g−2γ(x, −x) = g−1(g(x) + g(−x)) by (C1), and hence g(x) + g(−x) = 0. +(ii) The condition Img(γ) ⊆ Rad(γ) is a restatement of (C2). Any symmetric alternating +biadditive mapping satisfies 0 = γ(2x, y) = 2γ(x, y). +(iii) We have x ∈ Rad(γ) iff γ(x, y) = 0 for all y ∈ X iff g−1γ(x, y) = g−1(0) = 0 for all +y ∈ X iff γ(g(x), y) = 0 for all y ∈ X by (C3) iff g(x) ∈ Rad(γ). Similarly, z ∈ Img(γ) iff +8 + +z = γ(x, y) for some x, y ∈ X iff g−1(z) = g−1γ(x, y) = γ(g(x), y) for some x, y ∈ X by (C3) +iff g−1(z) ∈ Img(γ). +(iv) Suppose without loss of generality that x ∈ Rad(γ). Then γ(x, y) = 0 and, by (i), +giγ(x, y) = 0 for all i. Thus 0 = γ(x, y) + g−1γ(x, y) + g−2γ(x, y) = g−1(g(x) + g(y)) − x − y +by (C1), and g(x + y) = g(x) + g(y) follows. Suppose that the claim holds for some i. +By (iii), gk(x) ∈ Rad(γ) for every integer k. By the induction assumption, we then have +gi+1(x + y) = gi(g(x + y)) = gi(g(x) + g(y)) = gi(g(x)) + gi(g(y)) = gi+1(x) + gi+1(y). +Furthermore, gi(g−i(x) + g−i(y)) = gi(g−i(x)) + gi(g−i(y)) = x + y and hence g−i(x + y) = +g−i(x) + g−i(y). The claim therefore holds for every i. +Part (v) follows from (iii) and (iv). +□ +Lemma 3.2. Let (g, γ) be a construction pair on (X, +). Then +γ(gix, gjy) = g−i−jγ(x, y) +(3.1) +for all x, y ∈ X and i, j ∈ Z. +Proof. It suffices to show +γ(gix, y) = g−iγ(x, y) +(3.2) +for all i ∈ Z, since then γ(gix, gjy) = g−iγ(x, gjy) = g−iγ(gjy, x) = g−ig−jγ(y, x) = +g−i−jγ(x, y) by symmetry of γ. Let us prove (3.2). For i = 0 there is nothing to show and +for i = 1 we recover (C3). If (3.2) holds for i then γ(gi+1x, y) = γ(gigx, y) = g−iγ(gx, y) = +g−ig−1γ(x, y) = g−i−1γ(x, y) shows that (3.2) holds for i + 1. To prove (3.2) for i = −1, +substitute g−1(x) for x in (C3) to obtain g−1γ(g−1x, y) = γ(x, y), so γ(g−1x, y) = gγ(x, y). +Finally, if (3.2) holds for i then γ(gi−1x, y) = γ(gig−1x, y) = g−iγ(g−1x, y) = g−igγ(x, y) = +g−i+1γ(x, y) shows that (3.2) holds for i − 1. +□ +Lemma 3.3. Let (g, γ) be a construction pair on (X, +). Then (g−1, gγ) is a construction +pair on (X, +). +Proof. We have gγ(x, y) = gγ(y, x) by symmetry of γ. As g(0) = 0 by Lemma 3.1, we have +gγ(x, x) = g(0) = 0. Since g restricts to an automorphism of Rad(γ) and Img(γ) ⊆ Rad(γ) +by Lemma 3.1, we have gγ(x + y, z) = g(γ(x, z) + γ(y, z)) = gγ(x, z) + gγ(y, z). This proves +that that gγ is symmetric, alternating and biadditive. +Now, gγ(gγ(x, y), z) = gg−1γ(γ(x, y), z) = γ(γ(x, y), z) = 0 by (C3) and (C2). Hence +Img(gγ) ⊆ Rad(gγ). +Finally, g(x) + g(y) = g(x + y + γ(x, y) + g−1γ(x, y) + g−2γ(x, y)) = g(x + y) + gγ(x, y) + +γ(x, y) + g−1γ(x, y) by Lemma 3.1. Substituting g−1(x) for x, g−1(y) for y and using (3.1) +yields x + y = g(g−1x + g−1y) + g3γ(x, y) + g2γ(x, y) + gγ(x, y). Since γ(x, y) = −γ(x, y) +and g(−z) = −g(z) by Lemma 3.1, we deduce +g(g−1x + g−1y) = x + y + gγ(x, y) + g2γ(x, y) + g3γ(x, y), +which is the analog of (C1) for (g−1, gγ). +□ +Lemma 3.4 lists basic properties of the intervals I(i, j) of (1.1). For subsets U, V ⊆ Z, let +U ⊕ V = (U ∪ V ) \ (U ∩ V ) +be the symmetric difference of U an V . +Lemma 3.4. Let i, j, k be integers. Then: +9 + +(i) I(i, j) = I(j, i), +(ii) I(i, j) + k = I(i + k, j + k), +(ii) I(i, k) ⊕ I(j, k) = I(i, j). +Proof. Parts (i) and (ii) are clear from the definition of I(i, j). For (iii), thanks to symmetry +it suffices to discuss the three cases i ≤ j ≤ k, i ≤ k ≤ j and k ≤ i ≤ j. Note that then +I(i, j) = [i, j) = {i, i + 1, . . . , j − 1}. If i ≤ j ≤ k then I(i, k) ⊕ I(j, k) = [i, k) ⊕ [j, k) = +[i, j), if i ≤ k ≤ j then I(i, k) ⊕ I(j, k) = [i, k) ⊕ [k, j) = [i, j), and if k ≤ i ≤ j then +I(i, k) ⊕ I(j, k) = [k, i) ⊕ [k, j) = [i, j). +□ +Lemma 3.5. Let (g, γ) be a construction pair on (X, +). Then +g−i(gix + giy) = x + y + +� +k∈I(0,3i) +g−kγ(x, y) +(3.3) +for all x, y ∈ X and i, j ∈ Z. +Proof. There is nothing to prove for i = 0 since I(0, 0) = ∅. For i = 1 we get I(0, 3i) = +I(0, 3) = {0, 1, 2} and we recover (C1). If (3.3) holds for some i ≥ 0, then +g−i−1(gi+1x + gi+1y) = g−1g−i(gigx + gig(y)) = g−1� +gx + gy + +� +k∈I(0,3i) +g−kγ(gx, gy) +� +. +We have γ(gx, gy) = g−2γ(x, y) by (3.1). By Lemma 3.1, g−kγ(gx, gy) ∈ Img(γ) ⊆ Rad(γ), +and a power of g behaves as a homomorphism if at least one of the summands lies in Rad(γ). +Therefore g−i−1(gi+1x + gi+1y) is equal to +g−1(gx + gy) + +� +k∈I(0,3i) +g−1g−kg−2γ(x, y) = g−1(gx + gy) + +� +k∈I(0,3i) +g−k−3γ(x, y) += x + y + +� +k∈{0,1,2} +g−kγ(x, y) + (g−3γ(x, y) + · · · + g−3i−2γ(x, y)) = x + y + +� +k∈I(0,3(i+1)) +g−kγ(x, y), +which gives (3.3) for i + 1. +By Lemma 3.3, (g−1, gγ) is a construction pair on (X, +). For i = −1 we get I(0, 3i) = +I(0, −3) = {−3, −2, −1} and (3.3) becomes +g(g−1x + g−1y) = x + y + gγ(x, y) + g2γ(x, y) + g3γ(x, y), +which is the analog of (C1) for (g−1, gγ). If (3.3) holds for some i ≤ 0, then +g−i+1(gi−1x + gi−1y) = gg−i(gig−1x + gig−1y) = g +� +g−1x + g−1y + +� +k∈I(0,3i) +g−kγ(g−1x, g−1y) +� +. +We have γ(g−1x, g−1y) = g2γ(x, y) by (3.1). +Using Lemma 3.1 as above, we see that +g−i+1(gi−1x + gi−1y) is equal to +g(g−1x + g−1y) + +� +k∈I(0,3i) +gg−kg2γ(x, y) = g(g−1x + g−1y) + +� +k∈I(0,3i) +g−k+3γ(x, y) += x + y + +� +k∈{1,2,3} +gkγ(x, y) + (g4γ(x, y) + · · · + g3i+3γ(x, y)) = x + y + +� +k∈I(0,3(i−1)) +g−kγ(x, y), +which gives (3.3) for i − 1. +□ +10 + +Corollary 3.6. Let (g, γ) be a construction pair on (X, +) and let i ∈ Z. Then +g−i(x + y) = g−i(x) + g−i(y) + +� +k∈I(−2i,i) +g−kγ(x, y) +(3.4) +for every x, y ∈ X. +Proof. By (3.3) we have +g−i(x + y) = g−i(gig−ix + gig−iy) = g−i(x) + g−i(y) + +� +k∈I(0,3i) +g−kγ(g−ix, g−iy). +By Lemma 3.2, γ(g−ix, g−iy) = g2iγ(x, y). Therefore +� +k∈I(0,3i) +g−kγ(g−ix, g−iy) = +� +k∈I(0,3i) +g−k+2iγ(x, y) = +� +k∈I(0,3i)−2i +g−kγ(x, y). +As I(0, 3i) − 2i = I(−2i, i) by Lemma 3.4, we are done. +□ +4. Moufang loops obtained from construction pairs +In this section we introduce a multiplication formula on C × X for a cyclic group C and +a construction pair (g, γ) on (X, +), cf. (4.2). We give a necessary and sufficient condition +for the resulting loop to be a Moufang loop. +To further simplify the notation employed in Section 3, let +� +U +(x, y) = +� +k∈U +g−kγ(x, y), +for a fixed construction pair (g, γ) on (X, +) and any U ⊆ Z and x, y ∈ Z. +Lemma 4.1. Let (g, γ) be a construction pair on (X, +), U, V ⊆ Z, x, y ∈ X and i, j ∈ Z. +Then: +(i) � +U(gix, gjy) = � +U+i+j(x, y), +(ii) gi � +U(x, y) = � +U−i(x, y), +(iii) � +U(x, y) + � +V (x, y) = � +U⊕V (x, y) +Proof. For (i), we can use by (3.1) to get +� +U +(gix, gjy) = +� +k∈U +g−kγ(gix, gjy) = +� +k∈U +g−k−i−jγ(x, y) = +� +k∈U+i+j +g−kγ(x, y) = +� +U+i+j +(x, y). +By Lemma 3.1, g restricts to an automorphism of Rad(γ) and Img(γ) ⊆ Rad(γ). Hence +gi � +U +(x, y) = gi � +k∈U +g−kγ(x, y) = +� +k∈U +gig−kγ(x, y) = +� +k∈U +g−(k−i)γ(x, y) = +� +U−i +(x, y), +proving (ii). Part (iii) follows immediately from 2Img(γ) = 0 of Lemma 3.1. +□ +For a construction pair (g, γ) on (X, +), let r(g, γ) be the least positive integer r such that +� +0≤k 0. Now, � +I(0,n)(x, y) = � +0≤k 0. Note that if r is finite then Lemma 4.2 implies � +I(i,i+r)(x, y) = 0 for every i. +Suppose that (i) holds. +Then r < ∞ and n = qr for some positive integer q. +Since +I(0, n) = I(0, r) ∪ I(r, 2r) ∪ · · · ∪ I((q − 1)r, n), we have � +I(0,n)(x, y) = � +I(0,r)(x, y) + +� +I(r,2r)(x, y) + · · · + � +I((q−1)r,n)(x, y) = 0. Conversely, suppose that (ii) holds, which again +implies that r < ∞. Suppose for a contradiction that n = qr + t for some 0 < t < r. Then +� +I(0,t)(x, y) = � +I(0,n)(x, y)−� +I(t,n)(x, y) = � +I(0,n)(x, y)−g−t � +I(0,qr)(x, y) = 0−g−t(0) = 0 +for all x, y ∈ X. Then � +0≤k 0. Continuing with i > 0, we have f i(f −i(x) + f −i(y)) = f i(f −i(x)) + f i(f −i(y)) = x + y +and hence f −i(x + y) = f −i(x) + f −i(y). Part (v) follows from (iii) and (iv). +□ +Lemma 9.2. Let f be a Moufang pair on (X, +). Then +β(f ix, f iy) = f iβ(f 3ix, y), +(9.1) +β(f 3ix, f 3jy) = f −3(i+j)β(x, y) +(9.2) +for all integers i, j. +Proof. Let us first prove (9.1) for all nonnegative integers i by induction on i. There is +nothing to show for i = 0. For i = 1, (9.1) becomes the defining condition (P3). Suppose +that (9.1) holds for some i ≥ 0. Then +β(f i+1x, f i+1y) = β(f ifx, f ify) = f iβ(f 3ifx, fy) += f iβ(ff 3ix, fy) +(P3) += f ifβ(f 3f 3ix, y) = f i+1β(f 3(i+1)x, y). +Still with a nonnegative integer i, we then also have +f iβ(f −ix, f −iy) = f iβ(f 3if −4ix, f −iy) +(9.1) += β(f if −4ix, y) = β(f −3ix, y), +which implies β(f −ix, f −iy) = f −iβ(f −3ix, y), finishing the proof of (9.1). +By symmetry of β, we then have for any integer i +f iβ(f 3ix, y) +(9.1) += β(f ix, f iy) = β(f iy, f ix) +(9.1) += f iβ(f 3iy, x) = f iβ(x, f 3iy). +Applying f −i to both sides yields +β(f 3ix, y) = β(x, f 3iy). +(9.3) +For (9.2), first note that +f −3iβ(x, y) = f −3iβ(f 3(−3i)f 9ix, y) +(9.1) += β(f −3if 9ix, f −3iy) = β(f 6ix, f −3iy) +(9.3) += β(f 3ix, y), +which is (9.2) for the special case of j = 0. For any j, we then have +β(f 3ix, f 3jy) +(9.3) += β(f 3(i+j)x, y) = f −3(i+j)β(x, y), +finishing the proof. +□ +We proceed to show that if f is a Moufang permutation on (X, +) then f i is also a Moufang +permutation on (X, +). +24 + +Lemma 9.3. Let (f, β) be a Moufang pair on (X, +). Then for every integer i, the mapping +f iβ : X × X → X is symmetric, alternating and biadditive, and it satisfies Img(f iβ) = +Img(β) ⊆ Rad(β) = Rad(f iβ). +Proof. By Lemma 9.1, f i is an automorphism of Rad(β), Img(β) ⊆ Rad(β) and f i permutes +Img(β). Therefore Img(f iβ) = Img(β), f iβ is symmetric and biadditive (as β is symmetric +and biadditive), and also alternating (since β is alternating and f(0) = 0). Using f(0) = 0 +again, we see that f iβ(x, y) = 0 if and only if β(x, y) = 0. Hence Rad(f iβ) = Rad(β). +□ +Lemma 9.4. If (f, β) is a Moufang pair on (X, +) then (f −1, f 3β) is also a Moufang pair +on (X, +). +Proof. Recall that f(−x) = −f(x) and 2Img(β) = 0, by Lemma 9.1. By (P1) and Lemma +9.1, we have f(x) + f(y) = f(x + y + β(x, y)) = f(x + y) + fβ(x, y) = f(x + y) − fβ(x, y), +so fβ(x, y) = f(x + y) − f(x) − f(y) for any x, y ∈ X. Using this in the last step of the +following calculation, we see that +f 3β(x, y) +(9.2) += β(f −3x, y) +(9.1) += fβ(f −1x, f −1y) = f(f −1x + f −1y) − x − y, +which is the analog of (P1) for (f −1, f 3β). Moreover, +f 3β(f −1x, f −1y) +(9.1) += f 3f −1β(f −3x, y) = f 2β(f −3x, y) = f −1f 3β((f −1)3x, y), +which is the analog of (P3). By Lemma 9.3, f 3β is alternating and biadditive. Since f +permutes Img(β) ⊆ Rad(β) and f(0) = 0 by Lemma 9.1, we have f 3β(f 3β(x, y), z) = +f 3(0) = 0, the analog of (P2). +□ +Recall the intervals I(i, j) of (1.1). +Proposition 9.5. Let (f, β) be a Moufang pair on (X, +). Then for every integer i we have +f −i(f i(x) + f i(y)) = x + y + +� +k∈I(0,i) +f −3kβ(x, y) +(9.4) +for every x, y ∈ X. +Proof. Let us first prove (9.4) for all i ≥ 0, a situation in which I(0, i) = {0, 1, . . . , i − 1}. +The case i = 0 is clear since I(0, 0) = ∅, and the case i = 1 is (P1). If (9.4) holds for some +i ≥ 0 then +f −i−1(f i+1x + f i+1y) = f −1f −i(f ifx + f ify) = f −1� +fx + fy + +� +k∈I(0,i) +f −3kβ(fx, fy) +� +. +By Lemma 9.1, Img(β) ⊆ Rad(β) and f permutes Rad(β), which implies that every sum- +mand of � f −3kβ(fx, fy) as thus also the entire sum are elements of Rad(β). By Lemma +9.1(iv), we can the continue the above calculation as +f −1(fx + fy) + f −1 � +k∈I(0,i) +f −3kβ(fx, fy) = f −1(fx + fy) + +� +k∈I(0,i) +f −3k−1β(fx, fy). +Now, f −1(fx + fy) = x + y + β(x, y) by (P1). Also, β(fx, fy) = fβ(f 3x, y) = f −2β(x, y) +by (P3) and (9.2), so +� +k∈I(0,i) +f −3k−1β(fx, fy) = +� +k∈I(0,i) +f −3k−1f −2β(x, y) = +� +k∈I(0,i) +f −3(k+1)β(x, y) = +i +� +k=1 +f −3kβ(x, y). +25 + +Altogether, +f −i−1(f i+1x + f i+1y) = x + y + β(x, y) + +i +� +k=1 +f −3kβ(x, y) = x + y + +i +� +k=0 +f −3kβ(x, y), +which is (9.4) for i + 1. +Let now i < 0 so that I(0, i) = {−i, −i + 1, . . . , −1}. By Lemma 9.4, (f −1, f 3β) is a +Moufang pair. For i = −1, (9.4) reduces to f(f −1(x) + f −1(y)) = x + y + f 3β(x, y), which +is the analog of (P1) for (f −1, f 3β). If (9.4) holds for some i < 0 then +f −i+1(f i−1x + f i−1y) = ff −i(f if −1x + f if −1y) = f +� +f −1x+f −1y+ +� +k∈I(0,i) +f −3kβ(f −1x, f −1y) +� +, +which by Lemma 9.1 can be further written as +f(f −1x + f −1y) + f +� +k∈I(0,i) +f −3kβ(f −1x, f −1y) = f(f −1x + f −1y) + +� +k∈I(0,i) +f −3k+1β(f −1x, f −1y). +Now, f(f −1x + f −1y) = x + y + f 3β(x, y) by (P1) for the Moufang pair (f −1, f 3β). Also, +β(f −1x, f −1y) = f −1β(f −3x, y) = f 2β(x, y) by (9.1) and (9.2), so +� +k∈I(0,i) +f −3k+1β(f −1x, f −1y) = +� +k∈I(0,i) +f −3k+1f 2β(x, y) = +� +k∈I(0,i) +f −3(k−1)β(x, y) = +−2 +� +k=−(i−1) +f −3kβ(x, y). +Altogether, +f −i+1(f i−1x + f i−1y) = x + y + f 3β(x, y) + +−2 +� +k=−(i−1) +f −3kβ(x, y) = x + y + +−1 +� +k=−(i−1) +f −3kβ(x, y), +which is (9.4) for i − 1. +□ +Proposition 9.6. Let (f, β) be a Moufang pair on (X, +) and i ∈ Z. Let +βi = +� +k∈I(0,i) +f −3kβ. +Then (f i, βi) is a Moufang pair on (X, +). Furthermore, Img(βi) ⊆ Img(β) ⊆ Rad(β) ⊆ +Rad(βi). +Proof. By Lemma 9.3, βi is a symmetric alternating biadditive mapping such that Img(βi) ⊆ +Img(β) ⊆ Rad(β) ⊆ Rad(βi). The analog of (P1) for (f i, βi) holds by Proposition 9.5. By +(9.1) and Lemma 9.1, we have +βi(f ix, f iy) = +� +k∈I(0,i) +f −3kβ(f ix, f iy) = f i � +k∈I(0,i) +f −3kβ(f 3ix, y) = f iβi(f 3ix, y), +which is the analog of (P3). The analog of (P2) is satisfied by a similar argument as in the +proof of Lemma 9.4. +□ +Corollary 9.7. Let (f, β) be a Moufang pair on (X, +) and let βi be defined as in Proposition +9.6. Then +f −i(x + y) = f −i(x) + f −i(y) + f 2iβi(x, y). +(9.5) +for every x, y ∈ X and i ∈ Z. +26 + +Proof. By (P1) for the Moufang pair (f i, βi), we have +f −i(x + y) = f −i(f if −i(x) + f if −i(y)) = f −i(x) + f −i(y) + βi(f −ix, f −iy). +By (P3) and (9.2) for (f i, βi), we have +βi(f −ix, f −iy) = f −iβi(f −3ix, y) = f −if 3iβi(x, y) = f 2iβ(x, y). +□ +Finally, here is the relation between Moufang pairs and construction pairs announced in +the introduction. +Proposition 9.8. Let (f, β) be a Moufang pair on (X, +). Then (f 3, β) is a construction +pair on (X, +). +Proof. By definition, β is alternating and biadditive. It is clear from (P1) that β is also sym- +metric. Let (g, γ) = (f 3, β). Then (P2) becomes (C2). By (9.2), g−1γ(x, y) = f −3β(x, y) = +β(f 3x, y) = γ(gx, y), which is (C3). Finally, by (9.4) for i = 3, we have g−1(gx + gy) = +f −3(f 3x+f 3y) = x+y +�2 +k=0 f −3kβ(x, y) = x+y +γ(x, y)+g−1γ(x, y)+g−2γ(x, y), which +is (C1). +□ +10. Moufang loops constructed from Moufang permutations +In this section we consider the Moufang loops constructed from Moufang permutation and +we prove most of the main results. +Theorem 10.1. Let (X, +) be an abelian normal subgroup of a Moufang loop (Q, ·) and let +a ∈ Q. Denote by f the restriction of Ta to X. Then f is a Moufang permutation on (X, +) +with associated biadditive mapping β defined by (P1). Moreover, +a3ix · ya3j = a3(i+j)� +f −3j(x) + f −3j(y) + +� +k∈I(−2j,i) +f −3kβ(x, y) +� +, +(10.1) +a3ix · a3jy = a3(i+j)� +f −3j(x) + y + +� +k∈I(i+j,−j) +f −3kβ(x, y) +� +(10.2) +for all integers i, j and all x, y ∈ Q. +Proof. By Proposition 8.4, f is a Moufang permutation on (X, +) with associated biadditive +mapping β. Let δ = i − j and b = a3. By (2.10), we have +bix · ybj = bi+jf −i−2j(f i−jx + f i−jy) = bi+jf −3jf −δ(f δx + f δy). +Since (f δ, βδ) is a Moufang pair by Proposition 9.6, we obtain +bix · ybj = bi+jf −3jf −δ(f δx + f δy) = bi+jf −3j(x + y + βδ(x, y)). +By Lemma 9.1 and (9.5), f −3j(x + y + βδ(x, y)) is equal to +f −3j(x + y) + f −3jβδ(x, y) = f −3jx + f −3jy + f 6jβ3j(x, y) + f −3jβδ(x, y). +By Lemma 9.1 and Lemma 3.4(ii), we have +f 6jβ3j = f 6j � +k∈I(0,3j) +f −3kβ = +� +k∈I(0,3j) +f −3(k−2j)β = +� +k∈I(0,3j)−2j +f −3kβ = +� +k∈I(−2j,j) +f −3kβ +27 + +and +f −3jβδ = f −3j � +k∈I(0,δ) +f −3kβ = +� +k∈I(0,i−j) +f −3(k+j)β = +� +k∈I(0,i−j)+j +f −3kβ = +� +k∈I(j,i) +f −3kβ. +Since 2Img(β) = 0 by Lemma 9.1, Lemma 3.4 implies +f 6jβ3j + f −3jβδ = +� +k∈I(−2j,j) +f −3kβ + +� +k∈I(j,i) +f −3kβ = +� +k∈I(−2j,j)⊕I(j,i) +f −3kβ = +� +k∈I(−2j,i) +f −3kβ, +finishing the proof of (10.1). +For (10.2), note that Tb = f 3, so bix·bjy = bix·bjyb−jbj = bix·f 3j(y)bj. Now (10.1) yields +bix · bjy = bi+j� +f −3j(x) + y + +� +k∈I(−2j,i) +f −3kβ(x, f 3jy) +� +. +By (9.2) and Lemma 3.4, the sum in the above formula is equal to +� +k∈I(−2j,i) +f −3(k+j)β(x, y) = +� +k∈I(−2j,i)+j +f −3kβ(x, y) = +� +k∈I(−j,i+j) +f −3kβ(x, y) = +� +k∈I(i+j,−j) +f −3kβ(x, y), +finishing the proof. +□ +Corollary 10.2. Under the assumptions of Theorem 10.1, if Q = ⟨a3⟩X then each of the +formulas (10.1), (10.2) fully describes the multiplication in Q. +Recall the loops C ⋉(g,γ) X of Definition 4.6 that are Moufang by Theorem 4.7. Per our +convention, whenever we write C ⋉(g,γ) X, we assume that all assumptions of Definition 4.6 +are satisfied. Also recall the parameter r(g, γ) of (4.1) that appears in Definition 4.6. +Lemma 10.3. Let (X, +) be a 3-divisible abelian group and C a finite cyclic group. Let f be +a Moufang permutation on (X, +) with associated biadditive mapping β such that |f 3| divides +|C|. Then (g, γ) = (f 3, β) is a construction pair that satisfies all assumptions of Definition +4.6. +Proof. By Proposition 9.8, (g, γ) = (f 3, β) is a construction pair on (X, +). Let n = |C|. By +our assumption, |g| divides n. Let r = r(g, γ). Since X is 3-divisible, it contains no elements +of order 3. Then certainly Rad(γ) contains no elements of order 3. Since |g| < ∞, Lemma +4.5 implies that r divides |g|. Thus r divides n and all assumptions of Definition 4.6 are +satisfied. +□ +Theorem 10.4. Let Q = CX be a 3-divisible Moufang loop, where (X, +) is a normal +abelian subgroup of Q and C = ⟨a3⟩ for some a ∈ C. Let f be the restriction of Ta to X and +let β be the associated biadditive mapping. Then: +(i) (g, γ) = (f 3, β) is a construction pair satisfying all assumptions of Definition 4.6 and +the Moufang loop M = ⟨a3⟩ ⋉(g,γ) X is well defined, +(ii) ϕ : M → Q defined by ϕ(a3i, x) �→ a3ix is a surjective homomorphism, +(iii) the kernel of ϕ intersects both C × 0 and 1 × X trivially. +Proof. (i) Since C is 3-divisible, it is finite, say of order n, By Theorem 10.1, the restriction +f of Ta to X is a Moufang permutation on (X, +) and the multiplication in Q is fully +determined by (10.2). By Proposition 9.8, (g, γ) = (f 3, β) is a construction pair. On X, we +have f 3n = (Ta)3n = (Ta3n) = T1 = idX, so |f 3| divides n. By Lemma 10.3, all assumptions +28 + +of Definition 4.6 are satisfied. By Theorem 4.7, M = ⟨a3⟩⋉(f3,β) X is a well-defined Moufang +loop with multiplication (1.4). +(ii) and (iii): By (10.2) and (1.4), we have +ϕ(a3i, x)ϕ(a3j, y) = a3ix · a3jy = a3(i+j)(f −3j(x) + y + +� +k∈I(i+j,−j) +f −3kβ(x, y)) += ϕ(a3(i+j), f −3j(x) + y + +� +k∈I(i+j,−j) +f −3k(β(x, y))) = ϕ((a3i, x)(a3j, y)), +so ϕ is a homomorphism, clearly surjective. The kernel of ϕ consists of all (a3i, x) ∈ M with +0 ≤ i < n such that a3ix = 1. If x = 1, we get a3i = 1 and and hence a3 = 1, a = 1. If +a3i = 1, we get x = 1. +□ +Proposition 10.5. Let S be a nonempty subset of Q = C ⋉(g,γ) X = ⟨b⟩ ⋉(g,γ) X. Then S is +a normal subloop of Q that intersects both 1 × X and C × 0 trivially if and only if there are +k ∈ Z and z ∈ X such that S = ⟨(bk, z)⟩ = {(bki, iz) : i ∈ Z}, |bk| = |z| and (bk, z) ∈ Z(Q). +Proof. If (bk, z) ∈ Z(Q) then g(z) = z ∈ Rad(γ) by Proposition 5.5 and thus (bk, z)i = +(bki, iz) for all i ∈ Z. The converse implication is now clear since every central subloop is +normal and the condition |bk| = |z| guarantees that S intersects both 1 × X and C × 0 +trivially. +Let us prove the direct implication. +Suppose that S ⊴ Q intersects both 1 × X and +C × 0 trivially. +We claim that for every c ∈ C there is at most one x ∈ X such that +(c, x) ∈ S. Suppose that (c, x), (c, y) ∈ S for some x, y ∈ X. Then (c, x)(c, y)−1 ∈ S and +(c, x)(c, y)−1 = (1, u) for some u ∈ X. Since S intersects 1 × X trivially, we have u = 0 and +x = y. +Hence there are k ∈ Z and z ∈ X such that S = ⟨(bk, z)⟩. We have (bk, z) ∈ S and +also T(bi,x)(bk, z) ∈ S since S ⊴ Q. But T(bi,x)(bk, z) = (bi, x)(bk, z) · (bi, z)−1 = (bk, u) for +some u ∈ X and it follows that T(bi,x)(bk, z) = (bk, z). Similarly, L(bi,x),(bj,y)(bk, z) = (bk, z) = +R(bi,x),(bj,y)(bk, z). Hence (bk, z) ∈ Z(Q). +We also claim that for every x ∈ X there is at most one c ∈ C such that (c, x) ∈ S. Suppose +that (c, x), (d, x) ∈ S for some c, d ∈ C. We have (c, x) = (c, 0)(1, x) and (d, x) = (d, 0)(1, x). +Hence (cd−1, 0) = (c, 0)(d, 0)−1 ∈ S. Since S intersects C × 0 trivially, we have c = d. We +conclude that |bk| = |z|. +□ +Theorem 10.6 (Split 3-divisible abelian-by-cyclic Moufang loops). Suppose that (X, +) is +a 3-divisible abelian group and C a 3-divisible cyclic group. The following conditions are +equivalent: +(i) Q is a Moufang loop, X ⊴ Q, Q = CX and C ∩ X = 1. +(ii) Q is isomorphic to C ⋉(f3,β) X for some Moufang permutation f on (X, +) with +associated biadditive mapping β. +Proof. The converse implication (ii) ⇒ (i) is clear from Theorem 4.7. For the direct implica- +tion, Theorem 10.4 shows that Q is the image of the homomorphism ϕ : M = ⟨a3⟩⋉(f3,β)X → +Q, ϕ(a3i, x) = a3ix, where f is the restriction of Ta to X. Suppose that (a3i, x) is in the +kernel of ϕ. Then a3ix = 1 implies that x ∈ C ∩X = 1 and a3i = 1. 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Carolin. 55 (2014), no. 3, 411–420. +[24] Robert L. Griess, Jr., Code loops, J. Algebra 100 (1986), no. 1, 224–234. +[25] Tom´aˇs Kepka and Petr Nˇemec, Commutative Moufang loops and distributive groupoids of small orders, +Czechoslovak Math. J. 31(106) (1981), no. 4, 633–669. +[26] Michael K. Kinyon and Kenneth Kunen, The structure of extra loops, Quasigroups Related Systems 12 +(2004), 39–60. +[27] G´abor P. Nagy and Maurizio Valsecchi, On nilpotent Moufang loops with central associators, J. Algebra +307 (2007), no. 2, 547–564. +[28] Fook Leong and Andrew Rajah, Moufang loops of odd order pαq2 +1 · · · q2 +nr1 · · · rm, J. Algebra 190 (1997), +no. 2, 474–486. +[29] Fook Leong and Andrew Rajah, Split extension in Moufang loops, Publ. Math. Debrecen 52 (1998), +no. 1–2, 33–42. +[30] Martin W. Liebeck, The classification of finite simple Moufang loops, Math. Proc. Cambridge Philos. +Soc. 102 (1987), no. 1, 33–47. +30 + +[31] Ruth Moufang, Alternativk¨orper und der Satz vom vollst¨andigen Vierseit (D9), Abh. Math. Sem. Univ. +Hamburg 9 (1933), no. 1, 207–222. +[32] Ruth Moufang, Zur Struktur von Alternativk¨orpern, Math. Ann. 110 (1935), no. 1, 416–430. +[33] G´abor P. Nagy and Petr Vojtˇechovsk´y, LOOPS: Computing with quasigroups and loops in GAP, version +2.2.0. +[34] G´abor P. Nagy and Petr Vojtˇechovsk´y, The Moufang loops of order 64 and 81, J. Symbolic Comput. 42 +(2007), no. 9, 871–883. +[35] E.A. O’Brien and Petr Vojtˇechovsk´y, Code loops in dimension at most 8, J. Algebra 473 (2017), 607– +626. +[36] Lowell J. Paige, A class of simple Moufang loops, Proc. Amer. Math. Soc. 7 (1956), 471–482. +[37] H.O. Pflugfelder, Quasigroups and Loops: Introduction, Heldermann, Berlin (1990). +[38] J.D. Phillips and Petr Vojtˇechovsk´y, A scoop from groups: Equational foundations for loops, proceedings +of Loops ’07, Prague, published in Comment. Math. Univ. Carolin. 49 (2008), no. 2, 279–290. +[39] Andrew Rajah, Moufang loops of odd order pq3, J. Algebra 235 (2001), no. 1, 66–93. +[40] Michael C. Slattery and Ashley L. Zenisek, Moufang loops of order 243, Comment. Math. Univ. Carolin. +53 (2012), no. 3, 423–428. +[41] Tonny A. Springer and Ferdinand D. Veldkamp, Octonions, Jordan algebras and exceptional groups, +Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2000. +[42] David Stanovsk´y and Petr Vojtˇechovsk´y, Commutator theory for loops, J. Algebra 399 (2014), 290–322. +[43] David Stanovsk´y and Petr Vojtˇechovsk´y, Abelian extensions and solvable loops, Results Math. 66 (2014), +367–384. +[44] Max Zorn, Alternativk¨orper und quadratische Systeme, Abh. Math. Sem. Univ. Hamburg 9 (1933), +395–402. +(Dr´apal) Department of Mathematics, Charles University, Sokolovsk´a 83, 18675 Praha 8, +Czech Republic +Email address, Dr´apal: drapal@karlin.mff.cuni.cz +(Vojtˇechovsk´y) Dept. of Mathematics, University of Denver, 2390 S. York St., Denver, CO +80208, USA +Email address, Vojtˇechovsk´y: petr@math.du.edu +31 + diff --git a/k9E2T4oBgHgl3EQfIwYe/content/tmp_files/load_file.txt b/k9E2T4oBgHgl3EQfIwYe/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..47d808289059a55808fb27f7b014dd1f019c4095 --- /dev/null +++ b/k9E2T4oBgHgl3EQfIwYe/content/tmp_files/load_file.txt @@ -0,0 +1,1423 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf,len=1422 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='03683v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='GR] 9 Jan 2023 ON ABELIAN-BY-CYCLIC MOUFANG LOOPS ALEˇS DR´APAL AND PETR VOJTˇECHOVSK´Y Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' We study abelian-by-cyclic Moufang loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' We construct all split 3-divisible abelian-by-cyclic Moufang loops from so-called Moufang permutations on abelian groups (X, +), which are permutations that deviate from an automorphism of (X, +) by an al- ternating biadditive mapping (satisfying certain properties).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' More generally, we obtain additional abelian-by-cyclic Moufang loops from so-called construction pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' As an aside, we show that in the Moufang loops Q obtained from a construction pair on (X, +) the abelian normal subgroup (X, +) induces an abelian congruence of Q if and only if Q is a group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Introduction Moufang loops were introduced in 1935 [32] and studied ever since.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Yet there remain serious gaps in our understanding of basic structural concepts for Moufang loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' One of the reasons for the state of affairs is the lack of constructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' In this paper we start a systematic approach to constructions of abelian-by-cyclic Moufang loops, that is, Moufang loops Q possessing an abelian normal subgroup X such that Q/X is cyclic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' The main results are Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='7 (a construction of many abelian-by-cyclic Moufang loops from so-called construction pairs), Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='1 (every conjugation by a in a Moufang loop restricts to a so-called Moufang permutation on X and the multiplication on the subloop ⟨a3⟩X is just like the abstract multiplication formula of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='7), Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='4 (all 3-divisible abelian-by-cyclic Moufang loops are homomorphic images of the loops obtained from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='7 with construction pairs induced by Moufang permutations), and Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='6 (a characterization of split 3-divisible abelian-by-cyclic Moufang loops).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' A brief overview of constructions of Moufang loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' The motivating and most important examples of Moufang loops come from the multiplicative loops of nonzero oc- tonions [2, 11, 31, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' These octonionic Moufang loops have additional strong structural properties not shared by all Moufang loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' For instance, in the 16-element loop O16 of basic units of real octonions, every square associates with all elements (making O16 an extra loop [10, 16]) and there is in fact a unique nonidentity square (making O16 a code loop [9, 24]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' All nonassociative finite simple Moufang loops are obtained as central factors of the loop of unit elements in split octonion algebras over finite fields [30, 36, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Another interesting class of Moufang loops are Moufang p-loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Moufang p-loops are centrally nilpotent (see [20] for p odd, [21] for p = 2, and [14] for a recent elementary 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' 20N05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Abelian by cyclic Moufang loop, Moufang loop, conjugation in Moufang loops, Moufang permutation, solvability, congruence solvability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Dr´apal supported by the INTER-EXCELLENCE project LTAUSA19070 of MˇSMT Czech Republic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Vojtˇechovsk´y supported by the Simons Foundation Mathematics and Physical Sciences Collaboration Grant for Mathematicians no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' 855097.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' 1 proof of both cases).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Using geometric considerations, Bol [4] constructed a nonassociative commutative Moufang loop of order 34 (also see [25] for all commutative Moufang loops of order less than 36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Later, Bruck [5] constructed a nonassociative Moufang loop of order p5 for every prime p, and Nagy and Valsecchi [27] classified Moufang loops of order p5 for all primes p > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Using a computational approach to central extensions, all Moufang loops of orders 26 and 34 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' 35) were classified in [34] (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' [40]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' The aforementioned code loops are equivalently described as Moufang 2-loops Q possessing a central subloop Z of order 2 such that Q/Z is an elementary abelian 2-group, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' [1, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Code loops of order ≤ 29 were enumerated in [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' The constructions of Moufang p-loops are often of combinatorial character and they say little about Moufang loops that are not of prime power order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Yet another source of constructions is connected with the (still open) question for which integers n there exists a nonassociative Moufang loop of order n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Chein described all Moufang loops of order less than 32 in [7] and all Moufang loops of order less than 64 in [8] (see also [22, 33] for a catalog of all nonassociative Moufang loops of order less than 64).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' The constructions that appear in Chein’s classification of small Moufang loops include the well- known Chein double M(G, 2) of a group G as well as several detailed variations on the Chein double.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Leong and Rajah [28] proved that any Moufang loop of order pαqα1 1 · · · qαk k with p < q1 < · · · < qk odd primes and with α ≤ 3 and αi ≤ 2 is a group, and similarly for the case p > 3 and α = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' The intermediate order factorizations between this lower bound (on exponents) and the upper bound furnished by nonassociative Moufang p-loops was investigated extensively by Rajah and various coauthors, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=', for instance, [6, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' By the very nature of the question, the constructions in this area of research are mostly ad hoc since a single nonassociative example is required to settle the existence question for any given order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' As far as general constructions of Moufang loops are concerned, there are substantial results of Kinyon and Kunen on semidirect products in the context of extra loops [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' For (not necessarily extra) Moufang loops, the first step was a formula discovered by Gagola [19], cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' our (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='12), that gives a necessary condition for the existence of a semidirect product of a Moufang loop (which is normal in the product) and a cyclic group of order coprime to three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Sufficient conditions for the existence of such a semidirect product were then given by Dr´apal [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' The conditions of [13] are in the form of certain requirements on semiautomorphisms of the normal subloop and they are difficult to work with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' The present paper was inspired by [13] but is independent of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Finally, we wish to mention the papers [18, 23, 29] that deal with semidirect products and/or split extensions of Moufang loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' A summary of main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' The following definition is key to obtaining many abelian-by-cyclic Moufang loops: Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Let (X, +) be an abelian group, g a permutation of X and γ : X ×X → X a mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Then (g, γ) is a construction pair on (X, +) if γ is a symmetric alternating biadditive mapping, g−1(g(x) + g(y)) = x + y + γ(x, y) + g−1(γ(x, y)) + g−2(γ(x, y)) (C1) holds for all x, y ∈ X, γ(γ(x, y), z) = 0 (C2) 2 holds for all x, y, z ∈ X, and g−1(γ(x, y)) = γ(g(x), y) (C3) holds for all x, y ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Note that the condition (C2) may be equivalently expressed by stating that the image of γ is contained in the radical Rad(γ) = {x ∈ X : γ(x, y) = 0 for all y ∈ X} of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' For i, j ∈ Z, define the interval I(i, j) of Z by I(i, j) = \uf8f1 \uf8f2 \uf8f3 ∅, if i = j, {i, i + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' , j − 1}, if i < j, {j, j + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' , i − 1}, if j < i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='1) Given an abelian group (X, +) and a cyclic group C = ⟨b⟩, we show that the formula (bi, x) · (bj, y) = � bi+j, g−j(x) + y + � k∈I(i+j,−j) g−k(γ(x, y)) � (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='2) correctly defines a multiplication on C × X if and only if either C is infinite, or C is finite, |g| divides |C| and � 0≤k<|C| gk(x) ∈ Rad(γ) for all x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' (The condition � 0≤k<|C| gk(x) ∈ Rad(γ) is satisfied whenever |C| is finite, |g| divides |C| and Rad(γ) contains no elements of order 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=') In that case the resulting groupoid is in fact a Moufang loop Q = C ⋉(g,γ) X that contains a normal subloop 1 × X such that Q/(1 × X) is isomorphic to C, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' In addition, (C × 0) ∩ (1 × X) = 1, so Q is a split extension of X by C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Several key properties of the loops C ⋉(g,γ) X are established in Section 5, such as the formula for an elementwise commutator, elementwise associator, the associator subloop, the commutator subloop, the derived subloop, the nucleus and the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' In the brief Section 6 we show that in the Moufang loop Q = C ⋉(g,γ) X the normal subloop 1×X induces an abelian congruence of Q if and only if Q is a group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' This is related to another current line of investigation of ours concerned with the two notions of solvability in loops, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' [15, 42, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' The results of Section 6 are not used elsewhere in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' The rest of the paper is concerned with a partial converse of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' It is easy to see that Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='7 does not yield all abelian-by-cyclic Moufang loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' For instance, the quaternion group Q8 is abelian-by-cyclic but it is not split and hence it cannot be obtained from the construction of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' (A smallest nonassociative abelian-by- cyclic Moufang loop that is not split is of order 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=') In fact, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='7 does not yield all split abelian-by-cyclic Moufang loops either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' For instance, the nonassociative commutative Moufang loop of order 81 and exponent 3 is a split extension of X = Z3 × Z3 × Z3 by C = Z3, but Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='7 does not yield any nonassociative commutative Moufang loops, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' On the other hand, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='7 yields all abelian-by-cyclic Moufang loops that are split and 3-divisible, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' To prove this fact, we will need the following notion similar to construction pairs: Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Let (X, +) be an abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' A permutation f on X is said to be a Moufang permutation on (X, +) if the mapping β : X × X → X defined by β(x, y) = f −1(f(x) + f(y)) − x − y (P1) 3 is (symmetric) alternating and biadditive, β(β(x, y), z) = 0 (P2) holds for all x, y, z ∈ X, and β(f(x), f(y)) = f(β(f 3(x), y)) (P3) holds for all x, y ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' When f is a Moufang permutation and β is defined by (P1), we call β the biadditive mapping associated withf and the tuple (f, β) a Moufang pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' The relation between construction pairs and Moufang pairs is that if (f, β) is a Moufang pair then (f 3, β) is a construction pair, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' In particular, when the remaining assumptions of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='7 are satisfied, we can use the multiplication formula (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='2) with (g, γ) = (f 3, β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' If also b = a3 for some a ∈ C (which certainly holds when C is 3-divisible), the formula (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='2) becomes (a3i, x) · (a3j, y) = � a3(i+j), f −3j(x) + y + � k∈I(i+j,−j) f −3k(β(x, y)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='3) After deriving some rather general results on pseudoautomorphisms and semiautomor- phisms of Moufang loops induced by inner mappings in Sections 7 and 8, we prove that for every abelian normal subgroup X of a Moufang loop Q and every element a ∈ Q the restriction of the “conjugation” Ta to X is a Moufang permutation on X, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' In Section 9 we study abstract properties of Moufang permutations f and their powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Among other results, we derive a formula for the expression f −i(f i(x)+f i(y)), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' We then prove in Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='1 that if (X, +) is an abelian normal subgroup of a Moufang loop Q and if a ∈ Q then a3ix · a3jy = a3(i+j)� f −3j(x) + y + � k∈I(i+j,−j) f −3k(β(x, y)) � (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='4) holds for all i, j ∈ Z and x, y ∈ X, where f is the restriction of Ta to X, which we know from Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='4 is a Moufang permutation on (X, +).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Here and throughout the paper, if (X, +) is a normal subloop of (Q, ·), we write either x+ y or x· y for the product of x, y ∈ X in Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Note that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='4) does not necessarily describe the multiplication for all pairs of elements of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Nevertheless, it easily follows, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Corollary 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='2, that if (X, +) ⊴ Q and Q = ⟨a3⟩X for some a ∈ Q, then (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='4) does fully describe the multiplication of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' It is then not difficult to show that all 3-divisible abelian-by-cyclic Moufang loops Q = CX are homomorphic images of the loops C ⋉(f3,β) X, where (f, β) is a Moufang pair on (X, +).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Here, |f 3| must divide |C|, else C ⋉(f3,β) X is not even defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' The kernels of the homomorphisms are described in Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Finally, 3-divisible abelian-by-cyclic Moufang loops Q = CX that are split are precisely the the loops C ⋉(f3,β) X with (f, β) a Moufang pair on (X, +), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Let us conclude the summary of main results with a few comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' 4 The multiplication formula (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='4) might appear rather complicated and unnatural.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' A point of departure in deriving the formula is the important identity a3ix · ya3j = a3(i+j)T −i−2j a (T i−j a (x)T i−j a (y)) valid in all Moufang loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' We record this identity and all its variants in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='6, including the already mentioned identity (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='12) of Gagola [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' It turns out that in the finite case each construction pair (g, γ) on (X, +) induces an integer m and a mapping U ×U → V that is alternating and bilinear over R, where U = X/Rad(γ), V = ⟨Img(γ)⟩ and R = F2[x]/(xm − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' To describe construction pairs for a given abelian group (X, +) thus means to start from a classification of the relevant alternating bilinear mappings and then include several steps of lifting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' We intend to study the problem of obtaining construction pairs and of classifying the resulting Moufang loops up to isomorphism in a future paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Background 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Mappings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Let (X, +) be an abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Let us call a mapping γ : X × X → X symmetric if γ(x, y) = γ(y, x) for all x, y ∈ X, antisymmetric if γ(x, y) = −γ(y, x) for all x, y ∈ X, alternating if γ(x, x) = 0 for all x ∈ X, and biadditive if γ(x + y, z) = γ(x, z) + γ(y, z) and γ(x, y + z) = γ(x, y) + γ(x, z) for all x, y, z ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Note that a biadditive mapping satisfies γ(0, x) = 0 = γ(x, 0), γ(−x, y) = −γ(x, y) = γ(x, −y), γ(x−y, z) = γ(x, z) −γ(y, z) and γ(x, y −z) = γ(x, y) −γ(x, z) for all x, y, z ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Indeed, γ(0, x) = γ(0 + 0, x) = 2γ(0, x) yields γ(0, x) = 0, then 0 = γ(x + (−x), y) = γ(x, y)+γ(−x, y) implies γ(−x, y) = −γ(x, y), and finally γ(x−y, z) = γ(x, z)+γ(−y, z) = γ(x, z) − γ(y, z) An alternating biadditive mapping is antisymmetric since 0 = γ(x + y, x + y) = γ(x, x) + γ(x, y) + γ(y, x) + γ(y, y) = γ(x, y) + γ(y, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' A symmetric alternating biadditive mapping then satisfies 2γ(x, y) = γ(2x, y) = 0 because γ(x, y) = γ(y, x) = −γ(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' All these properties will be used without reference throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' The image of any mapping g will be denoted by Img(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' As in the introduction, for a symmetric mapping γ : X × X → X, the radical of γ is defined by Rad(γ) = {x ∈ X : γ(x, y) = 0 for all y ∈ X}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' If γ is symmetric and biadditive, Rad(γ) is a subgroup of (X, +).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' In order to improve legibility, we will often omit parentheses around arguments of unary mappings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' For instance, if g : X → X and γ : X ×X → X, we might write gγ(x, y) instead of g(γ(x, y)), and γ(gx, y) instead of γ(g(x), y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' See [5, 37] for an introduction to loop theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' A set Q with a binary operation · and an element 1 ∈ Q is a loop if for every x ∈ Q we have x · 1 = 1 · x = x and the translations Lx : Q → Q, Lx(y) = x · y and Rx : Q → Q, Rx(y) = y · x are permutations of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' The induced division operations will be denoted by x\\y = L−1 x (y) and x/y = R−1 y (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' 5 Associative subloops will be referred to as subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' A loop Q is power associative if every element of Q generates a subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' A loop Q is diassociative if every two elements of Q generate a subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' We will often write xy instead of x · y and use · to indicate priority of multipli- cation, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=', x · yz stands for x · (y · z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' The multiplication group of a loop Q is the permutation group Mlt(Q) = ⟨Lx, Rx : x ∈ Q⟩, and the inner mapping group Inn(Q) of Q is the stabilizer of 1 in Mlt(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' It is well known that Inn(Q) = ⟨Tx, Lx,y, Rx,y : x, y ∈ Q⟩, where Tx = R−1 x Lx, Lx,y = L−1 xy LxLy and Rx,y = R−1 xy RyRx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' We refer to the inner mappings Tx as conjugations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' The nucleus Nuc(Q) of Q consists of all x ∈ Q such that x(yz) = (xy)z, y(xz) = (yx)z and y(zx) = (yz)x for all y, z ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' The center Z(Q) of Q consists of all x ∈ Nuc(Q) such that xy = yx for all y ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' The commutator [x, y] of x, y ∈ Q is defined by (yx)[x, y] = xy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' The associator [x, y, z] of x, y, z ∈ Q is defined by (x · yz)[x, y, z] = xy · z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Morphisms and topisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Denote by Sym(Q) the symmetric group on Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' A tuple (c, f) ∈ Q × Sym(Q) is a (left) pseudoautomorphism of Q if cf(x) · f(y) = cf(xy) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='1) holds for every x, y ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' The element c is called a companion of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' The set of all pseudoau- tomorphisms of Q forms a group Psaℓ(Q) under the operations (c, f)(d, g) = (cf(d), fc) and (c, f)−1 = (f −1(c\\1), f −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='2) A permutation f ∈ Sym(Q) is a semiautomorphism of Q if f(1) = 1 and f(x · yx) = f(x) · f(y)f(x) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='3) holds for all x, y ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' If f is a semiautomorphism of a power associative loop Q then f(xi) = f(x)i for every i ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' The condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='3) can be written as fLxRx = Lf(x)Rf(x)f and therefore also as L−1 f(x)fLx = Rf(x)fR−1 x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='4) Thus any permutation f of a loop Q that satisfies f(1) = 1 and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='4) is a semiautomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' An autotopism of a loop Q is a triple (α, β, γ) of permutations of Q such that α(x)β(y) = γ(xy) holds for all x, y ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Autotopisms of Q may be composed and inverted componentwise, forming the autotopism group Atp(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Note that (c, f) is a pseudoautomorphism of Q if and only if (Lcf, f, Lcf) is an autotopism of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' We use this fact in the proof of the following observation about autotopisms in which the middle component fixes the identity element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Let (Q, ·, 1) be a loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Suppose that (α, β, γ) ∈ Atp(Q) and β(1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Then (α(1), β) ∈ Psaℓ(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Let c = α(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' By our assumption, α(x) = α(x)β(1) = γ(x · 1) = γ(x) and cβ(x) = α(1)β(x) = γ(1 · x) = γ(x) for every x ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Hence (α, β, γ) = (Lcβ, β, Lcβ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' □ 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Moufang loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' A loop Q is Moufang if it satisfies any one of the equivalent Moufang identities xy · zx = (x · yz)x, (M1) xy · zx = x(yz · x), (M2) x(y · zy) = (xy · z)y, (M3) x(y · xz) = (xy · x)z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' (M4) Let us summarize a few facts about Moufang loops, consisting of easy observations and standard results on Moufang loops [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' The identity (M1) can be stated as (Lx, Rx, RxLx) ∈ Atp(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' This is a useful characteri- zation of Moufang loops in terms of autotopisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' By Moufang Theorem [12, 32], if three elements x, y and z of a Moufang loop associate, that is, x(yz) = (xy)z, then the subloop ⟨x, y, z⟩ is a group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Consequently, Moufang loops are diassociative, power associative, satisfy the flexible law x(yx) = (xy)x, the inverse properties x−1(xy) = y = (yx)x−1, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' We take advantage of diassociativity in Moufang loops and write unambiguously xyx, xy = y−1xy, [x, y] = x−1y−1xy, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' [15] Let Q be a Moufang loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Then x−1(xy · z) = yx−1 · xz, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='5) (z · yx)x−1 = zx · x−1y (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='6) for every x, y, z ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' All inner mappings of a Moufang loop can be seen as pseudoautomorphisms, with suitable companions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' In particular, (x−3, Tx), ([y, x], Rx,y) and ([x−1, y−1], Lx,y) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='7) are pseudoautomorphisms in a Moufang loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Every pseudoautomorphism of a Moufang loop is a semiautomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Let Q be a Moufang loop, c ∈ Q and f ∈ Sym(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Then (c, f) ∈ Psaℓ(Q) if and only if xc−1 · cy = f(f −1(x)f −1(y)) for all x, y ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Substituting f −1(x) for x and f −1(y) for y into cf(x) · f(y) = cf(xy) yields cx · y = cf(f −1(x)f −1(y)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' We are done upon multiplying by c−1 on the left and applying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' □ Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Let Q be a Moufang loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Then xa−3 · a3y = T −1 a (Ta(x)Ta(y)) for all a, x, y ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' The equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='8) below is of crucial importance for this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Let Q be a Moufang loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Then a3ix · ya3j = a3i · T j−i a (T i−j a (x)T i−j a (y)) · a3j (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='8) for all a, x, y ∈ Q and all i, j ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' 7 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Let b = a3i and δ = i − j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' By diassociativity and (M1), bix · ybj = bj(bδx) · ybj = bj(bδx · y)bj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='5) and Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='4, bδx · y = bδ(xb−δ · bδy) = bδTa−δ(Taδ(x)Taδ(y)) = bδT −δ a (T δ a(x)T δ a(y)), where we have used Tak = T k a in the last step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Combining, we get bix · ybj = bj(bδx · y)bj = bj · bδT −δ a (T δ a(x)T δ a(y)) · bj = bi · T −δ a (T δ a(x)T δ a(y)) · bj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' □ Here are some variations on the identity (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='8): Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Let Q be a Moufang loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Then a3ix · a3jy = a3(i+j)T −i−2j a (T i−j a (x)T i+2j a (y)) = T 2i+j a (T i−j a (x)T i+2j a (y))a3(i+j), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='9) a3ix · ya3j = a3(i+j)T −i−2j a (T i−j a (x)T i−j a (y)) = T 2i+j a (T i−j a (x)T i−j a (y))a3(i+j), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='10) xa3i · a3jy = a3(i+j)T −i−2j a (T −2i−j a (x)T i+2j a (y)) = T 2i+j a (T −2i−j a (x)T i+2j a (y))a3(i+j), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='11) xa3i · ya3j = a3(i+j)T −i−2j a (T −2i−j a (x)T i−j a (y)) = T 2i+j a (T −2i−j a (x)T i−j a (y))a3(i+j) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='12) for all a, x, y ∈ Q and all i, j ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Note that akz = akza−kak = T k a (z)ak, so it suffices to establish the first equality in each (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='9)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' With u = T j−i a (T i−j a (x)T i−j a (y)), equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='8) yields a3ix · ya3j = a3iua3j = T 3i a (u)a3ia3j = T 2i+j a (T i−j a (x)T i−j a (y))a3(i+j), which is (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' The remaining identities then follow from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='10) as a3ix · a3jy = a3ix · T 3j a (y)a3j, xa3i · a3jy = a3iT −3i a (x) · T 3j a (y)a3j and xa3i · ya3j = a3iT −3i a (x) · ya3j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Construction pairs and their properties Construction pairs were defined in the Introduction, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' In this section we will establish basic properties of construction pairs (g, γ) on (X, +).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' The condition (C1) will be often used in the form g(x) + g(y) = g(x + y + γ(x, y) + g−1(γ(x, y)) + g−2(γ(x, y))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Let (g, γ) be a construction pair on (X, +) and let i be an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Then: (i) g(0) = 0, g(2x) = 2g(x) and g(−x) = −g(x) for all x ∈ X, (ii) Img(γ) ⊆ Rad(γ), 2X ⊆ Rad(γ) and 2Img(γ) = 0, (iii) gi permutes both Img(γ) and Rad(γ), (iv) gi(x + y) = gi(x) + gi(y) whenever {x, y} ∩ Rad(γ) ̸= ∅, (v) gi restricts to an automorphism of Rad(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' (i) We have γ(x, 0) = 0 thanks to biadditivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Hence g−1(0) = g−1γ(0, 0) = γ(g(0), 0) = 0 by (C3) and g(0) = 0 follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Since γ is alternating and g(0) = 0, we then have 0 = γ(x, x)+g−1γ(x, x)+g−2γ(x, x) = g−1(g(x)+g(x))−x−x = g−1(2g(x))−2x by (C1), which yields 2g(x) = g(2x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Finally, γ(x, −x) = −γ(x, x) = 0, so 0 = γ(x, −x) + g−1γ(x, −x) + g−2γ(x, −x) = g−1(g(x) + g(−x)) by (C1), and hence g(x) + g(−x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' (ii) The condition Img(γ) ⊆ Rad(γ) is a restatement of (C2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Any symmetric alternating biadditive mapping satisfies 0 = γ(2x, y) = 2γ(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' (iii) We have x ∈ Rad(γ) iff γ(x, y) = 0 for all y ∈ X iff g−1γ(x, y) = g−1(0) = 0 for all y ∈ X iff γ(g(x), y) = 0 for all y ∈ X by (C3) iff g(x) ∈ Rad(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Similarly, z ∈ Img(γ) iff 8 z = γ(x, y) for some x, y ∈ X iff g−1(z) = g−1γ(x, y) = γ(g(x), y) for some x, y ∈ X by (C3) iff g−1(z) ∈ Img(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' (iv) Suppose without loss of generality that x ∈ Rad(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Then γ(x, y) = 0 and, by (i), giγ(x, y) = 0 for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Thus 0 = γ(x, y) + g−1γ(x, y) + g−2γ(x, y) = g−1(g(x) + g(y)) − x − y by (C1), and g(x + y) = g(x) + g(y) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Suppose that the claim holds for some i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' By (iii), gk(x) ∈ Rad(γ) for every integer k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' By the induction assumption, we then have gi+1(x + y) = gi(g(x + y)) = gi(g(x) + g(y)) = gi(g(x)) + gi(g(y)) = gi+1(x) + gi+1(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Furthermore, gi(g−i(x) + g−i(y)) = gi(g−i(x)) + gi(g−i(y)) = x + y and hence g−i(x + y) = g−i(x) + g−i(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' The claim therefore holds for every i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Part (v) follows from (iii) and (iv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Let (g, γ) be a construction pair on (X, +).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Then γ(gix, gjy) = g−i−jγ(x, y) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='1) for all x, y ∈ X and i, j ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' It suffices to show γ(gix, y) = g−iγ(x, y) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='2) for all i ∈ Z, since then γ(gix, gjy) = g−iγ(x, gjy) = g−iγ(gjy, x) = g−ig−jγ(y, x) = g−i−jγ(x, y) by symmetry of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Let us prove (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' For i = 0 there is nothing to show and for i = 1 we recover (C3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' If (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='2) holds for i then γ(gi+1x, y) = γ(gigx, y) = g−iγ(gx, y) = g−ig−1γ(x, y) = g−i−1γ(x, y) shows that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='2) holds for i + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' To prove (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='2) for i = −1, substitute g−1(x) for x in (C3) to obtain g−1γ(g−1x, y) = γ(x, y), so γ(g−1x, y) = gγ(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Finally, if (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='2) holds for i then γ(gi−1x, y) = γ(gig−1x, y) = g−iγ(g−1x, y) = g−igγ(x, y) = g−i+1γ(x, y) shows that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='2) holds for i − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Let (g, γ) be a construction pair on (X, +).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Then (g−1, gγ) is a construction pair on (X, +).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' We have gγ(x, y) = gγ(y, x) by symmetry of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' As g(0) = 0 by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='1, we have gγ(x, x) = g(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Since g restricts to an automorphism of Rad(γ) and Img(γ) ⊆ Rad(γ) by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='1, we have gγ(x + y, z) = g(γ(x, z) + γ(y, z)) = gγ(x, z) + gγ(y, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' This proves that that gγ is symmetric, alternating and biadditive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Now, gγ(gγ(x, y), z) = gg−1γ(γ(x, y), z) = γ(γ(x, y), z) = 0 by (C3) and (C2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Hence Img(gγ) ⊆ Rad(gγ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Finally, g(x) + g(y) = g(x + y + γ(x, y) + g−1γ(x, y) + g−2γ(x, y)) = g(x + y) + gγ(x, y) + γ(x, y) + g−1γ(x, y) by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Substituting g−1(x) for x, g−1(y) for y and using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='1) yields x + y = g(g−1x + g−1y) + g3γ(x, y) + g2γ(x, y) + gγ(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Since γ(x, y) = −γ(x, y) and g(−z) = −g(z) by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='1, we deduce g(g−1x + g−1y) = x + y + gγ(x, y) + g2γ(x, y) + g3γ(x, y), which is the analog of (C1) for (g−1, gγ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='4 lists basic properties of the intervals I(i, j) of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' For subsets U, V ⊆ Z, let U ⊕ V = (U ∪ V ) \\ (U ∩ V ) be the symmetric difference of U an V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Let i, j, k be integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Then: 9 (i) I(i, j) = I(j, i), (ii) I(i, j) + k = I(i + k, j + k), (ii) I(i, k) ⊕ I(j, k) = I(i, j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Parts (i) and (ii) are clear from the definition of I(i, j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' For (iii), thanks to symmetry it suffices to discuss the three cases i ≤ j ≤ k, i ≤ k ≤ j and k ≤ i ≤ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Note that then I(i, j) = [i, j) = {i, i + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' , j − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' If i ≤ j ≤ k then I(i, k) ⊕ I(j, k) = [i, k) ⊕ [j, k) = [i, j), if i ≤ k ≤ j then I(i, k) ⊕ I(j, k) = [i, k) ⊕ [k, j) = [i, j), and if k ≤ i ≤ j then I(i, k) ⊕ I(j, k) = [k, i) ⊕ [k, j) = [i, j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Let (g, γ) be a construction pair on (X, +).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Then g−i(gix + giy) = x + y + � k∈I(0,3i) g−kγ(x, y) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='3) for all x, y ∈ X and i, j ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' There is nothing to prove for i = 0 since I(0, 0) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' For i = 1 we get I(0, 3i) = I(0, 3) = {0, 1, 2} and we recover (C1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' If (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='3) holds for some i ≥ 0, then g−i−1(gi+1x + gi+1y) = g−1g−i(gigx + gig(y)) = g−1� gx + gy + � k∈I(0,3i) g−kγ(gx, gy) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' We have γ(gx, gy) = g−2γ(x, y) by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='1, g−kγ(gx, gy) ∈ Img(γ) ⊆ Rad(γ), and a power of g behaves as a homomorphism if at least one of the summands lies in Rad(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Therefore g−i−1(gi+1x + gi+1y) is equal to g−1(gx + gy) + � k∈I(0,3i) g−1g−kg−2γ(x, y) = g−1(gx + gy) + � k∈I(0,3i) g−k−3γ(x, y) = x + y + � k∈{0,1,2} g−kγ(x, y) + (g−3γ(x, y) + · · · + g−3i−2γ(x, y)) = x + y + � k∈I(0,3(i+1)) g−kγ(x, y), which gives (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='3) for i + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='3, (g−1, gγ) is a construction pair on (X, +).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' For i = −1 we get I(0, 3i) = I(0, −3) = {−3, −2, −1} and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='3) becomes g(g−1x + g−1y) = x + y + gγ(x, y) + g2γ(x, y) + g3γ(x, y), which is the analog of (C1) for (g−1, gγ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' If (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='3) holds for some i ≤ 0, then g−i+1(gi−1x + gi−1y) = gg−i(gig−1x + gig−1y) = g � g−1x + g−1y + � k∈I(0,3i) g−kγ(g−1x, g−1y) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' We have γ(g−1x, g−1y) = g2γ(x, y) by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='1 as above, we see that g−i+1(gi−1x + gi−1y) is equal to g(g−1x + g−1y) + � k∈I(0,3i) gg−kg2γ(x, y) = g(g−1x + g−1y) + � k∈I(0,3i) g−k+3γ(x, y) = x + y + � k∈{1,2,3} gkγ(x, y) + (g4γ(x, y) + · · · + g3i+3γ(x, y)) = x + y + � k∈I(0,3(i−1)) g−kγ(x, y), which gives (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='3) for i − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' □ 10 Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Let (g, γ) be a construction pair on (X, +) and let i ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Then g−i(x + y) = g−i(x) + g−i(y) + � k∈I(−2i,i) g−kγ(x, y) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='4) for every x, y ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='3) we have g−i(x + y) = g−i(gig−ix + gig−iy) = g−i(x) + g−i(y) + � k∈I(0,3i) g−kγ(g−ix, g−iy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='2, γ(g−ix, g−iy) = g2iγ(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Therefore � k∈I(0,3i) g−kγ(g−ix, g−iy) = � k∈I(0,3i) g−k+2iγ(x, y) = � k∈I(0,3i)−2i g−kγ(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' As I(0, 3i) − 2i = I(−2i, i) by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='4, we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Moufang loops obtained from construction pairs In this section we introduce a multiplication formula on C × X for a cyclic group C and a construction pair (g, γ) on (X, +), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' We give a necessary and sufficient condition for the resulting loop to be a Moufang loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' To further simplify the notation employed in Section 3, let � U (x, y) = � k∈U g−kγ(x, y), for a fixed construction pair (g, γ) on (X, +) and any U ⊆ Z and x, y ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Let (g, γ) be a construction pair on (X, +), U, V ⊆ Z, x, y ∈ X and i, j ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Then: (i) � U(gix, gjy) = � U+i+j(x, y), (ii) gi � U(x, y) = � U−i(x, y), (iii) � U(x, y) + � V (x, y) = � U⊕V (x, y) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' For (i), we can use by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='1) to get � U (gix, gjy) = � k∈U g−kγ(gix, gjy) = � k∈U g−k−i−jγ(x, y) = � k∈U+i+j g−kγ(x, y) = � U+i+j (x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='1, g restricts to an automorphism of Rad(γ) and Img(γ) ⊆ Rad(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Hence gi � U (x, y) = gi � k∈U g−kγ(x, y) = � k∈U gig−kγ(x, y) = � k∈U g−(k−i)γ(x, y) = � U−i (x, y), proving (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' Part (iii) follows immediately from 2Img(γ) = 0 of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E2T4oBgHgl3EQfIwYe/content/2301.03683v1.pdf'} +page_content=' □ For a construction pair (g, γ) on (X, +), let r(g, γ) be the least positive integer r such that � 0≤k 90 b were identified [63] and +compared against the SILVA SSU database 128 (blastn, +min. length 90, min. identity 80%, e value 1e-5). To ver- +ify that the microbial community composition was in- +deed +mostly +prokaryotic, +we +did +a +more +general +screening of the metagenomics reads that identified also +candidate 18S rRNA fragments > 90 b (see Additional +file 1: Tables S4-S5). The complete trimmed read sets +were assembled into contigs ≥ 1 kb with MEGAHIT [64] +(v1.0.3–6-gc3983f9) using paired-end mode, k min = 21, +k max = 131, k step = 10. Genes were predicted using +Prodigal [65] (v.2.6.2) and RNAs with rna_hmm3 [66] +and tRNAscan-SE [67]. Assembled 16S rRNA sequences +were compared to a manually curated version from the +SILVA SSU database (e value ≥ 1e-5). Predicted protein +sequences +were +annotated +against +KEGG +with +GhostKOALA (genus_prokaryotes + family_eukaryotes ++ viruses) [68]. Marker genes for central metabolic +pathways and key environmental element transforma- +tions were identified based on K number assignments +[15, 69–71]. +Contigs ≥ 2.5 kb were binned with METABAT [72] +(superspecific mode) based on differential coverage +Vavourakis et al. Microbiome (2018) 6:168 +Page 13 of 18 + +values obtained by mapping all five trimmed readsets to +all five contig sets with Bowtie2 [73]. The bins were sub- +jected to post-binning (an overview of the workflow is +given in Additional file 2: Figure S13). Bins were +assessed with lineage-specific single copy genes using +CheckM [74] and further processed with the metage- +nomics workflow in Anvi’o [75] (v2.3.2). Since Candidate +Phyla Radiant (CPR) is not included in the CheckM ref- +erence trees and are likely to have low-genome com- +pleteness, we used an existing training file of 797 CPR +genomes to identify putative CPR bins [76]. Bins with +CheckM-completeness ≥ 50% (884 out of 1778) and an +additional four CPR bins were further processed. Coding +sequences +were +annotated +for +taxonomy +against +NCBI-nr (July, 2017) with USEARCH [77] (5.2.32) to +verify that most hits in each bin were to prokaryotic ref- +erences. Phage or viral contigs were manually removed. +Genome +contamination (redundancy) +was estimated +based on marker sets of universal single copy genes +identified for Bacteria [30] and Archaea [78] as imple- +mented in Anvi’o. Genome coverage was obtained by +mapping trimmed reads with BBMap [79] v36.x (kfilter +31, subfilter 15, maxindel 80). Bins with ≥ 5% redun- +dancy were further refined with Anvi’o using circle phy- +lograms +(guide +trees +tnf-cov: +euclidian +ward) +and +scanned again for CPR. Post-binning resulted in a total +of 2499 metagenome-assembled genomes (MAGs), of +which 871 were either medium-quality genome drafts +(CheckM estimated completeness ≥ 50% and contamin- +ation ≤ 10% [80], Additional file 4) or lower quality draft +genomes from CPR. +Phylogeny of the MAGs was assessed based on 16 +single-copy ribosomal proteins and representative refer- +ence genomes of major prokaryote lineages across the +tree of life [17]. Individual ribosomal proteins in our +MAGs were identified by K number assignments. Only +ribosomal proteins ≥ 80 aa were considered. Initial +maximum-likelihood (ML) trees were constructed to de- +termine which organisms belonged to the Archaea, Bac- +teria, or CPR with FastTree 2 [81] (WAG + CAT). Final +separate trees for the three distant evolutionary groups +were constructed in the same manner. Each ribosomal +protein set was aligned separately with MAFFT [82] +(v7.055b, − auto) and concatenated only if a MAG +encoded at least 8 out of 16 proteins. For all trees, a +100× posterior bootstraps +analysis +was +performed. +Phylogenetic trees were visualized together with gen- +ome statistics and abundance information using iTOL +[83]. We cross-checked the taxonomic assignments +based on the phylogeny of the ribosomal protein cas- +sette +with +the +top +hit +contig annotations +against +NCBI-nr and with the reference lineage obtained with +CheckM. Lastly, we manually corrected the MAGs for +misplaced 16S rRNA genes. The final trees presented +in the manuscript were redrawn using FigTree v1.4.3 +[84]. +Detailed genome analyses +CPR +MAGs +were +re-annotated +more +thoroughly: +genes were predicted with Prokka [85], and functional +predictions were performed by running InterProScan +5 locally on the supplied COG, CDD, TIGRFAMs, +HAMAP, Pfam, and SMART databases [86]. BLAST +Koala was used for KEGG pathway predictions [68]. +To find putative carbohydrate-active enzymes in all +final MAGs, we used the web-resource dbCAN [87] +to annotate all predicted proteins ≥ 80 aa against +CAZy [88]. +To identify the top ten abundant MAGs from each re- +spective dataset, ten million randomly sampled single- +tons were mapped onto each MAG with a cut-off of 95% +identity in minimum of 50 bases. Coverage values were +additionally normalized for genome size and expressed +as reads per kilobase of sequence per gigabase of +mapped reads (RPKG) [89]. A positive score (from 871 +to 1) was assigned to each MAG according to the rank- +ing of the summed RPKG of MAGs in the high-salinity +datasets (B1Sed10 and T1Sed) and a negative score ac- +cording to the ranking of the summed RPKGs in the +moderate salinity datasets (CSSed10, CSSed11, T3Se +d10). Both scores were summed to get a “salinity prefer- +ence score” with MAGs recruiting preferably from high +salinity datasets on the positive end, moderate salinity +datasets in the negative end, and those without prefer- +ence in the middle. +We determined species delineation for the most +abundant MAGs and their closest reference genomes +(NCBI-nr) by Average Nucleotide Identity (ANI) and +conserved DNA-matrices, as follows [90]: ANI ≥ 95%, +conDNA ≥ 69% = same species, ANI ≥ 95%, condDNA +< 69% = might be same species, ANI < 95%, condDNA +< 69% = different species. Single gene trees based on +maximum +likelihood +were +constructed +with +un- +trimmed alignments (MAFFT, L-INS-i model) and +FastTree 2 (WAG + CAT, increased accuracy, -spr4 +-mlacc 2 -slownni) using 100× bootstraps. References +were pulled from eggNOG (v4.5.1) [91] and supple- +mented +with +sequences +from +NCBI-nr +or +refined +according to [7, 33, 46, 92–94]. The curated MAGs +were +scanned +for +the +presence +of +rhodopsin +sequences with the hmmsearch software [95] and a +profile +hidden +Markov +model +(HMM) +of +the +bacteriorhodopsin-like protein family (Pfam accession +number +PF01036). +The +identified +sequences +with +significant similarity were aligned together with a +curated database composed of a collection of type-1 +rhodopsins, using MAFFT (L-INS-i accuracy model) +[82]. This protein alignment was further utilized to +Vavourakis et al. Microbiome (2018) 6:168 +Page 14 of 18 + +construct a maximum likelihood tree with 100× boot- +strap with FastTree 2 [81]. All other genes were +identified using the KEGG annotation. +Additional files +Additional file 1: Table S1. General features of the four sampled soda +lakes at time of sampling. Table S2. SILVA classification of the 16S rRNA +gene sequences found in all ≥1 kb contigs of five soda sediment +metagenomic datasets. Table S3. Enzymes involved in lipopolysaccharide +biosynthesis found among different members of the CPR. Table S4. +Sub-kingdom classification of candidate SSU rRNA gene fragments +found in subsamples of 10 million random forward reads from the +five soda sediment metagenomes. Table S5. Top-level taxonomic +classification of the 18S rRNA gene fragments found in subsamples +of 10 million random forward reads from the five soda sediment +metagenomes. Table S6. Description of the metagenomic datasets, +NCBI Sequence Read Archive (SRA) accession numbers and general +statistics of the assembled contigs. (PDF 740 kb) +Additional file 2: Figure S1. Taxonomic fingerprints determined by 16S +rRNA gene amplicon sequencing. Figure S2. Genome statistics of the +871 MAGs. Figure S3. Phylogeny of MAGs belonging to “Candidatus +Aenigmarchaeota” and “Ca. Nanohaloarchaeota”. Figure S4. Phylogeny of +MAGs related to “Candidatus Acetothermia”, candidate division WS1 and +“Candidatus Lindowbacteria”. Figure S5. Phylogeny of MAGs related to +candidate division KSB3 and “Candidatus Schekmanbacteria”. Figure S6. +Multiple sequence alignment of the V-type ATPase subunits K. Figure S7. +Multiple sequence alignment of the F-type ATPase subunits c. Figure S8. +Maximum likelihood tree of the large subunits of RuBisCo and RubisCo- +like proteins. Figure S9. Maximum likelihood tree of the putative +rhodopsins. Figure S10. Predicted isoelectric points (pI) profiles of all +MAGs from CPR members. Figure S11. Predicted isoelectric points +profiles for members of the “Ca. Nealsonbacteria” and “Ca. Vogelbacteria”. +Figure S12. Multiple sequence alignment of the dissimilatory +cytochrome c nitrite reductases (nrfA/TvNiR, K03385). Figure S13. +Overview of the post-binning workflow used for genome recovery. +(PDF 6548 kb) +Additional file 3: Dataset S1. Relative abundance of the OTUs assigned +to the genus-level within the Archaea, Bacteria and organelles from +Eukaryota detected by 16S rRNA gene amplicon sequencing. The OTUs +with less than 0.1% abundance accross all five datasets are not shown. +The names of highly abundant genera (≥1% in at least one of the data- +sets) are shown in bold. (XLSX 24 kb) +Additional file 4: Dataset S2. Organism names, statistics and general +description incl. Completeness and contamination estimates, phylogeny +and DDBJ/EMBL/Genbank accession numbers of the metagenome +assembled genomes (MAGs) described in this paper. All submitted +versions described in this paper are version XXXX01000000. Size = +recovered genome size, Completeness (Compl1), contamination (Cont), +strain heterogenity (Str het) and Taxon CheckM were inferred from +lineage-specific marker sets and a reference tree build with CheckM [74]. +Additional completeness (compl2) and redundancy (red) estimates were +inferred based on the presence of universal single copy genes for Bacteria +and Archaea [75]. Decision and confidence intervals from the Candidate +Phyla Radiation (CPR) scan [75] are given, as well as the taxonomy of the +besthit in SILVA when 16S rRNA genes were present. Phylum/class 16 +ribosomal proteins is the taxonomy derived from our ribosomal protein +trees (see main text: Figs. 2 and 3). OTU gives the inferred link of a +population genome with our 16S rRNA gene amplicon dataset +(Additional file 3). (XLSX 253 kb) +Additional file 5: Dataset S3. Estimated abundance and derived +salinity preference from each MAG in each metagenomic dataset +expressed as Reads per Kilobase of MAG per Gigabase of mapped reads +(RPKG) and “salinity preference score” (see Methods section), basis for +Fig. 4. (XLSX 143 kb) +Additional file 6: Dataset S4. Average Nucleotide Identity (ANI) and +conserved DNA (condna) matrices to determine species delineation +between the most abundant MAGs shown in Fig. 4, closely related +(less abundant) MAGs and NCBI reference genomes. Decision matrix +shows: 1 = same species, − 1 = might be same species, 0 = different +species (see Methods section). (XLSX 1161 kb) +Additional file 7: Dataset S5. Sheet 1 Presence and absence of marker +genes and putative carbohydrate-active enzymes in the MAGs to infer putative +roles in C, N and S element cycles based on K-number assignments and CAZy +annotations. Sheet 2 Summary basis for Fig. 4. (XLSX 41 kb) +Additional file 8: Information S1. More detailed description of the +main metabolisms encoded by Thioalkalivibrio-related MAGs. +Information S2 More detailed description of the main metabolisms +encoded by Deltaproteobacterial-related MAGs. (PDF 219 kb) +Additional file 9: Dataset 6. Sheet 1 shows the MAGs positive for the +marker gene acsB (K14138) encoding an acetyl-CoA synthase (ACS). The +basis for Fig. 6, namely presence and absence of key genes involved in +the Wood-Ljungdahly pathway, acetogenesis, methanogenesis, glycolysis +and pyruvate to CO2 conversion is shown for each MAG. Sheet 2 shows +the MAGs positive for the marker gene cdhC (K00193) encoding for the +beta subunit of an acetyl-CoA decarboxylase synthase complex. While +acsB and cdhC correspond roughly to the Bacterial-type and Archaeal- +type (methanogens) enzymes with the same function, we found few +discrepancies between marker gene and genome phylogeny within the +Methanomassiliicoccaceae and Chloroflexi. (XLSX 52 kb) +Acknowledgments +We thank Dr. Nikolai Chernych for his technical assistance during the +isolation and purification of metagenomics DNA. We also thank the +Department of Energy Joint Genome Institute for sequencing the +metagenomes. +Funding +CDV and GM were supported by the ERC Advanced Grant PARASOL (no. 322551). +A-SA and RG were supported by the research grant 17-04828S from the Grant +Agency of the Czech Republic. MM was supported by the Czech Academy of +Sciences (Postdoc program PPPLZ application number L200961651). DYS was +supported by the SIAM/Gravitation Program (Dutch Ministry of Education and +Science, grant 24002002) and by the Russian Science Foundation (grant 16–14- +00121). Sequencing was performed by the U.S. Department of Energy Joint +Genome Institute, a DOE Office of Science User Facility, as part of the Community +Sequencing Program (contract no. DE-AC02- 05CH11231). +Availability of data and materials +The raw sequence reads of the five metagenomes have been deposited to +the NCBI Sequence Read Archive (see Additional file 1: Table S6 for accession +numbers and read and contig statistics). The final 871 MAGs described in this +paper have been deposited as Whole Genome Shotgun projects at DDBJ/ +EMBL/GenBank, and accession numbers are listed in Additional file 4 +(BioProject ID PRJNA434545). All versions described in this paper are version +XXXX01000000. The cleaned and dereplicated amplicon sequence datasets +are available in FigShare (https://figshare.com/s/7684627445e3621aba24). +Maximum likelihood trees based on the concatenated alignment of 16 +ribosomal proteins, basis for Figs. 2 and 3, in newick format (.tre file) and +complementary datasets (used to plot completeness, contamination, +genome recovery size, G + C mol% and RPKG in iTOL), as well as K number +assignments for the predicted proteins of all MAGs (KEGG-orthologues, +Ghost Koala) and the fully annotated CPR MAGs supporting the conclusions +of this article are also available in FigShare (https://figshare.com/s/ +7684627445e3621aba24). +Authors’ contributions +GM and DYS initiated this study and were responsible for the fieldwork, +sample preparation, and sequencing effort. CDV conceptualized the research +goals under supervision of DYS and GM, and performed the bioinformatics +analysis under close guidance of A-SA and RG. CDV is the primary author of +this manuscript. MM, RG, and CDV prepared the main figures. All authors +read and approved the final manuscript. +Ethics approval and consent to participate +Not applicable. +Vavourakis et al. Microbiome (2018) 6:168 +Page 15 of 18 + +Consent for publication +Not applicable. +Competing interests +The authors declare that they have no competing interests. +Publisher’s Note +Springer Nature remains neutral with regard to jurisdictional claims in +published maps and institutional affiliations. +Author details +1Microbial Systems Ecology, Department of Freshwater and Marine Ecology, +Institute for Biodiversity and Ecosystem Dynamics, Faculty of Science, +University of Amsterdam, Postbus 94248, 1090, GE, Amsterdam, the +Netherlands. 2Department of Aquatic Microbial Ecology, Institute of +Hydrobiology, Biology Centre CAS, Na Sadkach 7, 370 05 Ceske Budejovice, +Czech Republic. 3Winogradsky Institute of Microbiology, Research Centre of +Biotechnology, Russian Academy of Sciences, 60 let Oktyabrya pr-t, 7, bld. 2, +Moscow, Russian Federation117312. 4Environmental Biotechnology, +Department of Biotechnology, Delft University of Technology, Van der +Maasweg 9, 2629, HZ, Delft, the Netherlands. +Received: 23 June 2018 Accepted: 3 September 2018 +References +1. +Sorokin DY, Berben T, Melton ED, Overmars L, Vavourakis CD, Muyzer G. +Microbial diversity and biogeochemical cycling in soda lakes. 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Microbiome (2018) 6:168 +Page 18 of 18 + diff --git a/kb_47/content/tmp_files/load_file.txt b/kb_47/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..eabb20eb978ce0d9e68e7f3644fccc69e2a8c85c --- /dev/null +++ b/kb_47/content/tmp_files/load_file.txt @@ -0,0 +1,1051 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf,len=1050 +page_content='RESEARCH Open Access A metagenomics roadmap to the uncultured genome diversity in hypersaline soda lake sediments Charlotte D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Vavourakis1 , Adrian-Stefan Andrei2†, Maliheh Mehrshad2†, Rohit Ghai2, Dimitry Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Sorokin3,4 and Gerard Muyzer1* Abstract Background: Hypersaline soda lakes are characterized by extreme high soluble carbonate alkalinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Despite the high pH and salt content, highly diverse microbial communities are known to be present in soda lake brines but the microbiome of soda lake sediments received much less attention of microbiologists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Here, we performed metagenomic sequencing on soda lake sediments to give the first extensive overview of the taxonomic diversity found in these complex, extreme environments and to gain novel physiological insights into the most abundant, uncultured prokaryote lineages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Results: We sequenced five metagenomes obtained from four surface sediments of Siberian soda lakes with a pH 10 and a salt content between 70 and 400 g L−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' The recovered 16S rRNA gene sequences were mostly from Bacteria, even in the salt-saturated lakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Most OTUs were assigned to uncultured families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' We reconstructed 871 metagenome-assembled genomes (MAGs) spanning more than 45 phyla and discovered the first extremophilic members of the Candidate Phyla Radiation (CPR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Five new species of CPR were among the most dominant community members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Novel dominant lineages were found within previously well-characterized functional groups involved in carbon, sulfur, and nitrogen cycling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Moreover, key enzymes of the Wood-Ljungdahl pathway were encoded within at least four bacterial phyla never previously associated with this ancient anaerobic pathway for carbon fixation and dissimilation, including the Actinobacteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Conclusions: Our first sequencing effort of hypersaline soda lake sediment metagenomes led to two important advances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' First, we showed the existence and obtained the first genomes of haloalkaliphilic members of the CPR and several hundred other novel prokaryote lineages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' The soda lake CPR is a functionally diverse group, but the most abundant organisms in this study are likely fermenters with a possible role in primary carbon degradation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Second, we found evidence for the presence of the Wood-Ljungdahl pathway in many more taxonomic groups than those encompassing known homo-acetogens, sulfate-reducers, and methanogens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Since only few environmental metagenomics studies have targeted sediment microbial communities and never to this extent, we expect that our findings are relevant not only for the understanding of haloalkaline environments but can also be used to set targets for future studies on marine and freshwater sediments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Keywords: Soda lake sediments, Metagenomics, Haloalkaliphilic extremophiles, Candidate Phyla Radiation, Wood-Ljungdahl pathway Correspondence: G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content='Muijzer@uva.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content='nl †Adrian-Stefan Andrei and Maliheh Mehrshad contributed equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' 1Microbial Systems Ecology, Department of Freshwater and Marine Ecology, Institute for Biodiversity and Ecosystem Dynamics, Faculty of Science, University of Amsterdam, Postbus 94248, 1090, GE, Amsterdam, the Netherlands Full list of author information is available at the end of the article © The Author(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' 2018 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content='0 International License (http://creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content='org/licenses/by/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content='0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' The Creative Commons Public Domain Dedication waiver (http://creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content='org/publicdomain/zero/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content='0/) applies to the data made available in this article, unless otherwise stated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Vavourakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Microbiome (2018) 6:168 https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content='1186/s40168-018-0548-7 MicrobiomeBackground Soda lakes are evaporative, athallasic salt lakes with low cal- cium and magnesium concentrations and a high-alkaline pH up to 11 buffered by dissolved (bi-) carbonate ions .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' They are constrained to arid regions across the globe, mainly the tropical East African Rift Valley , the Libyan Desert , the deserts in California and Nevada , and the dry steppe belt of Central Asia that spans to southern Si- beria, north-eastern Mongolia, and Inner Mongolia in China .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' On top of the extreme salinity and alkaline pH, the Eurasian soda lakes experience extreme seasonal temperature differences, causing highly unstable water re- gimes and fluctuating salinities .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Yet, soda lakes harbor diverse communities of haloalkaliphilic microbes, mostly prokaryotes that are well adapted to survive and grow in these extreme environments and consist of similar func- tional groups in soda lakes around the world .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' The relative abundance of different groups is typically governed by the salinity of the brine , and microbial-mediated nutrient cycles become partially hampered only at salt-saturating conditions .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' So far, all characterized prokaryotic lineages cultured from soda lakes comprise over 70 different species within more than 30 genera .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' From these, only a lim- ited number of genomes have been sequenced today, mostly from chemolithoautotrophic sulfur-oxidizing bac- teria belonging to the genus Thioalkalivibrio (class Gam- maproteobacteria) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' It is well established that metagenomics enables the recovery of genomes and the identification of novel genetic diversity where culturing ef- forts fail .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' In recent years, next-generation sequen- cing has recovered a massive number of genomes from previously unknown groups of prokaryotes , including a strikingly large and diverse group called “Candidate Phyla Radiation” (CPR), only distantly related to other cultured bacterial lineages .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Previously, we conducted a metagenomics study on soda lakes and re- constructed novel genomes from uncultured Bacteroidetes and “Candidatus Nanohaloarchaeaota” living in hypersa- line Siberian soda brines .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Here, we turned our atten- tion to the far more complex prokaryotic communities living in the sediments of the hypersaline soda lakes from the same region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' We give a broad overview of all the taxonomic groups sequenced and focus on the metabolic diversity found in the reconstructed genomes of the most abundant, uncultured organisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Results Overall prokaryote community structure The salinities from the studied soda lakes ranged from moderately hypersaline (between 70 and 110 g L−1) to salt-saturated (400 g L−1 salt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' The soluble carbonate al- kalinity was in the molar range, and the pH in all lakes was around ten (see Additional file 1: Table S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' To give an overview of the overall prokaryotic community com- position in each of the samples, we looked at the taxo- nomic classification of 16S rRNA genes recovered both by amplicon sequencing and direct metagenomics se- quencing (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' 1, see also Additional file 2: Figure S1; Additional file 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' The prokaryotic communities of all five sediment samples were highly diverse and consisted mostly of uncultured taxonomic groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Bacteria were more abundant than Archaea, regardless of the salinity of the overlaying brine (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Euryarchaeota were the second and third largest group in the sediments of the two salt-saturated lakes comprising ~ 10 and ~ 20% of the 16S rRNA genes in the metagenomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Most Euryarchaeota-related OTUs detected by amplicon se- quencing belonged either to the uncultured Thermoplas- mata group KTK 4A (SILVA classification) or the genera Halohasta and Halorubrum (class Halobacteria).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' In ac- cordance with cultivation-dependent studies , most OTUs assigned to methanogens were from the class Methanomicrobia, especially the lithotrophic genus Methanocalculus (up to ~ 3%) and the methylotrophic genus Methanosalsum (Additional file 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' The varying ratio of the three dominant bacterial groups, Firmicutes, Bacteroidetes (including the newly proposed phyla Rhodothermaeota and Balneolaeota ), and Gammaproteobacteria, showed no clear trend in relation to the salinity in the lakes, but when Firmicutes were domin- ant, Bacteroidetes were less abundant and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Most Firmicutes belonged to the order Clostridales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Uncultured members from the family Syntrophomonadaceae had a relative abundance of more than 5% in all five metagen- omes and comprised in two lakes even ~ 11–20% of the recovered amplicon sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' The second most abundant Firmicutes order was Halanaerobiales, particularly the genus Halanaerobium (family Halanaerobiaceae) and un- cultured members of the Halobacteroidaceae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' The majority of Bacteroidetes-related OTUs could not be assigned down to the genus level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' The uncultured ML635J-40 aquatic group (order Bacteroidales) comprised at least 5% of all five prokaryotic communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' This group has been previously found to be abundant in Mono Lake (a soda lake) and in an anoxic bioreactor degrading cyanobacterial biomass under haloalkaline conditions .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Two other highly abun- dant (up to ~ 8%) uncultured groups from the class Balneo- lia (proposed new phylum Balneolaeota ) were also detected in other soda lakes before .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Within the Gam- maproteobacteria, the genus Thioalkalivibrio was abundant (~ 3% of the total community), but also uncultured members of HOC36 were prevailing at moderate salinities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Members of the Deltaproteobacteria, Alphaproteobacteria, and Chloroflexi comprised up to ~ 10% of the detected 16S rRNA gene in some of the metagenomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' The GIF9 family of the class Dehalococcoidia was among the top three most abundant OTUs in two lakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' The extremely salt-tolerant Vavourakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Microbiome (2018) 6:168 Page 2 of 18 and alkaliphilic genera Desulfonatronobacter (order Desulfo- bacterales) and Desulfonatronospira (order Desulfovibrio- nales) were the dominant Deltaproteobacteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Highly abundant OTUs, within the Actinobacteria belonged to the class Nitriliruptoria and within the Alphaproteobacteria to the family Rhodobacteraceae and the genus Roseibaca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' The important nitrifying genus Nitrobacter (Alphaproteobacteria) was present in only one of the lakes with moderate salinity (Additional file 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Some bacterial top-level taxa appeared less dominant (< 5%) from the 16S rRNA genes recovered from the metagenomes but were represented mainly by a single highly abundant OTU in the amplicon sequences, in- cluding the haloalkaliphilic genus Truepera within the phylum Deinococcus-Thermus, the genus Spirochaeata within the phylum Spirochaetes, the family BSN166 within the phylum Ignavibacteriae, the BD2–11 terres- trial group within the Gemmatimonadetes, and the WCHB1–41 order within the Verrucomicrobia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' All OTUs within the Thermotogae and Lentisphaerae belonged to uncultured genera from the family Kosmoto- gaceae and Oligosphaeraceae, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' All Tenericu- tes-related OTUs belonged to the class Mollicutes, and especially the order NB1-n was dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' In contrast, the phylum Planctomycetes was relatively diverse, with at least 11 different genus-level OTUs spread over four class-level groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' High-throughput genome recovery We obtained 717 medium-quality (≥ 50% complete, < 10% contamination) and 154 near-complete (≥ 90% complete, < 5% contamination) metagenome-assembled genomes (MAGs) across three major prokaryote groups: Archaea, Bacteria, and CPR (see Additional file 4 and Additional file 2: Figure S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Figures 2 and 3 show the top-level phylogeny of all MAGs based on 16 ribosomal proteins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' The reference database used contains a repre- sentative for each major prokaryote lineage .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' We a b Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' 1 Abundant prokaryotic groups in five hypersaline soda lake sediments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' a Relative abundance of the top-level taxa (those with ≥ 1% abundance in at least one dataset) based on 16S rRNA reads in unassembled metagenomic datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' b Relative abundance of the 16S rRNA OTUs (those with sum of abundance in all datasets ≥ 3%) recovered by amplicon sequencing assigned where possible down to the genus-level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Three of the assessed soda lakes have a moderate salinity (70–110 g L−1), two are salt-saturated (400 g L− 1) Vavourakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Microbiome (2018) 6:168 Page 3 of 18 colored the different phyla from which we obtained a MAG in alternate blue and orange colors, and highlighted the MAGs obtained here in a darker shade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Many MAGs belonged to uncultured groups commonly detected in soda lake 16S rRNA gene surveys, over 100 MAGs still belonged to candidate prokaryote phyla and divisions that to our knowledge were never detected be- fore in soda lakes, including CPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Although only few MAGs had near-complete 16S rRNA genes, in most cases we were able to link available taxonomic gene an- notations and ribosomal protein phylogeny to the SILVA taxonomy of the OTUs assigned to the amplicon se- quences, while cross-checking the abundance profiles of both MAGs (Additional file 5) and OTUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' The soda lake CPR recovered from the metagenomes was restricted to a few distinct phyla within the Parcubacteria group, mostly affiliating with “Candidatus Nealsonbacteria” and “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Zambryskibacteria” (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' The first group of MAGs encompassed four different branches in our riboso- mal protein tree, suggesting a high-phylogenetic diversity, with 33 putative new species sampled here (ANI and con- DNA matrices given in Additional file 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' The “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Zambrys- kibacteria-”related MAGs consisted of at least five new species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Few MAGs were recovered from CPR groups also detected by amplicon sequencing (see Additional file 2: Figure S1), namely the “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Dojkabacteria” (former WS6), “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Saccharibacteria” (former TM7), CPR2, and “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Katanobacteria” (former WWE3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' 2 Maximum-likelihood phylogeny of the CPR and archaeal MAGs based on 16 ribosomal proteins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' The archaeal tree is unrooted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' The CPR tree is rooted to the Wirthbacteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Alternate orange and blue colors show phyla/classes from which we obtained MAGs (labeled as “Phyla present”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Reconstructed MAGs of this study are highlighted by darker shades (labeled as “MAG present”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Phyla/classes for which there was no representative in the reconstructed MAGs of this study are shown as gray cartoons (labeled as “Phyla not present”), and the numerical labels are represented at the bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Colored circles at the nodes show confidence percentage of the bootstraps analysis (100×) Vavourakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Microbiome (2018) 6:168 Page 4 of 18 Most archaeal MAGs belonged to the phylum Euryarch- aeota and the abundant classes Halobacteria, Methanomi- crobia, and Thermoplasmata (related to OTU KTK 4A) within.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' In addition, three Thermoplasmata-related MAGs that encoded for the key enzyme for methanogenesis (methyl-coenzyme M reductase, mcr) affiliated with refer- ence genomes from Methanomassilicoccales, the seventh order of methanogens have been recovered .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Another MCR-encoding MAG was closely related to the latest discovered group of poly-extremophilic, methyl-reducing methanogens from hypersaline lakes from the class Methanonatronarchaeia (related to OTU ST-12K10A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' We recovered also one MAG from the class Methanobacteria and a high-quality MAG from the WCHA1–57 group (“Candidatus Methanofastidiosa” ) in the candidate division WSA2 (Arc I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Several MAGs were recovered from the DPANN archaeal groups “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Diapherotrites,”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Aenigmarchaeota,”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' (see Additional file 2: Figure S3) and “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Woesearch- aeota” (former Deep Sea Hydrothermal Vent Group 6, DHVEG-6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Although we did not reconstruct any reasonable-sized MAGs from the TACK superphylum, we found several 16S rRNA genes on the assembled contigs that affiliated to the Thaumarchaeota (see Additional file 1: Table S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Nearly every known bacterial phylum had an extremo- philic lineage sampled from our hypersaline soda lake sediments (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' In most cases, the soda lake lineages clearly formed separate branches appearing as sister groups to known reference lineages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' The highest genome recovery was from the same top-level taxonomic groups that were also abundant in our 16S rRNA gene analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' From the Verrucomicrobia, most MAGs belonged to the order WCHB1-41 (16S rRNA gene identity 92–100%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' However, in our ribosomal protein tree, they branched within the phylum Lentisphaerae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Sixteen Tenericutes MAGs from at least 12 different species (Additional file 6) were closely related to the NB1-n group of Mollicutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Based on the recovered genome size and encoded meta- bolic potential, these organisms are free-living anaerobic fermenters of simple sugars, similar to what has recently been proposed for “Candidatus Izimaplasma” .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' 3 Maximum-likelihood phylogeny of the bacterial MAGs (CPR excluded) based on 16 ribosomal proteins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Alternate orange and blue colors show phyla/ classes from which we obtained MAGs (labeled as “Phyla present”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Reconstructed MAGs of this study are highlighted by darker shades (labeled as “MAG present”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Phyla/classes for which there was no representative in the reconstructed MAGs of this study are shown as gray cartoons (labeled as “Phyla not present”), and the numerical labels are represented at the bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Colored circles at the nodes show confidence percentage of the bootstraps analysis (100×) Vavourakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Microbiome (2018) 6:168 Page 5 of 18 Several MAGs belonged to the candidate phyla “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Omnitrophica,”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Atribacteria,”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' and “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Acetother- mia” (former OP1), which were moderately abundant also in some sediment (see Additional file 2: Figure S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' For the latter phylum, we suspect that four MAGs were more closely related to ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' div.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' WS1 and “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Lindow- bacteria” for which only few reference genomes are currently available in NCBI (see Additional file 2: Figure S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Due to a high-sequencing coverage, we also managed to reconstruct several MAGs from rare Bacteria (< 100 amplicon sequences detected, see Additional file 2: Figure S1), including the phyla “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Hydrogenedentes,”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Cloacimonetes,”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' div.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' BRC1, Elusimicrobia, Caldi- serica, and “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Latescibacteria.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' The MAGs from the latter phylum were more closely related to the recently proposed phylum “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Handelsmanbacteria” .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Two additional MAGs with 16S rRNA gene fragments with 94–95% identity to the class MD2898-B26 (Nitrospinae) were more likely members of ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' div.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' KSB3 (proposed “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Moduliflexus” , see Additional file 2: Figure S5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Draft genomes of haloalkaliphilic CPR Strikingly, members of the CPR related to “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Nealson- bacteria” and “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Vogelbacteria” were among the top 5% of abundant organisms in the surface sediments of the soda lakes, especially those with moderate salinity (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Like most members of the CPR, the MAGs of the four most abundant “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Nealsonbacteria” seem to be anaerobic fermenters .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' They lacked a complete TCA cycle and most complexes from the oxidative elec- tron transfer chain, except for the subunit F of a NADH-quinone oxidoreductase (complex I, nuoF, nuoG, nuoA) and coxB genes (complex II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' All CPR MAGs had a near-complete glycolysis pathway (Embden-Meyerhof- Parnas) encoded, but pentose phosphate pathways were severely truncated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' The commonly encoded F- and V-type ATPase can establish a membrane potential for symporter-antiporters by utilizing the ATP formed by substrate-level phosphorylation during fermentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' All CPR have V-type ATPases that can translocate Na+ in addition to H+ (see Additional file 2: Figure S6), while only two members of the “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Falkowbacteria” had puta- tive Na+-coupled F-type ATPases (see Additional file 2: Figure S7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' The coupling of ATP hydrolysis to sodium translocation is advantageous to maintain pH homeosta- sis in alkaline environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Interestingly, with only two exceptions , all CPR genomes recovered from other environments with neutral pH were reported to encode only F-type ATPases [28–32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' One low-abundant MAG affiliated to “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Peregrinibacteria” contained both the large subunit of a RuBisCO (type II/III, see Additional file 2: Figure S8) and a putative phosphoribu- lokinase (PRK, K00855) encoded in the same contig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' This is remarkable because PRK homologs were not previously identified among CPR, and RuBisCo form II/ III was inferred to function in a nucleoside salvage path- way .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' One “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Saccharibacteria” MAG encoded for a putative channelrhodopsin (see Additional file 2: Figure S9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' This is the first rhodopsin found among the CPR and suggests that this enigmatic group of organ- isms may have acquired evolutionary adaptations to a life in sunlit surface environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' A previous study showed that most CPR has coccoid cell morphotypes with a monoderm cell envelope resem- bling those from Gram-positives and Archaea but with a distinct S-layer .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Thick peptidoglycans coated with acidic surface polymers such as teichoic acids help pro- tect the cells of Gram-positives against reactive hydroxyl ions in highly alkaline environments (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' 5a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' All soda lake CPR had indeed the capability for peptidogly- can biosynthesis, but we found proteins typical for Gram-negatives for the biosynthesis of lipopolysaccha- rides (see Additional file 1: Table S3), homologous to the inner membrane proteins of type II secretion systems and to several proteins associated to the outer membrane and peptidoglycan, including OmpA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' It remains to be determined whether the soda lake CPR also lacks an outer membrane and perhaps anchor lipopolysaccharides, S-layer proteins, and lipoproteins to the inner cell membrane or peptidoglycan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' We also found gene encoding cardiolipin and squalene synthases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Increased levels of cardiolipin and the presence of squa- lene make the cytoplasmic membrane less leaky for protons .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' In addition, cation/proton exchangers are known to play a crucial role for pH homeostasis in alka- liphilic prokaryotes as they help acidify the cytoplasm during the extrusion of cations .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Putative Na+/H+ exchangers of the Nha-type and multi-subunit Mnh-type were found only within a few soda lake CPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Secondary active transport of K+ might be mediated in most soda lake CPR by KefB (COG0475)/kch Kef-type, glutathione- dependent K+ transport systems, with or without H+ antiport (67,68).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Various soda lake CPR had an acidic proteome, with pI curves resembling those found in extremely halophilic Bacteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Intracellular proteins enriched in acidic amino acids might be an adaptation to a “salt-in” strategy, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content='e., maintaining high intracellular potassium (K+) concentra- tions to keep osmotic balance (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' 5b; see Additional file 2: Figure S10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Such a strategy is energet- ically favorable over de novo synthesis or import of osmolytes such as ectoine and glycine betaine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' We did not find genes for the synthesis of organic osmolytes and homologs of ABC-type transporters for primary active uptake of proline/glycine betaine which were encoded only in one MAG (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' 5a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' For the “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Nealsonbac- teria” and “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Vogelbacteria,”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' the salt-in strategy might be a unique feature for the soda lake species explaining Vavourakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Microbiome (2018) 6:168 Page 6 of 18 their high abundance in the hypersaline soda lake sedi- ments, as we did not found an acidic proteome pre- dicted from genomes obtained from other non-saline environments (See Additional file 2: Figure S11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' The uptake of K+ ions remains enigmatic for most soda lake CPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Low-affinity Trk-type K+ uptake transporters (gen- erally with symport of H+) (67,68) were encoded only by a limited number of MAGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' We found three MAGs Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' 4 Relative abundance and metabolic potential of the dominant species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Abundance values, expressed as reads per kilobase of MAG per gigabase of mapped reads (RPKG), were averaged for the top ten abundant MAGs from each dataset that were (likely) the same species (Additional file 5, Additional file 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Population genomes were ranked by their “salinity preference scores”: those recruiting relatively more from moderate salinity datasets (cold colors) are drawn to the top, from high salinity datasets (warm colors) to the bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' The metabolic potential derived from functional marker genes (Additional file 7) is depicted by the colored symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' CBB = Calvin-Benson-Bassham cycle, DNRA = dissimilatory nitrite reduction to ammonia, fix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' = fixation, red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' = reduction, ox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' = oxidation, dis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' = disproportionation Vavourakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Microbiome (2018) 6:168 Page 7 of 18 a b Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' 5 (See legend on next page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=') Vavourakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Microbiome (2018) 6:168 Page 8 of 18 encoding for Kdp-type sensor kinases (kdpD) but no corresponding genes for the response regulator (kdpE) or for Kdp-ATPases that function as the inducible, high- affinity K+ transporters in other Bacteria (67,68).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Finally, mechanosensitive ion channels (mscS, mscL) and ABC- type multidrug transport systems (AcrAB, ccmA, EmrA, MdlB, NorM) and sodium efflux permeases (NatB) were encoded in almost every MAG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' The first might rapidly restore the turgor pressure under fluctuating salinity conditions by releasing cytoplasmic ions .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Novel abundant groups involved in sulfur, nitrogen, and carbon cycles A new species of Thioalkalivibrio (family Ectothiorhodospir- aceae) was by far the most abundant in the sediments of the two salt-saturated lakes (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' In the sediment of Bitter-1, also a purple sulfur bacterium from the same fam- ily was highly abundant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' It was closely related to Halorho- dospira, a genus also frequently cultured from hypersaline soda lakes .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' None of the abundant Ectothiorhodospira- ceae spp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' had already a species-representative genome sequenced (Additional file 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' The potential of the Thioalk- alivibrio spp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' for chemolithotrophic sulfur oxidation was evident (Additional file 7; see Additional file 8: Information S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Interestingly, the encoded nitrogen metabolisms were quite versatile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' While Thioalkalivibrio sp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' 1 had the poten- tial for nitrate reduction to nitrite, Thioalkalivibrio sp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' 2 might perform dissimilatory nitrite reduction to ammonia (DNRA; see Additional file 2: Figure S12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Two deltaproteobacterial lineages of dissimilatory sulfate-reducing bacteria (SRB) were highly abundant in the soda lake sediment of Bitter-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' One MAG from the family Desulfobacteraceae (order Desulfobacterales) is the first genome from the genus Desulfonatronobacter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' It encodes the genes for complete sulfate reduction to sul- fide using various electron donors, as well as for the complete oxidation of volatile fatty acids and alcohols, a unique feature for the genus Desulfonatronobacter among haloalkaliphilic SRB (see Additional file 8: Information S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Fumarate and nitrite (DNRA, NrfAH) could be used as alternative electron acceptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' The sec- ond dominant lineage was a new species from the genus Desulfonatronospira (family Desulfohalobiaceae, order Desulfovibrionales).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Like other members of this genus, it had the potential to reduce or disproportionate partially reduced sulfur compounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' In addition, it could also use nitrite as an alternative electron acceptor (NrfAH) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' A novel lineage of gammaproteobacterial SOB was highly abundant in the sediments of the moderately hy- persaline Cock Soda Lake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' It appeared as a sister group of the family Xanthomonadaceae in the ribosomal protein tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' This heterotrophic organism could conserve energy through aerobic respiration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' It might detoxify sulfide by oxidizing it to elemental sulfur (sqr) with subsequent re- duction or disproportionation of the polysulfides (psrA) chemically formed from the sulfur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' It also encoded the po- tential for DNRA (nrfA and napC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Genes likely involved in sulfide detoxification (sqr and psrA) were found also in several other abundant MAGs of heterotrophs, including one new abundant species from the family of Nitrilirup- toraceae (class Nitriliruptoria, phylum Actinobacteria).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' We found a wide variety of carbohydrate-active enzymes in these MAGs, such as cellulases (GH1 family) in addition to genes for glycolysis and TCA cycle and a chlorophyll/bacteriochlorophyll a/b synthase (bchG gene).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' The latter was also found in other Actinobacteria from the genus Rubrobacter .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' No evidence was found for nitrile-degrading potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' A second novel, uncultured lineage of Gammaproteo- bacteria that was highly abundant at moderate salinities branched in our ribosomal protein tree as a sister group to the family Halothiobacillaceae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' The MAGs encoded for a versatile metabolism typical for purple non-sulfur bacteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' The MAGs contained puf genes, bch genes, genes for carotenoid biosynthesis (not shown), and a Calvin cycle for photoautotrophic growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Alternatively, energy may be conserved through aerobic respiration, while acetate and proprionate could be taken up via an acetate permease (actP) and further used for acetyl-CoA biosynthesis and carbon assimilation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Since the sqr gene was present, but no dsr or sox genes, the organism might oxidize sulfide only to elemental sulfur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' One bin contained also nifDKH genes suggesting putative diazo- trophy, as well as a coenzyme F420 hydrogenase suggest- ing photoproduction of hydrogen .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' The abundant Euryarchaeota organism showed a clear preference for higher salinities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' We obtained one highly abundant MAG from the class Thermoplasmata that encoded a full-length 16S rRNA gene only distantly re- lated (91,2% identity, e value 0) to that of a member of the KTK 4A group found in a hypersaline endoevaporitic microbial mat .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' The abundant soda lake organism is likely a new genus and species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' All KTK 4A-related MAGs found here encoded for similar heterotrophic, fermentative metabolisms, with the potential for (See figure on previous page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=') Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' 5 Potential mechanisms for regulating the intracellular pH and cytoplasmic ion content in different CPR phyla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' a Membrane transporters, channels, and lipids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Peptidoglycan is depicted in gray and S-layer proteins in cyan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' b Predicted isoelectric points (bin width 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content='2) for the coding sequences of MAGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' A representative proteome is depicted for each phylum for which several members had a pronounced acidic peak (see also Additional file 2: Figure S11) Vavourakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Microbiome (2018) 6:168 Page 9 of 18 anaerobic formate and CO oxidation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' The KTK 4A might be also primary degraders since they encoded pu- tative cellulases (CAZY-families GH1, GH5) and chiti- nases (GH18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Interestingly, half of the MAGs encoded a putative chlorophyll/bacteriochlorophyll a/b synthase (bchG), which is highly unusual for Archaea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Although little can be inferred from the presence of only one marker gene, a functional bchG was previously also found in Crenarchaeota .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' The remaining two highly abundant Euryarchaeota-related MAGs belonged to a new species of Halorubrum (Additional file 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Key genes of the Wood-Ljungdahl pathway found in novel phylogenetic groups More than 50 MAGs were related to the family Syntro- phomonadaceae (class Clostridia, phylum Firmicutes) based on ribosomal protein phylogeny.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' All 16S rRNA gene sequences found in the MAGS had 86–95% iden- tity to sequences obtained from uncultured organisms related to the genus Dethiobacter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' While an isolated strain of Dethiobacter alkaliphilus is a facultative auto- troph that respires thiosulfate, elemental sulfur or polysulfides with hydrogen as an electron donor or disproportionates sulfur , other haloalkaliphilic members of the Syntrophomonadaceae are reverse acetogens, oxidizing acetate in syntrophy with a hydro- genotrophic partner .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Two populations (different species, Additional file 6) were especially abundant in Cock Soda Lake (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' They encoded for a full CODH/ACS complex, the key enzyme for the reductive acetyl-CoA or Wood-Ljungdahl pathway (WL) and a complete Eastern branch for CO2 conversion to 5-methyl-tetrahydrofolate (Additional file 9) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Acetogens use the WL to reduce CO2 to acetyl-CoA, which can be fixed into the cell or used to conserve en- ergy via acetogenesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Syntrophic acetate oxidizers, some sulfate reducing bacteria and aceticlastic methanogens run the WL in reverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Syntrophomonadaceae sp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' 2 encoded for a putative thiosulfate/polysulfide reductase as well as a phosphotransacetylase (pta) and an acetate kinase (ack) for the ATP-dependent conversion of acet- ate to acetyl-CoA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Although alternative pathways for the latter interconversion can exist, this second species has the complete potential for (reversed) acetogenesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Highly remarkable was the presence of a bacterial-type CODH/ACS complex and a near-complete eastern branch of the WL in a highly abundant species in Cock Soda Lake from the family Coriobacteriaceae (phylum Actinobacteria).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' This prompted us to scan all 871 MAGs for the presence of acsB encoding for the beta-subunit of the oxido-reductase module of CODH/ACS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' We con- firmed an encoded (near)-complete WL in several additional organisms belonging to phylogenetic groups not previously associated with this pathway (Additional file 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' We removed the Coriobacteriaceae acsB genes from the final dataset to construct a phylo- genetic tree since they were < 500 aa (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' 6) but found seven MAGs from the OPB41 class within the Actino- bacteria (16S rRNA gene fragment identity 94–96%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' The eastern branch of WL can function independently in folate-dependent C1 metabolism , but the pres- ence of the Western-branch in a phylum that comprises mostly aerobic isolates is very surprising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' The WL in combination with the potential for acetate to acetyl-CoA interconversion (pta/ack) and a glycolysis pathway were also present in the soda lake MAGs from the phyla “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Handelsmanbacteria,”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Atribacteria” (latter branched within the “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Acetothermia”), and the class LD1-PA32 (Chlamydiae), suggesting all these uncultured organisms might be heterotrophic acetogens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' However, it should be noted that a PFOR typically connecting glycolysis to the WL was only encoded in the LD1-PA32 MAGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' More- over, from the genetic make-up alone, it cannot be excluded that acetate is activated, and the WL run in reverse for syntrophic acetate oxidation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Finally, the novel acsB genes from soda lake Halanaerobiaceae, Natranaerobiaceae, and Halobacteroidaceae (Firmicutes) and from Brocadiaceae and Planctomycetaceae (Plancto- mycetes) disrupt the previously proposed dichotomy between Terrabacteria and Gracilicutes bacterial groups unifying 16S rRNA and acsB gene phylogenies and suggest a far more complex evolutionary history of the WL pathway than previously anticipated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Discussion Extensive classical microbiology efforts have been already undertaken to explore the unique extremophilic microbial communities inhabiting soda lakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' These un- covered the presence of most of the functional groups participating in carbon, nitrogen, sulfur, and minor element cycling at haloalkaline conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' The main re- sults of this work are summarized in several recent re- views .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Since most microbes, including those living in soda lakes, still evade all cultivation ef- forts, a very effective way to discover new microbes and assess their physiology based on their genetic repertoire is either through single cell genomics or by directly se- quenced environmental DNA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' This exploratory metage- nomics study performed on soda lake sediments effectively overcame the existing cultivation bottleneck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' First, we expanded the known diversity of CPR consider- ably with the first genomes of poly-extremophiles sam- pled from soda lake sediments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Although the presence of 16S rRNA genes from CPR in marine sediments and hy- persaline microbial mats was previously shown , until now, CPR MAGs were mainly obtained from deep, subsurface environments [15, 26, 29, 32, 49–52], and hu- man microbiota .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Despite being highly abundant Vavourakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Microbiome (2018) 6:168 Page 10 of 18 100 % 90-100 % 70-90 % 50-70 % some MAGs all MAGs Bootstraps Genes present Glycolysis (EMP) PFOR WL-Eastern branch H4MPT TH4 WL-Western branch CODH/ACS Acetogenesis/ acetate activation (pta/ack) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content='4 PVC group (Chlamydiae LD1-PA32) Syntrophorhabdus aromaticivorans PVC group bacterium CSSed11_184 Aerophobetes bacterium SCGC_AAA255-F10 Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Acetothermia Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Handelsmanbacteria Planctomycetaceae Anaerolineae Firmicutes Brocadiaceae Planctomycetes Methanomassiliicoccales Halobacteroidaceae Natranaerobiaceae Methanomicrobiales Desulfonatronospira Firmicutes Dehalococcoidia Armatimonadetes bacterium CSP1-3 Deltaproteobacteria Thermodesulfobacteria Desulfobulbaceae Halanaerobiaceae Nitrospirae Actinobacteria (OPB41) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' 6 Maximum likelihood phylogeny of the bacterial-type acetyl-coA synthases (acsB) found in the MAGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Only sequences ≥ 500 aa were included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Lineages for which we discovered the Wood-Ljungdahl (WL) in this study are highlighted in orange, and the presence of genes in the respective MAGs related to WL, glycolysis, pyruvate, and acetate conversion is indicated by the colored symbols (see also Additional file 9: Dataset S6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Additional lineages found in this study are marked in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' The three was rooted according to .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Circles at the nodes show confidence percentage of the bootstraps analysis (100×).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' EMP = Embden-Meyerhof-Parnas, PFOR = pyruvate:ferredoxin oxidoreductase complex, pta = phosphotransacetylase gene, ack = acetate kinase gene, H4MPT = tetrahydromethanopterin-linked pathway, TH4 = tetrahydrofolate pathway, CODH/ACS = carbon monoxide dehydrogenase/acetyl-CoA synthase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' PVC group bacterium CSSed11_184 is likely a member of the WCHB1-41 class within the Verrucomicrobia Vavourakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Microbiome (2018) 6:168 Page 11 of 18 here, CPR went unnoticed in previous amplicon sequen- cing studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' This might be due to the fact that many CPR representatives have random inserts of various length in their 16S rRNA genes or due to primer mis- matches .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' This illustrates also that direct metage- nomics should not only be preferred over amplicon sequencing to infer functional potential, but the former is far more effective for the discovery of novel organ- isms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Second, we obtained many more genomes from “traditional” bacterial phyla such as the Planctomycetes and Chloroflexi, as well as candidate phyla, for which no soda lake isolates, hence no genomes were previously obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Third, even within the sulfur cycle, the most active and frequently studied element cycle in soda lakes , we found considerable metabolic novelty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Finally, we found the Wood-Ljungdahl pathway in several novel phyla, not closely related to any known acetogens, methanogens, or sulfate-reducing bacteria .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' The lat- ter shows that our sequencing recovery effort has also significantly contributed to the discovery of metabolic novelty within various prokaryote phylogenetic groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Salinity is often considered to be the major factor shaping prokaryote community composition in diverse habitats .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Extreme halophilic Euryarchaeota seem to be always the dominant group in salt-saturated hypersaline brines, both those with neutral or alkaline pH .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Here, we found that although these haloarchaea are still relatively more abundant in the sed- iments exposed to brines with salt-saturating conditions, the clear majority of microbes in all investigated hyper- saline soda lake sediments are Bacteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' It could be hypothesized that the sediment is a hide-out for the extreme alkalinity and salinity governing the water column, and that sediment stratification, especially in the anoxic part, offers plenty of opportunities for niche diversification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' On the other hand, it should no longer be a surprise that soda lakes are such productive and biodiverse systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' First, it has been previously elaborated that soda lake organisms are exposed to approximately half the osmotic pressure in sodium carbonate-dominated brines compared to sodium chloride-dominated brines with the same Na+ molarity .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Second, nitrogen limitation in the community can be overcome when many members contribute to the fixation of atmospheric N2, and various forms of organic nitrogen are efficiently recycled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' The soda lakes exam- ined in this study were also eutrophic, and sulfur com- pounds were abundant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Sulfide is also far less toxic at high pH as it mostly occurs in the form of bisulfide (HS−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Besides the evident high metabolic and taxo- nomic diversity of dissimilatory sulfur-cycling bacteria, a diverse heterotrophic community can be sustained com- prising both generalist and very specialized carbon de- graders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Less eutrophic soda lakes might not suffer from carbon limitation either, due to a presence of high-bicarbonate concentrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' These effectively elim- inate the inorganic carbon limitation for primary pro- ducers who are highly active in soda lakes, especially Cyanobacteria .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Third, light that penetrates the surface of the sediment seems to stimulate oxygenic and anoxygenic phototrophic growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Moreover, various het- erotrophs, such as the rhodopsin-containing haloarchaea and Bacteroidetes, have the option to tap into this un- limited energy source for example to help sustain the costly maintenance of osmotic balance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Unexpectedly, we even found the first rhodopsin encoded by a member of the CPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Fourth, tight syntrophic relations, as pro- posed for CPR members and Syntrophomonadaceae spp., might be the solution to successful growth in an energetically challenging environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Since our metagenomes are snapshots in time and space, the failure to reconstruct specific MAGs gives no conclu- sive evidence for the absence of certain microbial-mediated element transformation in hypersaline soda lake sediments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Additionally, technical limitations of the assembly and bin- ning of highly micro-diverse genome populations might hamper genome recovery .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' More importantly, the abundance of a specific microbe is not necessarily corre- lated to the importance of its performance in an ecosystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Many metabolic capacities are redundant, and often key transformations are reserved for a few rare organisms that might proliferate for a short time-span when specific condi- tions allow for it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' For example, although no MAGs were re- covered from chemolithoautotrophic nitrifiers , we did detect a Nitrobacter-related OTU by amplicon sequencing and assembled 16S rRNA genes from Thaumarchaeota, suggesting bacterial and archaeal nitrifiers are present in the surface sediments of soda lakes at very low abundance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Finally, the method of DNA isolation might impact the community composition apparent in the final metagenome sequenced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Environmental samples contain complex mix- tures of different organisms, and it is impossible to find a protocol where the DNA from every single organism is ex- tracted as efficiently without compromising the final quality of the extracted DNA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' However, since we find all the im- portant taxonomic and functional groups known from pre- vious cultivation-dependent studies back in either our amplicon sequencing datasets or our directly sequenced metagenomes, we are confident that the community com- position and the MAGs presented here are representative for the microbiomes of the soda lake sediments in the Kulunda Steppe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Conclusion Years of intensive microbiological research on soda lakes seem to have paid off, since many of the described gen- era we could detect here have a cultured representative from soda lakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' However, as many of the abundant Vavourakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Microbiome (2018) 6:168 Page 12 of 18 lineages and groups found in soda lake sediments are still uncultured, metagenomics proved to be a helpful tool to gain primary insights in the potential physiology and ecology of these poly-extremophilic prokaryotes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' We reconstructed the first genomes for many of such organisms and proposed new functional roles for the most abundant ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Future studies should provide more in depth analyses of these genomes, especially from the less abundant organisms that might perform key ecological processes, such as methanogens and nitri- fiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' In addition, they should focus on gaining physio- logical culture-based evidence or proof for in situ activity for the abundant organisms described here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' The key metabolic insights provided by this metagenomics study can lead to the design of new cultivation strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' In general, sediment communities are far more complex than those found in the corresponding water column and are therefore often considered too complex for efficient metagenomic analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Many of the novel lineages found here may therefore have related neutro- philic lineages in marine and freshwater sediments that await discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' We demonstrate here that, by providing sufficient sequencing depth, the “state of the art metage- nomics toolbox” can effectively be used on sediments as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Methods Site description and sample collection The top 10 cm sediments from four hypersaline, eutrophic soda lakes located in the Kulunda Steppe (south-western Siberia, Altai, Russia) were sampled in July of 2010 and 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' General features and exact location of the sampled soda lakes are summarized in Additional file 1: Table S1; a map of the area was published previously .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Cock Soda Lake (a stand-alone lake, sampled both in 2010 and 2011) and Tanatar-3 (Tanatar system) were moderately hypersa- line (~ 100 g L−1) with sandy sediment, while Tanatar-1 and Bitter-1 (Bitter system) were salt-saturated (400 g L−1) with sulfide-rich sapropel sediments underlined by rock trona deposits .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Especially, Bitter-1 harbors a very active microbial community, probably due to its high- organic and -mineral content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Surface sediments were col- lected by a plastic corer into sterile glass containers and transported to the laboratory in a cooler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' DNA isolation, 16S rRNA amplicon, and metagenomic sequencing The colloidal fraction of each sediment sample (~ 10% of 50 g) was separated from the course sandy fraction by several short (30–60 s) low-speed (1–2,000 rpm in 50 mL Falcon tubes) centrifugation steps and washed with 1–2 M NaCl solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' The pelleted colloidal sedi- ment fraction was first subjected to 3 cycles of freezing in liquid nitrogen/thawing, then re-suspended in 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content='1 M Tris (pH 8)/10 mM EDTA, and then subjected to harsh bead beating treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Next, the samples were incu- bated with lysozyme (15 mg/mL) for 2 h at 37 °C followed by a SDS (10% w/v) and proteinase K (10 μg/ mL) treatment for 30 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' at 45 °C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' High molecular weight DNA was isolated using phenol/chloroform ex- traction, quality-checked, and sequenced as described previously .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Direct high-throughput sequencing of the DNA was performed on an Illumina HiSeq 2000 plat- form to generate 150 b paired-end reads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Amplification of the V4-V6 region of prokaryote 16S rRNA genes using barcoded 926F-1392R primers, amplicon purifica- tion, quantification, and Roche (454)-sequencing was performed together in a batch with brine samples from the same sampling campaigns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Barcodes and adapter se- quences were removed from de-multiplexed amplicon sequence reads and analyzed with the automated NGS analysis pipeline of the SILVA rRNA gene database pro- ject (SILVAngs 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content='3, database release version 128) using default parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' The OTUs (97% identity) assigned down to the genus level were only considered when they had a relative abundance ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content='1% in at least one of the five datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Processing metagenomics reads, assembly, binning, and post-binning Metagenomic raw reads were quality trimmed using Sickle (version 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content='33), and only reads ≥ 21 b were retained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' The prokaryotic community structure at taxo- nomic top levels was extrapolated from ten million ran- domly sampled singletons from each dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Candidate 16S rRNA fragments > 90 b were identified and compared against the SILVA SSU database 128 (blastn, min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' length 90, min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' identity 80%, e value 1e-5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' To ver- ify that the microbial community composition was in- deed mostly prokaryotic, we did a more general screening of the metagenomics reads that identified also candidate 18S rRNA fragments > 90 b (see Additional file 1: Tables S4-S5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' The complete trimmed read sets were assembled into contigs ≥ 1 kb with MEGAHIT (v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content='3–6-gc3983f9) using paired-end mode, k min = 21, k max = 131, k step = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Genes were predicted using Prodigal (v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content='2) and RNAs with rna_hmm3 and tRNAscan-SE .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Assembled 16S rRNA sequences were compared to a manually curated version from the SILVA SSU database (e value ≥ 1e-5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Predicted protein sequences were annotated against KEGG with GhostKOALA (genus_prokaryotes + family_eukaryotes + viruses) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Marker genes for central metabolic pathways and key environmental element transforma- tions were identified based on K number assignments [15, 69–71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Contigs ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content='5 kb were binned with METABAT (superspecific mode) based on differential coverage Vavourakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Microbiome (2018) 6:168 Page 13 of 18 values obtained by mapping all five trimmed readsets to all five contig sets with Bowtie2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' The bins were sub- jected to post-binning (an overview of the workflow is given in Additional file 2: Figure S13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Bins were assessed with lineage-specific single copy genes using CheckM and further processed with the metage- nomics workflow in Anvi’o (v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Since Candidate Phyla Radiant (CPR) is not included in the CheckM ref- erence trees and are likely to have low-genome com- pleteness, we used an existing training file of 797 CPR genomes to identify putative CPR bins .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Bins with CheckM-completeness ≥ 50% (884 out of 1778) and an additional four CPR bins were further processed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Coding sequences were annotated for taxonomy against NCBI-nr (July, 2017) with USEARCH (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content='32) to verify that most hits in each bin were to prokaryotic ref- erences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Phage or viral contigs were manually removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Genome contamination (redundancy) was estimated based on marker sets of universal single copy genes identified for Bacteria and Archaea as imple- mented in Anvi’o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Genome coverage was obtained by mapping trimmed reads with BBMap v36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content='x (kfilter 31, subfilter 15, maxindel 80).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Bins with ≥ 5% redun- dancy were further refined with Anvi’o using circle phy- lograms (guide trees tnf-cov: euclidian ward) and scanned again for CPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Post-binning resulted in a total of 2499 metagenome-assembled genomes (MAGs), of which 871 were either medium-quality genome drafts (CheckM estimated completeness ≥ 50% and contamin- ation ≤ 10% , Additional file 4) or lower quality draft genomes from CPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Phylogeny of the MAGs was assessed based on 16 single-copy ribosomal proteins and representative refer- ence genomes of major prokaryote lineages across the tree of life .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Individual ribosomal proteins in our MAGs were identified by K number assignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Only ribosomal proteins ≥ 80 aa were considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Initial maximum-likelihood (ML) trees were constructed to de- termine which organisms belonged to the Archaea, Bac- teria, or CPR with FastTree 2 (WAG + CAT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Final separate trees for the three distant evolutionary groups were constructed in the same manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Each ribosomal protein set was aligned separately with MAFFT (v7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content='055b, − auto) and concatenated only if a MAG encoded at least 8 out of 16 proteins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' For all trees, a 100× posterior bootstraps analysis was performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Phylogenetic trees were visualized together with gen- ome statistics and abundance information using iTOL .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' We cross-checked the taxonomic assignments based on the phylogeny of the ribosomal protein cas- sette with the top hit contig annotations against NCBI-nr and with the reference lineage obtained with CheckM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Lastly, we manually corrected the MAGs for misplaced 16S rRNA genes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' The final trees presented in the manuscript were redrawn using FigTree v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content='3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Detailed genome analyses CPR MAGs were re-annotated more thoroughly: genes were predicted with Prokka , and functional predictions were performed by running InterProScan 5 locally on the supplied COG, CDD, TIGRFAMs, HAMAP, Pfam, and SMART databases .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' BLAST Koala was used for KEGG pathway predictions .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' To find putative carbohydrate-active enzymes in all final MAGs, we used the web-resource dbCAN to annotate all predicted proteins ≥ 80 aa against CAZy .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' To identify the top ten abundant MAGs from each re- spective dataset, ten million randomly sampled single- tons were mapped onto each MAG with a cut-off of 95% identity in minimum of 50 bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Coverage values were additionally normalized for genome size and expressed as reads per kilobase of sequence per gigabase of mapped reads (RPKG) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' A positive score (from 871 to 1) was assigned to each MAG according to the rank- ing of the summed RPKG of MAGs in the high-salinity datasets (B1Sed10 and T1Sed) and a negative score ac- cording to the ranking of the summed RPKGs in the moderate salinity datasets (CSSed10, CSSed11, T3Se d10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Both scores were summed to get a “salinity prefer- ence score” with MAGs recruiting preferably from high salinity datasets on the positive end, moderate salinity datasets in the negative end, and those without prefer- ence in the middle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' We determined species delineation for the most abundant MAGs and their closest reference genomes (NCBI-nr) by Average Nucleotide Identity (ANI) and conserved DNA-matrices, as follows : ANI ≥ 95%, conDNA ≥ 69% = same species, ANI ≥ 95%, condDNA < 69% = might be same species, ANI < 95%, condDNA < 69% = different species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Single gene trees based on maximum likelihood were constructed with un- trimmed alignments (MAFFT, L-INS-i model) and FastTree 2 (WAG + CAT, increased accuracy, -spr4 mlacc 2 -slownni) using 100× bootstraps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' References were pulled from eggNOG (v4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content='1) and supple- mented with sequences from NCBI-nr or refined according to [7, 33, 46, 92–94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' The curated MAGs were scanned for the presence of rhodopsin sequences with the hmmsearch software and a profile hidden Markov model (HMM) of the bacteriorhodopsin-like protein family (Pfam accession number PF01036).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' The identified sequences with significant similarity were aligned together with a curated database composed of a collection of type-1 rhodopsins, using MAFFT (L-INS-i accuracy model) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' This protein alignment was further utilized to Vavourakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Microbiome (2018) 6:168 Page 14 of 18 construct a maximum likelihood tree with 100× boot- strap with FastTree 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' All other genes were identified using the KEGG annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Additional files Additional file 1: Table S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' General features of the four sampled soda lakes at time of sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Table S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' SILVA classification of the 16S rRNA gene sequences found in all ≥1 kb contigs of five soda sediment metagenomic datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Table S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Enzymes involved in lipopolysaccharide biosynthesis found among different members of the CPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Table S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Sub-kingdom classification of candidate SSU rRNA gene fragments found in subsamples of 10 million random forward reads from the five soda sediment metagenomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Table S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Top-level taxonomic classification of the 18S rRNA gene fragments found in subsamples of 10 million random forward reads from the five soda sediment metagenomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Table S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Description of the metagenomic datasets, NCBI Sequence Read Archive (SRA) accession numbers and general statistics of the assembled contigs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' (PDF 740 kb) Additional file 2: Figure S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Taxonomic fingerprints determined by 16S rRNA gene amplicon sequencing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Figure S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Genome statistics of the 871 MAGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Figure S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Phylogeny of MAGs belonging to “Candidatus Aenigmarchaeota” and “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Nanohaloarchaeota”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Figure S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Phylogeny of MAGs related to “Candidatus Acetothermia”, candidate division WS1 and “Candidatus Lindowbacteria”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Figure S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Phylogeny of MAGs related to candidate division KSB3 and “Candidatus Schekmanbacteria”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Figure S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Multiple sequence alignment of the V-type ATPase subunits K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Figure S7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Multiple sequence alignment of the F-type ATPase subunits c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Figure S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Maximum likelihood tree of the large subunits of RuBisCo and RubisCo- like proteins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Figure S9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Maximum likelihood tree of the putative rhodopsins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Figure S10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Predicted isoelectric points (pI) profiles of all MAGs from CPR members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Figure S11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Predicted isoelectric points profiles for members of the “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Nealsonbacteria” and “Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Vogelbacteria”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Figure S12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Multiple sequence alignment of the dissimilatory cytochrome c nitrite reductases (nrfA/TvNiR, K03385).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Figure S13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Overview of the post-binning workflow used for genome recovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' (PDF 6548 kb) Additional file 3: Dataset S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Relative abundance of the OTUs assigned to the genus-level within the Archaea, Bacteria and organelles from Eukaryota detected by 16S rRNA gene amplicon sequencing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' The OTUs with less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content='1% abundance accross all five datasets are not shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' The names of highly abundant genera (≥1% in at least one of the data- sets) are shown in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' (XLSX 24 kb) Additional file 4: Dataset S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Organism names, statistics and general description incl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Completeness and contamination estimates, phylogeny and DDBJ/EMBL/Genbank accession numbers of the metagenome assembled genomes (MAGs) described in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' All submitted versions described in this paper are version XXXX01000000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Size = recovered genome size, Completeness (Compl1), contamination (Cont), strain heterogenity (Str het) and Taxon CheckM were inferred from lineage-specific marker sets and a reference tree build with CheckM .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Additional completeness (compl2) and redundancy (red) estimates were inferred based on the presence of universal single copy genes for Bacteria and Archaea .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Decision and confidence intervals from the Candidate Phyla Radiation (CPR) scan are given, as well as the taxonomy of the besthit in SILVA when 16S rRNA genes were present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Phylum/class 16 ribosomal proteins is the taxonomy derived from our ribosomal protein trees (see main text: Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' 2 and 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' OTU gives the inferred link of a population genome with our 16S rRNA gene amplicon dataset (Additional file 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' (XLSX 253 kb) Additional file 5: Dataset S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Estimated abundance and derived salinity preference from each MAG in each metagenomic dataset expressed as Reads per Kilobase of MAG per Gigabase of mapped reads (RPKG) and “salinity preference score” (see Methods section), basis for Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' (XLSX 143 kb) Additional file 6: Dataset S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Average Nucleotide Identity (ANI) and conserved DNA (condna) matrices to determine species delineation between the most abundant MAGs shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' 4, closely related (less abundant) MAGs and NCBI reference genomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Decision matrix shows: 1 = same species, − 1 = might be same species, 0 = different species (see Methods section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' (XLSX 1161 kb) Additional file 7: Dataset S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Sheet 1 Presence and absence of marker genes and putative carbohydrate-active enzymes in the MAGs to infer putative roles in C, N and S element cycles based on K-number assignments and CAZy annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Sheet 2 Summary basis for Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' (XLSX 41 kb) Additional file 8: Information S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' More detailed description of the main metabolisms encoded by Thioalkalivibrio-related MAGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Information S2 More detailed description of the main metabolisms encoded by Deltaproteobacterial-related MAGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' (PDF 219 kb) Additional file 9: Dataset 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Sheet 1 shows the MAGs positive for the marker gene acsB (K14138) encoding an acetyl-CoA synthase (ACS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' The basis for Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' 6, namely presence and absence of key genes involved in the Wood-Ljungdahly pathway, acetogenesis, methanogenesis, glycolysis and pyruvate to CO2 conversion is shown for each MAG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Sheet 2 shows the MAGs positive for the marker gene cdhC (K00193) encoding for the beta subunit of an acetyl-CoA decarboxylase synthase complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' While acsB and cdhC correspond roughly to the Bacterial-type and Archaeal- type (methanogens) enzymes with the same function, we found few discrepancies between marker gene and genome phylogeny within the Methanomassiliicoccaceae and Chloroflexi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' (XLSX 52 kb) Acknowledgments We thank Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Nikolai Chernych for his technical assistance during the isolation and purification of metagenomics DNA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' We also thank the Department of Energy Joint Genome Institute for sequencing the metagenomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Funding CDV and GM were supported by the ERC Advanced Grant PARASOL (no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' 322551).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' A-SA and RG were supported by the research grant 17-04828S from the Grant Agency of the Czech Republic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' MM was supported by the Czech Academy of Sciences (Postdoc program PPPLZ application number L200961651).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' DYS was supported by the SIAM/Gravitation Program (Dutch Ministry of Education and Science, grant 24002002) and by the Russian Science Foundation (grant 16–14- 00121).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Sequencing was performed by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Department of Energy Joint Genome Institute, a DOE Office of Science User Facility, as part of the Community Sequencing Program (contract no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' DE-AC02- 05CH11231).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Availability of data and materials The raw sequence reads of the five metagenomes have been deposited to the NCBI Sequence Read Archive (see Additional file 1: Table S6 for accession numbers and read and contig statistics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' The final 871 MAGs described in this paper have been deposited as Whole Genome Shotgun projects at DDBJ/ EMBL/GenBank, and accession numbers are listed in Additional file 4 (BioProject ID PRJNA434545).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' All versions described in this paper are version XXXX01000000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' The cleaned and dereplicated amplicon sequence datasets are available in FigShare (https://figshare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content='com/s/7684627445e3621aba24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Maximum likelihood trees based on the concatenated alignment of 16 ribosomal proteins, basis for Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' 2 and 3, in newick format (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content='tre file) and complementary datasets (used to plot completeness, contamination, genome recovery size, G + C mol% and RPKG in iTOL), as well as K number assignments for the predicted proteins of all MAGs (KEGG-orthologues, Ghost Koala) and the fully annotated CPR MAGs supporting the conclusions of this article are also available in FigShare (https://figshare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content='com/s/ 7684627445e3621aba24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Authors’ contributions GM and DYS initiated this study and were responsible for the fieldwork, sample preparation, and sequencing effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' CDV conceptualized the research goals under supervision of DYS and GM, and performed the bioinformatics analysis under close guidance of A-SA and RG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' CDV is the primary author of this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' MM, RG, and CDV prepared the main figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' All authors read and approved the final manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Ethics approval and consent to participate Not applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Vavourakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Microbiome (2018) 6:168 Page 15 of 18 Consent for publication Not applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Competing interests The authors declare that they have no competing interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Author details 1Microbial Systems Ecology, Department of Freshwater and Marine Ecology, Institute for Biodiversity and Ecosystem Dynamics, Faculty of Science, University of Amsterdam, Postbus 94248, 1090, GE, Amsterdam, the Netherlands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' 2Department of Aquatic Microbial Ecology, Institute of Hydrobiology, Biology Centre CAS, Na Sadkach 7, 370 05 Ceske Budejovice, Czech Republic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' 3Winogradsky Institute of Microbiology, Research Centre of Biotechnology, Russian Academy of Sciences, 60 let Oktyabrya pr-t, 7, bld.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' 2, Moscow, Russian Federation117312.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' 4Environmental Biotechnology, Department of Biotechnology, Delft University of Technology, Van der Maasweg 9, 2629, HZ, Delft, the Netherlands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Received: 23 June 2018 Accepted: 3 September 2018 References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Sorokin DY, Berben T, Melton ED, Overmars L, Vavourakis CD, Muyzer G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Microbial diversity and biogeochemical cycling in soda lakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' Extremophiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' 2014;18:791–809.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_47/content/kb_47.pdf'} +page_content=' 2.' metadata={'source': 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a/o9E3T4oBgHgl3EQfLgkW/content/tmp_files/2301.04363v1.pdf.txt b/o9E3T4oBgHgl3EQfLgkW/content/tmp_files/2301.04363v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..4a5c012c62ed637b41b7998674604ac24c3e2125 --- /dev/null +++ b/o9E3T4oBgHgl3EQfLgkW/content/tmp_files/2301.04363v1.pdf.txt @@ -0,0 +1,1317 @@ +Systematic study of the effect of individual +rotational energy levels on the fusion cross-section +of 16O-based reactions of range 480 ≤ ZPZT ≤ 592 +Nishu Jain1, M. Bhuyan2 and Raj Kumar1 +1 School of Physics and Materials Science, Thapar Institute of Engineering and +Technology, Patiala-147004, Punjab, India +2 Center for Theoretical and Computational Physics, Department of Physics, Faculty +of Science, University of Malaya, Kuala Lumpur 50603, Malaysia +E-mail: nishujain1003@gmail.com +E-mail: bunuphy@um.edu.my +E-mail: rajkumar@thapar.edu +September 2022 +Abstract. +In heavy-ion fusion reactions, the enhancement in the sub-barrier fusion +cross-section has been observed as compared to the 1-Dimensional barrier penetration +model due to the coupling of many degrees of freedom to the relative motion. This +enhancement can be explained theoretically by including nuclear structure effects like +deformation and the coupling of relative motion among two colliding nuclei. +The +present work aims to investigate the effect of individual rotational energy levels on +the fusion cross-sections for 16O-based reaction systems, namely, 16O + 182,184,186W, +16O + 176,180Hf, 16O + 174,176Yb, 16O + 166Er, 16O + 148,152,154Sm, 16O + 150Nd +at energies below the fusion barrier. Using the CCFULL code, the effect of low-lying +rotational energy levels on the fusion cross-section for 16O induced reactions has been +investigated at energies below and around the Coulomb barrier. +The calculations +are performed by assuming the fixed value of diffuseness parameter a0 = 0.65 fm in +the Woods-Saxon nuclear potential and the other two parameters are optimised by +fitting the experimental data at the above barrier. Here we have determined the V0 +and r0 as a function of ZP ZT , where experimental cross-sections are available. From +our calculations, it is observed that the hexadecapole deformation (β4) with different +magnitudes has a significant influence on the fusion cross sections. For the case of the ++ve value of β4, beyond 10+, the rotational levels cease to contribute significantly and +also there is a significant difference between the contribution of sequential channels. +On the other hand, in the case of -ve β4, up to 6+ levels contribute significantly. +Furthermore, we have established an algebraic systematic of fitting, which one can use +to determine the parameters V0, r0 of Woods-Saxon nuclear potential within the range +of ZP ZT lie in between 480 ≤ ZP ZT ≤ 592. +arXiv:2301.04363v1 [nucl-th] 11 Jan 2023 + +Phys. Scr. (2022) +2 +1. Introduction +Over the last few decades, various theoretical and experimental efforts have been centred +on exploring the role of nuclear structure in the reaction dynamics [1, 2, 3]. During +the fusion process, the collision of an incident projectile and a target may lead to the +formation of a compound nucleus through a quantum tunnelling process across the +fusion barrier [1, 4, 5, 6]. Heavy-ion fusion reactions are accomplished in the crucial role +of extending the nuclear chart and synthesizing the heavy elements. Furthermore, some +heavy-ion fusion processes in lighter mass systems are pivotal to the reaction channels +that govern the elemental synthesis in stellar environments and energy production +[2, 3, 7]. Experimentally, it has been observed that there is a large enhancement in the +sub-barrier fusion cross-sections by considering the one-dimensional barrier penetration +model [8, 9, 10, 11, 12, 13] using bare nucleon-nucleon potentials. +Such sub-barrier +fusion enhancement could be elucidated by the coupling of relative motion degrees of +freedom with the internal degrees of freedom such as static deformations (rotational +nuclei) [9, 10], vibrational effect in the nuclear surface (spherical nuclei) [14, 15, 16], and +the neck formation [17]. However, the effects of nucleon transfer on fusion below the +barrier have not yet been completely separated [6, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27]. +Several works [1, 14, 28, 29, 30, 31, 32, 33, 34] have discussed the coupling between +the rotational state of the target nuclei. The internal degrees of freedom coupled with +the relative motion of colliding nuclei lowers the barrier. The significant decrease in +the fusion cross-section with respect to the experimental data, at energies below the +Coulomb barrier, is also known as fusion hindrance [4]. A certain amount of flux allows +the projectile to tunnel through the fusion barrier and fuse well into the target [1, 8] +mainly at below barrier energies. +Therefore, coupled-channel calculations become a +benchmark theoretical tool to understand fusion reaction dynamics. +Nuclear fusion is a complicated phenomenon because of the involvement of many +nucleons. +Therefore, it is tedious to handle the interaction potential between two +colliding nuclei, which play a key role in describing the fusion reaction dynamics. In +general, the interaction potential comprises long-range repulsive Coulomb potential, +centrifugal interaction, and short-range attractive nuclear potential terms. +Unlike +Coulomb and centrifugal potentials, the nuclear potential is not well established till +now. In the past few decades, various efforts have been made to provide a simple and +accurate form of the nuclear interaction potential [35, 36, 37, 38, 39, 40, 41]. Several +fits of the nuclear potentials exist in the literature [42, 43, 44, 45, 46] which either +kept the coupling off and/or included in the Woods-Saxon form while using the other +potentials. The traditional Woods-Saxon potential has been prominent and widely used +to probe the heavy-ion nuclear fusion dynamics [28, 47, 48, 49, 50, 51, 52, 53]. The +Woods-Saxon potential consists of three parameters namely; potential depth, range +and diffuseness parameter. +The diffuseness parameter is an essential component in +the parameterization of Woods-Saxon nuclear potential, as it describes the slopes of +the potential in the tail region, where fusion begins [54]. Remarkably, the small value + +Phys. Scr. (2022) +3 +of the diffuseness parameter a0 = 0.65fm is most suitable for a good description of +experimental data [12, 55] in the elastic scattering analysis while the large values of +the diffuseness parameter ranging from a = 0.75 to 1.5 fm provide the best fit for +several excitation functions in the above barrier region to explore the experimental +data [4, 5, 12, 55, 56]. As a result, the Woods-Saxon nuclear potential parameters are +employed to study the elastic scattering and heavy-ion fusion processes in the Coupled +channel approach (CCFULL), which provides a good description of below barrier fusion +[55, 56]. +It is known that the nuclear reactions are adequately affected by the entrance +channel parameters and the internal structure such as mass asymmetry, deformation, +and orientation of the colliding nuclei [57, 58, 59, 60], which are remarkably governed +to influence the anticipation of a compound nucleus system. +The significance of +shape degrees of freedom in sub-barrier fusion enhancement has been investigated +experimentally using the fusion of two colliding nuclei [61]. As we know, CCFULL is +one of the computational codes used to explore fusion dynamics, where a full description +of the Woods-Saxon nuclear potential parameters (V0, r0 and a0) for colliding nuclei is +essential. To understand the nuclear interaction potential for a recently synthesized or +anticipated target nuclei for 16O- induced reactions, one must fit the main ingredient +parameters for the nuclear potential as per the shape degrees of freedom. We will also +introduce an algebraic fitting of V0 and r0 by using the available known data. This +will be crucial for the theoretical calculations in predicting the fusion characteristics of +compound nuclei, which is an essential input for the upcoming experiments. +Furthermore, the study will include the significant role of each rotational energy +level (i.e. +2+, 4+, 6+, 8+, 10+ and 12+) in the enhancement of the fusion cross- +section mainly at energies below the Coulomb barrier for spherical and deformed nuclei. +Here, the fusion cross-section of 12 different 16O- induced reactions with rotational +target nuclei i.e., 16O+182,184,186W, 176,180Hf, 174,176Yb, 166Er, 148,152,154Sm, 150Nd will be +analyzed at sub-barrier energies within the static Woods-Saxon potential with standard +diffuseness parameter a0 = 0.65 fm. We have considered 16O as spherical [61, 62, 63] +to investigate the effect of the individual rotational energy level of target nuclei on the +fusion cross-section. It is worth mentioning that the low-lying rotational energy levels for +the chosen target nuclei follow the sequence of state with I = 0+, 2+, 4+, 6+, etc and the +excitation energies of I+ state is proportional to I(I +1)∗E2T/6 [64]. Moreover, a better +channel selection has been made for different signs of β-values [33, 59, 60, 65, 66] for +individual energy levels. In addition, a comparison will be made between the resulting +theoretical results and the available experimental data for the nuclear reactions under +consideration. +This paper is organized as follows: A brief description of the theoretical formalism used +in this work is given in Section 2. The results of the coupled channel calculations are +given in detail in Section 3. Section 4 summarizes and concludes this work. + +Phys. Scr. (2022) +4 +2. Theoretical Formalism +This section provides a brief description of Coupled Channel approach (CCFULL) used +in the present study, which provides a reasonable understanding of the nuclear fusion +dynamics at energies around the barrier. Within this approach, multidimensional barrier +penetration is considered instead of single barrier penetration. This method is used +under the effect of coupling of the relative motion with the intrinsic degrees of freedom +of the interacting nuclei [4, 34, 67], mainly for the calculation of mean angular momenta +and the fusion cross-sections of the compound nucleus. +The traditional method for +addressing the effects of the coupling between relative motion and intrinsic degrees of +freedom on fusion is to numerically solve the coupled-channels equations, which includes +all relevant channels [68, 69]. The CC equations solve numerically within the Coupled +channel approach are given as, +� +−ℏ2 +2µ +d2 +dr2 + J(J + 1)ℏ2 +2µr2 ++ ZPZTe2 +r ++ VN + ϵn − Ec.m. +� +× ψn(r) + +� +m +Vnm(r)ψm(r) = 0. +(1) +Here r defines the radial part of relative motion between the participating nuclei and +µ is known as the reduced mass of the colliding system. ϵn is the excitation energy for +the nth channel and Ec.m. is the bombarding energy in the centre of the mass frame. VN +represents the nuclear potential and Vnm symbolizes the matrix elements of the coupled +Hamiltonian. +Since there are several Coupled channel equations, their dimension is +also large. +Thus, the rotating frame approximation or no-Coriolis approximation is +employed to reduce the dimension of Coupled channel equations [34, 67, 70, 71]. The +CC equations with non-linear coupling are significant in studying the heavy-ion fusion +reactions mainly at sub-barrier energies. All these sets of non-linear coupling are taken +into account. +The incoming wave boundary conditions (IWBC) [72] are also essential for the +solution of Coupled channel equation because IWBC or ingoing wave conditions are +quite sensitive for the potential pocket of the interaction fusion barrier. The incoming +wave of the entrance channel is present inside the barrier at the minimum position (r += rmin) and the outgoing wave of other channels are present at an infinite position. +By including the effect of the dominant intrinsic channels, the fusion cross-sections are +calculated as given below: +σJ(E) = σfus(E) = π +k2 +0 +� +J +(2J + 1)PJ(E). +(2) +Here, the total angular momentum ‘J’ is substituted in place of ‘ℓ’ for each channel by + +Phys. Scr. (2022) +5 +applying iso-centrifugal approximation by using the following equation: +⟨ℓ⟩ = +� +J +JσJ(E)/ +� +J +σJ(E) = +� +π +k2 +0 +� +J +J(2J + 1)PJ(E) +� +�� +π +k2 +0 +� +J +(2J + 1)PJ(E) +� +, +(3) +where PJ(E) is the total transmission coefficient. The Woods-Saxon form of nuclear +potential is used to analyze the nuclear structure effects [34, 67, 70, 71] and it is defined +as +VN = +−V0 +1 + exp +�� +r0 − R0 +� +/a0 +�, +(4) +where V0, r0, and a0 are the nuclear potential parameter. +In the Coupled channel approach, the rotational coupling with a pure rotor is taken +into consideration. One can generate the nuclear coupling Hamiltonian by changing the +target radius in the nuclear potential to a dynamical operator, +R0 → R0 + ˆO = R0 + β2RTY20 + β4RTY40. +(5) +Here RT is rcoupA1/3 and β2 and β4 are the quadrupole and hexadecapole deformation +parameters of the deformed target nucleus, respectively. Thus, the nuclear coupling +Hamiltonian is given by +VN(r, ˆO) = +−V0 +1 + exp +�� +r0 − R0 − ˆO +� +/a0 +�, +(6) +To connect the |n⟩ = |I0⟩ and |m⟩ = |I′0⟩ states of the target’s ground rotational band, +we need matrix elements of the coupling Hamiltonian. These are readily accessible using +matrix algebra [73]. In this algebra, the eigenvalues and eigenvectors of the operator ˆO, +which satisfies +ˆO | α >= λα | α > +(7) +This is implemented in the CCFULL program by diagonalizing the matrix ˆO, whose +elements are given by +ˆOII′ = +� +5(2I + 1)(2I′ + 1) +4π +β2RT +� +I +2 +I′ +0 +0 +0 +�2 ++ +� +9(2I + 1)(2I′ + 1) +4π +β4RT +� +I +4 +I′ +0 +0 +0 +�2 +. +(8) +The nuclear coupling matrix elements are then evaluated as +V (N) +nm = < I0 | VN(r, ˆO) | I +′0 > −V (0) +N (r)δn,m, += +� +α +< I0 | α >< α | I +′0 > VN(r, λα) − V (0) +N (r)δn,m. +(9) + +Phys. Scr. (2022) +6 +Table 1. +The parameters of Woods-Saxon potential (V0 & r0), the deformation +parameters (β2 > 0, β4 < 0) and the excitation energy corresponding to quadrupole +deformation of the nuclei [74, 75] used in the coupled channel calculations. +System +V0(MeV ) +r0(fm) +Target +E+ +2 (MeV ) +β2 +β4 +16O+182W +63.899 +1.165 +0.100 +0.2500 +-0.066 +16O+184W +63.987 +1.178 +0.111 +0.236 +-0.093 +16O+186W +70.0 +1.18 +0.122 +0.221 +-0.095 +16O+176Hf +63.627 +1.18 +0.088 +0.295 +-0.057 +16O+180Hf +70.5 +1.17 +0.093 +0.274 +-0.068 +16O+174Yb +63.53 +1.18 +0.076 +0.325 +-0.042 +16O+176Yb +60.0 +1.165 +0.082 +0.304 +-0.068 +The last term is included in the equation to avoid the diagonal component from being +counted twice. The linear rotational coupling approximation is used to calculate the +Coulomb matrix and explained as +V C +R(I,I′) = 3ZPZTR2 +T +5r3 +� +5(2I + 1)(2I′ + 1) +4π +× +� +β2 + 2 +7β2 +2 +� +5 +π +� � +I +2 +I′ +0 +0 +0 +�2 ++3ZPZTR4 +T +9r5 +� +9(2I + 1)(2I′ + 1) +4π +× +� +β4 + 9 +7β2 +2 +� � +I +4 +I′ +0 +0 +0 +� +, +(10) +for rotational coupling. +These coupled channel equations are used to calculate the +fusion cross-section of the compound nucleus by considering the coupling of all orders +as discussed in Sec. 3. +3. Results and Discussions +The fusion cross-sections have been calculated for 16O-induced nuclear reactions with +different rotational target nuclei, namely, +16O + +182,184,186W, +16O + +176,180Hf, +16O ++ 174,176Yb, +16O + 166Er, +16O + 148,152,154Sm, +16O + 150Nd with Coupled channel +calculations by adopting CCFULL code. +Various theoretical approaches have been +developed to generate the attractive nuclear potential for the description of fusion data +over wide energy ranges [76, 77, 78]. In this work, Woods-Saxon parameterization has +been taken into account to compute the interaction potential. The standard value of +diffuseness parameter a0 = 0.65 fm is used for the considered nuclear reactions [12, 55]. +The deformation parameter βλ connected with the transition of multipolarity λ were + +Phys. Scr. (2022) +7 +Total Potential (MeV) +50 +55 +60 +65 +70 +75 +R (fm) +5 +6 +7 +8 +9 +10 11 12 13 14 15 +16O + 182W +16O + 184W +16O + 186W +16O + 176Hf +16O + 180Hf +16O + 174Yb +16O + 176Yb +Figure 1. +(Color online) The variation of total interaction potential in the respect +of separation distance r (fm) for 16O+182,184,186W, 16O+176,180Hf, and 16O+174,176Yb +reactions under β2 > 0, β4 < 0 condition. +determined from experimental transition probabilities B(E2) by using the following +relation: +βλ = +4π +3ZRλ +� +B(Eλ) ↑ +e2 +, +(11) +where R = 1.2A1/3fm, and B(Eλ) ↑ is in units of e2b2. +For λ = 2, the values of +B(E2) ↑ [74] are experimental and independent of nuclear models. On the other hand, +the parameter βλ depends upon the nuclear model and determines the calculation of +deformation parameter. +In the present study, we aim to understand the effect of each rotational energy +level up to 12+ levels i.e., 2+, 4+, 6+, 8+, 10+ and 12+ in the enhancement of the +fusion cross-sections at below barrier energies. The calculations for the fusion cross- +sections have been exercised in steps (increment of 2+ state in each step) from 0+ to 12+ +channels. Nonetheless, we observed that for negative and positive β4 values, beyond 6+ +and 10+ respectively, the higher-order channels cease to contribute significantly towards +the fusion cross-section around and below the Coulomb barrier. Further, two different +conditions based on the shape of nuclei: (1) β4 < 0, and (2) β4 > 0 are considered. The +quadrupole (β2) and/or hexadecapole (β4) deformations of deformed nuclei are generally +taken into account while characterizing the rotation of the nuclei. It has been suggested +that the reactions containing nuclei with β4 show an enhancement in the fusion cross- +section at energies below the barrier [59, 79]. Our present calculations assume the 16O +(projectile) as spherical [59, 61, 62, 63, 80, 81] to determine the rotational effect of the +target nucleus on fusion cross-section effectively. It is important to note here that the +nucleon transfer channels are not considered in the present calculations, which may play +a significant role at energies below the Coulomb barrier [18, 19, 20]. + +Phys. Scr. (2022) +8 +3.1. For Hexadecapole deformation β4 < 0 +The near barrier and sub-barrier fusion cross-sections for 16O+182,184,186W, 176,180Hf, +174,176Yb systems have been analyzed systematically by employing CCFULL code us- +ing the static Woods-Saxon potential with β2 > 0, β4 < 0 of the target nuclei. The +Woods-Saxon parameterizations of Aky¨uz-Winther potential (AW), the values of the +deformation parameters (β2, β4) and the excitation energy corresponding to the first +excitation state are given in Table 1. The values of the Woods-Saxon potential param- +eters (V0, r0 & a0) are chosen to fit the experimental fusion cross-section at the above +barrier energies for the case of the 1-D barrier penetration model. The variation of the +total interaction potential at ℓ = 0ℏ with the separation distance ‘r’ for these systems +is also shown in Fig. 1. Here the total interaction potential is the sum of Woods-Saxon +potential in Eq. (4), and the Coulomb potential VC= ZP ZT e2 +r +. From the figure, one can +notice that the pocket formed for 16O+180Hf reaction is much deeper in comparison to +the others, which may result in a larger fusion probability. +The fusion cross-sections have been calculated using a 1D barrier penetration model for +these reactions. The one-dimensional barrier penetration data is represented by a solid +black line as shown in Fig. 2. From the figure, we found the obtained results under- +estimate the experimental data, especially at low barrier energies. In order to address +the fusion cross-section of the above-mentioned reactions, rotational degrees of freedom +have been included in the CCFULL calculations. Only the quadrupole deformation (β2) +is initially considered, and the calculations are performed along with various values of +β2 ranging from 0.221 to 0.324. The deformation parameters used in these reactions +decreases as the mass number increases. Up to 12+ channels are incorporated in each +system to observe the effect of each rotational level in the enhancement of the fusion +cross-section and/or to predict the number of channels that are good enough to converge +to the experimental data and beyond it the higher order ceases to contribute. Within +the addition of rotational channels corresponding to these levels, the enhancement in +the fusion cross-sections has been observed as compared to the single barrier penetration +case. The contribution of the rotational energy levels on the fusion cross-section up to +6+ state has been found to be significant. It has been observed that the ground state +quadrupole deformation alone is unable to reproduce the experimental data across the +below barrier energy region. +To address these issues, the hexadecapole deformation (β4) is included in the CC calcu- +lations, as illustrated in Fig 3. The estimated results using 1-D BPM are comparatively +lower than the experimental data. The coupled channel computations are performed for +this system including rotational energy levels up to the 12+ state of the target nuclei. A +significant change in the fusion cross-section is observed for 0+ to 2+ state as compared +to the 1-D BPM, and a small change for 2+ to 6+ states. Furthermore, we observe +that the cross-section with the inclusion of higher-order channels, beyond 6+, overlaps +with that of up to the 6+ state. The fusion cross-section for 16O + 182,184,186W reactions +are calculated with the β4 values ranging from -0.066 to -0.095. The calculated results + +Phys. Scr. (2022) +9 +10−1 +100 +101 +102 +103 +70 +80 +90 +100 +inert +0+ - 2+ +0+ - 4+ +0+ - 6+ +0+ - 8+ +0+ - 10+ +0+ - 12+ +Exp. +(a) +16O+182W +10−1 +100 +101 +102 +103 +60 +70 +80 +90 +100 +16O+184W (b) +10−1 +100 +101 +102 +103 +65 +70 +75 +80 +85 +90 +16O+176Hf +(d) +Cross-section σ (mb) +10−1 +100 +101 +102 +103 +60 +70 +80 +90 +100 +16O+180Hf (e) +10−1 +100 +101 +102 +103 +60 +70 +80 +90 +16O+174Yb (f) +10−1 +100 +101 +102 +103 +Ec.m. (MeV) +60 +70 +80 +90 +16O+176Yb (g) +10−1 +100 +101 +102 +103 +60 +70 +80 +90 +100 +16O+186W (c) +Figure 2. +(Color online) The fusion cross-sections estimated up to 12+ channels in +sequential manner as a function of Ec.m.(MeV) for 16O+182,184,186W, 16O+176,180Hf, +and 16O+174,176Yb nuclear reactions having quadrupole deformation and compared +with experimental data [62, 79, 82]. See the text for details. + +Phys. Scr. (2022) +10 +10−1 +100 +101 +102 +103 +70 +80 +90 +100 + inert +0+ - 2+ +0+ - 4+ +0+ - 6+ +0+ - 8+ +0+ - 10+ +0+ - 12+ + Exp. +16O+182W (a) +10−1 +100 +101 +102 +103 +60 +70 +80 +90 +100 +16O+184W (b) +10−1 +100 +101 +102 +103 +60 +70 +80 +90 +16O+186W(c) +10−1 +100 +101 +102 +103 +65 +70 +75 +80 +85 +90 +(d) +Cross-section σ (mb) +10−1 +100 +101 +102 +103 +60 65 70 75 80 85 90 95 +16O+176Hf +16O+180Hf +16O+174Yb +(e) +10−1 +100 +101 +102 +103 +60 +70 +80 +90 +(f) +10−1 +100 +101 +102 +103 +60 +70 +80 +90 +16O+176Yb (g) +Ec.m. (MeV) +Figure 3. +(Color online) Same as Fig. 2, but with the inclusion of -ve hexadecupole +(β4) deformation + +Phys. Scr. (2022) +11 +are shown in Fig. 3(a), 3(b), 3(c), and the experimental data [79, 82] are given for +comparison. The theoretical results of CC calculations with 6+ channels enhance the +fusion cross-section more in comparison to the 1-D BPM for 16O + 182W reaction, as +represented in the Fig. 3(a). There is substantial variation in the β4 value as the mass +number increases from 182W to 184W i.e. the difference between their β4 values is 0.027. +As a result, there is slight change in the cross-section as we move from 2+ to 6+ channels +at -0.093 and -0.095 values of the β4 parameter, as illustrated in Fig. 3(b) and 3(c). For +the above barrier energies, the calculated cross-section is a reasonably good match with +the experimental data [79, 82]. +Similar observation can be pointed out for the case of 16O+176Hf and 16O+180Hf, where +the β4 values are -0.057 and -0.068, respectively. As demonstrated in Fig. 3(d) for +16O+176Hf reaction, energy levels up to 4+ channels (solid green line) show contribution +in the increment of the cross-section, whereas up to 6+ levels (solid blue line) are good +enough for the enhancement of the fusion cross-sections for 16O+180Hf in comparison to +the 1-D BPM as shown in Fig. 3(e). The results obtained for 16O+176Hf and 16O+180Hf +reactions provide a satisfactory fit to the data well above barrier experimental values +[79]. As shown in Fig. 3 (f), rotational energy levels up to 4+ (solid green line) con- +tribute to enhancing the fusion cross-section for the 16O+174Yb reaction. In contrast, +up to 6+ channels (solid blue line) contribute to an increase in the cross-section for the +16O+176Yb reaction, as shown in Fig. 3(g). The obtained results are good enough for +the convergence of available experimental data [62] mainly at the above barrier ener- +gies. However, at below and near-barrier energies there is no discernible change in fusion +cross-section after the inclusion of rotational energy levels beyond the 6+ state. The +above observation suggests that the fusion cross-sections are strongly influenced by the +negative β4 values as predicted in Refs. [59, 79]. Up to 6+ state has a considerable effect +on the fusion cross-section in the case of β4 deformation; however, the negligible impact +can be observed for the rest of the channels. Furthermore, for all discussed systems, the +β4 values range from -0.042 to -0.095. In addition, the result shows that if β4 lies in the +range of -0.042 to -0.057, up to 4+ levels show contribution in the enhancement of the +cross-section w.r.t 1-D BPM. The rotational levels up to 6+ state gives an acceptable fit +between -0.066 and -0.095 range at above barrier energies. At -0.093 and -0.095 values of +the deformation parameter, there is a considerable difference in the fusion cross-section +between 0+, 2+, and 4+ levels. Thus in general, with an increase in the magnitude of +-ve β4, there is an addition of a level or two, which starts contributing to the fusion +cross-section.The relative change in the fusion cross-section w.r.t. rotational channels +has been plotted in Fig. 4 to thoroughly investigate the effect of individual channels +on the cross-section corresponding to 16O+182W reaction. As the case illustrated for +16O+182W reaction, the relative change in the fusion cross-section of 1.06 % and 0.45 % +has been observed at E = 73 MeV, 94 MeV corresponding to 4+ - 6+ state. On the other +hand, the relative change in the cross-section observed is less than 1% at different Ec.m. +corresponding to 6+ - 8+ state. It shows that the rotational channel has a considerable +impact on the fusion cross-section up to the 6+ state, whereas the other higher channels + +Phys. Scr. (2022) +12 +Relative change (σ(%)) +0 +5 +10 +15 +20 +2 +3 +(a) + at E = 73 MeV + at E = 94 MeV +2+ - 4+ +6+ - 8+ +4+ - 6+ +β2 = 0.250 +β4 = -0.066 +16O+182W +Channels +Figure 4. +(Color online) The relative change in the fusion cross-section w.r.t. +different channels for 16O+182W. +have a negligible effect. Similarly, the decreasing trend is noticed in the rest of the +reactions (not shown here). +3.2. For Hexadecapole deformation β4 > 0 +The same procedure as discussed in the previous section 3.1 is followed to calculate +the fusion cross-sections for 16O+166Er, 16O+148,152,154Sm, 16O+150Nd reactions having +β2 > 0, β4 > 0 values using the static Woods-Saxon potential by employing the +CCFULL code. The Woods-Saxon parameterizations of Aky¨uz-Winther potential (AW), +deformation parameters β2, β4 and the excitation energy corresponding to the first +excitation state are mentioned in Table 2. The values of AW potential parameters are +chosen to fit the experimental data at the above barrier energies for the inert case or +1-D BPM. The variation of the total interaction potential i.e. the sum of the Woods- +Saxon and Coulomb potentials, at ℓ = 0ℏ with the separation distance ‘r’ for these +systems is shown in Fig. 5. In comparison with the other reactions, the pocket formed +in 16O+152Sm reaction is substantially deeper. The probability of fusion is expected to +be very significant for such relatively deeper pockets. +The solid black line in Fig. +6 represents the 1-D penetration case. +From the +figure, one can observe that at below-barrier energies, the theoretical cross-section +obtained using 1-D BPM underestimates the experimental values. As mentioned earlier, +rotational channels are taken into account to reduce the fusion hindrance at the below- + +Phys. Scr. (2022) +13 +Table 2. +Same as Table 1, but for the case of (β2 > 0, β4 > 0) +System +V0(MeV ) +r0(fm) +Target +E+ +2 (MeV ) +β2 +β4 +16O+166Er +67.0 +1.185 +0.080 +0.342 +0.007 +16O+148Sm +62.204 +1.177 +0.550 +0.1423 +0.060 +16O+152Sm +70.0 +1.18 +0.121 +0.3064 +0.097 +16O+154Sm +62.53 +1.17 +0.081 +0.341 +0.105 +16O+150Nd +60.0 +1.165 +0.130 +0.2853 +0.110 +Total Potential (MeV) +40 +45 +50 +55 +60 +65 +70 +R (fm) +4 +5 +6 +7 +8 +9 10 11 12 13 14 15 +16O + 166Er +16O + 148Sm +16O + 152Sm +16O + 154Sm +16O + 150Nd +Figure 5. +(Color online) Same as Fig. 1 but for 16O+166Er, 16O+148,152,154Sm, and +16O+150Nd reactions. +barrier energies. Initially, the calculations are performed by considering the β2 values +ranging from 0.142 to 0.342. +The contribution of the rotational energy levels in +the enhancement of the fusion cross-section obtained is the same as in the previous +Section (3.1) except for 16O+148Sm nuclear reactions. In these reactions, the target +nuclei have β2 value 0.1423, whereas, in other reactions, the target nuclei are highly +deformed (0.285 ≤ β2 ≤ 0.342). +Based upon these values, the rotational levels up +to 4+ show enhancement in the fusion cross-section as compared to the 1-D barrier +penetration model as shown in Fig. +6(b). +However, higher-order channels have a +negligible contribution toward the fusion cross-section. +The ground state β2 values +alone as presented in Fig. 6 are incapable of reproducing the experimental data over +the whole energy range, thus need to be included in the β4 along with β2. +The positive β4 plays an important role in the enhancement of the fusion cross- +sections mainly at below-barrier energies. For 16O+166Er, there is an enhancement in + +Phys. Scr. (2022) +14 +10−1 +100 +101 +102 +103 +55 +60 +65 +70 +75 +16O+148Sm (b) +10−1 +100 +101 +102 +103 +50 +60 +70 +80 +90 +16O+154Sm (d) +10−1 +100 +101 +102 +103 +50 +55 +60 +65 +70 +75 +80 +16O+152Sm (c) +Cross-section σ (mb) +10−1 +100 +101 +102 +103 +Ec.m. (MeV) +55 +60 +65 +70 +75 +80 +16O+150Nd (e) +10−1 +100 +101 +102 +103 +60 +70 +80 +90 +100 + inert +0+ - 2+ +0+ - 4+ +0+ - 6+ +0+ - 8+ +0+ - 10+ +0+ - 12+ + Exp. +(a) +16O+166Er +Figure 6. +(Color online) Same as Fig. 2 but for 16O+166Er, 16O+148,152,154Sm, and +16O+150Nd nuclear reactions and compared with experimental data [83, 84, 85]. See +the text for details. +the fusion cross-sections up to 6+ levels (solid blue line) and other higher channels up to +12+ have negligible impact on fusion cross-sections to converge towards the experimental +data [59] as shown in Fig. 7(a). The difference between 4+ and 6+ channel is quite +significant because of strong β2, and β4 value for 166Er target nuclei. For Sm targets, +there is a significant variation in the quadrupole deformation β2 ranging from 0.1423 to +0.306 and also in the value of β4 from 0.060 to 0.097 for 148Sm and 152Sm. For 16O+148Sm +reaction, the effect of hexadecapole deformation on the fusion cross-section obtained is +the same. Similarly, the outcomes of 16O+152Sm, and 16O+154Sm reactions are also +identical. The 6+ channels (solid blue line) contribute to the enhancement of the fusion +cross-section in 16O+148Sm reaction, as shown in Fig. 7 (b), and 7 (c). In contrast, the +10+ channels (solid magenta line) play a significant role in the increment of the fusion +cross-section or reduced fusion hindrance at below barrier energies in 16O+152Sm, and +16O+154Sm reactions, as shown in Fig. 7(c), and 7(d). Theoretical results obtained for +these reactions give the best fit with the experimental data [8, 83, 84]. This difference in + +Phys. Scr. (2022) +15 +10−1 +100 +101 +102 +103 +55 +60 +65 +70 +75 +16O+148Sm (b) +10−1 +100 +101 +102 +103 +50 +60 +70 +80 +16O+152Sm (c) +Cross-section σ (mb) +10−1 +100 +101 +102 +103 +50 +60 +70 +80 +90 +16O+154Sm (d) +16O+152Sm +10−1 +100 +101 +102 +103 +50 +60 +70 +80 +16O+150Nd (e) +Ec.m. (MeV) +10−1 +100 +101 +102 +103 +60 +70 +80 +90 +100 + inert +0+ - 2+ +0+ - 4+ +0+ - 6+ +0+ - 8+ +0+ - 10+ +0+ - 12+ + Exp. +16O+166Er (a) +Figure 7. +(Color online) Same as Fig. 6, but with the inclusion of −ve hexadecupole +(β4) deformation +the rotational energy levels is due to β4 values because these values change significantly +from 0.060 to 0.097. The significant change between each channel are observed because +of strong deformation (β2, β4) values in case of 154Sm. Also, it is well known that 154Sm +is a perfect rotor [84]. Further, with the inclusion of higher channels, a negligible effect +on the fusion cross-sections is observed. +The similar results are obtained for 16O+150Nd reactions system. In the case of +16O+150Nd, 10+ (solid magenta line) channels play a significant role in increasing the +cross-section, as illustrated in Fig. 7(e). These theoretical results provide a satisfactory +fit with the experimental values [85]. The involvement of higher-order channels up to +12+ does not affect the fusion cross-sections. There is a significant difference between +different channels w.r.t. +1-D barrier penetration model because of +ve deformation +values. From the above results, we conclude that the rotational levels up to 4+ are good +enough to converge the experimental data except for the reaction in which β2 values +are less than 0.142. The value of the β4 parameter for this system is 0.060. We can +conclude from the above-discussed systems, that when the value of β4 lie in a range + +Phys. Scr. (2022) +16 +Relative change (σ(%)) +0 +20 +40 +4 +5 +(a) +16O+154Sm + at E = 58 MeV + at E = 77 MeV +2+ - 4+ +6+ - 8+ +4+ - 6+ +Channels +β2 = 0.341 +β4 = 0.105 +10+ - 12+ +8+ - 10+ +Figure 8. +(Color online) Same as Fig. 4 but for 16O+154Sm. +between 0.007-0.060, rotational energy levels up to 6+ lead to an enhancement in the +fusion cross-section. However, when the β4 value lies in the range 0.097-0.110, 10+ levels +play a significant role in increasing the cross-section and also provide a satisfactory fit +with the experimental values. +As illustrated for 16O+154Sm reaction , Fig. 8 depicts the relative change in the +fusion cross-section with respect to rotational channels. The relative change in the fusion +cross-section at E = 58 MeV (below the Coulomb barrier region) and 77 MeV (above +the Coulomb barrier region) of energy is 2.21 % and 0.15 %, respectively, corresponding +to the 8+ - 10+ state. In contrast, at different Ec.m. corresponding to 10+ - 12+ states, +the relative change in the cross-section found is less than 1%. +However, Rowley et +al. +[28] had demonstrated that up to 3-channels of the rotational energy levels are +enough to address the experimental data for 154Sm. Nonetheless, certain discrepancies +were evident between the experimental and the theoretical results. It was suggested +that with the inclusion of phonon(s) or transfer channels, these discrepancies can be +overcome. Comparatively, here the relative change clearly shows that up to 10+ states +of the rotational channel have a significant effect on the fusion cross-section. Moreover, +our results indicate that when higher-order channels are used, there is no need to include +the phonon and/or the transfer coupling for reproducing the experimental data. The +rest of the reactions follow the same pattern as the first in terms of decreasing intensity. + +Phys. Scr. (2022) +17 +Figure 9. +(Color online) Variation of fitted V0 and r0 as a function of ZpZT . The +fitted polynomial are also shown. +3.3. Fitting Curve +The development of the radioactive beam makes it possible to synthesize a variety +of nuclei that lie in the valley of stability as well as far from the β-stability region +including the superheavy island. Determining the proper reaction dynamics necessitates +a thorough understanding of the synthesis and characteristics of these nuclei. Many +efforts are being devoted to the direction of theoretical and experimental studies. On +the other hand, experimental verification is too difficult. As a result, we need to execute +theoretical modelling to confirm their characteristics in terms of reaction dynamics. As +we know, CCFULL is one of the computational codes used to study fusion dynamics, +which requires a detailed description of Woods-Saxon nuclear potential parameters, i.e., +V0 and r0 of the colliding nuclei. It is difficult to extract the potential parameters for the +recently synthesized or predicted target(s) nuclei with 16O induced reactions. As a result, +for interacting nuclei that lie in the stable and unstable mass region, one must fit the free +parameters for the nuclear potential in CCFULL. Among the considered reactions, the +fusion cross-sections for twelve reaction system, namely, 182,184,186W, 176,180Hf, 174,176Yb, +166Er, 148,152,154Sm, 150Nd are in good agreement with the available experimental data +at above barrier energies. +In Woods-Saxon parameterization potential, the standard value of diffuseness + +V, (MeV) +80 +V。 (MeV)(fit linear) +Y, = 66.9+ 0.00625 ZpZ +Vo (MeV) +V。 (MeV)(constant) +Y, = 60 +70 +1.22 +ro (fm) +1.2 +ro (fm)(fit linear) +(fim) +Y, = 1.128 + 1.04e-04 ZpZ +ro (fm)(constant) +Yz = 1.165 +1.18 +1.16 +480500 +520 +540 +560 +580 +600 +ZpZrPhys. Scr. (2022) +18 +parameter a0 = 0.65 fm and the value of the parameters such as V0 and r0 are difficult +to extract for the unexplored reaction systems. The Woods-Saxon parameters available +on the NRV website [86] are unable to fit the experimental data even at the above +barrier energies. Thus the values of V0 and r0 used in this study are chosen to fit the +experimental data at above barrier energies for the inert case and later on the coupling +to various rotational energy levels is included. The motivation of the present study is to +extract the relative contribution of individual rotational energy levels up to higher-order +states (12+). Also, the results drawn are independent of the choice of nuclear potential. +In this direction, we have given an algebraic function by a linear fitting of the curve for +known systems as a function of ZPZT as shown in Fig. 9. Thus, one can generate the +value of V0 and r0 using the algebraic formula for V02 = 66.9 + 0.00625ZpZt, and V01 = +60 MeV and r02 = 1.128 + 1.04 ∗ 10−4ZpZt, and r01 = 1.165fm. Here the subscript 1, +and 2 stands for the lower and upper limit of the extracted band region for V0, and r0 +as shown in Fig. (9). These fitted values of V0, and r0 are valid from ZpZt = 480 to +ZpZt = 592 in the rotational region of the Periodic Table. Using these simple algebraic +formulas, one can extract the potential parameters for the limited range of target nuclei +interacting with 16O as a projectile, which will be proven essential for the upcoming +experiment. +4. Summary and Conclusions +In the present work, we have studied the effect of individual rotational energy levels on +the fusion cross-section at deep sub-barrier energies in heavy-ion nuclear reactions. Here, +we have considered 16O-induced reactions in which target nuclei chosen are rotational +in nature(i.e. +182,184,186W, 176,180Hf, 174,176Yb, 166Er, 148,152,154Sm, 150Nd) and projectile +is spherical. +As such, we have demonstrated the effect of nuclear shapes on fusion +cross-sections by considering the deformed target nuclei (rotational) only. For different +values of deformation parameters, the role of different rotational energy levels has been +described in the terms of the fusion cross-sections. The contribution of the rotational +energy levels up to 6+ levels has been observed on the fusion cross-section for quadrupole +deformation (β2). For -ve hexadecapole deformation, higher-order channels up to 6+ +are found suitable for the convergence of the cross-sections towards experimental data +whereas for +ve β4 deformation, 10+ levels fit the data well. It is noticed that channels +beyond 10+ have a negligible impact on the fusion cross-section. 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A 248, 356 +(1975). +[86] http://nrv.jinr.ru/nrv/webnrv/fusion/reactions.php. + diff --git a/o9E3T4oBgHgl3EQfLgkW/content/tmp_files/load_file.txt b/o9E3T4oBgHgl3EQfLgkW/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..cff8e65d4c1d6685c7ce107008f2ae4e2031262f --- /dev/null +++ b/o9E3T4oBgHgl3EQfLgkW/content/tmp_files/load_file.txt @@ -0,0 +1,1216 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf,len=1215 +page_content='Systematic study of the effect of individual rotational energy levels on the fusion cross-section of 16O-based reactions of range 480 ≤ ZPZT ≤ 592 Nishu Jain1, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Bhuyan2 and Raj Kumar1 1 School of Physics and Materials Science, Thapar Institute of Engineering and Technology, Patiala-147004, Punjab, India 2 Center for Theoretical and Computational Physics, Department of Physics, Faculty of Science, University of Malaya, Kuala Lumpur 50603, Malaysia E-mail: nishujain1003@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='com E-mail: bunuphy@um.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='my E-mail: rajkumar@thapar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='edu September 2022 Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' In heavy-ion fusion reactions, the enhancement in the sub-barrier fusion cross-section has been observed as compared to the 1-Dimensional barrier penetration model due to the coupling of many degrees of freedom to the relative motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' This enhancement can be explained theoretically by including nuclear structure effects like deformation and the coupling of relative motion among two colliding nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' The present work aims to investigate the effect of individual rotational energy levels on the fusion cross-sections for 16O-based reaction systems, namely, 16O + 182,184,186W, 16O + 176,180Hf, 16O + 174,176Yb, 16O + 166Er, 16O + 148,152,154Sm, 16O + 150Nd at energies below the fusion barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Using the CCFULL code, the effect of low-lying rotational energy levels on the fusion cross-section for 16O induced reactions has been investigated at energies below and around the Coulomb barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' The calculations are performed by assuming the fixed value of diffuseness parameter a0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='65 fm in the Woods-Saxon nuclear potential and the other two parameters are optimised by fitting the experimental data at the above barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Here we have determined the V0 and r0 as a function of ZP ZT , where experimental cross-sections are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' From our calculations, it is observed that the hexadecapole deformation (β4) with different magnitudes has a significant influence on the fusion cross sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' For the case of the +ve value of β4, beyond 10+, the rotational levels cease to contribute significantly and also there is a significant difference between the contribution of sequential channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' On the other hand, in the case of -ve β4, up to 6+ levels contribute significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Furthermore, we have established an algebraic systematic of fitting, which one can use to determine the parameters V0, r0 of Woods-Saxon nuclear potential within the range of ZP ZT lie in between 480 ≤ ZP ZT ≤ 592.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='04363v1 [nucl-th] 11 Jan 2023 Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Scr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' (2022) 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Introduction Over the last few decades, various theoretical and experimental efforts have been centred on exploring the role of nuclear structure in the reaction dynamics [1, 2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' During the fusion process, the collision of an incident projectile and a target may lead to the formation of a compound nucleus through a quantum tunnelling process across the fusion barrier [1, 4, 5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Heavy-ion fusion reactions are accomplished in the crucial role of extending the nuclear chart and synthesizing the heavy elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Furthermore, some heavy-ion fusion processes in lighter mass systems are pivotal to the reaction channels that govern the elemental synthesis in stellar environments and energy production [2, 3, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Experimentally, it has been observed that there is a large enhancement in the sub-barrier fusion cross-sections by considering the one-dimensional barrier penetration model [8, 9, 10, 11, 12, 13] using bare nucleon-nucleon potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Such sub-barrier fusion enhancement could be elucidated by the coupling of relative motion degrees of freedom with the internal degrees of freedom such as static deformations (rotational nuclei) [9, 10], vibrational effect in the nuclear surface (spherical nuclei) [14, 15, 16], and the neck formation [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' However, the effects of nucleon transfer on fusion below the barrier have not yet been completely separated [6, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Several works [1, 14, 28, 29, 30, 31, 32, 33, 34] have discussed the coupling between the rotational state of the target nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' The internal degrees of freedom coupled with the relative motion of colliding nuclei lowers the barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' The significant decrease in the fusion cross-section with respect to the experimental data, at energies below the Coulomb barrier, is also known as fusion hindrance [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' A certain amount of flux allows the projectile to tunnel through the fusion barrier and fuse well into the target [1, 8] mainly at below barrier energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Therefore, coupled-channel calculations become a benchmark theoretical tool to understand fusion reaction dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Nuclear fusion is a complicated phenomenon because of the involvement of many nucleons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Therefore, it is tedious to handle the interaction potential between two colliding nuclei, which play a key role in describing the fusion reaction dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' In general, the interaction potential comprises long-range repulsive Coulomb potential, centrifugal interaction, and short-range attractive nuclear potential terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Unlike Coulomb and centrifugal potentials, the nuclear potential is not well established till now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' In the past few decades, various efforts have been made to provide a simple and accurate form of the nuclear interaction potential [35, 36, 37, 38, 39, 40, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Several fits of the nuclear potentials exist in the literature [42, 43, 44, 45, 46] which either kept the coupling off and/or included in the Woods-Saxon form while using the other potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' The traditional Woods-Saxon potential has been prominent and widely used to probe the heavy-ion nuclear fusion dynamics [28, 47, 48, 49, 50, 51, 52, 53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' The Woods-Saxon potential consists of three parameters namely;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' potential depth, range and diffuseness parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' The diffuseness parameter is an essential component in the parameterization of Woods-Saxon nuclear potential, as it describes the slopes of the potential in the tail region, where fusion begins [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Remarkably, the small value Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Scr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' (2022) 3 of the diffuseness parameter a0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='65fm is most suitable for a good description of experimental data [12, 55] in the elastic scattering analysis while the large values of the diffuseness parameter ranging from a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='75 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='5 fm provide the best fit for several excitation functions in the above barrier region to explore the experimental data [4, 5, 12, 55, 56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' As a result, the Woods-Saxon nuclear potential parameters are employed to study the elastic scattering and heavy-ion fusion processes in the Coupled channel approach (CCFULL), which provides a good description of below barrier fusion [55, 56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' It is known that the nuclear reactions are adequately affected by the entrance channel parameters and the internal structure such as mass asymmetry, deformation, and orientation of the colliding nuclei [57, 58, 59, 60], which are remarkably governed to influence the anticipation of a compound nucleus system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' The significance of shape degrees of freedom in sub-barrier fusion enhancement has been investigated experimentally using the fusion of two colliding nuclei [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' As we know, CCFULL is one of the computational codes used to explore fusion dynamics, where a full description of the Woods-Saxon nuclear potential parameters (V0, r0 and a0) for colliding nuclei is essential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' To understand the nuclear interaction potential for a recently synthesized or anticipated target nuclei for 16O- induced reactions, one must fit the main ingredient parameters for the nuclear potential as per the shape degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' We will also introduce an algebraic fitting of V0 and r0 by using the available known data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' This will be crucial for the theoretical calculations in predicting the fusion characteristics of compound nuclei, which is an essential input for the upcoming experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Furthermore, the study will include the significant role of each rotational energy level (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' 2+, 4+, 6+, 8+, 10+ and 12+) in the enhancement of the fusion cross- section mainly at energies below the Coulomb barrier for spherical and deformed nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Here, the fusion cross-section of 12 different 16O- induced reactions with rotational target nuclei i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=', 16O+182,184,186W, 176,180Hf, 174,176Yb, 166Er, 148,152,154Sm, 150Nd will be analyzed at sub-barrier energies within the static Woods-Saxon potential with standard diffuseness parameter a0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='65 fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' We have considered 16O as spherical [61, 62, 63] to investigate the effect of the individual rotational energy level of target nuclei on the fusion cross-section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' It is worth mentioning that the low-lying rotational energy levels for the chosen target nuclei follow the sequence of state with I = 0+, 2+, 4+, 6+, etc and the excitation energies of I+ state is proportional to I(I +1)∗E2T/6 [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Moreover, a better channel selection has been made for different signs of β-values [33, 59, 60, 65, 66] for individual energy levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' In addition, a comparison will be made between the resulting theoretical results and the available experimental data for the nuclear reactions under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' This paper is organized as follows: A brief description of the theoretical formalism used in this work is given in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' The results of the coupled channel calculations are given in detail in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Section 4 summarizes and concludes this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Scr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' (2022) 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Theoretical Formalism This section provides a brief description of Coupled Channel approach (CCFULL) used in the present study, which provides a reasonable understanding of the nuclear fusion dynamics at energies around the barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Within this approach, multidimensional barrier penetration is considered instead of single barrier penetration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' This method is used under the effect of coupling of the relative motion with the intrinsic degrees of freedom of the interacting nuclei [4, 34, 67], mainly for the calculation of mean angular momenta and the fusion cross-sections of the compound nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' The traditional method for addressing the effects of the coupling between relative motion and intrinsic degrees of freedom on fusion is to numerically solve the coupled-channels equations, which includes all relevant channels [68, 69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' The CC equations solve numerically within the Coupled channel approach are given as, � −ℏ2 2µ d2 dr2 + J(J + 1)ℏ2 2µr2 + ZPZTe2 r + VN + ϵn − Ec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' � × ψn(r) + � m Vnm(r)ψm(r) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' (1) Here r defines the radial part of relative motion between the participating nuclei and µ is known as the reduced mass of the colliding system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' ϵn is the excitation energy for the nth channel and Ec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' is the bombarding energy in the centre of the mass frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' VN represents the nuclear potential and Vnm symbolizes the matrix elements of the coupled Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Since there are several Coupled channel equations, their dimension is also large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Thus, the rotating frame approximation or no-Coriolis approximation is employed to reduce the dimension of Coupled channel equations [34, 67, 70, 71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' The CC equations with non-linear coupling are significant in studying the heavy-ion fusion reactions mainly at sub-barrier energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' All these sets of non-linear coupling are taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' The incoming wave boundary conditions (IWBC) [72] are also essential for the solution of Coupled channel equation because IWBC or ingoing wave conditions are quite sensitive for the potential pocket of the interaction fusion barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' The incoming wave of the entrance channel is present inside the barrier at the minimum position (r = rmin) and the outgoing wave of other channels are present at an infinite position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' By including the effect of the dominant intrinsic channels, the fusion cross-sections are calculated as given below: σJ(E) = σfus(E) = π k2 0 � J (2J + 1)PJ(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' (2) Here, the total angular momentum ‘J’ is substituted in place of ‘ℓ’ for each channel by Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Scr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' (2022) 5 applying iso-centrifugal approximation by using the following equation: ⟨ℓ⟩ = � J JσJ(E)/ � J σJ(E) = � π k2 0 � J J(2J + 1)PJ(E) � �� π k2 0 � J (2J + 1)PJ(E) � , (3) where PJ(E) is the total transmission coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' The Woods-Saxon form of nuclear potential is used to analyze the nuclear structure effects [34, 67, 70, 71] and it is defined as VN = −V0 1 + exp �� r0 − R0 � /a0 �, (4) where V0, r0, and a0 are the nuclear potential parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' In the Coupled channel approach, the rotational coupling with a pure rotor is taken into consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' One can generate the nuclear coupling Hamiltonian by changing the target radius in the nuclear potential to a dynamical operator, R0 → R0 + ˆO = R0 + β2RTY20 + β4RTY40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' (5) Here RT is rcoupA1/3 and β2 and β4 are the quadrupole and hexadecapole deformation parameters of the deformed target nucleus, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Thus, the nuclear coupling Hamiltonian is given by VN(r, ˆO) = −V0 1 + exp �� r0 − R0 − ˆO � /a0 �, (6) To connect the |n⟩ = |I0⟩ and |m⟩ = |I′0⟩ states of the target’s ground rotational band, we need matrix elements of the coupling Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' These are readily accessible using matrix algebra [73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' In this algebra, the eigenvalues and eigenvectors of the operator ˆO, which satisfies ˆO | α >= λα | α > (7) This is implemented in the CCFULL program by diagonalizing the matrix ˆO, whose elements are given by ˆOII′ = � 5(2I + 1)(2I′ + 1) 4π β2RT � I 2 I′ 0 0 0 �2 + � 9(2I + 1)(2I′ + 1) 4π β4RT � I 4 I′ 0 0 0 �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' (8) The nuclear coupling matrix elements are then evaluated as V (N) nm = < I0 | VN(r, ˆO) | I ′0 > −V (0) N (r)δn,m, = � α < I0 | α >< α | I ′0 > VN(r, λα) − V (0) N (r)δn,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' (9) Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Scr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' (2022) 6 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' The parameters of Woods-Saxon potential (V0 & r0), the deformation parameters (β2 > 0, β4 < 0) and the excitation energy corresponding to quadrupole deformation of the nuclei [74, 75] used in the coupled channel calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' System V0(MeV ) r0(fm) Target E+ 2 (MeV ) β2 β4 16O+182W 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='899 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='165 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='2500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='066 16O+184W 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='987 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='178 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='111 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='236 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='093 16O+186W 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='122 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='221 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='095 16O+176Hf 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='627 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='088 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='295 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='057 16O+180Hf 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='093 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='274 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='068 16O+174Yb 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='53 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='076 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='325 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='042 16O+176Yb 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='165 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='082 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='304 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='068 The last term is included in the equation to avoid the diagonal component from being counted twice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' The linear rotational coupling approximation is used to calculate the Coulomb matrix and explained as V C R(I,I′) = 3ZPZTR2 T 5r3 � 5(2I + 1)(2I′ + 1) 4π × � β2 + 2 7β2 2 � 5 π � � I 2 I′ 0 0 0 �2 +3ZPZTR4 T 9r5 � 9(2I + 1)(2I′ + 1) 4π × � β4 + 9 7β2 2 � � I 4 I′ 0 0 0 � , (10) for rotational coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' These coupled channel equations are used to calculate the fusion cross-section of the compound nucleus by considering the coupling of all orders as discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Results and Discussions The fusion cross-sections have been calculated for 16O-induced nuclear reactions with different rotational target nuclei, namely, 16O + 182,184,186W, 16O + 176,180Hf, 16O + 174,176Yb, 16O + 166Er, 16O + 148,152,154Sm, 16O + 150Nd with Coupled channel calculations by adopting CCFULL code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Various theoretical approaches have been developed to generate the attractive nuclear potential for the description of fusion data over wide energy ranges [76, 77, 78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' In this work, Woods-Saxon parameterization has been taken into account to compute the interaction potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' The standard value of diffuseness parameter a0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='65 fm is used for the considered nuclear reactions [12, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' The deformation parameter βλ connected with the transition of multipolarity λ were Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Scr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' (2022) 7 Total Potential (MeV) 50 55 60 65 70 75 R (fm) 5 6 7 8 9 10 11 12 13 14 15 16O + 182W 16O + 184W 16O + 186W 16O + 176Hf 16O + 180Hf 16O + 174Yb 16O + 176Yb Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' (Color online) The variation of total interaction potential in the respect of separation distance r (fm) for 16O+182,184,186W, 16O+176,180Hf, and 16O+174,176Yb reactions under β2 > 0, β4 < 0 condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' determined from experimental transition probabilities B(E2) by using the following relation: βλ = 4π 3ZRλ � B(Eλ) ↑ e2 , (11) where R = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='2A1/3fm, and B(Eλ) ↑ is in units of e2b2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' For λ = 2, the values of B(E2) ↑ [74] are experimental and independent of nuclear models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' On the other hand, the parameter βλ depends upon the nuclear model and determines the calculation of deformation parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' In the present study, we aim to understand the effect of each rotational energy level up to 12+ levels i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=', 2+, 4+, 6+, 8+, 10+ and 12+ in the enhancement of the fusion cross-sections at below barrier energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' The calculations for the fusion cross- sections have been exercised in steps (increment of 2+ state in each step) from 0+ to 12+ channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Nonetheless, we observed that for negative and positive β4 values, beyond 6+ and 10+ respectively, the higher-order channels cease to contribute significantly towards the fusion cross-section around and below the Coulomb barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Further, two different conditions based on the shape of nuclei: (1) β4 < 0, and (2) β4 > 0 are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' The quadrupole (β2) and/or hexadecapole (β4) deformations of deformed nuclei are generally taken into account while characterizing the rotation of the nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' It has been suggested that the reactions containing nuclei with β4 show an enhancement in the fusion cross- section at energies below the barrier [59, 79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Our present calculations assume the 16O (projectile) as spherical [59, 61, 62, 63, 80, 81] to determine the rotational effect of the target nucleus on fusion cross-section effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' It is important to note here that the nucleon transfer channels are not considered in the present calculations, which may play a significant role at energies below the Coulomb barrier [18, 19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Scr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' (2022) 8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' For Hexadecapole deformation β4 < 0 The near barrier and sub-barrier fusion cross-sections for 16O+182,184,186W, 176,180Hf, 174,176Yb systems have been analyzed systematically by employing CCFULL code us- ing the static Woods-Saxon potential with β2 > 0, β4 < 0 of the target nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' The Woods-Saxon parameterizations of Aky¨uz-Winther potential (AW), the values of the deformation parameters (β2, β4) and the excitation energy corresponding to the first excitation state are given in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' The values of the Woods-Saxon potential param- eters (V0, r0 & a0) are chosen to fit the experimental fusion cross-section at the above barrier energies for the case of the 1-D barrier penetration model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' The variation of the total interaction potential at ℓ = 0ℏ with the separation distance ‘r’ for these systems is also shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Here the total interaction potential is the sum of Woods-Saxon potential in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' (4), and the Coulomb potential VC= ZP ZT e2 r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' From the figure, one can notice that the pocket formed for 16O+180Hf reaction is much deeper in comparison to the others, which may result in a larger fusion probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' The fusion cross-sections have been calculated using a 1D barrier penetration model for these reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' The one-dimensional barrier penetration data is represented by a solid black line as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' From the figure, we found the obtained results under- estimate the experimental data, especially at low barrier energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' In order to address the fusion cross-section of the above-mentioned reactions, rotational degrees of freedom have been included in the CCFULL calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Only the quadrupole deformation (β2) is initially considered, and the calculations are performed along with various values of β2 ranging from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='221 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='324.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' The deformation parameters used in these reactions decreases as the mass number increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Up to 12+ channels are incorporated in each system to observe the effect of each rotational level in the enhancement of the fusion cross-section and/or to predict the number of channels that are good enough to converge to the experimental data and beyond it the higher order ceases to contribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Within the addition of rotational channels corresponding to these levels, the enhancement in the fusion cross-sections has been observed as compared to the single barrier penetration case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' The contribution of the rotational energy levels on the fusion cross-section up to 6+ state has been found to be significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' It has been observed that the ground state quadrupole deformation alone is unable to reproduce the experimental data across the below barrier energy region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' To address these issues, the hexadecapole deformation (β4) is included in the CC calcu- lations, as illustrated in Fig 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' The estimated results using 1-D BPM are comparatively lower than the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' The coupled channel computations are performed for this system including rotational energy levels up to the 12+ state of the target nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' A significant change in the fusion cross-section is observed for 0+ to 2+ state as compared to the 1-D BPM, and a small change for 2+ to 6+ states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Furthermore, we observe that the cross-section with the inclusion of higher-order channels, beyond 6+, overlaps with that of up to the 6+ state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' The fusion cross-section for 16O + 182,184,186W reactions are calculated with the β4 values ranging from -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='066 to -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='095.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' The calculated results Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Scr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' (2022) 9 10−1 100 101 102 103 70 80 90 100 inert 0+ - 2+ 0+ - 4+ 0+ - 6+ 0+ - 8+ 0+ - 10+ 0+ - 12+ Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' (a) 16O+182W 10−1 100 101 102 103 60 70 80 90 100 16O+184W (b) 10−1 100 101 102 103 65 70 75 80 85 90 16O+176Hf (d) Cross-section σ (mb) 10−1 100 101 102 103 60 70 80 90 100 16O+180Hf (e) 10−1 100 101 102 103 60 70 80 90 16O+174Yb (f) 10−1 100 101 102 103 Ec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' (MeV) 60 70 80 90 16O+176Yb (g) 10−1 100 101 102 103 60 70 80 90 100 16O+186W (c) Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' (Color online) The fusion cross-sections estimated up to 12+ channels in sequential manner as a function of Ec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' (MeV) for 16O+182,184,186W, 16O+176,180Hf, and 16O+174,176Yb nuclear reactions having quadrupole deformation and compared with experimental data [62, 79, 82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' See the text for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Scr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' (2022) 10 10−1 100 101 102 103 70 80 90 100 inert 0+ - 2+ 0+ - 4+ 0+ - 6+ 0+ - 8+ 0+ - 10+ 0+ - 12+ Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' 16O+182W (a) 10−1 100 101 102 103 60 70 80 90 100 16O+184W (b) 10−1 100 101 102 103 60 70 80 90 16O+186W(c) 10−1 100 101 102 103 65 70 75 80 85 90 (d) Cross-section σ (mb) 10−1 100 101 102 103 60 65 70 75 80 85 90 95 16O+176Hf 16O+180Hf 16O+174Yb (e) 10−1 100 101 102 103 60 70 80 90 (f) 10−1 100 101 102 103 60 70 80 90 16O+176Yb (g) Ec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' (MeV) Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' (Color online) Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' 2, but with the inclusion of -ve hexadecupole (β4) deformation Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Scr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' (2022) 11 are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' 3(a), 3(b), 3(c), and the experimental data [79, 82] are given for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' The theoretical results of CC calculations with 6+ channels enhance the fusion cross-section more in comparison to the 1-D BPM for 16O + 182W reaction, as represented in the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' 3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' There is substantial variation in the β4 value as the mass number increases from 182W to 184W i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' the difference between their β4 values is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='027.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' As a result, there is slight change in the cross-section as we move from 2+ to 6+ channels at -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='093 and -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='095 values of the β4 parameter, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' 3(b) and 3(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' For the above barrier energies, the calculated cross-section is a reasonably good match with the experimental data [79, 82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Similar observation can be pointed out for the case of 16O+176Hf and 16O+180Hf, where the β4 values are -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='057 and -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='068, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' As demonstrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' 3(d) for 16O+176Hf reaction, energy levels up to 4+ channels (solid green line) show contribution in the increment of the cross-section, whereas up to 6+ levels (solid blue line) are good enough for the enhancement of the fusion cross-sections for 16O+180Hf in comparison to the 1-D BPM as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' 3(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' The results obtained for 16O+176Hf and 16O+180Hf reactions provide a satisfactory fit to the data well above barrier experimental values [79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' 3 (f), rotational energy levels up to 4+ (solid green line) con- tribute to enhancing the fusion cross-section for the 16O+174Yb reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' In contrast, up to 6+ channels (solid blue line) contribute to an increase in the cross-section for the 16O+176Yb reaction, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' 3(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' The obtained results are good enough for the convergence of available experimental data [62] mainly at the above barrier ener- gies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' However, at below and near-barrier energies there is no discernible change in fusion cross-section after the inclusion of rotational energy levels beyond the 6+ state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' The above observation suggests that the fusion cross-sections are strongly influenced by the negative β4 values as predicted in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' [59, 79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Up to 6+ state has a considerable effect on the fusion cross-section in the case of β4 deformation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' however, the negligible impact can be observed for the rest of the channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Furthermore, for all discussed systems, the β4 values range from -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='042 to -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='095.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' In addition, the result shows that if β4 lies in the range of -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='042 to -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='057, up to 4+ levels show contribution in the enhancement of the cross-section w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='t 1-D BPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' The rotational levels up to 6+ state gives an acceptable fit between -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='066 and -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='095 range at above barrier energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' At -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='093 and -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='095 values of the deformation parameter, there is a considerable difference in the fusion cross-section between 0+, 2+, and 4+ levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Thus in general, with an increase in the magnitude of ve β4, there is an addition of a level or two, which starts contributing to the fusion cross-section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='The relative change in the fusion cross-section w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' rotational channels has been plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' 4 to thoroughly investigate the effect of individual channels on the cross-section corresponding to 16O+182W reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' As the case illustrated for 16O+182W reaction, the relative change in the fusion cross-section of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='06 % and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='45 % has been observed at E = 73 MeV, 94 MeV corresponding to 4+ - 6+ state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' On the other hand, the relative change in the cross-section observed is less than 1% at different Ec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' corresponding to 6+ - 8+ state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' It shows that the rotational channel has a considerable impact on the fusion cross-section up to the 6+ state, whereas the other higher channels Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Scr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' (2022) 12 Relative change (σ(%)) 0 5 10 15 20 2 3 (a) at E = 73 MeV at E = 94 MeV 2+ - 4+ 6+ - 8+ 4+ - 6+ β2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='250 β4 = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='066 16O+182W Channels Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' (Color online) The relative change in the fusion cross-section w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' different channels for 16O+182W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' have a negligible effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Similarly, the decreasing trend is noticed in the rest of the reactions (not shown here).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' For Hexadecapole deformation β4 > 0 The same procedure as discussed in the previous section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='1 is followed to calculate the fusion cross-sections for 16O+166Er, 16O+148,152,154Sm, 16O+150Nd reactions having β2 > 0, β4 > 0 values using the static Woods-Saxon potential by employing the CCFULL code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' The Woods-Saxon parameterizations of Aky¨uz-Winther potential (AW), deformation parameters β2, β4 and the excitation energy corresponding to the first excitation state are mentioned in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' The values of AW potential parameters are chosen to fit the experimental data at the above barrier energies for the inert case or 1-D BPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' The variation of the total interaction potential i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' the sum of the Woods- Saxon and Coulomb potentials, at ℓ = 0ℏ with the separation distance ‘r’ for these systems is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' In comparison with the other reactions, the pocket formed in 16O+152Sm reaction is substantially deeper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' The probability of fusion is expected to be very significant for such relatively deeper pockets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' The solid black line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' 6 represents the 1-D penetration case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' From the figure, one can observe that at below-barrier energies, the theoretical cross-section obtained using 1-D BPM underestimates the experimental values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' As mentioned earlier, rotational channels are taken into account to reduce the fusion hindrance at the below- Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Scr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' (2022) 13 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Same as Table 1, but for the case of (β2 > 0, β4 > 0) System V0(MeV ) r0(fm) Target E+ 2 (MeV ) β2 β4 16O+166Er 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='185 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='080 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='342 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='007 16O+148Sm 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='204 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='177 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='550 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='1423 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='060 16O+152Sm 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='121 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='3064 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='097 16O+154Sm 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='53 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='081 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='341 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='105 16O+150Nd 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='165 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='130 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='2853 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='110 Total Potential (MeV) 40 45 50 55 60 65 70 R (fm) 4 5 6 7 8 9 10 11 12 13 14 15 16O + 166Er 16O + 148Sm 16O + 152Sm 16O + 154Sm 16O + 150Nd Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' (Color online) Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' 1 but for 16O+166Er, 16O+148,152,154Sm, and 16O+150Nd reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' barrier energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Initially, the calculations are performed by considering the β2 values ranging from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='142 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='342.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' The contribution of the rotational energy levels in the enhancement of the fusion cross-section obtained is the same as in the previous Section (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='1) except for 16O+148Sm nuclear reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' In these reactions, the target nuclei have β2 value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='1423, whereas, in other reactions, the target nuclei are highly deformed (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='285 ≤ β2 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='342).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Based upon these values, the rotational levels up to 4+ show enhancement in the fusion cross-section as compared to the 1-D barrier penetration model as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' 6(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' However, higher-order channels have a negligible contribution toward the fusion cross-section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' The ground state β2 values alone as presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' 6 are incapable of reproducing the experimental data over the whole energy range, thus need to be included in the β4 along with β2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' The positive β4 plays an important role in the enhancement of the fusion cross- sections mainly at below-barrier energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' For 16O+166Er, there is an enhancement in Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Scr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' (2022) 14 10−1 100 101 102 103 55 60 65 70 75 16O+148Sm (b) 10−1 100 101 102 103 50 60 70 80 90 16O+154Sm (d) 10−1 100 101 102 103 50 55 60 65 70 75 80 16O+152Sm (c) Cross-section σ (mb) 10−1 100 101 102 103 Ec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' (MeV) 55 60 65 70 75 80 16O+150Nd (e) 10−1 100 101 102 103 60 70 80 90 100 inert 0+ - 2+ 0+ - 4+ 0+ - 6+ 0+ - 8+ 0+ - 10+ 0+ - 12+ Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' (a) 16O+166Er Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' (Color online) Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' 2 but for 16O+166Er, 16O+148,152,154Sm, and 16O+150Nd nuclear reactions and compared with experimental data [83, 84, 85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' See the text for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' the fusion cross-sections up to 6+ levels (solid blue line) and other higher channels up to 12+ have negligible impact on fusion cross-sections to converge towards the experimental data [59] as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' 7(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' The difference between 4+ and 6+ channel is quite significant because of strong β2, and β4 value for 166Er target nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' For Sm targets, there is a significant variation in the quadrupole deformation β2 ranging from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='1423 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='306 and also in the value of β4 from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='060 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='097 for 148Sm and 152Sm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' For 16O+148Sm reaction, the effect of hexadecapole deformation on the fusion cross-section obtained is the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Similarly, the outcomes of 16O+152Sm, and 16O+154Sm reactions are also identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' The 6+ channels (solid blue line) contribute to the enhancement of the fusion cross-section in 16O+148Sm reaction, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' 7 (b), and 7 (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' In contrast, the 10+ channels (solid magenta line) play a significant role in the increment of the fusion cross-section or reduced fusion hindrance at below barrier energies in 16O+152Sm, and 16O+154Sm reactions, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' 7(c), and 7(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Theoretical results obtained for these reactions give the best fit with the experimental data [8, 83, 84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' This difference in Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Scr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' (2022) 15 10−1 100 101 102 103 55 60 65 70 75 16O+148Sm (b) 10−1 100 101 102 103 50 60 70 80 16O+152Sm (c) Cross-section σ (mb) 10−1 100 101 102 103 50 60 70 80 90 16O+154Sm (d) 16O+152Sm 10−1 100 101 102 103 50 60 70 80 16O+150Nd (e) Ec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' (MeV) 10−1 100 101 102 103 60 70 80 90 100 inert 0+ - 2+ 0+ - 4+ 0+ - 6+ 0+ - 8+ 0+ - 10+ 0+ - 12+ Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' 16O+166Er (a) Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' (Color online) Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' 6, but with the inclusion of −ve hexadecupole (β4) deformation the rotational energy levels is due to β4 values because these values change significantly from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='060 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='097.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' The significant change between each channel are observed because of strong deformation (β2, β4) values in case of 154Sm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Also, it is well known that 154Sm is a perfect rotor [84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Further, with the inclusion of higher channels, a negligible effect on the fusion cross-sections is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' The similar results are obtained for 16O+150Nd reactions system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' In the case of 16O+150Nd, 10+ (solid magenta line) channels play a significant role in increasing the cross-section, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' 7(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' These theoretical results provide a satisfactory fit with the experimental values [85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' The involvement of higher-order channels up to 12+ does not affect the fusion cross-sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' There is a significant difference between different channels w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' 1-D barrier penetration model because of +ve deformation values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' From the above results, we conclude that the rotational levels up to 4+ are good enough to converge the experimental data except for the reaction in which β2 values are less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='142.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' The value of the β4 parameter for this system is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='060.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' We can conclude from the above-discussed systems, that when the value of β4 lie in a range Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Scr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' (2022) 16 Relative change (σ(%)) 0 20 40 4 5 (a) 16O+154Sm at E = 58 MeV at E = 77 MeV 2+ - 4+ 6+ - 8+ 4+ - 6+ Channels β2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='341 β4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='105 10+ - 12+ 8+ - 10+ Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' (Color online) Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' 4 but for 16O+154Sm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='007-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='060, rotational energy levels up to 6+ lead to an enhancement in the fusion cross-section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' However, when the β4 value lies in the range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='097-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='110, 10+ levels play a significant role in increasing the cross-section and also provide a satisfactory fit with the experimental values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' As illustrated for 16O+154Sm reaction , Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' 8 depicts the relative change in the fusion cross-section with respect to rotational channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' The relative change in the fusion cross-section at E = 58 MeV (below the Coulomb barrier region) and 77 MeV (above the Coulomb barrier region) of energy is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='21 % and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='15 %, respectively, corresponding to the 8+ - 10+ state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' In contrast, at different Ec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' corresponding to 10+ - 12+ states, the relative change in the cross-section found is less than 1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' However, Rowley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' [28] had demonstrated that up to 3-channels of the rotational energy levels are enough to address the experimental data for 154Sm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Nonetheless, certain discrepancies were evident between the experimental and the theoretical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' It was suggested that with the inclusion of phonon(s) or transfer channels, these discrepancies can be overcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Comparatively, here the relative change clearly shows that up to 10+ states of the rotational channel have a significant effect on the fusion cross-section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Moreover, our results indicate that when higher-order channels are used, there is no need to include the phonon and/or the transfer coupling for reproducing the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' The rest of the reactions follow the same pattern as the first in terms of decreasing intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Scr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' (2022) 17 Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' (Color online) Variation of fitted V0 and r0 as a function of ZpZT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' The fitted polynomial are also shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Fitting Curve The development of the radioactive beam makes it possible to synthesize a variety of nuclei that lie in the valley of stability as well as far from the β-stability region including the superheavy island.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Determining the proper reaction dynamics necessitates a thorough understanding of the synthesis and characteristics of these nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Many efforts are being devoted to the direction of theoretical and experimental studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' On the other hand, experimental verification is too difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' As a result, we need to execute theoretical modelling to confirm their characteristics in terms of reaction dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' As we know, CCFULL is one of the computational codes used to study fusion dynamics, which requires a detailed description of Woods-Saxon nuclear potential parameters, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=', V0 and r0 of the colliding nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' It is difficult to extract the potential parameters for the recently synthesized or predicted target(s) nuclei with 16O induced reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' As a result, for interacting nuclei that lie in the stable and unstable mass region, one must fit the free parameters for the nuclear potential in CCFULL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Among the considered reactions, the fusion cross-sections for twelve reaction system, namely, 182,184,186W, 176,180Hf, 174,176Yb, 166Er, 148,152,154Sm, 150Nd are in good agreement with the available experimental data at above barrier energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' In Woods-Saxon parameterization potential, the standard value of diffuseness V, (MeV) 80 V。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' (MeV)(fit linear) Y, = 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='9+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='00625 ZpZ Vo (MeV) V。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' (MeV)(constant) Y, = 60 70 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='22 ro (fm) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='2 ro (fm)(fit linear) (fim) Y, = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='128 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='04e-04 ZpZ ro (fm)(constant) Yz = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='165 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='18 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='16 480500 520 540 560 580 600 ZpZrPhys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Scr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' (2022) 18 parameter a0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='65 fm and the value of the parameters such as V0 and r0 are difficult to extract for the unexplored reaction systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' The Woods-Saxon parameters available on the NRV website [86] are unable to fit the experimental data even at the above barrier energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Thus the values of V0 and r0 used in this study are chosen to fit the experimental data at above barrier energies for the inert case and later on the coupling to various rotational energy levels is included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' The motivation of the present study is to extract the relative contribution of individual rotational energy levels up to higher-order states (12+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Also, the results drawn are independent of the choice of nuclear potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' In this direction, we have given an algebraic function by a linear fitting of the curve for known systems as a function of ZPZT as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Thus, one can generate the value of V0 and r0 using the algebraic formula for V02 = 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='9 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='00625ZpZt, and V01 = 60 MeV and r02 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='128 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='04 ∗ 10−4ZpZt, and r01 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='165fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Here the subscript 1, and 2 stands for the lower and upper limit of the extracted band region for V0, and r0 as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' These fitted values of V0, and r0 are valid from ZpZt = 480 to ZpZt = 592 in the rotational region of the Periodic Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Using these simple algebraic formulas, one can extract the potential parameters for the limited range of target nuclei interacting with 16O as a projectile, which will be proven essential for the upcoming experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Summary and Conclusions In the present work, we have studied the effect of individual rotational energy levels on the fusion cross-section at deep sub-barrier energies in heavy-ion nuclear reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Here, we have considered 16O-induced reactions in which target nuclei chosen are rotational in nature(i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' 182,184,186W, 176,180Hf, 174,176Yb, 166Er, 148,152,154Sm, 150Nd) and projectile is spherical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' As such, we have demonstrated the effect of nuclear shapes on fusion cross-sections by considering the deformed target nuclei (rotational) only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' For different values of deformation parameters, the role of different rotational energy levels has been described in the terms of the fusion cross-sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' The contribution of the rotational energy levels up to 6+ levels has been observed on the fusion cross-section for quadrupole deformation (β2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' For -ve hexadecapole deformation, higher-order channels up to 6+ are found suitable for the convergence of the cross-sections towards experimental data whereas for +ve β4 deformation, 10+ levels fit the data well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' It is noticed that channels beyond 10+ have a negligible impact on the fusion cross-section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' For the determination of the free parameter of Woods-Saxon potential, the parameters are fitted as an algebraic function of ZPZT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' This work will be provided in identifying the combinations of the target nuclei with 16O projectile within the range of ZPZT from 480 to 592.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Acknowledgements This work has been supported by Science Engineering Research Board (SERB), File No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' CRG/2021/001229, FOSTECT Project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' FOSTECT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='2019B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content='04, FAPESP Project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Scr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' (2022) 19 2017/05660-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' References 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Hinde, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Mein, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Morton, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' C.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Liang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Kelly, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Sonzogni, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Vandenbosch, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' 76, 1587 (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' [10] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Stokstad and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Ackermann, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Corradi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' He, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Montagnoli, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' Beghini, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} +page_content=' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E3T4oBgHgl3EQfLgkW/content/2301.04363v1.pdf'} diff --git a/p9FPT4oBgHgl3EQf8DXy/content/tmp_files/2301.13207v1.pdf.txt b/p9FPT4oBgHgl3EQf8DXy/content/tmp_files/2301.13207v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..c8f673e1c547ea55f9deb907dfaddd2e0ddb8a9f --- /dev/null +++ b/p9FPT4oBgHgl3EQf8DXy/content/tmp_files/2301.13207v1.pdf.txt @@ -0,0 +1,1052 @@ +A quantum trajectory analysis of singular wave functions +Angel S. Sanz,1 Luis L. Sánchez-Soto,1, 2 and Andrea Aiello2 +1Departamento de Óptica, Facultad de Física, Universidad Complutense, 28040 Madrid, Spain +2Max-Planck-Institut für die Physik des Lichts, 91058 Erlangen, Germany +(Dated: February 1, 2023) +The Schrödinger equation admits smooth and finite solutions that spontaneously evolve into a singularity, even +for a free particle. This blowup is generally ascribed to the intrinsic dispersive character of the associated time +evolution. We resort to the notion of quantum trajectories to reinterpret this singular behavior. We show that the +blowup can be directly related to local phase variations, which generate an underlying velocity field responsible +for driving the quantum flux toward the singular region. +I. +INTRODUCTION +The Schrödinger equation is, perhaps, the prototype of a +dispersive equation; that is, if no boundary conditions are +imposed, its wave solutions spread out in space as they evolve +in time [1]. A frequent way to quantify this dispersion is by the +so-called dispersive estimates, a topic with a long history [2–4] +and whose main goal is to establish tight bounds on the decay +of the solutions. +Recently, it has been pointed out that the Schrödinger equa- +tion, even for a free particle, presents dispersive singulari- +ties [5, 6]: an initial square-integrable profile 𝜓(𝑥, 0) could +result in a solution 𝜓(𝑥, 𝑡) that blows up in a finite time. In +the remainder such profiles will be termed as singular wave +packets. While this singular behavior (sometimes denoted as +self-focusing or wave collapse) is well understood in presence +of nonlinearities [7–9], it is, at first sight, surprising in a pure +linear evolution. +From a mathematical viewpoint, this dispersive blowup can +be related to the fact that the linear Schrödinger equation is +ill-posed in the space 𝐿∞: the free propagator is not a Fourier +multiplier in 𝐿∞ [10]. In physical terms, dispersive blowup +is a focusing phenomenon due to both the unbounded do- +main of the problem and the propensity of the dispersion re- +lation to propagating energy at different speeds. Interestingly, +the same singular behavior has been described in for paraxial +beams [11–13], which is consequent with the complete equiv- +alence between the time-dependent Schrödinger equation and +the paraxial wave equation [14]. +In this paper, we address the physical interpretation of these +singularities from the perspective of quantum trajectories. In +this picture, quantum formalism is reinterpreted as describing +particles following definite trajectories, each with a precisely +defined position at each instant in time. However, in this ap- +proach, called Bohmian mechanics [15–17], the trajectories of +the particles are quite different from those of classical parti- +cles because they are guided by the wave function [18–21]. +Our analysis shows that the blowup can be directly related to +local phase variations, which generate an underlying velocity +field (the phase gradient) responsible for driving the quantum +flux toward the singular region. To shed light on this point, +we compare the blowup with the focusing of a Gaussian and +a rectangular wave packet: this demonstrates that imploding +solutions are distinguished by an initial phase factor. +Furthermore, for Gaussian wave packets, which can be +nicely analyzed in closed form, it is also observed that there +are two types of solutions with very different properties, de- +spite their initial density distributions being identical. One of +such solutions leads to a classical type of propagation because +the phase factor plays a minor role (or even no role at all). In +contradistinction, the other type of solution is characterized by +wide initial wave functions with an intrinsic highly oscillatory +behavior. This emphasizes the prominent role of the phase as +an active agent in the subsequent dynamics. +This article is organized as follows. In Sec. II we briefly +discuss the spontaneous generation of a singularity in the +Schrödinger equation and introduce the basic elements needed +to define a quantum trajectory. In terms of this notion, we +analyze the singularity and put forward the fundamental role +played by the quantum phase to understand that phenomenon. +In Sec. III we examine the behavior of a Gaussian and a rect- +angular packet and compare with the previous singular wave. +Finally, Sec. IV summarizes our conclusions. +II. +DISPERSIVE BLOWUP IN THE SCHRÖDINGER +EQUATION +A. +Spontaneous generation of a singularity +We first set the stage for our discussion. We will be consid- +ering the simplest case of the Schrödinger equation for a free +particle of mass 𝑚 in one dimension +𝑖ℏ𝜕𝜓(𝑥, 𝑡) +𝜕𝑡 += − ℏ2 +2𝑚 +𝜕2𝜓(𝑥, 𝑡) +𝜕𝑥2 +, +(1) +with the initial Cauchy problem 𝜓(𝑥, 0) ∈ 𝐿2(R). The unique +solution of (1) can be written in terms of the free-space prop- +agator as [22] +𝜓(𝑥, 𝑡) = +√︂ +𝑚 +2𝜋𝑖ℏ𝑡 +∫ +R +exp +� 𝑖𝑚 +2ℏ𝑡 (𝑥 − 𝑥′)2 +� +𝜓(𝑥′, 0) 𝑑𝑥′ , (2) +where the integral has to be understood in the improper Rie- +mann sense. In this way, the Schrödinger equation appears as +an integral equation, rather than a differential one, with the +advantage of being valid even if the wave function is not a +differentiable function. +arXiv:2301.13207v1 [quant-ph] 30 Jan 2023 + +2 +singular +regime +square-integrable +Lorentzian +1 moment +st +2 moment +nd +FIG. 1. For 𝜈 > 1/4 the wave function (3) is square integrable. The +red band indicates the range 1/4 < 𝜈 < 1/2 where the corresponding +𝜓(𝑥, 𝑡) exhibits a singularity at time 𝑡 = 𝜏. For 𝜈 > 1/2, the cor- +responding 𝜓(𝑥, 𝑡) is finite everywhere and the first moment ⟨𝑥⟩ of +the associated probability density |𝜓(𝑥, 𝑡)|2 exists and it is equal to 0. +Finally, the second moment ⟨𝑥2⟩ is finite for 𝜈 > 3/4. The case 𝜈 = 1 +corresponds to the Lorentzian function. +Slightly generalizing results from Peres [5], we choose the +initial data to be +𝜓(𝑥, 0) = +1 +√N𝜈 +exp �− 𝑖𝑚 +2ℏ𝜏 𝑥2� +� +1 + 𝑥2 +𝜎2 +�𝜈 +, +(3) +where N𝜈 is a normalization constant, and 𝜏 and 𝜎 are real +numbers fixing the time scale and the width of the distribution, +respectively. One can check that for 𝜈 > 1/4, this function is in +the space 𝐿2(R), and so it is a physically admissible solution. +When this holds true, the normalization constant is finite and +equal to N𝜈 = √𝜋𝜎Γ(2𝜈 − 1/2)/Γ(2𝜈). +For 𝑡 ≠ 𝜏, we can apply the Riemann-Lebesgue lemma [23] +to show that the resulting 𝜓(𝑥, 𝑡) is continuous in 𝑥 and 𝑡 and +tends to zero as |𝑥| → ∞ (although not necessarily uniformly +with respect to 𝑡). However, at 𝑡 = 𝜏 a discontinuity occurs: at +this time the wave function reads +𝜓(𝑥, 𝜏) = +√︂ +𝑚 +2𝜋𝑖ℏN𝜈 +𝑒 +𝑖𝑚 +2ℏ𝜏 𝑥2 ∫ +R +𝑒−𝑖 𝑚 +ℏ𝜏 𝑥𝑥′ +�1 + 𝑥′2/𝜎2�𝜈 𝑑𝑥′ . +(4) +This integral is the Fourier transform of a Bessel potential [24] +and can thus be expressed as +𝜓(𝑥, 𝜏) = +√︄ +𝑚𝜎2 +𝑖ℏN𝜈 +𝑒 +𝑖𝑚 +2ℏ𝜏 𝑥2 +2𝜈−1Γ(𝜈) +�𝑚𝜎 +ℏ𝜏 |𝑥| +�𝜈− 1 +2 𝐾𝜈− 1 +2 +�𝑚𝜎 +ℏ𝜏 |𝑥| +� +, +(5) +which is valid for 𝜈 > 0. Here, 𝐾𝜈 denotes the modified Bessel +function of order 𝜈 [25], which is infinite at the origin but is +nevertheless square integrable. The function 𝜓(𝑥, 𝜏) is thus +continuous, except perhaps at 𝑥 = 0. To check the behavior +around that point, we use the approximation of 𝐾𝜈 for small +values of the argument. This leads +|𝑧|𝜈− 1 +2 𝐾𝜈− 1 +2 (|𝑧|) ≈ +Γ +� +𝜈 − 1 +2 +� +2 +3 +2 −𝜈 ++ +1 +|𝑧|1−2𝜈 +Γ +� +1 +2 − 𝜈 +� +2𝜈− 1 +2 ++𝑂(|𝑧|2𝜈+1) , +(6) +which shows that the singularity in 𝜓(𝑥, 𝜏) thus arises for +𝜈 < 1/2. In summary, when +1 +4 < 𝜈 < 1 +2 +(7) +we get the aforementioned singularity. +A similar analysis can be performed with the moments of +the associated probability density |𝜓(𝑥, 𝑡)|2 [11]. +The first +moment ⟨𝑥⟩ is finite and equal to zero when 𝜈 > 1/2, whereas +the second moment ⟨𝑥2⟩ exists provided that 𝜈 > 3/4. All this +relevant information is concisely summarized in Fig. 1. +B. +Quantum trajectories at the singularity +To explore the physical meaning of the singularity and, more +particularly, its dynamical emergence, we resort to the concept +of quantum trajectory. Apart from providing us with informa- +tion on the probability density distribution, the wave function +𝜓(𝑥, 𝑡) also contains dynamical information relevant to un- +derstand its time evolution. +The Bohmian picture stresses +this latter aspect, which manifests as quantum trajectories, +which are in compliance with the evolution of the quan- +tum flux [21]. To this end, one first decomposes 𝜓(𝑥, 𝑡) as +𝜓(𝑥, 𝑡) = +√︁ +𝜚(𝑥, 𝑡) exp[𝑖𝑆(𝑥, 𝑡)]/ℏ, which allows us to split up +the density information from the phase information encoded +in the wave function. Quantum trajectories are directly related +to the local variations undergone by the phase term, 𝑆(𝑥, 𝑡), +according to the so-called Bohmian guiding condition (or local +velocity field) [26], +�𝑥 = 𝐽(𝑥, 𝑡) +𝜚(𝑥, 𝑡) = 1 +𝑚 Re +� ˆ𝑝𝜓 +𝜓 +� += 1 +𝑚 +𝜕𝑆(𝑥, 𝑡) +𝜕𝑥 +, +(8) +with ˆ𝑝 = −𝑖ℏ𝜕/𝜕𝑥 being the usual momentum operator in +the position representation and 𝐽(𝑥, 𝑡) the probability current +density or quantum flux [27]. We stress that Eq. (8) constitutes +a general result that goes beyond any particular interpretation, +as it involves quantities that are well defined in any picture of +quantum mechanics. +More importantly, Eq. (8) explicitly shows the important +role played by the phase, not as an indirect effect (e.g., in +the appearance of interference features), but as a fundamental +quantity that specifies the local dynamics exhibited by the +quantum system on each point of the configuration space at +each time. This action emerges in the form of the local velocity +field that governs the dynamical evolution of the probability +density at any time, making it to move from a region to another, +to spread out all over the place, or, as it is the case here, to +coalesce on a highly localized region at a very precise time. +After all, note that the above local velocity field is what +allows us to establish the connection between the probability +density, 𝜚(𝑥, 𝑡), and the quantum flux, 𝐽(𝑥, 𝑡), according to the +well-known transport relation 𝐽(𝑥, 𝑡) = 𝑣(𝑥, 𝑡)𝜚(𝑥, 𝑡). Quan- +tum trajectories simply arise after assuming that 𝑣(𝑥, 𝑡) defines +an equation of motion that can be integrated in time, render- +ing as a result such trajectories. Physically, these trajectories +describe the flow of probability at a more local level than the + +3 +probability density itself does (to some extent, we can say that +this latter quantity provides us with a global view of what is +going on). A more detailed discussion on the issue can be +found in Ref. [28]. +For definiteness, we take the initial state (3), with 𝜈 = 1/3, to +ensure a singular wave packet. However, to produce a numer- +ically reliable (and physically more realistic) wave function, +instead of the initial ansatz (3), we consider the following +modified one +𝜓(𝑥, 0) = +1 +√N𝜈 +exp �− 𝑖𝑚 +2ℏ𝜏 𝑥2� +� +1 + 𝑥2 +𝜎 +� 1 +3 +� +1 + tanh +�𝑥 + 𝑥𝑏 +𝜎 +�� +× +� +1 + tanh +�𝑥 − 𝑥𝑏 +𝜎 +�� +, +(9) +where 𝑥𝑏 > 0. The two smooth step functions represented by +the hyperbolic tangents produce a relatively soft decay or cut- +off at distance 𝑥𝑏/𝜎 from the origin, which somehow mimics +the effect of a limited aperture with soft boundaries, avoiding +the appearance of spurious frequencies associated with a sud- +den cutoff or Gibbs phenomenon [29]. Because of the cutoff +introduced, it is expected that there will not be time symmetry +with respect to 𝑡 = 𝜏, although the time-evolved of (9) will +behave close to the exact solution. +We next perform a numerical integration of the evolution +(2) using a standard pseudospectral method on a spatial mess +of size 50𝜎 with a total of 1,024 grid points, integrating in +time from 𝑡 = 0 to 𝑡 = 2𝜏 with a time step 𝛿𝑡 = 10−3, which +suffices for our purposes. The numerical solution 𝜓(𝑥, 𝑡) is +monitored through both density plots of the corresponding +probability density and the associated quantum trajectories. A +density plot of the probability density is shown in Fig. 2a), +with a set of 51 trajectories (white solid lines) with equidistant +initial conditions between 𝑥/𝜎 = −15 and 𝑥/𝜎 = 15 to cover a +wide region of the initial probability density. We have chosen +𝑥𝑏/𝜎 = 22.5. +As it can be noticed, as time approaches the critical value 𝜏, +the swarm of trajectories quickly evolves towards the origin, +which turns into a prominent increase of the density within a +very narrow spatial region, thus originating the singularity. +This behavior can be better appreciated in the zoomed ver- +sion around the singular region displayed in Fig. 2a’). In the +same manner, as time proceeds and becomes larger than 𝜏, +the swarm of trajectories gets dispersed quickly again. It is +worth noting that, while the quantum flux is quite laminar +before and after the singularity, as it is indicated by the rel- +ative smoothness of the trajectories (they evolve with nearly +uniform motion), in the region around the singularity there +is a turbulent flow led by the appearance of transient nodes. +In their attempt for avoiding these nodes (nodal regions), the +trajectories will be forced to undergo a whirling motion. +III. +SINGULAR VERSUS SMOOTH WAVE PACKET +EVOLUTION +To better understand the singularity, we will next examine a +few characteristics of simpler but illustrative cases of smoothly +focusing wave packets. +A. +Gaussian wave packet +As it is well known, the evolution of a Gaussian wave packet +undergoes an initial boost or acceleration, and then it reaches a +stationary linear expansion [30]. Consider the initial normal- +ized Gaussian ansatz +𝜓(𝑥, 0) = +1 +√N𝐺 +exp +� +− 𝑥2 +4𝜎2 +0 +� +, +(10) +where 𝜎0 > 0 is a real-valued parameter determining the width +of the wave packet and the normalization constant is N𝐺 = +√︃ +2𝜋𝜎2 +0. Substituting this into the free-space propagator leads +to its time-evolved form, +𝜓(𝑥, 𝑡) = +1 +√N𝐺 +√︂ 𝜎0 +˜𝜎(𝑡) exp +� +− +𝑥2 +4𝜎0 ˜𝜎(𝑡) +� +, +(11) +where the Gaussian complex-valued parameter +˜𝜎(𝑡) = 𝜎0 +� +1 + +𝑖ℏ𝑡 +2𝑚𝜎2 +0 +� +(12) +accounts for both the spreading in time of the wave packet, +given by +𝜎(𝑡) = | ˜𝜎𝑡 | = 𝜎0 +� +� +� +1 + +� +ℏ𝑡 +2𝑚𝜎2 +0 +�2 +, +(13) +and the development of a space-dependent phase factor. +From the hydrodynamical point of view, the evolution of the +above wave function maps onto the trajectories arising from +the equation of motion +�𝑥 = +ℏ2𝑡 +(2𝑚𝜎2 +0)2 +𝜎2 +0 +𝜎(𝑡)2 𝑥. +(14) +After integration, this equation of motion renders the hyper- +bolic trajectories +𝑥(𝑡) = 𝜎(𝑡) +𝜎0 +𝑥(0). +(15) +From Eq. (14), it is clear that, for 𝑡 > 0, the trajectories are +“repelled” from the region where they are initially confined, +namely, the waist of the wave packet, since the sign of �𝑥 directly +depends on the sign of 𝑥 and hence on the corresponding initial +conditions. Although the initial expansion is slow, later on, +for 𝑡 ≫ 𝑡𝑠, with 𝑡𝑠 = 2𝑚𝜎2 +0/ℏ being a characteristic spreading +time, it becomes essentially linear with time; for 𝑡 ∼ 𝑡𝑠, the +expansion is accelerated, although at different rates as time +proceeds [19]. +All this information is nicely conveyed by the trajecto- +ries (15), which separate at a rate proportional to their initial + +4 +-10 +-5 +0 +5 +10 +0.0 +0.5 +1.0 +1.5 +2.0 +x / s +t / t +0.00 +0.12 +0.24 +0.36 +0.48 +(a) +(a') +-0.50 +-0.25 +0.00 +0.25 +0.50 +0.96 +0.98 +1.00 +1.02 +1.04 +x / s +t / t +0.00 +0.12 +0.24 +0.36 +0.48 +-10 +-5 +0 +5 +10 +0.0 +0.5 +1.0 +1.5 +2.0 +x / s +0,- +t / t +0.00 +0.06 +0.12 +0.18 +0.24 +(b) +(b') +-1.0 +-0.5 +0.0 +0.5 +1.0 +0.8 +0.9 +1.0 +1.1 +1.2 +x / s +0,- +t / t +0.00 +0.06 +0.12 +0.18 +0.24 +-10 +-5 +0 +5 +10 +0.0 +0.5 +1.0 +1.5 +2.0 +x / a +t / t +0.00 +0.09 +0.18 +0.27 +0.36 +(c) +(c') +-0.50 +-0.25 +0.00 +0.25 +0.50 +0.96 +0.98 +1.00 +1.02 +1.04 +x / a +t / t +0.00 +0.18 +0.36 +0.54 +0.72 +FIG. 2. +(Top panels) Quantum trajectories (51) displayed on top of a density plot describing the time evolution of the probability associated +with (a) the the wave function (3) with 𝜈 = 1/3, (b) the Gaussian (11) with waist width 𝜎0,−, and (c) the rectangular wave packet (19) with +width 𝑎. For clarity in the density plot, due to the high values of the probability density around the singularity, it has been truncated to a tenth +of its maximum value. (Bottom panels) Zoomed version of top panels around the focal region within the time interval where the maximum +concentration of probability density is reached. The whirls in the trajectories denote the appearance and disappearance of nodes as the wave +function approaches its maximum focusing. +distance, 𝑑(0) = |𝑥2(0) − 𝑥1(0)|, since 𝑑(𝑡)/𝑑(0) = 𝜎(𝑡)/𝜎0, +where 𝑑(𝑡) = |𝑥2(𝑡) − 𝑥1(𝑡)|. Taking into account (13), for the +same 𝑑(0), the largest 𝜎0, the slowest the dispersion, and vice +versa, in compliance with what is expected in this case. +So far there are no novelties. However, we stress that the +above solution is reversible in time, which means that, in the +same way that the wave packet undergoes an expansion, it can +also be tracked backwards. If the wave packet is then prop- +agated ahead again, it will evolve imploding until reaching +a minimum width (waist width), and then expanding again. +Taking into account the translational time invariance of the so- +lutions of the Schrödinger equation, if we call 𝜏 the time when +waist occurs, we can define a generalized Gaussian coefficient +as ˜𝜎𝑔(𝑡) = 𝜎(𝑡 − 𝜏). In this way the width and the phase of +the wave packet at time 𝑡 are given by +𝜎𝑔(𝑡) = 𝜎0 +� +� +� +1 + +� +ℏ(𝑡 − 𝜏) +2𝑚𝜎2 +0 +�2 +, +𝜃𝑔(𝑡) = arctan +� +ℏ(𝑡 − 𝜏) +2𝑚𝜎2 +0 +� +. +(16) +It is clear from these expressions that, at 𝑡 = 𝜏, we will observe +a minimum waist, with 𝜎𝑔(𝜏) = 𝜎0, and zero phase, 𝜃𝑔(𝜏) = 0. +Now, contrary to the standard case, we note that there are two +factors ruling the expansion dynamics: one associated with the +initial width and another one related to a phase, which play +opposite roles. If 𝜎0 is too large, the phase factor decreases +very rapidly, while a small width leads to a prominent phase +factor. This dependence is shown in Fig. 3, where the phase and +modulus are separately represented for a better understanding. +As it can be seen, 𝜎𝑔(0) has a minimum for 𝜎0 = +√︁ +𝜏/2, +increasing linearly with 𝜎0 for large widths and as 1/𝜎0 when +𝜎0 goes to zero. The associated phase approaches −𝜋/2 as 𝜎0 +decreases, while tends to vanish rapidly as 𝜎0 increases above +the threshold for minimum 𝜎𝑔(0). +From the above discussion, we may now consider the initial +Gaussian ansatz as in (10), but replacing 𝜎0 with 𝜎𝑔(0). The +associated time evolution can be directly obtained and leads to +the trajectories +𝑥(𝑡) = 𝜎𝑔(𝑡) +𝜎0 +𝑥(0). +(17) +As before, these trajectories undergo an initial implosion, until +𝑡 = 𝜏, and then a subsequent expansion. +The question is +how important the effect is, particularly taking into account +that two different values of 𝜎0, as it can readily be noticed +from (16), can be associated with the same initial probability +density. These two values will lead to very different dynamical +behaviors. Thus, fixing the value of 𝜎𝑔(0), from (16) we obtain +the following two admissible values for the waist width +𝜎2 +0,± = 1 +2𝜎2 +𝑔(0) ± +√︄ +𝜎4𝑔(0) − +� ℏ𝜏 +2𝑚 +�2 +. +(18) +To quantify the above effect, we consider a Gaussian wave +packet with the (initial) width of its probability density at a + +5 +0 +1 +2 +3 +1 +2 +3 +s +g,0 + (a.u.) +s +0 + (a.u.) +-0.5p +-0.4p +-0.3p +-0.2p +-0.1p +0.0p +q +g,0 +FIG. 3. +Dependence of the phase (green line) and modulus (black +line) of the initial complex-valued Gaussian parameter ˜𝜎𝑔 on the waist +width, 𝜎0, for 𝑡/𝜏 = 1. The vertical blue dotted lines denote the values +of the phase and modulus of ˜𝜎𝑐 that correspond to Gaussians such +that their width at 0.1 of their maximum value equals the same value +of the probability density corresponding to the wave function. The +horizontal red dashed line shows that there are always two Gaussian +wave packets with the same initial width, but that lead to two different +waist widths (in this case, 𝜎𝑔 ≃ 2.579 is associated with 𝜎0,+ ≃ 2.571 +and 𝜎0,− ≃ 0.194). Despite having the same value for 𝜎𝑔, each +Gaussian wave packet has a very different initial phase, in particular, +𝜃𝑔,+ ≃ −0.024𝜋 versus 𝜃𝑔,− ≃ −0.477𝜋. +tenth of the maximum value; i.e., 𝜚𝐺(𝑠±, 0)/𝜚𝐺(0, 0) = 0.1, +equal to the corresponding value for the (modified) singular +wave function (9). This yields an initial width for both wave +packets given by 𝜎2 +𝑔(0) = (10 +√ +10 − 1)/(2 ln 10) ≃ 6.6512, +which gives the waist widths 𝜎0,+ ≃ 2.571 and 𝜎0,− ≃ 0.194. +When compared with the value for 𝜎𝑔(0), we notice that while +𝜎0,+ is practically the same [∼ 99% 𝜎𝑔(0)], which already +indicates a poor dynamics, 𝜎0,− is significantly different [∼ +7.5% 𝜎𝑔(0)] and hence a more relevant dynamical behavior is +expected. +The above expectations translate into the results displayed +in Fig. 2b) for 𝜎0,−. The characteristic time scale here is 𝑡𝑠,− ≃ +0.15, about a tenth of 𝜏 and hence with noticeable effects both +in the implosion and, afterwards, in the subsequent dispersion. +Note here that there is a more important phase contribution, +since 𝜃𝑔,−(0) ≃ −0.38𝜋, a value closer to the maximum bound +for the phase. Nonetheless, unlike the singular wave packet, +here near the singular region the flux is not turbulent, which is +consistent with the fact that the evolution of a Gaussian wave +packet is characterized by the absence of nodes. +In Fig. 4 we plot the reverse case of a Gaussian wave packet +for 𝜎0,+. We can appreciate that the wave packet remains un- +affected, with the flux described by the swarm of 51 Bohmian +trajectories being nearly stationary. The characteristic spread- +ing time scale is 𝑡𝑠,+ ≃ 26.4𝜏, which implies that neither the +evolution before 𝜏 nor afterwards is going to be importantly +affected. +Indeed. +the initial phase is 𝜃𝑔,+(0) ≃ −0.061𝜋, +which already indicates the rather small contribution of the +-10 +-5 +0 +5 +10 +0.0 +0.5 +1.0 +1.5 +2.0 +x / s +0, + +t / t +0.00 +0.02 +0.04 +0.06 +0.08 +0.0 +0.5 +1.0 +1.5 +2.0 +-10 +-5 +0 +5 +10 +FIG. 4. +Same trajectories as in Fig. 2b) for the probability asso- +ciated with a Gaussian wave packet with waist width 𝜎0,+. Note +that, because the waist width is relatively large compared to the ini- +tial width 𝜎𝑔(0), there is no apparent self-implosion (only a very +slight narrowing at 𝜏), as it is evidenced by the nearly parallel flux +trajectories. +phase factor in the dynamics. +In Fig. 5 we represent the probability densities associated +with these initial Gaussian wave packets. Interestingly, these +probability densities are indistinguishable in position space, +but they are completely different in momentum space: the +momentum distribution for 𝜎0,+ is rather wide, while for 𝜎0,− +it approaches a Dirac delta function. It is precisely this wider +momentum distribution that allows the second wave packet to +coalesce toward the origin as the time approaches 𝜏, similarly +to the singular wave function, while the first wave packet will +remain essentially the same. +B. +Rectangular wave packet +As our last example, we consider a rectangular wave +packet [31], with an initial profile +𝜓(𝑥, 0) = +1 +√N𝑟 +exp +� +− 𝑖𝑚 +2ℏ𝜏 𝑥2 +� +rect𝑎(𝑥) , +(19) +where the rectangle function rect𝑎(𝑥) is defined as 1 for |𝑥| ≤ +𝑎/2 and 0 for |𝑥| > 𝑎/2 and the normalization constant is +N𝑟 = 𝑎. The time evolution can be found using again (2), +finding [31] +𝜓(𝑥, 𝑡) = (−1)3/4 +√4𝑖N𝑟 +exp +� 𝑖𝑚 +2ℏ𝜏 𝑥2 +� � +erfi +� +(−1)1/4 +√︂ 𝑚 +2ℏ𝑡 +� +𝑥 − 𝑎 +2 +�� +− erfi +� +(−1)1/4 +√︂ 𝑚 +2ℏ𝑡 +� +𝑥 + 𝑎 +2 +��� +, +(20) +where erfi(𝑥) is the imaginary error function and this is valid +for 𝑡 > 0. +The wave packet is composed of an infinite number of plane +waves. +At time 𝑡 = 0 these plane waves interfere to give + +6 +-20 +-10 +0 +10 +20 +10 +-3 +10 +-2 +10 +-1 +10 +0 +10 +1 +-20 +-10 +0 +10 +20 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +|y(x,0)| +2 + / |y(0,0)| +2 +x / s +y +(a) +|y(k)| +2 +s +y + k +(b) +-25 +0 +25 +0.0 +0.5 +1.0 +FIG. 5. Probability density in the 𝑥-position space (upper panel) and +in the 𝑘-momentum space (lower panel) for the singular wave function +(black solid line), a Gaussian wave packet with waist width 𝜎0,+ (red +dashed line), and a Gaussian wave packet with waist width 𝜎0,− (blue +dotted line), and a rectangular wave packet (green dash-dotted line) +for 𝑡 = 0. The inset shows the same plots on a linear vertical scale. +The waist widths for both Gaussians have been adjusted to the width +of the probability density for the singular wave function at 0.1 of its +maximum value. +a rectangular shape. As time elapses, the component plane +waves travel, both to the right (𝑘 > 0) and to the left (𝑘 < 0), +at different phase velocities ℏ𝑘/2𝑚. Thus the pattern of the +interference of these plane waves gradually changes, resulting +in the dispersion of the wave packet. +In Fig. 2c), we plot the quantum trajectories associated with +this evolution. Near the time 𝜏, we appreciate the presence of +wiggles for both the singular and the rectangular wave packets, +which remind of a nonlaminar flux. Conversely, the Gaussian +profile looks perfectly laminar nearby the singular point. We +recall that a flood in a river occurs because at some point water +slows down and the quicker mass of water arriving from behind +finds this “potential barrier" created by the slow water and +tries to overcome it. In this case, the wiggles mark somehow +a slower light flow, so that energy accumulates nearby the +0.0 +0.5 +1.0 +1.5 +2.0 +0 +10 +20 +30 +40 +FWHM / s +y +t / t +0.0 +0.5 +1.0 +1.5 +2.0 +0 +2 +4 +6 +FIG. 6. Evolution temporal of the FWHM for the same wave packets +as in Fig. 5, with the same symbols: the singular wave function (black +solid line), a Gaussian wave packet with waist width 𝜎0,+ (red dashed +line), a Gaussian wave packet with waist width 𝜎0,− (blue dotted +line), and a rectangular wave packet (green dash-dotted line). +singularity and the density grows. +An alternative way to capture the degree of localization of +a wave function is by studying the behavior exhibited its full +width at half maximum (FWHM) [32]. +More specifically, +this quantity is computed in all cases determining the distance +between the two positions, 𝑥+ and 𝑥−, at which the correspond- +ing probability density reaches half its maximum value at any +time; that is +𝜚(𝑥±, 𝑡) +𝜚max(𝑥, 𝑡) = 1 +2. +(21) +Except for Gaussian wave packets, the above equation can- +not be solved analytically, so 𝑥+ and 𝑥− have been numer- +ically determined on the fly, during the time-evolution of +the corresponding wave functions. +From this, we obtain +FWHM(𝑡) = 𝑥+(𝑡) − 𝑥−(𝑡), which is shown in Fig. 6 for the +for cases here considered. +As it can be noticed, while the +FWHM is nearly constant for the Gaussian with waist width +𝜎0,+, it shows a linear decrease and increase, before and after +the waist, respectively, for the Gaussian with 𝜎0,−. A similar +trend is also observed for the square wave function, although +the FWHM shows a tiny asymmetry before and after the sin- +gularity, which is related to the limitations involved in the +numerical method (the spatial size of the grid sets a cutoff for +the high spatial frequencies). Finally, for the singular wave +function (3), the FWHM slowly decrease until 𝑡 is close to +𝜏, as it can be appreciated in the inset of Fig. 6. Near this +time, the FWHM undergoes a sudden decrease and then in- +crease afterwards; at any later time, the FWHM increase near +linearly, in a similar fashion to the Gaussian with 𝜎0,−. We no- +tice again a different behavior between the FWHM dynamics +before and after 𝑡 = 𝜏, which is related to the fact that the wave +function considered is not exactly the ansatz eqrefeq:psi0), but + +7 +the truncated version (9). All these characteristics concur with +the corresponding probability density and quantum trajectories +displayed in Fig. 2. +IV. +CONCLUDING REMARKS +To summarize, we have studied a family of solutions of the +Schrödinger equation that spontaneously develop a singular- +ity while propagating in free space. Due to the finiteness of +these solutions, their singularities do not require a nonphys- +ical infinite amount of energy to manifest. Nevertheless, the +local amplitude of the field at a singular point may grow un- +boundedly. We have given a physical interpretation in terms +of quantum trajectories. +While there is a widespread belief that extreme focusing +requires strong nonlinear effects, we have demonstrated that +this can be easily achieved with only linear propagation. This +promising field enhancement mechanism may foster further +interesting research in fields such as electron microscopy or +optics. +ACKNOWLEDGMENTS +Financial support is acknowledged to the Spanish Research +Agency (Grant No. PID2021-127781NB-I00). AA acknowl- +edges support from Deutsche Forschungsgemeinschaft (Grant +No. 429529648-TRR 306). +[1] T. Tao, Nonlinear dispersive equations. Local and global anal- +ysis, CBMS Regional Conference Series in Mathematics, Vol. +106 (AMS, Providence, RI, 2006). +[2] W. Schlag, Mathematical aspects on nonlinear dispersive equa- +tions (Princeton University Press, Princeton, 2007). +[3] R. Mandel, Dispersive estimates, blow-up and failure of +strichartz estimates for the schrödinger equation with slowly +decaying initial data, Pure Appl. Anal. 2, 519 (2020). +[4] C. Dietze, Dispersive estimates for nonlinear Schrödinger equa- +tions with external potentials, J. Math. Phys. 62, 111502 (2021). +[5] A. 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Miret-Artés, Quantum phase analysis with +quantum trajectories: A step towards the creation of a bohmian +thinking, Am. J. Phys. 80, 525 (2012). +[20] D. H. Mahler, L. Rozema, K. Fisher, L. Vermeyden, K. J. Resch, +H. M. Wiseman, and A. Steinberg, Experimental nonlocal and +surreal Bohmian trajectories, Sci. Adv. 2, e1501466 (2016). +[21] A. S. Sanz, Bohm’s approach to quantum mechanics: Alterna- +tive theory or practical picture?, Front. Phys. 14, 11301 (2019). +[22] E. Merzbacher, Quantum Mechanics, 3rd ed. (Wiley, New York, +1998). +[23] R. J. Iorio and V. M. Iorio, Fourier Analysis and Partial Dif- +ferential Equations (Cambridge University Press, Cambridge, +2001). +[24] N. Aronszajn and K. T. Smith, Theory of Bessel potentials. I, +Ann. Inst. Fourier 11, 385 (1961). +[25] DLMF, NIST Digital Library of Mathematical Functions, +http://dlmf.nist.gov/, Release 1.1.8 of 2022-12-15, f. W. J. Olver, +A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. +Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, +and M. A. McClain, eds. +[26] P. R. Holland, The Quantum Theory of Motion (Cambridge Uni- +versity Press, Cambridge, 1993). +[27] L. I. Schiff, Quantum Mechanics, 3rd ed. (McGraw-Hill, Singa- +pore, 1968). +[28] A. S. Sanz, Bohm’s quantum “non-mechanics”: An alterna- +tive quantum theory with its own ontology?, Ann. Fond. Louis +Broglie 46, 19 (2021). +[29] E. Hewitt and R. E. Hewitt, The Gibbs-Wilbraham phenomenon: +An episode in Fourier analysis, Arch. Hist. Exact Sci. 21, 129 +(1979). +[30] A. S. Sanz and S. Miret-Artés, A Trajectory Description of Quan- +tum Processes. II. Applications, Lecture Notes in Physics, Vol. +831 (Springer, Berlin, 2014). +[31] K. Mita, Dispersion of non-Gaussian free particle wave packets, +Am. J. Phys. 75, 950 (2007). +[32] A. García-Sánchez and A. S. Sanz, Analysis of the gradual +transition from the near to the far field in single-slit diffraction, +Phys. Scr. 97, 055507 (2022). + diff --git a/p9FPT4oBgHgl3EQf8DXy/content/tmp_files/load_file.txt b/p9FPT4oBgHgl3EQf8DXy/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..28cb4d8445131f81c962bbfb9a7376b9510b478c --- /dev/null +++ b/p9FPT4oBgHgl3EQf8DXy/content/tmp_files/load_file.txt @@ -0,0 +1,540 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf,len=539 +page_content='A quantum trajectory analysis of singular wave functions Angel S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' Sanz,1 Luis L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' Sánchez-Soto,1, 2 and Andrea Aiello2 1Departamento de Óptica, Facultad de Física, Universidad Complutense, 28040 Madrid, Spain 2Max-Planck-Institut für die Physik des Lichts, 91058 Erlangen, Germany (Dated: February 1, 2023) The Schrödinger equation admits smooth and finite solutions that spontaneously evolve into a singularity, even for a free particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' This blowup is generally ascribed to the intrinsic dispersive character of the associated time evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' We resort to the notion of quantum trajectories to reinterpret this singular behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' We show that the blowup can be directly related to local phase variations, which generate an underlying velocity field responsible for driving the quantum flux toward the singular region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' INTRODUCTION The Schrödinger equation is, perhaps, the prototype of a dispersive equation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' that is, if no boundary conditions are imposed, its wave solutions spread out in space as they evolve in time [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' A frequent way to quantify this dispersion is by the so-called dispersive estimates, a topic with a long history [2–4] and whose main goal is to establish tight bounds on the decay of the solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' Recently, it has been pointed out that the Schrödinger equa- tion, even for a free particle, presents dispersive singulari- ties [5, 6]: an initial square-integrable profile 𝜓(𝑥, 0) could result in a solution 𝜓(𝑥, 𝑡) that blows up in a finite time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' In the remainder such profiles will be termed as singular wave packets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' While this singular behavior (sometimes denoted as self-focusing or wave collapse) is well understood in presence of nonlinearities [7–9], it is, at first sight, surprising in a pure linear evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' From a mathematical viewpoint, this dispersive blowup can be related to the fact that the linear Schrödinger equation is ill-posed in the space 𝐿∞: the free propagator is not a Fourier multiplier in 𝐿∞ [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' In physical terms, dispersive blowup is a focusing phenomenon due to both the unbounded do- main of the problem and the propensity of the dispersion re- lation to propagating energy at different speeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' Interestingly, the same singular behavior has been described in for paraxial beams [11–13], which is consequent with the complete equiv- alence between the time-dependent Schrödinger equation and the paraxial wave equation [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' In this paper, we address the physical interpretation of these singularities from the perspective of quantum trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' In this picture, quantum formalism is reinterpreted as describing particles following definite trajectories, each with a precisely defined position at each instant in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' However, in this ap- proach, called Bohmian mechanics [15–17], the trajectories of the particles are quite different from those of classical parti- cles because they are guided by the wave function [18–21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' Our analysis shows that the blowup can be directly related to local phase variations, which generate an underlying velocity field (the phase gradient) responsible for driving the quantum flux toward the singular region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' To shed light on this point, we compare the blowup with the focusing of a Gaussian and a rectangular wave packet: this demonstrates that imploding solutions are distinguished by an initial phase factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' Furthermore, for Gaussian wave packets, which can be nicely analyzed in closed form, it is also observed that there are two types of solutions with very different properties, de- spite their initial density distributions being identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' One of such solutions leads to a classical type of propagation because the phase factor plays a minor role (or even no role at all).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' In contradistinction, the other type of solution is characterized by wide initial wave functions with an intrinsic highly oscillatory behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' This emphasizes the prominent role of the phase as an active agent in the subsequent dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' This article is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' II we briefly discuss the spontaneous generation of a singularity in the Schrödinger equation and introduce the basic elements needed to define a quantum trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' In terms of this notion, we analyze the singularity and put forward the fundamental role played by the quantum phase to understand that phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' III we examine the behavior of a Gaussian and a rect- angular packet and compare with the previous singular wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' Finally, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' IV summarizes our conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' DISPERSIVE BLOWUP IN THE SCHRÖDINGER EQUATION A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' Spontaneous generation of a singularity We first set the stage for our discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' We will be consid- ering the simplest case of the Schrödinger equation for a free particle of mass 𝑚 in one dimension 𝑖ℏ𝜕𝜓(𝑥, 𝑡) 𝜕𝑡 = − ℏ2 2𝑚 𝜕2𝜓(𝑥, 𝑡) 𝜕𝑥2 , (1) with the initial Cauchy problem 𝜓(𝑥, 0) ∈ 𝐿2(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' The unique solution of (1) can be written in terms of the free-space prop- agator as [22] 𝜓(𝑥, 𝑡) = √︂ 𝑚 2𝜋𝑖ℏ𝑡 ∫ R exp � 𝑖𝑚 2ℏ𝑡 (𝑥 − 𝑥′)2 � 𝜓(𝑥′, 0) 𝑑𝑥′ , (2) where the integral has to be understood in the improper Rie- mann sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' In this way, the Schrödinger equation appears as an integral equation, rather than a differential one, with the advantage of being valid even if the wave function is not a differentiable function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content='13207v1 [quant-ph] 30 Jan 2023 2 singular regime square-integrable Lorentzian 1 moment st 2 moment nd FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' For 𝜈 > 1/4 the wave function (3) is square integrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' The red band indicates the range 1/4 < 𝜈 < 1/2 where the corresponding 𝜓(𝑥, 𝑡) exhibits a singularity at time 𝑡 = 𝜏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' For 𝜈 > 1/2, the cor- responding 𝜓(𝑥, 𝑡) is finite everywhere and the first moment ⟨𝑥⟩ of the associated probability density |𝜓(𝑥, 𝑡)|2 exists and it is equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' Finally, the second moment ⟨𝑥2⟩ is finite for 𝜈 > 3/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' The case 𝜈 = 1 corresponds to the Lorentzian function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' Slightly generalizing results from Peres [5], we choose the initial data to be 𝜓(𝑥, 0) = 1 √N𝜈 exp �− 𝑖𝑚 2ℏ𝜏 𝑥2� � 1 + 𝑥2 𝜎2 �𝜈 , (3) where N𝜈 is a normalization constant, and 𝜏 and 𝜎 are real numbers fixing the time scale and the width of the distribution, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' One can check that for 𝜈 > 1/4, this function is in the space 𝐿2(R), and so it is a physically admissible solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' When this holds true, the normalization constant is finite and equal to N𝜈 = √𝜋𝜎Γ(2𝜈 − 1/2)/Γ(2𝜈).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' For 𝑡 ≠ 𝜏, we can apply the Riemann-Lebesgue lemma [23] to show that the resulting 𝜓(𝑥, 𝑡) is continuous in 𝑥 and 𝑡 and tends to zero as |𝑥| → ∞ (although not necessarily uniformly with respect to 𝑡).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' However, at 𝑡 = 𝜏 a discontinuity occurs: at this time the wave function reads 𝜓(𝑥, 𝜏) = √︂ 𝑚 2𝜋𝑖ℏN𝜈 𝑒 𝑖𝑚 2ℏ𝜏 𝑥2 ∫ R 𝑒−𝑖 𝑚 ℏ𝜏 𝑥𝑥′ �1 + 𝑥′2/𝜎2�𝜈 𝑑𝑥′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' (4) This integral is the Fourier transform of a Bessel potential [24] and can thus be expressed as 𝜓(𝑥, 𝜏) = √︄ 𝑚𝜎2 𝑖ℏN𝜈 𝑒 𝑖𝑚 2ℏ𝜏 𝑥2 2𝜈−1Γ(𝜈) �𝑚𝜎 ℏ𝜏 |𝑥| �𝜈− 1 2 𝐾𝜈− 1 2 �𝑚𝜎 ℏ𝜏 |𝑥| � , (5) which is valid for 𝜈 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' Here, 𝐾𝜈 denotes the modified Bessel function of order 𝜈 [25], which is infinite at the origin but is nevertheless square integrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' The function 𝜓(𝑥, 𝜏) is thus continuous, except perhaps at 𝑥 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' To check the behavior around that point, we use the approximation of 𝐾𝜈 for small values of the argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' This leads |𝑧|𝜈− 1 2 𝐾𝜈− 1 2 (|𝑧|) ≈ Γ � 𝜈 − 1 2 � 2 3 2 −𝜈 + 1 |𝑧|1−2𝜈 Γ � 1 2 − 𝜈 � 2𝜈− 1 2 +𝑂(|𝑧|2𝜈+1) , (6) which shows that the singularity in 𝜓(𝑥, 𝜏) thus arises for 𝜈 < 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' In summary, when 1 4 < 𝜈 < 1 2 (7) we get the aforementioned singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' A similar analysis can be performed with the moments of the associated probability density |𝜓(𝑥, 𝑡)|2 [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' The first moment ⟨𝑥⟩ is finite and equal to zero when 𝜈 > 1/2, whereas the second moment ⟨𝑥2⟩ exists provided that 𝜈 > 3/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' All this relevant information is concisely summarized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' Quantum trajectories at the singularity To explore the physical meaning of the singularity and, more particularly, its dynamical emergence, we resort to the concept of quantum trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' Apart from providing us with informa- tion on the probability density distribution, the wave function 𝜓(𝑥, 𝑡) also contains dynamical information relevant to un- derstand its time evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' The Bohmian picture stresses this latter aspect, which manifests as quantum trajectories, which are in compliance with the evolution of the quan- tum flux [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' To this end, one first decomposes 𝜓(𝑥, 𝑡) as 𝜓(𝑥, 𝑡) = √︁ 𝜚(𝑥, 𝑡) exp[𝑖𝑆(𝑥, 𝑡)]/ℏ, which allows us to split up the density information from the phase information encoded in the wave function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' Quantum trajectories are directly related to the local variations undergone by the phase term, 𝑆(𝑥, 𝑡), according to the so-called Bohmian guiding condition (or local velocity field) [26], �𝑥 = 𝐽(𝑥, 𝑡) 𝜚(𝑥, 𝑡) = 1 𝑚 Re � ˆ𝑝𝜓 𝜓 � = 1 𝑚 𝜕𝑆(𝑥, 𝑡) 𝜕𝑥 , (8) with ˆ𝑝 = −𝑖ℏ𝜕/𝜕𝑥 being the usual momentum operator in the position representation and 𝐽(𝑥, 𝑡) the probability current density or quantum flux [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' We stress that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' (8) constitutes a general result that goes beyond any particular interpretation, as it involves quantities that are well defined in any picture of quantum mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' More importantly, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' (8) explicitly shows the important role played by the phase, not as an indirect effect (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=', in the appearance of interference features), but as a fundamental quantity that specifies the local dynamics exhibited by the quantum system on each point of the configuration space at each time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' This action emerges in the form of the local velocity field that governs the dynamical evolution of the probability density at any time, making it to move from a region to another, to spread out all over the place, or, as it is the case here, to coalesce on a highly localized region at a very precise time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' After all, note that the above local velocity field is what allows us to establish the connection between the probability density, 𝜚(𝑥, 𝑡), and the quantum flux, 𝐽(𝑥, 𝑡), according to the well-known transport relation 𝐽(𝑥, 𝑡) = 𝑣(𝑥, 𝑡)𝜚(𝑥, 𝑡).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' Quan- tum trajectories simply arise after assuming that 𝑣(𝑥, 𝑡) defines an equation of motion that can be integrated in time, render- ing as a result such trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' Physically, these trajectories describe the flow of probability at a more local level than the 3 probability density itself does (to some extent, we can say that this latter quantity provides us with a global view of what is going on).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' A more detailed discussion on the issue can be found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' For definiteness, we take the initial state (3), with 𝜈 = 1/3, to ensure a singular wave packet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' However, to produce a numer- ically reliable (and physically more realistic) wave function, instead of the initial ansatz (3), we consider the following modified one 𝜓(𝑥, 0) = 1 √N𝜈 exp �− 𝑖𝑚 2ℏ𝜏 𝑥2� � 1 + 𝑥2 𝜎 � 1 3 � 1 + tanh �𝑥 + 𝑥𝑏 𝜎 �� × � 1 + tanh �𝑥 − 𝑥𝑏 𝜎 �� , (9) where 𝑥𝑏 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' The two smooth step functions represented by the hyperbolic tangents produce a relatively soft decay or cut- off at distance 𝑥𝑏/𝜎 from the origin, which somehow mimics the effect of a limited aperture with soft boundaries, avoiding the appearance of spurious frequencies associated with a sud- den cutoff or Gibbs phenomenon [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' Because of the cutoff introduced, it is expected that there will not be time symmetry with respect to 𝑡 = 𝜏, although the time-evolved of (9) will behave close to the exact solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' We next perform a numerical integration of the evolution (2) using a standard pseudospectral method on a spatial mess of size 50𝜎 with a total of 1,024 grid points, integrating in time from 𝑡 = 0 to 𝑡 = 2𝜏 with a time step 𝛿𝑡 = 10−3, which suffices for our purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' The numerical solution 𝜓(𝑥, 𝑡) is monitored through both density plots of the corresponding probability density and the associated quantum trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' A density plot of the probability density is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' 2a), with a set of 51 trajectories (white solid lines) with equidistant initial conditions between 𝑥/𝜎 = −15 and 𝑥/𝜎 = 15 to cover a wide region of the initial probability density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' We have chosen 𝑥𝑏/𝜎 = 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' As it can be noticed, as time approaches the critical value 𝜏, the swarm of trajectories quickly evolves towards the origin, which turns into a prominent increase of the density within a very narrow spatial region, thus originating the singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' This behavior can be better appreciated in the zoomed ver- sion around the singular region displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' 2a’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' In the same manner, as time proceeds and becomes larger than 𝜏, the swarm of trajectories gets dispersed quickly again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' It is worth noting that, while the quantum flux is quite laminar before and after the singularity, as it is indicated by the rel- ative smoothness of the trajectories (they evolve with nearly uniform motion), in the region around the singularity there is a turbulent flow led by the appearance of transient nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' In their attempt for avoiding these nodes (nodal regions), the trajectories will be forced to undergo a whirling motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' SINGULAR VERSUS SMOOTH WAVE PACKET EVOLUTION To better understand the singularity, we will next examine a few characteristics of simpler but illustrative cases of smoothly focusing wave packets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' Gaussian wave packet As it is well known, the evolution of a Gaussian wave packet undergoes an initial boost or acceleration, and then it reaches a stationary linear expansion [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' Consider the initial normal- ized Gaussian ansatz 𝜓(𝑥, 0) = 1 √N𝐺 exp � − 𝑥2 4𝜎2 0 � , (10) where 𝜎0 > 0 is a real-valued parameter determining the width of the wave packet and the normalization constant is N𝐺 = √︃ 2𝜋𝜎2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' Substituting this into the free-space propagator leads to its time-evolved form, 𝜓(𝑥, 𝑡) = 1 √N𝐺 √︂ 𝜎0 ˜𝜎(𝑡) exp � − 𝑥2 4𝜎0 ˜𝜎(𝑡) � , (11) where the Gaussian complex-valued parameter ˜𝜎(𝑡) = 𝜎0 � 1 + 𝑖ℏ𝑡 2𝑚𝜎2 0 � (12) accounts for both the spreading in time of the wave packet, given by 𝜎(𝑡) = | ˜𝜎𝑡 | = 𝜎0 � � � 1 + � ℏ𝑡 2𝑚𝜎2 0 �2 , (13) and the development of a space-dependent phase factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' From the hydrodynamical point of view, the evolution of the above wave function maps onto the trajectories arising from the equation of motion �𝑥 = ℏ2𝑡 (2𝑚𝜎2 0)2 𝜎2 0 𝜎(𝑡)2 𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' (14) After integration, this equation of motion renders the hyper- bolic trajectories 𝑥(𝑡) = 𝜎(𝑡) 𝜎0 𝑥(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' (15) From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' (14), it is clear that, for 𝑡 > 0, the trajectories are “repelled” from the region where they are initially confined, namely, the waist of the wave packet, since the sign of �𝑥 directly depends on the sign of 𝑥 and hence on the corresponding initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' Although the initial expansion is slow, later on, for 𝑡 ≫ 𝑡𝑠, with 𝑡𝑠 = 2𝑚𝜎2 0/ℏ being a characteristic spreading time, it becomes essentially linear with time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' for 𝑡 ∼ 𝑡𝑠, the expansion is accelerated, although at different rates as time proceeds [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' All this information is nicely conveyed by the trajecto- ries (15), which separate at a rate proportional to their initial 4 10 5 0 5 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content='02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content='04 x / a t / t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content='72 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' (Top panels) Quantum trajectories (51) displayed on top of a density plot describing the time evolution of the probability associated with (a) the the wave function (3) with 𝜈 = 1/3, (b) the Gaussian (11) with waist width 𝜎0,−, and (c) the rectangular wave packet (19) with width 𝑎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' For clarity in the density plot, due to the high values of the probability density around the singularity, it has been truncated to a tenth of its maximum value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' (Bottom panels) Zoomed version of top panels around the focal region within the time interval where the maximum concentration of probability density is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' The whirls in the trajectories denote the appearance and disappearance of nodes as the wave function approaches its maximum focusing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' distance, 𝑑(0) = |𝑥2(0) − 𝑥1(0)|, since 𝑑(𝑡)/𝑑(0) = 𝜎(𝑡)/𝜎0, where 𝑑(𝑡) = |𝑥2(𝑡) − 𝑥1(𝑡)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' Taking into account (13), for the same 𝑑(0), the largest 𝜎0, the slowest the dispersion, and vice versa, in compliance with what is expected in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' So far there are no novelties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' However, we stress that the above solution is reversible in time, which means that, in the same way that the wave packet undergoes an expansion, it can also be tracked backwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' If the wave packet is then prop- agated ahead again, it will evolve imploding until reaching a minimum width (waist width), and then expanding again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' Taking into account the translational time invariance of the so- lutions of the Schrödinger equation, if we call 𝜏 the time when waist occurs, we can define a generalized Gaussian coefficient as ˜𝜎𝑔(𝑡) = 𝜎(𝑡 − 𝜏).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' In this way the width and the phase of the wave packet at time 𝑡 are given by 𝜎𝑔(𝑡) = 𝜎0 � � � 1 + � ℏ(𝑡 − 𝜏) 2𝑚𝜎2 0 �2 , 𝜃𝑔(𝑡) = arctan � ℏ(𝑡 − 𝜏) 2𝑚𝜎2 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' (16) It is clear from these expressions that, at 𝑡 = 𝜏, we will observe a minimum waist, with 𝜎𝑔(𝜏) = 𝜎0, and zero phase, 𝜃𝑔(𝜏) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' Now, contrary to the standard case, we note that there are two factors ruling the expansion dynamics: one associated with the initial width and another one related to a phase, which play opposite roles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' If 𝜎0 is too large, the phase factor decreases very rapidly, while a small width leads to a prominent phase factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' This dependence is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' 3, where the phase and modulus are separately represented for a better understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' As it can be seen, 𝜎𝑔(0) has a minimum for 𝜎0 = √︁ 𝜏/2, increasing linearly with 𝜎0 for large widths and as 1/𝜎0 when 𝜎0 goes to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' The associated phase approaches −𝜋/2 as 𝜎0 decreases, while tends to vanish rapidly as 𝜎0 increases above the threshold for minimum 𝜎𝑔(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' From the above discussion, we may now consider the initial Gaussian ansatz as in (10), but replacing 𝜎0 with 𝜎𝑔(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' The associated time evolution can be directly obtained and leads to the trajectories 𝑥(𝑡) = 𝜎𝑔(𝑡) 𝜎0 𝑥(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' (17) As before, these trajectories undergo an initial implosion, until 𝑡 = 𝜏, and then a subsequent expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' The question is how important the effect is, particularly taking into account that two different values of 𝜎0, as it can readily be noticed from (16), can be associated with the same initial probability density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' These two values will lead to very different dynamical behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' Thus, fixing the value of 𝜎𝑔(0), from (16) we obtain the following two admissible values for the waist width 𝜎2 0,± = 1 2𝜎2 𝑔(0) ± √︄ 𝜎4𝑔(0) − � ℏ𝜏 2𝑚 �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' (18) To quantify the above effect, we consider a Gaussian wave packet with the (initial) width of its probability density at a 5 0 1 2 3 1 2 3 s g,0 (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=') s 0 (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=') 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content='5p 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content='4p 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content='3p 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content='2p 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content='1p 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content='0p q g,0 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' Dependence of the phase (green line) and modulus (black line) of the initial complex-valued Gaussian parameter ˜𝜎𝑔 on the waist width, 𝜎0, for 𝑡/𝜏 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' The vertical blue dotted lines denote the values of the phase and modulus of ˜𝜎𝑐 that correspond to Gaussians such that their width at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content='1 of their maximum value equals the same value of the probability density corresponding to the wave function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' The horizontal red dashed line shows that there are always two Gaussian wave packets with the same initial width, but that lead to two different waist widths (in this case, 𝜎𝑔 ≃ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content='579 is associated with 𝜎0,+ ≃ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content='571 and 𝜎0,− ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content='194).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' Despite having the same value for 𝜎𝑔, each Gaussian wave packet has a very different initial phase, in particular, 𝜃𝑔,+ ≃ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content='024𝜋 versus 𝜃𝑔,− ≃ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content='477𝜋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' tenth of the maximum value;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=', 𝜚𝐺(𝑠±, 0)/𝜚𝐺(0, 0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content='1, equal to the corresponding value for the (modified) singular wave function (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' This yields an initial width for both wave packets given by 𝜎2 𝑔(0) = (10 √ 10 − 1)/(2 ln 10) ≃ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content='6512, which gives the waist widths 𝜎0,+ ≃ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content='571 and 𝜎0,− ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content='194.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' When compared with the value for 𝜎𝑔(0), we notice that while 𝜎0,+ is practically the same [∼ 99% 𝜎𝑔(0)], which already indicates a poor dynamics, 𝜎0,− is significantly different [∼ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content='5% 𝜎𝑔(0)] and hence a more relevant dynamical behavior is expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' The above expectations translate into the results displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' 2b) for 𝜎0,−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' The characteristic time scale here is 𝑡𝑠,− ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content='15, about a tenth of 𝜏 and hence with noticeable effects both in the implosion and, afterwards, in the subsequent dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' Note here that there is a more important phase contribution, since 𝜃𝑔,−(0) ≃ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content='38𝜋, a value closer to the maximum bound for the phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' Nonetheless, unlike the singular wave packet, here near the singular region the flux is not turbulent, which is consistent with the fact that the evolution of a Gaussian wave packet is characterized by the absence of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' 4 we plot the reverse case of a Gaussian wave packet for 𝜎0,+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' We can appreciate that the wave packet remains un- affected, with the flux described by the swarm of 51 Bohmian trajectories being nearly stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' The characteristic spread- ing time scale is 𝑡𝑠,+ ≃ 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content='4𝜏, which implies that neither the evolution before 𝜏 nor afterwards is going to be importantly affected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' Indeed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' the initial phase is 𝜃𝑔,+(0) ≃ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content='061𝜋, which already indicates the rather small contribution of the 10 5 0 5 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content='0 x / s 0, + t / t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content='0 10 5 0 5 10 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' Same trajectories as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' 2b) for the probability asso- ciated with a Gaussian wave packet with waist width 𝜎0,+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' Note that, because the waist width is relatively large compared to the ini- tial width 𝜎𝑔(0), there is no apparent self-implosion (only a very slight narrowing at 𝜏), as it is evidenced by the nearly parallel flux trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' phase factor in the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' 5 we represent the probability densities associated with these initial Gaussian wave packets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' Interestingly, these probability densities are indistinguishable in position space, but they are completely different in momentum space: the momentum distribution for 𝜎0,+ is rather wide, while for 𝜎0,− it approaches a Dirac delta function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' It is precisely this wider momentum distribution that allows the second wave packet to coalesce toward the origin as the time approaches 𝜏, similarly to the singular wave function, while the first wave packet will remain essentially the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' Rectangular wave packet As our last example, we consider a rectangular wave packet [31], with an initial profile 𝜓(𝑥, 0) = 1 √N𝑟 exp � − 𝑖𝑚 2ℏ𝜏 𝑥2 � rect𝑎(𝑥) , (19) where the rectangle function rect𝑎(𝑥) is defined as 1 for |𝑥| ≤ 𝑎/2 and 0 for |𝑥| > 𝑎/2 and the normalization constant is N𝑟 = 𝑎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' The time evolution can be found using again (2), finding [31] 𝜓(𝑥, 𝑡) = (−1)3/4 √4𝑖N𝑟 exp � 𝑖𝑚 2ℏ𝜏 𝑥2 � � erfi � (−1)1/4 √︂ 𝑚 2ℏ𝑡 � 𝑥 − 𝑎 2 �� − erfi � (−1)1/4 √︂ 𝑚 2ℏ𝑡 � 𝑥 + 𝑎 2 ��� , (20) where erfi(𝑥) is the imaginary error function and this is valid for 𝑡 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' The wave packet is composed of an infinite number of plane waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' At time 𝑡 = 0 these plane waves interfere to give 6 20 10 0 10 20 10 3 10 2 10 1 10 0 10 1 20 10 0 10 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content='0 |y(x,0)| 2 / |y(0,0)| 2 x / s y (a) |y(k)| 2 s y k (b) 25 0 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content='0 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' Probability density in the 𝑥-position space (upper panel) and in the 𝑘-momentum space (lower panel) for the singular wave function (black solid line), a Gaussian wave packet with waist width 𝜎0,+ (red dashed line), and a Gaussian wave packet with waist width 𝜎0,− (blue dotted line), and a rectangular wave packet (green dash-dotted line) for 𝑡 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' The inset shows the same plots on a linear vertical scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' The waist widths for both Gaussians have been adjusted to the width of the probability density for the singular wave function at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content='1 of its maximum value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' a rectangular shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' As time elapses, the component plane waves travel, both to the right (𝑘 > 0) and to the left (𝑘 < 0), at different phase velocities ℏ𝑘/2𝑚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' Thus the pattern of the interference of these plane waves gradually changes, resulting in the dispersion of the wave packet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' 2c), we plot the quantum trajectories associated with this evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' Near the time 𝜏, we appreciate the presence of wiggles for both the singular and the rectangular wave packets, which remind of a nonlaminar flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' Conversely, the Gaussian profile looks perfectly laminar nearby the singular point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' We recall that a flood in a river occurs because at some point water slows down and the quicker mass of water arriving from behind finds this “potential barrier" created by the slow water and tries to overcome it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' In this case, the wiggles mark somehow a slower light flow, so that energy accumulates nearby the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content='0 0 10 20 30 40 FWHM / s y t / t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content='0 0 2 4 6 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' Evolution temporal of the FWHM for the same wave packets as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' 5, with the same symbols: the singular wave function (black solid line), a Gaussian wave packet with waist width 𝜎0,+ (red dashed line), a Gaussian wave packet with waist width 𝜎0,− (blue dotted line), and a rectangular wave packet (green dash-dotted line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' singularity and the density grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' An alternative way to capture the degree of localization of a wave function is by studying the behavior exhibited its full width at half maximum (FWHM) [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' More specifically, this quantity is computed in all cases determining the distance between the two positions, 𝑥+ and 𝑥−, at which the correspond- ing probability density reaches half its maximum value at any time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' that is 𝜚(𝑥±, 𝑡) 𝜚max(𝑥, 𝑡) = 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' (21) Except for Gaussian wave packets, the above equation can- not be solved analytically, so 𝑥+ and 𝑥− have been numer- ically determined on the fly, during the time-evolution of the corresponding wave functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' From this, we obtain FWHM(𝑡) = 𝑥+(𝑡) − 𝑥−(𝑡), which is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' 6 for the for cases here considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' As it can be noticed, while the FWHM is nearly constant for the Gaussian with waist width 𝜎0,+, it shows a linear decrease and increase, before and after the waist, respectively, for the Gaussian with 𝜎0,−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' A similar trend is also observed for the square wave function, although the FWHM shows a tiny asymmetry before and after the sin- gularity, which is related to the limitations involved in the numerical method (the spatial size of the grid sets a cutoff for the high spatial frequencies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' Finally, for the singular wave function (3), the FWHM slowly decrease until 𝑡 is close to 𝜏, as it can be appreciated in the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' Near this time, the FWHM undergoes a sudden decrease and then in- crease afterwards;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' at any later time, the FWHM increase near linearly, in a similar fashion to the Gaussian with 𝜎0,−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' We no- tice again a different behavior between the FWHM dynamics before and after 𝑡 = 𝜏, which is related to the fact that the wave function considered is not exactly the ansatz eqrefeq:psi0), but 7 the truncated version (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' All these characteristics concur with the corresponding probability density and quantum trajectories displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' CONCLUDING REMARKS To summarize, we have studied a family of solutions of the Schrödinger equation that spontaneously develop a singular- ity while propagating in free space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' Due to the finiteness of these solutions, their singularities do not require a nonphys- ical infinite amount of energy to manifest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' Nevertheless, the local amplitude of the field at a singular point may grow un- boundedly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' We have given a physical interpretation in terms of quantum trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' While there is a widespread belief that extreme focusing requires strong nonlinear effects, we have demonstrated that this can be easily achieved with only linear propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' This promising field enhancement mechanism may foster further interesting research in fields such as electron microscopy or optics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' ACKNOWLEDGMENTS Financial support is acknowledged to the Spanish Research Agency (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' PID2021-127781NB-I00).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' AA acknowl- edges support from Deutsche Forschungsgemeinschaft (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' 429529648-TRR 306).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' [1] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/p9FPT4oBgHgl3EQf8DXy/content/2301.13207v1.pdf'} +page_content=' Tao, Nonlinear dispersive equations.' metadata={'source': 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sha256:7fd268521b8bebc06873bc25c3b99c0ee5619e4cd682882c8c92d240fa852ca0 +size 150326 diff --git a/ptE3T4oBgHgl3EQf8As1/content/tmp_files/2301.04803v1.pdf.txt b/ptE3T4oBgHgl3EQf8As1/content/tmp_files/2301.04803v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..8d1fd2d9f498c0314dc14d53279474300fff42b5 --- /dev/null +++ b/ptE3T4oBgHgl3EQf8As1/content/tmp_files/2301.04803v1.pdf.txt @@ -0,0 +1,1144 @@ +1 + +Nucleation of Al Nanocrystals in Solute-Substituted Al Metallic Glasses I: Structural +characterization +Feng Yi, S. D. Imhoff, J. H. Perepezko and P. M. Voyles +Department of Materials Science and Engineering, University of Wisconsin-Madison, WI, USA + +Abstract +Primary crystallization in high Al-content metallic glasses is driven by nanometer-diameter +regions with internal structure similar to fcc Al. Comparison of fluctuation electron microscopy +(FEM) data to FEM simulations of fcc Al clusters dispersed in a dense-random packed matrix is +used to extract the diameter and volume fraction of the ordered regions in a Al88Y7Fe5 base glass +and in glasses with 1 at.% Cu substituted for Y or Al. The size and density of nanocrystals were +measured as a function of isothermal annealing time for the same alloys. The volume fraction of +crystalline material grows under isothermal annealing, so the phase transformation is not purely +grain coarsening, but the crystalline volume fraction is lower than the volume fraction of ordered +regions in the as-quenched samples, so not all of the ordered regions act as nuclei. Changes in +diameter and volume fraction of the ordered regions with alloying are correlated with changes in +the crystallization temperature, nucleation rate, and nanocrystal density. No evidence for phase +separation is observed, and FEM simulations from a molecular dynamics quenched structural +model of similar composition do not show the features observed in experiment. +Keywords: Primary crystallization, Al-based metallic glass, FEM, MRO + +2 + +1. Introduction +The primary crystallization reaction in high Al content metallic glasses [1, 2] is +interesting for fundamental studies of nucleation. Nucleation occurs at high density [3], but only +limited growth of the Al phase occurs due to impingement of the diffusion field of solute +expelled by the growing pure Al nanocrystal [4]. This enables experimental measurements of the +nucleation density and nucleation rate as a function of time and temperature that are difficult in +other systems. The amorphous/nanocrystal composite formed by primary crystallization also has +attractive mechanical properties [5-7], and it has similar microstructure to other systems of +interest for applications, such as ductile bulk metallic glass (BMG) composites [8] and soft- +magnetic Fe-based composites [9]. It is therefore of significant interest to control this structure. +Inoue [7] summarized the general pathway to produce Al alloys with nanocrystal dispersions and +pointed out the Al amorphous-nanocrystal composite exhibits superior mechanical properties +compared to single phase alloy. Foley et al. [3] discussed strategies to control the development +of nanocrystallites by alloying plus suitable processing routes. +Primary crystallization in Al-based metallic glasses is characterized by an incubation +time [10-12] and high composition sensitivity [13, 14]. Transient heterogeneous nucleation was +suggested as the mechanism driving the transformation [12], but the heterogeneous nucleation +site was not identified. Using isothermal DSC, Nakazato et al.[13] found that crystallization of +Al85Ni10Ce5 involved nucleation and growth, while crystallization of Al87Ni10Ce3 involved +growth only, demonstrating that primary crystallization is very sensitive to the alloy +composition. +Three theories have been proposed to explain this phase transformation. Perepezko and +Hebert [15] categorized Al-based glass as a growth controlled glass. In this type of glass, during +the quenching, the cooling curve on a time-temperature-transformation (TTT) diagram bypasses +the growth curve, but intercepts the nucleation curve. Therefore, some seeds nucleate, but due to +the dramatic increase of viscosity, their growth is halted by quenching. These clusters are called +quenched-in nuclei, and may act as seeds for primary nanocrystalization. Xing et al.[16] +suggested extremely fine α-Al nanocrystal embedded in amorphous matrix in the as-quenched +sample. This is a similar microstructure to quenched-in nuclei, but with the key difference that +the phase transformation occurs by constant volume fraction coarsening, not by growth at the +expense of aluminum from the matrix.. Unlike a nucleation + growth reaction, there is no delay +time in grain coarsening, and the total volume fraction of the Al phase is conserved. The third +theory is based on phase separation. Gangopadhyay et al. [17] reported that phase separation +occurs in the amorphous phase, and that nucleation occurs at the phase interface, which is +evidenced by contrast in a TEM of annealed sample. Sahu et al. [18] have performed atom probe +tomography (APT) on Al88Y7Fe5 and found gradients of Al, Y and Fe in concentration profiles, +which they believe is an indication of phase separation. Other APT measurements on binary Al- +Sm do not show large scale composition fluctuations [19, 20]. Additionally, the final +microstructures do not resemble those normally observed during solid state spinodal reactions. +Regardless of mechanism, the genesis of primary crystallization must depend in some +way on the structure of the Al-based metallic glass. A dense random packing (DRP) [21] was +used to explain the base glass atomic structure of metal-metalloid glasses, but it is not consistent +with experiment for Al-based glasses. Anomalous X-ray diffraction of Al87Y8Ni5 [22], and + +3 + +neutron diffraction of Al87Ni7Nd6[23] indicate bond shortening between Al and solute atoms, +which implies a strong interaction between the solvent atoms and solute atoms. Miracle and +Senkov [24-26] proposed the solute-centered cluster concept, and the efficient cluster packing +(ECP) of these clusters extends this model to longer length scales. In their model, in order to +achieve ECP, there are several types of polyhedral clusters in the atomic structure. Sheng et +al.[27] used computer simulation coordinated with experiments to investigate the atomic +structure and found three types of cluster packing, which are face-sharing, edge-sharing, and +vertex sharing clusters. Although these results give insight into the as-quenched atomic structure +in Al-based glasses, none of them shows an obvious connection with primary crystallization in +Al-based glass. Clearly, a coordinated study of the local atomic structure and the relationship of +this structure to the nanocrystallization kinetics is necessary to advance the understanding of +primary crystallization in amorphous Al alloys. +In the current study fluctuation electron microscopy (FEM) is used to measure the +structure that drives primary crystallization in as-quenched state of Al-based glasses. The FEM +technique is sensitive to many-body atom correlations. Therefore, it can be used to detect +nanometer-scale structure called medium-range-order (MRO) in amorphous materials[28]. FEM +samples spatial fluctuation in diffraction at nm resolution. The fluctuations are quantified by +using the normalized variance, + +( +) +( +) +( +) +2 +2 +, , +, +1 +, , +I +k R +V k R +I k R += +− +r +r + +(1) +where I is the annular average of DP in each image, k is the magnitude of scattering vector, R is +the resolution, r is the position over the sample, and the angular bracket means average over +many positions. In the post-processing, the annular average diffraction intensity was calculated +for each diffraction pattern. Stratton et al. [29] first applied this method to Al-based glass, and +found FCC Al-like MRO in Al92Sm8. “Al-like” means that this structure diffracts at the Bragg +conditions for FCC Al; thus, its internal atomic structure is similar to FCC Al. As shown below +and elsewhere [30], the structure may be strained, distorted, or defective. Simulation [29] shows +that nanoscale Al order reproduces the peak position and relative peak height compared to the +experiment, but that various icosahedral structures do not. Similar MRO was not found in a +Al92Sm8 sample amorphized by cold rolling. In differential scanning calorimeter (DSC) +measurement, primary crystallization was only observed in melt-quenched samples and not in +cold-rolled samples [31]. Additionally, an obvious glass transition, Tg and an exothermic signal +corresponding to eutectic phase formation were observed in DSC measurement of cold-rolled +sample. Thus, the presence of Al-like MRO is correlated with primary crystallization, and these +regions were identified as the quenched-in nuclei. The FEM data do not, however, exclude ECP +or any other model for the regions between the Al-like MRO regions. Moreover, the simulations +in [29] used Al-like MRO with diameter d of 3 nm only, so that they did not extract the size and +volume fraction of Al-like MRO. The diameter and volume fraction data are crucial for +quantitative nucleation modeling, and for distinguishing quenched-in nuclei from grain +coarsening on a structural basis. +Atom probe tomography performed on Al90Sm10 [32] supports the existence of nanosize +pure Al region in as-quenched sample, although there is no crystallography information in the + +4 + +APT data. FEM experiments by Wen et al. [33] and Daulton et al. [34] have confirmed MRO in +Al-based glass. In [33], it is proposed that the solute-centered quasi-equivalent clusters are the +structural building blocks in Al85Ni5Y10 and Al85Ni5Y8Co2, and that the MRO structure consists +of organized packing of such clusters. The authors do not exclude the possibility of quenched-in +nuclei, but neither do they connect any aspect of the structure to primary crystallization. The two +Al-based glasses in [33] exhibit a Tg in DSC. In [34], they adopted phenomenological model +developed by Stratton and Voyles [35], and used this model to estimate the correlation length of +local order. +Stratton and Voyles [35, 36] developed an analytical model of the variance V as a +function of MRO size d and volume fraction Φ. In their model, the sample is divided into many +columns of equal size R×R×t, in which R is the resolution and t is the sample thickness. The +column is further divided into bins of cubic shape with size R, which are randomly occupied by +randomly-oriented Al-like MRO clusters. Perfect Al nanocrystals are used to represent Al-like +MRO in the model, although the real Al-MRO may have some internal strain or disorder. How +many bins will be occupied by the Al-like MRO is determined by the volume fraction of MRO. +It is a simple model, but it incorporates the fundamental physical parameters of atomic structure +that dominate the variance from Al-based glasses. They found the variance goes up +monotonically with d, and goes through maximum as a function of Φ. The variance is inversely +proportional to R2 and t. Within the limits of thin sample required by kinematic scattering, the R +and t dependences have been experimentally confirmed [37, 38]. +In this model, the only factor that varies with k (i.e. from Bragg reflection to Bragg +reflection) is the Bragg active fraction Ahkl. Ahkl is the fraction of randomly-oriented population of +nanocrystals that will exhibit strong diffraction into one of a family of reflections {hkl}. Ahkl ≈ +(1/4)Mhkl∆θ, and ∆θ = (dhkl/d) + 2(α +β), where dhkl is the plane spacing for the reflection {hkl}, +d is the nanocrystal diameter, α is the pixel size in the diffraction pattern and β is the +illumination convergence angle [36]. Mhkl is the multiplicity of the {hkl} family, defined as the +number of unique planes (hkl) that belong to the family {hkl}. For the fcc structure, M111 = 12, M200 += 6, and M220 = 12 again. This means that reflections with different Mhkl have a different +dependence on d and Φ, so that, in principle, if V for two reflections hkl with different Mhkl is +known, d and Φ can be determined. +In the current work the physical insight from the Stratton/Voyles model is adopted with +the main focus on analysis of V111 and V200 in FEM data from Al-based glass. However, the +simplifying assumptions made to render the model analytical tractable are too severe. Some +improvements to the model have been developed [39], but it is still not sufficient for a +quantitative match to the experimental data. Instead, state-of-the-art multislice dynamical +diffraction simulations [40] from atomistic models of Al-based glass are employed which allow +to the modeling of the full complexity of the real interaction between the electron beam and +sample to reliably extract d and Φ from FEM data. +In the following, the FEM experiments and simulations are described and combined to +extract d and Φ for Al88Y7Fe5 metallic glass, and glasses with 1 at. % Cu substituted for Y and +substituted for Al. Finally, the results are compared with measurements of the volume fraction +and nucleation rate of nanocrystals after crystallization., All of the results are demonstrated to be + +5 + +qualitatively consistent with a primary crystallization driven by retained MRO.. They are +inconsistent with grain coarsening, and no evidence for phase separation is observed in any of +the microscopy. In part II a quantitative nucleation model is developed that connects the retained +MRO measured by FEM with the microstructure after primary nanocrystallization. +2. Methods +2.1 Experiment +Samples of amorphous Al88Y7Fe5, Al88Y6Fe5Cu1, and Al87Y7Fe5Cu1 were prepared by +rapid quenching in a single wheel melt spinner at a tangential wheel speed of 55 m/s. Each +segment of the ribbon used for further experiments was checked by XRD to confirm that it was +completely amorphous. Samples were thinned for TEM by electropolishing in 75% methanol ++25% nitric acid at around -42 ºC. After electropolishing, the sample was cleaned using +trichloroethylene, acetone, and methanol in sequence to remove organic contamination. +Subsequently, the sample was subjected to plasma cleaning for 3 minutes and 15 seconds before +loading the sample into the STEM. There is no crystallization induced by plasma cleaning for +this time period, which was confirmed by taking electron diffraction patterns (DPs) of plasma +cleaned sample and only observing broad, fuzzy rings in the DPs. + +STEM FEM experiments were performed using FEI Titan at 200 kV, which yielded a +substantial improvement over previous TEM FEM experiments [36-38]. The STEM mode is +employed to scan t the beam across the sample area of interest. At each position, a +nanodiffraction pattern is collected. In order to remove inelastically scattered electrons, energy +filtering with 10eV slit width was used in the FEM experiment. The nanodiffraction patterns +(DP) were collected on a 2048×2048 US1000 CCD in a Gatan 865 imaging filter. I(k,R,r) is the +annular average of each diffraction pattern. The electron transmittance ratio was used to estimate +the sample thickness in units of elastic mean free path, which for Al87Y7Fe5Cu1 is 84±1 nm [41]. +Samples with 0.70 transmittance, or a physical thickness of 29 nm were used. In order to +optimize the tradeoff between the probe coherence and acquisition time, a spot size 8 with a 10 +µm condenser aperture was selected for the experiments. Spot size 8 corresponds to the source +size of 0.46 nm with probe current of 3 pA. Based on previous results[29], a resolution of 1.88 +nm was used in the measurements, which at 200 kV is a convergence half angle of 0.81 mrad. +During STEM FEM experiments, at least ten thin separate areas were examined in each sample. +In each area, 10 by 10 DPs, or 100 DPs were acquired.. The V(k,R) that arises from shot noise +was calculated based on the analysis by Fan et al.[42] and subtracted from eqn.(1). Thickness +filtering was applied as well using the method described by Hwang and Voyles [37]. + +Both standard bright field (BF) and dark field (DF) TEM imaging were used to +characterize the population of nanocrystals after primary crystallization. The BF or DF images +were acquired using Philips CM200 Ultra-twin TEM under 200 kV. The images were obtained +from at least four areas in each sample to obtain statistically significant counting. The electron +beam transmittance was used to estimate the sample thickness. (Because of large objective +aperture in Philips CM200, a calibration was established from Titan STEM analysis on the same +sample). + +6 + +2.2 Simulation +The simulated V(k) values were obtained from models consisting of Al nanocrystals +embedded in a Al DRP matrix as a function of the nanocrystal diameter, volume fraction, and +strain state. Comprehensive simulation results are presented elsewhere [30]; the simulations that +best match the experiment are discussed here and were used to extract d + Φ from the +experimental data. The models were constructed and used to calculate the variance as follows. +Perfect Al nanocrystals were employed to approximate the MRO in the model. The Lennard- +Jones potential presented by Zhang and Xia [43] was adopted to build the DRP matrix in the +model. Since the density of amorphous Al glass is very close to its crystalline counterpart, the +DRP structure was constructed with same density as the Al FCC crystal, 0.0602/Å3. A +Metropolis Monte Carlo (MC) method was used to relax the DRP structure until the energy is +converged, which is established when the system energy change in 50000 random movements is +smaller than 0.1%. +After the DRP structure is built, spherical nanocrystals are constructed of the required +size with random orientation. The nanocrystals are inserted into the DRP matrix at random +positions with random orientation. The atoms in the matrix overlapping with nanocrystals are +removed and a few extra atoms are either added or subtracted randomly in the matrix to maintain +constant atom number density. Then the matrix atoms were relaxed using the MC method until +the system energy is converged without moving the atoms inside the crystal. (Because the +crystals are very small, the atoms are in a high energy state, and the crystal structure will be +destroyed during relaxation if those atoms are allowed to rearrange.) +In some models, both hydrostatic strain and disorder in the nanocrystals were considered. +The strain was applied to the nanocrystal before it was inserted in the matrix. After the atoms in +the matrix are relaxed, the atoms in the nanocrystal are relaxed at 300 K while fixing the matrix +atoms. A range of d from 1.20 nm to 1.65 nm corresponding to 55 to 135 atoms inside the +nanocrystals was simulated as well as a range of Φ from 0.0335 to 0.2343 for perfect crystals. A +range of d from 1.62 nm to 2.44 nm, and Φ from 0.0669 to 0.2343 was also simulated for +crystals including strain and disorder. To investigate the model size effect on the variance, +model sizes varying from 8.44 nm to 18.56 nm at constant d and Φ were simulated. (The model +is a cubic, and the model size refers to cubic edge length). +The variance of this atomic structure was calculated by computing many I(k, r) in +different orientations of the model, assuming global structural isotropy. A state-the-art multislice +dynamical diffraction algorithm [44] was used to calculate I(k, r). There is a background in V(k) +that arises from DRP matrix, which was subtracted. +Each model contains a limited number of nanocrystals, typically 300. In order to check +the effect of finite sampling of the random orientations and random positions of the embedded +nanocrystals, we generated five different instances of the models with d=1.36 nm but different +Φ. Then, the variance of the created atomic structure was calculated as a function of Φ. For all +the structures, the standard deviation of the variance is within 5% of the mean, which provides an +estimate of the statistical uncertainty in the simulated variance of ±5%. + +7 + +3. Results +3.1 Experiments + +Fig. +1 +compares +the +variance +of +Al88Y7Fe5, Al88Y6Fe5Cu1 and Al87Y7Fe5Cu1 as- +spun samples. We have subtracted the matrix +background using a Lorentzian function. All the +traces exhibit a major peak at 0.39 Å-1 with a +shoulder on the high-k side near 0.47 Å-1.The +height of the main peak, the height of the +shoulder, and the ratio of the two heights +changes with composition. Since the sample +thickness in the three alloy samples was +controlled carefully to 29.0±2.7 nm, there is a +significant difference among the magnitude of +the major peak in the variance. In addition, the +shoulder is different among these three alloys. + +The main peak is assigned to Al {111} +diffraction and the shoulder to Al {200}. To +decompose these contributions, the first peak +region was fitted to a sum of two Gaussians, one +for the peak and the other one for the shoulder. +The two Gaussian functions were required to +have the same width. The FEM measurement +and fitting results are shown in Fig. 2. The +amplitude and its uncertainty for each Gaussian +function are given in Table 1. Fig. 3 shows the +volume fraction of nanocrystal change as a +function of annealing time. The volume fraction +of nanocrystal increases as annealing time +increases for all compositions. +3.2 Simulation + +Fig. 4 shows the simulated variance of +the DRP matrix with no nanocrystals and a +model of Al89La6Ni6 constructed by Sheng et al. +[27] using molecular dynamics quenching with +an empirical potential. The Al89La6Ni6 model +consists of solute-centered quasi-equivalent +clusters. The Voronoi polyhedra that define +those clusters are not connected in an ordered +fashion. Fig. 4 shows that V(k) from the DRP +matrix and the MD model are virtually the same. +The small oscillations on top of the sloping + +Fig. 1: Variance for as-spun samples. + +Fig. 2 (a) Fitting results for Al88Y7Fe5 (b) +Al88Y6Fe5Cu1, and (c) Al87Y7Fe5Cu1. The green line +are the fitting to the major peak, the blue line is the +background using Lorentzian function, the pink line +is the Gaussian peak due to {111} Bragg reflection, +and the black line is the Gaussian peak due to +{200} Bragg reflection. + +0.010 +Al:Y,Fe +0.008 +Al88Y6FeCu1 +Variance +Al87Y-FeCu1 +0.006 +0.004 +0.002 +0.000. +0.2 +0.4 +0.6 +0.8 +1.0 +k(A-1(a) 0.012 +Al88Y{Fe5 +0.010 +fitting to Al88Y-Fe5 +background +[111] Gaussian +0.006 +[200} Gaussian +0.004 +0.002 +0.2 +0.4 +0.6 +0.8 +1.0 +k (A) +(b) 0.014 - +Al88Y6Fe5Cu +0.012 +fitting to Al88YsFesCu1 +0.010 +background +[111} Gaussian +0.008 +[200} Gaussian +0.006 +0.004 +0.002 +0.2 +0.4 +0.6 +0.8 +1.0 +k (A1) +Al87Y-Fe5Cu +0.010 +fitting to Alg7Y,FesCu. +background +9 0.008. +{111} Gaussian +[200} Gaussian +0.006 +0.004 +0.002 +0.2 +0.4 +0.6 +0.8 +1.0 +k (A-1)8 + +background found in both V(k) curves is likely to an +artifact of the limited model size. +The variance as a function of k for representative +models containing different sizes and volume fractions +of nanocrystals are shown Fig. 5. In these models, the +embedded nanocrystals are perfect crystals. Fig. 5 +shows that variance for {111} and {200} both generally +increases as a function of d and Φ. + +The experimental peak positions are shifted +systematically to lower k compared to the perfect Al +Bragg reflections, suggesting that the embedded Al-like +regions are under tensile strain. The hydrostatic tensile +strain systematically shifts the peaks in V(k) [30], and +the strain-induced disorder suppresses peaks at high k +[30]. + +From simulations for a series of strains, a 4.1% +tensile hydrostatic strain in the nanocrystal was found to +give the best match to experiment for both the +magnitude of the variance and the peak position. Fig. 6 shows the variance for the model with +4.1% tensile strain and disorder by relaxing atoms in the nanocrystal, without allowing the +nanocrystal volume to change. V111 and V200 generally increase as a function of d and Φ, but the +magnitude of peaks is suppressed by disorder. This level of disorder effectively suppresses the + +Figure 3 Volume fraction of nanocrystal +change as a function of annealing time (a) +Al88Y7Fe5 (b) Al88Y6Fe5Cu1 and (c) +Al87Y7Fe5Cu1. +Both +Al88Y7Fe5 +and +Al87Y7Fe5Cu1 were annealed at 245 ºC, +and Al88Y6Fe5Cu1 was annealed at 200 ºC. +Table 1 Fitting parameters for Al88Y7Fe5, Al88Y6Fe5Cu1 and +Al87Y7Fe5Cu1. + +{111}Gaussian +peak +{200}Gaussian +peak +Al88Y7Fe5 +0.0064±0.001 +0.0026±0.0002 +Al88Y6Fe5Cu1 +0.0088±0.0002 +0.0037±0.0002 +Al87Y7Fe5Cu1 +0.0076±0.0004 +0.0026±0.0003 + + +Figure 4: Simulated variance of the DRP structure used as the +matrix for the simulations presented here, and a model +Al89La6Ni5 consisting of quasi-equivalent clusters, created by +molecular dynamics quenching [27]. + +(a) +0.05 +0.04 +AlsY,Fes +Volume Fraction +0.02 +0.01 +0 +0.00 +0 +500 +1000 +1500 +2000 +2500 +3000 +3500 +Time (s) +(b) +0.04 +AlaeY,Fe,Cu, +1 +4 +1 +0.01 +4 +0 +500 +1000 +1500 +2000 +250030003500 +(s) swl +(C) +0.04 +Alg7Y,Fe,Cu, +0.01 +8 +L +0 +500 +1000 +1500 +2000 +2500 +0000 +3500 +Time0.05 +AlsLaNi, MD +AlDRP +0.04 +Variance +0.03 +0.02 +0.01 - +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +k(A')9 + +Al {220} peak at 0.67 Å-1, as seen experimentally in Fig. 1. The local maxima and minima are +noise in simulations due to the limited number of nanocrystals in the model. In the analytical +simulation, the variance is third order in r. Therefore, at constant Φ, a polynomial function of +form +( ) +3 +2 +1 +2 +3 +f r +a +r +a +r +a +r += +× ++ +× ++ +× was used to fit V as a function of r at constant Φ based on +non- linear least square method. This functional form is based on the Stratton and Voyles + +Figure 5 For number density ρ=0.06/Å3, t=118.12 Å +and R=18.8 Å (a) Variance for d = 14 Å at different +volume fraction Φ, and Variance as a function of +nanocrystal size d and volume fraction Φ for (b) +{111} plane and (c) {200} plane using femauto +algorithm. + +Figure 6 For number density ρ=0.06/Å3, and R=18.8 +Å (a) Variance for t=168.74 Å and d=16.2 Å at +different volume fraction Φ, and Variance as a +function of nanocrystal size d and volume fraction Φ +for (b) {111} plane and (c) {200} plane using +femauto algorithm. Both (b) and (c) are normalized to +t=29 nm. There is 4.1% hydrostatic strain in the +nanocrystal, and the atoms in the nanocrystals are +relaxed at 300 K. + +(a) +0.020- +Φ=0.2343 +Φ=0.1674 +0.015- +Φ=0.1004 +Variance +0.010 +0.005- +0.000- +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +k(A") +(b) +0.03 +0.02 +111 +0.01 +0 +2 +0.4 +1.5 +d (nm) +0.2 +1 0 +@ +(c) +0.03 +0.02 +200 +7 +0.01 +0. +2 +0.4 +1.5 +d (nm) +0.2 +1 0 +@(a) +0.012 +Φ=0.2343 +0.010. +Φ=0.1674 +Variance +0.008 +Φ=0.1004 +0.006. +0.004 +0.002 +0.000 +0.4 +0.6 +0.8 +1.0 +1.2 +k(A"l) +(b) +0.04 +0.03 += 0.02 +： +0.01 +0 +2.5 +0.4 +d (nm) +2 +0.2 +1.5~0 +(c) +0.015 +0.01 +200 +V +0.005 +0: +2.5 +0.4 +2 +d (nm) +0.2 +1.5°0 +@10 + +analytical model results [36]. The fit (and therefore +smoothed) V111(r, Φ) and V200(r, Φ) are shown in Fig. 7, +where a linear interpolation was employed to fill the +space at finer grid (Φ). +4. Discussion +4.1 The implication from simulation results + +The agreement of the simulated V(k) between +the DRP matrix of the current models and the MD +model of Al89La6Ni5 in Fig. 4 justifies our neglect of the +Y and Fe atoms in the matrix of our model, since +including atoms of similar scattering power does not +change the simulated V. Neither model agrees with the +experimental data in Fig. 1 or Fig. 2, so the MD model +does not capture the structure of the real material. As +shown by the simulations in Fig. 5, fcc-Al-like +nanoscale order is the missing ingredient in the MD +model. These results also show that FEM signal is +dominated by the nanoscale Al-like regions, which, in +this system, renders it insensitive to the structural +difference between the pure Al DRP and the MD model. +The FEM data are therefore silent on the question of +what structure lies between the Al-like regions. It may be quasi-equivalent clusters as in the MD +model, or it may not be. This insensitivity to subtle details in quasi-equivalent clustering but high +sensitivity to crystal-like order has been noted previously by Wen et al. [45]. + +The variance of Bragg reflections intensity at both {111} and {200} positions increases +monotonically as a function of nanocrystal size d and volume fraction Φ in the calculation range. +In the Stratton/Voyles model, V(Φ) reaches a maximum at fairly low Φ, less than 10%, then +decreases as Φ continues to increase [35, 36]. This maximum is not observed in the simulations, +up to Φ of 20%. Qualitatively, this behavior must be correct: A perfect single crystal, for which +Φ = 1 and Ahkl = 1 must have V = 0, since the structure and thus the diffracted intensity are the +same everywhere. However, the Stratton/Voyles model severely underestimates the Φ at which +this occurs. This is the result of the simplifying assumptions that omit several sources of +variability in the diffracted intensity from an ensemble of nanoparticles which are captured in the +simulations in Fig. 5 and 6. This is explored in greater detail elsewhere [30], but the most +important factor that was not included is the detailed dependence of the diffracted intensity on +the deviation parameter, s. that is the distance of the reciprocal lattice point corresponding to a +reflection (hkl) from the Ewald sphere. In the kinematic limit, the diffracted intensity oscillates +as a function of s [46]. In the Voyles/Stratton model, the diffraction from a given nanocrystal is +either on or off. The simulations also capture disregistry between the positions of the probes and +the positions of the nanocrystals, which is not allowed in the Stratton/Voyles model. If the probe +sometimes catches just a small portion of a nanocrystal, but other times illuminates the entire +nanocrystal, then the variability in diffraction intensity will increase from place to place and act +to increase V. Finally, the Stratton/Voyles model predicts V = 0 for the DRP matrix, which is not + +Figure 7 at ε=4.1% (a) V111 as a function +of d and Φ (b) V200 as a function of d and +Φ. + +(a) +0.03 +0.02 +0.01 +0> +2.5 +0.25 +0.2 +d (nm) +2 +0.15 +0.1 +d +1.5 +0.05 +(b) +0.012 +0.01 +0.008 +90000 +0.004 +0.002 +0 +2.5 +0.25 +2 +0.2 +d (nm) +0.15 +0.1 +Q +1.5 +0.0511 + +observed in the simulations. Thus, the simple +assumption that I ∝ N, where N is the number of matrix +atoms under the probe must not be correct. +The +remaining +discrepancy +between +the +simulations and experiment is that the width of the +peaks in V(k) in the experiment is wider than in +simulation. The peak width depends on instrumental +factors such as the probe convergence angle and on the +state of structural disorder in the sample, with more +disorder leading to broader peaks. Disorder also +changes the peak magnitude, however, and if a +sufficiently large disorder is introduced to match the +experimental peak widths, V200 drops to nearly zero, +which does not agree with the experiments. Since only +the peak magnitudes are employed for the current +analysis, further refinements were not used to match +the peak width exactly. +The disorder associated with strain decreases V +at all k, but the decrease is larger at higher k. This is +because disorder suppresses diffraction more for small +plane spacings and high k. Similar results were found in +models of amorphous silicon with embedded crystal- +like regions [47]. A perfect Si crystal results in much +high variance at high k value, and using strained +paracrystalline Si removed this effect. +4.2 Extracting (d, Φ) from experiments +Graphically, (d, Φ) can be extracted from the +experimental data by the following steps. First, the +experimental values for V111 and V200 define (d, Φ) +contours on the surface plots in Fig. 7. The intersection +of these two contours gives a single (d, Φ) for a given +(V111, V200). The best estimate of d, Φ and the +uncertainty of r and Φ arising from uncertainty in V111 +and V200 are calculated using a Monte Carlo approach. +First, 1.6×105 (V111G, V200G) pairs were created with a +Gaussian probability density function with the mean and +standard deviation given by the experimental values in +the Table 1. Then, V111(d, Φ) and V200(d, Φ) are searched to find (d, Φ) numerically in Fig. 7, +which generates 1.6×105 (d, Φ) pairs. 2D histograms of the (d, Φ) lists are shown in Fig. 8. +Lastly, the 2D Gaussian function of the form + +Fig. 8 The histogram for (a) Al88Y7Fe5 (b) +Al88Y6Fe5Cu1 and (c) Al87Y7Fe5Cu1. The +red line in the figure is contour of fitted +2D Gaussian function with value A1/e. + +(a) +0.08 +0.12 +0.16 +0.20 +0.24 +6. +8. +1 + 100 +80 +2 + 60 +d + 40 +2 +2' + 20 +4. +0 +2 +(b) +0.08 +0.12 +0.16 +0.20 +0.24 + 400 +-300 + 200 +2 +d + 100 +2. +2 +T0 +4.. +2i +(c) +@ +0.08 +0.12 +0.16 +0.20 +0.24 +:140 + 120 + 100 +d +- 80 +21 + 60 + 40 + 20 +41 +-012 + + +( +) +2 +2 +0 +1 +2 +2 +1 +exp +2 1 +d +r +cor d +d +d +d +A +A +cor +σ +σ +σ σ +− +− +− +− +Φ +Φ + + + + + + + +− +Φ − Φ + + + + + + + + + + + +− +− +Φ − Φ + + + + + + + + + ++ ++ +− + + + + + + +− + + + + + + + + + + + + + + + + + +(2) +was used to fit each histogram in Fig. 8. Table 2 lists d +− + and +− +Φ as the best estimates for each +composition, and +d +σ and σ Φ as the uncertainties. However, this ignores the correlation between +the uncertainties in d and Φ, which is shown graphically by the contour on each figure, which are +drawn at a value of 1/e of the maximum. The shape of the histograms shows that at large +confidence intervals, the uncertainties of r and Φ are strongly correlated. However, the +maximum of each histogram is well fitted by the 2D Gaussian model. +4.3 Connection with Tx and post annealing measurement +The experimental primary crystallization temperature Tx from DSC [14] and the +nanocrystal density after annealing treatment from TEM for these three alloys are shown in +Table 3. Both Al88Y7Fe5 and Al88Y6Fe5Cu1 were annealed at 245 ºC for 1 hour, and +Al87Y7Fe5Cu1 was annealed at 200 ºC for 1 hour. Tx represents the onset of detectable nucleation +and is determined by three factors: the density of nucleation sites, the potency of the nucleation +sites, and the transient nucleation delay time, which is governed by the relevant diffusivities. As +shown in Table 3, the diameter of the nuclei are almost constant from alloy to alloy, within the +experimental uncertainty. The density of MRO calculated from Table 2 based on +( +) +3 6 +MRO +MRO +d +ρ +π += Φ + is also shown for comparison. As indicated in Table 3, the rank order of +ρMRO is the same as that of ρcrystal, and opposite to the rank order of Tx, although the ρMRO are not +really different outside of experimental uncertainty between Al88Y7Fe5 and Al88Y6Fe5Cu1. This is +due entirely to the uncertainty in d for the base alloy. The rank order of the volume fraction Φ is +distinguishable and follows ρcrystal directly. This correlation indicates that MRO acts as +heterogeneous site for nanocrystal nucleation. During continuous heating, if there are more MRO +clusters and thus nucleation sites in the as-quenched sample, the nucleation rate is higher, +shifting Tx lower. During isothermal annealing, higher MRO density results in higher crystal +density by providing a higher density of nucleation sites. However, the difference in Φ between +the base alloy (Al88Y7Fe5) and 1 at.% substituted for Al (Al87Y7Fe5Cu1) is much smaller than the +change in Tx. +In addition, Fig. 3 demonstrates that the nanocrystal volume fraction in the base alloy and +the alloys with Cu substitution increases as the annealing time increases. If the grain coarsening +mechanism were valid for growth, the volume fraction of nanocrystal should be constant during +Table 2 d and Φ of MRO in Al88Y7Fe5, +Al88Y6Fe5Cu1 and Al87Y7Fe5Cu1.as- +spun sample. + +d(nm) +Φ +Al88Y7Fe5 +1.8±0.1 +0.16±0.03 +Al88Y6Fe5Cu1 +1.84±0.06 +0.20±0.02 +Al87Y7Fe5Cu1 +2.2±0.2 +0.10±0.02 + +Table 3 Tx, ρcrystal and ρMRO for Al88Y7Fe5, Al88Y6Fe5Cu1 and +Al87Y7Fe5Cu1.as-spun sample. + +ρMRO(1025/m3) +ρnanocrystal (1021/m3) +Τx (ºC) +Al88Y7Fe5 +5.2±1.0 +3.5±1 +265±5 +Al88Y6Fe5Cu1 +6.1±0.6 +13±4 +215±5 +Al87Y7Fe5Cu1 +1.8±0.4 +0.7±0.2 +275±5 + + +13 + +annealing treatment. However, Fig. 3 indicates this is not correct. Moreover, Φ in Table 2 is +much higher than the crystal volume fraction at the shortest time in Fig. 3. If Φ in the as- +quenched state is treated as the initial crystal fraction in an amorphous/nanocrystal composite, +then a larger drop in Φ should occur at the start of the reaction. This is also inconsistent with +grain coarsening. +There is no evidence of phase separation in any of the data. The Z-contrast STEM images +that were used as a guide for the STEM FEM experiments [37] are sensitive to composition and +thickness changes. In several hundred images, a thickness gradient is detected in towards the +holes in the samples introduced by electropolishing, but not the patterned structures proposed for +phase separation based on atom probe tomography. Moreover, the circular arrangements of +nanoparticles seen by Gangopadhyay [17] were not observed in any of the conventional TEM +images. +5. Conclusion + +By combining FEM experiments and simulations, the diameter and volume fraction of +retained MRO regions were determined in Al88Y7Fe5, Al87Y7Fe5Cu1 and Al88Y6Fe5Cu1 metallic +glasses. Comparing the diameter and volume fraction in the as-quenched state to the +crystallization temperature and nanocrystal volume fraction after devitrification for all three +alloys strongly supports the activity of MRO regions to catalyze the primary crystallization. The +data are inconsistent with a grain coarsening model and reveal no evidence for a precursor phase +separation reaction. +Acknowledgement +Work by F.Y. and P.M.V. was supported by the U.S. National Science Foundation (DMR- +0905793). Work by S.D.I. and J.H.P. was support by . NSF (DMR-1005334). The authors thank +H. W. Sheng for sharing the coordinates of his Al89La6Ni5 structural model. + +References +[1] Y. He, S.J. Poon, G.J. Shiflet, Synthesis and properties of metallic glasses that contain aluminum, +Science 241 (1988) 1640-1642. +[2] G.J. Shiflet, Y. He, S.J. Poon, Mechanical properties of a new class of metallic glasses based on +aluminum, J. Appl. Phys. 64(12) (1988) 6863-6865. +[3] J.C. Foley, D.R. Allen, J.H. 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Matter 19 +(2007) 455204. + + diff --git a/ptE3T4oBgHgl3EQf8As1/content/tmp_files/load_file.txt b/ptE3T4oBgHgl3EQf8As1/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9a1423bc507e00f57cb63ceeb9875f9358947a16 --- /dev/null +++ b/ptE3T4oBgHgl3EQf8As1/content/tmp_files/load_file.txt @@ -0,0 +1,938 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf,len=937 +page_content='1 Nucleation of Al Nanocrystals in Solute-Substituted Al Metallic Glasses I: Structural characterization Feng Yi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Imhoff, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Perepezko and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Voyles Department of Materials Science and Engineering, University of Wisconsin-Madison, WI, USA Abstract Primary crystallization in high Al-content metallic glasses is driven by nanometer-diameter regions with internal structure similar to fcc Al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Comparison of fluctuation electron microscopy (FEM) data to FEM simulations of fcc Al clusters dispersed in a dense-random packed matrix is used to extract the diameter and volume fraction of the ordered regions in a Al88Y7Fe5 base glass and in glasses with 1 at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='% Cu substituted for Y or Al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' The size and density of nanocrystals were measured as a function of isothermal annealing time for the same alloys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' The volume fraction of crystalline material grows under isothermal annealing, so the phase transformation is not purely grain coarsening, but the crystalline volume fraction is lower than the volume fraction of ordered regions in the as-quenched samples, so not all of the ordered regions act as nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Changes in diameter and volume fraction of the ordered regions with alloying are correlated with changes in the crystallization temperature, nucleation rate, and nanocrystal density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' No evidence for phase separation is observed, and FEM simulations from a molecular dynamics quenched structural model of similar composition do not show the features observed in experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Keywords: Primary crystallization, Al-based metallic glass, FEM, MRO 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Introduction The primary crystallization reaction in high Al content metallic glasses [1, 2] is interesting for fundamental studies of nucleation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Nucleation occurs at high density [3], but only limited growth of the Al phase occurs due to impingement of the diffusion field of solute expelled by the growing pure Al nanocrystal [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' This enables experimental measurements of the nucleation density and nucleation rate as a function of time and temperature that are difficult in other systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' The amorphous/nanocrystal composite formed by primary crystallization also has attractive mechanical properties [5-7], and it has similar microstructure to other systems of interest for applications, such as ductile bulk metallic glass (BMG) composites [8] and soft- magnetic Fe-based composites [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' It is therefore of significant interest to control this structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Inoue [7] summarized the general pathway to produce Al alloys with nanocrystal dispersions and pointed out the Al amorphous-nanocrystal composite exhibits superior mechanical properties compared to single phase alloy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Foley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' [3] discussed strategies to control the development of nanocrystallites by alloying plus suitable processing routes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Primary crystallization in Al-based metallic glasses is characterized by an incubation time [10-12] and high composition sensitivity [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Transient heterogeneous nucleation was suggested as the mechanism driving the transformation [12], but the heterogeneous nucleation site was not identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Using isothermal DSC, Nakazato et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' [13] found that crystallization of Al85Ni10Ce5 involved nucleation and growth, while crystallization of Al87Ni10Ce3 involved growth only, demonstrating that primary crystallization is very sensitive to the alloy composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Three theories have been proposed to explain this phase transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Perepezko and Hebert [15] categorized Al-based glass as a growth controlled glass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' In this type of glass, during the quenching, the cooling curve on a time-temperature-transformation (TTT) diagram bypasses the growth curve, but intercepts the nucleation curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Therefore, some seeds nucleate, but due to the dramatic increase of viscosity, their growth is halted by quenching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' These clusters are called quenched-in nuclei, and may act as seeds for primary nanocrystalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Xing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' [16] suggested extremely fine α-Al nanocrystal embedded in amorphous matrix in the as-quenched sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' This is a similar microstructure to quenched-in nuclei, but with the key difference that the phase transformation occurs by constant volume fraction coarsening, not by growth at the expense of aluminum from the matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='. Unlike a nucleation + growth reaction, there is no delay time in grain coarsening, and the total volume fraction of the Al phase is conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' The third theory is based on phase separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Gangopadhyay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' [17] reported that phase separation occurs in the amorphous phase, and that nucleation occurs at the phase interface, which is evidenced by contrast in a TEM of annealed sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Sahu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' [18] have performed atom probe tomography (APT) on Al88Y7Fe5 and found gradients of Al, Y and Fe in concentration profiles, which they believe is an indication of phase separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Other APT measurements on binary Al- Sm do not show large scale composition fluctuations [19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Additionally, the final microstructures do not resemble those normally observed during solid state spinodal reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Regardless of mechanism, the genesis of primary crystallization must depend in some way on the structure of the Al-based metallic glass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' A dense random packing (DRP) [21] was used to explain the base glass atomic structure of metal-metalloid glasses, but it is not consistent with experiment for Al-based glasses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Anomalous X-ray diffraction of Al87Y8Ni5 [22], and 3 neutron diffraction of Al87Ni7Nd6[23] indicate bond shortening between Al and solute atoms, which implies a strong interaction between the solvent atoms and solute atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Miracle and Senkov [24-26] proposed the solute-centered cluster concept, and the efficient cluster packing (ECP) of these clusters extends this model to longer length scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' In their model, in order to achieve ECP, there are several types of polyhedral clusters in the atomic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Sheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' [27] used computer simulation coordinated with experiments to investigate the atomic structure and found three types of cluster packing, which are face-sharing, edge-sharing, and vertex sharing clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Although these results give insight into the as-quenched atomic structure in Al-based glasses, none of them shows an obvious connection with primary crystallization in Al-based glass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Clearly, a coordinated study of the local atomic structure and the relationship of this structure to the nanocrystallization kinetics is necessary to advance the understanding of primary crystallization in amorphous Al alloys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' In the current study fluctuation electron microscopy (FEM) is used to measure the structure that drives primary crystallization in as-quenched state of Al-based glasses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' The FEM technique is sensitive to many-body atom correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Therefore, it can be used to detect nanometer-scale structure called medium-range-order (MRO) in amorphous materials[28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' FEM samples spatial fluctuation in diffraction at nm resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' The fluctuations are quantified by using the normalized variance, ( ) ( ) ( ) 2 2 , , , 1 , , I k R V k R I k R = − r r (1) where I is the annular average of DP in each image, k is the magnitude of scattering vector, R is the resolution, r is the position over the sample, and the angular bracket means average over many positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' In the post-processing, the annular average diffraction intensity was calculated for each diffraction pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Stratton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' [29] first applied this method to Al-based glass, and found FCC Al-like MRO in Al92Sm8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' “Al-like” means that this structure diffracts at the Bragg conditions for FCC Al;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' thus, its internal atomic structure is similar to FCC Al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' As shown below and elsewhere [30], the structure may be strained, distorted, or defective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Simulation [29] shows that nanoscale Al order reproduces the peak position and relative peak height compared to the experiment, but that various icosahedral structures do not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Similar MRO was not found in a Al92Sm8 sample amorphized by cold rolling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' In differential scanning calorimeter (DSC) measurement, primary crystallization was only observed in melt-quenched samples and not in cold-rolled samples [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Additionally, an obvious glass transition, Tg and an exothermic signal corresponding to eutectic phase formation were observed in DSC measurement of cold-rolled sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Thus, the presence of Al-like MRO is correlated with primary crystallization, and these regions were identified as the quenched-in nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' The FEM data do not, however, exclude ECP or any other model for the regions between the Al-like MRO regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Moreover, the simulations in [29] used Al-like MRO with diameter d of 3 nm only, so that they did not extract the size and volume fraction of Al-like MRO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' The diameter and volume fraction data are crucial for quantitative nucleation modeling, and for distinguishing quenched-in nuclei from grain coarsening on a structural basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Atom probe tomography performed on Al90Sm10 [32] supports the existence of nanosize pure Al region in as-quenched sample, although there is no crystallography information in the 4 APT data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' FEM experiments by Wen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' [33] and Daulton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' [34] have confirmed MRO in Al-based glass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' In [33], it is proposed that the solute-centered quasi-equivalent clusters are the structural building blocks in Al85Ni5Y10 and Al85Ni5Y8Co2, and that the MRO structure consists of organized packing of such clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' The authors do not exclude the possibility of quenched-in nuclei, but neither do they connect any aspect of the structure to primary crystallization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' The two Al-based glasses in [33] exhibit a Tg in DSC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' In [34], they adopted phenomenological model developed by Stratton and Voyles [35], and used this model to estimate the correlation length of local order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Stratton and Voyles [35, 36] developed an analytical model of the variance V as a function of MRO size d and volume fraction Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' In their model, the sample is divided into many columns of equal size R×R×t, in which R is the resolution and t is the sample thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' The column is further divided into bins of cubic shape with size R, which are randomly occupied by randomly-oriented Al-like MRO clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Perfect Al nanocrystals are used to represent Al-like MRO in the model, although the real Al-MRO may have some internal strain or disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' How many bins will be occupied by the Al-like MRO is determined by the volume fraction of MRO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' It is a simple model, but it incorporates the fundamental physical parameters of atomic structure that dominate the variance from Al-based glasses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' They found the variance goes up monotonically with d, and goes through maximum as a function of Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' The variance is inversely proportional to R2 and t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Within the limits of thin sample required by kinematic scattering, the R and t dependences have been experimentally confirmed [37, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' In this model, the only factor that varies with k (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' from Bragg reflection to Bragg reflection) is the Bragg active fraction Ahkl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Ahkl is the fraction of randomly-oriented population of nanocrystals that will exhibit strong diffraction into one of a family of reflections {hkl}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Ahkl ≈ (1/4)Mhkl∆θ, and ∆θ = (dhkl/d) + 2(α +β), where dhkl is the plane spacing for the reflection {hkl}, d is the nanocrystal diameter, α is the pixel size in the diffraction pattern and β is the illumination convergence angle [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Mhkl is the multiplicity of the {hkl} family, defined as the number of unique planes (hkl) that belong to the family {hkl}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' For the fcc structure, M111 = 12, M200 = 6, and M220 = 12 again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' This means that reflections with different Mhkl have a different dependence on d and Φ, so that, in principle, if V for two reflections hkl with different Mhkl is known, d and Φ can be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' In the current work the physical insight from the Stratton/Voyles model is adopted with the main focus on analysis of V111 and V200 in FEM data from Al-based glass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' However, the simplifying assumptions made to render the model analytical tractable are too severe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Some improvements to the model have been developed [39], but it is still not sufficient for a quantitative match to the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Instead, state-of-the-art multislice dynamical diffraction simulations [40] from atomistic models of Al-based glass are employed which allow to the modeling of the full complexity of the real interaction between the electron beam and sample to reliably extract d and Φ from FEM data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' In the following, the FEM experiments and simulations are described and combined to extract d and Φ for Al88Y7Fe5 metallic glass, and glasses with 1 at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' % Cu substituted for Y and substituted for Al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Finally, the results are compared with measurements of the volume fraction and nucleation rate of nanocrystals after crystallization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=', All of the results are demonstrated to be 5 qualitatively consistent with a primary crystallization driven by retained MRO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='. They are inconsistent with grain coarsening, and no evidence for phase separation is observed in any of the microscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' In part II a quantitative nucleation model is developed that connects the retained MRO measured by FEM with the microstructure after primary nanocrystallization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Methods 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='1 Experiment Samples of amorphous Al88Y7Fe5, Al88Y6Fe5Cu1, and Al87Y7Fe5Cu1 were prepared by rapid quenching in a single wheel melt spinner at a tangential wheel speed of 55 m/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Each segment of the ribbon used for further experiments was checked by XRD to confirm that it was completely amorphous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Samples were thinned for TEM by electropolishing in 75% methanol +25% nitric acid at around -42 ºC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' After electropolishing, the sample was cleaned using trichloroethylene, acetone, and methanol in sequence to remove organic contamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Subsequently, the sample was subjected to plasma cleaning for 3 minutes and 15 seconds before loading the sample into the STEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' There is no crystallization induced by plasma cleaning for this time period, which was confirmed by taking electron diffraction patterns (DPs) of plasma cleaned sample and only observing broad, fuzzy rings in the DPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' STEM FEM experiments were performed using FEI Titan at 200 kV, which yielded a substantial improvement over previous TEM FEM experiments [36-38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' The STEM mode is employed to scan t the beam across the sample area of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' At each position, a nanodiffraction pattern is collected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' In order to remove inelastically scattered electrons, energy filtering with 10eV slit width was used in the FEM experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' The nanodiffraction patterns (DP) were collected on a 2048×2048 US1000 CCD in a Gatan 865 imaging filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' I(k,R,r) is the annular average of each diffraction pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' The electron transmittance ratio was used to estimate the sample thickness in units of elastic mean free path, which for Al87Y7Fe5Cu1 is 84±1 nm [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Samples with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='70 transmittance, or a physical thickness of 29 nm were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' In order to optimize the tradeoff between the probe coherence and acquisition time, a spot size 8 with a 10 µm condenser aperture was selected for the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Spot size 8 corresponds to the source size of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='46 nm with probe current of 3 pA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Based on previous results[29], a resolution of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='88 nm was used in the measurements, which at 200 kV is a convergence half angle of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='81 mrad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' During STEM FEM experiments, at least ten thin separate areas were examined in each sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' In each area, 10 by 10 DPs, or 100 DPs were acquired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='. The V(k,R) that arises from shot noise was calculated based on the analysis by Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' [42] and subtracted from eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Thickness filtering was applied as well using the method described by Hwang and Voyles [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Both standard bright field (BF) and dark field (DF) TEM imaging were used to characterize the population of nanocrystals after primary crystallization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' The BF or DF images were acquired using Philips CM200 Ultra-twin TEM under 200 kV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' The images were obtained from at least four areas in each sample to obtain statistically significant counting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' The electron beam transmittance was used to estimate the sample thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' (Because of large objective aperture in Philips CM200, a calibration was established from Titan STEM analysis on the same sample).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='2 Simulation The simulated V(k) values were obtained from models consisting of Al nanocrystals embedded in a Al DRP matrix as a function of the nanocrystal diameter, volume fraction, and strain state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Comprehensive simulation results are presented elsewhere [30];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' the simulations that best match the experiment are discussed here and were used to extract d + Φ from the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' The models were constructed and used to calculate the variance as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Perfect Al nanocrystals were employed to approximate the MRO in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' The Lennard- Jones potential presented by Zhang and Xia [43] was adopted to build the DRP matrix in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Since the density of amorphous Al glass is very close to its crystalline counterpart, the DRP structure was constructed with same density as the Al FCC crystal, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='0602/Å3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' A Metropolis Monte Carlo (MC) method was used to relax the DRP structure until the energy is converged, which is established when the system energy change in 50000 random movements is smaller than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' After the DRP structure is built, spherical nanocrystals are constructed of the required size with random orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' The nanocrystals are inserted into the DRP matrix at random positions with random orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' The atoms in the matrix overlapping with nanocrystals are removed and a few extra atoms are either added or subtracted randomly in the matrix to maintain constant atom number density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Then the matrix atoms were relaxed using the MC method until the system energy is converged without moving the atoms inside the crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' (Because the crystals are very small, the atoms are in a high energy state, and the crystal structure will be destroyed during relaxation if those atoms are allowed to rearrange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=') In some models, both hydrostatic strain and disorder in the nanocrystals were considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' The strain was applied to the nanocrystal before it was inserted in the matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' After the atoms in the matrix are relaxed, the atoms in the nanocrystal are relaxed at 300 K while fixing the matrix atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' A range of d from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='20 nm to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='65 nm corresponding to 55 to 135 atoms inside the nanocrystals was simulated as well as a range of Φ from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='0335 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='2343 for perfect crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' A range of d from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='62 nm to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='44 nm, and Φ from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='0669 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='2343 was also simulated for crystals including strain and disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' To investigate the model size effect on the variance, model sizes varying from 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='44 nm to 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='56 nm at constant d and Φ were simulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' (The model is a cubic, and the model size refers to cubic edge length).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' The variance of this atomic structure was calculated by computing many I(k, r) in different orientations of the model, assuming global structural isotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' A state-the-art multislice dynamical diffraction algorithm [44] was used to calculate I(k, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' There is a background in V(k) that arises from DRP matrix, which was subtracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Each model contains a limited number of nanocrystals, typically 300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' In order to check the effect of finite sampling of the random orientations and random positions of the embedded nanocrystals, we generated five different instances of the models with d=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='36 nm but different Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Then, the variance of the created atomic structure was calculated as a function of Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' For all the structures, the standard deviation of the variance is within 5% of the mean, which provides an estimate of the statistical uncertainty in the simulated variance of ±5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Results 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='1 Experiments Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' 1 compares the variance of Al88Y7Fe5, Al88Y6Fe5Cu1 and Al87Y7Fe5Cu1 as- spun samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' We have subtracted the matrix background using a Lorentzian function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' All the traces exhibit a major peak at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='39 Å-1 with a shoulder on the high-k side near 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='47 Å-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='The height of the main peak, the height of the shoulder, and the ratio of the two heights changes with composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Since the sample thickness in the three alloy samples was controlled carefully to 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='0±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='7 nm, there is a significant difference among the magnitude of the major peak in the variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' In addition, the shoulder is different among these three alloys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' The main peak is assigned to Al {111} diffraction and the shoulder to Al {200}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' To decompose these contributions, the first peak region was fitted to a sum of two Gaussians, one for the peak and the other one for the shoulder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' The two Gaussian functions were required to have the same width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' The FEM measurement and fitting results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' The amplitude and its uncertainty for each Gaussian function are given in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' 3 shows the volume fraction of nanocrystal change as a function of annealing time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' The volume fraction of nanocrystal increases as annealing time increases for all compositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='2 Simulation Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' 4 shows the simulated variance of the DRP matrix with no nanocrystals and a model of Al89La6Ni6 constructed by Sheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' [27] using molecular dynamics quenching with an empirical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' The Al89La6Ni6 model consists of solute-centered quasi-equivalent clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' The Voronoi polyhedra that define those clusters are not connected in an ordered fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' 4 shows that V(k) from the DRP matrix and the MD model are virtually the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' The small oscillations on top of the sloping Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' 1: Variance for as-spun samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' 2 (a) Fitting results for Al88Y7Fe5 (b) Al88Y6Fe5Cu1, and (c) Al87Y7Fe5Cu1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' The green line are the fitting to the major peak, the blue line is the background using Lorentzian function, the pink line is the Gaussian peak due to {111} Bragg reflection, and the black line is the Gaussian peak due to {200} Bragg reflection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='010 Al:Y,Fe 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='008 Al88Y6FeCu1 Variance Al87Y-FeCu1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='0 k(A-1(a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='012 Al88Y{Fe5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='010 fitting to Al88Y-Fe5 background [111] Gaussian 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='006 [200} Gaussian 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='0 k (A) (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='014 - Al88Y6Fe5Cu 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='012 fitting to Al88YsFesCu1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='010 background [111} Gaussian 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='008 [200} Gaussian 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='0 k (A1) Al87Y-Fe5Cu 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='010 fitting to Alg7Y,FesCu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' background 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' {111} Gaussian [200} Gaussian 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='0 k (A-1)8 background found in both V(k) curves is likely to an artifact of the limited model size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' The variance as a function of k for representative models containing different sizes and volume fractions of nanocrystals are shown Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' In these models, the embedded nanocrystals are perfect crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' 5 shows that variance for {111} and {200} both generally increases as a function of d and Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' The experimental peak positions are shifted systematically to lower k compared to the perfect Al Bragg reflections, suggesting that the embedded Al-like regions are under tensile strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' The hydrostatic tensile strain systematically shifts the peaks in V(k) [30], and the strain-induced disorder suppresses peaks at high k [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' From simulations for a series of strains, a 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='1% tensile hydrostatic strain in the nanocrystal was found to give the best match to experiment for both the magnitude of the variance and the peak position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' 6 shows the variance for the model with 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='1% tensile strain and disorder by relaxing atoms in the nanocrystal, without allowing the nanocrystal volume to change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' V111 and V200 generally increase as a function of d and Φ, but the magnitude of peaks is suppressed by disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' This level of disorder effectively suppresses the Figure 3 Volume fraction of nanocrystal change as a function of annealing time (a) Al88Y7Fe5 (b) Al88Y6Fe5Cu1 and (c) Al87Y7Fe5Cu1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Both Al88Y7Fe5 and Al87Y7Fe5Cu1 were annealed at 245 ºC, and Al88Y6Fe5Cu1 was annealed at 200 ºC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Table 1 Fitting parameters for Al88Y7Fe5, Al88Y6Fe5Cu1 and Al87Y7Fe5Cu1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' {111}Gaussian peak {200}Gaussian peak Al88Y7Fe5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='0064±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='0026±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='0002 Al88Y6Fe5Cu1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='0088±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='0002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='0037±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='0002 Al87Y7Fe5Cu1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='0076±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='0004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='0026±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='0003 Figure 4: Simulated variance of the DRP structure used as the matrix for the simulations presented here, and a model Al89La6Ni5 consisting of quasi-equivalent clusters, created by molecular dynamics quenching [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='04 AlsY,Fes Volume Fraction 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='01 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='00 0 500 1000 1500 2000 2500 3000 3500 Time (s) (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='04 AlaeY,Fe,Cu, 1 4 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='01 4 0 500 1000 1500 2000 250030003500 (s) swl (C) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='04 Alg7Y,Fe,Cu, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='01 8 L 0 500 1000 1500 2000 2500 0000 3500 Time0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='05 AlsLaNi, MD AlDRP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='04 Variance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='01 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content="0 k(A')9 Al {220} peak at 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='67 Å-1, as seen experimentally in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' The local maxima and minima are noise in simulations due to the limited number of nanocrystals in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' In the analytical simulation, the variance is third order in r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Therefore, at constant Φ, a polynomial function of form ( ) 3 2 1 2 3 f r a r a r a r = × + × + × was used to fit V as a function of r at constant Φ based on non- linear least square method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' This functional form is based on the Stratton and Voyles Figure 5 For number density ρ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='06/Å3, t=118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='12 Å and R=18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='8 Å (a) Variance for d = 14 Å at different volume fraction Φ, and Variance as a function of nanocrystal size d and volume fraction Φ for (b) {111} plane and (c) {200} plane using femauto algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Figure 6 For number density ρ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='06/Å3, and R=18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='8 Å (a) Variance for t=168.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='74 Å and d=16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='2 Å at different volume fraction Φ, and Variance as a function of nanocrystal size d and volume fraction Φ for (b) {111} plane and (c) {200} plane using femauto algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Both (b) and (c) are normalized to t=29 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' There is 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='1% hydrostatic strain in the nanocrystal, and the atoms in the nanocrystals are relaxed at 300 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='020- Φ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='2343 Φ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='1674 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='015- Φ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='1004 Variance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='005- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='000- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='0 k(A") (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='02 111 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='01 0 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='5 d (nm) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='2 1 0 @ (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='02 200 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='5 d (nm) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='2 1 0 @(a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='012 Φ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='2343 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Φ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='1674 Variance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='008 Φ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='1004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='2 k(A"l) (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='03 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='02 ： 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='01 0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='4 d (nm) 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='5~0 (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='01 200 V 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='005 0: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='4 2 d (nm) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='5°0 @10 analytical model results [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' The fit (and therefore smoothed) V111(r, Φ) and V200(r, Φ) are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' 7, where a linear interpolation was employed to fill the space at finer grid (Φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Discussion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='1 The implication from simulation results The agreement of the simulated V(k) between the DRP matrix of the current models and the MD model of Al89La6Ni5 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' 4 justifies our neglect of the Y and Fe atoms in the matrix of our model, since including atoms of similar scattering power does not change the simulated V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Neither model agrees with the experimental data in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' 1 or Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' 2, so the MD model does not capture the structure of the real material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' As shown by the simulations in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' 5, fcc-Al-like nanoscale order is the missing ingredient in the MD model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' These results also show that FEM signal is dominated by the nanoscale Al-like regions, which, in this system, renders it insensitive to the structural difference between the pure Al DRP and the MD model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' The FEM data are therefore silent on the question of what structure lies between the Al-like regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' It may be quasi-equivalent clusters as in the MD model, or it may not be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' This insensitivity to subtle details in quasi-equivalent clustering but high sensitivity to crystal-like order has been noted previously by Wen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' The variance of Bragg reflections intensity at both {111} and {200} positions increases monotonically as a function of nanocrystal size d and volume fraction Φ in the calculation range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' In the Stratton/Voyles model, V(Φ) reaches a maximum at fairly low Φ, less than 10%, then decreases as Φ continues to increase [35, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' This maximum is not observed in the simulations, up to Φ of 20%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Qualitatively, this behavior must be correct: A perfect single crystal, for which Φ = 1 and Ahkl = 1 must have V = 0, since the structure and thus the diffracted intensity are the same everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' However, the Stratton/Voyles model severely underestimates the Φ at which this occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' This is the result of the simplifying assumptions that omit several sources of variability in the diffracted intensity from an ensemble of nanoparticles which are captured in the simulations in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' 5 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' This is explored in greater detail elsewhere [30], but the most important factor that was not included is the detailed dependence of the diffracted intensity on the deviation parameter, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' that is the distance of the reciprocal lattice point corresponding to a reflection (hkl) from the Ewald sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' In the kinematic limit, the diffracted intensity oscillates as a function of s [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' In the Voyles/Stratton model, the diffraction from a given nanocrystal is either on or off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' The simulations also capture disregistry between the positions of the probes and the positions of the nanocrystals, which is not allowed in the Stratton/Voyles model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' If the probe sometimes catches just a small portion of a nanocrystal, but other times illuminates the entire nanocrystal, then the variability in diffraction intensity will increase from place to place and act to increase V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Finally, the Stratton/Voyles model predicts V = 0 for the DRP matrix, which is not Figure 7 at ε=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='1% (a) V111 as a function of d and Φ (b) V200 as a function of d and Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='01 0> 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='2 d (nm) 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='1 d 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='05 (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='008 90000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='002 0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='25 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='2 d (nm) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='1 Q 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='0511 observed in the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Thus, the simple assumption that I ∝ N, where N is the number of matrix atoms under the probe must not be correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' The remaining discrepancy between the simulations and experiment is that the width of the peaks in V(k) in the experiment is wider than in simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' The peak width depends on instrumental factors such as the probe convergence angle and on the state of structural disorder in the sample, with more disorder leading to broader peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Disorder also changes the peak magnitude, however, and if a sufficiently large disorder is introduced to match the experimental peak widths, V200 drops to nearly zero, which does not agree with the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Since only the peak magnitudes are employed for the current analysis, further refinements were not used to match the peak width exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' The disorder associated with strain decreases V at all k, but the decrease is larger at higher k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' This is because disorder suppresses diffraction more for small plane spacings and high k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Similar results were found in models of amorphous silicon with embedded crystal- like regions [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' A perfect Si crystal results in much high variance at high k value, and using strained paracrystalline Si removed this effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='2 Extracting (d, Φ) from experiments Graphically, (d, Φ) can be extracted from the experimental data by the following steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' First, the experimental values for V111 and V200 define (d, Φ) contours on the surface plots in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' The intersection of these two contours gives a single (d, Φ) for a given (V111, V200).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' The best estimate of d, Φ and the uncertainty of r and Φ arising from uncertainty in V111 and V200 are calculated using a Monte Carlo approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' First, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='6×105 (V111G, V200G) pairs were created with a Gaussian probability density function with the mean and standard deviation given by the experimental values in the Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Then, V111(d, Φ) and V200(d, Φ) are searched to find (d, Φ) numerically in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' 7, which generates 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='6×105 (d, Φ) pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' 2D histograms of the (d, Φ) lists are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Lastly, the 2D Gaussian function of the form Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' 8 The histogram for (a) Al88Y7Fe5 (b) Al88Y6Fe5Cu1 and (c) Al87Y7Fe5Cu1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' The red line in the figure is contour of fitted 2D Gaussian function with value A1/e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='24 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=" 1 100 80 2 60 d 40 2 2' 20 4." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' 0 2 (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='24 400 300 200 2 d 100 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' 2 T0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='. 2i (c) @ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='24 :140 120 100 d 80 21 60 40 20 41 012 ( ) 2 2 0 1 2 2 1 exp 2 1 d r cor d d d d A A cor σ σ σ σ − − − − Φ Φ \uf8ee \uf8f9 \uf8ee \uf8f9 \uf8eb \uf8f6\uf8eb \uf8f6 − Φ − Φ \uf8eb \uf8f6 \uf8eb \uf8f6 \uf8ec \uf8f7\uf8ec \uf8f7 \uf8ef \uf8fa \uf8ef \uf8fa − − Φ − Φ \uf8ed \uf8f8\uf8ed \uf8f8 \uf8ec \uf8f7 \uf8ec \uf8f7 \uf8ef \uf8fa + + − \uf8ef \uf8fa \uf8ec \uf8f7 \uf8ec \uf8f7 − \uf8ef \uf8fa \uf8ef \uf8fa \uf8ed \uf8f8 \uf8ed \uf8f8 \uf8ef \uf8fa \uf8ef \uf8fa \uf8f0 \uf8fb \uf8f0 \uf8fb (2) was used to fit each histogram in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Table 2 lists d − and − Φ as the best estimates for each composition, and d σ and σ Φ as the uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' However, this ignores the correlation between the uncertainties in d and Φ, which is shown graphically by the contour on each figure, which are drawn at a value of 1/e of the maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' The shape of the histograms shows that at large confidence intervals, the uncertainties of r and Φ are strongly correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' However, the maximum of each histogram is well fitted by the 2D Gaussian model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='3 Connection with Tx and post annealing measurement The experimental primary crystallization temperature Tx from DSC [14] and the nanocrystal density after annealing treatment from TEM for these three alloys are shown in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Both Al88Y7Fe5 and Al88Y6Fe5Cu1 were annealed at 245 ºC for 1 hour, and Al87Y7Fe5Cu1 was annealed at 200 ºC for 1 hour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Tx represents the onset of detectable nucleation and is determined by three factors: the density of nucleation sites, the potency of the nucleation sites, and the transient nucleation delay time, which is governed by the relevant diffusivities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' As shown in Table 3, the diameter of the nuclei are almost constant from alloy to alloy, within the experimental uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' The density of MRO calculated from Table 2 based on ( ) 3 6 MRO MRO d ρ π = Φ is also shown for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' As indicated in Table 3, the rank order of ρMRO is the same as that of ρcrystal, and opposite to the rank order of Tx, although the ρMRO are not really different outside of experimental uncertainty between Al88Y7Fe5 and Al88Y6Fe5Cu1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' This is due entirely to the uncertainty in d for the base alloy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' The rank order of the volume fraction Φ is distinguishable and follows ρcrystal directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' This correlation indicates that MRO acts as heterogeneous site for nanocrystal nucleation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' During continuous heating, if there are more MRO clusters and thus nucleation sites in the as-quenched sample, the nucleation rate is higher, shifting Tx lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' During isothermal annealing, higher MRO density results in higher crystal density by providing a higher density of nucleation sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' However, the difference in Φ between the base alloy (Al88Y7Fe5) and 1 at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='% substituted for Al (Al87Y7Fe5Cu1) is much smaller than the change in Tx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' In addition, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' 3 demonstrates that the nanocrystal volume fraction in the base alloy and the alloys with Cu substitution increases as the annealing time increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' If the grain coarsening mechanism were valid for growth, the volume fraction of nanocrystal should be constant during Table 2 d and Φ of MRO in Al88Y7Fe5, Al88Y6Fe5Cu1 and Al87Y7Fe5Cu1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='as- spun sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' d(nm) Φ Al88Y7Fe5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='8±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='16±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='03 Al88Y6Fe5Cu1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='84±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='20±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='02 Al87Y7Fe5Cu1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='2±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='10±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='02 Table 3 Tx, ρcrystal and ρMRO for Al88Y7Fe5, Al88Y6Fe5Cu1 and Al87Y7Fe5Cu1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='as-spun sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' ρMRO(1025/m3) ρnanocrystal (1021/m3) Τx (ºC) Al88Y7Fe5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='2±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='5±1 265±5 Al88Y6Fe5Cu1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='1±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='6 13±4 215±5 Al87Y7Fe5Cu1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='8±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='7±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='2 275±5 13 annealing treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' However, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' 3 indicates this is not correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Moreover, Φ in Table 2 is much higher than the crystal volume fraction at the shortest time in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' If Φ in the as- quenched state is treated as the initial crystal fraction in an amorphous/nanocrystal composite, then a larger drop in Φ should occur at the start of the reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' This is also inconsistent with grain coarsening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' There is no evidence of phase separation in any of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' The Z-contrast STEM images that were used as a guide for the STEM FEM experiments [37] are sensitive to composition and thickness changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' In several hundred images, a thickness gradient is detected in towards the holes in the samples introduced by electropolishing, but not the patterned structures proposed for phase separation based on atom probe tomography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Moreover, the circular arrangements of nanoparticles seen by Gangopadhyay [17] were not observed in any of the conventional TEM images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Conclusion By combining FEM experiments and simulations, the diameter and volume fraction of retained MRO regions were determined in Al88Y7Fe5, Al87Y7Fe5Cu1 and Al88Y6Fe5Cu1 metallic glasses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Comparing the diameter and volume fraction in the as-quenched state to the crystallization temperature and nanocrystal volume fraction after devitrification for all three alloys strongly supports the activity of MRO regions to catalyze the primary crystallization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' The data are inconsistent with a grain coarsening model and reveal no evidence for a precursor phase separation reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Acknowledgement Work by F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' was supported by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' National Science Foundation (DMR- 0905793).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Work by S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' was support by .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' NSF (DMR-1005334).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' The authors thank H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE3T4oBgHgl3EQf8As1/content/2301.04803v1.pdf'} +page_content=' Sheng for sharing the coordinates of his Al89La6Ni5 structural model.' metadata={'source': 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Bekmambetova1 and Piero Triverio∗1 +1The Edward S. Rogers Sr. Department of Electrical & Computer Engineering, University of +Toronto +January 12, 2023 +Abstract +We study the probability and energy conservation properties of a leap-frog finite-difference time-domain +(FDTD) method for solving the Schr¨odinger equation. We propose expressions for the total numerical probability +and energy contained in a region, and for the flux of probability current and power through its boundary. We +show that the proposed expressions satisfy the conservation of probability and energy under suitable conditions. +We demonstrate their connection to the Courant-Friedrichs-Lewy condition for stability. We argue that these +findings can be used for developing a modular framework for stability analysis in advanced algorithms based on +FDTD for solving the Schr¨odinger equation. +Keywords: FDTD, Schr¨odinger equation, probability conservation, energy conservation, stability +1 +Introduction +The finite-difference time-domain (FDTD) algorithm is a popular numerical method for solving Maxwell’s equa- +tions [1]. The leap-frog FDTD approach has also been proposed for solving the Schr¨odinger equation [2–4]. This +scheme, which we will refer to as quantum FDTD (FDTD-Q), has since been used in various applications. For +example, in [5] it was applied to model two electrons in a quantum dot, in [6] it formed the core of the method for +determining the eigenstates of arbitrary nanoscale structures, in [7] it was used for studying anti-reflective coating +models, and in [8] for simulating an electron diffraction through a double slit. Different properties of the FDTD- +Q method have been studied, such as accuracy and stability [2, 4, 9, 10]. Other time-domain techniques for the +Schr¨odinger equation that are based on finite differences include non-leap-frog implicit [2,11] and explicit [12,13] +approaches, as well as higher order methods [14,15]. +In the continuous domain, solutions of the Schr¨odinger equation respect the principle of conservation for both +probability and energy. When one considers the entire space, the total probability of finding a particle must be +constant and, with proper normalization, equal to one [16]. Similarly, the amount of energy (or more precisely +the expectation value of energy) associated with the particle in a time-invariant potential must stay constant [16– +18]. The probability and energy would also stay constant when one considers a region that is isolated from the +surrounding space, for example an infinite potential well [16, 17]. In general, when the Schr¨odinger equation is +discretized, the conservation properties are not guaranteed to be preserved. Proving that a discretization method +conserves probability and energy can be done by finding discrete counterparts of these quantities and demonstrating +that they remain unchanged from one time step to the next [2,18–27]. This task is not trivial for the case of FDTD- +Q due to the staggered sampling of the real and imaginary parts of the wavefunction. In relation to the conservation +of probability, in [2] two approximations were proposed for the probability density in one-dimensional FDTD-Q, +which were argued, though without detailed proof, to conserve the principle of probability conservation. Regarding +energy conservation, we are not aware of any work that studies this property specifically in the context of leap-frog +FDTD-Q. However, many works have investigated the conservation of energy for other time-domain methods for +the linear [18,19,22,28] and nonlinear [18,20–22,25–27,29] Schr¨odinger equation. Approaches based on symplectic +∗E-mail: piero.triverio@utoronto.ca +1 +arXiv:2301.04142v1 [cs.CE] 10 Jan 2023 + +integrators [30,31] have been proposed for solving the Schr¨odinger equation [15,22,32–34]. Symplectic algorithms +can be constructed to conserve energy [22,30] by preserving the symplectic structure of the continuous equations. +Symplectic integrators give rise to a wide class of methods with different temporal discretization, including the +leap-frog approach [30,32,33]. However, to the best of our knowledge, they have not been applied to analyze the +conservation properties of the FDTD-Q scheme considered in this paper. +The works in the literature on conservation properties of numerical methods for the Schr¨odinger equation +typically assume either zero [2,24, 26, 27] or periodic boundary conditions [2,19, 21, 22, 25, 26,29]. Such boundary +conditions imply that the energy and probability contained in the region stay constant with time. However, there is +a motivation for studying the conservation properties for the general case where probability and energy can flow from +the region into the surrounding space and vice versa. For example, some simulation scenarios [6,10,12,14] involve +absorbing boundary conditions meant to model unbounded domains. One may also be interested in quantifying +the energy and probability in a sub-region of a larger system. Examples include studies of tunneling phenomena +in hydrogen transfer reactions [35] or in quantum dot potential wells [36]. Consistency with physical laws is an +important accuracy criterion for numerical methods. Hence, expressions approximating the probability and energy +in a sub-region should, ideally, obey the discrete counterparts of the principle of probability and energy conservation, +in addition to being close in values to the analytical solutions. +A mechanism for the numerical probability and energy to leave or enter the region introduces new challenges in +the analysis of the conservation properties. In particular, one needs to (i) quantify the rate at which the exchange +of probability and energy occurs with the surrounding space and (ii) show that this rate balances with the rate +of change of probability and energy stored in the region. Moreover, one needs to (iii) ensure that the region is +unable to provide indefinite amounts of energy and probability to the surrounding space. In this study, these three +challenges are systematically addressed. The work by Fei, P´erez-Garc´ıa, and V´azquez [20] should be mentioned, +as it provides an investigation of the form of the nonlinear Schr¨odinger equation that allows the total charge to +vary with time. In [20], questions similar to (i) and (ii) are addressed for a scheme that involves a variation of the +Crank-Nicholson discretization in time and centered finite-difference discretization in space. +Moreover, in view of scenarios where a region is part of a larger setup, one may wish to study the conservation +properties of a numerical method without knowing a priori what that region is connected to. +This approach +has proven useful in facilitating stability analysis and enforcement in FDTD for the Maxwell equations [37, 38]. +The basis for using conservation arguments for stability analysis lies in the fact that even a small violation of +energy or probability conservation provides the growth mechanism that can lead to numerical solutions that grow +indefinitely, which constitutes instability. This connection between the conservation properties and stability has +been recognized for the case of FDTD for Maxwell’s equations [37,39,40] but not for the case of the leap-frog FDTD- +Q scheme for the Schr¨odinger equation. By ensuring that an FDTD region respects the conservation properties, +one guarantees that this region would not contribute to instability when integrated in a larger setup. An advantage +of the conservation approach is that this guarantee can be made without having knowledge of the surrounding +space, which could involve a grid of different resolution [37,41,42], a reduced order model [38,43], a representation +of an open boundary [6,10,12,14], or another model. +Derivations of stability conditions for FDTD-Q have been performed using approaches related to the von +Neumann analysis, which involve the investigation of temporal growth of plane wave solutions [2, 4, 10]. These +methods are meant for simple scenarios involving constant potential and uniform discretization, where the plane +wave functions are valid solutions to the discretized equations. In [9], stability conditions were derived by studying +the time evolution of the norm of the error between two solutions. Both non-uniform and time-varying potentials +were considered, making the proofs more general than those obtained using von Neumann-type analyses. However, +the derivations in [9] are not applicable to schemes where an FDTD-Q region is finite and is coupled to models other +than a restricted set of boundary conditions. Another method involves analyzing the eigenvalues of the so-called +iteration matrix or system amplification matrix. The iteration matrix method has been used in [14] for deriving +stability limits of schemes closely related to FDTD-Q. The approach could be used to analyze scenarios where an +FDTD-Q region is part of a larger setup consisting of multiple parts. However, in general, the eigenvalues would +need to be studied for the matrix corresponding to the entire coupled scheme [44], which can make the analysis +challenging. The conservation approach to stability analysis can circumvent this issue. Lastly, it should be noted +that the conservation argument for stability has appeared in the literature on other time-domain numerical methods +for the linear and nonlinear Schr¨odinger equations [18,20,24]. +This work presents a systematic study of probability and energy conservation in FDTD-Q for an open region, +extending the energy conservation and dissipativity approaches developed previously for electromagnetics [37, 39, +40, 45, 46]. The concepts in this work take root in the theory of dissipative systems [47, 48]. We formulate the +FDTD-Q equations for a region, introducing unknowns on the boundary [49,50] that allow quantifying the energy +2 + +and probability exchange with the space outside the boundary. We propose expressions for discrete probability +and energy, as well as particle current and supplied power. Using these expressions, we derive the conditions for +the conservation of probability and energy and reveal that they are related to the Courant-Friedrichs-Lewy (CFL) +condition that is traditionally understood as a condition for ensuring stability of an isolated system [4]. For the +case of the basic FDTD-Q scheme in an isolated region, our approach can serve as an alternative derivation of +the CFL limit, which we illustrate in the paper. Moreover, in contrast to the traditional approaches of stability +analysis [2, 4, 9, 10, 14], the conservation approach allows making conclusions on whether the region is capable of +destabilizing a simulation, prior to having any knowledge of how the region is terminated or what model is used +to describe the space outside the region’s boundary [37]. Lastly, we verify that the discrete expressions serve as +an accurate approximation of their continuous counterparts, with the conservation properties being an obvious +advantage over other possible expressions. +This paper is organized as follows. Section 2 provides background information on the leap-frog FDTD-Q method +in the literature. Section 3 describes the equations for the region, with modifications on the boundary to allow +the probability and energy to travel through the boundary. Sections 4 and 5, respectively, analyze the discrete +conservation of probability and energy for the region. Section 6 discusses how the proposed theory could be used +for stability analysis and enforcement. Section 7 provides numerical examples and Section 8 concludes the paper. +2 +Background +This section describes the FDTD-Q method [2–4], which is taken as the starting point in this work. The method +solves the Schr¨odinger equation, which reads +ℏ∂ψR +∂t += − ℏ2 +2m∇2ψI + UψI , +(1a) +ℏ∂ψI +∂t = ℏ2 +2m∇2ψR − UψR , +(1b) +where ψR(x, y, z, t) and ψI(x, y, z, t) are the real and imaginary parts of the wavefunction, respectively, m is the +mass of the particle, ℏ is the reduced Planck’s constant, and U(x, y, z) is the potential energy profile. +A rectangular region divided into nx ×ny ×nz primary cells1 with dimensions ∆x×∆y×∆z, as shown in Fig. 1. +The edges of the primary cells are called primary edges, which are oriented in the +x, +y, and +z directions. The +nodes at the corners of the primary cells are referred to as primary nodes. The primary nodes are indexed as (i, j, k) +from (1, 1, 1) to (nx + 1, ny + 1, nz + 1), where (i, j, k) corresponds to coordinates x = (i − 1)∆x, y = (j − 1)∆y, +z = (k − 1)∆z. The time is divided into nt uniform time steps of size ∆t, with temporal index n denoting t = n∆t. +Both ψR and ψI are sampled at the primary nodes. The real part of the wavefunction ψR is sampled at the integer +time steps n in {0, 1, . . . , nt} and the imaginary part ψI is sampled at the time instances shifted by half a time step, +namely {−0.5, 0.5, . . . , nt − 0.5}. Using the centered differences to discretize the time derivatives and Laplacian +operators in (1a)–(1b), one obtains [2–4] +ℏ +ψR|n+1 +i,j,k − ψR|n +i,j,k +∆t += − ℏ2 +2m +� +ψI| +n+ 1 +2 +i+1,j,k − 2ψI| +n+ 1 +2 +i,j,k + ψI| +n+ 1 +2 +i−1,j,k +(∆x)2 ++ +ψI| +n+ 1 +2 +i,j+1,k − 2ψI| +n+ 1 +2 +i,j,k + ψI| +n+ 1 +2 +i,j−1,k +(∆y)2 ++ +ψI| +n+ 1 +2 +i,j,k+1 − 2ψI| +n+ 1 +2 +i,j,k + ψI| +n+ 1 +2 +i,j,k−1 +(∆z)2 +� ++ U|i,j,kψI| +n+ 1 +2 +i,j,k +(2a) +ℏ +ψI| +n+ 1 +2 +i,j,k − ψI| +n− 1 +2 +i,j,k +∆t += ℏ2 +2m +� +ψR|n +i+1,j,k − 2ψR|n +i,j,k + ψR|n +i−1,j,k +(∆x)2 ++ +ψR|n +i,j+1,k − 2ψR|n +i,j,k + ψR|n +i,j−1,k +(∆y)2 ++ +ψR|n +i,j,k+1 − 2ψR|n +i,j,k + ψR|n +i,j,k−1 +(∆z)2 +� +∇2ψR − U|i,j,kψR|n +i,j,k +(2b) +for the nodes strictly inside the region. +The superscripts in (2a)–(2b) represent the time instances when the +quantities are sampled and the subscripts represent the indices of the primary nodes. The samples involved in (2a) +1The concept of primary and secondary grids comes from FDTD in electromagnetics [51]. +3 + +primary +edges +primary +nodes +primary +cells +secondary +cells +Figure 1: Illustration of the geometrical quantities associated with the discretized region. In this example nx = 2, +ny = 4, nz = 3. All primary cells have dimensions ∆x × ∆y × ∆z. The secondary cells strictly inside the region +have dimensions ∆x × ∆y × ∆z. +The secondary cells adjacent to one face of the boundary are halved in the +dimension normal to the face of the boundary. Similarly, the secondary cells adjacent to two or three faces of the +boundary are halved in two or three dimensions, respectively. +are shown in Fig. 2(a). The numerical solution is obtained starting from initial conditions ψI|−0.5 and ψR|0 and +computing ψI|n+0.5 and ψR|n+1 from (2b) and (2a) in a leap-frog manner. In order to ensure that the scheme is +stable, the time step needs to be taken below the CFL limit [9] +∆t < ∆tCFL = +2 +2ℏ +m +� +1 +(∆x)2 + +1 +(∆y)2 + +1 +(∆z)2 +� ++ maxi,j,k +��U|i,j,k +�� +ℏ +. +(3) +The update procedure for a wavefunction at a particular node requires knowing the previous time step values +corresponding to the six surrounding nodes, which are only available when performing the updates at the nodes +strictly inside the region. +Hence, boundary conditions need to be assumed, such as zero Dirichlet or periodic +boundary conditions. The six faces of the boundary are referred to as west (W), east (E), south (S), north (N), +bottom (B), and top (T), corresponding to i = 1, i = nx + 1, j = 1, j = ny + 1, k = 1, and k = nz + 1, respectively. +3 +Equations for the region +This section presents the proposed discretization of (1a)–(1b), which facilitates the investigation of probability and +energy conservation in Sections 4 and 5. As a starting point, we take the FDTD-Q method outlined in Section 2. +The proposed equations consider a general scenario where the FDTD-Q region could either be terminated with +boundary conditions or constitute a portion of a larger domain. In this scenario, the staggered nature of spatial +sampling in FDTD-Q makes it difficult to precisely define the region’s boundary. As a result, quantification of the +probability and energy pertaining to the region becomes non-trivial. In this section, we propose equations on the +boundary involving the so-called hanging variables [37,49]. These equations allow for an unambiguous separation +between the region and the space outside the region’s boundary. The concept of hanging variables is related to the +mortar methods [52]. +3.1 +Equations at each node +The discussion below shows a detailed treatment of the discrete equations corresponding to (1a). Equation (1b) is +treated analogously. For the samples of ψR strictly inside the region, we take (2a), multiplied on both sides by the +4 + +(a) +(b) +(c) +(d) +Figure 2: Samples involved in the discrete equations updating the real part of the wavefunction. The hanging +variables are shown in red. (a) Samples in (4) for an internal node. (b) Samples in (6) for a node on the bottom +face of the boundary. (c) Samples in (7) for a node on the edge shared between the bottom and east faces of +the boundary. (d) Samples in (8) for the node on the corner formed by the bottom, south, and east faces of the +boundary. +factor ∆x∆y∆z: +ℏ ∆x∆y∆z +ψR|n+1 +i,j,k − ψR|n +i,j,k +∆t += − ℏ2 +2m +� +∆y∆z +ψI| +n+ 1 +2 +i+1,j,k − ψI| +n+ 1 +2 +i,j,k +∆x +− ∆y∆z +ψI| +n+ 1 +2 +i,j,k − ψI| +n+ 1 +2 +i−1,j,k +∆x ++∆x∆z +ψI| +n+ 1 +2 +i,j+1,k − ψI| +n+ 1 +2 +i,j,k +∆y +−∆x∆z +ψI| +n+ 1 +2 +i,j,k − ψI| +n+ 1 +2 +i,j−1,k +∆y ++∆x∆y +ψI| +n+ 1 +2 +i,j,k+1 − ψI| +n+ 1 +2 +i,j,k +∆z +−∆x∆y +ψI| +n+ 1 +2 +i,j,k − ψI| +n+ 1 +2 +i,j,k−1 +∆z +� ++ ∆x∆y∆z U|i,j,kψI| +n+ 1 +2 +i,j,k , +(4) +where the samples involved in the equation are shown in Fig. 2(a). +The factor of ∆x∆y∆z is introduced for +convenience, as will become clear in the subsequent derivations. This factor corresponds to the volume of the +∆x × ∆y × ∆z cell centered on the node (i, j, k). Such a cell is referred to as a “secondary cell”. The subdivision +of the region into the secondary cells is illustrated in Fig. 1 using the dotted lines. Equation (4) can be interpreted +as a discretization of the integral form of (1a), which reads2 +ℏ +� +∆V ′′ +∂ψR +∂t dV = − ℏ2 +2m +� +∂∆V ′′ ∇ψI · ⃗dS + +� +∆V ′′ UψI dV , +(5) +where the integrals are taken over the volume of the corresponding secondary cell ∆V ′′ and over its boundary +∂∆V ′′. The first term on the right hand side of (5) involves the flux of ∇ψI through the boundary of the secondary +cell, and is the continuous counterpart of the term in the square brackets in (4). +For nodes (i, j, 1) on the bottom boundary of the region, equation (4) would involve samples of the wavefunction +that are outside the boundary, namely ψI|i,j,k−1. The involvement of such samples is undesirable for two reasons. +First, it would make it difficult to distinguish the energy and probability corresponding to the region from the +energy and probability that should be attributed to the space outside the region. Second, the samples ψI|i,j,k−1, +in general, may not be available. For example, the space outside the boundary of the region may involve a grid +of different resolution or an entirely different model that does not involve FDTD-Q samples. Hence, we place +2The concept of discretizing partial differential equations via their integral form is related to finite volume methods [53]. +5 + +a hanging variable [49] representing the z-component of the gradient of ψI on the boundary of the region. The +hanging variable is shown in red in Fig. 2(b). +Using this variable, we write the discretization of (5) over the +corresponding secondary cell as +ℏ ∆x∆y ∆z +2 +ψR|n+1 +i,j,1 − ψR|n +i,j,1 +∆t += − ℏ2 +2m +� +∆y ∆z +2 +ψI| +n+ 1 +2 +i+1,j,1 − ψI| +n+ 1 +2 +i,j,1 +∆x +− ∆y ∆z +2 +ψI| +n+ 1 +2 +i,j,1 − ψI| +n+ 1 +2 +i−1,j,1 +∆x ++ ∆x∆z +2 +ψI| +n+ 1 +2 +i,j+1,1 − ψI| +n+ 1 +2 +i,j,1 +∆y +− ∆x∆z +2 +ψI| +n+ 1 +2 +i,j,1 − ψI| +n+ 1 +2 +i,j−1,1 +∆y ++ ∆x∆y +ψI| +n+ 1 +2 +i,j,2 − ψI| +n+ 1 +2 +i,j,1 +∆z +− ∆x∆y[∂zψI] +n+ 1 +2 +i,j,1 +� ++ ∆x∆y ∆z +2 +U|i,j,1ψI| +n+ 1 +2 +i,j,1 . +(6) +The differences with (4) are the dimensions of the secondary cell involved (which are ∆x × ∆y × ∆z/2 for the +cells adjacent to the bottom boundary), and the introduction of the hanging variable [∂zψI]i,j,1. The notation +∂z indicates that the variable represents a partial derivative with respect to z and the square brackets are used +to distinguish the hanging variables from the finite-difference approximation of the gradient of ψI. The hanging +variables can be used to couple the region with the model describing the space beyond the boundary of the +region [37,49]. As will become clear in the subsequent discussion, the hanging variables will help quantify the rate +at which the region exchanges the probability and energy with the surrounding space [37]. +For nodes on the edges of the boundary, the equations involve two hanging variables and are written over +secondary cells of volume ∆x∆y∆z/4. For example, for the nodes on the bottom-east edge of the boundary, such +as the node shown in Fig. 2(c), the proposed equation involves hanging variables [∂xψI] and [∂zψI] +ℏ ∆x +2 ∆y ∆z +2 +ψR|n+1 +nx+1,j,1 − ψR|n +nx+1,j,1 +∆t += − ℏ2 +2m +� +∆y ∆z +2 [∂xψI] +n+ 1 +2 +nx+1,j,1 − ∆y ∆z +2 +ψI| +n+ 1 +2 +nx+1,j,1 − ψI| +n+ 1 +2 +nx,j,1 +∆x ++ ∆x +2 +∆z +2 +ψI| +n+ 1 +2 +nx+1,j+1,1 − ψI| +n+ 1 +2 +nx+1,j,1 +∆y +− ∆x +2 +∆z +2 +ψI| +n+ 1 +2 +nx+1,j,1 − ψI| +n+ 1 +2 +nx+1,j−1,1 +∆y ++ ∆x +2 ∆y +ψI| +n+ 1 +2 +nx+1,j,2 − ψI| +n+ 1 +2 +nx+1,j,1 +∆z +− ∆x +2 ∆y[∂zψI] +n+ 1 +2 +nx+1,j,1 +� ++ ∆x +2 ∆y ∆z +2 +U|nx+1,j,1ψI| +n+ 1 +2 +nx+1,j,1 . +(7) +For nodes on the region’s corners, the proposed equations involve three hanging variables ([∂xψI], [∂yψI], and +[∂zψI]) and are written over secondary cells with dimensions ∆x/2 × ∆y/2 × ∆z/2. For example, on the bottom- +south-east corner illustrated in Fig. 2(d), the equation reads +ℏ ∆x +2 +∆y +2 +∆z +2 +ψR|n+1 +nx+1,1,1 − ψR|n +nx+1,1,1 +∆t += − ℏ2 +2m +� +∆y +2 +∆z +2 [∂xψI] +n+ 1 +2 +nx+1,1,1 − ∆y +2 +∆z +2 +ψI| +n+ 1 +2 +nx+1,1,1 − ψI| +n+ 1 +2 +nx,1,1 +∆x ++ ∆x +2 +∆z +2 +ψI| +n+ 1 +2 +nx+1,2,1 − ψI| +n+ 1 +2 +nx+1,1,1 +∆y +− ∆x +2 +∆z +2 [∂yψI] +n+ 1 +2 +nx+1,1,1 ++ ∆x +2 +∆y +2 +ψI| +n+ 1 +2 +nx+1,1,2 − ψI| +n+ 1 +2 +nx+1,1,1 +∆z +− ∆x +2 +∆y +2 [∂zψI] +n+ 1 +2 +nx+1,1,1 +� ++ ∆x +2 +∆y +2 +∆z +2 +U|nx+1,1,1ψI| +n+ 1 +2 +nx+1,1,1 . +(8) +The discretization of (1b) is performed analogously, resulting in equations similar to (4), (6), (7), and (8). +3.2 +Compact matrix form +In order to facilitate the subsequent derivations, we write the equations described in Section 3.1 in a compact +matrix form. Equations corresponding to (1a), such as (4), (6), (7), and (8), can be written as +ℏΛ′′ +V +ψn+1 +R +− ψn +R +∆t += − ℏ2 +2m +� +−DΛ′′ +S(Λ′ +l)−1DT ψ +n+ 1 +2 +I ++ LΛ(ˆn·)Λ′′ +S,b[∇ψI] +n+ 1 +2 +⊥ +� ++ Λ′′ +V ΛUψ +n+ 1 +2 +I +(9) +and an analogous matrix form can be written for the discrete equations corresponding to (1b). In (9), vectors ψR +and ψI contain samples of ψR and ψI at the primary nodes and vector [∇ψI]⊥ collects the hanging variables on +6 + +the boundary of the region. Matrix Λ′′ +V is a diagonal matrix containing the volumes of secondary cells depicted in +Fig. 1. Diagonal matrix ΛU contains the values of the potential U at primary nodes. The two terms in the brackets +on the right hand side of (9) correspond to the discrete outward flux of ∇ψI through the boundary of each of the +secondary cells. The first term is the contribution due to the finite-difference approximation of ∇ψI on the primary +edges. The second term is the contribution due to the hanging variables. The rows of matrix D correspond to the +primary nodes3 and its columns correspond to the primary edges. For each primary edge, the respective column of +D contains a +1 in the row corresponding to the primary node at the tail of the primary edge and a −1 in the row +corresponding to the head of the primary edge. Diagonal matrices Λ′ +l and Λ′′ +S contain, respectively, the length of +the primary edges and the area of the secondary cell faces pierced by these edges. With these definitions, +∇ψ +n+ 1 +2 +I += −(Λ′ +l)−1DT ψ +n+ 1 +2 +I +(10) +is a vector containing the finite difference approximations of ∇ψI on each of the primary edges, and the left +multiplication of this vector by DΛ′′ +S in (9) computes their contribution to the outward flux values for each +secondary cell. The columns of the matrix L correspond to the additional boundary edges where the hanging +variables are sampled. For each such column, L contains a +1 in the row corresponding to the node collocated +with the additional boundary edge. Diagonal matrix Λ(ˆn·) has the diagonal elements equal to −1 for the hanging +variables on the west, south, and bottom boundaries and to +1 for the variables on the east, north, and top +boundaries. Matrix Λ′′ +S,b contains on the diagonal the areas of the secondary cell faces pierced by the edges where +the hanging variables are sampled. Similarly to ∇ψn+0.5 +I +in (10), we also define a compact notation for the vector +containing the finite difference approximations of ∇ψR, which is given by +∇ψn +R = −(Λ′ +l)−1DT ψn +R +(11) +and will be useful in the subsequent discussion. The indexing convention and the corresponding matrix expressions +are detailed in Appendix A. +Equation (9) and its counterpart for the imaginary part of the wavefunction can be written more compactly as +ℏΛ′′ +V +ψn+1 +R +− ψn +R +∆t += Hψ +n+ 1 +2 +I +− H⊥[∇ψI] +n+ 1 +2 +⊥ +, +∀n = 0, 1, . . . , nt − 1 , +(12a) +ℏΛ′′ +V +ψ +n+ 1 +2 +I +− ψ +n− 1 +2 +I +∆t += −Hψn +R + H⊥[∇ψR]n +⊥ , +∀n = 0, 1, . . . , nt − 1 , +(12b) +where +H = ℏ2 +2mDΛ′′ +S(Λ′ +l)−1DT + Λ′′ +V ΛU , +(13) +H⊥ = ℏ2 +2mLΛ(ˆn·)Λ′′ +S,b . +(14) +Equations (12a)–(12b) can also be written as a single matrix equation, +ℏP ψn+1 − ψn +∆t += (J1 ⊗ H) ψn+1 + ψn +2 +− (J1 ⊗ H⊥)[∇ψ] +n+ 1 +2 +⊥ +, +n = 0, 1, . . . , nt − 1 , +(15) +where +P = I2 ⊗ Λ′′ +V − ∆t +2ℏ |J1| ⊗ H = +� Λ′′ +V +− ∆t +2ℏ H +− ∆t +2ℏ H +Λ′′ +V +� +, +(16) +ψn = +� +ψn +R +ψ +n− 1 +2 +I +� +, +n = 0, 1, . . . , nt , +(17) +[∇ψ] +n+ 1 +2 +⊥ += +� +[∇ψR]n +⊥ +[∇ψI] +n+ 1 +2 +⊥ +� +, +n = 0, 1, . . . , nt − 1 , +(18) +where Jm is a 2m × 2m matrix of the form +Jm = +� 0 +Im +−Im +0 +� +, +(19) +3Equivalently, we can say that the rows of D correspond to the secondary cells. +7 + +and matrix Im is an m×m identity matrix. Brackets “|·|” denote an element-wise absolute value operation and “⊗” +is the Kronecker product [54]. Matrix P in (16) will lead to an expression for the probability of finding the particle in +the region. Equation (15) can be also seen as a discrete-time dynamical system [55] that approximates the solution +of the Schr¨odinger equation describing the evolution in time of the real and imaginary parts of the wavefunction. +The evolution of the system depends on the values of the hanging variables on the boundary [∇ψ]n+0.5 +⊥ +, which act +as an excitation to (15). This excitation is referred to as the input of the dynamical system [55]. +4 +Probability conservation +In this section we propose expressions for the total probability in the region and for the probability current leaving +the region through the boundary. We show that these expressions satisfy probability conservation under a condition +on ∆t, which is recognized to be a generalized CFL limit. Furthermore, we prove that the conventional CFL limit (3) +is a sufficient condition for the probability conservation. +4.1 +Total probability and probability current +In the continuous domain, the probability of finding a particle in a volume V is given by [16] +P(t) = +� +V |ψ|2 dV = +� +V (ψ2 +R + ψ2 +I) dV . +(20) +The probability can leave the region through the boundary ∂V at the rate dictated by the outward probability +current +IP (t) ≈ +� +∂V +⃗JP (x, y, z, t) · ˆn dS , +(21) +where ˆn is the outward normal vector and ⃗JP (x, y, z, t) is the probability current density, also known in the literature +as the particle current density [16] +⃗JP = iℏ +2m(ψ∇ψ∗ − ψ∗∇ψ) = ℏ +m (ψR∇ψI − ψI∇ψR) . +(22) +In order to analyze the probability conservation properties of FDTD-Q, we define discrete expressions that approx- +imate (20) and (21). +The most obvious approach to defining the probability associated with ψn in (17) would be to directly discretize +the integral in (20) with the use of ψn +R and ψn−0.5 +I +, obtaining +Pn +simple = +nx+1 +� +i=1 +ny+1 +� +j=1 +nz+1 +� +k=1 +∆V ′′|i,j,k +� +(ψR|n +i,j,k)2 + (ψI| +n− 1 +2 +i,j,k )2� += (ψn +R)T Λ′′ +V ψn +R + (ψ +n− 1 +2 +I +)T Λ′′ +V ψ +n− 1 +2 +I +, +(23) +where ∆V ′′|i,j,k is the volume of the secondary cell associated with the node (i, j, k). However, as we will demon- +strate in Section 7, this expression does not respect the principle of probability conservation even for an isolated +region. Instead, we propose an expression for the total probability that involves the matrix P, which appears in +equation (15) describing the time evolution of the wavefunction. +Definition 4.1 (Total probability). The total probability of finding the particle in a region described by FDTD-Q +equations (12a)–(12b) is given by +Pn = (ψn)T Pψn , +n = 0, 1, . . . , nt . +(24) +To see the connection between (24) and (20), we expand (24) using the definitions of P and ψn in (16) and +(17), respectively +Pn = (ψn +R)T Λ′′ +V ψn +R + (ψ +n− 1 +2 +I +)T Λ′′ +V ψ +n− 1 +2 +I +− ∆t +ℏ (ψ +n− 1 +2 +I +)T Hψn +R += (ψn +R)T Λ′′ +V ψn +R + (ψ +n− 1 +2 +I +)T +� +Λ′′ +V ψ +n− 1 +2 +I +− ∆t +ℏ Hψn +R +� +, +n = 0, 1, . . . , nt , +(25) +and with the use of (12b) obtain +Pn = (ψn +R)T Λ′′ +V ψn +R + (ψ +n− 1 +2 +I +)T Λ′′ +V ψ +n+ 1 +2 +I +− ∆t +ℏ (ψ +n− 1 +2 +I +)T H⊥[∇ψR]n +⊥ , +∀n = 0, 1, . . . , nt − 1 . +(26) +8 + +When ∆t is small, the last term in (26) can be neglected and Pn can be approximated as +Pn ≈ (ψn +R)T Λ′′ +V ψn +R + (ψ +n− 1 +2 +I +)T Λ′′ +V ψ +n+ 1 +2 +I +, +(27) +which involves the samples of the real part of the wavefunction at time t = n∆t and the product of the samples of +the imaginary part at (n − 0.5)∆t and (n + 0.5)∆t. Hence, Pn can be seen as an approximation of P(t) in (20) at4 +t = n∆t. +The combination of staggered samples in (27) (and (29) in the footnote) has been used by Visscher [2] to pro- +pose expressions for the probability density in 1D leap-frog FDTD-Q, where periodic or zero Dirichlet boundary +conditions were assumed. Visscher argued that those expressions ensure that the total probability stays constant +with time. This combination of samples has also been shown effective in defining energy in FDTD for electromag- +netics [40]. For the case of zero Dirichlet or zero Neumann boundary conditions, the last term in (26) becomes +zero and Pn reduces to (27), becoming analogous to the expressions proposed in [2]. Hence, Definition 4.1 can be +thought of as a generalization of (27) for a 3D region that could be either isolated or open to the flow of probability +through the boundary. The proposed Definition 4.1 has an advantage of being a quadratic form, which will be +important for the analysis of the conservation properties [37,40]. +Next, we propose an expression approximating (21) in FDTD-Q. This expression quantifies the probability +current through the boundary of the region and, as will be shown in Section 4.2, respects the principle of probability +conservation. +Definition 4.2 (Probability current). The probability current flowing out of a region described by FDTD-Q +equations (12a)–(12b) is given by +I +n+ 1 +2 +P += 2 +ℏ +�ψn+1 + ψn +2 +�T +(J1 ⊗ H⊥)[∇ψ] +n+ 1 +2 +⊥ +, +∀n = 0, 1, . . . , nt − 1 . +(30) +In order to recognize that (30) approximates (21), we expand (30) using (14), (17), (18), and (19) to obtain +I +n+ 1 +2 +P += 2 +ℏ +� +ψn+1 +R ++ ψn +R +2 +�T +H⊥[∇ψI] +n+ 1 +2 +⊥ +− 2 +ℏ +� +ψ +n+ 1 +2 +I ++ ψ +n− 1 +2 +I +2 +�T +H⊥[∇ψR]n +⊥ += ℏ +m +� +LT ψn+1 +R ++ ψn +R +2 +�T +Λ(ˆn·)Λ′′ +S,b[∇ψI] +n+ 1 +2 +⊥ +− ℏ +m +� +LT ψ +n+ 1 +2 +I ++ ψ +n− 1 +2 +I +2 +�T +Λ(ˆn·)Λ′′ +S,b[∇ψR]n +⊥ . +(31) +In (31), each row of LT selects from ψR or ψI the sample on the boundary node collocated with the corresponding +hanging variable [∇ψI]⊥ or [∇ψR]⊥. This way, expressions of the form +� +LT ψR +�T Λ(ˆn·)Λ′′ +S,b[∇ψI]⊥ or +� +LT ψI +�T × +Λ(ˆn·)Λ′′ +S,b[∇ψR]⊥ are summations of terms representing the contribution of each secondary cell face on the boundary +to the flux of ψR∇ψI or ψI∇ψR, respectively. With that, we can explicitly write the different terms in In+0.5 +P +by +first splitting (31) into the contributions from the six faces of the region’s boundary +I +n+ 1 +2 +P += I +n+ 1 +2 +P,W + I +n+ 1 +2 +P,E + I +n+ 1 +2 +P,S ++ I +n+ 1 +2 +P,N + I +n+ 1 +2 +P,B + I +n+ 1 +2 +P,T , +(32) +The contribution from the west face is +I +n+ 1 +2 +P,W = +ny+1 +� +j=1 +nz+1 +� +k=1 +ℏ +m[ˆnW · ˆx]∆S′′ +x|1,j,k +� +�ψR|n+1 +1,j,k + ψR|n +1,j,k +2 +[∂xψI] +n+ 1 +2 +1,j,k − +ψI| +n+ 1 +2 +1,j,k + ψI| +n− 1 +2 +1,j,k +2 +[∂xψR]n +1,j,k +� +� , +(33) +where [ˆnW · ˆx] = −1, with ˆnW representing the outward normal vector on the west face of the boundary. The +contributions from the other five faces have analogous expressions. From (32) and (33), one can see that the first +term in (31) approximates the flux of (ℏ/m)ψR∇ψI at t = (n + 0.5)∆t and the second term approximates the flux +of −(ℏ/m)ψI∇ψR at t = n∆t, which are the fluxes involved in (21). +4Alternatively, Pn can be interpreted as an approximation of P(t) performed at t = (n − 0.5)∆t. To see this, we can use (12a) +instead of (12b) to write Pn as +Pn = (ψn +R)T Λ′′ +V ψn−1 +R ++ (ψ +n− 1 +2 +I +)T Λ′′ +V ψ +n− 1 +2 +I +− ∆t +ℏ (ψn +R)T H⊥[∇ψI] +n− 1 +2 +⊥ +, +∀n = 1, 2, . . . , nt , +(28) +to obtain +Pn ≈ (ψn +R)T Λ′′ +V ψn−1 +R ++ (ψ +n− 1 +2 +I +)T Λ′′ +V ψ +n− 1 +2 +I +. +(29) +9 + +4.2 +Probability conservation +In order to ensure that expressions in (24) and (30) respect the principle of probability conservation in the discrete +domain, we need to satisfy the following conditions: +1. The total probability (24) should be non-negative. +2. The rate of change of the total probability should equal the rate at which the probability is absorbed through +the boundary via the probability current (30). +These conditions are analogous to the conditions for a lossless system in the context of dissipative systems theory [47, +48]. The importance of a condition such as Condition 1 for defining lossless systems has been discussed in [47]. +This condition has the same significance for studying the probability conservation. If Condition 2 holds without +imposing Condition 1, the probability contained in the region is allowed to become infinitely negative. If that +occurs, the region will supply an infinite amount of probability to the surrounding space, akin to a bottomless well. +This would clearly indicate a violation in the principle of probability conservation. The following theorem provides +a restriction on the time step in FDTD-Q which ensures that the proposed expressions for the total probability (24) +and probability current (30) satisfy the two conditions above. +Theorem 4.1. Consider a region described by FDTD-Q equations (12a)–(12b) with the time step taken below the +following generalized CFL limit +∆t < ∆tCFL,gen = +2 +ρ +�1 +ℏ(Λ′′ +V )− 1 +2 H(Λ′′ +V )− 1 +2 +� , +(34) +where ρ(.) is the spectral radius of a matrix and (.)− 1 +2 denotes the inverse of the principal square root5 [54]. For +this region, the total probability (24) is bounded below by zero +Pn ≥ 0, +∀n = 0, . . . , nt . +(35) +Moreover, the total probability (24) and the probability current (30) satisfy the following relation: +Pn+1 − Pn +∆t += −I +n+ 1 +2 +P +, +∀n = 0, 1, . . . , nt − 1 . +(36) +Prior to showing the proof of the theorem, we elaborate on its meaning. When the generalized CFL condi- +tion (34) holds, the largest amount of probability that the region can supply to the surrounding space via In+0.5 +P +over the course of the simulation is equal to the probability stored in the region at the beginning of the simu- +lation [47]. In contrast, when the time step exceeds the generalized CFL limit, there is no bound on how much +probability can leave the region. Hence, violation of the generalized CFL condition allows the region to provide an +infinite amount of spurious probability to the surrounding space. This behavior would be unphysical and would +distort calculations of any quantity one wishes to obtain from the simulation. +The matrices involved in the expression for the generalized CFL limit in (34) depend only on the cell dimensions +and on the potential profile, similarly to the conventional CFL limit (3). Moreover, the following relation holds +between the conventional and generalized CFL limits (3) and (34). +Theorem 4.2. Consider a region described by (12a)–(12b). Let ∆tCFL be the CFL limit in (3) and let ∆tCFL,gen +be the generalized CFL limit in (34). Then, +∆tCFL ≤ ∆tCFL,gen . +(37) +Proof. See Appendix B. +Hence, the CFL limit (3) can be used in place of (34) in Theorem 4.1 as a sufficient condition to ensure +probability conservation. The proof of Theorem 4.2 is based on showing that the total probability can be written +as the sum of probabilities associated with each cell and proving the statement of Theorem 4.2 for each single-cell +region. A similar approach has been used in [40] in the context of electromagnetic energy. In essence, the condition +∆t < ∆tCFL ensures that probability is conserved in each primary cell and, consequently, in any region composed by +multiple primary cells. Theorems 4.1 and 4.2 give a new meaning to the CFL condition for stability (3). Specifically, +we recognize that the same condition can be used to also ensure the conservation of probability of a general open +region in FDTD-Q. Next, we provide a proof of Theorem 4.1, starting with the following lemma. +5For Λ′′ +V , (Λ′′ +V )− 1 +2 is simply a diagonal matrix containing the reciprocals of square roots of the diagonal elements of Λ′′ +V [54]. +10 + +Lemma 4.3. Matrix P in (16) has the following property: +P ≻ 0 +⇐⇒ +∆t < ∆tCFL,gen , +(38) +where “≻ 0” denotes a positive definite matrix. +Proof. Consider matrix P defined by (16). Using the properties of the Schur complement [56], the condition P ≻ 0 +holds if and only if +� +� +� +� +� +Λ′′ +V ≻ 0 +Λ′′ +V − +�∆t +2ℏ +�2 +H(Λ′′ +V )−1H ≻ 0 +. +(39) +The first condition in (39) holds for any time step. The second condition can be simplified by writing the equiva- +lent [54] condition +(Λ′′ +V )− 1 +2 +� +Λ′′ +V − +�∆t +2ℏ +�2 +H(Λ′′ +V )−1H +� +(Λ′′ +V )− 1 +2 ≻ 0 , +(40) +which reduces to +I − +�∆t +2 +�2 +Σ2 ≻ 0 , +(41) +where +Σ = 1 +ℏ(Λ′′ +V )− 1 +2 H(Λ′′ +V )− 1 +2 . +(42) +Let +Σ = QΛQH +(43) +be a Schur decomposition of the symmetric real (hence normal) matrix Σ [54], where Q is a square unitary matrix, +(.)H denotes a conjugate transpose, and Λ is a diagonal matrix containing the real eigenvalues of Σ. Then [54], +I − +�∆t +2 +�2 +Σ2 ≻ 0 +⇐⇒ +I − +�∆t +2 +�2 +Λ2 ≻ 0 +⇐⇒ +∆t < +2 +ρ(Σ) , +(44) +proving (38). +Proof of Theorem 4.1. Assume the time step is taken below the generalized CFL limit (34). From Lemma 4.3, this +implies that P is positive definite. With this, (35) follows directly from Definition 4.1. +The relation (36) can be shown by expanding the left hand side using Definition 4.1 +Pn+1 − Pn +∆t += (ψn+1)T Pψn − (ψn)T Pψn +∆t += 2(ψn+1 + ψn)T +2 +P ψn+1 − ψn +∆t +. +(45) +Using (15), +Pn+1 − Pn +∆t += 2 +ℏ +(ψn+1 + ψn)T +2 +(J1 ⊗ H) ψn+1 + ψn +2 +− 2 +ℏ +(ψn+1 + ψn)T +2 +(J1 ⊗ H⊥)[∇ψ] +n+ 1 +2 +⊥ +, +(46) +and using the fact that J1 ⊗ H is a skew-symmetric matrix, +Pn+1 − Pn +∆t += −2 +ℏ +(ψn+1 + ψn)T +2 +(J1 ⊗ H⊥)[∇ψ] +n+ 1 +2 +⊥ += −I +n+ 1 +2 +P +, +(47) +which proves (36). +5 +Energy conservation +In this section, we propose expressions for the total energy in the region and the power supplied through its +boundary and study conditions under which these expressions satisfy the principle of energy conservation. We find +that energy conservation can be demonstrated under the generalized CFL limit (34) if the total probability (24) is +bounded from above. We further argue that the existence of the upper bound on probability is guaranteed as long +as the model of the space outside the region conserves probability as well. +11 + +5.1 +Total energy and supplied power +The following expression can be used to describe the energy associated with the region +H(t) = +� +V W(x, y, z, t) dV , +(48) +where W is the energy density6 [17] +W(x, y, z, t) = ℏ2 +2m∇ψ∗ · ∇ψ + Uψ∗ψ = ℏ2 +2m∇ψR · ∇ψR + ℏ2 +2m∇ψI · ∇ψI + Uψ2 +R + Uψ2 +I . +(49) +The corresponding power entering the region through the boundary is given by +s(t) = +� +S +⃗S(x, y, z, t) · (−ˆn) dS , +(50) +where ⃗S is the energy flux density given by [17] +⃗S(x, y, z, t) = iℏ +2m +�� +− ℏ2 +2m∇2ψ + Uψ +� +∇ψ∗ − +� +− ℏ2 +2m∇2ψ + Uψ +�∗ +∇ψ +� += −ℏ2 +m +∂ψR +∂t ∇ψR − ℏ2 +m +∂ψI +∂t ∇ψI . +(51) +The proposed definitions of the total energy and energy flux in an FDTD-Q region serve as discrete counterparts +of (48) and (50). +Similarly to the case of probability, one could define the total energy in FDTD-Q by directly discretizing the +volume integration and the gradient of the wavefunction in (48), arriving at +Hn +simple = ℏ2 +2m(∇ψn +R)T Λ′′ +SΛ′ +l(∇ψn +R) + ℏ2 +2m(∇ψ +n− 1 +2 +I +)T Λ′′ +SΛ′ +l(∇ψ +n− 1 +2 +I +) + (ψn +R)T Λ′′ +V ΛUψn +R + (ψ +n− 1 +2 +I +)T Λ′′ +V ΛUψ +n− 1 +2 +I +, +(52) +where ∇ψn +R and ∇ψn−0.5 +I +are defined in (11) and (10), respectively. Using (13), (52) can be written more compactly +as +Hn +simple = (ψn +R)T Hψn +R + (ψ +n− 1 +2 +I +)T Hψ +n− 1 +2 +I +. +(53) +Similarly to Pn +simple, Hn +simple does not respect the energy conservation principle, as demonstrated in Section 7. +Instead, we propose expressions for the total energy and the corresponding supplied power for which the energy +conservation can be shown. +Definition 5.1 (Total energy). The total energy stored in a region described by FDTD-Q equations (12a)–(12b) +is given by +Hn = (ψn +R)T Hψn +R + (ψ +n− 1 +2 +I +)T Hψ +n− 1 +2 +I ++ ∆t(ψn +R − ψn−1 +R +)T +∆t +(ℏΛ′′ +V )ψ +n+ 1 +2 +I +− ψ +n− 1 +2 +I +∆t +, +n = 1, 2, . . . , nt − 1 . +(54) +The expression for Hn consists of Hn +simple and a term proportional to ∆t, which would vanish when the time +step approaches zero. In that respect, the expression (54) somewhat resembles the expressions for energy in FDTD +for Maxwell’s equations [40]. The last term in (54) is needed to ensure that the principle of energy conservation is +respected, as will be shown in the subsequent discussion. +In order to see why Hn approximates (48), we rewrite (54) using (12a) as +Hn = (ψn +R)T Hψn +R + (ψ +n− 1 +2 +I +)T Hψ +n− 1 +2 +I ++ ∆t +� +Hψ +n− 1 +2 +I +− H⊥[∇ψI] +n− 1 +2 +⊥ +�T ψ +n+ 1 +2 +I +− ψ +n− 1 +2 +I +∆t +, +n = 1, 2, . . . , nt − 1 , +(55) +which simplifies to +Hn = (ψn +R)T Hψn +R + (ψ +n− 1 +2 +I +)T Hψ +n+ 1 +2 +I +− ∆t +� +H⊥[∇ψI] +n− 1 +2 +⊥ +�T ψ +n+ 1 +2 +I +− ψ +n− 1 +2 +I +∆t +, +n = 1, 2, . . . , nt − 1 . +(56) +6Different expressions can be chosen to represent the kinetic energy contribution Wkin to the energy density (49). In this work we +choose W (1) +kin = +ℏ2 +2m ∇ψ∗ · ∇ψ, which appears in [17, 18, 57]. Expression W (2) +kin = − ℏ2 +2m ψ∗∇2ψ from [16, 57] is one possible alternative. +When considering the entire space in (48), the two expressions W (1) +kin and W (2) +kin can be shown to give the same value of H(t) [57]. +However, this is not the case when a finite region V is considered. Various expressions for Wkin, including W (1) +kin and W (2) +kin, have been +studied in [57]. +12 + +Assuming ∆t is small +Hn ≈ (ψn +R)T Hψn +R + (ψ +n− 1 +2 +I +)T Hψ +n+ 1 +2 +I +(57) +and using the definition of matrix H in (13), +Hn ≈ ℏ2 +2m(∇ψn +R)T Λ′′ +SΛ′ +l(∇ψn +R)+ ℏ2 +2m(∇ψ +n− 1 +2 +I +)T Λ′′ +SΛ′ +l(∇ψ +n+ 1 +2 +I +)+(ψn +R)T Λ′′ +V ΛUψn +R +(ψ +n− 1 +2 +I +)T Λ′′ +V ΛUψ +n+ 1 +2 +I +. (58) +From (58), Hn can be seen as an approximation of (48) at7 t = n∆t. +Definition 5.2 (Supplied power). The power supplied through the boundary to a region described by (12a)–(12b) +is given by +sn+ 1 +2 = 2(ψn+1 +R +− ψn +R)T +∆t +H⊥ +[∇ψR]n+1 +⊥ ++ [∇ψR]n +⊥ +2 ++ 2(ψ +n+ 1 +2 +I +− ψ +n− 1 +2 +I +)T +∆t +H⊥ +[∇ψI] +n+ 1 +2 +⊥ ++ [∇ψI] +n− 1 +2 +⊥ +2 +, +n = 1, 2, . . . , nt − 2 . +(61) +In order to reveal the similarity between (61) and (50), we expand (61) using the definition of H⊥ in (14) to +obtain +sn+ 1 +2 = ℏ2 +m +� +LT ψn+1 +R +− ψn +R +∆t +�T +Λ(ˆn·)Λ′′ +S,b +[∇ψR]n+1 +⊥ ++ [∇ψR]n +⊥ +2 ++ ℏ2 +m +� +LT ψ +n+ 1 +2 +I +− ψ +n− 1 +2 +I +∆t +�T +Λ(ˆn·)Λ′′ +S,b +[∇ψI] +n+ 1 +2 +⊥ ++ [∇ψI] +n− 1 +2 +⊥ +2 +. +(62) +The first term in (62) is an approximation of the part of (50) associated with ψR at t = (n + 0.5)∆t. The second +term is an approximation of the part of (50) associated with ψI at t = n∆t. +5.2 +Energy conservation +The energy conservation properties of (54) and (61) are verified in a similar way to the probability conservation. +In particular, using concepts from dissipative systems theory [47, 48], we investigate the conditions for which the +energy is bounded from below and show that the rate of change of energy equals the power supplied to the region +through the boundary. +Theorem 5.1. Consider a region described by FDTD-Q equations (12a)–(12b). Assume that ∆t < ∆tCFL,gen in +(34) and that the total probability (24) is bounded from above by some finite value Pmax +Pn ≤ Pmax , +n = 0, 1, . . . , nt . +(63) +For this region, total energy (54) is bounded from below as follows +Hn ≥ ∆x∆y∆z +Pmax +λmin(P) +� +min +� +min +i,j,k U|i,j,k, 0 +� +− 4ℏ +∆t +� +, +∀n = 1, 2, . . . , nt − 1 , +(64) +where λmin denotes the smallest eigenvalue of a symmetric real matrix. Moreover, the total energy (54) and the +supplied power (61) satisfy the following relation: +Hn+1 − Hn +∆t += sn+ 1 +2 , +∀n = 1, 2, . . . , nt − 2 . +(65) +7Energy Hn can also be interpreted as an approximation of (48) at t = (n − 0.5)∆t by rewriting (54) using (12b) as +Hn = (ψn−1 +R +)T Hψn +R + (ψ +n− 1 +2 +I +)T Hψ +n− 1 +2 +I ++ ∆t (ψn +R − ψn−1 +R +)T +∆t +H⊥[∇ψR]n +⊥, +n = 1, 2, . . . , nt − 1 +(59) +and neglecting the last term. After substituting the definition of H in (13), one would obtain +Hn ≈ ℏ2 +2m (∇ψn−1 +R +)T Λ′′ +SΛ′ +l(∇ψn +R) + ℏ2 +2m (∇ψ +n− 1 +2 +I +)T Λ′′ +SΛ′ +l(∇ψ +n− 1 +2 +I +) + (ψn−1 +R +)T Λ′′ +V ΛUψn +R + (ψ +n− 1 +2 +I +)T Λ′′ +V ΛUψ +n− 1 +2 +I +. +(60) +13 + +Before proving Theorem 5.1, we argue that the condition (63) can be assumed if the model of the space outside +the region obeys the principle of probability conservation. +Lemma 5.2. Consider the region in Fig. 1 described by (12a)–(12b). Let the time step be taken below the generalized +CFL limit (34). Let +Pn +† ≥ 0 , +∀n = 0, 1, . . . , nt +(66) +be the probability associated with the space outside the region and let In+0.5 +P † +, the probability current leaving that +space, satisfy +Pn+1 +† +− Pn +† +∆t += −I +n+ 1 +2 +P † +, +∀n = 0, 1, . . . , nt − 1 . +(67) +Furthermore, assume that the probability current leaving the region equals the current entering the space surrounding +the region: +I +n+ 1 +2 +P += −I +n+ 1 +2 +P † +, +∀n = 0, 1, . . . , nt − 1 . +(68) +Then the probability associated with the region has a finite a priori upper bound +Pn ≤ Pmax , +∀n = 0, 1, . . . nt , +(69) +where Pmax = P0 + P0 +† . +Proof. From Theorem 4.1, Pn and In+0.5 +P +satisfy (36). Adding (36) and (67) and using (68), +(Pn+1 + Pn+1 +† +) − (Pn + Pn +† ) +∆t += −I +n+ 1 +2 +P +− I +n+ 1 +2 +P † += 0 , +∀n = 0, 1, . . . , nt − 1 . +(70) +From (70), +Pn + Pn +† = P0 + P0 +† , +∀n = 0, 1, . . . , nt . +(71) +Hence, using (66) we conclude +Pn ≤ P0 + P0 +† , +∀n = 0, 1, . . . , nt . +(72) +proving (69). +Typically, the simulation setup would be such that Pmax in Lemma 5.2 equals to one. A region terminated in +zero Dirichlet or zero Neumann boundary conditions would constitute a trivial case of the lemma, with Pn +† = 0 and +In+0.5 +P † += 0. Section 6.2 considers in more detail an example of a setup in Lemma 5.2 consisting of two connected +regions, as well as the boundary conditions at their interface that ensure (68). Next, we prove Theorem 5.1. +Proof of Theorem 5.1. First, let us derive the bound (64). Under the generalized CFL limit (34), all eigenvalues of +P are strictly positive (Lemma 4.3). This allows us to derive an upper bound on ||ψn||2, which will then be used +to derive (64): +λmin(P)||ψn||2 +2 ≤ (ψn)Pψn ≤ Pmax , +n = 0, 1, . . . , nt , +(73) +||ψn||2 ≤ +� +Pmax +λmin(P) , +n = 0, 1, . . . , nt . +(74) +The first two terms in (54) can be written using (13) and (17) as +(ψn +R)T Hψn +R + (ψ +n− 1 +2 +I +)T Hψ +n− 1 +2 +I += (ψn)T +� +I2 ⊗ ℏ2 +2mDΛ′′ +S(Λ′ +l)−1DT +� +ψn + (ψn)T (I2 ⊗ Λ′′ +V ΛU)ψn . +(75) +The first term on the right hand side of (75) is nonnegative and the second term is bounded from below as follows +� +(ψn)T (I2 ⊗ Λ′′ +V ΛU)ψn ≥ 0 , +if mini,j,k U|i,j,k ≥ 0 +(ψn)T (I2 ⊗ Λ′′ +V ΛU)ψn ≥ ∆x∆y∆z mini,j,k U|i,j,k ||ψn||2 +2 , +if mini,j,k U|i,j,k < 0 . +(76) +Thus, with the use of (74), the first two terms in (54) are bounded from below as +(ψn +R)T Hψn +R + (ψ +n− 1 +2 +I +)T Hψ +n− 1 +2 +I +≥ ∆x∆y∆z min +� +min +i,j,k U|i,j,k, 0 +� +||ψn||2 +2 +≥ ∆x∆y∆z min +� +min +i,j,k U|i,j,k, 0 +� +Pmax +λmin(P) . +(77) +14 + +The last term in (54) has a following lower bound: +∆t(ψn +R − ψn−1 +R +)T +∆t +(ℏΛ′′ +V )ψ +n+ 1 +2 +I +− ψ +n− 1 +2 +I +∆t +≥ − ℏ +∆t +���(ψn +R − ψn−1 +R +)T Λ′′ +V (ψ +n+ 1 +2 +I +− ψ +n− 1 +2 +I +) +��� +≥ − ℏ +∆t||ψn +R − ψn−1 +R +||2||Λ′′ +V ||2||ψ +n+ 1 +2 +I +− ψ +n− 1 +2 +I +||2 +≥ − ℏ +∆t∆x∆y∆z(||ψn +R||2 + ||ψn−1 +R +||2)(||ψ +n+ 1 +2 +I +|| + ||ψ +n− 1 +2 +I +||2) . +(78) +Using (74), +∆t(ψn +R − ψn−1 +R +)T +∆t +(ℏΛ′′ +V )ψ +n+ 1 +2 +I +− ψ +n− 1 +2 +I +∆t +≥ −∆x∆y∆z 4ℏ +∆t +Pmax +λmin(P) . +(79) +Finally, combining (77) and (79), +Hn ≥ ∆x∆y∆z min +� +min +i,j,k U|i,j,k, 0 +� +Pmax +λmin(P) − ∆x∆y∆z 4ℏ +∆t +Pmax +λmin(P) , +(80) +which proves (64). +To show (65), we use (54) to expand the left hand side of (65) as +Hn+1 − Hn +∆t += (ψn+1 +R +)T Hψn+1 +R +− (ψn +R)T Hψn +R +∆t ++ (ψ +n+ 1 +2 +I +)T Hψ +n+ 1 +2 +I +− (ψ +n− 1 +2 +I +)T Hψ +n− 1 +2 +I +∆t ++ (ψn+1 +R +− ψn +R)T +∆t +(ℏΛ′′ +V )ψ +n+ 3 +2 +I +− ψ +n+ 1 +2 +I +∆t +− (ψ +n+ 1 +2 +I +− ψ +n− 1 +2 +I +)T +∆t +(ℏΛ′′ +V )ψn +R − ψn−1 +R +∆t +, +(81) +where the last term, being a scalar, has been written as its own transpose. The first two terms on the right hand +side of (81) can be written as +(ψn+1 +R +)T Hψn+1 +R +− (ψn +R)T Hψn +R +∆t += (ψn+1 +R +− ψn +R)T +∆t +(Hψn+1 +R ++ Hψn +R) , +(82) +(ψ +n+ 1 +2 +I +)T Hψ +n+ 1 +2 +I +− (ψ +n− 1 +2 +I +)T Hψ +n− 1 +2 +I +∆t += (ψ +n+ 1 +2 +I +− ψ +n− 1 +2 +I +)T +∆t +(Hψ +n+ 1 +2 +I ++ Hψ +n− 1 +2 +I +) . +(83) +With that, we can write (81) as +Hn+1 − Hn +∆t += (ψn+1 +R +− ψn +R)T +∆t +� +Hψn +R + Hψn+1 +R ++ ℏΛ′′ +V +ψ +n+ 3 +2 +I +− ψ +n+ 1 +2 +I +∆t +� ++ (ψ +n+ 1 +2 +I +− ψ +n− 1 +2 +I +)T +∆t +� +Hψ +n+ 1 +2 +I ++ Hψ +n− 1 +2 +I +− ℏΛ′′ +V +ψn +R − ψn−1 +R +∆t +� +. +(84) +Using (12a) and (12b), we can make the following substitutions +Hψn +R = −ℏΛ′′ +V +ψ +n+ 1 +2 +I +− ψ +n− 1 +2 +I +∆t ++ H⊥[∇ψR]n +⊥ , +∀n = 0, 1, . . . , nt − 1 , +(85) +Hψn+1 +R ++ ℏΛ′′ +V +ψ +n+ 3 +2 +I +− ψ +n+ 1 +2 +I +∆t += H⊥[∇ψR]n+1 +⊥ +, +∀n = −1, 1, . . . , nt − 2 , +(86) +Hψ +n+ 1 +2 +I += ℏΛ′′ +V +ψn+1 +R +− ψn +R +∆t ++ H⊥[∇ψI] +n+ 1 +2 +⊥ +, +∀n = 0, 1, . . . , nt − 1 , +(87) +Hψ +n− 1 +2 +I +− ℏΛ′′ +V +ψn +R − ψn−1 +R +∆t += H⊥[∇ψI] +n− 1 +2 +⊥ +, +∀n = 1, . . . , nt , +(88) +15 + +and write the left hand side of (65) as +Hn+1 − Hn +∆t += (ψn+1 +R +− ψn +R)T +∆t +� +−ℏΛ′′ +V +ψ +n+ 1 +2 +I +− ψ +n− 1 +2 +I +∆t ++ H⊥[∇ψR]n +⊥ + H⊥[∇ψR]n+1 +⊥ +� ++ (ψ +n+ 1 +2 +I +− ψ +n− 1 +2 +I +)T +∆t +� +ℏΛ′′ +V +ψn+1 +R +− ψn +R +∆t ++ H⊥[∇ψI] +n+ 1 +2 +⊥ ++ H⊥[∇ψI] +n− 1 +2 +⊥ +� += (ψn+1 +R +− ψn +R)T +∆t +� +H⊥[∇ψR]n +⊥ + H⊥[∇ψR]n+1 +⊥ +� ++ (ψ +n+ 1 +2 +I +− ψ +n− 1 +2 +I +)T +∆t +� +H⊥[∇ψI] +n+ 1 +2 +⊥ ++ H⊥[∇ψI] +n− 1 +2 +⊥ +� +, +(89) +which is equal to the right hand side of (65). +6 +A new framework to create FDTD-Q Schemes with guaranteed stability +In many applications it is desirable to create FDTD schemes for the Schr¨odinger equation where different models +are coupled together. Improper coupling is a notorious cause for instabilities in FDTD-type schemes [58]. Hence, +careful analysis is required to ensure that a coupled scheme is stable, which is challenging with existing methods. +The theory proposed in this work provides a rigorous way to construct new stable schemes in a modular and +constructive fashion. The approach is based on ensuring that the models and the coupling between them respect +the principle of probability conservation. +Conservation-based approaches have been previously used for analyzing stability of FDTD schemes in electro- +magnetics [37, 40]. The effectiveness of this type of analysis was demonstrated in FDTD for Maxwell’s equations +by creating a subgridding scheme [37] and embedding reduced models with extended CFL limit [38]. Here, as a +proof of concept, we show how the conservation approach can be used for analyzing stability of FDTD schemes for +the Schr¨odinger equation using two examples: a region isolated to the flow of probability current and two regions +that are coupled via boundary conditions. Application to more advanced scenarios is left for the future work. +6.1 +Region with no probability current through the boundary +Lemma 6.1. Consider a region described by FDTD-Q equations (12a)–(12b). Assume that ∆t < ∆tCFL,gen in +(34) and that +I +n+ 1 +2 +P += 0 , +∀n = 0, 2, . . . , nt − 1 . +(90) +Then +||ψn||2 ≤ +� +κ(P)||ψ0||2, +∀n = 0, 1, . . . , nt , +(91) +where κ(P) is the condition number of P. +Examples of boundary conditions where no net probability current exists include zero Dirichlet (ψ = 0) and +zero Neumann [∇ψ]⊥ = 0 boundary conditions. +Proof of Lemma 6.1. Assume ∆t < ∆tCFL,gen. From Theorem 4.1 and from (90), Pn stays constant over the course +of the simulation +(ψn)T Pψn = (ψ0)T Pψ0 +∀n = 0, 1, . . . , nt . +(92) +Moreover, +λmin(P)||ψn||2 +2 ≤ (ψn)T Pψn +(93) +and +(ψ0)T Pψ0 ≤ λmax(P)||ψ0||2 +2 , +(94) +giving +λmin(P)||ψn||2 +2 ≤ λmax(P)||ψ0||2 +2 . +(95) +Since, from Lemma 4.3, λmax(P) is strictly positive, +||ψn||2 +2 ≤ λmax(P) +λmin(P) ||ψ0||2 +2 . +(96) +The ratio of eigenvalues for a symmetric positive definite matrix is the condition number [54], which proves the +statement of the lemma. +16 + +Region A +Region B +Figure 3: Two FDTD-Q regions joined together by equating the quantities on the adjacent boundary as described +in (97a)–(97d). +Hence, the 2-norm of the vector ψn, which contains the samples of the real and imaginary parts of the wave- +function, has a bound that is known prior to running the simulation. The existence of such bound guarantees +stability. +From Theorem 4.2, the conventional CFL limit ∆tCFL in (3) can be used in place of ∆tCFL, gen in +Lemma 6.1. Hence, Lemma 6.1 in conjunction with Theorem 4.2 provide an alternative proof for the previously +known result [4,9] on FDTD-Q stability under the CFL limit (3). Generalized stability limits that could be relaxed +to simpler conditions such as (3) have been derived in the past using the iteration matrix approach for a method +similar to FDTD-Q [14] and for FDTD for the Maxwell equations [51,59]. +6.2 +Connection of FDTD-Q regions +In this section, we discuss how the proposed theory can be used to couple FDTD-Q models in a stable manner. +We consider a simple but representative scenario in Fig. 3, involving two adjacent regions discretized using FDTD- +Q with the same grid size and time step. The two regions need to be appropriately coupled at the interface to +maintain probability conservation and hence stability of the scheme. The north face of the first region (Region A) +is adjacent to the south face of the second region (Region B). To couple the two regions, we equate the samples of +the wavefunction and the hanging variables at the interface between Region A and Region B +ψA +R|n +i,nA +y +1,k = ψB +R|n +i,1,k, +2 ≤ i ≤ nx, 2 ≤ k ≤ nz, 0 ≤ n ≤ nt , +(97a) +ψA +I +��n− 1 +2 +i,nA +y +1,k = ψB +I +��n− 1 +2 +i,1,k , +2 ≤ i ≤ nx, 2 ≤ k ≤ nz, 0 ≤ n ≤ nt , +(97b) +[∂yψA +R]n +i,nA +y +1,k = [∂yψB +R]n +i,1,k, +2 ≤ i ≤ nx, 2 ≤ k ≤ nz, 0 ≤ n ≤ nt − 1 , +(97c) +[∂yψA +I ] +n+ 1 +2 +i,nA +y +1,k = [∂yψB +I ] +n+ 1 +2 +i,1,k , +2 ≤ i ≤ nx, 2 ≤ k ≤ nz, 0 ≤ n ≤ nt − 1 , +(97d) +where “A” and “B” denote the region a quantity corresponds to. +At all other nodes on the boundary of the +two regions, zero Dirichlet boundary condition is imposed. Next, we derive the update equations resulting from +(97a)–(97d) and show that the coupled scheme is stable if the generalized CFL limit is satisfied in each of the +regions. +17 + +6.2.1 +Update equations at the interface +From Section 3, the equation discretizing (1a) for the nodes on the north face of Region A is given by +ℏ ∆x∆y +2 ∆z +ψA +R|n+1 +i,nA +y +1,k − ψA +R|n +i,nA +y +1,k +∆t += − ℏ2 +2m +� +∆y +2 ∆z +ψA +I | +n+ 1 +2 +i+1,nA +y +1,k − ψA +I | +n+ 1 +2 +i,nA +y +1,k +∆x +− ∆y +2 ∆z +ψA +I | +n+ 1 +2 +i,nA +y +1,k − ψA +I | +n+ 1 +2 +i−1,nA +y +1,k +∆x ++ ∆x∆z[∂yψA +I ] +n+ 1 +2 +i,nA +y +1,k − ∆x∆z +ψA +I | +n+ 1 +2 +i,nA +y +1,k − ψA +I | +n+ 1 +2 +i,nA +y ,k +∆y ++∆x∆y +2 +ψA +I | +n+ 1 +2 +i,nA +y +1,k+1 − ψA +I | +n+ 1 +2 +i,nA +y +1,k +∆z +−∆x∆y +2 +ψA +I | +n+ 1 +2 +i,nA +y +1,k − ψA +I | +n+ 1 +2 +i,nA +y +1,k−1 +∆z +� ++∆x∆y +2 ∆z U A|i,nA +y +1,kψA +I | +n+ 1 +2 +i,nA +y +1,k . +(98) +The corresponding equation for the south face of Region B is +ℏ ∆x∆y +2 ∆z +ψB +R|n+1 +i,1,k − ψB +R|n +i,1,k +∆t += − ℏ2 +2m +� +∆y +2 ∆z +ψB +I | +n+ 1 +2 +i+1,1,k − ψB +I | +n+ 1 +2 +i,1,k +∆x +− ∆y +2 ∆z +ψB +I | +n+ 1 +2 +i,1,k − ψB +I | +n+ 1 +2 +i−1,1,k +∆x ++ ∆x∆z +ψB +I | +n+ 1 +2 +i,2,k − ψB +I | +n+ 1 +2 +i,1,k +∆y +− ∆x∆z[∂yψB +I ] +n+ 1 +2 +i,1,k + ∆x∆y +2 +ψB +I | +n+ 1 +2 +i,1,k+1 − ψB +I | +n+ 1 +2 +i,1,k +∆z +− ∆x∆y +2 +ψB +I | +n+ 1 +2 +i,1,k − ψB +I | +n+ 1 +2 +i,1,k−1 +∆z +� ++ ∆x∆y +2 ∆z U B|i,1,kψB +I | +n+ 1 +2 +i,1,k . +(99) +Adding (98) and (99) and using (97a), (97b), and (97d), +ℏ ∆x∆y∆z +ψB +R|n+1 +i,1,k − ψB +R|n +i,1,k +∆t += − ℏ2 +2m +� +∆y∆z +ψB +I | +n+ 1 +2 +i+1,1,k − ψB +I | +n+ 1 +2 +i,1,k +∆x +− ∆y∆z +ψB +I | +n+ 1 +2 +i,1,k − ψB +I | +n+ 1 +2 +i−1,1,k +∆x ++ ∆x∆z +ψB +I | +n+ 1 +2 +i,2,k − ψB +I | +n+ 1 +2 +i,1,k +∆y +− ∆x∆z +ψB +I | +n+ 1 +2 +i,1,k − ψA +I | +n+ 1 +2 +i,nA +y ,k +∆y ++ ∆x∆y +ψB +I | +n+ 1 +2 +i,1,k+1 − ψB +I | +n+ 1 +2 +i,1,k +∆z +− ∆x∆y +ψB +I | +n+ 1 +2 +i,1,k − ψB +I | +n+ 1 +2 +i,1,k−1 +∆z +� ++ ∆x∆y∆z +U A|i,nA +y +1,k + U B|i,1,k +2 +ψB +I | +n+ 1 +2 +i,1,k , +(100) +which has the exact same form as the FDTD equation for internal nodes (4), with the potential taken as the average +of the two adjacent nodes8. The same can be shown for the equations corresponding to (1b). +6.2.2 +Stability analysis +Assume that the time step satisfies the generalized CFL limit (34) for each region +∆t < min +� +∆tA +CFL, gen, ∆tB +CFL, gen +� +. +(101) +From Theorem 4.1, the probability in each region evolves according to +Pn+1 +A +− Pn +A +∆t += −I +n+ 1 +2 +P,A , +(102) +Pn+1 +B +− Pn +B +∆t += −I +n+ 1 +2 +P,B . +(103) +8The derivation of (100) is analogous to the treatment of material interfaces in FDTD for electromagnetics [51]. We also remark that +the similarity between (100) at the interface and (4) for internal nodes in FDTD-Q makes existing results on FDTD-Q stability [4, 9] +applicable to this particular scenario and the specific boundary conditions (97a)–(97d) selected at the interface. However, as we discuss +in Section 6.2.2 the proposed approach could be applied to developing other schemes and this example serves as an illustration. +18 + +Adding (102) and (103), +� +Pn+1 +A ++ Pn+1 +B +� +− (Pn +B + Pn +B) +∆t += −IP,A|n+ 1 +2 − IP,B|n+ 1 +2 . +(104) +As can be seen from (31), (32), and (33), the conditions (97a)–(97d) equating the wavefunction and the hanging +variables on the adjacent boundaries of the two regions ensure that the probability currents on the right hand side +of (104) cancel out. Hence, +Pn +A + Pn +B = P0 +A + P0 +B . +(105) +From Theorem 4.1, under the CFL limit Pn +A, Pn +B are both non-negative. Hence, from (105), Pn +A, and Pn +B are each +at most P0 +A + P0 +B. Repeating the reasoning in the proof of Lemma 6.1, +||ψn +A||2 +2 ≤ P0 +A + P0 +B +λmin(P) , +(106a) +||ψn +B||2 +2 ≤ P0 +A + P0 +B +λmin(P) . +(106b) +This means that the system is stable, as the values of the wavefunction samples cannot grow without bound. In +Section 7.3 we investigate the consequences of taking the time step beyond (101). In particular, we show that +violating the generalized CFL limit in a region allows that region to provide infinite probability to the surrounding +space and thus destabilize the simulation. Lastly, we remark that under condition (101), the coupled scheme can +be shown to also conserve energy using similar arguments. +This example, although simple, shows how with the proposed theory one can create composite FDTD schemes +obtained by coupling different models discretizing the Schr¨odinger equation. If each of the models satisfies the con- +servation of probability and the models are coupled in the probability-conserving manner, the resulting scheme will +by construction satisfy the probability conservation. The resulting scheme will be stable under the most restrictive +value of the generalized CFL limit (101), which is also known by construction. The choice of coupling between +the models, such as the boundary conditions (97a)–(97d), is essential for ensuring the probability conservation and +stability. In general, a different coupling scheme is not guaranteed to be probability-conserving. +The proposed approach could be applied to developing more advanced schemes in the future. For example, +subgridding scenarios [41, 42] could be analyzed as a connection of grids of different resolution [37], where one +needs to ensure that the grids exchange probability in a conserving manner to achieve the cancellation on the +right hand side of (104). Another approach that could be used to analyze general schemes is the iteration matrix +method [14, 51, 59]. The proposed conservation-based approach provides an intuitive physical interpretation that +the root cause of instability is the generation of spurious energy or probability. Moreover, the proposed approach +provides a very natural way to determine whether an FDTD-Q region or any other part of the setup is capable +of introducing instability, prior to knowing anything about the overall setup. In general, such modular stability +analysis is not trivial, since stability is a property of the entire system and coupling of equations corresponding to +stable schemes does not automatically achieve stability. The use of concepts of conservation provides a systematic, +modular, and constructive strategy to create stable composite FDTD schemes for the Schr¨odinger equation with a +guarantee of probability and energy conservation. +7 +Numerical examples +In order to investigate the validity of the results in Section 4 and Section 5, the proposed method was implemented +in Matlab and the time evolution of numerical probability and energy was investigated for different simulation +scenarios. +7.1 +Infinite well +First, we demonstrate the validity of the proposed theory for an isolated region. In particular, we consider an +electron trapped in a potential well with the potential U equal to zero inside the region and tending to infinity on +the boundary of the region and outside the region. The infinite potential well results in zero Dirichlet boundary +conditions [16]. The region has a side length of a = 30 nm in each dimension. +The region was discretized into nx = ny = nz = 30 primary cells. The CFL limit ∆tCFL and the generalized +CFL limit ∆tCFL,gen were both equal to 2.879 fs, with the relative difference (1.370×10−15) comparable to machine +19 + +(b) +(a) +simple +simple +Figure 4: Infinite well example in Section 7.1 for nx = ny = nz = 30: (a) the total probability Pn computed with +the proposed formula (24) and Pn +simple computed with the simpler expression (23) (b) the total energy Hn in (54) +and Hn +simple in (53). The dashed line shows the analytic value of energy (110). +Table 1: Error of energy expressions vs grid refinement in the example in Section 7.1 (a = 30 nm, ∆t = 0.999∆tCFL, +nt∆t = 28.76 ps). +nx = ny = nz +maxn |Pn − 1| +maxn |Pn +simple − 1| +minn |Hn − E1|/E1 +maxn |Hn +simple − E1|/E1 +10 +8.88×10−16 +2.51×10−2 +8.20×10−3 +3.19×10−2 +20 +5.55×10−16 +6.19×10−3 +2.05×10−3 +8.15×10−3 +30 +2.22×10−15 +2.74×10−3 +9.14×10−4 +3.64×10−3 +40 +1.55×10−15 +1.54×10−3 +5.14×10−4 +2.05×10−3 +50 +3.44×10−15 +9.87×10−4 +3.29×10−4 +1.31×10−3 +precision. The time step ∆t was taken as 0.999 ∆tCFL. The initial conditions ψR|0 and ψI|−0.5 were set by sampling +the following particular solution of the Schr¨odinger equation [16] +ψ(x, y, z, t) = A sin (kxx) sin (kyy) sin (kzz) exp +� +−i +�E1 +ℏ t + π +3 +�� +(107) +at the primary nodes. In (107), +i = +√ +−1 , +(108) +kx = ky = kz = π +a , +(109) +E1 = ℏ2 +2m(k2 +x + k2 +y + k2 +z) , +(110) +and A is the real positive normalization constant ensuring that P0 in (24) is equal to 1. This choice of normalization +will be discussed shortly. +Since the region is isolated to the flow of power and probability current, the energy and probability contained +inside the region are constant in the continuous domain. From the blue curves in Fig. 4, it is evident that the +proposed expressions for probability and energy respect this property in the discrete domain, in accordance with +Theorem 4.1 and Theorem 5.1. Indeed, the range of values of Pn and Hn was 3.997×10−15 and 3.228 aeV, which +could be explained by finite machine precision. In contrast, the values of Pn +simple and Hn +simple shown in red in the +figure exhibit some fluctuations. Moreover, the values of Pn +simple exceed 1 in some of the time intervals, which is +inconsistent with physics. +The continuous expression for energy (48) can be shown to equal E1 = 1.2534 meV, which is plotted in Fig. 4(b) +with the dashed line. The value of the proposed expression, Hn = 1.2523 meV, is in good agreement with E1, +corresponding to a relative error of 9.14×10−4. Table 1 shows deviations of the values of different probability +and energy expressions from the analytic solution. The values were obtained for different grid resolutions, while +maintaining the infinite well geometry and keeping the constant ratio ∆t/∆tCFL. As evident from the table, the +discrepancy between Hn and E1 can be reduced by refining the cell size and the corresponding time step. The error +in Pn +simple and Hn +simple also reduces with improved spatial resolution. However, both Pn +simplen and Hn +simple exhibit +larger errors than the proposed expressions. +Hence the proposed expressions of probability and energy provide a more accurate approximation of the analyt- +ical quantities and respect the principle of probability and energy conservation, in contrast to Pn +simple and Hn +simple. +20 + +Figure 5: Scenario considered in Section 7.2, where a Gaussian wavepacket impinges on a potential barrier. +The computational cost associated with evaluating Pn is increased compared to Pn +simple due to the off-diagonal +blocks in P in (16). However, the cost of the added computations is on par with evaluating the terms associated +with the diagonal blocks in P, even if one evaluates the additional terms directly. +Similarly, the overhead in +computing the additional terms in Hn is comparable to the cost of computing the terms that are in common with +Hn +simple. Lastly, the proposed expression of the total probability is very convenient for computing the normalization +constant A in (107), since the value of Pn stays constant in an isolated region. In contrast, the value Pn +simple varies +from time step to time step. As evident from Fig. 4(a), if the initial values of the wavefunction were normalized +such that P0 +simple equaled 1, both Pn and Pn +simple would have been centered around a value that greater than 1. +7.2 +Reflection from a potential barrier +We consider the scenario in Fig. 5, where a Gaussian wavepacket impinges on a potential barrier. The expression +for the incident pulse is [16] +ψinc(x, t) = +� +p +Ape−i(ωpt−kp(x−x0)) , +(111) +where p is an integer index, x0 is the center of the wavepacket at t = 0, kp are evenly spaced real scalars, ωp are +the corresponding angular frequencies given by ℏk2 +p/(2m), and the coefficients Ap are given by +Ap = exp +� +−1 +4 +�kp − ¯k +σ +�2� +, +(112) +where ¯k determines the center of the wavepacket in the k-space and σ determines its width. The wavepacket +impinges on a barrier with potential given by +U(x) = +� +� +� +� +� +� +� +0 , +x < a +U0 +2 , +x = a +U0 , +x > a +. +(113) +The solution can be found in quantum mechanics textbooks [16] and is given by +ψ(x, t) = +� +ψinc(x, t) + ψref(x, t) , +x < a +ψtran(x, t) +x > a , +(114) +where ψref and ψtran are reflected and transmitted waves given by +ψref(x, t) = +� +p +ApRpe−i(ωpt−kp(a−x0+a−x)) , +(115) +ψtran(x, t) = +� +p +ApTpe−i(ωpt−kp(a−x0)−Kp(x−a)) , +(116) +where Kp = +� +2m(ℏωp − V0)/ℏ, which can be real or imaginary, depending on the sign of ℏωp − V0. When Kp is +real, the corresponding wave in ψtran propagates forward in the x > a region. When Kp is imaginary, the wave +decays with x and hence cannot propagate. The reflection and transmission coefficients are given by +Rp = kp − Kp +kp + Kp +, +(117) +21 + +(a) +(b) +simple +simple +Figure 6: Results of the test in Section 7.2: (a) total probability in the region (b) total energy associated with the +region. +simple +simple +Figure 7: Results of the test in Section 7.2: (a) relative error in (119) (b) relative error in (120). +Tp = +2kp +kp + Kp +. +(118) +In this test, m is the mass of an electron, x0 = −200 nm, ¯k = 2π/¯λ, where ¯λ is 30 nm, and σ = ¯k/10. The values of +kp range between kmin = ¯k − 10σ and kmax = ¯k + 10σ, with a spacing of ∆k = σ/100. The height of the potential +barrier is U0 = 1.5 meV and a = 100 nm. +We select the space from x = 0 to x = 200 nm to be modeled with the proposed method. The dimensions +of the region were selected as 2 nm in the y and z dimensions. +The region was discretized into cells of size +∆x = ∆y = ∆z = 1 nm. The initial conditions in the region were set by sampling the analytical solution (114). +The update equations on the boundary were the modified FDTD-Q equations described in Section 3.1, such as +(6), (7), and (8), which involved the hanging variables. In order to obtain the values of the hanging variables, +expressions were found for the normal component of the gradient of the analytical solution (114). The expressions +were evaluated at integer time points n∆t for the real part and at (n + 0.5)∆t for the imaginary part. The values +of the hanging variables were zero on all faces of the boundary except for the east and west, where their values +were uniform in the y and z direction. This, in essence, rendered the problem one-dimensional, which was the +reason why choosing the y and z dimensions to be small (2 nm) was possible. The generalized CFL limit in (34) +was ∆tCFL,gen = 2.869968 fs. The CFL limit ∆tCFL in (3) was slightly lower (2.869915 fs), which is in agreement +with the theory. The simulation time step was set to 0.999∆tCFL. +Fig. 6(a) shows the probability of finding the particle in the region obtained via analytical computation, via the +proposed expression Pn in (24), and the simple expression Pn +simple in (23). The analytic values were obtained from +22 + +(20), applied to the analytical solution (114). The integration in (20) was approximated via a Riemann sum using +the midpoint rule with 1000 intervals between x = 0 nm and x = 200 nm. When using the analytical solution as a +reference, the accuracy of Pn and Pn +simple was comparable. The maximum relative errors were 8.827×10−3 for Pn, +8.978×10−3 for Pn +simple. Fig. 6(b) shows the energy stored in the region computed using Hn and Hn +simple, as well as +the analytic values approximated via the Riemann sum. The accuracy of Hn and Hn +simple was also comparable, with +a maximum relative error associated with Hn of 1.188×10−2 and the error associated with Hn +simple of 1.197×10−2. +As predicted by Theorem 4.1, the total probability Pn was non-negative, with the smallest value of 3.184×10−27. +The smallest value of energy Hn was 4.812×10−30 eV, which was higher than the bound of −6.193×10−17 eV +predicted by Theorem 5.1 when taking Pmax in (64) to be the highest value of Pn over the course of the simulation. +Based on Theorem 4.1, the values of the proposed expression Pn must satisfy +Pn = P0 − +n−1 +� +n′=0 +I +n′+ 1 +2 +P +∆t , +∀n = 1, . . . , nt , +(119) +where the right hand side is the sum of the initial probability and the probability that has entered the region due to +the flow of the probability current. The equality in (119) is evident from Fig. 6(a), showing both the left and right +hand sides of (119). The relative error between the two sides of (119) is plotted in Fig. 7(a). The largest relative +error was 4.514×10−15, which is comparable to the machine precision and hence corroborates the prediction. The +relative error in Fig. 7(a) was normalized to the largest value of the theoretical probability over all time steps +max {P(n∆t)} = 3.000×10−20. In contrast to the proposed expression for the total probability, when using Pn +simple +in place of Pn in (119), the left and right hand sides of the relation are no longer equal, as can be seen from +Fig. 7(a). The maximum value of the relative error was 1.519×10−3, which is much larger than the error in (119) +associated with Pn. Hence, at least when In+0.5 +P +is used as the representation of the probability current, Pn +simple +does not possess the conservation properties exhibited by Pn. These errors in conservation when using the simpler +expression Pn +simple also manifested themselves in the small fluctuations visible in the inset of Fig. 6(a). +Similarly, from Fig. 6(b) and Fig. 7(b) one can observe that the following relation is satisfied by the proposed +expressions for the total energy and supplied power: +Hn = H1 + +n−1 +� +n′=1 +sn′+ 1 +2 ∆t , +∀n = 2, . . . , nt − 1 , +(120) +with the largest relative error of 9.111×10−15, which is comparable to machine precision. In contrast, replacing Hn +with Hn +simple results in the relative error of 1.428×10−3 between the left and right hand sides of (120), which can no +longer be explained by the finite machine precision. The normalization constant for the relative error in Fig. 7(b) +was max{H(n∆t)} = 5.064×10−23 eV. Small fluctuations that were exhibited by Pn +simple can also be observed for +Hn +simple in the inset of Fig. 6(b). Hence, the results confirm that both Pn and Hn satisfy (119) and (120) and thus +(36) and (65), respectively. This property makes the proposed expressions preferable to Pn +simple and Hn +simple. +7.3 +Proton tunneling +In this section we consider the scenario illustrated in Fig. 8, which was taken from [35] and can be used as a +simplified model for hydrogen transfer reactions. The problem consists of three regions: reactant (r), barrier (b), +and product (p), and a proton that can transfer between the reactant and product regions via tunneling. The size of +the reactant, barrier, and product regions is lr +x ×ly ×lz, lb +x ×ly ×lz, and lp +x ×ly ×lz, respectively. The barrier region +has a potential of U0 and the other two regions have the potential equal to zero. Dirichlet zero boundary conditions +are imposed on the external boundaries of the regions. On the interfaces, the continuity of the wavefunction and +the normal component of its gradient is imposed. Using the separation of variables, the analytical solution in each +region can be found as [35] +ψr,b,p(x, y, z, t) = +N +� +m=1 +Mmf r,b,p +m +(x)gm(y)hm(z) exp +� +−iEm +ℏ t +� +, +0 ≤ x ≤ lr,b,p +x +, +(121) +23 + +where N is the number of eigenfunctions considered. The expression for f r,b,p +m +(x) in each region is +f r +m(x) = Am sin (kxmx) , +(122a) +f b +m(x) = Bm cosh +� +Kxm +� +x − lb +x +2 +�� ++ Cm sinh +� +Kxm +� +x − lb +x +2 +�� +, +(122b) +f p +m(x) = Dm sin (kxm (x − lp +x)) , +(122c) +where the choice between Bm = 0 and Cm = 0 determines the symmetry of the eigenfunction. The coefficients Am, +Bm, Cm, and Dm are chosen to ensure that the wavefunction is continuous across the region interfaces and that +� +r |f r +m(x)|2dx + +� +b |f b +m(x)|2dx + +� +p |f p +m(x)|2dx = 1 . +(123) +The expressions for gm(y) and hm(z) are given by +gm(y) = +� +2 +ly +sin (kymy) , +(124) +hm(z) = +� +2 +lz +sin (kzmz) . +(125) +The values of kym and kzm are such that the zero Dirichlet boundary conditions are satisfied for y = ly and z = lz. +The values of kxm and Kxm are related through +Exm = ℏ2k2 +xm +2mP += V0 − ℏ2K2 +xm +2mP +, +(126) +where mP = 1 dalton ≈ 1.661×10−27 kg. +The value of Ex,m can be obtained by imposing the continuity of +the x derivative of the wavefunction across the region interfaces. The energy Em in (121) corresponding to each +eigenfunction is given by +Em = Exm + Eym + Ezm , +(127) +where +Eym = ℏ2k2 +ym +2mP +, +(128) +Ezm = ℏ2k2 +zm +2mP +. +(129) +Following [35], the coefficients Mm in (121) are assumed to have the following dependence on the temperature +T: +Mm = MM ′ +m , +(130) +where +M ′ +m = exp +� +− Em +2TkB ++ iδm +� +, +(131) +where kB is the Boltzmann constant, and δm are phases that can be chosen arbitrarily. Scalar M in (130) is a +normalization constant given by +M = +1 +��N +m=1 |M ′m|2 +. +(132) +The temperature and the number of eigenfunctions were taken from [35] as T = 298 K and N = 8, respectively. +The dimensions of the regions, also from [35], are shown in Fig. 8. Unlike [35], we only consider a particular solution +with the phases δm in Table 2, as opposed to an ensemble of tunneling systems with many sets of randomized phases +δm. In [35], the model was further extended to include the effect of thermal vibrations. +We use this scenario to examine the conservation of probability and energy when multiple FDTD-Q models +are connected and to study the transfer of these quantities from one region’s model to another. Three FDTD-Q +models were defined: one discretizing the reactant region, another discretizing the barrier, and the third discretizing +the product region. The theoretical treatment of this scenario is described in Section 6.2 for two regions and an +24 + +Reactant +region (r) +Product +region (p) +Barrier +(b) +Figure 8: The scenario from [35] considered in Section 7.3, modeling proton tunneling through a barrier with +U = U0. +Table 2: Description of the solution modes considered (see [35] for details). +m +δm +Exm (meV) +Symmetry of f b +m(x) +kym +kzm +1 +0.15π +18.858713602402805 +Bm ̸= 0, Cm = 0 +π/ly +π/lz +2 +0.95π +18.858842481348997 +Bm = 0, Cm ̸= 0 +π/ly +π/lz +3 +0.25π +75.369931622707597 +Bm ̸= 0, Cm = 0 +π/ly +π/lz +4 +1.1π +75.370616971460279 +Bm = 0, Cm ̸= 0 +π/ly +π/lz +5 +0 +Ex1 +Bm ̸= 0, Cm = 0 +2π/ly +π/lz +6 +1.3π +Ex2 +Bm = 0, Cm ̸= 0 +2π/ly +π/lz +7 +0 +Ex1 +Bm ̸= 0, Cm = 0 +π/ly +2π/lz +8 +0.7π +Ex2 +Bm = 0, Cm ̸= 0 +π/ly +2π/lz +extension to tree regions is analogous. The cell dimensions in each region were ∆x = ∆y = ∆z = (1/30) ˚A. The +three FDTD-Q models were coupled by equating the wavefunction values and hanging variables on the interfaces, +as described in Section 6.2. The CFL limits and the generalized CFL limits in each region are shown in Table 3. +Their values were the same, apart from round-off error. The simulation time step was taken as 0.999 of the CFL +limit of the barrier region (∆t ≈ 55.79 as), which automatically satisfied the CFL limit condition in the reactant +and product regions. This time step ensures stability, according to [9] or based on the discussion in Section 6.2. +The initial conditions were set by sampling the analytical solution (121) and normalizing the result to achieve the +total probability in the three regions P0 +r + P0 +b + P0 +p = 1. +One simulation run was performed to study the first 1 ps of the particle’s evolution and a longer simulation +was done to observe the behavior of the system over 35 ns. In the longer simulation, probability and energy were +computed once in ten time steps (10∆t = 0.5579 fs). This allowed reducing the memory usage while providing +sufficient information, considering that the shortest period of a mode in (121) was 29.26 fs. The results are shown +in Fig. 9. During the first 1 ps, the simulated results match well with the analytical prediction. For the 35 ns +simulation, the results deviate from the analytical solution, which is expected due to dispersion errors caused by +the finite discretization of the Schr¨odinger equation. Nevertheless, the simulation was able to correctly model the +overall trend of the solution. The accuracy of Pn +simple and Hn +simple for each region was comparable to that of Pn +and Hn. +From Theorem 5.1, using 1 as a bound on probability, the energy Hn cannot take on values below -1.608 keV +for the reactant or product regions and below -54.10 keV for the barrier region. The values of energy in each region +were positive, which is in agreement with these lower bounds. Probability in each region was also positive, which +is consistent with Theorem 4.1. +Table 3: CFL limit and generalized CFL limit in each region in the test of Section 7.3 +Region +Reactant +Barrier +Product +∆tCFL,gen (as) +58.318879645754564 +55.844895610996367 +58.318879645754564 +∆tCFL (as) +58.318879645754819 +55.844895610996282 +58.318879645754819 +25 + +Probability +Time (ps) +Energy (meV) +Time (ns) +Total +r +p +b +0 +10 +20 +30 +Total +r +p +b +0 +0.2 +0.4 +0.6 +0.8 +1 +0 +20 +40 +60 +80 +100 +r +Total +b +p +0 +0.2 +0.4 +0.6 +0.8 +1 +r +Total +b +p +simple +simple +Figure 9: Energy and probability computed in the test of Section 7.3. Left panels: 1 ps simulation. Right panels: +35 ns simulation. +0 +10 +20 +30 +Time (ns) +10-10 +100 +Deviation of total probability +Proposed +Simple +0 +10 +20 +30 +Time (ns) +10-10 +100 +Relative deviation of total energy +Proposed +Simple +Figure 10: Relative deviation of the total probability and energy from their initial values (a) deviation of probability +computed from (135) and (137) (b) normalized deviation of energy computed from (136) and (138). +26 + +Total +b +r +p +Figure 11: Results of the test in Section 7.3 for the time step exceeding the CFL limit ∆tb +CFL and the generalized +CFL limit ∆tb +CFL, gen in the barrier region. +Based on the discussion in Section 6.2, the sum of the total probabilities over the three regions satisfies +� +Pn+1 +r ++ Pn+1 +b ++ Pn+1 +p +� +− +� +Pn +r + Pn +b + Pn +p +� +∆t += −IP,r|n+ 1 +2 − IP,b|n+ 1 +2 − IP,p|n+ 1 +2 = 0 +(133) +and hence the total probability must stay constant throughout the simulation. Similarly, one can show an analogous +result for the sum of the total energies +� +Hn+1 +r ++ Hn+1 +b ++ Hn+1 +p +� +− +� +Hn +r + Hn +b + Hn +p +� +∆t += s +n+ 1 +2 +r ++ s +n+ 1 +2 +b ++ s +n+ 1 +2 +p += 0 . +(134) +Thus, the total energy must likewise stay constant. In order to assess these predictions, we compute the relative +deviation of the probability and energy as +��� +Pn +r + Pn +b + Pn +p +� +− +� +P0 +r + P0 +b + P0 +p +��� , +n = 0, 10, 20, . . . , +(135) +1 +H(n∆t) +��� +Hn +r + Hn +b + Hn +p +� +− +� +H1 +r + H1 +b + H1 +p +��� , +n = 1, 11, 21, . . . , +(136) +where H(t) is the energy associated with the analytical solution and is equal to 77.5 meV. The values of (135) +and (136) are shown in Fig. 10(a) and Fig. 10(b). The largest values of (135) and (136) were 4.13×10−14 and +4.16×10−14, respectively. As expected, these fluctuations are extremely small and can be attributed to the finite +machine precision. For comparison, we also compute +��� +Pn +r,simple + Pn +b,simple + Pn +p,simple +� +− +� +P0 +r,simple + P0 +b,simple + P0 +p,simple +��� , +n = 0, 10, 20, . . . , +(137) +1 +H(n∆t) +��� +Hn +r,simple + Hn +b,simple + Hn +p,simple +� +− +� +H1 +r,simple + H1 +b,simple + H1 +p,simple +��� , +n = 1, 11, 21, . . . , +(138) +which are also shown in Fig. 10(a) and Fig. 10(b). The largest values of (137) and (138) are, respectively, 3.92×10−3 +and 4.21×10−3, meaning that the simple expressions do not abide the conservation of probability and energy exactly. +In order to illustrate a possible mechanism for the breakdown of conservation properties beyond the CFL +limit, we also run a simulation with a time step that violates the CFL limit condition in the barrier region +(∆t = 1.005∆tb +CFL = 0.9624∆tr +CFL = 0.9624∆tp +CFL). The resulting probability is shown in Fig. 11. Beyond the +CFL limit, the inequality (35) can be violated in the barrier region and the probability associated with that region is +permitted to indefinitely grow negative. This allows the barrier region to provide an infinite amount of probability +to the reactant and product regions, causing their probabilities to indefinitely grow positive. The total probability +remains unchanged, until the exponential growth of the wavefunction values due to instability eventually causes +the the total probability to deviate from 1, as a result of the finite machine precision. +8 +Conclusions +In this work, we investigated the probability and energy conservation properties of the finite-difference time-domain +scheme for solving the Schr¨odinger equation. Existing works on the conservation properties of numerical schemes for +27 + +Table 4: Indexing convention for vectors and matrices in Section 3.2. +Sample description +Scalar index +Vector index +Example(s) +Primary nodes +(i, j, k) +i + (j − 1)(nx + 1) + (k − 1)(nx + 1)(ny + 1) +ψn +R, ψn−0.5 +I +in (9) +Primary edges +x-directed +(i + 0.5, j, k) +i + (j − 1)nx + (k − 1)nx(ny + 1) +∂xψn +R in (141) +y-directed +(i, j + 0.5, k) +i + (j − 1)(nx + 1) + (k − 1)(nx + 1)ny +∂yψn +R in (141) +z-directed +(i, j, k + 0.5) +i + (j − 1)(nx + 1) + (k − 1)(nx + 1)(ny + 1) +∂zψn +R in (141) +Additional primary edges normal to the boundary +x-directed, W +(1, j, k) +j + (k − 1)(ny + 1) +[∂xψI]n+0.5 +W +in (147) +x-directed, E +(nx + 1, j, k) +j + (k − 1)(ny + 1) +[∂xψI]n+0.5 +E +in (147) +y-directed, S +(i, 1, k) +i + (k − 1)(nx + 1) +[∂yψI]n+0.5 +S +in (147) +y-directed, N +(i, ny + 1, k) +i + (k − 1)(nx + 1) +[∂yψI]n+0.5 +N +in (147) +z-directed, B +(i, j, 1) +i + (j − 1)(nx + 1) +[∂zψI]n+0.5 +B +in (147) +z-directed, T +(i, j, nz + 1) +i + (j − 1)(nx + 1) +[∂zψI]n+0.5 +T +in (147) +the Schr¨odinger equation consider a region that is terminated with periodic or zero Dirichlet boundary conditions, +which do not allow any net exchange of probability and energy with the surrounding space. +In contrast, we +considered a general scenario where a region can either be terminated using boundary conditions or form a portion +of a larger domain, where probability and energy may enter or leave the region through the boundary. We introduced +modified equations on the boundary of the region in order to write a self-contained model that allows analyzing +the properties of the region without making assumptions on the nature of the discretization beyond the boundary +of the region. We proposed expressions for the total probability and energy contained in a region discretized using +FDTD-Q, as well as expressions for the probability current and supplied power. Using these expressions, we showed +that the FDTD-Q method conserves probability under the CFL limit that has been traditionally used for ensuring +stability. We provided an illustration of the mechanism in which a violation of the CFL condition can result in +violation of the conservation of probability for an example involving three connected FDTD regions. Furthermore, +we have shown that the CFL condition ensures that the energy is conserved, under an assumption that the region +is coupled to other probability-conserving models. +The proposed expressions for computing probability and energy were compared to a more straightforward +approach and several advantages were found. First, the proposed expressions respect the probability and energy +conservation exactly. The exact conservation properties avoid spurious fluctuations exhibited by the values of the +simpler expressions, which are especially evident when a system is isolated. Second, the proposed expressions for +probability and energy tend to be slightly more accurate than the simple expressions. Lastly, in isolated systems, +the value of the proposed probability is convenient for normalizing initial values, since it is guaranteed to stay +constant. +The proposed theory sheds light on the energy and probability conservation in situations where the FDTD-Q +model of the region can exchange these quantities with the surrounding space. This insight can serve as a basis for +a stability analysis and enforcement framework in scenarios where the region is coupled to other models that can +exchange energy and/or probability with the region. As a proof of concept, we considered the case of multiple regions +with the same FDTD-Q discretization that are coupled to each other. We envision other possible applications, such +as creation of stable subgridding schemes [41,42], stable incorporation of reduced order models [38,43], and stability +analysis of advanced boundary conditions [6, 10, 12, 14]. Moreover, multi-physics simulations [60–64] can also be +considered as a type of scenario where an exchange occurs between the energy associated with the quantum particle +and the energy stored in another form, such as the energy of electromagnetic field. Hence, the approach proposed +in this paper could be very convenient if extended to such scenarios, allowing to prove stability by ensuring that +each part of the system conserves the quantities of interest. +28 + +Appendices +A +Indexing convention and expressions for the matrices in Section 3.2 +This appendix provides expressions for the matrices in Section 3.2. These expressions assume a specific order of +the samples in the vectors in Section 3.2. In particular, vectors of quantities sampled at primary nodes, such as ψn +R +or ψn−0.5 +I +in (9) have the elements ordered based on the indexing convention in Table 4. For example, a sample +ψR|n +i,j,k would be placed in the element i + (j − 1)(nx + 1) + (k − 1)(nx + 1)(ny + 1) in ψn +R. With this convention, +matrix Λ′′ +V in (9) is defined as +Λ′′ +V = ∆x∆y∆z ˜Inz+1 ⊗ ˜Iny+1 ⊗ ˜Inx+1 , +(139) +where ˜Im is an m × m matrix given by +˜Im = diag +�1 +2, 1, 1, . . . , 1, 1 +2 +� +. +(140) +Diagonal matrix ΛU contains the samples U|i,j,k placed on the diagonal elements according to the indexing con- +vention in Table 4. +Vectors with samples on the primary edges, such as ∇ψn +R in (11), contain, in that order, the samples on the +x-directed, y-directed, and z-directed edges. +∇ψn +R = +� +� +∂xψn +R +∂yψn +R +∂zψn +R +� +� , +(141) +where the ordering of samples in ∂xψn +R, ∂yψn +R, and ∂zψn +R is shown in Table 4. With this, D is defined as +D = +�Dx +Dy +Dz +� +, +(142) +where +Dx = −Inz+1 ⊗ Iny+1 ⊗ WT +nx , +Dy = −Inz+1 ⊗ WT +ny ⊗ Inx+1 , +Dz = −WT +nz ⊗ Iny+1 ⊗ Inx+1 , +(143) +where Wm is an m × (m + 1) matrix given by +Wm = +�0m×1 +Im +� +− +�Im +0m×1 +� +. +(144) +Matrices Λ′′ +S and Λ′ +l are given by +Λ′′ +S = diag +� +∆y∆z ˜Inz+1 ⊗ ˜Iny+1 ⊗ Inx, ∆x∆z ˜Inz+1 ⊗ Iny ⊗ ˜Inx+1, ∆x∆y Inz ⊗ ˜Iny+1 ⊗ ˜Inx+1 +� +, +(145) +Λ′ +l = diag +� +∆x Inz ⊗ Iny ⊗ Inz, ∆y Inz ⊗ Iny ⊗ Inz, ∆z Inz ⊗ Iny ⊗ Inz +� +. +(146) +The hanging variables corresponding to the six faces of the boundary are ordered as follows: west, east, south, +north, bottom, top. For example [∇ψI]n+0.5 +⊥ +in (9) has the following structure +[∇ψI] +n+ 1 +2 +⊥ += +� +���������� +[∂xψI] +n+ 1 +2 +W +[∂xψI] +n+ 1 +2 +E +[∂yψI] +n+ 1 +2 +S +[∂yψI] +n+ 1 +2 +N +[∂zψI] +n+ 1 +2 +B +[∂zψI] +n+ 1 +2 +T +� +���������� +, +(147) +where the indexing convention of each of the vectors on the right hand side of (147) is shown in Table 4. Based on +this ordering of the hanging variables, L is defined as +L = +�LW +LE +LS +LN +LB +LT +� +, +(148) +29 + +where +LW = Inz+1 ⊗ Iny+1 ⊗ e{1, nx + 1} , +LE = Inz+1 ⊗ Iny+1 ⊗ e{nx + 1, nx + 1} , +LS = Inz+1 ⊗ e{1, ny + 1} ⊗ Inx+1 , +LN = Inz+1 ⊗ e{ny + 1, ny + 1} ⊗ Inx+1 , +LB = e{1, nz + 1} ⊗ Iny+1 ⊗ Inx+1 , +LT = e{nz + 1, nz + 1} ⊗ Iny+1 ⊗ Inx+1 , +(149) +where e{p, m} a vector of size m × 1 with 1 in position p and zeroes in all other positions. Similarly, matrix Λ(ˆn·) +is given by +Λ(ˆn·) = diag +� +Λ(ˆn·)W, Λ(ˆn·)E, Λ(ˆn·)S, Λ(ˆn·)N, Λ(ˆn·)B, Λ(ˆn·)T +� +, +(150) +where +Λ(ˆn·)W = [ˆnW · ˆx] Inz+1 ⊗ Iny+1 , +Λ(ˆn·)E = [ˆnE · ˆx] Inz+1 ⊗ Iny+1 , +Λ(ˆn·)S = [ˆnS · ˆy] Inz+1 ⊗ Inx+1 , +Λ(ˆn·)N = [ˆnN · ˆy] Inz+1 ⊗ Inx+1 , +Λ(ˆn·)B = [ˆnB · ˆz] Iny+1 ⊗ Inx+1 , +Λ(ˆn·)T = [ˆnT · ˆz] Iny+1 ⊗ Inx+1 , +(151) +where [ˆnW · ˆx] = [ˆnS · ˆy] = [ˆnB · ˆz] = −1 and [ˆnE · ˆx] = [ˆnN · ˆy] = [ˆnT · ˆz] = 1. Matrix Λ′′ +S,b is given by +Λ′′ +S,b = diag +� +Λ′′ +S,b,W, Λ′′ +S,b,E, Λ′′ +S,b,S, Λ′′ +S,b,N, Λ′′ +S,b,B, Λ′′ +S,b,T +� +, +(152) +where +Λ′′ +S,b,W = ∆y∆z ˜Inz+1 ⊗ ˜Iny+1 , +Λ′′ +S,b,E = ∆y∆z ˜Inz+1 ⊗ ˜Iny+1 , +Λ′′ +S,b,S = ∆x∆z ˜Inz+1 ⊗ ˜Inx+1 , +Λ′′ +S,b,N = ∆x∆z ˜Inz+1 ⊗ ˜Inx+1 , +Λ′′ +S,b,B = ∆x∆y ˜Iny+1 ⊗ ˜Inx+1 , +Λ′′ +S,b,T = ∆x∆y ˜Iny+1 ⊗ ˜Inx+1 . +(153) +B +Proof of Theorem 4.2 +The proof draws inspiration from the approach in [40], where positive definiteness was shown for a matrix analogous +to P, but arising from FDTD for Maxwell’s equations and associated with stored energy. In particular, in [40], +the contribution of individual primary cells to a quadratic form analogous to (24) was considered in order to show +that the conventional CFL limit for FDTD for Maxwell’s equations guarantees positive definiteness of the matrix +associated with the entire region. +Lemma B.1. Consider a region described by FDTD-Q equations (12a)–(12b) with nx ×ny ×nz primary cells. Let +P be the matrix corresponding to the entire region. Let ψ be given by +ψ = +�ψR +ψI +� +, +(154) +where ψR, ψI ∈ R(nx+1)(ny+1)(nz+1)×1 are arbitrary vectors with each element corresponding to a primary node +in the region. Let Pijk be the matrix defined in the same way as P but for a single-cell region formed by an +individual primary cell with the bottom-south-west corner at the node (i, j, k) and the top-north-east corner at the +node (i + 1, j + 1, k + 1). Let ψijk be a vector formed by selecting the elements of ψ that correspond to the nodes +on the corners of that cell. Then, +ψT Pψ = +nx +� +i=1 +ny +� +j=1 +nz +� +k=1 +ψT +ijkPijkψijk . +(155) +The proof involves expanding both sides of (155) as a summation over primary nodes and primary edges and +collecting terms on the right hand side associated with the same primary node or primary edge. Then, each term +on the left hand side can be shown to have a corresponding term on the right hand side and vice versa. +Lemma B.2. Let ∆tCFL,gen and ∆t(i,j,k) +CFL,gen be the generalized CFL limits corresponding, respectively, to the entire +region and to a single-cell region formed by the primary cell with the bottom-south-west corner at the node (i, j, k). +Then, +min +i,j,k ∆t(i,j,k) +CFL,gen ≤ ∆tCFL,gen . +(156) +Proof. Consider matrices P and Pijk and vectors ψ and ψijk as described in Lemma B.1. Consider a time step +∆t such that +∆t < min +i,j,k ∆t(i,j,k) +CFL,gen . +(157) +30 + +With this time step, by Lemma 4.3 applied to the single-cell regions, +Pijk ≻ 0, +∀i ∈ 1, 2, . . . , nx, ∀j ∈ 1, 2, . . . , ny, ∀k ∈ 1, 2, . . . , nz , +(158) +and for each i, j, and k, we have +� +ψT +ijkPijkψijk > 0, +∀ψijk ∈ R16×1 where ψijk ̸= 0 +ψT +ijkPijkψijk = 0, +if ψijk = 0 +. +(159) +By Lemma B.1, +ψT Pψ > 0 , +∀ψ ∈ R2(nx+1)(ny+1)(nz+1)×1 with ψ ̸= 0 , +(160) +which means that P is positive definite. Hence, by Lemma 4.3, ∆t < ∆tCFL,gen. +In conclusion, the following implication holds +∆t < min +i,j,k ∆t(i,j,k) +CFL,gen +=⇒ +∆t < ∆tCFL,gen . +(161) +This is only possible if (156) is true. +Lemma B.3. Let ∆t(i,j,k) +CFL +and ∆t(i,j,k) +CFL,gen be the CFL limit and the generalized CFL limit for a single-cell region +composed of a primary cell with the bottom-south-west corner at the node (i, j, k). Then, +∆t(i,j,k) +CFL +≤ ∆t(i,j,k) +CFL,gen . +(162) +Proof. With reference to Appendix A, +Λ′′ +V,ijk = ∆x∆y∆z +8 +I8 , +(163) +Λ′′ +S,ijk = diag +�∆y∆z +4 +I4, ∆x∆z +4 +I4, ∆x∆y +4 +I4 +� +, +(164) +Λ′ +l,ijk = diag (∆x I4, ∆y I4, ∆z I4) , +(165) +Dijk = − +� +I2 ⊗ I2 ⊗ WT +1 +I2 ⊗ WT +1 ⊗ I2 +WT +1 ⊗ I2 ⊗ I2 +� +, +(166) +where subscripts ijk indicate that a matrix corresponds to the single-cell region. Define matrix Σijk as +Σijk = 1 +ℏ(Λ′′ +V,ijk)− 1 +2 Hijk(Λ′′ +V,ijk)− 1 +2 = +ℏ +2m(Λ′′ +V,ijk)− 1 +2 DijkΛ′′ +S(Λ′ +l,ijk)−1DT +ijk(Λ′′ +V )− 1 +2 + 1 +ℏΛU,ijk += ℏ +m +� +1 +(∆x)2 (I2 ⊗ I2 ⊗ WT +1 W1) + +1 +(∆y)2 (I2 ⊗ WT +1 W1 ⊗ I2) + +1 +(∆z)2 (WT +1 W1 ⊗ I2 ⊗ I2) +� ++ 1 +ℏΛU,ijk . +(167) +Let V be the following matrix: +V = +�v0 +vx +vy +vz +vyz +vxz +vxy +vxyz +� +, +(168) +where +v0 = ( +√ +8)−1|WT +1 | ⊗ |WT +1 | ⊗ |WT +1 | , +vx = ( +√ +8)−1|WT +1 | ⊗ |WT +1 | ⊗ WT +1 , +vy = ( +√ +8)−1|WT +1 | ⊗ WT +1 ⊗ |WT +1 | , +vz = ( +√ +8)−1WT +1 ⊗ |WT +1 | ⊗ |WT +1 | , +vyz = ( +√ +8)−1WT +1 ⊗ WT +1 ⊗ |WT +1 | , +vxz = ( +√ +8)−1WT +1 ⊗ |WT +1 | ⊗ WT +1 , +vxy = ( +√ +8)−1|WT +1 | ⊗ WT +1 ⊗ WT +1 , +vxyz = ( +√ +8)−1WT +1 ⊗ WT +1 ⊗ WT +1 . +(169) +Vectors |WT +1 | and WT +1 are eigenvectors of WT +1 W1 with eigenvalues 0 and 2, respectively. Using this fact, +ΣijkV = 2ℏ +m V diag +� +0, +1 +(∆x)2 , +1 +(∆y)2 , +1 +(∆z)2 , +1 +(∆y)2 + +1 +(∆z)2 , +1 +(∆x)2 + +1 +(∆z)2 , +1 +(∆x)2 + +1 +(∆y)2 , +1 +(∆x)2 + +1 +(∆y)2 + +1 +(∆z)2 +� ++ 1 +ℏΛU,ijkV . +(170) +31 + +Table 5: Relative difference of the CFL limit (3) and the generalized CFL limit (34) in the scenarios considered in +Appendix B. +Potential +nx = ny = nz = 1 +nx = ny = nz = 10 +U|i,j,k = 0 +2.49×10−16 +0 +U|i,j,k = 0.1 eV +−1.84×10−16 +1.47×10−15 +U|i,j,k = 1 eV +−1.81×10−16 +−1.81×10−16 +U|i,j,k = 10 eV +1.91×10−16 +9.56×10−16 +U|i,j,k = −0.1 eV +−6.51×10−1 +−6.51×10−1 +U|i,j,k = −1 eV +−1.72×10−1 +−1.72×10−1 +U|i,j,k = −10 eV +−2.03×10−2 +−2.03×10−2 +0 < U|i,j,k < 0.1 eV (randomized) +−5.13×10−2 +−9.18×10−2 +0 < U|i,j,k < 1 eV (randomized) +−8.05×10−2 +−4.96×10−2 +0 < U|i,j,k < 10 eV (randomized) +−1.15×10−2 +−1.01×10−2 +−0.1 eV < U|i,j,k < 0 (randomized) +−3.69×10−1 +−4.24×10−1 +−1 eV < U|i,j,k < 0 (randomized) +−2.55×10−1 +−2.28×10−1 +−10 eV < U|i,j,k < 0 (randomized) +−3.52×10−2 +−3.04×10−2 +Matrix V can be easily shown to satisfy VT V = I = VVT . Thus, +Σijk = 2ℏ +m V diag +� +0, +1 +(∆x)2 , +1 +(∆y)2 , +1 +(∆z)2 , +1 +(∆y)2 + +1 +(∆z)2 , +1 +(∆x)2 + +1 +(∆z)2 , +1 +(∆x)2 + +1 +(∆y)2 , +1 +(∆x)2 + +1 +(∆y)2 + +1 +(∆z)2 +� +VT + 1 +ℏΛU,ijk . +(171) +Hence Σijk is a sum of two symmetric matrices. The first matrix has the 2-norm equal to (2ℏ/m)((∆x)−2 + +(∆y)−2 + (∆z)−2). As a result [54], +ρ(Σijk) = ||Σijk||2 ≤ 2ℏ +m +� +1 +(∆x)2 + +1 +(∆y)2 + +1 +(∆z)2 +� ++ 1 +ℏ ||ΛU,ijk||2 +(172) +and +2 +2ℏ +m +� +1 +(∆x)2 + +1 +(∆y)2 + +1 +(∆z)2 +� ++ 1 +ℏ ||ΛU,ijk||2 +≤ +2 +ρ (Σijk) , +(173) +where +||ΛU,ijk||2 = max (|U|i,j,k| , |U|i+1,j,k| , |U|i,j+1,k| , |U|i+1,j+1,k| , |U|i,j,k+1| , |U|i+1,j,k+1| , |U|i,j+1,k+1| , |U|i+1,j+1,k+1|) . +(174) +Inequality (173) proves (162) by the definition of the two time steps in (162). +Corollary B.3.1. +∆tCFL ≤ min +i,j,k ∆t(i,j,k) +CFL,gen , +(175) +where ∆tCFL is the CFL limit for the entire region given by (3). +Proof. +∆tCFL ≤ ∆t(i,j,k) +CFL +≤ ∆t(i,j,k) +CFL,gen +∀i ∈ 1, 2, . . . , nx, ∀j ∈ 1, 2, . . . , ny, ∀k ∈ 1, 2, . . . , nz . +(176) +The statement of the corollary follows directly from (176). +Proof of Theorem 4.2. Combining the statements of Lemma B.2 and Corollary B.3.1, +∆tCFL ≤ min +i,j,k ∆t(i,j,k) +CFL,gen ≤ ∆tCFL,gen , +(177) +which proves the theorem. +32 + +Table 5 shows the relative difference between the two time steps in (3) and (34), computed as (∆tCFL − +∆tCFL,gen)/∆tCFL,gen. The cell dimensions were taken as ∆x = 1 nm, ∆y = 2 nm, and ∆z = 3 nm. The mass +of the particle was that of an electron. +As expected, for the regions consisting of a single cell with constant +nonnegative potential U at each of the eight nodes, the two time steps ∆tCFL and ∆tCFL, gen were the same, with +small discrepancy due to machine precision. Interestingly, the two time steps were also equal for the multi-cell +regions in Table 5 with constant nonnegative potential. However, when the potential either had a spacial variation +or was negative, the two time steps deviated, although the difference tended to be small. 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Plasma Sci., vol. 49, no. 11, pp. 3459–3471, 2021. +36 + diff --git a/qNE2T4oBgHgl3EQf0giO/content/tmp_files/load_file.txt b/qNE2T4oBgHgl3EQf0giO/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8aa182d6edcbb5562d06855d886ec73b392e5bb3 --- /dev/null +++ b/qNE2T4oBgHgl3EQf0giO/content/tmp_files/load_file.txt @@ -0,0 +1,1981 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf,len=1980 +page_content='Conservation properties of a leapfrog finite-difference time-domain method for the Schr¨odinger equation Fadime Bekmambetova1 and Piero Triverio∗1 1The Edward S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Rogers Sr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Department of Electrical & Computer Engineering, University of Toronto January 12, 2023 Abstract We study the probability and energy conservation properties of a leap-frog finite-difference time-domain (FDTD) method for solving the Schr¨odinger equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' We propose expressions for the total numerical probability and energy contained in a region, and for the flux of probability current and power through its boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' We show that the proposed expressions satisfy the conservation of probability and energy under suitable conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' We demonstrate their connection to the Courant-Friedrichs-Lewy condition for stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' We argue that these findings can be used for developing a modular framework for stability analysis in advanced algorithms based on FDTD for solving the Schr¨odinger equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Keywords: FDTD, Schr¨odinger equation, probability conservation, energy conservation, stability 1 Introduction The finite-difference time-domain (FDTD) algorithm is a popular numerical method for solving Maxwell’s equa- tions [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The leap-frog FDTD approach has also been proposed for solving the Schr¨odinger equation [2–4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' This scheme, which we will refer to as quantum FDTD (FDTD-Q), has since been used in various applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' For example, in [5] it was applied to model two electrons in a quantum dot, in [6] it formed the core of the method for determining the eigenstates of arbitrary nanoscale structures, in [7] it was used for studying anti-reflective coating models, and in [8] for simulating an electron diffraction through a double slit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Different properties of the FDTD- Q method have been studied, such as accuracy and stability [2, 4, 9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Other time-domain techniques for the Schr¨odinger equation that are based on finite differences include non-leap-frog implicit [2,11] and explicit [12,13] approaches, as well as higher order methods [14,15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' In the continuous domain, solutions of the Schr¨odinger equation respect the principle of conservation for both probability and energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' When one considers the entire space, the total probability of finding a particle must be constant and, with proper normalization, equal to one [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Similarly, the amount of energy (or more precisely the expectation value of energy) associated with the particle in a time-invariant potential must stay constant [16– 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The probability and energy would also stay constant when one considers a region that is isolated from the surrounding space, for example an infinite potential well [16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' In general, when the Schr¨odinger equation is discretized, the conservation properties are not guaranteed to be preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Proving that a discretization method conserves probability and energy can be done by finding discrete counterparts of these quantities and demonstrating that they remain unchanged from one time step to the next [2,18–27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' This task is not trivial for the case of FDTD- Q due to the staggered sampling of the real and imaginary parts of the wavefunction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' In relation to the conservation of probability, in [2] two approximations were proposed for the probability density in one-dimensional FDTD-Q, which were argued, though without detailed proof, to conserve the principle of probability conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Regarding energy conservation, we are not aware of any work that studies this property specifically in the context of leap-frog FDTD-Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' However, many works have investigated the conservation of energy for other time-domain methods for the linear [18,19,22,28] and nonlinear [18,20–22,25–27,29] Schr¨odinger equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Approaches based on symplectic ∗E-mail: piero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='triverio@utoronto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='ca 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='04142v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='CE] 10 Jan 2023 integrators [30,31] have been proposed for solving the Schr¨odinger equation [15,22,32–34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Symplectic algorithms can be constructed to conserve energy [22,30] by preserving the symplectic structure of the continuous equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Symplectic integrators give rise to a wide class of methods with different temporal discretization, including the leap-frog approach [30,32,33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' However, to the best of our knowledge, they have not been applied to analyze the conservation properties of the FDTD-Q scheme considered in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The works in the literature on conservation properties of numerical methods for the Schr¨odinger equation typically assume either zero [2,24, 26, 27] or periodic boundary conditions [2,19, 21, 22, 25, 26,29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Such boundary conditions imply that the energy and probability contained in the region stay constant with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' However, there is a motivation for studying the conservation properties for the general case where probability and energy can flow from the region into the surrounding space and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' For example, some simulation scenarios [6,10,12,14] involve absorbing boundary conditions meant to model unbounded domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' One may also be interested in quantifying the energy and probability in a sub-region of a larger system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Examples include studies of tunneling phenomena in hydrogen transfer reactions [35] or in quantum dot potential wells [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Consistency with physical laws is an important accuracy criterion for numerical methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Hence, expressions approximating the probability and energy in a sub-region should, ideally, obey the discrete counterparts of the principle of probability and energy conservation, in addition to being close in values to the analytical solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' A mechanism for the numerical probability and energy to leave or enter the region introduces new challenges in the analysis of the conservation properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' In particular, one needs to (i) quantify the rate at which the exchange of probability and energy occurs with the surrounding space and (ii) show that this rate balances with the rate of change of probability and energy stored in the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Moreover, one needs to (iii) ensure that the region is unable to provide indefinite amounts of energy and probability to the surrounding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' In this study, these three challenges are systematically addressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The work by Fei, P´erez-Garc´ıa, and V´azquez [20] should be mentioned, as it provides an investigation of the form of the nonlinear Schr¨odinger equation that allows the total charge to vary with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' In [20], questions similar to (i) and (ii) are addressed for a scheme that involves a variation of the Crank-Nicholson discretization in time and centered finite-difference discretization in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Moreover, in view of scenarios where a region is part of a larger setup, one may wish to study the conservation properties of a numerical method without knowing a priori what that region is connected to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' This approach has proven useful in facilitating stability analysis and enforcement in FDTD for the Maxwell equations [37, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The basis for using conservation arguments for stability analysis lies in the fact that even a small violation of energy or probability conservation provides the growth mechanism that can lead to numerical solutions that grow indefinitely, which constitutes instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' This connection between the conservation properties and stability has been recognized for the case of FDTD for Maxwell’s equations [37,39,40] but not for the case of the leap-frog FDTD- Q scheme for the Schr¨odinger equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' By ensuring that an FDTD region respects the conservation properties, one guarantees that this region would not contribute to instability when integrated in a larger setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' An advantage of the conservation approach is that this guarantee can be made without having knowledge of the surrounding space, which could involve a grid of different resolution [37,41,42], a reduced order model [38,43], a representation of an open boundary [6,10,12,14], or another model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Derivations of stability conditions for FDTD-Q have been performed using approaches related to the von Neumann analysis, which involve the investigation of temporal growth of plane wave solutions [2, 4, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' These methods are meant for simple scenarios involving constant potential and uniform discretization, where the plane wave functions are valid solutions to the discretized equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' In [9], stability conditions were derived by studying the time evolution of the norm of the error between two solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Both non-uniform and time-varying potentials were considered, making the proofs more general than those obtained using von Neumann-type analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' However, the derivations in [9] are not applicable to schemes where an FDTD-Q region is finite and is coupled to models other than a restricted set of boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Another method involves analyzing the eigenvalues of the so-called iteration matrix or system amplification matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The iteration matrix method has been used in [14] for deriving stability limits of schemes closely related to FDTD-Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The approach could be used to analyze scenarios where an FDTD-Q region is part of a larger setup consisting of multiple parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' However, in general, the eigenvalues would need to be studied for the matrix corresponding to the entire coupled scheme [44], which can make the analysis challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The conservation approach to stability analysis can circumvent this issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Lastly, it should be noted that the conservation argument for stability has appeared in the literature on other time-domain numerical methods for the linear and nonlinear Schr¨odinger equations [18,20,24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' This work presents a systematic study of probability and energy conservation in FDTD-Q for an open region, extending the energy conservation and dissipativity approaches developed previously for electromagnetics [37, 39, 40, 45, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The concepts in this work take root in the theory of dissipative systems [47, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' We formulate the FDTD-Q equations for a region, introducing unknowns on the boundary [49,50] that allow quantifying the energy 2 and probability exchange with the space outside the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' We propose expressions for discrete probability and energy, as well as particle current and supplied power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Using these expressions, we derive the conditions for the conservation of probability and energy and reveal that they are related to the Courant-Friedrichs-Lewy (CFL) condition that is traditionally understood as a condition for ensuring stability of an isolated system [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' For the case of the basic FDTD-Q scheme in an isolated region, our approach can serve as an alternative derivation of the CFL limit, which we illustrate in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Moreover, in contrast to the traditional approaches of stability analysis [2, 4, 9, 10, 14], the conservation approach allows making conclusions on whether the region is capable of destabilizing a simulation, prior to having any knowledge of how the region is terminated or what model is used to describe the space outside the region’s boundary [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Lastly, we verify that the discrete expressions serve as an accurate approximation of their continuous counterparts, with the conservation properties being an obvious advantage over other possible expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Section 2 provides background information on the leap-frog FDTD-Q method in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Section 3 describes the equations for the region, with modifications on the boundary to allow the probability and energy to travel through the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Sections 4 and 5, respectively, analyze the discrete conservation of probability and energy for the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Section 6 discusses how the proposed theory could be used for stability analysis and enforcement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Section 7 provides numerical examples and Section 8 concludes the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' 2 Background This section describes the FDTD-Q method [2–4], which is taken as the starting point in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The method solves the Schr¨odinger equation, which reads ℏ∂ψR ∂t = − ℏ2 2m∇2ψI + UψI , (1a) ℏ∂ψI ∂t = ℏ2 2m∇2ψR − UψR , (1b) where ψR(x, y, z, t) and ψI(x, y, z, t) are the real and imaginary parts of the wavefunction, respectively, m is the mass of the particle, ℏ is the reduced Planck’s constant, and U(x, y, z) is the potential energy profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' A rectangular region divided into nx ×ny ×nz primary cells1 with dimensions ∆x×∆y×∆z, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The edges of the primary cells are called primary edges, which are oriented in the +x, +y, and +z directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The nodes at the corners of the primary cells are referred to as primary nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The primary nodes are indexed as (i, j, k) from (1, 1, 1) to (nx + 1, ny + 1, nz + 1), where (i, j, k) corresponds to coordinates x = (i − 1)∆x, y = (j − 1)∆y, z = (k − 1)∆z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The time is divided into nt uniform time steps of size ∆t, with temporal index n denoting t = n∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Both ψR and ψI are sampled at the primary nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The real part of the wavefunction ψR is sampled at the integer time steps n in {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' , nt} and the imaginary part ψI is sampled at the time instances shifted by half a time step, namely {−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='5, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' , nt − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='5}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Using the centered differences to discretize the time derivatives and Laplacian operators in (1a)–(1b),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' one obtains [2–4] ℏ ψR|n+1 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k − ψR|n i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k ∆t = − ℏ2 2m � ψI| n+ 1 2 i+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k − 2ψI| n+ 1 2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k + ψI| n+ 1 2 i−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k (∆x)2 + ψI| n+ 1 2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='j+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k − 2ψI| n+ 1 2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k + ψI| n+ 1 2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='j−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k (∆y)2 + ψI| n+ 1 2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k+1 − 2ψI| n+ 1 2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k + ψI| n+ 1 2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k−1 (∆z)2 � + U|i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='kψI| n+ 1 2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k (2a) ℏ ψI| n+ 1 2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k − ψI| n− 1 2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k ∆t = ℏ2 2m � ψR|n i+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k − 2ψR|n i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k + ψR|n i−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k (∆x)2 + ψR|n i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='j+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k − 2ψR|n i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k + ψR|n i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='j−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k (∆y)2 + ψR|n i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k+1 − 2ψR|n i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k + ψR|n i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k−1 (∆z)2 � ∇2ψR − U|i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='kψR|n i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k (2b) for the nodes strictly inside the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The superscripts in (2a)–(2b) represent the time instances when the quantities are sampled and the subscripts represent the indices of the primary nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The samples involved in (2a) 1The concept of primary and secondary grids comes from FDTD in electromagnetics [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' 3 primary edges primary nodes primary cells secondary cells Figure 1: Illustration of the geometrical quantities associated with the discretized region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' In this example nx = 2, ny = 4, nz = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' All primary cells have dimensions ∆x × ∆y × ∆z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The secondary cells strictly inside the region have dimensions ∆x × ∆y × ∆z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The secondary cells adjacent to one face of the boundary are halved in the dimension normal to the face of the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Similarly, the secondary cells adjacent to two or three faces of the boundary are halved in two or three dimensions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The numerical solution is obtained starting from initial conditions ψI|−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='5 and ψR|0 and computing ψI|n+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='5 and ψR|n+1 from (2b) and (2a) in a leap-frog manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' In order to ensure that the scheme is stable, the time step needs to be taken below the CFL limit [9] ∆t < ∆tCFL = 2 2ℏ m � 1 (∆x)2 + 1 (∆y)2 + 1 (∆z)2 � + maxi,j,k ��U|i,j,k �� ℏ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (3) The update procedure for a wavefunction at a particular node requires knowing the previous time step values corresponding to the six surrounding nodes, which are only available when performing the updates at the nodes strictly inside the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Hence, boundary conditions need to be assumed, such as zero Dirichlet or periodic boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The six faces of the boundary are referred to as west (W), east (E), south (S), north (N), bottom (B), and top (T), corresponding to i = 1, i = nx + 1, j = 1, j = ny + 1, k = 1, and k = nz + 1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' 3 Equations for the region This section presents the proposed discretization of (1a)–(1b), which facilitates the investigation of probability and energy conservation in Sections 4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' As a starting point, we take the FDTD-Q method outlined in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The proposed equations consider a general scenario where the FDTD-Q region could either be terminated with boundary conditions or constitute a portion of a larger domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' In this scenario, the staggered nature of spatial sampling in FDTD-Q makes it difficult to precisely define the region’s boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' As a result, quantification of the probability and energy pertaining to the region becomes non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' In this section, we propose equations on the boundary involving the so-called hanging variables [37,49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' These equations allow for an unambiguous separation between the region and the space outside the region’s boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The concept of hanging variables is related to the mortar methods [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1 Equations at each node The discussion below shows a detailed treatment of the discrete equations corresponding to (1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Equation (1b) is treated analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' For the samples of ψR strictly inside the region, we take (2a), multiplied on both sides by the 4 (a) (b) (c) (d) Figure 2: Samples involved in the discrete equations updating the real part of the wavefunction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The hanging variables are shown in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (a) Samples in (4) for an internal node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (b) Samples in (6) for a node on the bottom face of the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (c) Samples in (7) for a node on the edge shared between the bottom and east faces of the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (d) Samples in (8) for the node on the corner formed by the bottom, south, and east faces of the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' factor ∆x∆y∆z: ℏ ∆x∆y∆z ψR|n+1 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k − ψR|n i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k ∆t = − ℏ2 2m � ∆y∆z ψI| n+ 1 2 i+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k − ψI| n+ 1 2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k ∆x − ∆y∆z ψI| n+ 1 2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k − ψI| n+ 1 2 i−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k ∆x +∆x∆z ψI| n+ 1 2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='j+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k − ψI| n+ 1 2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k ∆y −∆x∆z ψI| n+ 1 2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k − ψI| n+ 1 2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='j−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k ∆y +∆x∆y ψI| n+ 1 2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k+1 − ψI| n+ 1 2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k ∆z −∆x∆y ψI| n+ 1 2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k − ψI| n+ 1 2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k−1 ∆z � + ∆x∆y∆z U|i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='kψI| n+ 1 2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (4) where the samples involved in the equation are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The factor of ∆x∆y∆z is introduced for convenience, as will become clear in the subsequent derivations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' This factor corresponds to the volume of the ∆x × ∆y × ∆z cell centered on the node (i, j, k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Such a cell is referred to as a “secondary cell”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The subdivision of the region into the secondary cells is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' 1 using the dotted lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Equation (4) can be interpreted as a discretization of the integral form of (1a), which reads2 ℏ � ∆V ′′ ∂ψR ∂t dV = − ℏ2 2m � ∂∆V ′′ ∇ψI · ⃗dS + � ∆V ′′ UψI dV , (5) where the integrals are taken over the volume of the corresponding secondary cell ∆V ′′ and over its boundary ∂∆V ′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The first term on the right hand side of (5) involves the flux of ∇ψI through the boundary of the secondary cell, and is the continuous counterpart of the term in the square brackets in (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' For nodes (i, j, 1) on the bottom boundary of the region, equation (4) would involve samples of the wavefunction that are outside the boundary, namely ψI|i,j,k−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The involvement of such samples is undesirable for two reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' First, it would make it difficult to distinguish the energy and probability corresponding to the region from the energy and probability that should be attributed to the space outside the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Second, the samples ψI|i,j,k−1, in general, may not be available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' For example, the space outside the boundary of the region may involve a grid of different resolution or an entirely different model that does not involve FDTD-Q samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Hence, we place 2The concept of discretizing partial differential equations via their integral form is related to finite volume methods [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' 5 a hanging variable [49] representing the z-component of the gradient of ψI on the boundary of the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The hanging variable is shown in red in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' 2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Using this variable,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' we write the discretization of (5) over the corresponding secondary cell as ℏ ∆x∆y ∆z 2 ψR|n+1 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1 − ψR|n i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1 ∆t = − ℏ2 2m � ∆y ∆z 2 ψI| n+ 1 2 i+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1 − ψI| n+ 1 2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1 ∆x − ∆y ∆z 2 ψI| n+ 1 2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1 − ψI| n+ 1 2 i−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1 ∆x + ∆x∆z 2 ψI| n+ 1 2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='j+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1 − ψI| n+ 1 2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1 ∆y − ∆x∆z 2 ψI| n+ 1 2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1 − ψI| n+ 1 2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='j−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1 ∆y + ∆x∆y ψI| n+ 1 2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='2 − ψI| n+ 1 2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1 ∆z − ∆x∆y[∂zψI] n+ 1 2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1 � + ∆x∆y ∆z 2 U|i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1ψI| n+ 1 2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (6) The differences with (4) are the dimensions of the secondary cell involved (which are ∆x × ∆y × ∆z/2 for the cells adjacent to the bottom boundary), and the introduction of the hanging variable [∂zψI]i,j,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The notation ∂z indicates that the variable represents a partial derivative with respect to z and the square brackets are used to distinguish the hanging variables from the finite-difference approximation of the gradient of ψI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The hanging variables can be used to couple the region with the model describing the space beyond the boundary of the region [37,49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' As will become clear in the subsequent discussion, the hanging variables will help quantify the rate at which the region exchanges the probability and energy with the surrounding space [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' For nodes on the edges of the boundary, the equations involve two hanging variables and are written over secondary cells of volume ∆x∆y∆z/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' For example, for the nodes on the bottom-east edge of the boundary, such as the node shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' 2(c),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' the proposed equation involves hanging variables [∂xψI] and [∂zψI] ℏ ∆x 2 ∆y ∆z 2 ψR|n+1 nx+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1 − ψR|n nx+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1 ∆t = − ℏ2 2m � ∆y ∆z 2 [∂xψI] n+ 1 2 nx+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1 − ∆y ∆z 2 ψI| n+ 1 2 nx+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1 − ψI| n+ 1 2 nx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1 ∆x + ∆x 2 ∆z 2 ψI| n+ 1 2 nx+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='j+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1 − ψI| n+ 1 2 nx+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1 ∆y − ∆x 2 ∆z 2 ψI| n+ 1 2 nx+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1 − ψI| n+ 1 2 nx+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='j−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1 ∆y + ∆x 2 ∆y ψI| n+ 1 2 nx+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='2 − ψI| n+ 1 2 nx+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1 ∆z − ∆x 2 ∆y[∂zψI] n+ 1 2 nx+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1 � + ∆x 2 ∆y ∆z 2 U|nx+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1ψI| n+ 1 2 nx+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (7) For nodes on the region’s corners, the proposed equations involve three hanging variables ([∂xψI], [∂yψI], and [∂zψI]) and are written over secondary cells with dimensions ∆x/2 × ∆y/2 × ∆z/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' For example, on the bottom- south-east corner illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' 2(d), the equation reads ℏ ∆x 2 ∆y 2 ∆z 2 ψR|n+1 nx+1,1,1 − ψR|n nx+1,1,1 ∆t = − ℏ2 2m � ∆y 2 ∆z 2 [∂xψI] n+ 1 2 nx+1,1,1 − ∆y 2 ∆z 2 ψI| n+ 1 2 nx+1,1,1 − ψI| n+ 1 2 nx,1,1 ∆x + ∆x 2 ∆z 2 ψI| n+ 1 2 nx+1,2,1 − ψI| n+ 1 2 nx+1,1,1 ∆y − ∆x 2 ∆z 2 [∂yψI] n+ 1 2 nx+1,1,1 + ∆x 2 ∆y 2 ψI| n+ 1 2 nx+1,1,2 − ψI| n+ 1 2 nx+1,1,1 ∆z − ∆x 2 ∆y 2 [∂zψI] n+ 1 2 nx+1,1,1 � + ∆x 2 ∆y 2 ∆z 2 U|nx+1,1,1ψI| n+ 1 2 nx+1,1,1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (8) The discretization of (1b) is performed analogously, resulting in equations similar to (4), (6), (7), and (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='2 Compact matrix form In order to facilitate the subsequent derivations, we write the equations described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1 in a compact matrix form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Equations corresponding to (1a), such as (4), (6), (7), and (8), can be written as ℏΛ′′ V ψn+1 R − ψn R ∆t = − ℏ2 2m � −DΛ′′ S(Λ′ l)−1DT ψ n+ 1 2 I + LΛ(ˆn·)Λ′′ S,b[∇ψI] n+ 1 2 ⊥ � + Λ′′ V ΛUψ n+ 1 2 I (9) and an analogous matrix form can be written for the discrete equations corresponding to (1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' In (9), vectors ψR and ψI contain samples of ψR and ψI at the primary nodes and vector [∇ψI]⊥ collects the hanging variables on 6 the boundary of the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Matrix Λ′′ V is a diagonal matrix containing the volumes of secondary cells depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Diagonal matrix ΛU contains the values of the potential U at primary nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The two terms in the brackets on the right hand side of (9) correspond to the discrete outward flux of ∇ψI through the boundary of each of the secondary cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The first term is the contribution due to the finite-difference approximation of ∇ψI on the primary edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The second term is the contribution due to the hanging variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The rows of matrix D correspond to the primary nodes3 and its columns correspond to the primary edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' For each primary edge, the respective column of D contains a +1 in the row corresponding to the primary node at the tail of the primary edge and a −1 in the row corresponding to the head of the primary edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Diagonal matrices Λ′ l and Λ′′ S contain, respectively, the length of the primary edges and the area of the secondary cell faces pierced by these edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' With these definitions, ∇ψ n+ 1 2 I = −(Λ′ l)−1DT ψ n+ 1 2 I (10) is a vector containing the finite difference approximations of ∇ψI on each of the primary edges, and the left multiplication of this vector by DΛ′′ S in (9) computes their contribution to the outward flux values for each secondary cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The columns of the matrix L correspond to the additional boundary edges where the hanging variables are sampled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' For each such column, L contains a +1 in the row corresponding to the node collocated with the additional boundary edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Diagonal matrix Λ(ˆn·) has the diagonal elements equal to −1 for the hanging variables on the west, south, and bottom boundaries and to +1 for the variables on the east, north, and top boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Matrix Λ′′ S,b contains on the diagonal the areas of the secondary cell faces pierced by the edges where the hanging variables are sampled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Similarly to ∇ψn+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='5 I in (10), we also define a compact notation for the vector containing the finite difference approximations of ∇ψR, which is given by ∇ψn R = −(Λ′ l)−1DT ψn R (11) and will be useful in the subsequent discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The indexing convention and the corresponding matrix expressions are detailed in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Equation (9) and its counterpart for the imaginary part of the wavefunction can be written more compactly as ℏΛ′′ V ψn+1 R − ψn R ∆t = Hψ n+ 1 2 I − H⊥[∇ψI] n+ 1 2 ⊥ , ∀n = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' , nt − 1 , (12a) ℏΛ′′ V ψ n+ 1 2 I − ψ n− 1 2 I ∆t = −Hψn R + H⊥[∇ψR]n ⊥ , ∀n = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' , nt − 1 , (12b) where H = ℏ2 2mDΛ′′ S(Λ′ l)−1DT + Λ′′ V ΛU , (13) H⊥ = ℏ2 2mLΛ(ˆn·)Λ′′ S,b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (14) Equations (12a)–(12b) can also be written as a single matrix equation, ℏP ψn+1 − ψn ∆t = (J1 ⊗ H) ψn+1 + ψn 2 − (J1 ⊗ H⊥)[∇ψ] n+ 1 2 ⊥ , n = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' , nt − 1 , (15) where P = I2 ⊗ Λ′′ V − ∆t 2ℏ |J1| ⊗ H = � Λ′′ V − ∆t 2ℏ H − ∆t 2ℏ H Λ′′ V � , (16) ψn = � ψn R ψ n− 1 2 I � , n = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' , nt , (17) [∇ψ] n+ 1 2 ⊥ = � [∇ψR]n ⊥ [∇ψI] n+ 1 2 ⊥ � , n = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' , nt − 1 , (18) where Jm is a 2m × 2m matrix of the form Jm = � 0 Im −Im 0 � , (19) 3Equivalently, we can say that the rows of D correspond to the secondary cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' 7 and matrix Im is an m×m identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Brackets “|·|” denote an element-wise absolute value operation and “⊗” is the Kronecker product [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Matrix P in (16) will lead to an expression for the probability of finding the particle in the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Equation (15) can be also seen as a discrete-time dynamical system [55] that approximates the solution of the Schr¨odinger equation describing the evolution in time of the real and imaginary parts of the wavefunction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The evolution of the system depends on the values of the hanging variables on the boundary [∇ψ]n+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='5 ⊥ , which act as an excitation to (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' This excitation is referred to as the input of the dynamical system [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' 4 Probability conservation In this section we propose expressions for the total probability in the region and for the probability current leaving the region through the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' We show that these expressions satisfy probability conservation under a condition on ∆t, which is recognized to be a generalized CFL limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Furthermore, we prove that the conventional CFL limit (3) is a sufficient condition for the probability conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1 Total probability and probability current In the continuous domain, the probability of finding a particle in a volume V is given by [16] P(t) = � V |ψ|2 dV = � V (ψ2 R + ψ2 I) dV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (20) The probability can leave the region through the boundary ∂V at the rate dictated by the outward probability current IP (t) ≈ � ∂V ⃗JP (x, y, z, t) · ˆn dS , (21) where ˆn is the outward normal vector and ⃗JP (x, y, z, t) is the probability current density, also known in the literature as the particle current density [16] ⃗JP = iℏ 2m(ψ∇ψ∗ − ψ∗∇ψ) = ℏ m (ψR∇ψI − ψI∇ψR) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (22) In order to analyze the probability conservation properties of FDTD-Q, we define discrete expressions that approx- imate (20) and (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The most obvious approach to defining the probability associated with ψn in (17) would be to directly discretize the integral in (20) with the use of ψn R and ψn−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='5 I , obtaining Pn simple = nx+1 � i=1 ny+1 � j=1 nz+1 � k=1 ∆V ′′|i,j,k � (ψR|n i,j,k)2 + (ψI| n− 1 2 i,j,k )2� = (ψn R)T Λ′′ V ψn R + (ψ n− 1 2 I )T Λ′′ V ψ n− 1 2 I , (23) where ∆V ′′|i,j,k is the volume of the secondary cell associated with the node (i, j, k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' However, as we will demon- strate in Section 7, this expression does not respect the principle of probability conservation even for an isolated region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Instead, we propose an expression for the total probability that involves the matrix P, which appears in equation (15) describing the time evolution of the wavefunction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1 (Total probability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The total probability of finding the particle in a region described by FDTD-Q equations (12a)–(12b) is given by Pn = (ψn)T Pψn , n = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' , nt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (24) To see the connection between (24) and (20), we expand (24) using the definitions of P and ψn in (16) and (17), respectively Pn = (ψn R)T Λ′′ V ψn R + (ψ n− 1 2 I )T Λ′′ V ψ n− 1 2 I − ∆t ℏ (ψ n− 1 2 I )T Hψn R = (ψn R)T Λ′′ V ψn R + (ψ n− 1 2 I )T � Λ′′ V ψ n− 1 2 I − ∆t ℏ Hψn R � , n = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' , nt , (25) and with the use of (12b) obtain Pn = (ψn R)T Λ′′ V ψn R + (ψ n− 1 2 I )T Λ′′ V ψ n+ 1 2 I − ∆t ℏ (ψ n− 1 2 I )T H⊥[∇ψR]n ⊥ , ∀n = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' , nt − 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (26) 8 When ∆t is small, the last term in (26) can be neglected and Pn can be approximated as Pn ≈ (ψn R)T Λ′′ V ψn R + (ψ n− 1 2 I )T Λ′′ V ψ n+ 1 2 I , (27) which involves the samples of the real part of the wavefunction at time t = n∆t and the product of the samples of the imaginary part at (n − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='5)∆t and (n + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='5)∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Hence, Pn can be seen as an approximation of P(t) in (20) at4 t = n∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The combination of staggered samples in (27) (and (29) in the footnote) has been used by Visscher [2] to pro- pose expressions for the probability density in 1D leap-frog FDTD-Q, where periodic or zero Dirichlet boundary conditions were assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Visscher argued that those expressions ensure that the total probability stays constant with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' This combination of samples has also been shown effective in defining energy in FDTD for electromag- netics [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' For the case of zero Dirichlet or zero Neumann boundary conditions, the last term in (26) becomes zero and Pn reduces to (27), becoming analogous to the expressions proposed in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Hence, Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1 can be thought of as a generalization of (27) for a 3D region that could be either isolated or open to the flow of probability through the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The proposed Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1 has an advantage of being a quadratic form, which will be important for the analysis of the conservation properties [37,40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Next, we propose an expression approximating (21) in FDTD-Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' This expression quantifies the probability current through the boundary of the region and, as will be shown in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='2, respects the principle of probability conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='2 (Probability current).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The probability current flowing out of a region described by FDTD-Q equations (12a)–(12b) is given by I n+ 1 2 P = 2 ℏ �ψn+1 + ψn 2 �T (J1 ⊗ H⊥)[∇ψ] n+ 1 2 ⊥ , ∀n = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' , nt − 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (30) In order to recognize that (30) approximates (21), we expand (30) using (14), (17), (18), and (19) to obtain I n+ 1 2 P = 2 ℏ � ψn+1 R + ψn R 2 �T H⊥[∇ψI] n+ 1 2 ⊥ − 2 ℏ � ψ n+ 1 2 I + ψ n− 1 2 I 2 �T H⊥[∇ψR]n ⊥ = ℏ m � LT ψn+1 R + ψn R 2 �T Λ(ˆn·)Λ′′ S,b[∇ψI] n+ 1 2 ⊥ − ℏ m � LT ψ n+ 1 2 I + ψ n− 1 2 I 2 �T Λ(ˆn·)Λ′′ S,b[∇ψR]n ⊥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (31) In (31), each row of LT selects from ψR or ψI the sample on the boundary node collocated with the corresponding hanging variable [∇ψI]⊥ or [∇ψR]⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' This way, expressions of the form � LT ψR �T Λ(ˆn·)Λ′′ S,b[∇ψI]⊥ or � LT ψI �T × Λ(ˆn·)Λ′′ S,b[∇ψR]⊥ are summations of terms representing the contribution of each secondary cell face on the boundary to the flux of ψR∇ψI or ψI∇ψR, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' With that, we can explicitly write the different terms in In+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='5 P by first splitting (31) into the contributions from the six faces of the region’s boundary I n+ 1 2 P = I n+ 1 2 P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='W + I n+ 1 2 P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='E + I n+ 1 2 P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='S + I n+ 1 2 P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='N + I n+ 1 2 P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='B + I n+ 1 2 P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (32) The contribution from the west face is I n+ 1 2 P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='W = ny+1 � j=1 nz+1 � k=1 ℏ m[ˆnW · ˆx]∆S′′ x|1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k � �ψR|n+1 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k + ψR|n 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k 2 [∂xψI] n+ 1 2 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k − ψI| n+ 1 2 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k + ψI| n− 1 2 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k 2 [∂xψR]n 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (33) where [ˆnW · ˆx] = −1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' with ˆnW representing the outward normal vector on the west face of the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The contributions from the other five faces have analogous expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' From (32) and (33), one can see that the first term in (31) approximates the flux of (ℏ/m)ψR∇ψI at t = (n + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='5)∆t and the second term approximates the flux of −(ℏ/m)ψI∇ψR at t = n∆t, which are the fluxes involved in (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' 4Alternatively, Pn can be interpreted as an approximation of P(t) performed at t = (n − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='5)∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' To see this, we can use (12a) instead of (12b) to write Pn as Pn = (ψn R)T Λ′′ V ψn−1 R + (ψ n− 1 2 I )T Λ′′ V ψ n− 1 2 I − ∆t ℏ (ψn R)T H⊥[∇ψI] n− 1 2 ⊥ , ∀n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' , nt , (28) to obtain Pn ≈ (ψn R)T Λ′′ V ψn−1 R + (ψ n− 1 2 I )T Λ′′ V ψ n− 1 2 I .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (29) 9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='2 Probability conservation In order to ensure that expressions in (24) and (30) respect the principle of probability conservation in the discrete domain, we need to satisfy the following conditions: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The total probability (24) should be non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The rate of change of the total probability should equal the rate at which the probability is absorbed through the boundary via the probability current (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' These conditions are analogous to the conditions for a lossless system in the context of dissipative systems theory [47, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The importance of a condition such as Condition 1 for defining lossless systems has been discussed in [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' This condition has the same significance for studying the probability conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' If Condition 2 holds without imposing Condition 1, the probability contained in the region is allowed to become infinitely negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' If that occurs, the region will supply an infinite amount of probability to the surrounding space, akin to a bottomless well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' This would clearly indicate a violation in the principle of probability conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The following theorem provides a restriction on the time step in FDTD-Q which ensures that the proposed expressions for the total probability (24) and probability current (30) satisfy the two conditions above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Consider a region described by FDTD-Q equations (12a)–(12b) with the time step taken below the following generalized CFL limit ∆t < ∆tCFL,gen = 2 ρ �1 ℏ(Λ′′ V )− 1 2 H(Λ′′ V )− 1 2 � , (34) where ρ(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=') is the spectral radius of a matrix and (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' )− 1 2 denotes the inverse of the principal square root5 [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' For this region, the total probability (24) is bounded below by zero Pn ≥ 0, ∀n = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' , nt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (35) Moreover, the total probability (24) and the probability current (30) satisfy the following relation: Pn+1 − Pn ∆t = −I n+ 1 2 P , ∀n = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' , nt − 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (36) Prior to showing the proof of the theorem, we elaborate on its meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' When the generalized CFL condi- tion (34) holds, the largest amount of probability that the region can supply to the surrounding space via In+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='5 P over the course of the simulation is equal to the probability stored in the region at the beginning of the simu- lation [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' In contrast, when the time step exceeds the generalized CFL limit, there is no bound on how much probability can leave the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Hence, violation of the generalized CFL condition allows the region to provide an infinite amount of spurious probability to the surrounding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' This behavior would be unphysical and would distort calculations of any quantity one wishes to obtain from the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The matrices involved in the expression for the generalized CFL limit in (34) depend only on the cell dimensions and on the potential profile, similarly to the conventional CFL limit (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Moreover, the following relation holds between the conventional and generalized CFL limits (3) and (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Consider a region described by (12a)–(12b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Let ∆tCFL be the CFL limit in (3) and let ∆tCFL,gen be the generalized CFL limit in (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Then, ∆tCFL ≤ ∆tCFL,gen .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (37) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' See Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Hence, the CFL limit (3) can be used in place of (34) in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1 as a sufficient condition to ensure probability conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='2 is based on showing that the total probability can be written as the sum of probabilities associated with each cell and proving the statement of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='2 for each single-cell region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' A similar approach has been used in [40] in the context of electromagnetic energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' In essence, the condition ∆t < ∆tCFL ensures that probability is conserved in each primary cell and, consequently, in any region composed by multiple primary cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='2 give a new meaning to the CFL condition for stability (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Specifically, we recognize that the same condition can be used to also ensure the conservation of probability of a general open region in FDTD-Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Next, we provide a proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1, starting with the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' 5For Λ′′ V , (Λ′′ V )− 1 2 is simply a diagonal matrix containing the reciprocals of square roots of the diagonal elements of Λ′′ V [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' 10 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Matrix P in (16) has the following property: P ≻ 0 ⇐⇒ ∆t < ∆tCFL,gen , (38) where “≻ 0” denotes a positive definite matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Consider matrix P defined by (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Using the properties of the Schur complement [56], the condition P ≻ 0 holds if and only if � � � � � Λ′′ V ≻ 0 Λ′′ V − �∆t 2ℏ �2 H(Λ′′ V )−1H ≻ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (39) The first condition in (39) holds for any time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The second condition can be simplified by writing the equiva- lent [54] condition (Λ′′ V )− 1 2 � Λ′′ V − �∆t 2ℏ �2 H(Λ′′ V )−1H � (Λ′′ V )− 1 2 ≻ 0 , (40) which reduces to I − �∆t 2 �2 Σ2 ≻ 0 , (41) where Σ = 1 ℏ(Λ′′ V )− 1 2 H(Λ′′ V )− 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (42) Let Σ = QΛQH (43) be a Schur decomposition of the symmetric real (hence normal) matrix Σ [54], where Q is a square unitary matrix, (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' )H denotes a conjugate transpose, and Λ is a diagonal matrix containing the real eigenvalues of Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Then [54], I − �∆t 2 �2 Σ2 ≻ 0 ⇐⇒ I − �∆t 2 �2 Λ2 ≻ 0 ⇐⇒ ∆t < 2 ρ(Σ) , (44) proving (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Assume the time step is taken below the generalized CFL limit (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' From Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='3, this implies that P is positive definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' With this, (35) follows directly from Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The relation (36) can be shown by expanding the left hand side using Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1 Pn+1 − Pn ∆t = (ψn+1)T Pψn − (ψn)T Pψn ∆t = 2(ψn+1 + ψn)T 2 P ψn+1 − ψn ∆t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (45) Using (15), Pn+1 − Pn ∆t = 2 ℏ (ψn+1 + ψn)T 2 (J1 ⊗ H) ψn+1 + ψn 2 − 2 ℏ (ψn+1 + ψn)T 2 (J1 ⊗ H⊥)[∇ψ] n+ 1 2 ⊥ , (46) and using the fact that J1 ⊗ H is a skew-symmetric matrix, Pn+1 − Pn ∆t = −2 ℏ (ψn+1 + ψn)T 2 (J1 ⊗ H⊥)[∇ψ] n+ 1 2 ⊥ = −I n+ 1 2 P , (47) which proves (36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' 5 Energy conservation In this section, we propose expressions for the total energy in the region and the power supplied through its boundary and study conditions under which these expressions satisfy the principle of energy conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' We find that energy conservation can be demonstrated under the generalized CFL limit (34) if the total probability (24) is bounded from above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' We further argue that the existence of the upper bound on probability is guaranteed as long as the model of the space outside the region conserves probability as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' 11 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1 Total energy and supplied power The following expression can be used to describe the energy associated with the region H(t) = � V W(x, y, z, t) dV , (48) where W is the energy density6 [17] W(x, y, z, t) = ℏ2 2m∇ψ∗ · ∇ψ + Uψ∗ψ = ℏ2 2m∇ψR · ∇ψR + ℏ2 2m∇ψI · ∇ψI + Uψ2 R + Uψ2 I .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (49) The corresponding power entering the region through the boundary is given by s(t) = � S ⃗S(x, y, z, t) · (−ˆn) dS , (50) where ⃗S is the energy flux density given by [17] ⃗S(x, y, z, t) = iℏ 2m �� − ℏ2 2m∇2ψ + Uψ � ∇ψ∗ − � − ℏ2 2m∇2ψ + Uψ �∗ ∇ψ � = −ℏ2 m ∂ψR ∂t ∇ψR − ℏ2 m ∂ψI ∂t ∇ψI .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (51) The proposed definitions of the total energy and energy flux in an FDTD-Q region serve as discrete counterparts of (48) and (50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Similarly to the case of probability, one could define the total energy in FDTD-Q by directly discretizing the volume integration and the gradient of the wavefunction in (48), arriving at Hn simple = ℏ2 2m(∇ψn R)T Λ′′ SΛ′ l(∇ψn R) + ℏ2 2m(∇ψ n− 1 2 I )T Λ′′ SΛ′ l(∇ψ n− 1 2 I ) + (ψn R)T Λ′′ V ΛUψn R + (ψ n− 1 2 I )T Λ′′ V ΛUψ n− 1 2 I , (52) where ∇ψn R and ∇ψn−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='5 I are defined in (11) and (10), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Using (13), (52) can be written more compactly as Hn simple = (ψn R)T Hψn R + (ψ n− 1 2 I )T Hψ n− 1 2 I .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (53) Similarly to Pn simple, Hn simple does not respect the energy conservation principle, as demonstrated in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Instead, we propose expressions for the total energy and the corresponding supplied power for which the energy conservation can be shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1 (Total energy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The total energy stored in a region described by FDTD-Q equations (12a)–(12b) is given by Hn = (ψn R)T Hψn R + (ψ n− 1 2 I )T Hψ n− 1 2 I + ∆t(ψn R − ψn−1 R )T ∆t (ℏΛ′′ V )ψ n+ 1 2 I − ψ n− 1 2 I ∆t , n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' , nt − 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (54) The expression for Hn consists of Hn simple and a term proportional to ∆t, which would vanish when the time step approaches zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' In that respect, the expression (54) somewhat resembles the expressions for energy in FDTD for Maxwell’s equations [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The last term in (54) is needed to ensure that the principle of energy conservation is respected, as will be shown in the subsequent discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' In order to see why Hn approximates (48), we rewrite (54) using (12a) as Hn = (ψn R)T Hψn R + (ψ n− 1 2 I )T Hψ n− 1 2 I + ∆t � Hψ n− 1 2 I − H⊥[∇ψI] n− 1 2 ⊥ �T ψ n+ 1 2 I − ψ n− 1 2 I ∆t , n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' , nt − 1 , (55) which simplifies to Hn = (ψn R)T Hψn R + (ψ n− 1 2 I )T Hψ n+ 1 2 I − ∆t � H⊥[∇ψI] n− 1 2 ⊥ �T ψ n+ 1 2 I − ψ n− 1 2 I ∆t , n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' , nt − 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (56) 6Different expressions can be chosen to represent the kinetic energy contribution Wkin to the energy density (49).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' In this work we choose W (1) kin = ℏ2 2m ∇ψ∗ · ∇ψ, which appears in [17, 18, 57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Expression W (2) kin = − ℏ2 2m ψ∗∇2ψ from [16, 57] is one possible alternative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' When considering the entire space in (48), the two expressions W (1) kin and W (2) kin can be shown to give the same value of H(t) [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' However, this is not the case when a finite region V is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Various expressions for Wkin, including W (1) kin and W (2) kin, have been studied in [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' 12 Assuming ∆t is small Hn ≈ (ψn R)T Hψn R + (ψ n− 1 2 I )T Hψ n+ 1 2 I (57) and using the definition of matrix H in (13), Hn ≈ ℏ2 2m(∇ψn R)T Λ′′ SΛ′ l(∇ψn R)+ ℏ2 2m(∇ψ n− 1 2 I )T Λ′′ SΛ′ l(∇ψ n+ 1 2 I )+(ψn R)T Λ′′ V ΛUψn R +(ψ n− 1 2 I )T Λ′′ V ΛUψ n+ 1 2 I .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (58) From (58), Hn can be seen as an approximation of (48) at7 t = n∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='2 (Supplied power).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The power supplied through the boundary to a region described by (12a)–(12b) is given by sn+ 1 2 = 2(ψn+1 R − ψn R)T ∆t H⊥ [∇ψR]n+1 ⊥ + [∇ψR]n ⊥ 2 + 2(ψ n+ 1 2 I − ψ n− 1 2 I )T ∆t H⊥ [∇ψI] n+ 1 2 ⊥ + [∇ψI] n− 1 2 ⊥ 2 , n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' , nt − 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (61) In order to reveal the similarity between (61) and (50), we expand (61) using the definition of H⊥ in (14) to obtain sn+ 1 2 = ℏ2 m � LT ψn+1 R − ψn R ∆t �T Λ(ˆn·)Λ′′ S,b [∇ψR]n+1 ⊥ + [∇ψR]n ⊥ 2 + ℏ2 m � LT ψ n+ 1 2 I − ψ n− 1 2 I ∆t �T Λ(ˆn·)Λ′′ S,b [∇ψI] n+ 1 2 ⊥ + [∇ψI] n− 1 2 ⊥ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (62) The first term in (62) is an approximation of the part of (50) associated with ψR at t = (n + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='5)∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The second term is an approximation of the part of (50) associated with ψI at t = n∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='2 Energy conservation The energy conservation properties of (54) and (61) are verified in a similar way to the probability conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' In particular, using concepts from dissipative systems theory [47, 48], we investigate the conditions for which the energy is bounded from below and show that the rate of change of energy equals the power supplied to the region through the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Consider a region described by FDTD-Q equations (12a)–(12b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Assume that ∆t < ∆tCFL,gen in (34) and that the total probability (24) is bounded from above by some finite value Pmax Pn ≤ Pmax , n = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' , nt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (63) For this region, total energy (54) is bounded from below as follows Hn ≥ ∆x∆y∆z Pmax λmin(P) � min � min i,j,k U|i,j,k, 0 � − 4ℏ ∆t � , ∀n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' , nt − 1 , (64) where λmin denotes the smallest eigenvalue of a symmetric real matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Moreover, the total energy (54) and the supplied power (61) satisfy the following relation: Hn+1 − Hn ∆t = sn+ 1 2 , ∀n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' , nt − 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (65) 7Energy Hn can also be interpreted as an approximation of (48) at t = (n − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='5)∆t by rewriting (54) using (12b) as Hn = (ψn−1 R )T Hψn R + (ψ n− 1 2 I )T Hψ n− 1 2 I + ∆t (ψn R − ψn−1 R )T ∆t H⊥[∇ψR]n ⊥, n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' , nt − 1 (59) and neglecting the last term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' After substituting the definition of H in (13), one would obtain Hn ≈ ℏ2 2m (∇ψn−1 R )T Λ′′ SΛ′ l(∇ψn R) + ℏ2 2m (∇ψ n− 1 2 I )T Λ′′ SΛ′ l(∇ψ n− 1 2 I ) + (ψn−1 R )T Λ′′ V ΛUψn R + (ψ n− 1 2 I )T Λ′′ V ΛUψ n− 1 2 I .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (60) 13 Before proving Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1, we argue that the condition (63) can be assumed if the model of the space outside the region obeys the principle of probability conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Consider the region in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' 1 described by (12a)–(12b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Let the time step be taken below the generalized CFL limit (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Let Pn † ≥ 0 , ∀n = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' , nt (66) be the probability associated with the space outside the region and let In+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='5 P † , the probability current leaving that space, satisfy Pn+1 † − Pn † ∆t = −I n+ 1 2 P † , ∀n = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' , nt − 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (67) Furthermore, assume that the probability current leaving the region equals the current entering the space surrounding the region: I n+ 1 2 P = −I n+ 1 2 P † , ∀n = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' , nt − 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (68) Then the probability associated with the region has a finite a priori upper bound Pn ≤ Pmax , ∀n = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' nt , (69) where Pmax = P0 + P0 † .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' From Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1, Pn and In+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='5 P satisfy (36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Adding (36) and (67) and using (68), (Pn+1 + Pn+1 † ) − (Pn + Pn † ) ∆t = −I n+ 1 2 P − I n+ 1 2 P † = 0 , ∀n = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' , nt − 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (70) From (70), Pn + Pn † = P0 + P0 † , ∀n = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' , nt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (71) Hence, using (66) we conclude Pn ≤ P0 + P0 † , ∀n = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' , nt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (72) proving (69).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Typically, the simulation setup would be such that Pmax in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='2 equals to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' A region terminated in zero Dirichlet or zero Neumann boundary conditions would constitute a trivial case of the lemma, with Pn † = 0 and In+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='5 P † = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='2 considers in more detail an example of a setup in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='2 consisting of two connected regions, as well as the boundary conditions at their interface that ensure (68).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Next, we prove Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' First, let us derive the bound (64).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Under the generalized CFL limit (34), all eigenvalues of P are strictly positive (Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' This allows us to derive an upper bound on ||ψn||2, which will then be used to derive (64): λmin(P)||ψn||2 2 ≤ (ψn)Pψn ≤ Pmax , n = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' , nt , (73) ||ψn||2 ≤ � Pmax λmin(P) , n = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' , nt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (74) The first two terms in (54) can be written using (13) and (17) as (ψn R)T Hψn R + (ψ n− 1 2 I )T Hψ n− 1 2 I = (ψn)T � I2 ⊗ ℏ2 2mDΛ′′ S(Λ′ l)−1DT � ψn + (ψn)T (I2 ⊗ Λ′′ V ΛU)ψn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (75) The first term on the right hand side of (75) is nonnegative and the second term is bounded from below as follows � (ψn)T (I2 ⊗ Λ′′ V ΛU)ψn ≥ 0 , if mini,j,k U|i,j,k ≥ 0 (ψn)T (I2 ⊗ Λ′′ V ΛU)ψn ≥ ∆x∆y∆z mini,j,k U|i,j,k ||ψn||2 2 , if mini,j,k U|i,j,k < 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (76) Thus, with the use of (74), the first two terms in (54) are bounded from below as (ψn R)T Hψn R + (ψ n− 1 2 I )T Hψ n− 1 2 I ≥ ∆x∆y∆z min � min i,j,k U|i,j,k, 0 � ||ψn||2 2 ≥ ∆x∆y∆z min � min i,j,k U|i,j,k, 0 � Pmax λmin(P) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (77) 14 The last term in (54) has a following lower bound: ∆t(ψn R − ψn−1 R )T ∆t (ℏΛ′′ V )ψ n+ 1 2 I − ψ n− 1 2 I ∆t ≥ − ℏ ∆t ���(ψn R − ψn−1 R )T Λ′′ V (ψ n+ 1 2 I − ψ n− 1 2 I ) ��� ≥ − ℏ ∆t||ψn R − ψn−1 R ||2||Λ′′ V ||2||ψ n+ 1 2 I − ψ n− 1 2 I ||2 ≥ − ℏ ∆t∆x∆y∆z(||ψn R||2 + ||ψn−1 R ||2)(||ψ n+ 1 2 I || + ||ψ n− 1 2 I ||2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (78) Using (74), ∆t(ψn R − ψn−1 R )T ∆t (ℏΛ′′ V )ψ n+ 1 2 I − ψ n− 1 2 I ∆t ≥ −∆x∆y∆z 4ℏ ∆t Pmax λmin(P) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (79) Finally, combining (77) and (79), Hn ≥ ∆x∆y∆z min � min i,j,k U|i,j,k, 0 � Pmax λmin(P) − ∆x∆y∆z 4ℏ ∆t Pmax λmin(P) , (80) which proves (64).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' To show (65), we use (54) to expand the left hand side of (65) as Hn+1 − Hn ∆t = (ψn+1 R )T Hψn+1 R − (ψn R)T Hψn R ∆t + (ψ n+ 1 2 I )T Hψ n+ 1 2 I − (ψ n− 1 2 I )T Hψ n− 1 2 I ∆t + (ψn+1 R − ψn R)T ∆t (ℏΛ′′ V )ψ n+ 3 2 I − ψ n+ 1 2 I ∆t − (ψ n+ 1 2 I − ψ n− 1 2 I )T ∆t (ℏΛ′′ V )ψn R − ψn−1 R ∆t , (81) where the last term, being a scalar, has been written as its own transpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The first two terms on the right hand side of (81) can be written as (ψn+1 R )T Hψn+1 R − (ψn R)T Hψn R ∆t = (ψn+1 R − ψn R)T ∆t (Hψn+1 R + Hψn R) , (82) (ψ n+ 1 2 I )T Hψ n+ 1 2 I − (ψ n− 1 2 I )T Hψ n− 1 2 I ∆t = (ψ n+ 1 2 I − ψ n− 1 2 I )T ∆t (Hψ n+ 1 2 I + Hψ n− 1 2 I ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (83) With that, we can write (81) as Hn+1 − Hn ∆t = (ψn+1 R − ψn R)T ∆t � Hψn R + Hψn+1 R + ℏΛ′′ V ψ n+ 3 2 I − ψ n+ 1 2 I ∆t � + (ψ n+ 1 2 I − ψ n− 1 2 I )T ∆t � Hψ n+ 1 2 I + Hψ n− 1 2 I − ℏΛ′′ V ψn R − ψn−1 R ∆t � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (84) Using (12a) and (12b), we can make the following substitutions Hψn R = −ℏΛ′′ V ψ n+ 1 2 I − ψ n− 1 2 I ∆t + H⊥[∇ψR]n ⊥ , ∀n = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' , nt − 1 , (85) Hψn+1 R + ℏΛ′′ V ψ n+ 3 2 I − ψ n+ 1 2 I ∆t = H⊥[∇ψR]n+1 ⊥ , ∀n = −1, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' , nt − 2 , (86) Hψ n+ 1 2 I = ℏΛ′′ V ψn+1 R − ψn R ∆t + H⊥[∇ψI] n+ 1 2 ⊥ , ∀n = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' , nt − 1 , (87) Hψ n− 1 2 I − ℏΛ′′ V ψn R − ψn−1 R ∆t = H⊥[∇ψI] n− 1 2 ⊥ , ∀n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' nt ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (88) 15 and write the left hand side of (65) as Hn+1 − Hn ∆t = (ψn+1 R − ψn R)T ∆t � −ℏΛ′′ V ψ n+ 1 2 I − ψ n− 1 2 I ∆t + H⊥[∇ψR]n ⊥ + H⊥[∇ψR]n+1 ⊥ � + (ψ n+ 1 2 I − ψ n− 1 2 I )T ∆t � ℏΛ′′ V ψn+1 R − ψn R ∆t + H⊥[∇ψI] n+ 1 2 ⊥ + H⊥[∇ψI] n− 1 2 ⊥ � = (ψn+1 R − ψn R)T ∆t � H⊥[∇ψR]n ⊥ + H⊥[∇ψR]n+1 ⊥ � + (ψ n+ 1 2 I − ψ n− 1 2 I )T ∆t � H⊥[∇ψI] n+ 1 2 ⊥ + H⊥[∇ψI] n− 1 2 ⊥ � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (89) which is equal to the right hand side of (65).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' 6 A new framework to create FDTD-Q Schemes with guaranteed stability In many applications it is desirable to create FDTD schemes for the Schr¨odinger equation where different models are coupled together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Improper coupling is a notorious cause for instabilities in FDTD-type schemes [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Hence, careful analysis is required to ensure that a coupled scheme is stable, which is challenging with existing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The theory proposed in this work provides a rigorous way to construct new stable schemes in a modular and constructive fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The approach is based on ensuring that the models and the coupling between them respect the principle of probability conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Conservation-based approaches have been previously used for analyzing stability of FDTD schemes in electro- magnetics [37, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The effectiveness of this type of analysis was demonstrated in FDTD for Maxwell’s equations by creating a subgridding scheme [37] and embedding reduced models with extended CFL limit [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Here, as a proof of concept, we show how the conservation approach can be used for analyzing stability of FDTD schemes for the Schr¨odinger equation using two examples: a region isolated to the flow of probability current and two regions that are coupled via boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Application to more advanced scenarios is left for the future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1 Region with no probability current through the boundary Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Consider a region described by FDTD-Q equations (12a)–(12b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Assume that ∆t < ∆tCFL,gen in (34) and that I n+ 1 2 P = 0 , ∀n = 0, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' , nt − 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (90) Then ||ψn||2 ≤ � κ(P)||ψ0||2, ∀n = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' , nt , (91) where κ(P) is the condition number of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Examples of boundary conditions where no net probability current exists include zero Dirichlet (ψ = 0) and zero Neumann [∇ψ]⊥ = 0 boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Proof of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Assume ∆t < ∆tCFL,gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' From Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1 and from (90), Pn stays constant over the course of the simulation (ψn)T Pψn = (ψ0)T Pψ0 ∀n = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' , nt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (92) Moreover, λmin(P)||ψn||2 2 ≤ (ψn)T Pψn (93) and (ψ0)T Pψ0 ≤ λmax(P)||ψ0||2 2 , (94) giving λmin(P)||ψn||2 2 ≤ λmax(P)||ψ0||2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (95) Since, from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='3, λmax(P) is strictly positive, ||ψn||2 2 ≤ λmax(P) λmin(P) ||ψ0||2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (96) The ratio of eigenvalues for a symmetric positive definite matrix is the condition number [54], which proves the statement of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' 16 Region A Region B Figure 3: Two FDTD-Q regions joined together by equating the quantities on the adjacent boundary as described in (97a)–(97d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Hence, the 2-norm of the vector ψn, which contains the samples of the real and imaginary parts of the wave- function, has a bound that is known prior to running the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The existence of such bound guarantees stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' From Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='2, the conventional CFL limit ∆tCFL in (3) can be used in place of ∆tCFL, gen in Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Hence, Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1 in conjunction with Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='2 provide an alternative proof for the previously known result [4,9] on FDTD-Q stability under the CFL limit (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Generalized stability limits that could be relaxed to simpler conditions such as (3) have been derived in the past using the iteration matrix approach for a method similar to FDTD-Q [14] and for FDTD for the Maxwell equations [51,59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='2 Connection of FDTD-Q regions In this section, we discuss how the proposed theory can be used to couple FDTD-Q models in a stable manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' We consider a simple but representative scenario in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' 3, involving two adjacent regions discretized using FDTD- Q with the same grid size and time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The two regions need to be appropriately coupled at the interface to maintain probability conservation and hence stability of the scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The north face of the first region (Region A) is adjacent to the south face of the second region (Region B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' To couple the two regions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' we equate the samples of the wavefunction and the hanging variables at the interface between Region A and Region B ψA R|n i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='nA y +1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k = ψB R|n i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' 2 ≤ i ≤ nx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' 2 ≤ k ≤ nz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' 0 ≤ n ≤ nt ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (97a) ψA I ��n− 1 2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='nA y +1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k = ψB I ��n− 1 2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' 2 ≤ i ≤ nx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' 2 ≤ k ≤ nz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' 0 ≤ n ≤ nt ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (97b) [∂yψA R]n i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='nA y +1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k = [∂yψB R]n i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' 2 ≤ i ≤ nx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' 2 ≤ k ≤ nz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' 0 ≤ n ≤ nt − 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (97c) [∂yψA I ] n+ 1 2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='nA y +1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k = [∂yψB I ] n+ 1 2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' 2 ≤ i ≤ nx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' 2 ≤ k ≤ nz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' 0 ≤ n ≤ nt − 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (97d) where “A” and “B” denote the region a quantity corresponds to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' At all other nodes on the boundary of the two regions, zero Dirichlet boundary condition is imposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Next, we derive the update equations resulting from (97a)–(97d) and show that the coupled scheme is stable if the generalized CFL limit is satisfied in each of the regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' 17 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1 Update equations at the interface From Section 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' the equation discretizing (1a) for the nodes on the north face of Region A is given by ℏ ∆x∆y 2 ∆z ψA R|n+1 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='nA y +1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k − ψA R|n i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='nA y +1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k ∆t = − ℏ2 2m � ∆y 2 ∆z ψA I | n+ 1 2 i+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='nA y +1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k − ψA I | n+ 1 2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='nA y +1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k ∆x − ∆y 2 ∆z ψA I | n+ 1 2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='nA y +1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k − ψA I | n+ 1 2 i−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='nA y +1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k ∆x + ∆x∆z[∂yψA I ] n+ 1 2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='nA y +1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k − ∆x∆z ψA I | n+ 1 2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='nA y +1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k − ψA I | n+ 1 2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='nA y ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k ∆y +∆x∆y 2 ψA I | n+ 1 2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='nA y +1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k+1 − ψA I | n+ 1 2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='nA y +1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k ∆z −∆x∆y 2 ψA I | n+ 1 2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='nA y +1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k − ψA I | n+ 1 2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='nA y +1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k−1 ∆z � +∆x∆y 2 ∆z U A|i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='nA y +1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='kψA I | n+ 1 2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='nA y +1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (98) The corresponding equation for the south face of Region B is ℏ ∆x∆y 2 ∆z ψB R|n+1 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k − ψB R|n i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k ∆t = − ℏ2 2m � ∆y 2 ∆z ψB I | n+ 1 2 i+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k − ψB I | n+ 1 2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k ∆x − ∆y 2 ∆z ψB I | n+ 1 2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k − ψB I | n+ 1 2 i−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k ∆x + ∆x∆z ψB I | n+ 1 2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k − ψB I | n+ 1 2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k ∆y − ∆x∆z[∂yψB I ] n+ 1 2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k + ∆x∆y 2 ψB I | n+ 1 2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k+1 − ψB I | n+ 1 2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k ∆z − ∆x∆y 2 ψB I | n+ 1 2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k − ψB I | n+ 1 2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k−1 ∆z � + ∆x∆y 2 ∆z U B|i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='kψB I | n+ 1 2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (99) Adding (98) and (99) and using (97a),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (97b),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' and (97d),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' ℏ ∆x∆y∆z ψB R|n+1 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k − ψB R|n i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k ∆t = − ℏ2 2m � ∆y∆z ψB I | n+ 1 2 i+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k − ψB I | n+ 1 2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k ∆x − ∆y∆z ψB I | n+ 1 2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k − ψB I | n+ 1 2 i−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k ∆x + ∆x∆z ψB I | n+ 1 2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k − ψB I | n+ 1 2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k ∆y − ∆x∆z ψB I | n+ 1 2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k − ψA I | n+ 1 2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='nA y ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k ∆y + ∆x∆y ψB I | n+ 1 2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k+1 − ψB I | n+ 1 2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k ∆z − ∆x∆y ψB I | n+ 1 2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k − ψB I | n+ 1 2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k−1 ∆z � + ∆x∆y∆z U A|i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='nA y +1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k + U B|i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k 2 ψB I | n+ 1 2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='k ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (100) which has the exact same form as the FDTD equation for internal nodes (4),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' with the potential taken as the average of the two adjacent nodes8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The same can be shown for the equations corresponding to (1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='2 Stability analysis Assume that the time step satisfies the generalized CFL limit (34) for each region ∆t < min � ∆tA CFL, gen, ∆tB CFL, gen � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (101) From Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1, the probability in each region evolves according to Pn+1 A − Pn A ∆t = −I n+ 1 2 P,A , (102) Pn+1 B − Pn B ∆t = −I n+ 1 2 P,B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (103) 8The derivation of (100) is analogous to the treatment of material interfaces in FDTD for electromagnetics [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' We also remark that the similarity between (100) at the interface and (4) for internal nodes in FDTD-Q makes existing results on FDTD-Q stability [4, 9] applicable to this particular scenario and the specific boundary conditions (97a)–(97d) selected at the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' However, as we discuss in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='2 the proposed approach could be applied to developing other schemes and this example serves as an illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' 18 Adding (102) and (103), � Pn+1 A + Pn+1 B � − (Pn B + Pn B) ∆t = −IP,A|n+ 1 2 − IP,B|n+ 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (104) As can be seen from (31), (32), and (33), the conditions (97a)–(97d) equating the wavefunction and the hanging variables on the adjacent boundaries of the two regions ensure that the probability currents on the right hand side of (104) cancel out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Hence, Pn A + Pn B = P0 A + P0 B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (105) From Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1, under the CFL limit Pn A, Pn B are both non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Hence, from (105), Pn A, and Pn B are each at most P0 A + P0 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Repeating the reasoning in the proof of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1, ||ψn A||2 2 ≤ P0 A + P0 B λmin(P) , (106a) ||ψn B||2 2 ≤ P0 A + P0 B λmin(P) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (106b) This means that the system is stable, as the values of the wavefunction samples cannot grow without bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' In Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='3 we investigate the consequences of taking the time step beyond (101).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' In particular, we show that violating the generalized CFL limit in a region allows that region to provide infinite probability to the surrounding space and thus destabilize the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Lastly, we remark that under condition (101), the coupled scheme can be shown to also conserve energy using similar arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' This example, although simple, shows how with the proposed theory one can create composite FDTD schemes obtained by coupling different models discretizing the Schr¨odinger equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' If each of the models satisfies the con- servation of probability and the models are coupled in the probability-conserving manner, the resulting scheme will by construction satisfy the probability conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The resulting scheme will be stable under the most restrictive value of the generalized CFL limit (101), which is also known by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The choice of coupling between the models, such as the boundary conditions (97a)–(97d), is essential for ensuring the probability conservation and stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' In general, a different coupling scheme is not guaranteed to be probability-conserving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The proposed approach could be applied to developing more advanced schemes in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' For example, subgridding scenarios [41, 42] could be analyzed as a connection of grids of different resolution [37], where one needs to ensure that the grids exchange probability in a conserving manner to achieve the cancellation on the right hand side of (104).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Another approach that could be used to analyze general schemes is the iteration matrix method [14, 51, 59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The proposed conservation-based approach provides an intuitive physical interpretation that the root cause of instability is the generation of spurious energy or probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Moreover, the proposed approach provides a very natural way to determine whether an FDTD-Q region or any other part of the setup is capable of introducing instability, prior to knowing anything about the overall setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' In general, such modular stability analysis is not trivial, since stability is a property of the entire system and coupling of equations corresponding to stable schemes does not automatically achieve stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The use of concepts of conservation provides a systematic, modular, and constructive strategy to create stable composite FDTD schemes for the Schr¨odinger equation with a guarantee of probability and energy conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' 7 Numerical examples In order to investigate the validity of the results in Section 4 and Section 5, the proposed method was implemented in Matlab and the time evolution of numerical probability and energy was investigated for different simulation scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1 Infinite well First, we demonstrate the validity of the proposed theory for an isolated region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' In particular, we consider an electron trapped in a potential well with the potential U equal to zero inside the region and tending to infinity on the boundary of the region and outside the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The infinite potential well results in zero Dirichlet boundary conditions [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The region has a side length of a = 30 nm in each dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The region was discretized into nx = ny = nz = 30 primary cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The CFL limit ∆tCFL and the generalized CFL limit ∆tCFL,gen were both equal to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='879 fs, with the relative difference (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='370×10−15) comparable to machine 19 (b) (a) simple simple Figure 4: Infinite well example in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1 for nx = ny = nz = 30: (a) the total probability Pn computed with the proposed formula (24) and Pn simple computed with the simpler expression (23) (b) the total energy Hn in (54) and Hn simple in (53).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The dashed line shows the analytic value of energy (110).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Table 1: Error of energy expressions vs grid refinement in the example in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1 (a = 30 nm, ∆t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='999∆tCFL, nt∆t = 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='76 ps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' nx = ny = nz maxn |Pn − 1| maxn |Pn simple − 1| minn |Hn − E1|/E1 maxn |Hn simple − E1|/E1 10 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='88×10−16 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='51×10−2 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='20×10−3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='19×10−2 20 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='55×10−16 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='19×10−3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='05×10−3 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='15×10−3 30 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='22×10−15 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='74×10−3 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='14×10−4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='64×10−3 40 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='55×10−15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='54×10−3 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='14×10−4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='05×10−3 50 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='44×10−15 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='87×10−4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='29×10−4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='31×10−3 precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The time step ∆t was taken as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='999 ∆tCFL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The initial conditions ψR|0 and ψI|−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='5 were set by sampling the following particular solution of the Schr¨odinger equation [16] ψ(x, y, z, t) = A sin (kxx) sin (kyy) sin (kzz) exp � −i �E1 ℏ t + π 3 �� (107) at the primary nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' In (107), i = √ −1 , (108) kx = ky = kz = π a , (109) E1 = ℏ2 2m(k2 x + k2 y + k2 z) , (110) and A is the real positive normalization constant ensuring that P0 in (24) is equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' This choice of normalization will be discussed shortly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Since the region is isolated to the flow of power and probability current, the energy and probability contained inside the region are constant in the continuous domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' From the blue curves in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' 4, it is evident that the proposed expressions for probability and energy respect this property in the discrete domain, in accordance with Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1 and Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Indeed, the range of values of Pn and Hn was 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='997×10−15 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='228 aeV, which could be explained by finite machine precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' In contrast, the values of Pn simple and Hn simple shown in red in the figure exhibit some fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Moreover, the values of Pn simple exceed 1 in some of the time intervals, which is inconsistent with physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The continuous expression for energy (48) can be shown to equal E1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='2534 meV, which is plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' 4(b) with the dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The value of the proposed expression, Hn = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='2523 meV, is in good agreement with E1, corresponding to a relative error of 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='14×10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Table 1 shows deviations of the values of different probability and energy expressions from the analytic solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The values were obtained for different grid resolutions, while maintaining the infinite well geometry and keeping the constant ratio ∆t/∆tCFL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' As evident from the table, the discrepancy between Hn and E1 can be reduced by refining the cell size and the corresponding time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The error in Pn simple and Hn simple also reduces with improved spatial resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' However, both Pn simplen and Hn simple exhibit larger errors than the proposed expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Hence the proposed expressions of probability and energy provide a more accurate approximation of the analyt- ical quantities and respect the principle of probability and energy conservation, in contrast to Pn simple and Hn simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' 20 Figure 5: Scenario considered in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='2, where a Gaussian wavepacket impinges on a potential barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The computational cost associated with evaluating Pn is increased compared to Pn simple due to the off-diagonal blocks in P in (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' However, the cost of the added computations is on par with evaluating the terms associated with the diagonal blocks in P, even if one evaluates the additional terms directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Similarly, the overhead in computing the additional terms in Hn is comparable to the cost of computing the terms that are in common with Hn simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Lastly, the proposed expression of the total probability is very convenient for computing the normalization constant A in (107), since the value of Pn stays constant in an isolated region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' In contrast, the value Pn simple varies from time step to time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' As evident from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' 4(a), if the initial values of the wavefunction were normalized such that P0 simple equaled 1, both Pn and Pn simple would have been centered around a value that greater than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='2 Reflection from a potential barrier We consider the scenario in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' 5, where a Gaussian wavepacket impinges on a potential barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The expression for the incident pulse is [16] ψinc(x, t) = � p Ape−i(ωpt−kp(x−x0)) , (111) where p is an integer index, x0 is the center of the wavepacket at t = 0, kp are evenly spaced real scalars, ωp are the corresponding angular frequencies given by ℏk2 p/(2m), and the coefficients Ap are given by Ap = exp � −1 4 �kp − ¯k σ �2� , (112) where ¯k determines the center of the wavepacket in the k-space and σ determines its width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The wavepacket impinges on a barrier with potential given by U(x) = � � � � � � � 0 , x < a U0 2 , x = a U0 , x > a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (113) The solution can be found in quantum mechanics textbooks [16] and is given by ψ(x, t) = � ψinc(x, t) + ψref(x, t) , x < a ψtran(x, t) x > a , (114) where ψref and ψtran are reflected and transmitted waves given by ψref(x, t) = � p ApRpe−i(ωpt−kp(a−x0+a−x)) , (115) ψtran(x, t) = � p ApTpe−i(ωpt−kp(a−x0)−Kp(x−a)) , (116) where Kp = � 2m(ℏωp − V0)/ℏ, which can be real or imaginary, depending on the sign of ℏωp − V0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' When Kp is real, the corresponding wave in ψtran propagates forward in the x > a region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' When Kp is imaginary, the wave decays with x and hence cannot propagate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The reflection and transmission coefficients are given by Rp = kp − Kp kp + Kp , (117) 21 (a) (b) simple simple Figure 6: Results of the test in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='2: (a) total probability in the region (b) total energy associated with the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' simple simple Figure 7: Results of the test in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='2: (a) relative error in (119) (b) relative error in (120).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Tp = 2kp kp + Kp .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (118) In this test, m is the mass of an electron, x0 = −200 nm, ¯k = 2π/¯λ, where ¯λ is 30 nm, and σ = ¯k/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The values of kp range between kmin = ¯k − 10σ and kmax = ¯k + 10σ, with a spacing of ∆k = σ/100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The height of the potential barrier is U0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='5 meV and a = 100 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' We select the space from x = 0 to x = 200 nm to be modeled with the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The dimensions of the region were selected as 2 nm in the y and z dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The region was discretized into cells of size ∆x = ∆y = ∆z = 1 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The initial conditions in the region were set by sampling the analytical solution (114).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The update equations on the boundary were the modified FDTD-Q equations described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1, such as (6), (7), and (8), which involved the hanging variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' In order to obtain the values of the hanging variables, expressions were found for the normal component of the gradient of the analytical solution (114).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The expressions were evaluated at integer time points n∆t for the real part and at (n + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='5)∆t for the imaginary part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The values of the hanging variables were zero on all faces of the boundary except for the east and west, where their values were uniform in the y and z direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' This, in essence, rendered the problem one-dimensional, which was the reason why choosing the y and z dimensions to be small (2 nm) was possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The generalized CFL limit in (34) was ∆tCFL,gen = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='869968 fs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The CFL limit ∆tCFL in (3) was slightly lower (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='869915 fs), which is in agreement with the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The simulation time step was set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='999∆tCFL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' 6(a) shows the probability of finding the particle in the region obtained via analytical computation, via the proposed expression Pn in (24), and the simple expression Pn simple in (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The analytic values were obtained from 22 (20), applied to the analytical solution (114).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The integration in (20) was approximated via a Riemann sum using the midpoint rule with 1000 intervals between x = 0 nm and x = 200 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' When using the analytical solution as a reference, the accuracy of Pn and Pn simple was comparable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The maximum relative errors were 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='827×10−3 for Pn, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='978×10−3 for Pn simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' 6(b) shows the energy stored in the region computed using Hn and Hn simple, as well as the analytic values approximated via the Riemann sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The accuracy of Hn and Hn simple was also comparable, with a maximum relative error associated with Hn of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='188×10−2 and the error associated with Hn simple of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='197×10−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' As predicted by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1, the total probability Pn was non-negative, with the smallest value of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='184×10−27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The smallest value of energy Hn was 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='812×10−30 eV, which was higher than the bound of −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='193×10−17 eV predicted by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1 when taking Pmax in (64) to be the highest value of Pn over the course of the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Based on Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1, the values of the proposed expression Pn must satisfy Pn = P0 − n−1 � n′=0 I n′+ 1 2 P ∆t , ∀n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' , nt , (119) where the right hand side is the sum of the initial probability and the probability that has entered the region due to the flow of the probability current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The equality in (119) is evident from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' 6(a), showing both the left and right hand sides of (119).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The relative error between the two sides of (119) is plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' 7(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The largest relative error was 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='514×10−15, which is comparable to the machine precision and hence corroborates the prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The relative error in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' 7(a) was normalized to the largest value of the theoretical probability over all time steps max {P(n∆t)} = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='000×10−20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' In contrast to the proposed expression for the total probability, when using Pn simple in place of Pn in (119), the left and right hand sides of the relation are no longer equal, as can be seen from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' 7(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The maximum value of the relative error was 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='519×10−3, which is much larger than the error in (119) associated with Pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Hence, at least when In+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='5 P is used as the representation of the probability current, Pn simple does not possess the conservation properties exhibited by Pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' These errors in conservation when using the simpler expression Pn simple also manifested themselves in the small fluctuations visible in the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' 6(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Similarly, from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' 6(b) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' 7(b) one can observe that the following relation is satisfied by the proposed expressions for the total energy and supplied power: Hn = H1 + n−1 � n′=1 sn′+ 1 2 ∆t , ∀n = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' , nt − 1 , (120) with the largest relative error of 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='111×10−15, which is comparable to machine precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' In contrast, replacing Hn with Hn simple results in the relative error of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='428×10−3 between the left and right hand sides of (120), which can no longer be explained by the finite machine precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The normalization constant for the relative error in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' 7(b) was max{H(n∆t)} = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='064×10−23 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Small fluctuations that were exhibited by Pn simple can also be observed for Hn simple in the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' 6(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Hence, the results confirm that both Pn and Hn satisfy (119) and (120) and thus (36) and (65), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' This property makes the proposed expressions preferable to Pn simple and Hn simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='3 Proton tunneling In this section we consider the scenario illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' 8, which was taken from [35] and can be used as a simplified model for hydrogen transfer reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The problem consists of three regions: reactant (r), barrier (b), and product (p), and a proton that can transfer between the reactant and product regions via tunneling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The size of the reactant, barrier, and product regions is lr x ×ly ×lz, lb x ×ly ×lz, and lp x ×ly ×lz, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The barrier region has a potential of U0 and the other two regions have the potential equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Dirichlet zero boundary conditions are imposed on the external boundaries of the regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' On the interfaces, the continuity of the wavefunction and the normal component of its gradient is imposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Using the separation of variables, the analytical solution in each region can be found as [35] ψr,b,p(x, y, z, t) = N � m=1 Mmf r,b,p m (x)gm(y)hm(z) exp � −iEm ℏ t � , 0 ≤ x ≤ lr,b,p x , (121) 23 where N is the number of eigenfunctions considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The expression for f r,b,p m (x) in each region is f r m(x) = Am sin (kxmx) , (122a) f b m(x) = Bm cosh � Kxm � x − lb x 2 �� + Cm sinh � Kxm � x − lb x 2 �� , (122b) f p m(x) = Dm sin (kxm (x − lp x)) , (122c) where the choice between Bm = 0 and Cm = 0 determines the symmetry of the eigenfunction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The coefficients Am, Bm, Cm, and Dm are chosen to ensure that the wavefunction is continuous across the region interfaces and that � r |f r m(x)|2dx + � b |f b m(x)|2dx + � p |f p m(x)|2dx = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (123) The expressions for gm(y) and hm(z) are given by gm(y) = � 2 ly sin (kymy) , (124) hm(z) = � 2 lz sin (kzmz) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (125) The values of kym and kzm are such that the zero Dirichlet boundary conditions are satisfied for y = ly and z = lz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The values of kxm and Kxm are related through Exm = ℏ2k2 xm 2mP = V0 − ℏ2K2 xm 2mP , (126) where mP = 1 dalton ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='661×10−27 kg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The value of Ex,m can be obtained by imposing the continuity of the x derivative of the wavefunction across the region interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The energy Em in (121) corresponding to each eigenfunction is given by Em = Exm + Eym + Ezm , (127) where Eym = ℏ2k2 ym 2mP , (128) Ezm = ℏ2k2 zm 2mP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (129) Following [35], the coefficients Mm in (121) are assumed to have the following dependence on the temperature T: Mm = MM ′ m , (130) where M ′ m = exp � − Em 2TkB + iδm � , (131) where kB is the Boltzmann constant, and δm are phases that can be chosen arbitrarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Scalar M in (130) is a normalization constant given by M = 1 ��N m=1 |M ′m|2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (132) The temperature and the number of eigenfunctions were taken from [35] as T = 298 K and N = 8, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The dimensions of the regions, also from [35], are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Unlike [35], we only consider a particular solution with the phases δm in Table 2, as opposed to an ensemble of tunneling systems with many sets of randomized phases δm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' In [35], the model was further extended to include the effect of thermal vibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' We use this scenario to examine the conservation of probability and energy when multiple FDTD-Q models are connected and to study the transfer of these quantities from one region’s model to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Three FDTD-Q models were defined: one discretizing the reactant region, another discretizing the barrier, and the third discretizing the product region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The theoretical treatment of this scenario is described in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='2 for two regions and an 24 Reactant region (r) Product region (p) Barrier (b) Figure 8: The scenario from [35] considered in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='3, modeling proton tunneling through a barrier with U = U0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Table 2: Description of the solution modes considered (see [35] for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' m δm Exm (meV) Symmetry of f b m(x) kym kzm 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='15π 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='858713602402805 Bm ̸= 0, Cm = 0 π/ly π/lz 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='95π 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='858842481348997 Bm = 0, Cm ̸= 0 π/ly π/lz 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='25π 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='369931622707597 Bm ̸= 0, Cm = 0 π/ly π/lz 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1π 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='370616971460279 Bm = 0, Cm ̸= 0 π/ly π/lz 5 0 Ex1 Bm ̸= 0, Cm = 0 2π/ly π/lz 6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='3π Ex2 Bm = 0, Cm ̸= 0 2π/ly π/lz 7 0 Ex1 Bm ̸= 0, Cm = 0 π/ly 2π/lz 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='7π Ex2 Bm = 0, Cm ̸= 0 π/ly 2π/lz extension to tree regions is analogous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The cell dimensions in each region were ∆x = ∆y = ∆z = (1/30) ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The three FDTD-Q models were coupled by equating the wavefunction values and hanging variables on the interfaces, as described in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The CFL limits and the generalized CFL limits in each region are shown in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Their values were the same, apart from round-off error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The simulation time step was taken as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='999 of the CFL limit of the barrier region (∆t ≈ 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='79 as), which automatically satisfied the CFL limit condition in the reactant and product regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' This time step ensures stability, according to [9] or based on the discussion in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The initial conditions were set by sampling the analytical solution (121) and normalizing the result to achieve the total probability in the three regions P0 r + P0 b + P0 p = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' One simulation run was performed to study the first 1 ps of the particle’s evolution and a longer simulation was done to observe the behavior of the system over 35 ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' In the longer simulation, probability and energy were computed once in ten time steps (10∆t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='5579 fs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' This allowed reducing the memory usage while providing sufficient information, considering that the shortest period of a mode in (121) was 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='26 fs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' During the first 1 ps, the simulated results match well with the analytical prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' For the 35 ns simulation, the results deviate from the analytical solution, which is expected due to dispersion errors caused by the finite discretization of the Schr¨odinger equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Nevertheless, the simulation was able to correctly model the overall trend of the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The accuracy of Pn simple and Hn simple for each region was comparable to that of Pn and Hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' From Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1, using 1 as a bound on probability, the energy Hn cannot take on values below -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='608 keV for the reactant or product regions and below -54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='10 keV for the barrier region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The values of energy in each region were positive, which is in agreement with these lower bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Probability in each region was also positive, which is consistent with Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Table 3: CFL limit and generalized CFL limit in each region in the test of Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='3 Region Reactant Barrier Product ∆tCFL,gen (as) 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='318879645754564 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='844895610996367 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='318879645754564 ∆tCFL (as) 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='318879645754819 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='844895610996282 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='318879645754819 25 Probability Time (ps) Energy (meV) Time (ns) Total r p b 0 10 20 30 Total r p b 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='8 1 0 20 40 60 80 100 r Total b p 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='8 1 r Total b p simple simple Figure 9: Energy and probability computed in the test of Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Left panels: 1 ps simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Right panels: 35 ns simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' 0 10 20 30 Time (ns) 10-10 100 Deviation of total probability Proposed Simple 0 10 20 30 Time (ns) 10-10 100 Relative deviation of total energy Proposed Simple Figure 10: Relative deviation of the total probability and energy from their initial values (a) deviation of probability computed from (135) and (137) (b) normalized deviation of energy computed from (136) and (138).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' 26 Total b r p Figure 11: Results of the test in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='3 for the time step exceeding the CFL limit ∆tb CFL and the generalized CFL limit ∆tb CFL, gen in the barrier region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Based on the discussion in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='2, the sum of the total probabilities over the three regions satisfies � Pn+1 r + Pn+1 b + Pn+1 p � − � Pn r + Pn b + Pn p � ∆t = −IP,r|n+ 1 2 − IP,b|n+ 1 2 − IP,p|n+ 1 2 = 0 (133) and hence the total probability must stay constant throughout the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Similarly, one can show an analogous result for the sum of the total energies � Hn+1 r + Hn+1 b + Hn+1 p � − � Hn r + Hn b + Hn p � ∆t = s n+ 1 2 r + s n+ 1 2 b + s n+ 1 2 p = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (134) Thus, the total energy must likewise stay constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' In order to assess these predictions, we compute the relative deviation of the probability and energy as ��� Pn r + Pn b + Pn p � − � P0 r + P0 b + P0 p ��� , n = 0, 10, 20, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' , (135) 1 H(n∆t) ��� Hn r + Hn b + Hn p � − � H1 r + H1 b + H1 p ��� , n = 1, 11, 21, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' , (136) where H(t) is the energy associated with the analytical solution and is equal to 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='5 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The values of (135) and (136) are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' 10(a) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' 10(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The largest values of (135) and (136) were 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='13×10−14 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='16×10−14, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' As expected, these fluctuations are extremely small and can be attributed to the finite machine precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' For comparison, we also compute ��� Pn r,simple + Pn b,simple + Pn p,simple � − � P0 r,simple + P0 b,simple + P0 p,simple ��� , n = 0, 10, 20, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' , (137) 1 H(n∆t) ��� Hn r,simple + Hn b,simple + Hn p,simple � − � H1 r,simple + H1 b,simple + H1 p,simple ��� , n = 1, 11, 21, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' , (138) which are also shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' 10(a) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' 10(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The largest values of (137) and (138) are, respectively, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='92×10−3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='21×10−3, meaning that the simple expressions do not abide the conservation of probability and energy exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' In order to illustrate a possible mechanism for the breakdown of conservation properties beyond the CFL limit, we also run a simulation with a time step that violates the CFL limit condition in the barrier region (∆t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='005∆tb CFL = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='9624∆tr CFL = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='9624∆tp CFL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The resulting probability is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Beyond the CFL limit, the inequality (35) can be violated in the barrier region and the probability associated with that region is permitted to indefinitely grow negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' This allows the barrier region to provide an infinite amount of probability to the reactant and product regions, causing their probabilities to indefinitely grow positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The total probability remains unchanged, until the exponential growth of the wavefunction values due to instability eventually causes the the total probability to deviate from 1, as a result of the finite machine precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' 8 Conclusions In this work, we investigated the probability and energy conservation properties of the finite-difference time-domain scheme for solving the Schr¨odinger equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Existing works on the conservation properties of numerical schemes for 27 Table 4: Indexing convention for vectors and matrices in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Sample description Scalar index Vector index Example(s) Primary nodes (i, j, k) i + (j − 1)(nx + 1) + (k − 1)(nx + 1)(ny + 1) ψn R, ψn−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='5 I in (9) Primary edges x-directed (i + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='5, j, k) i + (j − 1)nx + (k − 1)nx(ny + 1) ∂xψn R in (141) y-directed (i, j + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='5, k) i + (j − 1)(nx + 1) + (k − 1)(nx + 1)ny ∂yψn R in (141) z-directed (i, j, k + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='5) i + (j − 1)(nx + 1) + (k − 1)(nx + 1)(ny + 1) ∂zψn R in (141) Additional primary edges normal to the boundary x-directed, W (1, j, k) j + (k − 1)(ny + 1) [∂xψI]n+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='5 W in (147) x-directed, E (nx + 1, j, k) j + (k − 1)(ny + 1) [∂xψI]n+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='5 E in (147) y-directed, S (i, 1, k) i + (k − 1)(nx + 1) [∂yψI]n+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='5 S in (147) y-directed, N (i, ny + 1, k) i + (k − 1)(nx + 1) [∂yψI]n+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='5 N in (147) z-directed, B (i, j, 1) i + (j − 1)(nx + 1) [∂zψI]n+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='5 B in (147) z-directed, T (i, j, nz + 1) i + (j − 1)(nx + 1) [∂zψI]n+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='5 T in (147) the Schr¨odinger equation consider a region that is terminated with periodic or zero Dirichlet boundary conditions, which do not allow any net exchange of probability and energy with the surrounding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' In contrast, we considered a general scenario where a region can either be terminated using boundary conditions or form a portion of a larger domain, where probability and energy may enter or leave the region through the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' We introduced modified equations on the boundary of the region in order to write a self-contained model that allows analyzing the properties of the region without making assumptions on the nature of the discretization beyond the boundary of the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' We proposed expressions for the total probability and energy contained in a region discretized using FDTD-Q, as well as expressions for the probability current and supplied power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Using these expressions, we showed that the FDTD-Q method conserves probability under the CFL limit that has been traditionally used for ensuring stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' We provided an illustration of the mechanism in which a violation of the CFL condition can result in violation of the conservation of probability for an example involving three connected FDTD regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Furthermore, we have shown that the CFL condition ensures that the energy is conserved, under an assumption that the region is coupled to other probability-conserving models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The proposed expressions for computing probability and energy were compared to a more straightforward approach and several advantages were found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' First, the proposed expressions respect the probability and energy conservation exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The exact conservation properties avoid spurious fluctuations exhibited by the values of the simpler expressions, which are especially evident when a system is isolated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Second, the proposed expressions for probability and energy tend to be slightly more accurate than the simple expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Lastly, in isolated systems, the value of the proposed probability is convenient for normalizing initial values, since it is guaranteed to stay constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The proposed theory sheds light on the energy and probability conservation in situations where the FDTD-Q model of the region can exchange these quantities with the surrounding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' This insight can serve as a basis for a stability analysis and enforcement framework in scenarios where the region is coupled to other models that can exchange energy and/or probability with the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' As a proof of concept, we considered the case of multiple regions with the same FDTD-Q discretization that are coupled to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' We envision other possible applications, such as creation of stable subgridding schemes [41,42], stable incorporation of reduced order models [38,43], and stability analysis of advanced boundary conditions [6, 10, 12, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Moreover, multi-physics simulations [60–64] can also be considered as a type of scenario where an exchange occurs between the energy associated with the quantum particle and the energy stored in another form, such as the energy of electromagnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Hence, the approach proposed in this paper could be very convenient if extended to such scenarios, allowing to prove stability by ensuring that each part of the system conserves the quantities of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' 28 Appendices A Indexing convention and expressions for the matrices in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='2 This appendix provides expressions for the matrices in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' These expressions assume a specific order of the samples in the vectors in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' In particular, vectors of quantities sampled at primary nodes, such as ψn R or ψn−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='5 I in (9) have the elements ordered based on the indexing convention in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' For example, a sample ψR|n i,j,k would be placed in the element i + (j − 1)(nx + 1) + (k − 1)(nx + 1)(ny + 1) in ψn R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' With this convention, matrix Λ′′ V in (9) is defined as Λ′′ V = ∆x∆y∆z ˜Inz+1 ⊗ ˜Iny+1 ⊗ ˜Inx+1 , (139) where ˜Im is an m × m matrix given by ˜Im = diag �1 2, 1, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' , 1, 1 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (140) Diagonal matrix ΛU contains the samples U|i,j,k placed on the diagonal elements according to the indexing con- vention in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Vectors with samples on the primary edges, such as ∇ψn R in (11), contain, in that order, the samples on the x-directed, y-directed, and z-directed edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' ∇ψn R = � � ∂xψn R ∂yψn R ∂zψn R � � , (141) where the ordering of samples in ∂xψn R, ∂yψn R, and ∂zψn R is shown in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' With this, D is defined as D = �Dx Dy Dz � , (142) where Dx = −Inz+1 ⊗ Iny+1 ⊗ WT nx , Dy = −Inz+1 ⊗ WT ny ⊗ Inx+1 , Dz = −WT nz ⊗ Iny+1 ⊗ Inx+1 , (143) where Wm is an m × (m + 1) matrix given by Wm = �0m×1 Im � − �Im 0m×1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (144) Matrices Λ′′ S and Λ′ l are given by Λ′′ S = diag � ∆y∆z ˜Inz+1 ⊗ ˜Iny+1 ⊗ Inx, ∆x∆z ˜Inz+1 ⊗ Iny ⊗ ˜Inx+1, ∆x∆y Inz ⊗ ˜Iny+1 ⊗ ˜Inx+1 � , (145) Λ′ l = diag � ∆x Inz ⊗ Iny ⊗ Inz, ∆y Inz ⊗ Iny ⊗ Inz, ∆z Inz ⊗ Iny ⊗ Inz � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (146) The hanging variables corresponding to the six faces of the boundary are ordered as follows: west, east, south, north, bottom, top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' For example [∇ψI]n+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='5 ⊥ in (9) has the following structure [∇ψI] n+ 1 2 ⊥ = � ���������� [∂xψI] n+ 1 2 W [∂xψI] n+ 1 2 E [∂yψI] n+ 1 2 S [∂yψI] n+ 1 2 N [∂zψI] n+ 1 2 B [∂zψI] n+ 1 2 T � ���������� , (147) where the indexing convention of each of the vectors on the right hand side of (147) is shown in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Based on this ordering of the hanging variables, L is defined as L = �LW LE LS LN LB LT � , (148) 29 where LW = Inz+1 ⊗ Iny+1 ⊗ e{1, nx + 1} , LE = Inz+1 ⊗ Iny+1 ⊗ e{nx + 1, nx + 1} , LS = Inz+1 ⊗ e{1, ny + 1} ⊗ Inx+1 , LN = Inz+1 ⊗ e{ny + 1, ny + 1} ⊗ Inx+1 , LB = e{1, nz + 1} ⊗ Iny+1 ⊗ Inx+1 , LT = e{nz + 1, nz + 1} ⊗ Iny+1 ⊗ Inx+1 , (149) where e{p, m} a vector of size m × 1 with 1 in position p and zeroes in all other positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Similarly, matrix Λ(ˆn·) is given by Λ(ˆn·) = diag � Λ(ˆn·)W, Λ(ˆn·)E, Λ(ˆn·)S, Λ(ˆn·)N, Λ(ˆn·)B, Λ(ˆn·)T � , (150) where Λ(ˆn·)W = [ˆnW · ˆx] Inz+1 ⊗ Iny+1 , Λ(ˆn·)E = [ˆnE · ˆx] Inz+1 ⊗ Iny+1 , Λ(ˆn·)S = [ˆnS · ˆy] Inz+1 ⊗ Inx+1 , Λ(ˆn·)N = [ˆnN · ˆy] Inz+1 ⊗ Inx+1 , Λ(ˆn·)B = [ˆnB · ˆz] Iny+1 ⊗ Inx+1 , Λ(ˆn·)T = [ˆnT · ˆz] Iny+1 ⊗ Inx+1 , (151) where [ˆnW · ˆx] = [ˆnS · ˆy] = [ˆnB · ˆz] = −1 and [ˆnE · ˆx] = [ˆnN · ˆy] = [ˆnT · ˆz] = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Matrix Λ′′ S,b is given by Λ′′ S,b = diag � Λ′′ S,b,W, Λ′′ S,b,E, Λ′′ S,b,S, Λ′′ S,b,N, Λ′′ S,b,B, Λ′′ S,b,T � , (152) where Λ′′ S,b,W = ∆y∆z ˜Inz+1 ⊗ ˜Iny+1 , Λ′′ S,b,E = ∆y∆z ˜Inz+1 ⊗ ˜Iny+1 , Λ′′ S,b,S = ∆x∆z ˜Inz+1 ⊗ ˜Inx+1 , Λ′′ S,b,N = ∆x∆z ˜Inz+1 ⊗ ˜Inx+1 , Λ′′ S,b,B = ∆x∆y ˜Iny+1 ⊗ ˜Inx+1 , Λ′′ S,b,T = ∆x∆y ˜Iny+1 ⊗ ˜Inx+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (153) B Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='2 The proof draws inspiration from the approach in [40], where positive definiteness was shown for a matrix analogous to P, but arising from FDTD for Maxwell’s equations and associated with stored energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' In particular, in [40], the contribution of individual primary cells to a quadratic form analogous to (24) was considered in order to show that the conventional CFL limit for FDTD for Maxwell’s equations guarantees positive definiteness of the matrix associated with the entire region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Consider a region described by FDTD-Q equations (12a)–(12b) with nx ×ny ×nz primary cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Let P be the matrix corresponding to the entire region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Let ψ be given by ψ = �ψR ψI � , (154) where ψR, ψI ∈ R(nx+1)(ny+1)(nz+1)×1 are arbitrary vectors with each element corresponding to a primary node in the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Let Pijk be the matrix defined in the same way as P but for a single-cell region formed by an individual primary cell with the bottom-south-west corner at the node (i, j, k) and the top-north-east corner at the node (i + 1, j + 1, k + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Let ψijk be a vector formed by selecting the elements of ψ that correspond to the nodes on the corners of that cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Then, ψT Pψ = nx � i=1 ny � j=1 nz � k=1 ψT ijkPijkψijk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (155) The proof involves expanding both sides of (155) as a summation over primary nodes and primary edges and collecting terms on the right hand side associated with the same primary node or primary edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Then, each term on the left hand side can be shown to have a corresponding term on the right hand side and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Let ∆tCFL,gen and ∆t(i,j,k) CFL,gen be the generalized CFL limits corresponding, respectively, to the entire region and to a single-cell region formed by the primary cell with the bottom-south-west corner at the node (i, j, k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Then, min i,j,k ∆t(i,j,k) CFL,gen ≤ ∆tCFL,gen .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (156) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Consider matrices P and Pijk and vectors ψ and ψijk as described in Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Consider a time step ∆t such that ∆t < min i,j,k ∆t(i,j,k) CFL,gen .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (157) 30 With this time step, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='3 applied to the single-cell regions, Pijk ≻ 0, ∀i ∈ 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' , nx, ∀j ∈ 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' , ny, ∀k ∈ 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' , nz , (158) and for each i, j, and k, we have � ψT ijkPijkψijk > 0, ∀ψijk ∈ R16×1 where ψijk ̸= 0 ψT ijkPijkψijk = 0, if ψijk = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (159) By Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1, ψT Pψ > 0 , ∀ψ ∈ R2(nx+1)(ny+1)(nz+1)×1 with ψ ̸= 0 , (160) which means that P is positive definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Hence, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='3, ∆t < ∆tCFL,gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' In conclusion, the following implication holds ∆t < min i,j,k ∆t(i,j,k) CFL,gen =⇒ ∆t < ∆tCFL,gen .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (161) This is only possible if (156) is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Let ∆t(i,j,k) CFL and ∆t(i,j,k) CFL,gen be the CFL limit and the generalized CFL limit for a single-cell region composed of a primary cell with the bottom-south-west corner at the node (i, j, k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Then, ∆t(i,j,k) CFL ≤ ∆t(i,j,k) CFL,gen .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (162) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' With reference to Appendix A, Λ′′ V,ijk = ∆x∆y∆z 8 I8 , (163) Λ′′ S,ijk = diag �∆y∆z 4 I4, ∆x∆z 4 I4, ∆x∆y 4 I4 � , (164) Λ′ l,ijk = diag (∆x I4, ∆y I4, ∆z I4) , (165) Dijk = − � I2 ⊗ I2 ⊗ WT 1 I2 ⊗ WT 1 ⊗ I2 WT 1 ⊗ I2 ⊗ I2 � , (166) where subscripts ijk indicate that a matrix corresponds to the single-cell region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Define matrix Σijk as Σijk = 1 ℏ(Λ′′ V,ijk)− 1 2 Hijk(Λ′′ V,ijk)− 1 2 = ℏ 2m(Λ′′ V,ijk)− 1 2 DijkΛ′′ S(Λ′ l,ijk)−1DT ijk(Λ′′ V )− 1 2 + 1 ℏΛU,ijk = ℏ m � 1 (∆x)2 (I2 ⊗ I2 ⊗ WT 1 W1) + 1 (∆y)2 (I2 ⊗ WT 1 W1 ⊗ I2) + 1 (∆z)2 (WT 1 W1 ⊗ I2 ⊗ I2) � + 1 ℏΛU,ijk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (167) Let V be the following matrix: V = �v0 vx vy vz vyz vxz vxy vxyz � , (168) where v0 = ( √ 8)−1|WT 1 | ⊗ |WT 1 | ⊗ |WT 1 | , vx = ( √ 8)−1|WT 1 | ⊗ |WT 1 | ⊗ WT 1 , vy = ( √ 8)−1|WT 1 | ⊗ WT 1 ⊗ |WT 1 | , vz = ( √ 8)−1WT 1 ⊗ |WT 1 | ⊗ |WT 1 | , vyz = ( √ 8)−1WT 1 ⊗ WT 1 ⊗ |WT 1 | , vxz = ( √ 8)−1WT 1 ⊗ |WT 1 | ⊗ WT 1 , vxy = ( √ 8)−1|WT 1 | ⊗ WT 1 ⊗ WT 1 , vxyz = ( √ 8)−1WT 1 ⊗ WT 1 ⊗ WT 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (169) Vectors |WT 1 | and WT 1 are eigenvectors of WT 1 W1 with eigenvalues 0 and 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Using this fact, ΣijkV = 2ℏ m V diag � 0, 1 (∆x)2 , 1 (∆y)2 , 1 (∆z)2 , 1 (∆y)2 + 1 (∆z)2 , 1 (∆x)2 + 1 (∆z)2 , 1 (∆x)2 + 1 (∆y)2 , 1 (∆x)2 + 1 (∆y)2 + 1 (∆z)2 � + 1 ℏΛU,ijkV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (170) 31 Table 5: Relative difference of the CFL limit (3) and the generalized CFL limit (34) in the scenarios considered in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Potential nx = ny = nz = 1 nx = ny = nz = 10 U|i,j,k = 0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='49×10−16 0 U|i,j,k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1 eV −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='84×10−16 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='47×10−15 U|i,j,k = 1 eV −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='81×10−16 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='81×10−16 U|i,j,k = 10 eV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='91×10−16 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='56×10−16 U|i,j,k = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1 eV −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='51×10−1 −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='51×10−1 U|i,j,k = −1 eV −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='72×10−1 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='72×10−1 U|i,j,k = −10 eV −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='03×10−2 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='03×10−2 0 < U|i,j,k < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1 eV (randomized) −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='13×10−2 −9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='18×10−2 0 < U|i,j,k < 1 eV (randomized) −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='05×10−2 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='96×10−2 0 < U|i,j,k < 10 eV (randomized) −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='15×10−2 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='01×10−2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1 eV < U|i,j,k < 0 (randomized) −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='69×10−1 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='24×10−1 −1 eV < U|i,j,k < 0 (randomized) −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='55×10−1 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='28×10−1 −10 eV < U|i,j,k < 0 (randomized) −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='52×10−2 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='04×10−2 Matrix V can be easily shown to satisfy VT V = I = VVT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Thus, Σijk = 2ℏ m V diag � 0, 1 (∆x)2 , 1 (∆y)2 , 1 (∆z)2 , 1 (∆y)2 + 1 (∆z)2 , 1 (∆x)2 + 1 (∆z)2 , 1 (∆x)2 + 1 (∆y)2 , 1 (∆x)2 + 1 (∆y)2 + 1 (∆z)2 � VT + 1 ℏΛU,ijk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (171) Hence Σijk is a sum of two symmetric matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The first matrix has the 2-norm equal to (2ℏ/m)((∆x)−2 + (∆y)−2 + (∆z)−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' As a result [54], ρ(Σijk) = ||Σijk||2 ≤ 2ℏ m � 1 (∆x)2 + 1 (∆y)2 + 1 (∆z)2 � + 1 ℏ ||ΛU,ijk||2 (172) and 2 2ℏ m � 1 (∆x)2 + 1 (∆y)2 + 1 (∆z)2 � + 1 ℏ ||ΛU,ijk||2 ≤ 2 ρ (Σijk) , (173) where ||ΛU,ijk||2 = max (|U|i,j,k| , |U|i+1,j,k| , |U|i,j+1,k| , |U|i+1,j+1,k| , |U|i,j,k+1| , |U|i+1,j,k+1| , |U|i,j+1,k+1| , |U|i+1,j+1,k+1|) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (174) Inequality (173) proves (162) by the definition of the two time steps in (162).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Corollary B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' ∆tCFL ≤ min i,j,k ∆t(i,j,k) CFL,gen , (175) where ∆tCFL is the CFL limit for the entire region given by (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' ∆tCFL ≤ ∆t(i,j,k) CFL ≤ ∆t(i,j,k) CFL,gen ∀i ∈ 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' , nx, ∀j ∈ 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' , ny, ∀k ∈ 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' , nz .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' (176) The statement of the corollary follows directly from (176).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Combining the statements of Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='2 and Corollary B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='1, ∆tCFL ≤ min i,j,k ∆t(i,j,k) CFL,gen ≤ ∆tCFL,gen , (177) which proves the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' 32 Table 5 shows the relative difference between the two time steps in (3) and (34), computed as (∆tCFL − ∆tCFL,gen)/∆tCFL,gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The cell dimensions were taken as ∆x = 1 nm, ∆y = 2 nm, and ∆z = 3 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' The mass of the particle was that of an electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' As expected, for the regions consisting of a single cell with constant nonnegative potential U at each of the eight nodes, the two time steps ∆tCFL and ∆tCFL, gen were the same, with small discrepancy due to machine precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Interestingly, the two time steps were also equal for the multi-cell regions in Table 5 with constant nonnegative potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' However, when the potential either had a spacial variation or was negative, the two time steps deviated, although the difference tended to be small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' In all regions where the CFL limit (3) and the generalized CFL limit (34) deviated, the CFL limit (3) had a lower value, confirming the statement of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE2T4oBgHgl3EQf0giO/content/2301.04142v1.pdf'} +page_content=' Acknowledgments The work was in part funded by the Natural Sciences and Engineering Research Council of Canada (NSERC) Discovery Grant Program [funding reference number RGPIN-2019-05060], in part by the Canada Research Chairs Program [funding reference number 950-232062], and in part by the School of Graduate Studies and 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0000000000000000000000000000000000000000..f84246c232dc13866bf6166a21a6e7a9e2ad1b2a --- /dev/null +++ b/s9E4T4oBgHgl3EQfVwyP/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:90c182d95968cb4509b154633de921298876533e1889dedd279252b73c3c39e7 +size 151841 diff --git a/stFJT4oBgHgl3EQfcixn/content/tmp_files/2301.11544v1.pdf.txt b/stFJT4oBgHgl3EQfcixn/content/tmp_files/2301.11544v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..de3c0fb8c95a8ece9d34a442add463677dd20a91 --- /dev/null +++ b/stFJT4oBgHgl3EQfcixn/content/tmp_files/2301.11544v1.pdf.txt @@ -0,0 +1,1008 @@ +TARGETED ATTACKS ON TIMESERIES FORECASTING +Yuvaraj Govindarajulu, Avinash Amballa, Pavan Kulkarni, and Manojkumar Parmar +AIShield +Bosch Global Software Technologies, Bengaluru, India +{govindarajulu.yuvaraj, avinash.amballa}@in.bosch.com +ABSTRACT +Real-world deep learning models developed for Time Series Forecasting are used in several critical +applications ranging from medical devices to the security domain. Many previous works have shown +how deep learning models are prone to adversarial attacks and studied their vulnerabilities. However, +the vulnerabilities of time series models for forecasting due to adversarial inputs are not extensively +explored. While the attack on a forecasting model might aim to deteriorate the performance of the +model, it is more effective, if the attack is focused on a specific impact on the model’s output. In +this paper, we propose a novel formulation of Directional, Amplitudinal, and Temporal targeted +adversarial attacks on time series forecasting models. These targeted attacks create a specific impact +on the amplitude and direction of the output prediction. We use the existing adversarial attack +techniques from the computer vision domain and adapt them for time series. Additionally, we propose +a modified version of the Auto Projected Gradient Descent attack for targeted attacks. We examine +the impact of the proposed targeted attacks versus untargeted attacks. We use KS-Tests to statistically +demonstrate the impact of the attack. Our experimental results show how targeted attacks on time +series models are viable and are more powerful in terms of statistical similarity. It is, hence difficult +to detect through statistical methods. We believe that this work opens a new paradigm in the time +series forecasting domain and represents an important consideration for developing better defenses. +1 +Introduction +Time Series Forecasting (TSF) tasks are seen in many real-world problems across several domains. The wide range of +domains of applications include demand forecasting [1], anomaly detection [2], stock price prediction [3], electrical +pricing [4] and weather forecasting [5]. Improved availability of data and computation resources has been reflected +in the recent efforts [6, 7, 8] of applying deep learning techniques for forecasting tasks. The wide applications of +such deep learning models have led to threats due to adversaries and hence also the work towards exploration and +prevention [9, 10, 11] of such adversarial attacks. For a given classification model, the goal of the adversary could be +either targeted or untargeted. In targeted attacks, the adversary tries to misguide the model to a particular class other +than the true class. In an untargeted attack, the adversary tries to misguide the model to predict any of the incorrect +classes. The definition of targeted is well-defined for classification tasks and has been used in several previous works +[12, 13, 14]. These definitions are not applicable to regression tasks such as TSF. In the adversarial machine learning +domain, time series tasks have received significantly less attention as compared to those of computer vision. Also, the +adversarial attacks and defenses studied in the computer vision domain are not always useful for time series, requiring +specific adaptations and re-definitions. +In this paper, we address the above-mentioned shortcomings by providing a formulation for targeted attacks on TSF. To +do this, we extend the popularly known adversarial attacks from the computer vision domain to time series forecasting. +Together with popular attacks such as Fast Gradient Sign Method (FGSM) and Projected Gradient Descent (PGD), we +also propose a modified variant of Auto PGD attacks. We perform KS-tests on the loss attributes of the output forecasts +(predictions) to study the statistical properties of the proposed targeted attacks. +The contributions of our work are as follows: +1. We define and formalize targeted attacks on deep learning time series forecasting. +arXiv:2301.11544v1 [cs.LG] 27 Jan 2023 + +2. We propose a modified Auto PGD attack for Time Series Forecasting (mAPGD-TSF), an extension of the +AutoPGD algorithm for targeted time series attacks, which can be extended to any regression task. +3. Through statistical tests, we show that inputs with targeted perturbations are much more indistinguishable than +untargeted attacks through empirical studies on the Google Stock and Household electric power consumption +datasets. +2 +Related Work +Most of the approaches on adversarial attacks were first started on image classification in the deep learning domain. +Christian et al. [15] proposed adversarial examples for image recognition, which initiated a direction to investigate +adversarial attacks in various domains. Ian et al. [12] proposed the Fast Gradient Sign Method (FGSM) which is a +single-step attack. In a similar line, Alexey et al. [16] presented an iterative version of FGSM called the Basic Iterative +Method (BIM). These attacks pave the way for the current state-of-art adversarial attacks in computer vision. Samer et +al. [17] summarizes the existing white-box and black-box adversarial attacks and compares them on the basis of time +taken, strength, and transferability of the attacks. Additionally, by focusing on how they are employed in real-world +applications, they have offered a thorough description of the most popular defense mechanisms against adversarial +attacks. +Izaskun et al. [18] introduced the first adversarial attacks on Time Series Classification (TSC). Hassan et al. [19] used +existing adversarial attack mechanisms such as the FGSM, BIM to decrease the accuracy of residual networks for Time +Series Classification. Pradeep et al. [9] brings the first notion of targeted attack on time series data. The targeted attacks +are however focused on TSC tasks. Wenbo et al. [20] brings the notion of a black-box method called TSadv on the TSC +task. This work suggests a gradient-free black-box strategy to attack DNNs for TSC with local perturbations based on +time series shapelets and differential evolution, which is in contrast to prior work that required gradient information and +global perturbations. +Gautier et al. [21] introduced Smooth Gradient Method (SGM) attack based on a gradient method and shows how +adversarial training is a good way to improve a time series classifier’s (TSC) robustness against smoothed perturbations +by enforcing a smoothness condition on generated perturbations that contains spike and sawtooth patterns. The work +by Fazle et al. [22] takes into consideration that the time series models are sensitive to abnormal perturbations in the +input and stringent requirements on perturbations. To address this, the work crafts time series adversarial based on the +importance of measurement. The adversarial inputs are subjected to models performing time series prediction tasks +such as LSTNet, CNN-, RNN- and MHANET-based models. An importance-based adversarial attack needs much +smaller perturbations compared to other existing adversarial attacks. The work, however, does not formulate or address +the targeted attacks in time series forecasting. +Another work by Gautam et al. [23] on time series forecasting, explores the vulnerabilities of deep learning multi-time +series regression models to adversarial samples. The work also focuses on gradient-based white box attacks on deep +learning models such as CNNs, Gated-Recurrent Units (GRU), and Long-Short Term Memory (LSTM) models. The +vulnerabilities are shown to be transferable and have the ultimate consequences. This work also focuses on untargeted +adversarial attacks with an aim to increase the error of the deep learning model’s output. Raphaël et al. [24] considers +an adversarial setting in a probabilistic framework on auto-regressive forecasting models. This work uses Monte-Carlo +estimation in approximating the gradient of expectation and addresses the challenge of effectively differentiating through +Monte-Carlo estimation using reparametrization and score-function estimators. This also proposes an under-estimation +attack and an over-estimation attack on electricity consumption prediction for reparametrization and score-function +estimators. +3 +Settings and formulations +3.1 +Threat Model +Time Series Forecasting. Given a time series X = [x1, x2, ..xT ], a time series forecasting task predicts the value of +xT +1 based on the previous samples [xT −w, xT −w+1, ...xT ], where w is the window size under consideration. The +sample xT +1 corresponds to the forecasted value and is often represented by ˆY . +Adversarial Time Series. An adversarial perturbation η, typically superposed on a given input time series X, to +construct ˆX given by [ ˆx1, ˆx2, ..., ˆ +xT ]. The adversarial time series ˆX (Xadv) is intended to significantly worsen the +output prediction ˆY of a TSF model. +2 + +Goal of Adversary The goal of the attacker is to create a targeted output impact on the time series. We consider +L∞-bounded perturbation that causes a targeted attack. We consider the definition of white-box access, where the +gradients of the model are available for loss calculation. The Threat model can also be extended in a transfer attack +scenario, where an Oracle model created through extraction in a black-box setting can be used as white-box. Further, +the range of the input after the perturbation is assumed to be accepted by the model. The inputs are otherwise, limited +to the acceptable input range of the model. We denote the regressor f : R(M) �→ R(N) with parameters θ, the predicted +output for input x ∈ R(M) is represented as y = f(x). +Properties of the perturbation. In general, adding a perturbation to the input should detoriate the performance of the +model prediction. Additionally, It is also important for the perturbation to satisfy additional requirements including: 1. +Small changes to the input to create bigger performance degradation on the output. Larger perturbations to the inputs are +also easily detectable by the input plausibility check modules. It is notable that it is more expensive to achieve higher +input perturbation. 2. Imperceptible perturbations, attributing to reduced risk due to detection of the input perturbation. +The perturbation is hence formulated as a constrained optimization problem, where f is the Deep TSF model under +consideration and ϵ indicates the strength of the attack, +η = maximize||Y − ˆY || +s.t ||X − ˆX||≤ ϵ, where Y = f(X) and ˆY = f( ˆX) +(1) +3.2 +KS Tests in Timeseries +Kolmogorov-Smirnov Test (KS Test) [25] is a popular non-parametric test method in statistics, to test whether a sample +follows a reference probability distribution (one sample KS-Test) or where two samples follow a given probability +distribution. Given a single sample, the distance between the reference probability distribution and the empirical +distribution of the given sample is measured. Corresponding to the above distance, a p-value is calculated. Given a +significance value α. the null hypothesis that the sample follows the reference distribution is acceptable, if the p-value +is acceptable if it is greater than the significance value α. +In order to compare the attributes of actual prediction and outputs due to adversarial inputs, we consider the error of a +TSF model output within a given window. We consider the Root Mean Squared Error (RMSE) between the original +prediction with the outputs due to targeted and untargeted attacks. The distribution of the MSE across all the windows +in the given dataset roughly follows a normal distribution, making KS-test usable for statistical analysis. +4 +Targeted Time series attacks +Considering the limited number of work in the time series adversarial area and the need for the definition of targeted +attacks in the time series domain, we go about the formulation. We take into account the practical aspects that create +maximum impact on the output. Depending on the type of impact targeted on the forecasting output, the adversarial +attacks are classified as follows: +• DIRECTIONAL (DTA). In a Directional Targeted Attack (DTA), the attacker crafts the attack in a way, a +minor perturbation on the input causes a shift in the direction of the output. The forecasted output could be +shifted upwards or downwards. +• AMPLITUDINAL (ATA). In an Amplitudinal Targeted Attack (ATA), the attacker crafts the attack such that +a minor perturbation in the input causes the output amplitude to be limited within a prescribed threshold. In +such a given scenario, the attacker would want to hide any high-impact areas on the output for his benefit. +• TEMPORAL (TTA). In a Temporal Targeted Attack (TTA), the attacker crafts the attack such that a minor +perturbation in the input causes a specific time region in the output to be manipulated. In the given region, the +attacker would want to change the direction of the output (DTA) or the amplitude of the output (ATA) in a +specific time window (t1, t2). The attacker performs DTA or ATA specifically in the target time window to +realise TTA. TTA can hence be considered as a sub-category of DTA or ATA. +The attacks are formalized in the context of the respective adversarial attacks (FGSM, PGD, and APGD) in the further +sections. In the further equations, the input time series without perturbation is denoted as x0 and the adversarial time +series as xadv. For amplitudinal attacks (ATA), τ denotes the threshold to limit the output amplitude. In the case of +directional attacks (DTA), the factor α attributes to the direction of the attack and is either +1 or -1 accordingly. For +temporal attack (TTA), the type of impact to be created within a given time window (t1, t2) could be either DTA or +ATA and is referred as att(.) in the below equations. All the below attacks are defined for L∞ norm. +3 + +4.1 +Targeted FGSM +Fast Gradient Sign Method (FGSM). The FGSM attack [12] introduces as notion of gradient-based adversarial attack. +The adversarial input 2 is crafted by adding a small amount of perturbation that increases the loss of the true class +making the model misclassify the input xadv. For a classification example, the perturbation noise is calculated as the +gradient of the loss function L with respect to the input image x for the given true output class. On the other hand, +targeted attacks decrease the loss with respect to the target. +xadv = x + ϵ · sign(∇xL(θ, x, ytrue) +(2) +Targeted FGSM. Targeted Attacks for TSF using FGSM are defined as in equation 3, where L represents the loss or +cost function that was used as an optimization function during model training. +xadv,dta = x0 − ϵ · sign(∇x(L(θ, x, y + α · |y|))) +xadv,ata = x0 − ϵ · sign(∇x(L(θ, x, lim(τ, y)))) +(3) +4.2 +Targeted PGD +Projected Gradient Descent (PGD). The PGD attack is an extension of Iterative-FGSM (popularly called Basic +Iterative Method (BIM)) [13]. In this iterative method, after each step of perturbation, the adversarial example is +projected back into the ball of x using the projection function Π. The perturbed image after n iterations is denoted by +xn +adv. +xn +adv = Πϵ(xn−1 + ϵ · ∇x(L(θ, xn−1, ytrue))) +(4) +Targeted PGD. The targeted PGD attacks for TSF extend the targeted attack techniques of FGSM in the PGD context. +xn +adv = Πϵ(xn−1 − ϵ · ∇x(L(θ, xn−1, y′))) +where, y′ ∈ {y + α · |y|, lim(τ, y), att(y(t))} +(5) +4.3 +Targeted APGD +Auto Projected Gradient Descent (APGD). Auto PGD (or simply APGD) is an extension of the PGD attack addressing +the sub-optimal step size and objective function issues. APGD [14] proposes 1. adding momentum to the gradient step +while progressively reducing step size, 2. two conditions to update the step size when there is a successful increase in +the objective function and to early-stop when there is no improvement towards the objective function. In algorithm 1, +we propose mAPGD-TSF, an extension of the APGD algorithm and is discussed in detail in section:4.3. +In the targeted APGD attacks, we first go about presenting modified Auto PGD attacks for the Time Series forecasting +(mAPGD-TSF) algorithm 1. The modified variant applies to any regression model f in general. The following +modifications are made to the algorithm in comparison to AutoPGD to account for targeted time series applications. +The changes are namely: +• Loss function L. The confidence or the output accuracy is replaced with loss function L. The goal to increase +the confidence function is replaced with a goal to minimize the loss with respect to the target. +• Output Target. The output target is adapted as per the targeted attacks DTA, ATA and TTA respectively: +y′ ∈ {y + α · |y|, lim(τ, y), att(y(t))} +• Initialization. One-step PGD is performed before the start of an iterative reduction of loss. The one-step PGD +is considered as an initialization, as the step initializes the minimal loss fmin. +The conditions related to step size selection are retained from the original AutoPGD Work [14]. The step size is halved +if one of the conditions is true. With f (n) +min is the lowest objective value with the least loss in the first n-iterations, below +are the two conditions: +1. �wj−1 +i=wj−1 1L(f(x(i+1)),y′)) ? +@ ABC +JD-240000 +O-C [min] +Fig. 2. O−C diagrams of studied systems according to linear ephemeris in Section 3. +package dedicated to modeling close eclipsing binaries including surface fea- +tures such as spots and pulsations. ELISa utilizes modern approaches to the +EB modeling with an emphasis on computational speed while maintaining a +sufficient level of precision to process a ground-based and space-based obser- +vation. It was designed for easy use even by a not very experienced user. In +this paper, we take advantage of its capability to model the light curves of +close eclipsing binaries with the builtin capability to solve an inverse prob- +lem using least squares thrust region reflective algorithm and Markov Chain +Monte-Carlo (MCMC) methods (for references see ˇCokina et al. (2021)). +At the beginning of the fitting process, it is necessary to prepare the input +data. ELISa requires phased light curves with normalized flux. All our phased +observations in all passbands were transformed to flux and normalized accord- +ing to flux in the maxima and were simultaneously fitted by the least-square +method to find the global optimal solution. Subsequently, MCMC sampling +was used to produce 1σ confidence intervals of the fitted system’s parameters. +Each system was fitted with model containing 5 free parameters: orbital +inclination i, photometric mass ratio qp, surface potentials of both components +Ω1 and Ω2 and the effective temperature of the secondary component T eff +2 +. +Temperature of the primary component T eff +1 +for all systems were adopted +from Babusiaux et al. (2022) and was fixed during fitting process, while tem- +peratures of the secondary component were fitted with no restrictions. +For the components with convective envelopes (effective temperatures bel- +low ∼7000 K), the albedos A1, A2 of components were set to 0.6 (Ruci´nski +1969) and gravity darkening factors, g1 and g2 to 0.32 (Lucy 1967). In the +case of radiative envelope (above ∼7000K), the values of albedo and gravity + +8 +KUDAK ET AL. +TABLE 4 +PHOTOMETRIC PARAMETERS OF THE STUDIED SYSTEMS +RU UMi +VY UMi +GSC 04364-00648 +Primary +Secondary +Primary +Secondary +Primary +Secondary +i [deg] +83.4+0.15 +−0.22 +85.1+0.22 +−0.25 +76.9+0.20 +−0.21 +qp (M2/M1) +0.341+0.010 +−0.006 +0.536+0.023 +−0.025 +0.440+0.014 +−0.015 +T [K] +7420a +4885+148 +−201 +5340a +4850+240 +−250 +7970a +4065+44 +−49 +Ω +2.632+0.013 +−0.007 2.568+0.037 +−0.019 +2.821+0.034 +−0.036 +4.138+0.033 +−0.033 2.767+0.032 +−0.030 +lV /lV +tot +0.93+0.02 +−0.03 +0.07+0.09 +−0.10 +0.68+0.09 +−0.10 +0.32+0.06 +−0.07 +0.92+0.05 +−0.06 +0.08+0.03 +−0.03 +Ωcrit +2.553+0.022 +−0.012 +2.943+0.032 +−0.00.035 +2.760+0.029 +−0.028 +Req[SMA] +0.457+0.001 +−0.002 0.286+0.001 +−0.001 0.465+0.002 +−0.001 0.357+0.001 +−0.001 0.273+0.002 +−0.001 0.308+0.002 +−0.002 +a - Temperature of the primary component for all systems were adopted from +Babusiaux et al. (2022) +darkening factor were both set to 1.0. Castelli & Kurucz (2003) models of +stellar atmospheres were used. The linear limb darkening coefficients for each +component were interpolated from the van Hamme (1993) tables. +The weights of individual data points were established as 1/σ2, where σ is +the standard error of point derived during photometric measurement. Initially, +the least-square algorithm was used with suitable initial parameters to find +an approximate solution and then the parameter space near the solution was +explored with MCMC sampler with 500 walkers and 500 iterations with prior +300 iterations discarded as it belonged to the thermalization stage of the +sampling. +The resulting as well as derived parameters of all systems, like relative +luminosities of the components in V filter lV +1,2/lV +tot, a critical potential Ωcrit, +corresponding equivalent radius Req in SMA units (semi-major axis) are listed +in Tab. 4. The best-fit models with observed LCs and resulting flat chains +displayed in the form of the corner plot are shown in Fig. 3 and 3D models +with the surface temperature distributions are shown in Fig. 4. +5. ABSOLUTE PARAMETERS OF THE SYSTEMS +The absolute parameters of the binary components, like their masses M1,2, +radii R1,2, luminosities L1,2 and semi-major axis of the orbit a, can be mainly +determined by the combination of photometric solution and analysis of ra- +dial velocity curve. Radial velocities are available only for RU UMi system +(Okazaki et al. 1988; Maxted & Hilditch 1996). We used their measurements +and re-analyze them with ELISa code (assuming circular orbit) and deter- +mined orbital and absolute parameters of the components as listed in Tab.5. +But if we know the distance to object from independent measurement +(e.g parallaxes from GAIA measurements), we can find absolute parameters +from properties of binary derived from photometric solution and using basic +relationships, described in the following text. + +ANALYSIS OF 3 NEGLECTED BINARIES +9 +TABLE 5 +PARAMETERS OF THE SPECTROSCOPIC ORBIT AND ABSOLUTE +PARAMETERS OF RU UMI SYSTEM DERIVED FROM RADIAL +VELOCITY AND LIGHT CURVE SOLUTIONS. +M1 [M⊙] 2.65+0.11 +−0.16 K1 [km/s] 96.5+6.4 +−5.9 +M2 [M⊙] 0.85+0.12 +−0.10 K1 [km/s] 301.6+10.8 +−8.3 +R1 [R⊙] +1.89+0.7 +−0.4 +q 0.32+0.02 +−0.02 +R2 [R⊙] +1.18+0.4 +−0.3 a sin i [R⊙] 4.13+0.11 +−0.08 +L1 [L⊙] +9.61+2.51 +−1.48 +γ [km/s] −21.0+3.8 +−4.0 +L2 [L⊙] +0.71+0.16 +−0.12 +a [R⊙] +4.15+0.12 +−0.09 +Let us assume that we have calculated qp, i, T1,2, Req +1,2 and lV +1,2/lV +tot from +analysis of the light curves and we know standard V magnitude of system in +phase 0.25. Absolute magnitude MV of the system we can find from equation +M V = V − 5 log(d) + 5 − AV , +(4) +where d is distance and extinction coefficient AV can be determined from the +dust map in Green et al. (2019). The absolute magnitudes of each component +can be find from relation +M V +1,2 − M V = −2.5 log lV +1,2 +lV +tot +. +(5) +Corresponding bolometric magnitude is +M Bol +1,2 = M V +1,2 + BC, +(6) +where bolometric corection is from Eker et al. (2020). If we assume that bolo- +metric magnitude of the Sun is M Bol +⊙ +=4.73 mag (Torres 2010), the luminosities +of components can be found from +M Bol +1,2 − M Bol +⊙ += −2.5 log L1,2 +L⊙ +, +(7) +and corresponding radii +R1,2 = +� +L1,2 +4πσT 4 +1,2 +. +(8) +The distance between components can be found using their equivalent radii +Req +1,2 +a = 1 +2 +� R1 +Req +1 ++ R2 +Req +2 +� +. +(9) +Total mass M1 + M2 of the system we can derive using Kepler’s 3rd law +a3 +P 2 = G(M1 + M2) +4π2 +, +(10) + +10 +KUDAK ET AL. +TABLE 6 +ABSOLUTE PARAMETERS OF THE STUDIED SYSTEMS DERIVED +FROM THEIR DISTANCES AND PHOTOMETRIC SOLUTIONS. +RU UMi +VY UMi +GSC 04364-00648 +Primary Secondary Primary Secondary Primary Secondary +M [M⊙] +1.90(21) +0.64(17) +0.97(15) 0.52(18) +2.54(42) +1.12(31) +R [R⊙] +1.55(11) +1.16(9) +0.96(2) +0.87(2) +1.22(3) +2.25(5) +L [L⊙] +6.61(1.47) 0.69(14) +0.67(10) 0.38(11) +5.40(1.34) 1.24(23) +a [R⊙] +3.74(37) +2.27(21) +5.88(79) +AV [mag] +0.0 +0.0 +0.03(3) +M Bol [mag] +2.67(7) +5.12(13) +5.16(8) +5.75(14) +2.89(8) +4.49(11) +BC [mag] +0.06 +-0.31 +-0.082 +-0.30 +0.03 +-1.03 +d [pc] +283.0(1.2) +164.5(3) +512.5(4.8) +and individual masses can be found from mass ratio qp. +The absolute parameters for the studied objects determined by the method +described above are listed in Tab. 6. The uncertainties of the parameters were +calculated considering the errors of the light curve solutions of the systems +and errors in their distances. +6. DISCUSSION AND CONCLUSIONS +In our study, we have presented the photometric analysis of multi-color +BV R and TESS photometry of three eclipsing binaries, RU UMi, VY UMi +and GSC 04364-00648, for the first time for the last two systems. We have also +analyzed their period variations considering archival data and our new minima +times. The presented photometry solutions, mainly for VY UMi and GSC +04364-00648, has some small disagreements with observations, the residuals +at some phases show up to 0.1 deviations in normalized flux value. These +issues can be caused by spots, weather conditions, etc. They are really small, +and we can not even try to explain what phenomena they are caused by. +Future observation is needed for this purpose. +RU UMi has been studied in the past years by several authors. Recently, +Lee et al. (2008) analyzed period variation, fitted light-curve and determined +absolute parameters of the components from radial velocities solution. From +their period analysis authors concluded that long-term period changes can +be caused by the combination of angular momentum loss (AML) and mass +transfer from the less massive secondary to the more massive primary. In our +period analysis, we used only minima times obtained from ground-based pho- +toelectric and CCD observations as well as satellite observations from TESS, +where we can expect higher precision with respect to older visual and pho- +tographic observations. We detected wave-like variations with low-amplitude +(∼ 5 minutes) in O − C residua. It can be interpreted as a consequence of the +light-time effect caused by the 3rd invisible component. From the parameters + +ANALYSIS OF 3 NEGLECTED BINARIES +11 +listed in Tab. 3 we can see that the orbital period of the 3rd body is 7370 +days and the orbit is slightly eccentric. According to the mass function of the +3rd body f(m3) and masses of the binary components (see Tab. 6), we can +find that the minimum mass of the 3rd component in the case of the edge-on +orbit (sin i3 = 1) should be M3=0.063(16)M⊙ ∼ 60MJ. It corresponds to a +low massive red dwarf or more probably (due to its mass) it is a brown dwarf +(Joergens 2014) with very low luminosity. It is supported also by the results +of the photometric solution, where no 3rd light was detected. Photometric +analysis of BV R and TESS light curves confirmed previous findings that +RU UMi is a near contact system with a secondary component that almost +fulfills its Roche lobe. We were not able to find any satisfactory LC solution +with spot(s) (not evenTESS LC) as was done by Lee et al. (2008), although +some wave-like variation is visible in residuals in Fig. 3. We can explain it by +the temporal evolution of the spot, when spot’s parameters such as diameter, +temperature and position on the star’s surface are changing during decades. +Our absolute parameters of components determined from radial velocity so- +lution are little bigger than presented in Lee et al. (2008). On the other side, +absolute parameters determined from GAIA parallax are smaller than previ- +ously determined and correspond to A6V primary component and evolved K5 +star secondary. This differences deserves deeper analysis, because many fac- +tors can affect results. One of them is a mass ratio which has strong influence +to partial dimensions of the components as well as inclination due to q − i +correlation. Terrell & Wilson (2005) showed that the photometric mass ratio +for semi-detached and over-contact binaries is often overestimated for partial +eclipses. Recently Terrell (2022) noted that not properly modeled third light +will lead to mass ratios that are too low. Our solution of RU UMi show no +third light and no total eclipses on the LCs (just like other stars). Presented +photometric mass ratios qp have to be considered as a high estimates and this +affects also determination of absolute parameters from distances. +Photometric analysis of VY UMi showed that the system is a typical +W UMa type overcontact binary with a more massive primary component +(qp = 0.535). Its orbital period and determined temperatures of both com- +ponents place the system in a W-type subclass of overcontact binaries. The +detected parabolic period change reflected on the O − C diagram can be +explained by the mass transfer from a less massive star to a more massive +one. Period increase with rate 2.56(9)×10−7 d/yr−1 detected in the VY UMi +system corresponds to mass transfer from the secondary to the primary com- +ponent. +The first photometric solution of GSC 04364-00648 light curves revealed +that the system is semi-detached binary, where a cool secondary component +almost fills its Roche lobe as detected in some other near-contact systems, like +EG Cep (Djuraˇsevi´c et al. 2013) or CR Tau (Kudak et al. 2021). Although +we can see some quadratic changes on the O − C diagram, which corresponds +to a period decrease with a high rate of −2.26(5) × 10−5 d/yr−1, we cannot +make strict conclusions about period variation in the system, mainly due to + +12 +KUDAK ET AL. +short time (2019-2021) and uneven coverage of O − C diagram. We have to +wait for other observations to confirm or disprove this trend. +Acknowledgments +This work was supported by Ukrainian national grant 0122U000937 and by +the Slovak Research and Development Agency under contract No. APVV-20- +0148. The research of P.G. was supported by the internal grant No. VVGS- +PF-2021-2087 of the Faculty of Science, P. J. ˇSaf´arik University in Koˇsice. +REFERENCES +Applegate, J. H. 1992, ApJ, 385, 621 +Babusiaux, C., Fabricius, C., Khanna, S., Muraveva, T., Reyl´e, C., Spoto, F., Val- +lenari, A., Luri, X., Arenou, F., Alvarez, M. A., Anders, F., Antoja, T., Bal- +binot, E., Barache, C., Bauchet, N., Bossini, D., Busonero, D., Cantat-Gaudin, +T., Carrasco, J. M., Dafonte, C., Diakite, S., Figueras, F., Garcia-Gutierrez, A., +Garofalo, A., Helmi, A., Jimenez-Arranz, O., Jordi, C., Kervella, P., Kostrzewa- +Rutkowska, Z., Leclerc, N., Licata, E., Manteiga, M., Masip, A., Monguio, M., +Ramos, P., Robichon, N., Robin, A. 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Gajdoˇs and ˇS.Parimucha: Institute of Physics, Faculty of Sci- +ence, P.J. ˇSaf´arik University, Koˇsice, Park Angelinum 9, 04001, Slovakia +(stefan.parimucha@upjs.sk) + +14 +KUDAK ET AL. +Ω +D= 2.63+ 0.01 +− 0.01 +4600 +4800 +5000 +5200 +Teff +2 +Teff +2 +E 4885+ 143 +− 96 +K +2.550 +2.575 +2.600 +2.625 +F +2 +H2 +I 2.57+ 0.04 +− 0.02 +0.336 +0.344 +0.352 +q +q +J 0.34+ 0.01 +− 0.01 +83.00 +83.25 +83.50 +83.75 +i +2.610 +2.625 +2.640 +2.655 +K1 +4400 +4600 +4800 +5000 +5200 +Teff +2 +2.525 +2.550 +2.575 +2.600 +2.625 +L2 +0.328 +0.336 +0.344 +0.352 +0.360 +q +83.00 +83.25 +83.50 +83.75 +i +i +M 83.30+ 0.10 +− 0.20 deg +0.328 +0.360 +2.525 +4400 +q= 2.10+ 0.03 +N 0.03 +5.12 +5.20 +5.28 +5.36 +O +1 +P1= 5.28+ 0.04 +Q 0.04 +5500 +6000 +6500 +7000 +Teff +1 +Teff +1 = 6520+ 240 +R 270 K +5200 +5600 +6000 +6400 +Teff +2 +2.00 +2.05 +2.10 +2.15 +q +5.12 +5.20 +5.28 +5.36 +S1 +5500 +6000 +6500 +7000 +Teff +1 +5200 +5600 +6000 +6400 +Teff +2 +Teff +2 = 5850+ 200 +T 220 K +Ω1= 4.14+ 0.03 +− 0.04 +3960 +4020 +4080 +4140 +Teff +2 +Teff +2 = 4063+ 42 +− 49 K +2.70 +2.76 +2.82 +2.88 +Ω2 +Ω2= 2.77+ 0.03 +− 0.03 +0.400 +0.425 +0.450 +0.475 +0.500 +q +q= 0.44+ 0.02 +− 0.01 +76.4 +76.8 +77.2 +77.6 +i +4.02 +4.08 +4.14 +4.20 +4.26 +Ω1 +3960 +4020 +4080 +4140 +Teff +2 +2.70 +2.76 +2.82 +2.88 +Ω2 +0.400 +0.425 +0.450 +0.475 +0.500 +q +76.4 +76.8 +77.2 +77.6 +i +i= 76.90+ 0.20 +− 0.20 deg +Fig. 3. The synthetic model fitted on observational data of (from top) RU UMi, +VY UMi and GSC 04364-0064 together with the corresponding results of the MCMC +sampling displayed in the form of the corner plot. + +1.2 +1.0 +0.8 +Normalized Flux +0.6 +0.4 +TESS +0.2 +0.0 +GSC04364-0064 +-0.2 +-0.4 +0.00 +S +B-0.2 +Residual +-0.25 +V-0.4 +-0.50 +R-0.6 +-0.6 +-0.4 +0.2 +0.0 +0.2 +0.4 +0.6 +PhaseVY UMi +1.2 +B +1.0 +0.8 +Normalized Flux +0.6 +R +0.4 +0.2 +TESS +0.0 +-0.2 +0.00 +B-0.2 +S +Residuals +0.25 +V-0.4 +-0.50 +R-0.6 +-0.75 +-0.6 +-0.4 +-0.2 +0.0 +0.2 +0.4 +0.6 +PhaseRU UMi +1.0 +B +0.8 +Normalized Flux +0.6 +R +TESS +0.4 +0.2 +0.0 +B-0.2 +Residuals +-0.2 +V-0.4 +-0.4 +R-0.6 +-0.6 +-0.6 +-0.4 +-0.2 +0.0 +0.2 +0.4 +0.6 +PhaseANALYSIS OF 3 NEGLECTED BINARIES +15 +x +−0.8 +−0.6 +−0.4 +−0.2 +0.0 +0.2 +0.4 +y +−0.6 +−0.4 +−0.2 +0.0 +0.2 +0.4 +0.6 +z +−0.6 +−0.4 +−0.2 +0.0 +0.2 +0.4 +0.6 +6800 +7000 +7200 +7400 +7600 +primary T/[K] +4900 +5000 +5100 +5200 +5300 +5400 +5500 +5600 +secondary T/[K] +x +−0.75 +−0.50 +−0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +y +−0.75 +−0.50 +−0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +z +−1.00 +−0.75 +−0.50 +−0.25 +0.00 +0.25 +0.50 +0.75 +4700 +4800 +4900 +5000 +5100 +5200 +5300 +5400 + T/[K] +x +−0.8 +−0.6 +−0.4 +−0.2 +0.0 +0.2 +0.4 +y +−0.6 +−0.4 +−0.2 +0.0 +0.2 +0.4 +0.6 +z +−0.6 +−0.4 +−0.2 +0.0 +0.2 +0.4 +7875 +7900 +7925 +7950 +7975 +8000 +8025 +8050 +primary T/[K] +4000 +4200 +4400 +4600 +4800 +secondary T/[K] +Fig. 4. 3D models with the surface temperature distributions of (from top) RU UMi, +VY UMi and GSC 04364-0064 + +16 +KUDAK ET AL. +APPENDICES +TABLE 7: RU UMi times of minima determined from TESS lightcurves. Errors are in parenthesis. +BJD +BJD +BJD +BJD +BJD +BJD +BJD +58683.4591(5) +58706.2924(2) +58730.7012(1) +58886.8666(2) +58910.7460(11) +59428.8517(1) +59585.0166(2) +58683.7210(10) +58706.5541(1) +58730.9632(2) +58887.1296(1) +58911.0137(7) +59429.1138(2) +59585.5416(2) +58683.9827(1) +58706.8163(2) +58731.2262(9) +58887.3914(2) +58911.2710(10) +59429.3768(1) +59585.8049(2) +58684.2447(2) +58707.0794(1) +58731.4882(2) +58887.6544(1) +58911.5387(7) +59429.6388(2) +59586.0665(2) +58684.5074(1) +58707.3413(2) +58731.7511(1) +58887.9164(2) +58911.7950(20) +59429.9018(1) +59586.3312(2) +58684.7694(2) +58707.6043(1) +58732.0130(21) +58888.1793(1) +58913.9016(3) +59430.1637(2) +59586.5911(2) +58685.0323(1) +58707.8672(2) +58732.2761(1) +58888.4413(2) +58914.1627(2) +59430.4267(1) +59586.8546(2) +58685.2944(2) +58708.1292(1) +58732.5380(20) +58888.7046(1) +58914.4261(2) +59430.6886(2) +59587.1164(2) +58685.5573(1) +58708.3912(2) +58732.8008(1) +58888.9664(2) +58914.6877(2) +59430.9515(1) +59587.3796(2) +58685.8194(2) +58708.6541(1) +58733.0630(21) +58889.2293(1) +58914.9513(3) +59431.2135(2) +59587.6413(2) +58686.0823(1) +58708.9161(2) +58733.3253(2) +58889.4912(2) +58915.2128(2) +59431.4764(1) +59587.9044(2) +58686.3443(2) +58709.1791(1) +58733.5904(2) +58889.7543(1) +58915.4765(3) +59431.7385(2) +59588.1662(2) +58686.6071(1) +58709.4411(2) +58733.8509(1) +58890.0161(2) +58915.7377(1) +59432.0014(1) +59588.4297(2) +58686.8692(2) +58709.7040(1) +58734.1127(2) +58890.2793(1) +58916.0014(3) +59432.2631(1) +59588.6911(2) +58687.1321(1) +58709.9671(2) +58734.3757(1) +58890.5410(21) +58916.2626(1) +59432.5255(6) +59588.9543(2) +58687.3941(2) +58711.8037(1) +58734.6377(2) +58890.8042(1) +58916.5259(4) +59433.8383(2) +59589.2161(2) +58687.6569(1) +58712.0657(2) +58734.9006(1) +58891.0661(2) +58916.7789(8) +59434.1010(10) +59589.4791(2) +58688.1820(12) +58712.3286(1) +58735.1626(2) +58891.3293(1) +58917.0450(9) +59434.3631(2) +59589.7410(20) +58688.4440(23) +58712.5906(2) +58735.4257(1) +58891.5913(1) +58917.3119(7) +59434.6260(10) +59590.0042(2) +58688.7068(1) +58712.8536(1) +58735.6874(2) +58891.8539(1) +58917.5720(11) +59434.8880(21) +59590.2660(20) +58688.9690(21) +58713.1156(2) +58735.9504(1) +58892.1159(2) +58917.8376(6) +59435.1510(9) +59590.5291(2) +58689.2316(1) +58713.3785(1) +58736.2125(2) +58892.3788(1) +58918.0950(21) +59435.4130(22) +59590.7908(2) +58689.4937(2) +58713.6405(2) +58736.4755(1) +58892.6407(2) +58918.3627(7) +59435.6760(10) +59591.0541(2) +58689.7567(1) +58713.9036(1) +58736.7374(2) +58892.9037(1) +58918.8876(7) +59435.9379(2) +59591.3158(2) +58690.0188(2) +58714.1654(2) +58737.0004(1) +58893.1657(2) +58919.1440(20) +59436.2008(1) +59591.5790(20) +58690.2813(1) +58714.4286(1) +58870.8572(3) +58893.4289(1) +58919.4125(7) +59436.4628(2) +59591.8407(2) +58690.5437(2) +58714.6905(2) +58871.1191(2) +58893.6906(2) +58920.2051(9) +59436.7257(1) +59592.1039(2) +58690.8065(1) +58714.9533(1) +58871.3818(1) +58893.9536(1) +58920.7190(20) +59436.9877(2) +59592.3656(2) +58691.0687(2) +58715.2165(2) +58871.6437(2) +58894.2156(2) +58920.9868(7) +59437.2507(1) +59592.6290(20) +58691.3313(1) +58715.4772(3) +58871.9065(1) +58894.4786(1) +58921.2450(21) +59437.5127(2) +59594.2037(2) +58691.5935(2) +58716.0030(10) +58872.1686(2) +58894.7404(2) +58921.5123(7) +59437.7756(1) +59594.4653(2) +58691.8563(1) +58716.2651(2) +58872.4315(1) +58895.0033(1) +58921.7690(20) +59438.0377(2) +59594.7287(2) +58692.1184(2) +58716.5281(1) +58872.6935(2) +58895.2654(2) +58922.0371(7) +59438.3006(1) +59594.9902(2) +58692.3850(20) +58716.7901(2) +58872.9569(2) +58895.5285(1) +58922.2950(21) +59438.5625(2) +59595.2537(2) +58692.6441(2) +58717.0529(1) +58873.2186(2) +58895.7904(2) +58922.5621(7) +59438.8254(1) +59595.5152(2) +58692.9063(1) +58717.3150(24) +58873.4816(1) +58896.0534(1) +58922.8190(20) +59439.0874(2) +59595.7789(2) +58693.1684(2) +58717.5779(1) +58873.7435(2) +58896.3154(1) +58923.0864(7) +59439.3504(1) +59596.0401(2) +58693.4313(1) +58717.8400(21) +58874.0064(1) +58896.5783(1) +58923.3430(22) +59439.6124(2) +59596.3035(2) +58693.6934(2) +58718.1031(1) +58874.2683(2) +58896.8402(2) +58923.8690(20) +59439.8754(1) +59596.5651(2) +58693.9559(1) +58718.3648(2) +58874.5313(1) +58897.1032(1) +58924.1364(7) +59440.1374(2) +59596.8285(1) +58694.2182(2) +58718.6278(1) +58874.7932(2) +58897.3653(2) +58924.3940(21) +59440.4003(1) +59597.0899(2) +58694.4807(1) +58718.8898(2) +58875.0564(1) +58897.6277(1) +58924.6619(7) +59440.6622(2) +59597.3534(2) +58694.7431(2) +58719.1527(1) +58875.3182(2) +58899.4663(1) +58924.9190(20) +59440.9250(10) +59597.6149(2) +58695.0062(1) +58719.4159(2) +58875.5813(1) +58899.7285(3) +58925.1867(7) +59441.1872(2) +59597.8782(2) +58695.2681(2) +58719.6771(2) +58875.8430(20) +58899.9897(2) +58925.4440(21) +59441.7120(20) +59598.1398(2) +58695.5308(1) +58719.9590(20) +58876.1061(1) +58900.2535(3) +58925.9690(20) +59441.9750(10) +59598.4031(2) +58695.7930(22) +58720.2025(1) +58876.3679(1) +58900.5156(2) +58926.2364(8) +59442.2370(20) +59598.6647(2) +58696.0559(1) +58720.4646(2) +58876.6317(6) +58900.7786(3) +58926.4860(10) +59442.7619(2) +59598.9282(2) +58697.6307(1) +58720.7277(1) +58876.8921(3) +58901.0399(1) +59420.1901(2) +59443.0250(10) +59599.1897(2) +58697.8928(2) +58720.9896(2) +58877.1560(12) +58901.3032(3) +59420.4530(10) +59443.2868(2) +59599.4528(2) +58698.1556(9) +58721.2525(1) +58877.4178(2) +58901.5648(1) +59420.7150(20) +59443.5498(1) +59599.7146(2) +58698.4177(2) +58721.5145(2) +58877.6809(1) +58901.8268(4) +59420.9779(9) +59443.8117(2) +59599.9780(20) +58698.6805(1) +58721.7773(1) +58877.9427(2) +58902.0899(1) +59421.2399(2) +59444.0746(1) +59600.2408(3) +58698.9425(2) +58722.0394(2) +58878.2058(1) +58902.3519(3) +59421.5030(10) +59444.3367(2) +59600.7660(10) +58699.2056(1) +58722.3024(1) +58878.4677(2) +58902.6069(6) +59421.7650(20) +59444.5996(1) +59601.0278(2) +58699.4674(2) +58722.5642(2) +58878.7308(1) +58902.8732(8) +59422.0278(1) +59444.8616(2) +59601.2894(2) +58699.7303(1) +58722.8273(1) +58878.9927(2) +58903.1397(6) +59422.2898(2) +59445.1246(1) +59601.5529(2) +58699.9923(2) +58723.0893(2) +58879.2555(1) +58903.4005(9) +59422.5527(1) +59445.3865(2) +59601.8143(2) +58700.2553(1) +58723.3520(12) +58879.5176(2) +58903.6752(8) +59422.8147(2) +59445.6495(1) +59602.0777(2) +58700.5173(2) +58723.6155(2) +58879.7806(1) +58903.9267(4) +59423.0777(1) +59445.9114(2) +59602.3393(2) +58700.7803(1) +58725.1890(23) +58880.0425(2) +58904.1896(1) +59423.3397(2) +59446.1744(1) +59602.6026(2) +58701.0422(2) +58725.4521(1) +58880.3056(1) +58904.4516(4) +59423.6027(1) +59446.4364(2) +59602.8642(2) +58701.3053(1) +58725.7154(2) +58880.5674(2) +58904.7147(1) +59423.8646(2) +59580.0305(1) +59603.1274(2) +58701.5570(11) +58725.9739(9) +58880.8305(1) +58904.9773(3) +59424.1275(1) +59580.2920(20) +59603.3891(2) +58701.8292(3) +58726.2402(1) +58881.0923(2) +58905.2393(1) +59424.3894(2) +59580.5558(2) +59603.6525(2) +58702.0927(2) +58726.5019(1) +58881.3554(1) +58905.5015(4) +59424.6525(1) +59580.8172(2) +59603.9141(2) +58702.3552(1) +58726.7639(2) +58881.6173(2) +58906.0240(9) +59424.9144(2) +59581.0806(2) +59604.1773(2) +58702.6170(20) +58727.0268(1) +58881.8804(1) +58906.5460(22) +59425.1773(1) +59581.3422(2) +59604.4390(20) +58702.8797(1) +58727.2888(2) +58882.1422(2) +58906.8143(7) +59425.4394(2) +59581.6057(2) +59604.7026(1) +58703.1419(2) +58727.5517(1) +58882.4053(1) +58907.0720(11) +59425.7022(1) +59581.8670(20) +59604.9638(2) +58703.4050(1) +58727.8136(2) +58882.6668(1) +58907.3392(6) +59425.9640(10) +59582.1305(2) +59605.2275(2) +Continued on next page + +ANALYSIS OF 3 NEGLECTED BINARIES +17 +Table 7 – continued from previous page +BJD +BJD +BJD +BJD +BJD +BJD +BJD +58703.6669(2) +58728.0766(1) +58882.9380(10) +58907.5960(22) +59426.2276(7) +59582.3920(20) +59605.4888(2) +58703.9298(1) +58728.3386(2) +58883.1923(2) +58907.8639(6) +59426.4890(20) +59582.6552(2) +59605.7526(2) +58704.1918(2) +58728.6015(1) +58883.4553(1) +58908.1210(19) +59426.7521(1) +59582.9169(2) +59606.0138(2) +58704.4548(1) +58728.8633(2) +58883.7169(1) +58908.3891(6) +59427.0141(2) +59583.1802(2) +59606.2774(2) +58704.7168(2) +58729.1258(4) +58885.2924(1) +58908.6470(10) +59427.2771(1) +59583.4419(2) +59606.5387(2) +58704.9797(1) +58729.3882(3) +58885.5549(1) +58909.1720(11) +59427.5390(19) +59583.7054(2) +58705.2418(2) +58729.6514(1) +58885.8167(2) +58909.6960(22) +59427.8019(1) +59583.9668(2) +58705.5044(1) +58729.9134(2) +58886.0798(1) +58909.9638(6) +59428.0640(20) +59584.2301(2) +58705.7660(1) +58730.1763(1) +58886.3415(2) +58910.2220(10) +59428.3270(10) +59584.4918(2) +58706.0289(2) +58730.4383(2) +58886.6049(1) +58910.4888(7) +59428.5889(2) +59584.7552(2) +TABLE 8: VY UMi times of minima determined from TESS lightcurves. Errors are in parenthesis. +BJD +BJD +BJD +BJD +BJD +BJD +BJD +59390.7777(4) +59406.5601(1) +59423.1556(1) +59439.1004(1) +59586.9968(2) +59602.7798(1) +59619.5379(2) +59390.9408(1) +59406.7234(1) +59423.3189(1) +59439.2638(1) +59587.1606(1) +59602.9422(1) +59619.7013(1) +59391.1038(1) +59406.8855(1) +59423.4809(1) +59439.4256(1) +59587.3229(2) +59603.1054(1) +59619.8635(2) +59391.2662(9) +59407.0486(1) +59423.6442(1) +59439.5892(1) +59587.4858(1) +59603.2671(1) +59620.0265(1) +59391.4289(2) +59407.2108(1) +59423.8064(1) +59439.7511(1) +59587.6475(2) +59603.4307(1) +59620.1888(2) +59391.5915(1) +59407.3742(6) +59423.9697(1) +59439.9134(6) +59587.8113(1) +59603.5931(2) +59620.3517(1) +59391.7547(1) +59407.5363(1) +59424.1318(1) +59440.0763(3) +59587.9738(2) +59603.7562(1) +59620.5133(2) +59391.9169(1) +59407.6994(1) +59424.2951(1) +59440.2385(5) +59588.1367(1) +59603.9184(1) +59620.6776(1) +59392.0801(1) +59407.8616(1) +59424.4572(1) +59440.4021(1) +59588.2990(20) +59604.0816(1) +59620.8388(2) +59392.2424(1) +59408.0249(1) +59424.6205(1) +59440.5653(1) +59588.4624(1) +59604.2432(2) +59621.0027(1) +59392.4057(1) +59408.1872(1) +59424.7823(1) +59440.7273(1) +59588.6246(2) +59604.4070(10) +59621.1641(2) +59392.5679(1) +59408.3502(1) +59424.9458(1) +59440.8908(4) +59588.7877(1) +59604.5693(2) +59621.3282(1) +59392.7308(1) +59408.5126(1) +59425.1076(1) +59441.0528(1) +59588.9499(2) +59604.7324(1) +59623.7672(3) +59392.8932(1) +59408.6759(1) +59425.2713(1) +59441.2162(1) +59589.1131(1) +59604.8948(2) +59623.9314(1) +59393.0563(1) +59408.8379(1) +59425.4330(1) +59441.3783(8) +59589.2746(2) +59605.0579(1) +59624.0937(2) +59393.2186(1) +59409.0012(1) +59425.5967(1) +59441.5415(1) +59589.4384(1) +59605.2201(2) +59624.2567(1) +59393.3817(1) +59409.1633(7) +59425.7585(1) +59441.7035(1) +59589.6001(2) +59605.3833(1) +59624.4181(2) +59393.5440(1) +59409.3265(5) +59425.9221(1) +59441.8670(1) +59589.7640(10) +59605.5454(1) +59624.5821(1) +59393.7071(1) +59409.4887(1) +59426.0814(7) +59442.0290(1) +59589.9254(2) +59605.7087(1) +59624.7435(2) +59393.8695(1) +59409.8141(8) +59426.2465(6) +59442.1924(1) +59590.0894(1) +59605.8704(1) +59624.9080(10) +59394.0326(1) +59409.9773(1) +59426.4062(7) +59442.3543(9) +59590.2510(20) +59606.0341(1) +59625.0699(2) +59394.1949(1) +59410.1395(1) +59426.5724(2) +59442.5177(1) +59590.4148(1) +59606.1958(1) +59625.2333(1) +59394.3579(6) +59410.3029(1) +59426.7349(1) +59442.6798(1) +59590.5763(2) +59606.3593(1) +59625.3954(2) +59394.5203(1) +59410.4649(1) +59426.8984(1) +59442.8432(1) +59590.7402(1) +59606.5210(21) +59625.5582(1) +59394.6834(1) +59410.6282(1) +59427.0602(1) +59443.0053(8) +59590.9025(2) +59606.6845(1) +59625.7198(2) +59394.8458(1) +59410.7903(1) +59427.2239(2) +59443.1686(1) +59591.0655(1) +59606.8457(6) +59625.8842(1) +59395.0089(1) +59410.9536(1) +59427.3855(8) +59443.3307(1) +59591.2270(22) +59608.7994(5) +59626.0452(2) +59395.1711(1) +59411.1158(8) +59427.5491(1) +59443.4936(2) +59591.3909(1) +59608.9619(6) +59626.2091(1) +59395.3340(1) +59411.2790(1) +59427.7110(1) +59443.6561(1) +59591.5524(2) +59609.1250(20) +59626.3716(2) +59395.4964(1) +59411.4411(1) +59427.8745(1) +59443.8194(1) +59591.7167(1) +59609.2881(1) +59626.5349(1) +59395.6595(1) +59411.6045(1) +59428.0365(1) +59443.9814(1) +59591.8786(2) +59609.4504(2) +59626.6960(20) +59395.8218(1) +59411.7662(1) +59428.1999(1) +59444.1450(20) +59592.0418(1) +59609.6136(1) +59626.8599(1) +59395.9849(7) +59411.9293(5) +59428.3618(1) +59444.3068(1) +59592.2041(2) +59609.7759(2) +59627.0214(2) +59396.1472(1) +59412.2538(5) +59428.5254(4) +59444.4702(1) +59592.3671(1) +59609.9389(1) +59627.1854(1) +59396.3105(6) +59412.4174(1) +59428.6871(1) +59444.6322(1) +59592.5294(2) +59610.1014(2) +59627.3469(2) +59396.4726(1) +59412.5807(1) +59428.8502(2) +59444.7956(4) +59592.6923(1) +59610.2643(1) +59627.5107(1) +59396.6359(1) +59412.7423(1) +59429.0128(1) +59444.9575(1) +59592.8531(7) +59610.4258(2) +59627.6733(2) +59396.7979(1) +59412.9060(1) +59429.1762(1) +59445.1211(1) +59594.1564(2) +59610.5896(10) +59627.8365(1) +59396.9612(1) +59413.0683(1) +59429.3381(1) +59445.2829(1) +59594.3195(1) +59610.7512(2) +59627.9986(3) +59397.1235(1) +59413.2315(1) +59429.5013(1) +59445.4465(1) +59594.4818(2) +59610.9151(1) +59628.1617(1) +59397.2868(8) +59413.3933(1) +59429.6636(1) +59445.6085(1) +59594.6448(1) +59611.0776(2) +59628.3240(20) +59397.4484(8) +59413.5568(1) +59429.8271(1) +59445.7719(1) +59594.8071(2) +59611.2406(1) +59628.4874(1) +59397.6121(5) +59413.7187(1) +59429.9891(1) +59445.9338(1) +59594.9701(1) +59611.4029(2) +59628.6495(2) +59397.7769(8) +59413.8823(1) +59430.1524(1) +59446.0972(1) +59595.1324(1) +59611.5659(1) +59628.8123(1) +59397.9374(1) +59414.0444(1) +59430.3143(1) +59446.2594(1) +59595.2957(1) +59611.7274(2) +59628.9748(3) +59398.0996(1) +59414.2076(1) +59430.4778(6) +59446.4218(3) +59595.4574(1) +59611.8914(1) +59629.1382(1) +59398.2627(1) +59414.3698(1) +59430.6397(1) +59579.8384(5) +59595.6210(1) +59612.0537(2) +59629.3015(7) +59398.4249(1) +59414.5330(1) +59430.8032(1) +59580.0012(5) +59595.7833(2) +59612.2168(1) +59629.6234(8) +59398.5883(1) +59414.6951(1) +59430.9652(1) +59580.1634(2) +59595.9464(1) +59612.3791(2) +59629.7886(1) +59398.7504(1) +59414.8587(2) +59431.1286(1) +59580.3274(1) +59596.1089(2) +59612.5422(1) +59630.1141(1) +59398.9137(1) +59415.0204(1) +59431.2905(1) +59580.4896(2) +59596.2717(1) +59612.7036(2) +59630.2754(2) +59399.0757(1) +59415.1837(1) +59431.4540(1) +59580.6528(1) +59596.4342(1) +59612.8676(1) +59630.4395(1) +59399.2389(1) +59415.5093(1) +59431.6159(1) +59580.8150(20) +59596.5971(1) +59613.0299(2) +59630.6018(2) +59399.4011(1) +59415.6712(1) +59431.7795(1) +59580.9779(1) +59596.7596(1) +59613.1929(1) +59630.7651(1) +59399.5645(1) +59415.8346(1) +59431.9413(1) +59581.1396(2) +59596.9225(1) +59613.3554(2) +59630.9274(2) +59399.7266(1) +59415.9966(1) +59432.1048(1) +59581.3035(1) +59597.0850(21) +59613.5184(1) +59631.0902(1) +59399.8894(1) +59416.1601(1) +59432.2669(1) +59581.4649(2) +59597.2479(1) +59613.6799(2) +59631.2528(3) +59400.0519(9) +59416.3219(1) +59432.4311(4) +59581.6291(1) +59597.4104(1) +59613.8436(8) +59631.4158(1) +59400.2153(1) +59416.4851(2) +59433.8933(5) +59581.7912(2) +59597.5735(1) +59614.0052(3) +59631.5770(29) +59400.3774(1) +59416.6475(1) +59434.0574(1) +59581.9541(1) +59597.7352(1) +59614.1695(1) +59631.7413(1) +Continued on next page + +18 +KUDAK ET AL. +Table 8 – continued from previous page +BJD +BJD +BJD +BJD +BJD +BJD +BJD +59400.5406(1) +59416.8109(1) +59434.2193(1) +59582.1166(2) +59597.8988(1) +59614.3307(3) +59631.9035(2) +59400.7028(1) +59416.9728(8) +59434.3823(2) +59582.2799(1) +59598.0612(2) +59614.4938(2) +59632.0666(1) +59401.0281(1) +59417.1365(2) +59434.5448(1) +59582.4412(2) +59598.2242(1) +59614.6582(4) +59632.2277(3) +59401.1915(1) +59417.2979(1) +59434.7082(1) +59582.6052(1) +59598.3865(1) +59615.1456(1) +59632.3921(1) +59401.3535(1) +59417.4616(1) +59434.8701(1) +59582.7674(2) +59598.5496(9) +59615.3078(2) +59632.5532(3) +59401.5168(1) +59417.6237(1) +59435.0337(1) +59582.9306(1) +59598.7114(1) +59615.4707(1) +59632.7176(1) +59401.6789(1) +59417.7871(1) +59435.1954(1) +59583.0928(2) +59598.8751(1) +59615.6333(2) +59632.8798(2) +59401.8423(1) +59417.9489(1) +59435.3589(1) +59583.2558(1) +59599.0375(2) +59615.7960(10) +59633.0428(1) +59402.0044(1) +59418.1126(2) +59435.5209(1) +59583.4182(2) +59599.2005(1) +59615.9575(2) +59633.2052(3) +59402.1675(1) +59418.2743(1) +59435.6843(1) +59583.5813(1) +59599.3628(1) +59616.1219(1) +59633.3679(1) +59402.3297(1) +59418.4379(1) +59435.8461(1) +59583.7427(2) +59599.5258(1) +59616.2839(2) +59633.5306(3) +59402.4929(1) +59418.5990(20) +59436.0097(1) +59583.9066(1) +59599.6876(2) +59616.4469(1) +59633.6938(1) +59402.6551(1) +59420.2257(4) +59436.1718(1) +59584.0691(2) +59599.8513(1) +59616.6093(2) +59633.8548(2) +59402.8185(1) +59420.3904(1) +59436.3351(1) +59584.2322(1) +59600.0136(1) +59616.7726(1) +59634.0191(1) +59402.9807(1) +59420.5522(1) +59436.4972(1) +59584.3944(2) +59600.1756(3) +59616.9348(2) +59634.1814(3) +59403.1442(2) +59420.7157(4) +59436.6606(1) +59584.5575(1) +59600.3387(4) +59617.0980(10) +59634.3442(1) +59403.3060(1) +59420.8776(1) +59436.8224(1) +59584.7199(2) +59600.5006(5) +59617.2602(2) +59634.5068(3) +59403.4694(1) +59421.0412(1) +59436.9859(1) +59584.8830(10) +59600.6648(5) +59617.4234(1) +59634.6695(1) +59403.6312(1) +59421.2028(1) +59437.1478(1) +59585.0452(2) +59600.8274(1) +59617.5856(2) +59634.8311(2) +59403.7945(1) +59421.3664(1) +59437.3113(1) +59585.2083(1) +59600.9891(2) +59617.7485(9) +59634.9950(10) +59403.9568(1) +59421.5285(1) +59437.4734(1) +59585.3706(2) +59601.1529(1) +59617.9110(20) +59635.1576(2) +59404.1201(1) +59421.6919(1) +59437.6367(1) +59585.5337(1) +59601.3146(2) +59618.0741(1) +59635.3204(1) +59404.2819(8) +59421.8538(1) +59437.7987(1) +59585.6960(21) +59601.4783(1) +59618.2355(2) +59635.4831(3) +59405.4201(5) +59422.0172(1) +59437.9621(1) +59585.8591(1) +59601.6406(2) +59618.3995(1) +59635.6457(1) +59405.5834(1) +59422.1790(1) +59438.1242(1) +59586.0213(2) +59601.8038(1) +59618.5617(2) +59635.8090(10) +59405.7470(1) +59422.3428(1) +59438.2875(1) +59586.1834(5) +59601.9651(2) +59618.7251(1) +59405.9091(1) +59422.5048(1) +59438.4496(1) +59586.3450(28) +59602.1290(1) +59618.8873(2) +59406.0724(6) +59422.6680(1) +59438.6129(1) +59586.5115(5) +59602.2913(1) +59619.0501(1) +59406.2347(1) +59422.8298(1) +59438.7748(1) +59586.6721(2) +59602.4545(9) +59619.2117(2) +59406.3979(1) +59422.9935(1) +59438.9384(1) +59586.8353(1) +59602.6169(2) +59619.3758(1) +TABLE 9: GSC 04364-0064 times of minima determined from TESS lightcurves. +Errors are in +parenthesis. +BJD +BJD +BJD +BJD +BJD +BJD +BJD +58842.925(2) +58848.963(1) +58851.983(4) +58858.454(2) +58864.494(2) +59014.199(3) +59019.375(4) +58843.356(4) +58849.391(4) +58851.985(3) +58858.886(3) +58864.926(3) +59014.632(1) +59019.809(2) +58843.786(1) +58849.394(4) +58852.414(1) +58859.749(4) +58865.789(3) +59014.634(1) +59020.241(3) +58845.512(2) +58849.826(2) +58852.415(2) +58860.609(4) +58867.085(2) +59015.498(2) +59020.672(1) +58845.513(1) +58849.827(1) +58852.844(4) +58861.473(4) +59011.179(2) +59015.923(4) +59020.674(2) +58845.937(4) +58850.259(4) +58853.709(4) +58862.338(4) +59012.045(2) +59016.789(4) +59021.534(2) +58845.945(4) +58850.687(2) +58854.140(8) +58862.767(2) +59012.476(4) +59017.648(4) +59026.712(1) +58846.374(2) +58850.688(2) +58856.727(2) +58862.768(2) +59012.906(2) +59018.085(2) +59029.729(4) +58846.376(2) +58851.119(4) +58857.158(4) +58863.199(4) +59012.907(2) +59018.086(2) +59031.888(2) +58846.804(4) +58851.552(1) +58857.593(2) +58863.634(2) +59013.335(4) +59018.514(3) +59033.614(2) +58848.101(2) +58851.553(2) +58858.022(4) +58864.065(4) +59013.773(2) +59018.946(2) +59034.481(2) +TABLE 10: Observed times of minima of selected EB systems. Errors are in parenthesis. +Name +BJD +Name +BJD +RU UMi +59277.4101(2) +VY UMi +59516.3846(2) +RU UMi +59467.4327(1) +VY UMi +59517.3604(1) +VY UMi +59298.3633(2) +VY UMi +59517.5219(1) +VY UMi +59298.5245(2) +GSC 04364-0064 +59343.3781(2) +VY UMi +59516.2200(3) +GSC 04364-0064 +59374.4401(1) + diff --git a/ydAyT4oBgHgl3EQfa_fL/content/tmp_files/2301.00255v1.pdf.txt b/ydAyT4oBgHgl3EQfa_fL/content/tmp_files/2301.00255v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..e6d2aea00fed6f66ef0f7fb6a93f5f0d93f80b81 --- /dev/null +++ b/ydAyT4oBgHgl3EQfa_fL/content/tmp_files/2301.00255v1.pdf.txt @@ -0,0 +1,1274 @@ +IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION - DO NOT DISTRIBUTE. DOI 10.1109/LRA.2022.3231831 +© 2022 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in +any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating +new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in +other works.” +IEEE +ROBOTICS +AND +AUTOMATION +LETTERS. +PREPRINT +VERSION +- +DO +NOT +DISTRIBUTE. +DOI +10.1109/LRA.2022.3231831 +arXiv:2301.00255v1 [cs.RO] 31 Dec 2022 + +IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION - DO NOT DISTRIBUTE. DOI 10.1109/LRA.2022.3231831 +IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION - DO NOT DISTRIBUTE. DOI 10.1109/LRA.2022.3231831 +Landing a UAV in harsh winds and turbulent open waters +Parakh M. Gupta, `Eric Pairet, Tiago Nascimento, and Martin Saska +Abstract— Landing an unmanned aerial vehicle (UAV) on top +of an unmanned surface vehicle (USV) in harsh open waters +is a challenging problem, owing to forces that can damage the +UAV due to a severe roll and/or pitch angle of the USV during +touchdown. To tackle this, we propose a novel model predictive +control (MPC) approach enabling a UAV to land autonomously +on a USV in these harsh conditions. The MPC employs a novel +objective function and an online decomposition of the oscillatory +motion of the vessel in order to predict, attempt, and accomplish +the landing during near-zero tilt of the landing platform. The +nonlinear prediction of the motion of the vessel is performed +using visual data from an onboard camera. Therefore, the +system does not require any communication with the USV +or a control station. The proposed method was analyzed in +numerous robotics simulations in harsh and extreme conditions +and further validated in various real-world scenarios. +Supplemental material - MRS Webpage +I. INTRODUCTION +Heterogeneous robot teams that are composed of UAVs +and USVs are aimed to provide higher efficiency and de- +crease the high risk posed to human life in marine appli- +cations. An example of such an application is the process +of cleaning oceans to rid them of oil spills and non- +biodegradable waste [1]. While the UAVs can act as the eyes +in the sky for surveying, identifying, and localizing the clean- +up targets, the USVs are much better suited to the actual +clean-up as this task requires heavy equipment and lifting +capabilities close to the water surface. +These clean-up missions can be performed autonomously +by UAVs and can be conducted several dozen kilometers +away from a harbor or shore. Although UAVs have short +battery lives to be able to fly long distances, their strength +lies in their agility and their ability to perform short-duration +hover missions [2]. We can compensate for this short battery +life by making a UAV and USV behave as a team, where-in +the UAV can charge quickly during the mission for rapid +redeployment. However, the precipitous and violent nature +of the sea poses daunting challenges for landing on the USV +deck, especially due to the precision required for recharging +operations. +When landing on a USV, the first challenge is estimating +and predicting the movement of the deck of the USV before +landing. A fast-moving deck can damage the UAV during +landing through high impulse impacts, while a tilted deck +can result in the UAV rolling or falling off the deck before +the landing is complete. Additionally, a tilted deck can +cause an erroneous response from the controller of the UAV +during landing, which would jeopardize the landing position +since multi-rotors are under-actuated vehicles with coupled +angular and linear acceleration vectors. The second challenge +Fig. 1. +UAV landing on USV in real-world experiments. +that we focus on is attempting a landing without active +communication between the UAV and the USV. Relying on a +required communication channel with a high frame rate and +low latency would introduce a significant source of failure in +real open-water applications. In order to increase reliability +and applicability, we attempt to build a decentralized solution +that does not rely on communication between the agents. +Thus, we aim to study various aspects of the dynamics of +UAVs and USVs to develop a framework for predicting and +landing on the USV with high precision and reliability in +demanding conditions including wind and waves, often seen +in harsh environments. Finally, in this work, we can define +harsh environments as those that contain open water with +waves with a height of up to 4 meters, and a wind velocity +of up to 12m/s, which corresponds to a Beaufort scale of 6. +For intended applications, this would produce a tilt in the +range of [-0.5,0.5] radians for the USV. +II. RELATED WORKS +Riola et al. [3] show that the behavior of a ship can be +predicted based on its past motion up to short prediction +horizons if corrected by measured ship motion. Unsurpris- +ingly, the topic of wave predictions is highly relevant to +the shipping industry as it is needed to prevent cables from +slacking while trying to offload cargo from ships using port- +side cranes. Both K¨uchler et al. [4] and Neupert et al. [5] +describe an active heave compensation for port-side cranes +using a periodic oscillation model that proves to be effective. +Building on a similar model, Marconi et al. [6] and Lee et +al. [7] present sophisticated approaches for fixed-wing UAVs +landing on vessels. Both works adapt the model using a +Kalman filter and use this heave motion of the ship to predict +the altitude of the landing pad. However, these works do not +focus on a rolling and pitching deck. Meng et al. [8] take +a different approach and use an auto-regressive-model on +the fixed-wing UAV to observe and predict the ship motion +by breaking it into sinusoidal components. In addition, Ngo + +IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION - DO NOT DISTRIBUTE. DOI 10.1109/LRA.2022.3231831 +and Sultan [9] predict the quiescent periods for landing a +helicopter on a ship based on the model of the vessel, but they +do not tackle the problem of predictions of the motion for +landing on an untilted deck. This leads to short opportunistic +windows that have to be adhered to, even if the conditions +change rapidly. All of the above-mentioned works present +results only in simulation environments that are not harsh or +extreme. +The research on solutions for multi-rotor aerial vehicles +landing on marine vessels is recent. One of the first works +by Polvara et al. [10] uses a fiducial marker located on the +platform and an extended Kalman filter (EKF) that estimates +the position of the USV. In contrast, the approach presented +by Abujoub et al. [11] relies on a LiDAR onboard the UAV +to find the pose of the landing pad in order to learn the +behavior of the platform by hovering above it. However, they +classify the window of landing into go or no-go intervals. +Both preliminary works were validated on simulations in +conditions that were not harsh. +More recently, researchers have begun testing their ap- +proaches through real-world experiments. For example, Xu +et al. [12] use a fiducial marker for a decentralized approach, +so as to follow the USV and use a PD controller for landing +once the USV is discovered. For the second challenge +of achieving decentralization, Lee et al. [13] present an +interesting solution to finding a ship and its pose using +classical vision algorithms. Zhanhg et al. [14] take a different +approach and present a learning-based linear controller that +receives inputs from a fiducial marker in order to land the +UAV on a USV that is subject to the waves of a lake. +Furthermore, some works also present the application of an +MPC controller that enables a flexible-blade helicopter to +land on a marine vessel [15], [16]. These works use a non- +linear MPC to achieve near-perfect performance but do so +using a numerical benchmark that doesn’t run in real-time or +in a real-world experiment. Our work differs from these by +using simplifications and a new approach that fills these gaps +of real-time computing and applicability without a significant +drop in landing performance. We use these for comparison in +our experimental section to demonstrate the same. The most +advanced research presented with real-world flight data is the +work by Pearsson et al. [17], which presents a linear MPC +for autonomous landing of a UAV on the deck of a moving +boat. +For the purpose of our work, we assume that the USV +can be found by the UAV by ascending to a given altitude +during the mission without the need for conducting a planned +search which is beyond the scope of this paper. We also +assume that the motion of the USV perpendicular to the water +surface is minimal (the USV is waiting for the UAV to land +while controlling its global positioning on the water in order +to remain stationary, rather than drifting with the waves). +Furthermore, we assume that the USV is under the influence +of waves, which results in periodic oscillations of the USV +deck in each axis of a coordinated system with an origin at +the USV center of mass. For hardware, we assume that the +UAV is equipped with a 2MP downward facing camera, an +onboard computer for image processing and computing the +MPC, and that the USV is equipped with a landing pattern +to recognize relative pose. +The main difference between our proposed approach and +[17] is that our controller uses the non-linear model of +the USV for landing on a rapidly tilting deck and does +not employ any communication between the UAV and the +USV, as motivated by real-world applicability. To the best +of the authors’ knowledge, it is the first approach using +USV motion prediction in control feedback of a decentralized +controller. In summary, our contributions are as follows: +• We present a novel objective function for finding an +optimum landing trajectory that utilizes an MPC algo- +rithm to predict the future of the UAV and USV, without +communication. +• We propose a decentralized vision-based method for +observing and predicting the motion of a USV through +the use of an online observer that adapts the USV +motion model using observations from a downward- +facing camera. +• Our proposed approach enables landing on a highly +undulating platform with no prior knowledge of the +dimensions of the USV. +• We propose a prediction algorithm that is designed to +prevent a velocity overshoot at the set point for landing +with minimal impulse transfer from the surface upon +touchdown. +III. PROPOSED NON-LINEAR ESTIMATOR-BASED MPC +In this section, we present our proposed approach which +consists of a UAV prediction model and a simplified USV +prediction model. Our proposed controller must satisfy two +hard constraints imposed by real-world conditions, which +are: 1) The controller must perform its computation under +a time constraint of 50 ms (20 Hz); and 2) There is no +communication between the UAV and the USV and so, the +only method for estimating the state of the USV motion +is by visual pose estimation enabled by the AprilTag on +the landing platform. Thus, for the sake of clarity, we will +call our approach MPC-NE (Model Predictive Controler - +Non-linear Estimator). Figure 2 presents the control pipeline +used in this work; the contribution is encapsulated in Fig- +ure 2. For the UAV prediction model, a discrete linear +time-invariant system is used, while the USV model uses +a more complex linearised model to be described subse- +quently. The 6-degrees of freedom (DOF) USV pose b = +�b1 +b2 +b3 +b4 +b5 +b6 +�T is estimated in the world frame +through the detection of the fiducial tag in the center of the +landing pad from the on-board camera of the UAV. The pose +of the UAV is fused and accounted for to estimate the correct +world frame pose of the USV. This pose information is fed +to a fast Fourier transform (FFT) node (based on [18]) which +identifies the frequencies, amplitudes, and phases of the N +periodic oscillations that make up the USV motion in pitch +and roll axes. These identified modes are used to initialize a +linear Kalman observer node that corrects the observed state +and predicts future motion. These predictions are sent to the + +IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION - DO NOT DISTRIBUTE. DOI 10.1109/LRA.2022.3231831 +MPC Solver +Fast Fourrier +Transform +KF wave +Prediction +Setpoint +Generator +UAV Model +Reference +Tracker +Position/Attitude +Controller +Vision-based +Detector +Attitude rate +Controller +IMU +UAV +Actuators +Onboard +Sensors +State +Estimator +Odometry & +Localisation +ˆx +rd, ηd +FFT +accuracy +fj,i, +Aj,i, +φj,i +[b4,n, b5,n] +n = 1..Mp +˙rd, ˙ηd +ˆ˙¨rd, ˆ˙¨ηd +χd +100 Hz +ωd +Td +100 Hz +ad +τd +≈1 kHz +x +100 Hz +initialisation +only +x, R, ω +100 Hz +R, ω +b +UAV plant +Pixhawk autopilot +MPC-NE Architecture +USV Prediction Model +UAV Prediction Model +Fig. 2. +System architecture: As a primary contribution of this work, the MPC landing controller (yellow block) is integrated into the MRS system +(blocks in grey) and supplies the desired reference (velocity ˙rd = +� ˙x +˙y +˙z�T and heading rate ˙ηd). Within the MRS system, the first layer containing +a Reference tracker processes the desired reference and gives a full-state reference χ to the position/attitude controller. The feedback Position/Attitude +controller produces the desired thrust and angular velocities (Td, ωd) for the Pixhawk embedded flight controller (Attitude rate controller). The State +estimator fuses data from Onboard sensors and Odometry & localization methods to create an estimate of the UAV translation and rotation (x, R). Finally, +the Vision-based Detector obtains the visual data from the camera and sends the pose information b of the USV to the MPC. +MPC controller, which uses them to estimate the feasibility +of landing in the near future, i.e., if a sufficiently low tilt +of the USV can be found inside the predefined prediction +horizon. In turn, it generates the desired linear velocities for +x, y, and z axes, as well as the desired angular velocity in +heading η. The MPC also receives the estimated UAV state +vector x = +�x +˙x +¨x +y +˙y +¨y +z +˙z +¨z +η +˙η +¨η�T +using onboard state estimation proposed by our team in [19]. +A finite-state automaton-based approach is used to direct +our mission. Based on this, a setpoint generator node com- +mands the aircraft to increase its altitude until the vision +marker can be found. Once it is found, the reference of +the MPC is changed by the setpoint generator, such that it +can hover at a preset altitude above the identified marker. +Subsequently, the UAV waits for enough data to be gathered +so that the FFT accuracy threshold requirement can be met. +Once it is met, the setpoint generator sets the global reference +for landing. Then, the MPC begins to use the future motion +predictions of the wave to determine a suitable time for +landing. +A. USV Prediction Model +USV models can be classified into two different types: +Maneuvering Theory and Seakeeping Theory [20]. Owing +to the assumptions made in Section II, we choose to focus +on the Seakeeping theory since it concerns near-stationary +vessels. In addition, the use of a decentralized approach +brings challenges in estimating the true odometry of the +USV, as converging to reliable estimates of linear and +angular velocities of the USV is infeasible. Therefore, we +leverage the pose estimate from the camera efficiently by +focusing only on the kinematics of the USV. +Our USV prediction model is composed of three parts: +a fast Fourier transform, a Kalman observer, and a wave +prediction model. First, the FFT performs a decomposition +on the pose data obtained from the vision pipeline. The +identified modes of these oscillations are used to initialize +a Kalman observer that adapts the amplitude and the phase +of the wave using the observed values online. Finally, the +amplitudes and phases from the Kalman observer are sent to +the wave prediction model to enable future wave predictions. +1) FFT-based Modelling: We assume that the motion of +the USV is composed of Nj periodic waves and a non- +periodic term that accounts for random noise in tracking the +various components for each jth axis. Thus, let the state vec- +tor b be represented by the linear pose bj for j ∈ {1, 2, 3} for +x, y, z axes, respectively, and the angular pose be represented +by j ∈ {4, 5, 6} about these axes in the same order. Note +here that, a sufficiently large ship/boat (intended application) +would exhibit sufficiently low amplitude oscillations in Z- +axis such that they can be handled by changing the reference +at every camera frame (as shown here). Thus, the periodic +motion of the USV in an axis can be represented as a function +of time such that: +bj(t) = bj,off + +Nj +� +i=1 +Aj,i sin (2πfj,it + φj,i) +� +�� +� +Φj,i +, +(1) +with fj,i denoting the frequency, Aj,i the amplitude, and +φj,i the phase. Additionally, bj,off is the non-periodic term +accounting for random noise. For the initial condition, Φj,i(t) +is equal to Φj,i(tF F T ), which is the phase obtained as +the output of FFT at the time of identification tF F T . In +sea conditions, these frequency components can change +frequently due to changing winds. Therefore, the pose is +sampled continuously and an FFT is run every ∆TF F T +seconds. For each axis, we discard the modes that are below + +IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION - DO NOT DISTRIBUTE. DOI 10.1109/LRA.2022.3231831 +a certain threshold amplitude Aj,threshold, where +Aj,threshold = Agate · max{Aj,0, Aj,1, . . . , Aj,Nj}. +(2) +For reliable performance, and upon tunning on real-world +data, we assume Agate(= 0.02) to be a suitable cutoff. +This prevents us from identifying noise components as low- +amplitude periodic oscillations without losing more than 2% +of the accuracy. These erroneous components cause a loss +of performance in the Kalman observer, which is explained +in the next section. +2) Kalman Observer: The Kalman observer uses a linear +model to refine the estimate of identified amplitude and phase +of each mode. The observer is necessary because, while the +FFT accurately identifies the frequency components, the am- +plitude and phase outputs are averages for the entire ∆TF F T +sampling interval. Therefore, the observer receives new pa- +rameters for all identified modes every ∆TF F T seconds. In +order to allow sufficient time for the observer parameters +to converge to true values, we do not reinitialize the pre- +identified modes with the new parameters. Instead, only the +newly identified modes are initialized, while discarding the +old modes that no longer exist. +To assemble the model, we first write the ordinary differ- +ential equation (ODE) for each mode for a given axis at any +time t. We use vj,i to denote the ith mode of the USV state +vector component bj in the jth axis. Thus, the derivative of +the mode (∀j, j ∈ 1 . . . 6) is +˙vj,i = +� +0 +1 +−(2πfj,i)2 +0 +� +� +�� +� +B(tF F T ) +vj,i, +(3) +and the mode at time t is +vj,i = +� +Aj,i sin (Φj,i(t)) +2πAj,ifj,i cos (Φj,i(t)) +� +. +(4) +Next, we derive the observer model by adding the ODEs +of each mode. Thus, +˙vj(t) = +� +����� +Bj,1 +0 +. . . +0 +0 +0 +Bj,2 +. . . +0 +0 +... +... +... +... +... +0 +0 +. . . +Bj,N +0 +0 +0 +· · · +0 +0 +� +����� +� +�� +� +Bj +� +����� +vj,1 +vj,2 +... +vj,Nj +vj,off +� +����� +� +�� +� +vj(t) +. +(5) +Hence, the output for each axis is +bj(t) = +�Cj,1 +Cj,2 +· · · +Cj,N +Cj,off +� +� +�� +� +Cj +vj(t). +(6) +Note, that each component of the output vector of the mode +can be found as +bj,i = +�1 +0� +� �� � +Cj,i +vj,i. +(7) +Now, for the brevity of explanation and the readability of the +equations, we write the following relation for only one axial +DOF of the USV in discrete-time. In addition, we clarify that +it can be applied to all 6 of the DOF. Furthermore, notice that +a time instance t = k∆T +tF F T , wherein ∆T is the discrete +sampling time for new pose observations. Thus, we have a +straightforward change in notation such that, for example, +vj(t) ≡ v(k) +j +. Thus, by using the integral approximation +method, we have that +v(k+1) +j += exp(Bj∆T) +� +�� +� +Ψj +v(k) +j +, and +b(k) +j += Cj v(k) +j +. +(8) +Then, we continuously estimate the amplitude Aj,i and phase +φj,i of each mode every ∆T using the Kalman Filter. First, +Q is initialised using a diagonal matrix QI = λI, such that +Q = 1 +2(ΨQIΨT + QI)∆T, +(9) +where λ is the gain parameter for the process noise observed +in the model. +Meanwhile, the observation noise matrix R is set to +the mean amplitude of the observed noise in the system. +Thereafter, we apply the filter equations as follows: +ˆv(k) +j += Ψjv(k−1) +j +, +ˆP(k) = ΨjP(k−1)ΨT +j + Q, +ˆb(k) +j += Cj ˆv(k) +j +, +L(k) = ˆP(k)C +T +j (Cj ˆP(k)C +T +j + R)−1, +v(k) +j += ˆv(k) +j ++ L(k)(bj,m − ˆb(k) +j +), +P(k) = (I − L(k)Cj)ˆP(k), +(10) +where ˆ shows the predicted value of the vector/matrix, I +is an identity matrix, bj,m is the measured value of bj, +L ∈ R2(N+1) is the Kalman gain matrix of the system, +P and Q ∈ R2(N+1)×2(N+1) are the process co-variance +and system noise matrices, respectively, and R ∈ R is +observation noise. +At every identification, the relevant elements correspond- +ing to both of the modes that are no longer present, as +well as the newly identified modes of the Ψ matrix, are +re-initialized. The corresponding co-variance terms for these +modes are reset to maintain a consistent prediction without +large deviations. +3) Wave prediction: Let us now define tobs as the time +instant where the last observation was performed, since the +prediction algorithm is not run when there are no new obser- +vations. Thus, by running the Kalman observer at tobs we find +the new amplitude Aj,i(tobs) and phase Φj,i(tobs). At the +same instant in time tobs, we can extract the corresponding +vj,i and use (4) to acquire: +Φj,i(tobs) = arctan +�2πfj,i[vj,i]1,1 +[vj,i]2,1 +� +, +and +Aj,i(tobs) = +[vj,i]1,1 +sin (Φj,i(tobs)), +(11) +where [vj,i]m,n represents the element corresponding to the +mth row and nth column of the vector. +This enables us to predict the wave behavior at a future +time t > tobs as +bj(t) = +Nj +� +i=1 +Aj,i(tobs) sin [2πfj,i(t − tobs) + Φj,i(tobs)] ++ [vj,off]1,1. +(12) + +IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION - DO NOT DISTRIBUTE. DOI 10.1109/LRA.2022.3231831 +B. UAV Prediction Model +The UAV prediction model used in the proposed MPC is +based on the Euler approximation of a set of single particle +kinematics equations. Here, we employ the following +discrete linear time-invariant system: +x(k+1) = Dx(k) + Eu(k), with +u(k) = +� ˙¨x +˙¨y +˙¨z +˙¨η +�T . +(13) +In the model represented by (13), the state matrix D and +the input matrix E can be found through the Kronecker +product (⊗), such that: +D +12×12 = I +4×4 ⊗ D′ +3×3, with +D′ = +� +�1 +∆tpred +∆t2 +pred +2 +0 +1 +∆tpred +0 +0 +1 +� +� , (14) +E +12×4 = I +4×4 ⊗ E′ +3×1, with +E′ = +� +�� +∆t3 +pred +6 +∆t2 +pred +2 +∆tpred +� +�� , +(15) +where I is an identity matrix, with a prediction made every +∆tpred = 0.01 seconds. +Hence, the state vector represents the states of the system +and their derivatives up to acceleration in each axis, and the +control input is the jerk experienced in those axes. +C. MPC Objective Function +Once we have defined a prediction model of the UAV +and the USV, we can formulate an objective function to +enable both waypoint navigation and landing. For the sake +of simplification, we will omit the superscript (.)(k), which +represents a discrete instant in time. Therefore, we can define +the objective function J as: +min +u1,...,uMc +J(x, u) = +Mp +� +m=1 +˜xT +mS˜xm + hT +mThm +� +�� +� +J1 ++ +Mp +� +m=1 +αL × g(˜zm, b4,m, b5,m) +� +�� +� +J2 +, +subject to : +˜xm = xm − +∗xm, +˜zm = zm − +∗zm, +hm = um − um−1, +xm+1 = Dxm + Eum ∀ m ≤ Mc, +xm+1 = Dxm + EuMc ∀ m > Mc, +umin ≤ um ≤ umax, +x0 = xinitial, +u0 = uinitial, +∀ {m : m ∈ N, 1 ≤ m ≤ Mp}, +(16) +where +∗xm is the desired state, ˜xm is the error vector, ˜zm is +the error in zm position, hm is the rate of control input +change to ensure smooth input to the UAV, Mp(= 100) +is the prediction horizon, and Mc(= 40) is the control +horizon. S and T are the corresponding penalty matrices with +configurable weights for performance tuning, while αL(= +1200) is a weight chosen for the tuning of the objective +function g(.). Additionally, b4,m, b5,m are the pitch and roll +angles in discrete time of the USV about its x and y axes, +according to (12). +We emphasize that +∗xm (including +∗z) can either be a series +of points (trajectory) or a single point (step input). This +would enable the UAV to keep up with a drifting USV if +the XY-position state of the usv is estimated independently. +However, a slowly drifting USV is within the dynamic +limits of the UAV so as to be compensated by the single- +point reference that can be updated after every observation +(depending on the camera frame rate). We demonstrate and +test this in this linked video. While we do not constrain +the output of the MPC, we apply soft constraints to the +velocity and acceleration states of the model, such that v ≥ +vmax and a ≥ amax incur a high penalty in the objective +function. +Herein, we have introduced a novel objective function J2 +(described in the next section) which can account for the +predicted motion of the USV, producing a smooth control +input to change the altitude of the UAV without any abrupt +maneuvers. Using this function, we are able to incorporate +the finite state automaton approach using a sigmoid activation +function without explicitly describing the possible landing +condition. The UAV is able to follow the descend trajectory +generated by the MPC by autonomously adjusting its hover +distance above the USV. Additionally, it enables us to tune +the parameters to control the variance of the resulting landing +angles about the mean value of zero-tilt. It is important to +mention that the term J1 in our cost function, is a classical +quadratic objective function largely used in robotics and +well documented for its feasibility and stability. On the +other hand, the J2 term is different from usual works in +the literature because it tackles the terminal cost of the +optimization step as a potential barrier. +We employ a non-linear optimization library (NLOPT +[21] [22]) which provides the near-optimum solution for +the objective function. In order to exercise velocity-based +control, the first input from the series of optimum control +inputs calculated by the solver is then used to calculate the +next state using (13). The velocities for this predicted next +state are then passed to the system as the velocity references +for the UAV to track (as seen in Figure 2). +Since the term J1 primarily contributes to the position +control and J2 contributes to the landing approach, the term +J2 remains disabled until the conditions for the landing +approach are satisfied. +D. Landing approach +We define the function for landing cost as a combination +of sigmoids, such that: +g(˜zm, b4,m, b5,m) = f(˜zm) · ((b4,m)2 + (b5,m)2), +(17) +where f(˜zm) is such that + +IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION - DO NOT DISTRIBUTE. DOI 10.1109/LRA.2022.3231831 +0 +1 +2 +3 +4 +5 +ez +0 +100 +200 +300 +400 +500 +600 +Cost +J1 +J2 +J(t = 0.07) +J(t = 0.15) +J(t = 0.23) +J(t = 0.5) +zero tilt +waiting region +optimum landing +Fig. 3. An example illustration of the effective cost function values obtained +during the landing approach. +f(˜zm) = +� +� +� +� +� +� +� +� +1.0 + exp +� +− ˜zm − hd +−0.15 +��−1 +, +if ˜zm ≥ 0.16 +� +1.0 + exp +� ˜zm − 0.1 +−0.01 +��−1 +, +otherwise, +(18) +where hd controls the waiting region (see Figure 3) during +a landing attempt. Empirically, hd = 1.1 was chosen for our +experiments. For the scope of this paper, we assume that the +USV has relatively negligible motion in its x and y axes, +which is a fair assumption for the problem of landing. The +propulsion of the USV may easily compensate for the drift +generated by the water currents in order to facilitate landing. +It is also safe to assume that ˜z ≥ 0, as the UAV cannot +approach from beneath the USV. In order to activate J2 to +start the landing phase, two conditions must be met. First, +FFT accuracy is higher than a given threshold to detect +slow oscillations. Second, The position errors in x and +y are below a predefined threshold (i.e., ˜x, ˜y ≈ 0) and +horizontal velocities vx, vy are also minimal. +To demonstrate the interaction of J2 with J1 during the +landing approach, we present a highly-simplified plot of the +objective function (see Figure 3) using one mode each for +pitch and roll axes. When J2 is activated, we acquire a +combined plot governed by both the equation (17) and the +residual error ˜z in J1. In Figure 3, the value of the objective +function encounters a peak that continuously evolves as a +function of time. This peak acts as a potential barrier. The +higher cost associated with the peak holds the aircraft in the +waiting region (as marked in the plot). Meanwhile the USV +model generates predictions for the future of the USV motion +during every iteration of the MPC. The USV sometimes +gets close enough to a zero tilt wherein a feasible solution +appears, as shown by the zero-tilt points in the plot. The UAV +is then able to insert itself into the time-varying trajectory +of these special feasible points by reducing its altitude and +approaching in such a way that the cost continues to decrease +along the locus of these points. Therefore, the UAV is able to +follow the zero-tilt points and finish at the optimum landing +point, where touchdown is confirmed by the system based +on thrust and other information from onboard sensors. +IV. SIMULATION EXPERIMENTS +We demonstrate our simulation results in two scenarios: +with a numerical simulation, and with a realistic ROS- +based Gazebo simulator [19]. The SHMPC presented in +0 +5 +10 +15 +20 +25 +30 +35 +40 +45 +50 +55 +60 +65 +Time(s) +−0.8 +−0.6 +−0.4 +−0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +Pitch(rad) +predicted_future +observed_value +220 +225 +230 +235 +240 +245 +250 +255 +260 +265 +270 +275 +Time(s) +−0.8 +−0.6 +−0.4 +−0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +Pitch(rad) +predicted_future +observed_value +Fig. 4. +Comparison between the predictions made by the system using the +onboard IMU data (left) and using the vision data (right). +[15] is shown to work numerically. Thus, we use a similar +numerical implementation of our work (MPC-NE) to allow +us to perform a fair comparison with the state-of-the-art. +In this comparison, the non-linear optimization problem is +solved by [22] for a landing maneuver of 3 meters and +assumes true knowledge of the future motion of the USV. +The second comparison is performed using real-time flight +with our proposed MPC-NE inside the Gazebo simulator. +For this comparison, we use a standard MPC [19] designed +for waypoint navigation. For this standard approach, the UAV +attempts to locate the target, and lands after a programmed, +uniformly randomly distributed delay between 0 and 100 +seconds. We select this duration owing to the periodicity of +the tilt angle of the USV. We use a T650 quadrotor frame +weighing 3.6 kg carrying a Garmin LiDAR for laser-ranging +of altitude and an Intel Realsense D435 camera for live in- +simulation video. The video output of the Realsense camera +is sent to our system to enable processing on the vision +node. (Sec. III). The 3D model of the USV is similar to +our real-world experiments and is affixed with an AprilTag +[23] marker for pose estimation. We note that in both the +comparisons, we push the boundary of performance and test +our work in rough sea states, and drive our USV model +using a wave generator with 4-5 components of oscillation +in both pitch and roll axes, and tilt angles up to 30◦(0.5 +radians). The frequency components are set such that brief +windows for feasible landing exist. Wind disturbances are not +considered in this environment since it is tackled by body +disturbances estimated by the low-level control feedback +pipeline (see Figure 2). Finally, several experimental runs +are conducted and aggregated results are presented for these +two comparisons using Figure 5. +A. Prediction results +First, we demonstrate the ability of our system to predict +the wave motion up to 1 second into the future based on +the observed frequency components and our model of the +system. The performance of the system is tested in two +scenarios: a 100 Hz odometry output from the simulated +USV IMU (used as ground truth), and a 30 Hz stream from +the AprilTag. +As we see in Figure 4, the high-frequency IMU-based +predictions are able to match the observed wave reliably +without introducing noise. The observer is able to adapt +the observed frequency, amplitude, and phase of the modes +of the oscillations and converge reliably. As opposed to + +IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION - DO NOT DISTRIBUTE. DOI 10.1109/LRA.2022.3231831 +0 ∘ - 5 ∘ +5 ∘ - 10 ∘ +10 ∘ - 15 ∘ +15 ∘ - 20 ∘ +20 ∘ - 25 ∘ +25 ∘ - 30 ∘ +30 ∘ - 35 ∘ +Relative tilt angle upon landing ( ∘ ) +0 +10 +20 +30 +40 +50 +Percentage of the landings (%) +SHMPC (Numerical) +MPC-NE (Numerical) +MPC-NE (Realistic Sim) +Standard MPC (Realistic Sim) +Fig. 5. +Histogram comparison between the proposed approach and the +standard approach during the touchdown of the UAV on the USV deck. +that, we see slight deviations in the vision-based predictions +compared to the IMU results. This deviation in performance +can be explained by two factors. First, the linearisation +of the model in the time-domain causes inaccuracies that +grow as the sampling time increases. Due to this, the three- +fold sampling rate of the IMU leads to faster and more +accurate convergence. Second, the output rate of the AprilTag +identification node fluctuates around 30 Hz, depending on +the computational load of the onboard computer of the UAV +(or simulation computer, in this case). This leads to the +misidentification of modes, as the FFT algorithm requires a +fixed sample rate for observations. However, this sufficiently +proves the ability of the proposed system to reliably predict +wave behavior, which will be used for the USV landing +further down the pipeline. +B. Landing results +To continue, we present the ability of our system to land +on a platform while tilt angles are sufficiently close to zero. +Thus, we present Figure 5 that shows the results of the +numerical comparison between our MPC-NE and the state- +of-the-art SHMPC. Note here that the MPC-NE lands ≈ 94% +within 10◦ of tilt, while the SHMPC lands ≈ 71% of the +same tilt interval. In this same comparison, the solution time +per iteration of our MPC-NE was 9 times lower at 102 ms +compared to 917 ms for SH-MPC. +Furthermore, the same figure presents the results of the re- +alistic simulations using Gazebo. It is important to highlight +the difference between a numerical simulation result and a +realistic simulation result. This is explained by the existing +constraints of processing time that demand the algorithms +to be processed in real time. Note here that the MPC-NE is +able to conduct 72% of its landings within 15◦(0.26 rad) +of tilt compared to 23% of landings using the standard +MPC approach. In addition, the proposed approach reduces +the 80th percentile result by 9◦(0.16 rad) in comparison to +the standard approach. For this comparison, we classify a +landing conducted at a tilt angle of less than 20◦(0.35 rad) +as successful. Therefore, even in challenging tilts of up to +0.5 radians, the proposed approach had only three failures, +while the standard approach fails approximately 50% of +the landings. Finally, we also highlight that, even in an +unrealistic and challenging scenario, our system is able to +conduct 70% of the landings within 50 seconds of reaching +its FFT accuracy threshold. +V. REAL-WORLD EXPERIMENTS +To test the contributions and proposed algorithms in the +real world, we performed landings on an oscillating target at +an open water reservoir. For the purpose of this experiment, +we employed a 4.5 kg T650 quadrotor equipped with vertical +pontoons [24] for safety over water (see Figure 1). In addi- +tion, the sensor stack included a Garmin LiDAR for laser- +ranging of altitude, a Basler camera for the live video feed, +and an Intel NUC for onboard real-time processing of the +algorithms, data, and video. The target is a special custom- +made USV [25] equipped with a 2m × 2m landing zone, +affixed with an AprilTag [23] for 6-DOF pose-estimation. +The experimental conditions subjected the UAV to a wind +of 7m/s and and a USV oscillating with an amplitude of 0.3 +radians. +A. Prediction results +Here we demonstrate our prediction pipeline in two sce- +narios: a 30 Hz stream from AprilTag, and a 100 Hz stream +from the IMU. The prediction results for the real-world +experiments are presented in Figure 6 and discussed below. +For predictions based on vision-based pose estimation, as +seen in Figure 6(a), the near-term future correlates well +with the observed motion. However, Figure 6(b) indicates +that the predictions for the long-term future can suffer in +accuracy. This correlates well with the simulation results +as shown in Figure 4 and can be attributed to the higher +sampling time-step and its higher variability. Occasionally, +it also exhibits convergence and consecutive divergence as +more data is fed into the pipeline. For ground truth, we use +Figure 6(c) to demonstrate the effectiveness of the pipeline +in robustly predicting the future of the USV. However, +since MPC exhibits a higher reliance on the predictions that +are temporally proximal, the predictions for 0.25 and 0.50 +seconds into the future offer robust support for preventing a +landing during an infeasible window. The chosen angle for +landing is also sufficiently low in order to demonstrate the +prediction capabilities and the selection of a feasible landing +window. +B. Landing results +We demonstrate the real-world landing process through +Figure 7. In these experiments, the UAV was able to land +within 50 seconds of acquiring the required FFT accuracy. +This coincides with our findings in simulation experiments. +Additionally, the tilt angles upon touchdown were less than +5◦(0.09 rad). +VI. CONCLUSION +In this paper, we proposed an MPC that enables a UAV +to land autonomously on a tilting USV. The MPC employs +a novel objective function and an online decomposition of +the motion of the vessel in order to attempt and complete +the landing during a near-zero tilt of the landing platform. +We successfully demonstrated that we are able to model and +predict the behavior of the UAV and USV without active +communication between them. Further, we establish a novel + +IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION - DO NOT DISTRIBUTE. DOI 10.1109/LRA.2022.3231831 +80 +85 +90 +95 +100 +105 +110 +115 +120 +time - secs +−0.4 +−0.2 +0.0 +0.2 +0.4 +roll tilt - radians +future_0.25 secs +landing +observed +(a) Vision—0.25 s into the future. +80 +85 +90 +95 +100 +105 +110 +115 +120 +time - secs +−0.4 +−0.2 +0.0 +0.2 +0.4 +roll tilt - radians +future_1.0 secs +observed +landing +(b) Vision—1.0 s into the future. +100 +110 +120 +130 +140 +time - secs +−0.4 +−0.2 +0.0 +0.2 +roll tilt - radians +future_1.0 secs +observed +landing time +(c) IMU—1.0 s into the future. +Fig. 6. +Comparison between the predictions made using vision (a-b) and using the onboard IMU of the USV (c). +0 +20 +40 +60 +80 +100 +120 +140 +Time(s) +0 +2 +4 +6 +8 +Position(m) +z-position +Finding tag +Accuracy threshold wait +Attempting landing +Landed +−1.5 +−1.0 +−0.5 +0.0 +0.5 +1.0 +1.5 +Radians +roll angle +Fig. 7. +Plot of a selected real-world open-water experiment (video). +approach for landing on the USV using these predictions, +which autonomously adjusts the relative altitude for the UAV +to ensure that the landing occurs as close to the zero-tilt +state of the landing deck as possible, increasing safeness +of the landing phase and reducing impact forces on the +landing UAV. In comparison to state-of-the-art approaches, +we achieved significant improvement in the case of landing +in demanding conditions with high waves and high winds +without knowing the dimensions of the USV. +REFERENCES +[1] L. C. Lebreton, J. Van Der Zwet, J.-W. Damsteeg, B. Slat, A. Andrady, +and J. Reisser, “River plastic emissions to the world’s oceans,” Nature +communications, vol. 8, no. 1, pp. 1–10, 2017. +[2] T. Nascimento and M. Saska, “Position and attitude control of multi- +rotor aerial vehicles: A survey,” Annu. Rev. Control, vol. 48, pp. 129– +146, 2019. +[3] J. M. Riola, J. J. Diaz, and J. M. Giron-Sierra, “The prediction of +calm opportunities for landing on a ship: Aspects of the problem,” in +OCEANS 2011 IEEE - Spain, 2011, pp. 1–8. +[4] S. K¨uchler, T. Mahl, J. Neupert, K. Schneider, and O. Sawodny, +“Active control for an offshore crane using prediction of the vessels +motion,” IEEE-ASME T MECH, vol. 16, pp. 297–309, 2011. +[5] J. Neupert, T. Mahl, B. Haessig, O. 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De Masi, “Nukhada usv: a robot for autonomous +surveying and support to underwater operations,” in OCEANS 2022 - +Chennai, 2022, pp. 1–6. + diff --git a/ydAyT4oBgHgl3EQfa_fL/content/tmp_files/load_file.txt b/ydAyT4oBgHgl3EQfa_fL/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..31e2d9d9596236b8bc8ac3c7c3fd0d6f4f342668 --- /dev/null +++ b/ydAyT4oBgHgl3EQfa_fL/content/tmp_files/load_file.txt @@ -0,0 +1,696 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf,len=695 +page_content='IEEE ROBOTICS AND AUTOMATION LETTERS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' PREPRINT VERSION - DO NOT DISTRIBUTE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' DOI 10.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='RO] 31 Dec 2022 IEEE ROBOTICS AND AUTOMATION LETTERS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' PREPRINT VERSION - DO NOT DISTRIBUTE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='1109/LRA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='3231831 IEEE ROBOTICS AND AUTOMATION LETTERS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' PREPRINT VERSION - DO NOT DISTRIBUTE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='1109/LRA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='3231831 Landing a UAV in harsh winds and turbulent open waters Parakh M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Gupta, `Eric Pairet, Tiago Nascimento, and Martin Saska Abstract— Landing an unmanned aerial vehicle (UAV) on top of an unmanned surface vehicle (USV) in harsh open waters is a challenging problem, owing to forces that can damage the UAV due to a severe roll and/or pitch angle of the USV during touchdown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' To tackle this, we propose a novel model predictive control (MPC) approach enabling a UAV to land autonomously on a USV in these harsh conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' The MPC employs a novel objective function and an online decomposition of the oscillatory motion of the vessel in order to predict, attempt, and accomplish the landing during near-zero tilt of the landing platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' The nonlinear prediction of the motion of the vessel is performed using visual data from an onboard camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Therefore, the system does not require any communication with the USV or a control station.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' The proposed method was analyzed in numerous robotics simulations in harsh and extreme conditions and further validated in various real-world scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Supplemental material - MRS Webpage I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' INTRODUCTION Heterogeneous robot teams that are composed of UAVs and USVs are aimed to provide higher efficiency and de- crease the high risk posed to human life in marine appli- cations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' An example of such an application is the process of cleaning oceans to rid them of oil spills and non- biodegradable waste [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' While the UAVs can act as the eyes in the sky for surveying, identifying, and localizing the clean- up targets, the USVs are much better suited to the actual clean-up as this task requires heavy equipment and lifting capabilities close to the water surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' These clean-up missions can be performed autonomously by UAVs and can be conducted several dozen kilometers away from a harbor or shore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Although UAVs have short battery lives to be able to fly long distances, their strength lies in their agility and their ability to perform short-duration hover missions [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' We can compensate for this short battery life by making a UAV and USV behave as a team, where-in the UAV can charge quickly during the mission for rapid redeployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' However, the precipitous and violent nature of the sea poses daunting challenges for landing on the USV deck, especially due to the precision required for recharging operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' When landing on a USV, the first challenge is estimating and predicting the movement of the deck of the USV before landing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' A fast-moving deck can damage the UAV during landing through high impulse impacts, while a tilted deck can result in the UAV rolling or falling off the deck before the landing is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Additionally, a tilted deck can cause an erroneous response from the controller of the UAV during landing, which would jeopardize the landing position since multi-rotors are under-actuated vehicles with coupled angular and linear acceleration vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' The second challenge Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' UAV landing on USV in real-world experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' that we focus on is attempting a landing without active communication between the UAV and the USV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Relying on a required communication channel with a high frame rate and low latency would introduce a significant source of failure in real open-water applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' In order to increase reliability and applicability, we attempt to build a decentralized solution that does not rely on communication between the agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Thus, we aim to study various aspects of the dynamics of UAVs and USVs to develop a framework for predicting and landing on the USV with high precision and reliability in demanding conditions including wind and waves, often seen in harsh environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Finally, in this work, we can define harsh environments as those that contain open water with waves with a height of up to 4 meters, and a wind velocity of up to 12m/s, which corresponds to a Beaufort scale of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' For intended applications, this would produce a tilt in the range of [-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='5,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='5] radians for the USV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' RELATED WORKS Riola et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' [3] show that the behavior of a ship can be predicted based on its past motion up to short prediction horizons if corrected by measured ship motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Unsurpris- ingly, the topic of wave predictions is highly relevant to the shipping industry as it is needed to prevent cables from slacking while trying to offload cargo from ships using port- side cranes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Both K¨uchler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' [4] and Neupert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' [5] describe an active heave compensation for port-side cranes using a periodic oscillation model that proves to be effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Building on a similar model, Marconi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' [6] and Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' [7] present sophisticated approaches for fixed-wing UAVs landing on vessels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Both works adapt the model using a Kalman filter and use this heave motion of the ship to predict the altitude of the landing pad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' However, these works do not focus on a rolling and pitching deck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Meng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' [8] take a different approach and use an auto-regressive-model on the fixed-wing UAV to observe and predict the ship motion by breaking it into sinusoidal components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' In addition, Ngo IEEE ROBOTICS AND AUTOMATION LETTERS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' PREPRINT VERSION - DO NOT DISTRIBUTE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='1109/LRA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='3231831 and Sultan [9] predict the quiescent periods for landing a helicopter on a ship based on the model of the vessel, but they do not tackle the problem of predictions of the motion for landing on an untilted deck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' This leads to short opportunistic windows that have to be adhered to, even if the conditions change rapidly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' All of the above-mentioned works present results only in simulation environments that are not harsh or extreme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' The research on solutions for multi-rotor aerial vehicles landing on marine vessels is recent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' One of the first works by Polvara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' [10] uses a fiducial marker located on the platform and an extended Kalman filter (EKF) that estimates the position of the USV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' In contrast, the approach presented by Abujoub et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' [11] relies on a LiDAR onboard the UAV to find the pose of the landing pad in order to learn the behavior of the platform by hovering above it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' However, they classify the window of landing into go or no-go intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Both preliminary works were validated on simulations in conditions that were not harsh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' More recently, researchers have begun testing their ap- proaches through real-world experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' For example, Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' [12] use a fiducial marker for a decentralized approach, so as to follow the USV and use a PD controller for landing once the USV is discovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' For the second challenge of achieving decentralization, Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' [13] present an interesting solution to finding a ship and its pose using classical vision algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Zhanhg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' [14] take a different approach and present a learning-based linear controller that receives inputs from a fiducial marker in order to land the UAV on a USV that is subject to the waves of a lake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Furthermore, some works also present the application of an MPC controller that enables a flexible-blade helicopter to land on a marine vessel [15], [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' These works use a non- linear MPC to achieve near-perfect performance but do so using a numerical benchmark that doesn’t run in real-time or in a real-world experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Our work differs from these by using simplifications and a new approach that fills these gaps of real-time computing and applicability without a significant drop in landing performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' We use these for comparison in our experimental section to demonstrate the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' The most advanced research presented with real-world flight data is the work by Pearsson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' [17], which presents a linear MPC for autonomous landing of a UAV on the deck of a moving boat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' For the purpose of our work, we assume that the USV can be found by the UAV by ascending to a given altitude during the mission without the need for conducting a planned search which is beyond the scope of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' We also assume that the motion of the USV perpendicular to the water surface is minimal (the USV is waiting for the UAV to land while controlling its global positioning on the water in order to remain stationary, rather than drifting with the waves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Furthermore, we assume that the USV is under the influence of waves, which results in periodic oscillations of the USV deck in each axis of a coordinated system with an origin at the USV center of mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' For hardware, we assume that the UAV is equipped with a 2MP downward facing camera, an onboard computer for image processing and computing the MPC, and that the USV is equipped with a landing pattern to recognize relative pose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' The main difference between our proposed approach and [17] is that our controller uses the non-linear model of the USV for landing on a rapidly tilting deck and does not employ any communication between the UAV and the USV, as motivated by real-world applicability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' To the best of the authors’ knowledge, it is the first approach using USV motion prediction in control feedback of a decentralized controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' In summary, our contributions are as follows: We present a novel objective function for finding an optimum landing trajectory that utilizes an MPC algo- rithm to predict the future of the UAV and USV, without communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' We propose a decentralized vision-based method for observing and predicting the motion of a USV through the use of an online observer that adapts the USV motion model using observations from a downward- facing camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Our proposed approach enables landing on a highly undulating platform with no prior knowledge of the dimensions of the USV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' We propose a prediction algorithm that is designed to prevent a velocity overshoot at the set point for landing with minimal impulse transfer from the surface upon touchdown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' PROPOSED NON-LINEAR ESTIMATOR-BASED MPC In this section, we present our proposed approach which consists of a UAV prediction model and a simplified USV prediction model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Our proposed controller must satisfy two hard constraints imposed by real-world conditions, which are: 1) The controller must perform its computation under a time constraint of 50 ms (20 Hz);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' and 2) There is no communication between the UAV and the USV and so, the only method for estimating the state of the USV motion is by visual pose estimation enabled by the AprilTag on the landing platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Thus, for the sake of clarity, we will call our approach MPC-NE (Model Predictive Controler - Non-linear Estimator).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Figure 2 presents the control pipeline used in this work;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' the contribution is encapsulated in Fig- ure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' For the UAV prediction model, a discrete linear time-invariant system is used, while the USV model uses a more complex linearised model to be described subse- quently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' The 6-degrees of freedom (DOF) USV pose b = �b1 b2 b3 b4 b5 b6 �T is estimated in the world frame through the detection of the fiducial tag in the center of the landing pad from the on-board camera of the UAV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' The pose of the UAV is fused and accounted for to estimate the correct world frame pose of the USV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' This pose information is fed to a fast Fourier transform (FFT) node (based on [18]) which identifies the frequencies, amplitudes, and phases of the N periodic oscillations that make up the USV motion in pitch and roll axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' These identified modes are used to initialize a linear Kalman observer node that corrects the observed state and predicts future motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' These predictions are sent to the IEEE ROBOTICS AND AUTOMATION LETTERS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' PREPRINT VERSION - DO NOT DISTRIBUTE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='1109/LRA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='3231831 MPC Solver Fast Fourrier Transform KF wave Prediction Setpoint Generator UAV Model Reference Tracker Position/Attitude Controller Vision-based Detector Attitude rate Controller IMU UAV Actuators Onboard Sensors State Estimator Odometry & Localisation ˆx rd, ηd FFT accuracy fj,i, Aj,i, φj,i [b4,n, b5,n] n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='.Mp ˙rd, ˙ηd ˆ˙¨rd, ˆ˙¨ηd χd 100 Hz ωd Td 100 Hz ad τd ≈1 kHz x 100 Hz initialisation only x, R, ω 100 Hz R, ω b UAV plant Pixhawk autopilot MPC-NE Architecture USV Prediction Model UAV Prediction Model Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' System architecture: As a primary contribution of this work, the MPC landing controller (yellow block) is integrated into the MRS system (blocks in grey) and supplies the desired reference (velocity ˙rd = � ˙x ˙y ˙z�T and heading rate ˙ηd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Within the MRS system, the first layer containing a Reference tracker processes the desired reference and gives a full-state reference χ to the position/attitude controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' The feedback Position/Attitude controller produces the desired thrust and angular velocities (Td, ωd) for the Pixhawk embedded flight controller (Attitude rate controller).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' The State estimator fuses data from Onboard sensors and Odometry & localization methods to create an estimate of the UAV translation and rotation (x, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Finally, the Vision-based Detector obtains the visual data from the camera and sends the pose information b of the USV to the MPC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' MPC controller, which uses them to estimate the feasibility of landing in the near future, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=', if a sufficiently low tilt of the USV can be found inside the predefined prediction horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' In turn, it generates the desired linear velocities for x, y, and z axes, as well as the desired angular velocity in heading η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' The MPC also receives the estimated UAV state vector x = �x ˙x ¨x y ˙y ¨y z ˙z ¨z η ˙η ¨η�T using onboard state estimation proposed by our team in [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' A finite-state automaton-based approach is used to direct our mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Based on this, a setpoint generator node com- mands the aircraft to increase its altitude until the vision marker can be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Once it is found, the reference of the MPC is changed by the setpoint generator, such that it can hover at a preset altitude above the identified marker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Subsequently, the UAV waits for enough data to be gathered so that the FFT accuracy threshold requirement can be met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Once it is met, the setpoint generator sets the global reference for landing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Then, the MPC begins to use the future motion predictions of the wave to determine a suitable time for landing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' USV Prediction Model USV models can be classified into two different types: Maneuvering Theory and Seakeeping Theory [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Owing to the assumptions made in Section II, we choose to focus on the Seakeeping theory since it concerns near-stationary vessels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' In addition, the use of a decentralized approach brings challenges in estimating the true odometry of the USV, as converging to reliable estimates of linear and angular velocities of the USV is infeasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Therefore, we leverage the pose estimate from the camera efficiently by focusing only on the kinematics of the USV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Our USV prediction model is composed of three parts: a fast Fourier transform, a Kalman observer, and a wave prediction model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' First, the FFT performs a decomposition on the pose data obtained from the vision pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' The identified modes of these oscillations are used to initialize a Kalman observer that adapts the amplitude and the phase of the wave using the observed values online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Finally, the amplitudes and phases from the Kalman observer are sent to the wave prediction model to enable future wave predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' 1) FFT-based Modelling: We assume that the motion of the USV is composed of Nj periodic waves and a non- periodic term that accounts for random noise in tracking the various components for each jth axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Thus, let the state vec- tor b be represented by the linear pose bj for j ∈ {1, 2, 3} for x, y, z axes, respectively, and the angular pose be represented by j ∈ {4, 5, 6} about these axes in the same order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Note here that, a sufficiently large ship/boat (intended application) would exhibit sufficiently low amplitude oscillations in Z- axis such that they can be handled by changing the reference at every camera frame (as shown here).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Thus, the periodic motion of the USV in an axis can be represented as a function of time such that: bj(t) = bj,off + Nj � i=1 Aj,i sin (2πfj,it + φj,i) � �� � Φj,i , (1) with fj,i denoting the frequency, Aj,i the amplitude, and φj,i the phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Additionally, bj,off is the non-periodic term accounting for random noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' For the initial condition, Φj,i(t) is equal to Φj,i(tF F T ), which is the phase obtained as the output of FFT at the time of identification tF F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' In sea conditions, these frequency components can change frequently due to changing winds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Therefore, the pose is sampled continuously and an FFT is run every ∆TF F T seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' For each axis, we discard the modes that are below IEEE ROBOTICS AND AUTOMATION LETTERS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' PREPRINT VERSION - DO NOT DISTRIBUTE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='1109/LRA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='3231831 a certain threshold amplitude Aj,threshold, where Aj,threshold = Agate · max{Aj,0, Aj,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' , Aj,Nj}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' (2) For reliable performance, and upon tunning on real-world data, we assume Agate(= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='02) to be a suitable cutoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' This prevents us from identifying noise components as low- amplitude periodic oscillations without losing more than 2% of the accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' These erroneous components cause a loss of performance in the Kalman observer, which is explained in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' 2) Kalman Observer: The Kalman observer uses a linear model to refine the estimate of identified amplitude and phase of each mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' The observer is necessary because, while the FFT accurately identifies the frequency components, the am- plitude and phase outputs are averages for the entire ∆TF F T sampling interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Therefore, the observer receives new pa- rameters for all identified modes every ∆TF F T seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' In order to allow sufficient time for the observer parameters to converge to true values, we do not reinitialize the pre- identified modes with the new parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Instead, only the newly identified modes are initialized, while discarding the old modes that no longer exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' To assemble the model, we first write the ordinary differ- ential equation (ODE) for each mode for a given axis at any time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' We use vj,i to denote the ith mode of the USV state vector component bj in the jth axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Thus, the derivative of the mode (∀j, j ∈ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' 6) is ˙vj,i = � 0 1 −(2πfj,i)2 0 � � �� � B(tF F T ) vj,i, (3) and the mode at time t is vj,i = � Aj,i sin (Φj,i(t)) 2πAj,ifj,i cos (Φj,i(t)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' (4) Next, we derive the observer model by adding the ODEs of each mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Thus, ˙vj(t) = � ����� Bj,1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' 0 0 0 Bj,2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Bj,N 0 0 0 · · 0 0 � ����� � �� � Bj � ����� vj,1 vj,2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' vj,Nj vj,off � ����� � �� � vj(t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' (5) Hence, the output for each axis is bj(t) = �Cj,1 Cj,2 · · Cj,N Cj,off � � �� � Cj vj(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' (6) Note, that each component of the output vector of the mode can be found as bj,i = �1 0� � �� � Cj,i vj,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' (7) Now, for the brevity of explanation and the readability of the equations, we write the following relation for only one axial DOF of the USV in discrete-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' In addition, we clarify that it can be applied to all 6 of the DOF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Furthermore, notice that a time instance t = k∆T +tF F T , wherein ∆T is the discrete sampling time for new pose observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Thus, we have a straightforward change in notation such that, for example, vj(t) ≡ v(k) j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Thus, by using the integral approximation method, we have that v(k+1) j = exp(Bj∆T) � �� � Ψj v(k) j , and b(k) j = Cj v(k) j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' (8) Then, we continuously estimate the amplitude Aj,i and phase φj,i of each mode every ∆T using the Kalman Filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' First, Q is initialised using a diagonal matrix QI = λI, such that Q = 1 2(ΨQIΨT + QI)∆T, (9) where λ is the gain parameter for the process noise observed in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Meanwhile, the observation noise matrix R is set to the mean amplitude of the observed noise in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Thereafter,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' we apply the filter equations as follows: ˆv(k) j = Ψjv(k−1) j ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' ˆP(k) = ΨjP(k−1)ΨT j + Q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' ˆb(k) j = Cj ˆv(k) j ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' L(k) = ˆP(k)C T j (Cj ˆP(k)C T j + R)−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' v(k) j = ˆv(k) j + L(k)(bj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='m − ˆb(k) j ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' P(k) = (I − L(k)Cj)ˆP(k),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' (10) where ˆ shows the predicted value of the vector/matrix,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' I is an identity matrix,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' bj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='m is the measured value of bj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' L ∈ R2(N+1) is the Kalman gain matrix of the system,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' P and Q ∈ R2(N+1)×2(N+1) are the process co-variance and system noise matrices,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' and R ∈ R is observation noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' At every identification, the relevant elements correspond- ing to both of the modes that are no longer present, as well as the newly identified modes of the Ψ matrix, are re-initialized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' The corresponding co-variance terms for these modes are reset to maintain a consistent prediction without large deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' 3) Wave prediction: Let us now define tobs as the time instant where the last observation was performed, since the prediction algorithm is not run when there are no new obser- vations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Thus, by running the Kalman observer at tobs we find the new amplitude Aj,i(tobs) and phase Φj,i(tobs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' At the same instant in time tobs, we can extract the corresponding vj,i and use (4) to acquire: Φj,i(tobs) = arctan �2πfj,i[vj,i]1,1 [vj,i]2,1 � , and Aj,i(tobs) = [vj,i]1,1 sin (Φj,i(tobs)), (11) where [vj,i]m,n represents the element corresponding to the mth row and nth column of the vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' This enables us to predict the wave behavior at a future time t > tobs as bj(t) = Nj � i=1 Aj,i(tobs) sin [2πfj,i(t − tobs) + Φj,i(tobs)] + [vj,off]1,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' (12) IEEE ROBOTICS AND AUTOMATION LETTERS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' PREPRINT VERSION - DO NOT DISTRIBUTE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='1109/LRA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='3231831 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' UAV Prediction Model The UAV prediction model used in the proposed MPC is based on the Euler approximation of a set of single particle kinematics equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Here, we employ the following discrete linear time-invariant system: x(k+1) = Dx(k) + Eu(k), with u(k) = � ˙¨x ˙¨y ˙¨z ˙¨η �T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' (13) In the model represented by (13), the state matrix D and the input matrix E can be found through the Kronecker product (⊗), such that: D 12×12 = I 4×4 ⊗ D′ 3×3, with D′ = � �1 ∆tpred ∆t2 pred 2 0 1 ∆tpred 0 0 1 � � , (14) E 12×4 = I 4×4 ⊗ E′ 3×1, with E′ = � �� ∆t3 pred 6 ∆t2 pred 2 ∆tpred � �� , (15) where I is an identity matrix, with a prediction made every ∆tpred = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='01 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Hence, the state vector represents the states of the system and their derivatives up to acceleration in each axis, and the control input is the jerk experienced in those axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' MPC Objective Function Once we have defined a prediction model of the UAV and the USV, we can formulate an objective function to enable both waypoint navigation and landing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' For the sake of simplification, we will omit the superscript (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' )(k), which represents a discrete instant in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Therefore, we can define the objective function J as: min u1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='uMc J(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' u) = Mp � m=1 ˜xT mS˜xm + hT mThm � �� � J1 + Mp � m=1 αL × g(˜zm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' b4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' b5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='m) � �� � J2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' subject to : ˜xm = xm − ∗xm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' ˜zm = zm − ∗zm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' hm = um − um−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' xm+1 = Dxm + Eum ∀ m ≤ Mc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' xm+1 = Dxm + EuMc ∀ m > Mc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' umin ≤ um ≤ umax,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' x0 = xinitial,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' u0 = uinitial,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' ∀ {m : m ∈ N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' 1 ≤ m ≤ Mp},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' (16) where ∗xm is the desired state,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' ˜xm is the error vector,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' ˜zm is the error in zm position,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' hm is the rate of control input change to ensure smooth input to the UAV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Mp(= 100) is the prediction horizon,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' and Mc(= 40) is the control horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' S and T are the corresponding penalty matrices with configurable weights for performance tuning, while αL(= 1200) is a weight chosen for the tuning of the objective function g(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Additionally, b4,m, b5,m are the pitch and roll angles in discrete time of the USV about its x and y axes, according to (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' We emphasize that ∗xm (including ∗z) can either be a series of points (trajectory) or a single point (step input).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' This would enable the UAV to keep up with a drifting USV if the XY-position state of the usv is estimated independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' However, a slowly drifting USV is within the dynamic limits of the UAV so as to be compensated by the single- point reference that can be updated after every observation (depending on the camera frame rate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' We demonstrate and test this in this linked video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' While we do not constrain the output of the MPC, we apply soft constraints to the velocity and acceleration states of the model, such that v ≥ vmax and a ≥ amax incur a high penalty in the objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Herein, we have introduced a novel objective function J2 (described in the next section) which can account for the predicted motion of the USV, producing a smooth control input to change the altitude of the UAV without any abrupt maneuvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Using this function, we are able to incorporate the finite state automaton approach using a sigmoid activation function without explicitly describing the possible landing condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' The UAV is able to follow the descend trajectory generated by the MPC by autonomously adjusting its hover distance above the USV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Additionally, it enables us to tune the parameters to control the variance of the resulting landing angles about the mean value of zero-tilt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' It is important to mention that the term J1 in our cost function, is a classical quadratic objective function largely used in robotics and well documented for its feasibility and stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' On the other hand, the J2 term is different from usual works in the literature because it tackles the terminal cost of the optimization step as a potential barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' We employ a non-linear optimization library (NLOPT [21] [22]) which provides the near-optimum solution for the objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' In order to exercise velocity-based control, the first input from the series of optimum control inputs calculated by the solver is then used to calculate the next state using (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' The velocities for this predicted next state are then passed to the system as the velocity references for the UAV to track (as seen in Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Since the term J1 primarily contributes to the position control and J2 contributes to the landing approach, the term J2 remains disabled until the conditions for the landing approach are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Landing approach We define the function for landing cost as a combination of sigmoids, such that: g(˜zm, b4,m, b5,m) = f(˜zm) · ((b4,m)2 + (b5,m)2), (17) where f(˜zm) is such that IEEE ROBOTICS AND AUTOMATION LETTERS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' PREPRINT VERSION - DO NOT DISTRIBUTE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='1109/LRA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='3231831 0 1 2 3 4 5 ez 0 100 200 300 400 500 600 Cost J1 J2 J(t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='07) J(t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='15) J(t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='23) J(t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='5) zero tilt waiting region optimum landing Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' An example illustration of the effective cost function values obtained during the landing approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' f(˜zm) = � � � � � � � � 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='0 + exp � − ˜zm − hd −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='15 ��−1 , if ˜zm ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='16 � 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='0 + exp � ˜zm − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='01 ��−1 , otherwise, (18) where hd controls the waiting region (see Figure 3) during a landing attempt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Empirically, hd = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='1 was chosen for our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' For the scope of this paper, we assume that the USV has relatively negligible motion in its x and y axes, which is a fair assumption for the problem of landing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' The propulsion of the USV may easily compensate for the drift generated by the water currents in order to facilitate landing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' It is also safe to assume that ˜z ≥ 0, as the UAV cannot approach from beneath the USV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' In order to activate J2 to start the landing phase, two conditions must be met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' First, FFT accuracy is higher than a given threshold to detect slow oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Second, The position errors in x and y are below a predefined threshold (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=', ˜x, ˜y ≈ 0) and horizontal velocities vx, vy are also minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' To demonstrate the interaction of J2 with J1 during the landing approach, we present a highly-simplified plot of the objective function (see Figure 3) using one mode each for pitch and roll axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' When J2 is activated, we acquire a combined plot governed by both the equation (17) and the residual error ˜z in J1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' In Figure 3, the value of the objective function encounters a peak that continuously evolves as a function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' This peak acts as a potential barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' The higher cost associated with the peak holds the aircraft in the waiting region (as marked in the plot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Meanwhile the USV model generates predictions for the future of the USV motion during every iteration of the MPC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' The USV sometimes gets close enough to a zero tilt wherein a feasible solution appears, as shown by the zero-tilt points in the plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' The UAV is then able to insert itself into the time-varying trajectory of these special feasible points by reducing its altitude and approaching in such a way that the cost continues to decrease along the locus of these points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Therefore, the UAV is able to follow the zero-tilt points and finish at the optimum landing point, where touchdown is confirmed by the system based on thrust and other information from onboard sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' SIMULATION EXPERIMENTS We demonstrate our simulation results in two scenarios: with a numerical simulation, and with a realistic ROS- based Gazebo simulator [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' The SHMPC presented in 0 5 10 15 20 25 30 35 40 45 50 55 60 65 Time(s) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='8 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='6 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='8 Pitch(rad) predicted_future observed_value 220 225 230 235 240 245 250 255 260 265 270 275 Time(s) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='8 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='6 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='8 Pitch(rad) predicted_future observed_value Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Comparison between the predictions made by the system using the onboard IMU data (left) and using the vision data (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' [15] is shown to work numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Thus, we use a similar numerical implementation of our work (MPC-NE) to allow us to perform a fair comparison with the state-of-the-art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' In this comparison, the non-linear optimization problem is solved by [22] for a landing maneuver of 3 meters and assumes true knowledge of the future motion of the USV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' The second comparison is performed using real-time flight with our proposed MPC-NE inside the Gazebo simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' For this comparison, we use a standard MPC [19] designed for waypoint navigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' For this standard approach, the UAV attempts to locate the target, and lands after a programmed, uniformly randomly distributed delay between 0 and 100 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' We select this duration owing to the periodicity of the tilt angle of the USV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' We use a T650 quadrotor frame weighing 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='6 kg carrying a Garmin LiDAR for laser-ranging of altitude and an Intel Realsense D435 camera for live in- simulation video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' The video output of the Realsense camera is sent to our system to enable processing on the vision node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' III).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' The 3D model of the USV is similar to our real-world experiments and is affixed with an AprilTag [23] marker for pose estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' We note that in both the comparisons, we push the boundary of performance and test our work in rough sea states, and drive our USV model using a wave generator with 4-5 components of oscillation in both pitch and roll axes, and tilt angles up to 30◦(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='5 radians).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' The frequency components are set such that brief windows for feasible landing exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Wind disturbances are not considered in this environment since it is tackled by body disturbances estimated by the low-level control feedback pipeline (see Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Finally, several experimental runs are conducted and aggregated results are presented for these two comparisons using Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Prediction results First, we demonstrate the ability of our system to predict the wave motion up to 1 second into the future based on the observed frequency components and our model of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' The performance of the system is tested in two scenarios: a 100 Hz odometry output from the simulated USV IMU (used as ground truth), and a 30 Hz stream from the AprilTag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' As we see in Figure 4, the high-frequency IMU-based predictions are able to match the observed wave reliably without introducing noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' The observer is able to adapt the observed frequency, amplitude, and phase of the modes of the oscillations and converge reliably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' As opposed to IEEE ROBOTICS AND AUTOMATION LETTERS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' PREPRINT VERSION - DO NOT DISTRIBUTE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='1109/LRA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='3231831 0 ∘ - 5 ∘ 5 ∘ - 10 ∘ 10 ∘ - 15 ∘ 15 ∘ - 20 ∘ 20 ∘ - 25 ∘ 25 ∘ - 30 ∘ 30 ∘ - 35 ∘ Relative tilt angle upon landing ( ∘ ) 0 10 20 30 40 50 Percentage of the landings (%) SHMPC (Numerical) MPC-NE (Numerical) MPC-NE (Realistic Sim) Standard MPC (Realistic Sim) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Histogram comparison between the proposed approach and the standard approach during the touchdown of the UAV on the USV deck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' that, we see slight deviations in the vision-based predictions compared to the IMU results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' This deviation in performance can be explained by two factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' First, the linearisation of the model in the time-domain causes inaccuracies that grow as the sampling time increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Due to this, the three- fold sampling rate of the IMU leads to faster and more accurate convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Second, the output rate of the AprilTag identification node fluctuates around 30 Hz, depending on the computational load of the onboard computer of the UAV (or simulation computer, in this case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' This leads to the misidentification of modes, as the FFT algorithm requires a fixed sample rate for observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' However, this sufficiently proves the ability of the proposed system to reliably predict wave behavior, which will be used for the USV landing further down the pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Landing results To continue, we present the ability of our system to land on a platform while tilt angles are sufficiently close to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Thus, we present Figure 5 that shows the results of the numerical comparison between our MPC-NE and the state- of-the-art SHMPC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Note here that the MPC-NE lands ≈ 94% within 10◦ of tilt, while the SHMPC lands ≈ 71% of the same tilt interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' In this same comparison, the solution time per iteration of our MPC-NE was 9 times lower at 102 ms compared to 917 ms for SH-MPC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Furthermore, the same figure presents the results of the re- alistic simulations using Gazebo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' It is important to highlight the difference between a numerical simulation result and a realistic simulation result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' This is explained by the existing constraints of processing time that demand the algorithms to be processed in real time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Note here that the MPC-NE is able to conduct 72% of its landings within 15◦(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='26 rad) of tilt compared to 23% of landings using the standard MPC approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' In addition, the proposed approach reduces the 80th percentile result by 9◦(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='16 rad) in comparison to the standard approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' For this comparison, we classify a landing conducted at a tilt angle of less than 20◦(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='35 rad) as successful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Therefore, even in challenging tilts of up to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='5 radians, the proposed approach had only three failures, while the standard approach fails approximately 50% of the landings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Finally, we also highlight that, even in an unrealistic and challenging scenario, our system is able to conduct 70% of the landings within 50 seconds of reaching its FFT accuracy threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' REAL-WORLD EXPERIMENTS To test the contributions and proposed algorithms in the real world, we performed landings on an oscillating target at an open water reservoir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' For the purpose of this experiment, we employed a 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='5 kg T650 quadrotor equipped with vertical pontoons [24] for safety over water (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' In addi- tion, the sensor stack included a Garmin LiDAR for laser- ranging of altitude, a Basler camera for the live video feed, and an Intel NUC for onboard real-time processing of the algorithms, data, and video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' The target is a special custom- made USV [25] equipped with a 2m × 2m landing zone, affixed with an AprilTag [23] for 6-DOF pose-estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' The experimental conditions subjected the UAV to a wind of 7m/s and and a USV oscillating with an amplitude of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='3 radians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Prediction results Here we demonstrate our prediction pipeline in two sce- narios: a 30 Hz stream from AprilTag, and a 100 Hz stream from the IMU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' The prediction results for the real-world experiments are presented in Figure 6 and discussed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' For predictions based on vision-based pose estimation, as seen in Figure 6(a), the near-term future correlates well with the observed motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' However, Figure 6(b) indicates that the predictions for the long-term future can suffer in accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' This correlates well with the simulation results as shown in Figure 4 and can be attributed to the higher sampling time-step and its higher variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Occasionally, it also exhibits convergence and consecutive divergence as more data is fed into the pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' For ground truth, we use Figure 6(c) to demonstrate the effectiveness of the pipeline in robustly predicting the future of the USV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' However, since MPC exhibits a higher reliance on the predictions that are temporally proximal, the predictions for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='25 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='50 seconds into the future offer robust support for preventing a landing during an infeasible window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' The chosen angle for landing is also sufficiently low in order to demonstrate the prediction capabilities and the selection of a feasible landing window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Landing results We demonstrate the real-world landing process through Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' In these experiments, the UAV was able to land within 50 seconds of acquiring the required FFT accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' This coincides with our findings in simulation experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Additionally, the tilt angles upon touchdown were less than 5◦(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='09 rad).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' CONCLUSION In this paper, we proposed an MPC that enables a UAV to land autonomously on a tilting USV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' The MPC employs a novel objective function and an online decomposition of the motion of the vessel in order to attempt and complete the landing during a near-zero tilt of the landing platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' We successfully demonstrated that we are able to model and predict the behavior of the UAV and USV without active communication between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Further, we establish a novel IEEE ROBOTICS AND AUTOMATION LETTERS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' PREPRINT VERSION - DO NOT DISTRIBUTE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='1109/LRA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='3231831 80 85 90 95 100 105 110 115 120 time - secs −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='4 roll tilt - radians future_0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='25 secs landing observed (a) Vision—0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='25 s into the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' 80 85 90 95 100 105 110 115 120 time - secs −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='4 roll tilt - radians future_1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='0 secs observed landing (b) Vision—1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='0 s into the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' 100 110 120 130 140 time - secs −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='2 roll tilt - radians future_1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='0 secs observed landing time (c) IMU—1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='0 s into the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Comparison between the predictions made using vision (a-b) and using the onboard IMU of the USV (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' 0 20 40 60 80 100 120 140 Time(s) 0 2 4 6 8 Position(m) z-position Finding tag Accuracy threshold wait Attempting landing Landed −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='5 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content='5 Radians roll angle Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' Plot of a selected real-world open-water experiment (video).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAyT4oBgHgl3EQfa_fL/content/2301.00255v1.pdf'} +page_content=' approach for landing on the USV using these predictions, which autonomously adjusts the relative altitude for the UAV to ensure that the landing occurs as close to the zero-tilt state of the landing deck as possible, increasing safeness of the landing phase 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