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+  
+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. 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: 
+[email protected]). 
+Dmytro Vovchuk, Anna Mikhailovskaya, Dmitry Dobrykh, Pavel Ginzburg 
+School of Electrical Engineering, Tel Aviv University, Ramat Aviv, Tel Aviv, 
+69978, Israel (e-mail: [email protected], [email protected] ). 
+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: [email protected]) 
+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”.  
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+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.
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+

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+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 <[email protected]>.
+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
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+0
+20000
+40000
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+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.
+

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+arXiv:2301.03079v1  [math.FA]  8 Jan 2023
+Lp simulation for measures
+L. De Carli and E. Liflyand
+Abstract. Being motivated by general interest as well as by certain concrete problems of Fourier
+Analysis, we construct analogs of the Lp spaces for measures. It turns out that most of standard
+properties of the usual Lp spaces for functions are extended to the measure setting. 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.
+1. 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. This is not the case for the theories of the corresponding transforms for measures, see, e.g.,
+[3]. A simple curiosity may force one to wonder where the analogs for measures are hidden. We
+have not succeeded to find such a machinery in the literature. However, we have a more concrete
+reason to be interested in the depository of such treasures. Let us consider the following example,
+somewhat sketchy. 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.
+(1)
+Let f be locally absolutely continuous on (0, ∞); 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. Let in addition, lim
+t→∞ f(t) = 0 and Hof ′ ∈ L1(R+). 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; of course, understood in the principle
+value sense. When it is integrable, we will denote the corresponding Hardy space of such functions
+g by H1
+0(R+). 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+).
+(3)
+2020 Mathematics Subject Classification. Primary: 28A33; Secondary: 42A38.
+Key words and phrases. Measure; Fourier transform; Hausdorff-Young inequality; Young inequality; uncertainty
+principle.
+1
+
+2
+L. DE CARLI AND E. LIFLYAND
+For this result as well as many other more advanced ones, see [12] (cf. [5] and [7]; see also [8, Ch.3]
+or more recent [13]). Recall that the derivative of a function of bounded variation exists almost
+everywhere and is Lebesgue integrable. Here and in what follows ϕ ≲ ψ means that ϕ ≤ Cψ with
+C being an absolute constant.
+A natural question arises whether we can relax the assumption of absolute continuity. 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).
+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. The point is that the Hilbert transform of df
+does exist almost everywhere (see, e.g., [3, §8.1 ]) but its integrability leads to absolute continuity,
+the property that we aimed to remove (see, e.g., [4] and references therein).
+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). More precisely, for 1 < p < ∞, set
+∥g∥Op =
+� ∞
+0
+�1
+x
+�
+x≤t≤2x
+|g(t)|pdt
+� 1
+p
+dx.
+Further, for p = ∞, let
+∥g∥O∞ =
+� ∞
+0
+ess sup
+x≤t≤2x
+|g(t)| dx.
+Known are (see, e.g., the above sources) the following relations:
+O∞ ֒→ Op1 ֒→ Op2 ֒→ H1
+0 ֒→ L1
+(p1 > 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 Hausdor���-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: [email protected]
+Department of Mathematics, Bar-Ilan University, 52900 Ramat-Gan, Israel
+Email address: [email protected]
+

7NE1T4oBgHgl3EQfTgOa/content/tmp_files/load_file.txt

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+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': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'}
+page_content='  References  [1] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'}
+page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'}
+page_content=' Amrein and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQfTgOa/content/2301.03079v1.pdf'}
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+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(𝑡��),⋅)
+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. 
+ 
+ 
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+ 
+

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+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.
+
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+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;
+[email protected]
+† [email protected]; 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
+‡ [email protected]; 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; [email protected]
+¶ Moscow Institute of Physics and Technology - 1 “A” Kerchen-
+skaya st., Moscow, 117303, Russian Federation
+Lebedev Physical Institute RAS; [email protected]
+∗∗ 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-
+[email protected]
+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. This allows simultaneous and cor-
+related observation of TGF or TGE gamma rays from
+the top and from the bottom of a thundercloud. 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.
+ACKNOWLEDGEMENTS
+The work of E. Stadnichuk was supported by the Foun-
+dation for the Advancement of Theoretical Physics and
+Mathematics “BASIS”. The work of E. Svechnikova was
+supported by a grant from the Government of the Rus-
+sian Federation (contract no. 075-15-2019-1892).
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+

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
+*[email protected]
+†[email protected]
+‡[email protected]
+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<j
+� ui−u j
+ui+u j
+�2
+(�n
+j=1(u j +1/u j))2
+du1
+u1
+··· dun
+un
+(11)
+The integral Dn provides great insights into the symmetry breaking at low-temperature phase and finds great use in
+Quantum Field Theories and condensed matter physics. However, it is difficult to evaluate these integrals computa-
+tionally and analytically. On the other hand, the Cn (Cn = Cn,1) class integrals which are closely related to the Dn
+class, are easier to tackle and can produce closed-form expressions.
+The general Ising integrals Cn,k is defined as
+Cn,k = 4
+n!
+�∞
+0
+···
+�∞
+0
+1
+(�n
+j=1(u j +1/u j))k+1
+du1
+u1
+··· dun
+un
+(12)
+3
+
+The above expression can also be expressed as the moments of power of Bessel Function K0 as
+Cn,k = 2n−k+1
+n! k! cn,k := 2n−k+1
+n! k!
+�∞
+0
+tkKn
+0 (t)dt
+(13)
+We will now analyze the special case of the Cn,k family with k = 1 using the Method of Brackets [2,3,20] and Mellin-
+Barnes representations. After this, each general integral with Cn,k will be treated using the same procedure. The
+C1,k and C2,k integrals are easily tractable, and the results for them have been given just for completeness’ sake. The
+problem occurs when one considers Cn,k for n ≥ 3. Below we use the MMOB [24] and show that for the evaluation of
+the integrals requiring the use of regulators, it is better to use the MMOB and solve the corresponding integral using
+the CHMB method. The main utility of the method is that the limiting procedure is automatically taken care of while
+finding the residue in the case of CHMB, which is at times difficult, especially when there is more than 1 regulator,
+as in the case of C4,k.
+3.1
+C1,k
+For n = 1, we have
+C1,k = 4
+1!
+�∞
+0
+1
+(u1 +1/u1)k+1
+du1
+u1
+(14)
+The integral can simply be evaluated to give the general closed form:
+C1,k =
+�π21−kΓ
+�
+k+1
+2
+�
+Γ
+�
+k
+2 +1
+�
+(15)
+3.2
+C2,k
+For k = 1, we get:
+C2,k = 4
+2!
+�∞
+0
+�∞
+0
+1
+(u1 +1/u1 + u2 +1/u2)k+1
+du1
+u1
+du2
+u2
+(16)
+This evaluation using the MOB, for k = 1, gives:
+C2,1 = 1
+(17)
+The integral for the general value of k can also be evaluated to give the following closed form:
+C2,k =
+Γ
+�
+k
+2 + 1
+2
+�4
+Γ(k+1)2
+(18)
+3.3
+C3,k and C3,k(α,β,γ)
+For k = 1, we get:
+C3,1 = 4
+3!
+�∞
+0
+�∞
+0
+�∞
+0
+1
+(u1 +1/u1 + u2 +1/u2 + u3 +1/u3)2
+du1
+u1
+du2
+u2
+du3
+u3
+(19)
+We will illustrate the problem encountered in OMOB by writing the bracket series for the generalized case C3,k.
+Taking k = 1 will give us the result for C3,1.
+The following form of the integrand is motivated to maximize the number of brackets series in the expansion, which
+in turn reduces the number of variables:
+C3,k = 2
+3
+�∞
+0
+�∞
+0
+�∞
+0
+(u1u2u3)k
+(u1u2u3(u1 + u2)+ u3(u1 + u2)+ u1u2u2
+3 + u1u2)k+1 du1du2du3
+(20)
+Expanding the denominator using the rules of MOB,
+�
+{n}
+φ{n}(u1u2)n1+n3+n4 zn1+n2+2n3(u1 + u2)n1+n2 〈k+1+ n1 + n2 + n3 + n4〉
+Γ(k+1)
+(21)
+Now, (u1 + u2)n1+n2 has to be further expanded as:
+(u1 + u2)n1+n2 =
+�
+n5,n6
+φn5,n6un5
+1 un6
+2
+〈−n1 − n2 + n5 + n6〉
+Γ(−n1 − n2)
+(22)
+4
+
+Combining the expansions, the C3,k integral takes the form:
+C3,k =
+2
+3Γ(k+1)
+�
+{n}
+φ{n}
+〈−n1 − n2 + n5 + n6〉
+Γ(−n1 − n2)
+(23)
+×〈k+1+ n1 + n3 + n4 + n5〉〈k+1+ n1 + n3 + n4 + n6〉
+×〈k+1+ n1 + n2 +2n3〉〈k+1+ n1 + n2 + n3 + n4〉
+Now, the rules of MOB demand that we solve the linear equations of the brackets, but that poses the problem of
+giving rise to divergent terms like Γ(−n) and renders the whole procedure useless. To solve the issue, it is suggested
+to introduce regulators. For the case of C3,k, one regulator is enough. In particular, ǫ(→ 0) is introduced in the bracket
+as 〈k+1+n1+n2+2n3〉 → 〈k+1+n1+n2+2n3+ǫ〉 which mimics the effect of introducing a factor of uǫ
+3 in the integrand.
+Now, with this “new" bracket series, the divergent terms take the form of Γ(−n −ǫ) and are easier to work with. In
+the regime of OMOB, one requires the expansion of Γ(x) around integers to deal with the problem, which increases
+the complexity of the task.
+As n increases, the number of regulators increases monotonically and complicates the limiting procedure. On
+the other hand, MMOB doesn’t call for any regulators and is very computationally friendly. Using the MMOB in the
+above bracket series, we get the following MB representation for the C3,1
+C3,1 = 1
+3
+c+i∞
+�
+c−i∞
+dz
+2πi
+Γ(−z)4 Γ(1+ z)2
+Γ(−2z)
+(24)
+This evaluates to
+C3,1 = 2
+27
+�
+6i
+�
+3
+�
+Li2
+�
+1
+4 − i
+�
+3
+4
+�
+−Li2
+�
+i
+�
+3
+4
++ 1
+4
+��
++π
+�
+3log(4)−ψ(1)
+�1
+3
+�
++ψ(1)
+�2
+3
+��
+(25)
+where ψ(1) is the polygamma function of order 1.
+The generalized integral C3,k can be similarly obtained using the MMOB to give the following MB representation:
+C3,k =
+1
+3Γ(k+1)
+c+i∞
+�
+c−i∞
+dz
+2πi
+Γ(−z)4 Γ
+� 1
+2(k+2z +1)
+�2
+Γ(−2z)
+(26)
+The above integral can be evaluated to give
+C3,k =
+2
+3k!
+�
+πG2,3
+3,3
+�1
+4
+����
+1,1,1
+k+1
+2 , k+1
+2 , 1
+2
+�
+(27)
+where G is the Meijer-G function.
+A further generalization of C3,k integral namely C3,k(α,β,γ) is given in [3] where the following integral is considered
+C3,k(α,β,γ) =
+�∞
+0
+�∞
+0
+�∞
+0
+xα−1yβ−1zγ−1
+(x+1/x+ y+1/y+ z +1/z)k+1 dxdydz
+(28)
+Using the MMOB, we get the following MB representation
+C3,k(α,β,γ) =
+1
+3Γ(k+1)
+c+i∞
+�
+c−i∞
+dz
+2πi
+Γ(−z)Γ(−z +α−1)Γ(−z −β+1)Γ(−z +α−β)Γ
+�1
+2(k+2z −α+β−γ+2)
+�
+Γ
+� 1
+2(k+2z −α+β+γ)
+�
+Γ(−2z +α−β)
+(29)
+The result is given in the MATHEMATICA file Ising.nb and is found to be :
+= − 1
+3k!π3/2 csc(πγ)2−γ−k−1�
+4γΓ
+�1
+2(k−α−β−γ+4)
+�
+Γ
+�1
+2(k+α−β−γ+2)
+�
+Γ
+�1
+2(k−α+β−γ+2)
+�
+Γ
+�1
+2(k+α+β−γ)
+�
+(30)
+× 4 ˜F3
+�1
+2(k+α+β−γ), 1
+2(k−α−β−γ+4), 1
+2(k+α−β−γ+2), 1
+2(k−α+β−γ+2); 1
+2(k−γ+2), 1
+2(k−γ+3),2−γ; 1
+4
+�
+−4Γ
+�1
+2(k−α−β+γ+2)
+�
+Γ
+�1
+2(k+α−β+γ)
+�
+Γ
+�1
+2(k−α+β+γ)
+�
+Γ
+�1
+2(k+α+β+γ−2)
+�
+× 4 ˜F3
+�1
+2(k−α+β+γ), 1
+2(k+α+β+γ−2), 1
+2(k−α−β+γ+2), 1
+2(k+α−β+γ); k+γ
+2
+, 1
+2(k+γ+1),γ; 1
+4
+��
+5
+
+3.4
+C4,k and C4,k(α,β,γ,δ)
+For k = 1:
+C4,1 = 4
+4!
+�∞
+0
+�∞
+0
+�∞
+0
+�∞
+0
+1
+(u1 +1/u1 + u2 +1/u2 + u3 +1/u3 + u4 +1/u4)2
+du1
+u1
+du2
+u2
+du3
+u3
+du4
+u4
+(31)
+If one proceeds with the OMOB as in the case of C3,1, one is now required to use 2 regulators, namely ǫ and A [3]. The
+result for C4,1 is then obtained by taking the limit ǫ → 0, A→ 1. The use of two regulators significantly complicates
+the task of doing the limiting procedure. So we again proceed with the use of the MMOB. Using the MOB, we get the
+following MB representation for C4,1:
+C4,1 = 1
+12
+c+i∞
+�
+c−i∞
+dz
+2πi
+Γ(−z)4 Γ(1+ z)4
+Γ(−2z) Γ(2+2z)
+(32)
+This can be evaluated to give
+C4,1 = 7ζ(3)
+12
+(33)
+The general case for n = 4 can be simplified to the following MB representation:
+C4,k =
+1
+12Γ(k+1)
+c+i∞
+�
+c−i∞
+dz
+2πi
+Γ(−z)4 Γ
+�
+k+1
+2 + z
+�4
+Γ(−2z) Γ(k+2z +1)
+(34)
+This can be evaluated to give the closed-form expression:
+C4,k = π 2−k−1
+3Γ(k+1)G3,3
+4,4
+�
+1
+����
+1,1,1, k+2
+2
+k+1
+2 , k+1
+2 , k+1
+2 , 1
+2
+�
+(35)
+The given expression is of particular interest, as seen from its values when evaluated for any odd values of k. When
+C4,k is evaluated for any odd k, it takes the form of aζ(3) + b function, where a and b are some rational numbers.
+Some of the values are provided for reference in Table 1.
+A further generalization of C4,k integral namely C4,k(α,β,γ,δ) can be considered as follows
+C4,k(α,β,γ,δ) =
+�∞
+0
+�∞
+0
+�∞
+0
+�∞
+0
+xα−1yβ−1zγ−1wδ−1
+(x+1/x+ y+1/y+ z +1/z + w+1/w)k+1 dxdydzdw
+(36)
+Using the MMOB, we get the following MB representation
+C4,k(α,β,γ) =
+1
+12Γ(k+1)
+c+i∞
+�
+c−i∞
+dz
+2πi
+Γ(−z)Γ(−z +γ−1)Γ(−z −δ+1)Γ(−z +γ−δ)Γ
+�1
+2(k+2z −α−β−γ+δ+3)
+�
+12Γ(k+1)Γ(−2z +γ−δ)Γ
+� 1
+2(k+2z −α−β−γ+δ+3)+ 1
+2(k+2z +α+β−γ+δ−1)
+�
+×Γ
+�1
+2(k+2z +α−β−γ+δ+1)
+�
+Γ
+�1
+2(k+2z −α+β−γ+δ+1)
+�
+Γ
+�1
+2(k+2z +α+β−γ+δ−1)
+�
+(37)
+The above integral can be evaluated as before, and the solution has been provided in the accompanying MATHEMAT-
+ICA file Ising.nb.
+We end this section by noting that given an integral, the evaluation of its MB representation obtained using
+the MMOB [24] is more efficient than using the OMOB and its rules to evaluate the same. The regulators and the
+limiting procedure in the OMOB are automatically taken care of in the evaluation of MB integrals while evaluating
+the residue. Alternatively, this suggests that one can try to find a better rule that concerns the elimination of the
+bracket for the OMOB so that one does not require regulators and the result is obtained with their use.
+4
+An attempt at C5,k
+Using the machinery developed so far, we now attempt to evaluate the C5 integral in the same spirit. Using the MOB,
+we get the following MB representation for C5,k
+C5,k =
+1
+60Γ(k+1)
+c1+i∞
+�
+c1−i∞
+dz1
+2πi
+c2+i∞
+�
+c2−i∞
+dz2
+2πi
+Γ(−z1)4Γ(−z2)4Γ
+� 1
+2 (k+2z1 +2z2 +1)
+�2
+Γ(−2z1)Γ(−2z2)
+(38)
+6
+
+k
+C4,k
+0
+1
+6πG3,3
+4,4
+�
+1
+����
+1,1,1,1
+1
+2, 1
+2, 1
+2, 1
+2
+�
+1
+7ζ(3)
+12
+2
+1
+48πG3,3
+4,4
+�
+1
+����
+1,1,1,2
+3
+2, 3
+2, 3
+2, 1
+2
+�
+3
+7ζ(3)−6
+1152
+4
+1
+2304πG3,3
+4,4
+�
+1
+����
+1,1,1,3
+5
+2, 5
+2, 5
+2, 1
+2
+�
+5
+49ζ(3)−54
+368640
+6
+1
+276480πG3,3
+4,4
+�
+1
+����
+1,1,1,4
+7
+2, 7
+2, 7
+2, 1
+2
+�
+7
+63ζ(3)−74
+15482880
+Table 1: Values of C4,k for k = 0,··· ,7
+Evaluation of the above integral, when done directly using the MBConicHulls.wl, would result in the divergent
+series. A suitable way to approach such evaluation would be by taking two parameters that serve as the variables
+for the series that appear and then evaluating the results with these parameters. For the C5,k integral we have the
+following evaluation
+C5,k(α,β) =
+1
+60Γ(k+1)
+c1+i∞
+�
+c1−i∞
+dz1
+2πi
+c2+i∞
+�
+c2−i∞
+dz2
+2πi (α)z1(β)z2 Γ(−z1)4Γ(−z2)4Γ
+� 1
+2 (k+2z1 +2z2 +1)
+�2
+Γ(−2z1)Γ(−2z2)
+(39)
+We notice that the integral Eq.(39) has a more general structure than the integral Eq.(38) with the introduction of
+the two parameters. The C5,k can be obtained by putting α = β = 1. The evaluation of the Eq.(39) has been done in
+the accompanying MATHEMATICA file Ising.nb.
+We also note that though we have a result for integral (39), the result is not convergent for the value of interest
+α = β = 1. Proper analytic continuation techniques have to be used to achieve this goal. At present, with the form of
+series that we obtain, the task is not achievable using Olsson.wl . With the form of series at hand we believe that it
+can be written as a derivative of ‘some’ hypergeometric function. Then Olsson.wl can be used to find the ACs of this
+hypergeometric function so that it converges for α = β = 1, and then the derivative can be performed to get the final
+result.
+7
+
+5
+Box Integrals
+For dimension n, we define the box integral as the expected distance from a fixed point q (can be origin also) of point
+r chosen randomly and independently over the unit n-cube, with parameter s,
+Bn(s) =
+�1
+0
+···
+�1
+0
+�
+(r1)2 +···+(rn)2�s/2
+dr1 ···drn
+(40)
+∆n(s) =
+�1
+0
+···
+�1
+0
+�
+(r1 − q1)2 +···+(rn − qn)2�s/2
+dr1 ···drndq1 ···dqn
+(41)
+For certain special values of parameter s, the above integrals give the following interpretation:
+1. Bn(1): It gives the expected distance from the origin for a random point of the n-cube.
+2. ∆n(1): It gives the expected distance between two random points of the n-cube.
+Due to the physical significance of the box integrals and also their use in the electrostatic potential calculations, we
+wanted to evaluate these integrals and give closed-form expressions using the Method of Brackets that has been im-
+plemented throughout the paper.
+Using the quadrature formulae for all complex powers [25,26,29,33,34], we use the functions:
+b(u) =
+�1
+0
+e−u2x2dx =
+�πerf(u)
+2u
+(42)
+d(u) =
+�1
+0
+�1
+0
+e−u2(x−y)2dydx =
+�πuerf(u)+ e−u2 −1
+u2
+(43)
+which gives us the relation:
+Bn(s) =
+2
+Γ(−s/2)
+�∞
+0
+u−s−1bn(u)du
+(44)
+∆n(s) =
+2
+Γ(−s/2)
+�∞
+0
+u−s−1dn(u)du
+(45)
+5.1
+Bn(s)
+Now, for the method of brackets to be operational, we need integrals of the form with limits from 0 to ∞. We need to
+make an Euler substitution. The following substitution has been found to be the most efficient:
+x →
+a
+1+ a
+(46)
+which makes the integral
+b(u) =
+�1
+0
+e−u2x2dx =
+�∞
+0
+e−u2�
+a
+1+a
+�2
+1
+(1+ a)2 da
+(47)
+b(u) =
+�∞
+0
+∞
+�
+n=0
+1
+n!
+� −u2a2
+(1+ a)2
+�n
+1
+(1+ a)2 da
+(48)
+Substituting this back in Bn(u) and applying MMOB, it is obtained that Bn(s) has a pole at s = −n and we finally
+get:
+B1(s) =
+1
+s+1,s ̸= −1
+(49)
+B2(s) =
+2
+s+2 2F1
+�1
+2,− s
+2; 3
+2;−1
+�
+,s ̸= −2
+(50)
+The first two cases were easy to handle. The first non-trivial evaluation is that of B3(s). We found two different
+results for the same by using two different methods. Firstly we consider the following representation of B3 [26]:
+B3(s) =
+3
+3+ s C2,0(s,1) =
+6
+(3+ s)(2+ s)
+�π/4
+0
+��
+1+sec2 t
+�s/2+1 −1
+�
+(51)
+8
+
+The above can interestingly be evaluated in MATHEMATICA using Integrate command.
+Using it, we get the
+following evaluation for the B3(s) integral
+B3(s) =
+6
+(s+2)(s+3)
+�
+iF1
+�
+1; 1
+2,− s
+2;2;2,−2
+�
+− 2
+s+1
+2
+s+1 F1
+�1
+2(−s−1);−1
+2,− s
+2; 1− s
+2
+; 1
+2,−1
+2
+�
+− i 2F1
+�
+1,− s
+2; 3
+2;−1
+�
++2s/2 2F1
+�1
+2,− s
+2; 3
+2;−1
+2
+�
+−
+�π
+4Γ
+�
+1− s
+2
+� 2F1
+�
+− s
+2 − 1
+2,− s
+2;1− s
+2;−1
+�
+Γ
+�
+− s
+2 − 1
+2
+�
+− π
+4
+�
+(52)
+where F1(a;b1,b2;c;x, y) is the Appell F1 function which is defined for |x| < 1∧|y| < 1 as:
+F1(a;b1,b2;c;x, y) =
+∞
+�
+m,n=0
+(a)m+n(b1)m(b2)n
+(c)m+nm!n!
+xmyn
+(53)
+where (q)n is the Pochhammer symbol.
+The Eq.(52) requires the evaluation of the Appell F1 outside its region of convergence. Such evaluation requires
+the use of analytic continuation of F1, which has been done by Olsson [35].
+Though we got the result using MATHEMATICA , it doesn’t provide many insights so as to aid the computations of
+other Bn(s). So we proceed to a more systematic evaluation of the B3(s) so that the results can be generalized to other
+values of n. Using the MMOB [24] we get the following Mellin-Barnes integral for the B3(s)
+B3(s) =
+c1+i∞
+�
+c1−i∞
+c2+i∞
+�
+c2−i∞
+Γ(−z1)Γ(−z2)Γ(2z1 +1)Γ(2z2 +1)Γ(s−2z1 −2z2 +1)Γ
+�
+− s
+2 + z1 + z2
+�
+Γ
+�
+− s
+2
+�
+Γ(2z1 +2)Γ(2z2 +2)Γ(s−2z1 −2z2 +2)
+dz2
+2πi
+dz1
+2πi
+(54)
+We evaluate the above integral using the MBConicHulls.wl package [14]. The evaluation gives the following result:
+B3(s) = −
+π
+2
+�
+s2 +5s+6
+� +
+�π
+�
+(s+2)2F1
+� 1
+2,− s
+2 − 1
+2; 3
+2;−1
+�
++ 2F1
+�
+− s
+2 −1,− s
+2 − 1
+2;− s
+2;−1
+��
+Γ
+�
+− s
+2 − 1
+2
+�
+Γ(s+2)
+2(s+3)Γ
+�
+− s
+2
+�
+Γ(s+3)
++
+1
+1+ s F2:1:1
+1:1:1
+
+
+−1− s
+2
+, −s
+2 : 1
+2; 1
+2
+1− s
+2
+, 1
+2
+: 1
+2;−−
+�������
+−1,−1
+
+
+(55)
+Where F2:1:1
+1:1:1(x, y) is the KdF function which converges for |�x| + |�y| < 1. So to evaluate it at (−1,−1), one needs
+its analytic continuations. In the MATHEMATICA file Box.nb, we provide a systematic derivation of the analytic
+continuation for the same so that it converges at (−1,1).
+For general Bn(s) we get the following MB-representation
+Bn(s) =
+1
+Γ
+�
+− s
+2
+�
+c1+i∞
+�
+c1−i∞
+···
+cn−1+i∞
+�
+cn−1−i∞
+�
+n−1
+�
+p=1
+dzp
+2πi
+� ��n−1
+i=1 Γ(2zi +1)
+�
+Γ
+�
+s−2�n−1
+j=1 z j +1
+�
+Γ
+��n−1
+k=1 zk − s
+2
+�
+��n−1
+l=1 Γ(2zl +2)
+�
+Γ
+�
+s−2�n−1
+m=1 zm +2
+�
+(56)
+Using the Eq. (56) we obtain following representation for B4(s)
+B4(s,α,β,γ) =
+1
+Γ
+�
+− s
+2
+�
+c1+i∞
+�
+c1−i∞
+c2+i∞
+�
+c2−i∞
+c3+i∞
+�
+c3−i∞
+Γ(−z1)Γ(2z1 +1)Γ(−z2)Γ(2z2 +1)Γ(−z3)Γ(2z3 +1)Γ(s−2z1 −2z2 −2z3 +1)
+Γ(2z1 +2)Γ(2z2 +2)Γ(2z3 +2)Γ(s−2z1 −2z2 −2z3 +2)
+×Γ
+�
+− s
+2 + z1 + z2 + z3
+�
+(α)z1(β)z2(γ)z3 dz1dz2dz3
+(2πi)3
+(57)
+The above integral can be again evaluated readily using the MbConicHull.wl package. For the case of B4(s), due to
+the occurrence of a 3-variable hypergeometric function, the region of convergence analysis is difficult. In the OMOB
+all the series which converges in the same region of convergence are kept together. For 3 or more variables this
+analysis becomes complicated and is not always straightforward [9]. Here the CHMB method plays an important role
+in that it clubs the series converging in the same region of convergence together without prior knowledge of their
+region of convergence. The evaluation has been provided in the file Ising.nb .
+9
+
+5.2
+∆n(s)
+We now move on to the evaluation of ∆n integrals (41). Instead of directly doing the evaluation of the δn(s) integral,
+we refer to [26], to exploit the relation between Bn(s) and ∆n(s). A few instances of the same are as follows:
+∆1(s) = 2
+1
+(s+1)(s+2)
+(58)
+∆2(s) = 8
+2
+s
+2 +1(s+3)+1
+(s+2)(s+3)(s+4) +4B2(s)− 4(s+4)
+s+2 B2(s+2)
+(59)
+∆3(s) = 24
+�
+(s+5)
+�
+2
+s
+2 +3 −3
+s
+2 +2�
++1
+�
+(s+2)(s+4)(s+5)(s+6) + 24
+s+2 B2(s+2)−
+24(s+6)
+(s+2)(s+4) B2(s+4)− 12(s+5)
+s+2
+B3(s+2)
+(60)
++ 4(s+6)(s+7)
+(s+2)(s+4) B3(s+4)+8B3(s)
+(61)
+where B2(s) and B3(s) are given by Eq.(50) and Eq.(55). The results for ∆4 and ∆5 are provided in the appendix B.
+5.3
+Jelium Potential
+As an application of the evaluations done in the previous section, we refer to one more application of such evaluations,
+the Jellium potential [29]. It arises in the problem of electrostatics. The problem concerns finding the electrostatic
+potential energy of an electron (having charge -1) at the cube center, given an n-cube of uniformly charged jelly of
+total charge +1. For the problem, usually one takes the radial potential at a distance r from the electron as Vn(r) as
+follows
+V1(r) := r −1/2,
+V2(r) := log(2r),
+Vn(r) := 2n−2 −
+�1
+r
+�n−2
+,
+n > 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
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+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
+

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@@ -0,0 +1,3034 @@
+ON THE LIMIT SPECTRUM OF A DEGENERATE OPERATOR IN THE FRAMEWORK OF
+PERIODIC HOMOGENIZATION OR SINGULAR PERTURBATION PROBLEMS
+KAÏS AMMARI AND ALI SILI
+ABSTRACT. 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. 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. 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.
+CONTENTS
+1.
+Introduction, setting of the problem and statement of the results
+1
+2.
+Proof of the results in the case of a single thin structure: the reduction of dimension 3d − 1d
+8
+2.1.
+Apriori estimate on the sequence of eigenvalues and eigenvectors
+8
+2.2.
+The limit problem associated to (1.9)
+9
+2.3.
+The strong convergence of the eigenvectors
+11
+2.4.
+Proof of Theorem 1.3
+14
+3.
+Proof of Theorem 1.5
+18
+References
+20
+1. 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; more precisely, we consider the following problem
+(1.1)
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+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. ∂u
+∂nχ∂Fε) denotes the outer normal
+to the lateral boundary of Mε (resp. Fε). The space Vh (h stands for homogenization) is defined by
+(1.2)
+Vh :=
+�
+u ∈ H1(Ω), u(x′, 0) = u(x′, L) = 0 a.e. x′ = (x1, x2) ∈ ω
+�
+,
+2010 Mathematics Subject Classification. 35B25; 35B27; 35B40; 35B45; 35J25; 35J57; 35J70; 35P20.
+Key words and phrases. Spectrum, Degenerate, High contrast, Homogenization, Singular perturbation.
+1
+arXiv:2301.04226v1  [math.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.
+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. The gradient and the Laplacian with respect to the horizontal variables will be denoted
+respectively by ∇′ and ∆′.
+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. 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. The characteristic functions of Fε (resp. Mε) are denoted by χFε (resp. χMε). 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 . Throughout
+the paper, for a measurable set B we denote by |B| its Lebesgue measure and by χB its characteristic function. A
+generic positive constant the value of which may change from a line to another will be denoted by K.
+Let C be a square of R2 and let D be a disk strictly contained in C. The complement of D in C will be
+denoted by M ′ in such a way that C = M ′ ∪ D. The geometry of the domain is described as follows.
+(1.3)
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+Ci
+ε = (εC + εi) × (0, L); ω =
+�
+i∈Iε
+(εC + εi); 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.
+FIGURE 1. The composite structure after dilation which is also the reference cell in the homoge-
+nization setting.
+In Figure 1 we have represented the representative cell C = D ∪ M ′ which represents also the composite
+structure after dilation.
+When dealing with the homogenization of a problem posed on a domain Ω with a geometry given by (1.3), a
+reduction of dimension 3d − 1d appears locally in each cell; 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. 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; 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.
+In this setting, the spectral problem (1.1) takes the following form
+(1.4)
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+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
+(1.5)
+V ε
+s :=
+�
+v ∈ H1(Ωε), v(x′, 0) = v(x′, L) = 0 a.e. x′ = (x1, x2) ∈ εC
+�
+.
+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
+(1.6)
+∇′
+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). This change of variables transforms the problem (1.4) into the following singular perturbation problem,
+(1.7)
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+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 (1.5).
+Note that the study of the asymptotic behavior of (1.4) 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).
+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]). 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.
+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.
+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]. Other settings have been studied in [2], [13], [14], [18].
+Of course, this rule also fits for spectral problems, see for instance [28], [17], [23].
+To describe the behavior of the eigenvalues of (1.7) ( 3d − 1d problem) or (1.1) (homogenization) we use
+the variational formulation. Note that for a fixed ε, Aε defined either by (1.1) or by (1.7) is a bounded selfadjoint
+operator with compact resolvent so that one can state the following well known result.
+
+4
+KAÏS AMMARI AND ALI SILI
+Proposition 1.1. Problem (1.1) (or problem (1.7)) admits a sequence of eigenvalues (λk
+ε)k, 0 < λ1
+ε ≤ λ2
+ε ≤ ... ≤
+λn
+ε ≤ ..., with lim
+k→∞ λk
+ε = +∞ while the associate eigenvectors (uk
+ε)k may be chosen as an orthonormal basis of
+L2(Ω).
+Taking into account this result, the variational formulation of (1.1) and of (1.7) are respectively the following
+ones
+(1.8)
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+uk
+ε ∈ Vh,
+�
+Ω
+(ε2∇uk
+ε∇φχFε + ∇uk
+ε∇φχMε
+�
+dx = λk
+ε
+�
+Ω
+uk
+εφ dx,
+∀ φ ∈ Vh,
+(1.9)
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+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).
+We prove in Theorem 1.3 below that for each k, the limit λk of the sequence of eigenvalues (λk
+ε)ε of (1.7)
+( 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; 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; more precisely, λk is a solution of the following system
+(1.10)
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+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).
+Similar results (Theorem 1.5) are obtained for the homogenisation problem; the limit λk of the sequence of
+eigenvalues (λk
+ε)ε of (1.1) 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
+(1.11)
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+uk
+0 ∈ H1
+#(C),
+−∆′
+yuk
+0 = λkuk
+0 + 1 in D,
+uk
+0 = 0 on ∂D,
+vk ∈ L2(ω; H1
+0(0, L)),
+−∂2vk
+∂x2
+3
+= λk
+�
+1 +
+|D|
+|C \ D| +
+λk
+|C \ D|
+�
+D
+uk
+0 dy
+�
+vk
+in Ω.
+Let us notice the very close analogy between the two limit problems. The first equation of (1.11) is exactly
+the first one in (1.10) (the boundary condition uk
+0 = 0 on ∂D allows to consider uk
+0 as an element of H1
+#(C) in
+(1.11)) so that the only difference between (1.11) and (1.10) lies in vk arising in (1.11) is a function depending also
+
+ON THE LIMIT SPECTRUM OF A DEGENERATE OPERATOR
+5
+on the variable x′ ∈ ω while in (1.10) it depends only on the vertical variable x3 ∈ (0, L). The dependence of vk
+with respect to x′ in (1.11) 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.
+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; 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. 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. 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].
+Note also that the existence and the uniqueness of uk
+0 in (1.10) is ensured by the fact that λk belongs to the
+resolvent ρ(−∆′
+y) of −∆′
+y since λk < µ1.
+The fact that
+�
+D
+uk
+0 dy ̸= 0 will be proved in section 2, see (2.17), using the constant 1
+µ1
+in the Poincaré
+inequality (in fact such value is the best constant for the Poincaré inequality).
+Remark 1.2. It is natural to ask what is the relationship between the problem (1.10) (or the problem (1.11)) and
+the classical formulation of eigenvalue problems. In fact, (1.10) is derived from the system (2.10) which in turn is
+derived from the equation (2.9) satisfied by the pair (uk, vk), see the details of the proof in section 2 below. If one
+integrates the first equation of (2.10) over D, we get an equivalent formulation of (2.10) as follows
+(1.12)
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+uk(y, x3) ∈ L2((0, L); 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).
+Another equivalent formulation of (1.12) is the following
+(1.13)
+A0
+�uk
+vk
+�
+= λk
+�uk
+vk
+�
+where the operator A0 is defined by A0 : D(A0) → H := L2(Ω) × L2(0, L) with
+(1.14)
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+D(A0) =
+��u
+v
+�
+∈ L2(0, L; H1(D) × H1
+0(0, L); 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).
+We see from (1.13) and (1.14) the sharp difference between the bounded selfadjoint operator Aε and the limit
+operator A0 which is no more a bounded selfadjoint operator.
+Of course, the same remark may be made about the homogenized problem given by (1.11).
+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(��) (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.e. ̸= 0) is not straightforward. To overcome this difficulty, we will use
+an extension technique (see [10], [29]) combined with another slightly more intricate argument.
+From now on and based on the previous comments, we will focus on the asymptotic analysis of the singular
+perturbation problem (1.9) (the study of the reduction of dimension occurring in each cell). This kind of problems
+is usually encountered in the study of thin structures, see for instance [16] and [21].
+
+6
+KAÏS AMMARI AND ALI SILI
+Our main results may be stated as follows.
+Theorem 1.3. For each k = 1, 2, ..., the sequence of eigenvalues (λk
+ε)ε of (1.9) 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; H1(C)); 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; H1(C)) ×
+H1
+0(0, L) of (1.10) with vk ̸= 0 such that for the whole sequence ε, one has
+(1.15)
+λk
+ε → λk,
+(1.16)
+uk
+ε −→ uk(y, x3) := (λkuk
+0 + 1)vk strongly in L2(0, L; H1(C)),
+(1.17)
+uk
+εχM −→ vkχM strongly in L2(C; H1
+0(0, L)).
+Any λk such that 0 < λk < µ1 is a simple eigenvalue of the limit operator A0.
+Conversely, problem (1.10) admits non trivial solutions such that 0 < λk < µ1 and any λ ∈ (0, µ1[ which is
+an eigenvalue of (1.10) is a limit of a sequence (λk
+ε)ε of eigenvalues of (1.9).
+The unique accumulation point of the sequence (λk)k is the first eigenvalue µ1 of −∆′
+y; hence
+lim
+k→+∞ λk =
+µ1.
+Remark 1.4. The property vk ̸= 0 may be deduced from the strong convergence (1.16) 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.
+Regarding the homogenization problem, the result is in all respects similar to that of 3d − 1d. We state it
+through the following theorem which is the homogenized version of Theorem 1.3. To state the results, we need
+the use of the two-scale convergence, see [1], [20], [28]. We use the notation
+2−sc
+⇀ (resp.
+2−sc
+−→) for the two-scale
+convergence (resp. the strong two-scale convergence).
+Theorem 1.5. For each k = 1, 2, ..., the sequence of eigenvalues (λk
+ε)ε of (1.8) 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; H1(ω)); 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; H1
+#(C))×
+L2(ω; H1
+0(0, L)) of (1.11) with vk ̸= 0 such that for the whole sequence ε, one has
+(1.18)
+λk
+ε → λk,
+(1.19)
+uk
+ε
+2−sc
+−→ uk(x, y) := (λkuk
+0 + 1)vk,
+with the following corrector result
+(1.20)
+�
+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.
+Any λk such that 0 < λk < µ1 is a simple eigenvalue of the limit operator A0.
+Conversely, any eigenvalue λ ∈ (0, µ1[ of problem (1.11) is a limit of a sequence (λk
+ε)ε of eigenvalues of
+(1.8).
+The sequence (λk)k converges to µ1.
+Remark 1.6. 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; 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.
+
+ON THE LIMIT SPECTRUM OF A DEGENERATE OPERATOR
+7
+However, we will prove (see (2.41)) that for 0 < λk < µ1, the second equation describing the vibrations of the
+string (0, L) may be written as
+(1.21)
+− 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. Of course, the spectrum σ0 of the limit operator A0 contains eigenvalues on the right of µ1; in
+particular, (1.21) shows that any eigenvalue µn of −∆′ such that cn =
+�
+D
+fndy ̸= 0 is an accumulation point of
+σ0. 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.
+Remark also that in the homogenization setting, the analogous result of the convergence (1.17) is the conver-
+gence
+�
+Ω
+����
+∂uk
+ε
+∂x3
+− ∂vk
+∂x3
+����
+2
+χMε(x′) dx → 0 obtained from the corrector result (1.20). However, the latter does not
+mean that the sequence uk
+εχMε converges strongly in L2(ω; 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 (1.17). Unfortunately, because of the
+oscillations induced by the homogenization process, such exact analogue of (1.17) is false. This is one of the few
+differences between the 3d − 1d problem and the homogenization problem.
+Finally we point out other possible scalings of the form εγχFε + εδχMε as addressed in [11], [12], [25]. For
+instance in the static case, one can refer to [11]. 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.
+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. 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.
+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. 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]. On the other
+hand, one can consider other scalings of the form εγχFε + εδχMε as addressed in [11], [12], [25] in the static case.
+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, +∞[. Here we address the critical case in the framework of the Laplacian operator for the sake
+of simplicity and brevity.
+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. 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.
+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].
+2. PROOF OF THE RESULTS IN THE CASE OF A SINGLE THIN STRUCTURE: THE REDUCTION OF DIMENSION
+3d − 1d
+2.1. Apriori estimate on the sequence of eigenvalues and eigenvectors.
+Proposition 2.1. For each k = 1, 2, ..., the sequence (λk
+ε, uk
+ε) of eigenpairs of (1.9) is bounded in R×L2(0, L; H1(C)).
+There exist (λk, uk, vk) ∈ (0, µ1) × L2(0, L; H1(C)) × H1
+0(0, L) and a subsequence of ε still denoted by ε such
+that
+(2.1)
+uk
+ε ⇀ uk weakly in L2(0, L; H1(C)) and uk(y, x3) = vk(x3) in M = (C \ D) × (0, L),
+(2.2)
+∂uk
+ε
+∂x3
+χM ⇀ dvk
+dx3
+χM
+weakly in L2(Ω),
+(2.3)
+λk
+ε → λk.
+Proof. We first prove an apriori estimate on the sequence of eigenvalues which will play a key role in the sequel.
+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.
+We claim that
+(2.4)
+∀ ε,
+∀ k = 1, 2, ...,
+λk
+ε ≤ µ1 + ε2λ0
+k.
+Indeed, we use the well known min-max formula giving the k-th eigenvalue λk
+ε of (1.9),
+(2.5)
+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 (1.5) (with ε = 1) and the min runs over all subspaces V k of Vs with finite
+dimension k.
+Let φ(y) be an eigenvector associated to µ1 extended by zero in C \ D. Then φ(y)ψ(x3) belongs to Vs for
+any ψ ∈ H1
+0(0, L) and φψ = 0 in M := (C \ D) × (0, L).
+Let V k be the subspace of Vs spanned by
+�
+φv1, φv2, ..., φvk�
+where v1, v2, ..., vk denote the associated
+eigenvectors to the first k eigenvalues λ0
+1, λ0
+2, ..., λ0
+k of − d2
+dx2
+3
+with homogeneous Dirichlet boundary conditions.
+
+ON THE LIMIT SPECTRUM OF A DEGENERATE OPERATOR
+9
+For any u = α1φv1 + ... + αkφvk ∈ V k, we have u = 0 in M and since v1, v2, ..., vk, ..., is an orthonormal
+basis in H1
+0(0, L) we also have
+(2.6)
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+Ω
+u2dy dx3 =
+�
+D
+φ2 dy
+� L
+0
+�
+α2
+1(v1)2 + ... + α2
+k(vk)2�
+dx3
+=
+�
+α2
+1 + ... + α2
+k
+� �
+D
+φ2 dy,
+�
+Ω
+|∇′
+yu|2dy dx3 =
+�
+α2
+1 + ... + α2
+k
+� �
+D
+|∇′
+yφ|2 dy
+=
+�
+α2
+1 + ... + α2
+k
+�
+µ1
+�
+D
+|φ|2 dy,
+�
+Ω
+ε2
+����
+∂u
+∂x3
+����
+2
+dy dx3 = ε2
+� L
+0
+�
+α2
+1
+����
+dv1
+dx3
+����
+2
++ ... + α2
+k
+����
+dvk
+dx3
+����
+2�
+dx3
+�
+D
+|φ|2 dy,
+= ε2�
+α2
+1λ0
+1 + ... + α2
+kλ0
+k
+� �
+D
+|φ|2 dy ≤ ε2λ0
+k
+�
+α2
+1 + ... + α2
+k
+� �
+D
+|φ|2 dy.
+Note that the equality occurring in the fifth line of (2.6) is a consequence of the equation −∆′
+yφ = µ1φ
+in D.
+Hence, using (2.6) in the min-max formula above, we get estimate (2.4).
+We obtain that λk ∈ (0, µ1) by passing to the limit (for a subsequence of ε) in (2.4).
+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.
+Turning back to (1.9) and taking uk
+ε (with ∥ uk
+ε ∥L2(Ω)= 1) as a test function, we get
+(2.7)
+�
+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.
+The last estimate implies that ∇′uk
+ε is bounded in L2(Ω) and thus uk
+ε is bounded in L2(0, L; H1(C)). Hence, there
+exist a sequence of ε and uk ∈ L2(C; H1
+0(0, L)) such that the convergence (2.1) holds true.
+One has ∇′uk
+εχM(y) ⇀ ∇′ukχM weakly in L2(Ω). But ∇′uk
+εχM which is bounded in L2(Ω) by Cε
+strongly converges to zero in L2(Ω). Hence, ∇′ukχM = 0 which means that uk = vk(x3) for some vk ∈ L2(0, L)
+a.e. in M. 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; 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). Hence vk ∈ H1
+0(0, L) and the convergence (2.2) holds true. The proof of
+Proposition 2.1 is complete.
+□
+2.2. The limit problem associated to (1.9). Choosing a test function in (1.9) in the form φ = ¯u with ¯u =
+¯v(x3) in M and (¯u, ¯v) ∈ Vs × H1
+0(0, L), we get from (1.9)
+(2.8)
+�
+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.
+
+10
+KAÏS AMMARI AND ALI SILI
+Passing to the limit in this equation, we get with the help of (2.1)
+(2.9)
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+(uk, vk) ∈ L2(0, L; 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.
+Finally a density argument allows to extend (2.9) to all test functions ¯u ∈ L2(0, L; H1(C)) such that ¯u = ¯v in M
+and ¯v ∈ H1
+0(0, L).
+Choosing successively in (2.9) ¯u ∈ L2(0, L; H1(C)) such that ¯u = 0 in M and then ¯u ∈ L2(0, L; 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 (2.8) may be split into two equations leading to the following
+equivalent system
+(2.10)
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+uk(y, x3) ∈ L2((0, L); 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).
+Remark 2.2. Eigenvectors of (2.10) corresponding to eigenvalues λk < µ1 are pairs (uk, vk) made up of two
+inseparable elements. In particular, if vk = 0 then uk = 0 as shown by (2.10). Indeed, otherwise uk should be
+an eigenvector of −∆′
+y associated to the eigenvalue λk < µ1 which is a contradiction. Conversely if uk = 0 then
+vk = 0 since almost everywhere in (0, L), we have vk = uk on the boundary of D. Hence, the eigenvectors (uk, vk)
+of the limit operator are such that uk ̸= 0 and vk ̸= 0.
+We now prove that (1.10) and (2.10) are equivalent if one defines uk by (2.11) and then we will improve the
+lower bound of the limit eigenvalues using (1.10).
+Proposition 2.3. If (λk, uk, vk) solves the system (2.10) with 0 < λk < µ1, then vk ̸= 0 and uk writes as
+(2.11)
+uk(y, x3) = (λkuk
+0(y) + 1)vk(x3)
+where (λk, uk
+0, vk) solves (1.10). 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.
+Proof. Assume that (uk, vk) is a non trivial solution of (2.10), i.e, (uk, vk) is an eigenvector of the limit operator.
+Then according to the Remark 2.2 above, vk ̸= 0 and uk ̸= 0.
+Dividing by vk in the first system of (2.10), one can check easily that wk := uk
+vk − 1 is the unique solution of
+(2.12)
+�
+�
+�
+−∆′
+ywk = λkwk + λk in D
+wk = 0
+on ∂D.
+Note that the uniqueness of wk is ensured since λk < µ1 belongs to the resolvent of −∆′
+y. On the other hand, the
+function λkuk
+0 where uk
+0 is defined in (1.10) is also a solution of (2.12) so that the equality wk := uk
+vk − 1 = λkuk
+0
+holds true and therefore (2.11) follows. Using (2.11) in (2.10) we get (1.10).
+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.
+
+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,
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+Engrg. 187 (2000), 1-2, 91-117,
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+SIAM J. Math. Anal. 21 (1990), 823-836,
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+Paris, Série I, 346 (2008), 807-812,
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+Verlag, New York., (1999),
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+47 (2015), 3, 1671–1692,
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+[14] S. KESAVAN & N. SABU, Two-dimensional approximation of eigenvalue problems in shell theory: Flexural shells, Chin. Anna. of Math.,
+21 B:1 (2000), 1-16,
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+(1962),
+[16] H. LE DRET, Problèmes variationnels dans les multi-domaines: modélisation des jonctions et applications, Research in Applied Mathemat-
+ics, 19, Masson, Paris, (1991),
+[17] G. LEUGERING, S.A. NAZAROV & J. TASKINEN, The band-gap structures of the spectrum in a periodic medium of Masonry type,
+Networks and Het. Media., 15, 4 (2020), 555-580,
+[18] T. A. MEL’NYK & S. A. NAZAROV, Asymptotics of the Neumann spectral problem solution in a domain of “thick comb” type, J. Math.
+Sci. 85 (1997), 6, 2326-2346,
+[19] F. MURAT & A. SILI, A remark about the periodic homogenization of certain composite fibered media, Netw. Heterog. Media 15 (2020),1,
+125-142,
+[20] G. NGUETSENG, A General Convergence Result for a Functional Related to the Theory of Homogenization. SIAM J. Math Anal. 20 (1989),
+3, 608-623,
+[21] G. PANASENKO, Multi-scale modelling for structures and composites. Springer, (2005),
+[22] R. PARONI & A. SILI, Nonlocal effects by homogenization or 3D-1D dimension reduction in elastic materials reinforced by stiff fibers, J.
+Differential Equations, 260 (2016), no. 3, 2026-2059,
+[23] A. SILI, On the limit spectrum of a degenerate operator in the framework of periodic homogenization or singular perturbation problems,
+Comptes Rend. Math. 360 (2022), 1-23.
+[24] A. SILI, Homogenization of a nonlinear monotone problem in an anisotropic medium, Math. Models Methods Appl. Sci. 14 (2004), 3,
+329-353,
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+239-271,
+[27] D. YIHONG, Order Structure and Topological Methods in Nonlinear Partial Differential Equations, Vol.1: Maximum Principles and Appli-
+cations, World Scientific (2006),
+
+ON THE LIMIT SPECTRUM OF A DEGENERATE OPERATOR
+21
+[28] V.V. ZHIKOV, On an extension and application of the two-scale convergence method, Mat. Sb. 191, (2000), 973-1014,
+[29] V.V. ZHIKOV, S.M. KOZLOV & 0.A. OLEINIK, Homogenization of differential operators and integral functionals, Translated from the
+Russian by G.A. 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: [email protected]
+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: [email protected]
+

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+Email: Mandana Kariminejad 
+            [email protected] 
+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 = ���̅ + (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 
+ 
+

HNFJT4oBgHgl3EQfuC2G/content/tmp_files/load_file.txt

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+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) + (𝐵��𝑥 − 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 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFJT4oBgHgl3EQfuC2G/content/2301.11620v1.pdf'}
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+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'}
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