Journal articles on the topic 'Infinite-width limit'
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1
Pastur, L. "Eigenvalue distribution of large random matrices arising in deep neural networks: Orthogonal case." Journal of Mathematical Physics 63, no. 6 (June 1, 2022): 063505. http://dx.doi.org/10.1063/5.0085204.
Abstract:
This paper deals with the distribution of singular values of the input–output Jacobian of deep untrained neural networks in the limit of their infinite width. The Jacobian is the product of random matrices where the independent weight matrices alternate with diagonal matrices whose entries depend on the corresponding column of the nearest neighbor weight matrix. The problem has been considered in the several recent studies of the field for the Gaussian weights and biases and also for the weights that are Haar distributed orthogonal matrices and Gaussian biases. Based on a free probability argument, it was claimed in those papers that, in the limit of infinite width (matrix size), the singular value distribution of the Jacobian coincides with that of the analog of the Jacobian with special random but weight independent diagonal matrices, the case well known in random matrix theory. In this paper, we justify the claim for random Haar distributed weight matrices and Gaussian biases. This, in particular, justifies the validity of the mean field approximation in the infinite width limit for the deep untrained neural networks and extends the macroscopic universality of random matrix theory to this new class of random matrices.
2
Pacelli, R., S. Ariosto, M. Pastore, F. Ginelli, M. Gherardi, and P. Rotondo. "A statistical mechanics framework for Bayesian deep neural networks beyond the infinite-width limit." Nature Machine Intelligence 5, no. 12 (December 18, 2023): 1497–507. http://dx.doi.org/10.1038/s42256-023-00767-6.
3
Thorkildsen, Gunnar, and Helge B. Larsen. "X-ray diffraction in perfect t × l crystals. Rocking curves." Acta Crystallographica Section A Foundations of Crystallography 55, no. 5 (September 1, 1999): 840–54. http://dx.doi.org/10.1107/s0108767399002986.
Abstract:
A general formalism, based on the Takagi–Taupin equations, for calculating rocking curves in perfect t\times l crystals is presented. It includes nonsymmetrical scattering, refraction, and ordinary and anomalous absorption. t and l may be varied independently. In the limit of a semi-infinite crystal, the standard results from the fundamental theory are retrieved. For crystal dimensions less than the extinction length, the theory converges to the kinematical limit. Simulations for germanium and silicon show significant influence of crystal finiteness. When dynamical effects are prominent, the curves exhibit various degrees of asymmetry and the full width at half-maximum is generally larger than the corresponding Darwin width. This is attributed to combined Laue and Bragg contributions which are shifted with respect to each other owing to refraction.
4
Karr, D. G., J. C. Watson, and M. HooFatt. "Three-Dimensional Analysis of Ice Sheet Indentation: Limit Analysis Solutions." Journal of Offshore Mechanics and Arctic Engineering 111, no. 1 (February 1, 1989): 63–69. http://dx.doi.org/10.1115/1.3257141.
Abstract:
A method is presented for determining the collapse pressures of an ice sheet subjected to a uniformly distributed edge load by applying the upper-bound theorem of limit analysis. The ice sheet is idealized as a semi-infinite layer of elastic-perfectly plastic material. A quadratic anisotropic yield criterion is used to calculate the indentation pressures. The ice sheet consists of columnar ice and is assumed isotropic in the plane of the ice sheet. Upper-bound solutions are found by optimizing a three-dimensional discontinuous velocity field representing an assumed collapse pattern of the ice sheet. Solutions are based on various ratios of indentor width to ice thickness, thereby providing an envelope of indentation pressures over a range of aspect ratios, from conditions of plane strain to plane stress. Solutions are then compared with corresponding two and three-dimensional lower-bound analyses.
5
Landa, Haggai, Cecilia Cormick, and Giovanna Morigi. "Static Kinks in Chains of Interacting Atoms." Condensed Matter 5, no. 2 (May 13, 2020): 35. http://dx.doi.org/10.3390/condmat5020035.
Abstract:
We theoretically analyse the equation of topological solitons in a chain of particles interacting via a repulsive power-law potential and confined by a periodic lattice. Starting from the discrete model, we perform a gradient expansion and obtain the kink equation in the continuum limit for a power-law exponent n ≥ 1 . The power-law interaction modifies the sine-Gordon equation, giving rise to a rescaling of the coefficient multiplying the second derivative (the kink width) and to an additional integral term. We argue that the integral term does not affect the local properties of the kink, but it governs the behaviour at the asymptotics. The kink behaviour at the center is dominated by a sine-Gordon equation and its width tends to increase with the power law exponent. When the interaction is the Coulomb repulsion, in particular, the kink width depends logarithmically on the chain size. We define an appropriate thermodynamic limit and compare our results with existing studies performed for infinite chains. Our formalism allows one to systematically take into account the finite-size effects and also slowly varying external potentials, such as for instance the curvature in an ion trap.
6
AKHMEDIEV, N., J. M. SOTO-CRESPO, M. GRAPINET, and Ph GRELU. "DISSIPATIVE SOLITON PULSATIONS WITH PERIODS BEYOND THE LASER CAVITY ROUND TRIP TIME." Journal of Nonlinear Optical Physics & Materials 14, no. 02 (June 2005): 177–94. http://dx.doi.org/10.1142/s0218863505002645.
Abstract:
We review recent results on periodic pulsations of the soliton parameters in a passively mode-locked fiber laser. Solitons change their shape, amplitude, width and velocity periodically in time. These pulsations are limit cycles of a dissipative nonlinear system in an infinite-dimensional phase space. Pulsation periods can vary from a few to hundreds of round trips. We present a continuous model of a laser as well as a model with parameter management. The results of the modeling are supported with experimental results obtained using a fiber laser.
7
Zeng, Y., and S. Weinbaum. "Stokes flow through periodic orifices in a channel." Journal of Fluid Mechanics 263 (March 25, 1994): 207–26. http://dx.doi.org/10.1017/s0022112094004088.
Abstract:
This paper develops a three-dimensional infinite series solution for the Stokes flow through a parallel walled channel which is obstructed by a thin planar barrier with periodically spaced rectangular orifices of arbitrary aspect ratio B’/d’ and spacing D’. Here B’ is the half-height of the channel and d’ is the half-width of the orifice. The problem is motivated by recent electron microscopic studies of the intercellular channel between vascular endothelial cells which show a thin junction strand barrier with discontinuities or breaks whose spacing and width vary with the tissue. The solution for this flow is constructed as a superposition of Hasimoto's (1958) general solution for the two-dimensional flow through a periodic slit array in an infinite plane wall and a new three-dimensional solution which corrects for the top and bottom boundaries. In contrast to the well-known solutions of Sampson (1891) and Hasimoto (1958) for the flow through zero-thickness orifices of circular or elliptic cross-section or periodic slits in an infinite plane wall, which exhibit characteristic viscous velocity profiles, the present bounded solutions undergo a fascinating change in behaviour as the aspect ratio B’/d’ of the orifice opening is increased. For B’/d’ [Lt ] 1 and (D’ –- d’)/B’ of O(1) or greater, which represents a narrow channel, the velocity has a minimum at the orifice centreline, rises sharply near the orifice edges and then experiences a boundary-layer-like correction over a thickness of O(B’) to satisfy no-slip conditions. For B’/d’ of O(1) the profiles are similar to those in a rectangular duct with a maximum on the centreline, whereas for B’/d’ [Gt ] 1, which describes widely separated channel walls, the solution approaches Hasimoto's solution for the periodic infinite-slit array. In the limit (D’ –- d’)/B’ [Lt ] 1, where the width of the intervening barriers is small compared with the channel height, the solutions exhibit the same behaviour as Lee & Fung's (1969) solution for the flow past a single cylinder. The drag on the zero-thickness barriers in this case is nearly the same as for the cylinder for all aspect ratios.
8
DELEBECQUE, FANNY. "AN ASYMPTOTIC MODEL FOR THE TRANSPORT OF AN ELECTRON GAS IN A SLAB." Mathematical Models and Methods in Applied Sciences 21, no. 07 (July 2011): 1443–78. http://dx.doi.org/10.1142/s0218202511005453.
Abstract:
We study the limiting behavior of a Schrödinger–Poisson system describing a three-dimensional quantum gas that is confined along the vertical z-direction in a fine slab. The starting point is the three-dimensional Schrödinger–Poisson system with Dirichlet conditions on two horizontal planes z = 0 and z = ε, where the small parameter ε is the scale width of the slab. The limit ε → 0 appears to be an infinite system of two-dimensional nonlinear Schrödinger equations. Our strategy combines a refined analysis of the Poisson kernel acting on strongly confined densities and a time-averaging process that allows us to deal with the fast time oscillations.
9
VOJTA, MATTHIAS, YING ZHANG, and SUBIR SACHDEV. "RENORMALIZATION GROUP ANALYSIS OF QUANTUM CRITICAL POINTS IN d-WAVE SUPERCONDUCTORS." International Journal of Modern Physics B 14, no. 29n31 (December 20, 2000): 3719–34. http://dx.doi.org/10.1142/s0217979200004271.
Abstract:
We describe a search for renormalization group fixed points which control a second-order quantum phase transition between a dx2-y2-superconductor and some other superconducting ground state. Only a few candidate fixed points are found. In the finite temperature (T) quantum-critical region of some of these fixed points, the fermion quasiparticle lifetime is very short and the spectral function has an energy width of order kBT near the Fermi points. Under the same conditions, the thermal conductivity is infinite in the scaling limit. We thus provide simple, explicit, examples of quantum theories in two dimensions for which a purely fermionic quasiparticle description of transport is badly violated.
10
Jagannathan, Arjun, Kraig Winters, and Laurence Armi. "Stratified Flows over and around Long Dynamically Tall Mountain Ridges." Journal of the Atmospheric Sciences 76, no. 5 (May 1, 2019): 1265–87. http://dx.doi.org/10.1175/jas-d-18-0145.1.
Abstract:
Abstract Uniformly stratified flows approaching long and dynamically tall ridges develop two distinct flow components over disparate time scales. The fluid upstream and below a “blocking level” is stagnant in the limit of an infinite ridge and flows around the sides when the ridge extent is finite. The streamwise half-width of the obstacle at the blocking level arises as a natural inner length scale for the flow, while the excursion time over this half-width is an associated short time scale for the streamwise flow evolution. Over a longer time scale, low-level horizontal flow splitting leads to the establishment of an upstream layerwise potential flow beneath the blocking level. We demonstrate through numerical experiments that for sufficiently long ridges, crest control and streamwise asymmetry are seen on both the short and long time scales. On the short time scale, upstream blocking is established quickly and the flow is well described as a purely infinite-ridge overflow. Over the long time scale associated with flow splitting, low-level flow escapes around the sides, but the overflow continues to be hydraulically controlled and streamwise asymmetric in the neighborhood of the crest. We quantify this late-time overflow by estimating its volumetric transport and then briefly demonstrate how this approach can be extended to predict the overflow across nonuniform ridge shapes.
11
Ghosal, Sandip, and John D. Sherwood. "Screened Coulomb interactions with non-uniform surface charge." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 473, no. 2199 (March 2017): 20160906. http://dx.doi.org/10.1098/rspa.2016.0906.
Abstract:
The screened Coulomb interaction between a pair of infinite parallel planes with spatially varying surface charge is considered in the limit of small electrical potentials for arbitrary Debye lengths. A simple expression for the disjoining pressure is derived in terms of a two-dimensional integral in Fourier space. The integral is evaluated for periodic and random charge distributions and the disjoining pressure is expressed as a sum over Fourier–Bloch reciprocal lattice vectors or in terms of an integral involving the autocorrelation function, respectively. The force between planes with a finite area of uniform charge, a model for the DLVO interaction between finite surfaces, is also calculated. It is shown that the overspill of the charge cloud beyond the region immediately between the charged areas results in a reduction of the disjoining pressure, as reported by us recently in the long Debye length limit for planes of finite width.
12
Yang, Mimi X., Fuqian Yang, and Sanboh Lee. "Dielectric breakdown sizes of conducting plates." IMA Journal of Applied Mathematics 86, no. 3 (May 20, 2021): 502–13. http://dx.doi.org/10.1093/imamat/hxab013.
Abstract:
Abstract In this work, we propose mathematical formulations that detail the effect of the dielectric strength of dielectric material on the spatial distribution of electric field in an infinite space with a conducting plate. Using the dielectric strength of air as the maximum limit for the magnitude of electric field intensity and the equivalence of stored charge between two different zones, we determine the size of the dielectric breakdown region (the extended region with ionized material) for the conducting strip and the conducting disk charged to an electric voltage. The size of dielectric breakdown is proportional to the square of the applied voltage, and decreases with the increase of the width/radius of the conducting strip/disk.
13
MELLO, D. F. DE, and G. G. CABRERA. "LOCAL ORDER AND MAGNETIC FIELD EFFECTS ON THE ELECTRONIC PROPERTIES OF DISORDERED BINARY ALLOYS IN THE QUANTUM SITE PERCOLATION LIMIT." International Journal of Modern Physics B 13, no. 32 (December 30, 1999): 3861–77. http://dx.doi.org/10.1142/s0217979299004057.
Abstract:
Electronic properties of disordered binary alloys are studied via the calculation of the average Density of States (DOS) in two and three dimensions. We propose a new approximate scheme that allows for the inclusion of local order effects in finite geometries and extrapolates the behavior of infinite systems following finite-size scaling ideas. We particularly investigate the limit of the Quantum Site Percolation regime described by a tight-binding Hamiltonian. This limit was chosen to probe the role of short range order (SRO) properties under extreme conditions. The method is numerically highly efficient and asymptotically exact in important limits, predicting the correct DOS structure as a function of the SRO parameters. Magnetic field effects can also be included in our model to study the interplay of local order and the shifted quantum interference driven by the field. The average DOS is highly sensitive to changes in the SRO properties and striking effects are observed when a magnetic field is applied near the segregated regime. The new effects observed are twofold: there is a reduction of the band width and the formation of a gap in the middle of the band, both as a consequence of destructive interference of electronic paths and the loss of coherence for particular values of the magnetic field. The above phenomena are periodic in the magnetic flux. For other limits that imply strong localization, the magnetic field produces minor changes in the structure of the average DOS.
14
Karr, D. G. "Three-Dimensional Analysis of Ice Sheet Indentation: Lower-Bound Solutions." Journal of Offshore Mechanics and Arctic Engineering 110, no. 1 (February 1, 1988): 81–86. http://dx.doi.org/10.1115/1.3257128.
Abstract:
The methods of plastic limit analysis are used to determine the indentation pressures of a flat rigid punch on a columnar ice sheet. The ice sheet is idealized as a semi-infinite layer of elastic-perfectly plastic material. Representative strength parameters of columnar sea ice are used to define anisotropic yield criteria for the ice sheet. The anisotropic yield criteria reflect the variations in mechanical properties caused by the horizontal orientation of the c-axis of sea ice in the columnar zone. Numerical results are obtained by applying the lower-bound theorem of plastic limit analysis. A three-dimensional stress field is optimized for a given ice condition for various indentor sizes. The effects of varying the aspect ratio (defined as the ratio of indentor width to ice thickness) are then addressed. A comparison of results for intermediate aspect ratios to results for extremely high (plane stress) and extremely low (plane strain) aspect ratios is presented. It is found that the transition from plane stress to plane strain is governed by the tensile strength of the ice medium.
15
Erbin, H., V. Lahoche, and D. Ousmane Samary. "Non-perturbative renormalization for the neural network-QFT correspondence." Machine Learning: Science and Technology 3, no. 1 (February 21, 2022): 015027. http://dx.doi.org/10.1088/2632-2153/ac4f69.
Abstract:
Abstract In a recent work (Halverson et al 2021 Mach. Learn.: Sci. Technol. 2 035002), Halverson, Maiti and Stoner proposed a description of neural networks (NNs) in terms of a Wilsonian effective field theory. The infinite-width limit is mapped to a free field theory while finite N corrections are taken into account by interactions (non-Gaussian terms in the action). In this paper, we study two related aspects of this correspondence. First, we comment on the concepts of locality and power-counting in this context. Indeed, these usual space-time notions may not hold for NNs (since inputs can be arbitrary), however, the renormalization group (RG) provides natural notions of locality and scaling. Moreover, we comment on several subtleties, for example, that data components may not have a permutation symmetry: in that case, we argue that random tensor field theories could provide a natural generalization. Second, we improve the perturbative Wilsonian renormalization from Halverson et al (2021 Mach. Learn.: Sci. Technol. 2 035002) by providing an analysis in terms of the non-perturbative RG using the Wetterich-Morris equation. An important difference with usual non-perturbative RG analysis is that only the effective infrared 2-point function is known, which requires setting the problem with care. Our aim is to provide a useful formalism to investigate NNs behavior beyond the large-width limit (i.e. far from Gaussian limit) in a non-perturbative fashion. A major result of our analysis is that changing the standard deviation of the NN weight distribution can be interpreted as a renormalization flow in the space of networks. We focus on translations invariant kernels and provide preliminary numerical results.
16
Spetzler, Jesper, and Roel Snieder. "The Fresnel volume and transmitted waves." GEOPHYSICS 69, no. 3 (May 2004): 653–63. http://dx.doi.org/10.1190/1.1759451.
Abstract:
In seismic imaging experiments, it is common to use a geometric ray theory that is an asymptotic solution of the wave equation in the high‐frequency limit. Consequently, it is assumed that waves propagate along infinitely narrow lines through space, called rays, that join the source and receiver. In reality, recorded waves have a finite‐frequency content. The band limitation of waves implies that the propagation of waves is extended to a finite volume of space around the geometrical ray path. This volume is called the Fresnel volume. In this tutorial, we introduce the physics of the Fresnel volume and we present a solution of the wave equation that accounts for the band limitation of waves. The finite‐frequency wave theory specifies sensitivity kernels that linearly relate the traveltime and amplitude of band‐limited transmitted and reflected waves to slowness variations in the earth. The Fresnel zone and the finite‐frequency sensitivity kernels are closely connected through the concept of constructive interference of waves. The finite‐frequency wave theory leads to the counterintuitive result that a pointlike velocity perturbation placed on the geometric ray in three dimensions does not cause a perturbation of the phase of the wavefield. Also, it turns out that Fermat's theorem in the context of geometric ray theory is a special case of the finite‐frequency wave theory in the limit of infinite frequency. Last, we address the misconception that the width of the Fresnel volume limits the resolution in imaging experiments.
17
Hewitt, R. E., P. W. Duck, and A. J. Williams. "Injection into boundary layers: solutions beyond the classical form." Journal of Fluid Mechanics 822 (June 7, 2017): 617–39. http://dx.doi.org/10.1017/jfm.2017.288.
Abstract:
This theoretical and numerical study presents three-dimensional boundary-layer solutions for laminar incompressible flow adjacent to a semi-infinite flat plate, subject to a uniform free-stream speed and injection through the plate surface. The novelty in this case arises from a fully three-dimensional formulation, which also allows for slot injection over a spanwise length scale comparable to the boundary-layer thickness. This approach retains viscous effects in both the spanwise and transverse directions, and effectively results in a parabolised Navier–Stokes system (sometimes referred to as the ‘boundary-region equations’). Any injection profile can be described in this approach, but we restrict attention to three-dimensional states driven by a finite-width slot aligned with the flow direction and self-similar in their downstream development. The classical two-dimensional states are known to only exist up to a critical (‘blow off’) injection amplitude, but the three-dimensional solutions here appear possible for any injection velocity. These new states take the form of low-speed streamwise-aligned streaks whose geometry depends on the amplitude of injection and the spanwise width of the injection slot; intriguingly, although very low wall shear is typically obtained, streamwise flow reversal is not observed, however hard the blowing. Asymptotic descriptions are provided in the limit of increasing slot width and fixed injection velocity, which allow for classification of the solutions according to two bounding injection rates.
18
Babeshko, V. A., O. V. Evdokimova, O. M. Babeshko, V. S. Evdokimov, and M. V. Zaretskaya. "EXACT SOLUTION OF CONTACT PROBLEMS IN A FINITE-WIDTH BAND ON A MULTILAYER MEDIUM." Problems of Strength and Plasticity 85, no. 1 (2023): 36–44. http://dx.doi.org/10.32326/1814-9146-2023-85-1-36-44.
Abstract:
In this paper, for the first time, an exact solution of contact problems for rigid or deformable stamps in a strip of finite width is obtained. It is assumed that the strip is located on a multilayer base of finite thickness. The universal modeling method previously developed by the authors is used. With its help, the solutions of complex boundary value problems for systems of partial differential equations are reduced, using the Galerkin transform, to solving individual differential equations, among which the Helmholtz equations are the simplest. In an earlier work of the authors published, when studying the problem for a deformable stamp in a strip, we had to limit ourselves to an asymptotic solution that is valid only for strips of large relative width. The solution for a strip of any finite size was constrained by the impossibility of constructing an exact solution to the contact problem for a rigid stamp in a strip of any finite width. As a result of the exact solution of the Wiener-Hopf integral equation in a finite-width band for the case of a multilayer medium, this contact problem was solved. The approach applied to the solution consists in constructing an exact operator equation of an infinite system of algebraic equations for large-width bands, using the operator formula of functions from matrices and investigating the constructed solution in the range of small relative bandwidth. The solution obtained in this way coincides with the solution obtained another method, namely, the singular integral method for the case of a small relative bandwidth. The constructed solution brings closer to the problems of research for materials of complex rheologies of contact problems with a deformable stamp, the description of cracks of a new type in limited bodies, modeling of nano particles, the study of tectonic plates of limited dimensions.
19
CASTRO, IAN P. "Weakly stratified laminar flow past normal flat plates." Journal of Fluid Mechanics 454 (March 10, 2002): 21–46. http://dx.doi.org/10.1017/s0022112001007248.
Abstract:
Numerical computations of the steady, two-dimensional, incompressible, uniform velocity but stably stratified flow past a normal flat plate (of unit half-width) in a channel are presented. Attention is restricted to cases in which the stratification is weak enough to avoid occurrence of the gravity wave motions familiar in more strongly stratified flows over obstacles. The nature of the flow is explored for channel half-widths, H, in the range 5 [les ] H [les ] 100, for Reynolds numbers, Re, (based on body half-width and the upstream velocity, U) up to 600 and for stratification levels between zero (i.e. neutral flow) and the limit set by the first appearance of waves. The fourth parameter governing the flow is the Schmidt number, Sc, the ratio of the molecular diffusion of the agent providing the stratification to the molecular viscosity. For cases of very large (in the limit, infinite) Sc a novel technique is used, which avoids solving the density equation explicitly. Results are compared with the implications of the asymptotic theory of Chernyshenko & Castro (1996) and with earlier computations of neutral flows over both flat plates and circular cylinders. The qualitative behaviour in the various flow regimes identified by the theory is demonstrated, but it is also shown that in some cases a flow zone additional to those identified by the theory appears and that, in any case, precise agreement would, for most regimes, require very much higher Re and/or H. Some examples of multiple (i.e. non-unique) solutions are shown and we discuss the likelihood of these being genuine, rather than an artefact of the numerical scheme.
20
Borcea, L., J. Garnier, and K. Sølna. "Onset of energy equipartition among surface and body waves." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 477, no. 2246 (February 2021): 20200775. http://dx.doi.org/10.1098/rspa.2020.0775.
Abstract:
We derive a radiative transfer equation that accounts for coupling from surface waves to body waves and the other way around. The model is the acoustic wave equation in a two-dimensional waveguide with reflecting boundary. The waveguide has a thin, weakly randomly heterogeneous layer near the top surface, and a thick homogeneous layer beneath it. There are two types of modes that propagate along the axis of the waveguide: those that are almost trapped in the thin layer, and thus model surface waves, and those that penetrate deep in the waveguide, and thus model body waves. The remaining modes are evanescent waves. We introduce a mathematical theory of mode coupling induced by scattering in the thin layer, and derive a radiative transfer equation which quantifies the mean mode power exchange. We study the solution of this equation in the asymptotic limit of infinite width of the waveguide. The main result is a quantification of the rate of convergence of the mean mode powers toward equipartition.
21
Onofrei, Daniel, and Andrew E. Thaler. "Anomalous Localized Resonance Phenomena in the Nonmagnetic, Finite-Frequency Regime." Advances in Mathematical Physics 2016 (2016): 1–28. http://dx.doi.org/10.1155/2016/4156072.
Abstract:
The phenomenon of anomalous localized resonance (ALR) is observed at the interface between materials with positive and negative material parameters and is characterized by the fact that when a given source is placed near the interface, the electric and magnetic fields start to have very fast and large oscillations around the interface as the absorption in the materials becomes very small while they remain smooth and regular away from the interface. In this paper, we discuss the phenomenon of anomalous localized resonance (ALR) in the context of an infinite slab of homogeneous, nonmagnetic material (μ=1) with permittivityϵs=-1-iδfor some small lossδ≪1surrounded by positive, nonmagnetic, homogeneous media. We explicitly characterize the limit value of the product between frequency and the width of slab beyond which the ALR phenomenon does not occur and analyze the situation when the phenomenon is observed. In addition, we also construct sources for which the ALR phenomenon never appears.
22
Ren, He, and Wei-Feng Sun. "Characterizing Dielectric Permittivity of Nanoscale Dielectric Films by Electrostatic Micro-Probe Technology: Finite Element Simulations." Sensors 19, no. 24 (December 7, 2019): 5405. http://dx.doi.org/10.3390/s19245405.
Abstract:
Finite element simulations for detecting the dielectric permittivity of planar nanoscale dielectrics by electrostatic probe are performed to explore the microprobe technology of characterizing nanomaterials. The electrostatic force produced by the polarization of nanoscale dielectrics is analyzed by a capacitance gradient between the probe and nano-sample in an electrostatic detection system, in which sample thickness is varied in the range of 1 nm–10 μm, the width (diameter) encompasses from 100 nm to 10 μm, the tilt angle of probe alters between 0° and 20°, and the relative dielectric constant covers 2–1000 to represent a majority of dielectric materials. For dielectric thin films with infinite lateral dimension, the critical diameter is determined, not only by the geometric shape and tilt angle of detecting probe, but also by the thickness of the tested nanofilm. Meanwhile, for the thickness greater than 100 nm, the critical diameter is almost independent on the probe geometry while being primarily dominated by the thickness and dielectric permittivity of nanomaterials, which approximately complies a variation as exponential functions. For nanofilms with a plane size which can be regarded as infinite, a pertaining analytical formalism is established and verified for the film thickness in an ultrathin limit of 10–100 nm, with the probe axis being perpendicular and tilt to film plane, respectively. The present research suggests a general testing scheme for characterizing flat, nanoscale, dielectric materials on metal substrates by means of electrostatic microscopy, which can realize an accurate quantitative analysis of dielectric permittivity.
23
Fujioka, Hideki, and James B. Grotberg. "Steady Propagation of a Liquid Plug in a Two-Dimensional Channel." Journal of Biomechanical Engineering 126, no. 5 (October 1, 2004): 567–77. http://dx.doi.org/10.1115/1.1798051.
Abstract:
In this study, we investigate the steady propagation of a liquid plug within a two-dimensional channel lined by a uniform, thin liquid film. The Navier-Stokes equations with free-surface boundary conditions are solved using the finite volume numerical scheme. We examine the effect of varying plug propagation speed and plug length in both the Stokes flow limit and for finite Reynolds number (Re). For a fixed plug length, the trailing film thickness increases with plug propagation speed. If the plug length is greater than the channel width, the trailing film thickness agrees with previous theories for semi-infinite bubble propagation. As the plug length decreases below the channel width, the trailing film thickness decreases, and for finite Re there is significant interaction between the leading and trailing menisci and their local flow effects. A recirculation flow forms inside the plug core and is skewed towards the rear meniscus as Re increases. The recirculation velocity between both tips decreases with the plug length. The macroscopic pressure gradient, which is the pressure drop between the leading and trailing gas phases divided by the plug length, is a function of U and U2, where U is the plug propagation speed, when the fluid property and the channel geometry are fixed. The U2 term becomes dominant at small values of the plug length. A capillary wave develops at the front meniscus, with an amplitude that increases with Re, and this causes large local changes in wall shear stresses and pressures.
24
Kilani, M. I., A. Al-Salaymeh, and A. T. Al-Halhouli. "Effect of Channel Aspect Ratio on the Flow Performance of a Spiral-Channel Viscous Micropump." Journal of Fluids Engineering 128, no. 3 (November 1, 2005): 618–27. http://dx.doi.org/10.1115/1.2175169.
Abstract:
The paper investigates the effect of channel aspect ratio on the flow performance of a newly introduced spiral-channel viscous micropump. An approximate 2D analytical solution for the flow field, which ignores channel curvature but accounts for a finite wall height, is first developed at the lubrication limit. A number of 3D models for spiral pumps with different aspect ratios are then built and analyzed using the finite volume method. Numerical and analytical results are in good agreement and tend to support one another. The results are compared with an approximate 2D analytical solution developed for infinite aspect ratio, which neglects the effect of side walls, and assumes uniform velocity distribution across the channel width. The error in this approximation was found to exceed 5% for aspect ratios less than 10. Pressure and drag shape factors were introduced in the present work to express the effect of the pressure difference and boundary velocity on the flow rate at various aspect ratios for both moving and stationary walls. Also, it has been found numerically that the flow rate varies linearly with both the pressure difference and boundary velocity, which supports the validity of the linear lubrication model employed.
25
Román, Krishna, Andy Cumbicus, Saba Infante, and Rigoberto Fonseca-Delgado. "Deep Gaussian processes and infinite neural networks for the analysis of EEG signals in Alzheimer’s diseases." Revista de Matemática: Teoría y Aplicaciones 29, no. 2 (June 30, 2022): 289–312. http://dx.doi.org/10.15517/rmta.v29i2.48885.
Abstract:
Deep neural network models (DGPs) can be represented hierarchically by a sequential composition of layers. When the prior distribution over the weights and biases are independently identically distributed, there is an equivalence with Gaussian processes (GP) in the limit of an infinite net[1]work width. DGPs are non-parametric statistical models used to character[1]ize patterns of complex non-linear systems due to their flexibility, greater generalization capacity, and a natural way of making inferences about the parameters and states of the system. This article proposes a hierarchi[1]cal Bayesian structure to model the weights and biases of a deep neural network. We deduce a general formula to calculate the integrals of Gaussian processes with non-linear transfer densities and obtain a kernel to estimate the covariance functions. In the methodology, we conduct an empirical study analyzing an electroencephalogram (EEG) database for diagnosing Alzheimer’s disease. Additionally, the DGPs models are esti[1]mated and compared with the NN models for 5, 10, 50, 100, 500, and 1000 neurons in the hidden layer, considering two transfer functions: Recti[1]fied Linear Unit (ReLU) and hyperbolic Tangent (Tanh). The results show good performance in the classification of the signals. Finally, we use the mean square error as a goodness of fit measure to validate the proposed models, obtaining low estimation errors.
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DELLAR, PAUL J. "Planar channel flow in Braginskii magnetohydrodynamics." Journal of Fluid Mechanics 667 (January 14, 2011): 520–43. http://dx.doi.org/10.1017/s0022112010004507.
Abstract:
Braginskii magnetohydrodynamics (MHD) is a single-fluid description of large-scale motions in strongly magnetised plasmas. The ion Larmor radius in these plasmas is much shorter than the mean free path between collisions, so momentum transport across magnetic field lines is strongly suppressed. The relation between the strain rate and the viscous stress becomes highly anisotropic, with the viscous stress being predominantly aligned parallel to the magnetic field. We present an analytical study of the steady planar flow across an imposed uniform magnetic field driven by a uniform pressure gradient along a straight channel, the configuration known as Hartmann flow, in Braginskii MHD. The global momentum balance cannot be satisfied by just the parallel viscous stress, so we include the viscous stress perpendicular to magnetic field lines as well. The ratio of perpendicular to parallel viscosities is the key small parameter in our analysis. When another parameter, the Hartmann number, is large the flow is uniform across most of the channel, with boundary layers on either wall that are modifications of the Hartmann layers in standard isotropic MHD. However, the Hartmann layer solution predicts an infinite current and infinite shear at the wall, consistent with a local series solution of the underlying differential equation that is valid for all Hartmann numbers. These singularities are resolved by inner boundary layers whose width scales as the three-quarters power of the viscosity ratio, while the maximum velocity scales as the inverse one-quarter power of the viscosity ratio. The inner wall layers fit between the Hartmann layers, if present, and the walls. The solution thus does not approach a limit as the viscosity ratio tends to zero. Essential features of the solution, such as the maximum current and maximum velocity, are determined by the size of the viscosity ratio, which is the regularising small parameter.
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Yariv, Ehud, and John D. Sherwood. "Application of Schwarz–Christoffel mapping to the analysis of conduction through a slot." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 471, no. 2181 (September 2015): 20150292. http://dx.doi.org/10.1098/rspa.2015.0292.
Abstract:
We consider the generic problem of steady conduction through a slot traversing a non-conducting plate that separates two semi-infinite conducting regions. The current-density field is conservative; the dimensionless problem governing its potential depends upon a single geometric parameter, h , the ratio of the slot length (i.e. the plate thickness) to its width. We construct a Schwarz–Christoffel transformation to handle this two-dimensional transport problem. The transformation is expressed in terms of two parameters which are related to h through two implicit equations; in the limit h →0, it becomes explicit. Because of the slow decay of the current density at large distances from the slot, the integral representing the slot resistance diverges. The excess resistance of a finite-length slot relative to that of a zero-length slot is, however, finite. This excess resistance depends only upon the asymptotic behaviour of the potential far from the slot; it may therefore be directly obtained as a function of the two transformation parameters. Asymptotic approximations are found for the excess resistance at small and large h , respectively, scaling as h ln h and h . The single-slot solution is used to analyse conduction through a periodic array of widely spaced slots.
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Dong, Hao, Jean-Baptiste Doc, and Simon Félix. "Directivity of horns mounted in finite enclosures: A multimodal formulation." Journal of the Acoustical Society of America 155, no. 3 (March 1, 2024): 2270–79. http://dx.doi.org/10.1121/10.0025389.
Abstract:
The beamwidth is a primary directivity metric for the design of a constant directivity horn. To date, investigations on this property have predominantly been restricted to the half-space radiation or idealized geometries. This paper examines the beamwidth behavior of an axisymmetric horn mounted in a finite cylindrical enclosure by proposing an elegant multimodal solution to the far-field directivity pattern. The variation of beamwidth is examined for the frequency, dimensions of the enclosure, and shape of the horn. At low frequencies, a fitted model is proposed to precisely depict the intrinsic beam narrowing governed by the enclosure diffraction. The asymptotic behavior of the beamwidth is explored as the flange width increases. In the high-frequency range, the horn profile is a determinant of the directivity characteristics. We report the possibility of extending the bandwidth of a constant directivity horn by leveraging the enclosure diffraction effects. The proposed analytical method is highly accurate and much faster than the finite element method for wideband analysis. It allows for an arbitrary velocity distribution at the mouth of the horn and incorporates idealized flange configurations such as an infinite baffle, a zero-thickness closed baffle, and an infinitely long enclosure as limit cases.
29
Segadlo, Kai, Bastian Epping, Alexander van Meegen, David Dahmen, Michael Krämer, and Moritz Helias. "Unified field theoretical approach to deep and recurrent neuronal networks." Journal of Statistical Mechanics: Theory and Experiment 2022, no. 10 (October 1, 2022): 103401. http://dx.doi.org/10.1088/1742-5468/ac8e57.
Abstract:
Abstract Understanding capabilities and limitations of different network architectures is of fundamental importance to machine learning. Bayesian inference on Gaussian processes has proven to be a viable approach for studying recurrent and deep networks in the limit of infinite layer width, n → ∞. Here we present a unified and systematic derivation of the mean-field theory for both architectures that starts from first principles by employing established methods from statistical physics of disordered systems. The theory elucidates that while the mean-field equations are different with regard to their temporal structure, they yet yield identical Gaussian kernels when readouts are taken at a single time point or layer, respectively. Bayesian inference applied to classification then predicts identical performance and capabilities for the two architectures. Numerically, we find that convergence towards the mean-field theory is typically slower for recurrent networks than for deep networks and the convergence speed depends non-trivially on the parameters of the weight prior as well as the depth or number of time steps, respectively. Our method exposes that Gaussian processes are but the lowest order of a systematic expansion in 1/n and we compute next-to-leading-order corrections which turn out to be architecture-specific. The formalism thus paves the way to investigate the fundamental differences between recurrent and deep architectures at finite widths n.
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Karakida, Ryo, and Kazuki Osawa. "Understanding approximate Fisher information for fast convergence of natural gradient descent in wide neural networks*." Journal of Statistical Mechanics: Theory and Experiment 2021, no. 12 (December 1, 2021): 124010. http://dx.doi.org/10.1088/1742-5468/ac3ae3.
Abstract:
Abstract Natural gradient descent (NGD) helps to accelerate the convergence of gradient descent dynamics, but it requires approximations in large-scale deep neural networks because of its high computational cost. Empirical studies have confirmed that some NGD methods with approximate Fisher information converge sufficiently fast in practice. Nevertheless, it remains unclear from the theoretical perspective why and under what conditions such heuristic approximations work well. In this work, we reveal that, under specific conditions, NGD with approximate Fisher information achieves the same fast convergence to global minima as exact NGD. We consider deep neural networks in the infinite-width limit, and analyze the asymptotic training dynamics of NGD in function space via the neural tangent kernel. In the function space, the training dynamics with the approximate Fisher information are identical to those with the exact Fisher information, and they converge quickly. The fast convergence holds in layer-wise approximations; for instance, in block diagonal approximation where each block corresponds to a layer as well as in block tri-diagonal and K-FAC approximations. We also find that a unit-wise approximation achieves the same fast convergence under some assumptions. All of these different approximations have an isotropic gradient in the function space, and this plays a fundamental role in achieving the same convergence properties in training. Thus, the current study gives a novel and unified theoretical foundation with which to understand NGD methods in deep learning.
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Wise, Anthony, Chris W. Hughes, and Jeff A. Polton. "Bathymetric Influence on the Coastal Sea Level Response to Ocean Gyres at Western Boundaries." Journal of Physical Oceanography 48, no. 12 (December 2018): 2949–64. http://dx.doi.org/10.1175/jpo-d-18-0007.1.
Abstract:
AbstractIt is our aim with this paper to investigate how the presence of a continental shelf and slope alters the relationship between interior ocean dynamics and western boundary (coastal) sea level. The assumption of a flat-bottomed basin with vertical sidewall at the coast is shown to hide the role that depth plays in the net force acting on the coast. A linear β-plane theory is then developed describing the transmission of sea level over variable depth bathymetry as analogous to the steady advection–diffusion of a thermal fluid. The parameter , relating the friction parameter r to the bathymetry depth H and width , is found to determine the contribution of interior sea level to coastal sea level, with small giving maximum penetration and large maximum insulation. In the small (infinite friction) limit the frictional boundary layer extends far offshore, and coastal sea level tends toward the vertical sidewall solution. Adding simple stratification produces exactly the same result but with reduced effective depth and hence enhanced penetration. Penetration can be further enhanced by permitting weakly nonlinear variations of thermocline depth. Wider and shallower shelves relative to the overall scales are also shown to maximize penetration for realistic values of . The theory implies that resolution of bathymetry and representation of friction can have a large impact on simulated coastal sea level, calling into question the ability of coarse-resolution models to accurately represent processes determining the dynamic coastal sea level.
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LEPPINGTON, F. G., and R. A. SISSON. "On the interaction of a moving hollow vortex with an aerofoil, with application to sound generation." Journal of Fluid Mechanics 345 (August 25, 1997): 203–26. http://dx.doi.org/10.1017/s0022112097006253.
Abstract:
A hollow vortex in the form of a straight tube, parallel to the z-axis, and of radius a, moves in a uniform stream of fluid with velocity U in the x-direction, with U small compared with the sound speed c. This steady flow is disturbed by the presence of a thin symmetric fixed aerofoil. With a change of x-coordinate, the problem is equivalent to that of a moving aerofoil cutting through an initially fixed vortex in still fluid. The aim of this work is to determine the resulting perturbed flow, and to estimate the distant sound field. A detailed calculation is given for the perturbed velocity potential in the incompressible flow case, for the linearized equations in the limit of small aerofoil thickness. A formally exact solution involves a four-fold integral and an infinite sum over all mode numbers. For the important special case where the vortex tube has small radius a compared with the aerofoil width, the deformed vortex is characterized by a hypothetical vortex filament located at the ‘mean centre’ x¯(z, t), y¯(z, t) of the tube. Explicit results are given for x¯(z, t), y¯(z, t) for the case where the aerofoil has the elementary rectangular profile; results can then be obtained for more general and realistic cylindrical aerofoils by a single integral weighted with the derivative of the aerofoil thickness function. Finally the distant sound field is estimated, representing the aerofoil by a distribution of moving monopole sources and representing the effect of the deformed vortex in terms of compressible dipoles along the mean centre of the vortex.
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Lee, J., and J. M. Vanden-Broeck. "Bubbles rising in an inclined two-dimensional tube and jets falling along a wall." Journal of the Australian Mathematical Society. Series B. Applied Mathematics 39, no. 3 (January 1998): 332–49. http://dx.doi.org/10.1017/s0334270000009437.
Abstract:
AbstractThe motion of a two-dimensional bubble rising at a constant velocity U in an inclined tube of width H is considered. The bubble extends downwards without limit, and is bounded on the right by a wall of the tube, and on the left by a free surface. The same flow configuration describes also a jet emerging from a nozzle and falling down along an inclined wall. The acceleration of gravity g and the surface tension T are included in the free surface condition. The problem is characterized by the Froude number the angle β between the left wall and the horizontal, and the angle γ between the free surface and the right wall at the separation point. Numerical solutions are obtained via series truncation for all values of 0 < β < π. The results extend previous calculations of Vanden-Broeck [12–14] for β = π/2 and of Couët and Strumolo [3] for 0 < β < π/2. It is found that the behavior of the solutions depends on whether 0 < β 2π/3 or 2π/3 ≤ β < π. When T = 0, it is shown that there is a critical value F of Froude number for each 0 < β 2π/3 such that solutions with γ = 0, π/3 and π - β occur for F > Fc F = Fc and F < Fc respectively, and that all solutions are characterized by γ = 0 for 2π/3 ≤ β < π. When a small amount of surface tension T is included in the free surface condition, it is found that for each 0 < β < π there exists an infinite discrete set of values of F for which γ = π - β. A particular value F* of the Froude number for which T = 0 and γ = π - β is selected by taking the limit as T approaches zero. The numerical values of F* and the corresponding free surface profiles are found to be in good agreement with experimental data for bubbles rising in an inclined tube when 0 < β < π/2.
34
Ghosh, S., H. C. Chang, and M. Sen. "Heat-transfer enhancement due to slender recirculation and chaotic transport between counter-rotating eccentric cylinders." Journal of Fluid Mechanics 238 (May 1992): 119–54. http://dx.doi.org/10.1017/s0022112092001666.
Abstract:
Using Stokes flow between eccentric, counter-rotating cylinders as a prototype for bounded, nearly parallel lubrication flow, we investigate the effect of a slender recirculation region within the flow field on cross-stream heat or mass transport in the important limit of high Péclet number Pe where the enhancement over pure conduction heat transfer without recirculation is most pronounced. The steady enhancement is estimated with a matched asymptotic expansion to resolve the diffusive boundary layers at the separatrices which bound the recirculation region. The enhancement over pure conduction is shown to vary as ε½ at infinite Pe, where ε½ is the characteristic width of the recirculation region. The enhancement decays from this asymptote as Pe−½. If one perturbs the steady flow by a time-periodic forcing, fast relative to the convective and diffusive times, the separatrices undergo a homoclinic entanglement which allows fluid elements to cross the separatrices. We establish the existence of this homoclinic entanglement and show that the resulting chaotic particle transport further enhances the cross-stream flux. We estimate the penetration of the fluid elements across the separatrices and their effective diffusivity due to this chaotic transport by a Melnikov analysis for small-amplitude forcing. These and the steady results then provide quantitative estimates of the timeaveraged transport enhancement and allow optimization with respect to system parameters. An optimum forcing frequency which induces maximum heat-transfer enhancement is predicted and numerically verified. The predicted optimum frequency remains valid at strong forcing and large Pe where chaotic transport is as important as the recirculation mechanism. Since most heat and mass transport devices operate at high Pe, our analysis suggests that chaotic enhancement can improve their performance and that a small amplitude theory can be used to optimize its application.
35
Волосюк, Валерий Константинович, Семён Сергеевич Жила, Эдуард Алексеевич Цернэ, and Александр Иванович Стороженко. "МАТЕМАТИЧЕСКОЕ ОПИСАНИЕ ПРОЦЕДУР ПОСТРОЕНИЯ КОГЕРЕНТНЫХ ИЗОБРАЖЕНИЙ ПРИРОДНЫХ СРЕД В ЗОНЕ ФРАУНГОФЕРА МНОГОКАНАЛЬНЫМИ РАДИОТЕХНИЧЕСКИМИ СИСТЕМАМИ." Aerospace technic and technology, no. 4 (October 14, 2018): 92–97. http://dx.doi.org/10.32620/aktt.2018.4.11.
Abstract:
The structure of the electromagnetic field in the domain of its registration is considered in the case of the solution of problems of remote sensing of the underlying surfaces on the basis of the phenomenological approach. This approach is mainly based on the theory of ray optics and the Huygens-Fresnel principle. It allows to determine the radiated and scattered fields for complex types of surfaces. Analysis of the structure of the electromagnetic field shows that it can be regarded as a mathematical transformation over the true image of the surface. In this case, the basic procedures for the coherent imaging in the far-field Fraunhofer region by multichannel radio-engineering systems should be based on the inverse transformation. For incomplete restoration of the desired image, without the phase and attenuation due to propagation, the basic operation is the inverse Fourier transform on the angular coordinates. The quality of the imaging in the Fraunhofer zone is determined by the ambiguity function. In a simple case of a rectangular receiving domain, ambiguity function has the form of two sinc functions which width is proportional to wavelength, to height of sounding and the linear sizes of receiving domain. If the distance to each point of the surface is known, then it is possible to completely reconstruct the coherent image. In this case, it is necessary to apply sliding short-scale Fourier transform to the received electromagnetic field. Obtained results correspond to the classical theory of resonance scattering. While ambiguity function is constant in the infinite limits of integration for a specific fixed value of the direction, only one spectral component (spatial harmonic) can be extracted from the desired image. it Is possible to allocate an ever wider range of spatial frequencies with the narrowing of the ambiguity function. In the limit, when the ambiguity function is a delta function, the full spectrum of frequencies of the desired image can be extracted, i.e. this function can be completely restored. If it is not possible to create a system with narrow ambiguity function then the higher-quality coherent image can be obtained by the same receiving domain by scanning or movement in space
36
Copeland, David Watabe. "Fundamental Performance Limits of Heatsinks." Journal of Electronic Packaging 125, no. 2 (June 1, 2003): 221–25. http://dx.doi.org/10.1115/1.1569262.
Abstract:
Dedicated fan-duct-heatsink combinations have become a standard means of cooling computer processors. Most previous studies have considered optimization of fin geometry (pitch and thickness) with overall heatsink dimensions (width, height, length) fixed. The present study considers size requirements for the constraints of fixed air volume flow rate and pressure drop, fixed fan/blower power, and fixed thermal conductance. First, an ideal heatsink with infinite fin thermal conductivity is considered, providing simple power-law prediction of performance. Next, fins of finite thermal conductivity and thickness, as well as effects of developing flow are included in the analysis, permitting prediction and minimization of weight. Models of both levels of complexity can be used, previous to more detailed numerical and/or experimental studies, to design optimized heatsinks.
37
Raz, Shalom. "Beam stacking: A generalized preprocessing technique." GEOPHYSICS 52, no. 9 (September 1987): 1199–210. http://dx.doi.org/10.1190/1.1442383.
Abstract:
Gaussian beams are well understood frequency‐ domain entities combining the directional properties of plane waves with an effectively finite region of support. These outstanding properties are retained not only on a prescribed observation plane, but throughout the propagation path. A preprocessing sequence aimed at transforming raw seismic data into beam stacks is proposed. That is, time‐harmonic Gaussian beams are synthesized, replacing the plane waves generated by conventional slant‐stacking procedures. The suggested scheme is characterized by an open parameter, essentially the beam width, whose selection is critical to ultimate success. Specific criteria for choosing this parameter can be given. In the limits of zero and infinite beam widths, beam stacks degenerate to the original raw data and to the conventional slant stacks, respectively. Although beam stacking is basically a frequency‐domain procedure, a transformation into the time domain, using frequency constituents within selected bands, may be accomplished without losing finite spatial support. Advantages of choosing beam‐stacked data as a starting point for subsequent inversion may be cited on two levels. The intrinsic property of finite spatial support overcomes edge effects. In addition, the degree of localization achieved by beam stacking may point the way to new approaches to seismic imaging.
38
Moukhtari, Fatima-Ezzahra, and Brice Lecampion. "A semi-infinite hydraulic fracture driven by a shear-thinning fluid." Journal of Fluid Mechanics 838 (January 25, 2018): 573–605. http://dx.doi.org/10.1017/jfm.2017.900.
Abstract:
We use the Carreau rheological model which properly accounts for the shear-thinning behaviour between the low and high shear rate Newtonian limits to investigate the problem of a semi-infinite hydraulic fracture propagating at a constant velocity in an impermeable linearly elastic material. We show that the solution depends on four dimensionless parameters: a dimensionless toughness (function of the fracture velocity, confining stress, material and fluid parameters), a dimensionless transition shear stress (related to both fluid and material behaviour), the fluid shear-thinning index and the ratio between the high and low shear rate viscosities. We solve the complete problem numerically combining a Gauss–Chebyshev method for the discretization of the elasticity equation, the quasi-static fracture propagation condition and a finite difference scheme for the width-averaged lubrication flow. The solution exhibits a complex structure with up to four distinct asymptotic regions as one moves away from the fracture tip: a region governed by the classical linear elastic fracture mechanics behaviour near the tip, a high shear rate viscosity asymptotic and power-law asymptotic region in the intermediate field and a low shear rate viscosity asymptotic far away from the fracture tip. The occurrence and order of magnitude of the extent of these different viscous asymptotic regions are estimated analytically. Our results also quantify how shear thinning drastically reduces the size of the fluid lag compared to a Newtonian fluid. We also investigate simpler rheological models (power law, Ellis) and establish the small domain where they can properly reproduce the response obtained with the complete rheology.
39
Kulyavtseva, Svetlana, and Boris Pevchenko. "TO THE APPLICATION ASSESSMENT OF MASSTRANSFER ANALYTICAL CORRELATIONS TO DEFINE SORPTION CONSTANTS IN SPRM POLYMERIC MATERIALS." Perm National Research Polytechnic University Aerospace Engineering Bulletin, no. 66 (2021): 15–23. http://dx.doi.org/10.15593/2224-9982/2021.66.02.
Abstract:
The use of analytical solutions for Frick`s diffusion equation to define the main sorption constants leading to the necessity to solve the united problem is shown. To be more exact this is the application assessment of analytical solutions for more often used experimental results and the determination of the representative sample dimensions in the experiments which are need to construct the sorption/desorption curves. By numerical experiments it is established that the most exact values of masstransfer constants are determined for small and middle values of diffuser absorption temporary process with the help of analytical solutions. Because of the difference in real sample dimensions and structural models used for analytical solutions which as a rule are presented in the form of semi-infinite plates it is necessary to minimize the influence of side surfaces in real samples on the accuracy of masstransfer parameters determination. The numerical-analytic study according to the geometry of polymeric samples from the point of view their representation to define diffusion, solubility and swelling coefficients is conducted. Herewith it is brought out that to obtain the values of masstransfer constants to an accuracy of not more 3 %, the sample geometric features must be chosen with relative dimensions L/h >10 (L – width, h – height of sample, accordingly) – the conditions of plane omnidirectional absorption of the diffuser into the right-angle-formed plates. To obtain the diffusion, solubility and swelling coefficients to an accuracy of not more 3 % the cylinder samples with relative dimensions D/h >15 (D – diameter, h – height, accordingly) should be preferably used. The algorithm of masstransfer constants determination when jointly use the sorption curves and the numerical method of diffusion analysis – finite element method is proposed. Herewith the advantages of such approach removing the sample size and shape limits are shown in comparison with the analytical methods of diffusion analysis.
40
Chizat, Lénaïc, Maria Colombo, Xavier Fernández‐Real, and Alessio Figalli. "Infinite‐width limit of deep linear neural networks." Communications on Pure and Applied Mathematics, May 6, 2024. http://dx.doi.org/10.1002/cpa.22200.
Abstract:
AbstractThis paper studies the infinite‐width limit of deep linear neural networks (NNs) initialized with random parameters. We obtain that, when the number of parameters diverges, the training dynamics converge (in a precise sense) to the dynamics obtained from a gradient descent on an infinitely wide deterministic linear NN. Moreover, even if the weights remain random, we get their precise law along the training dynamics, and prove a quantitative convergence result of the linear predictor in terms of the number of parameters. We finally study the continuous‐time limit obtained for infinitely wide linear NNs and show that the linear predictors of the NN converge at an exponential rate to the minimal ‐norm minimizer of the risk.
41
KHORUNZHY, A., and W. KIRSCH. "Limit of infinite band width for product of two random matrices." Random Operators and Stochastic Equations 5, no. 4 (1997). http://dx.doi.org/10.1515/rose.1997.5.4.325.
42
Hanin, Boris. "Random neural networks in the infinite width limit as Gaussian processes." Annals of Applied Probability 33, no. 6A (December 1, 2023). http://dx.doi.org/10.1214/23-aap1933.
43
Demirtas, Mehmet, James Halverson, Anindita Maiti, Matthew D. Schwartz, and Keegan Stoner. "Neural Network Field Theories: Non-Gaussianity, Actions, and Locality." Machine Learning: Science and Technology, December 21, 2023. http://dx.doi.org/10.1088/2632-2153/ad17d3.
Abstract:
Abstract Both the path integral measure in field theory and ensembles of neural networks describe distributions over functions. When the central limit theorem can be applied in the infinite-width (infinite-$N$) limit, the ensemble of networks corresponds to a free field theory. Although an expansion in $1/N$ corresponds to interactions in the field theory, others, such as in a small breaking of the statistical independence of network parameters, can also lead to interacting theories. These other expansions can be advantageous over the $1/N$-expansion, for example by improved behavior with respect to the universal approximation theorem. Given the connected correlators of a field theory, one can systematically reconstruct the action order-by-order in the expansion parameter, using a new Feynman diagram prescription whose vertices are the connected correlators. This method is motivated by the Edgeworth expansion and allows one to derive actions for neural network field theories. Conversely, the correspondence allows one to engineer architectures realizing a given field theory by representing action deformations as deformations of neural network parameter densities. As an example, $\phi^4$ theory is realized as an infinite-$N$ neural network field theory.
44
Chen, Linxiao, and Joonas Turunen. "Ising Model on Random Triangulations of the Disk: Phase Transition." Communications in Mathematical Physics, December 20, 2022. http://dx.doi.org/10.1007/s00220-022-04508-5.
Abstract:
AbstractIn Chen and Turunen (Commun Math Phys 374(3):1577–1643, 2020), we have studied the Boltzmann random triangulation of the disk coupled to an Ising model on its faces with Dobrushin boundary condition at its critical temperature. In this paper, we investigate the phase transition of this model by extending our previous results to arbitrary temperature: We compute the partition function of the model at all temperatures, and derive several critical exponents associated with the infinite perimeter limit. We show that the model has a local limit at any temperature, whose properties depend drastically on the temperature. At high temperatures, the local limit is reminiscent of the uniform infinite half-planar triangulation decorated with a subcritical percolation. At low temperatures, the local limit develops a bottleneck of finite width due to the energy cost of the main Ising interface between the two spin clusters imposed by the Dobrushin boundary condition. This change can be summarized by a novel order parameter with a nice geometric meaning. In addition to the phase transition, we also generalize our construction of the local limit from the two-step asymptotic regime used in Chen and Turunen (2020) to a more natural diagonal asymptotic regime. We obtain in this regime a scaling limit related to the length of the main Ising interface, which coincides with predictions from the continuum theory of quantum surfaces (a.k.a. Liouville quantum gravity).
45
Urso, Vittoria, and Lucian A. Constantin. "Quasi-dimensional models applied to kinetic and exchange energy density functionals." European Physical Journal B 94, no. 7 (July 2021). http://dx.doi.org/10.1140/epjb/s10051-021-00159-y.
Abstract:
AbstractWe investigate the behavior of three-dimensional 3D exchange energy functional of density-functional theory in anisotropic systems with two-dimensional 2D character and 1D character. The local density approximation (LDA), the generalized gradient approximation (GGA), and the meta-GGA behave as functions of quantum well width. We use the infinite-barrier model (IBM) for the quantum well. In the first section, we describe the problem of three-dimensional exchange functional, in the second section we introduce the quasi-2D IBM system, in the third section we introduce the quasi-1D IBM system. Using that an exact-exchange functional provides the correct approach to the true two-dimensional limit, we want to show that the 2D limit can be considered as a constraint on approximate functionals. For the 1D limit case we also propose a new functional obtained with methods completely similar to those of 2D limit.
46
Sell, Torben, and Sumeetpal Sidhu Singh. "Trace-class Gaussian priors for Bayesian learning of neural networks with MCMC." Journal of the Royal Statistical Society Series B: Statistical Methodology, January 31, 2023. http://dx.doi.org/10.1093/jrsssb/qkac005.
Abstract:
Abstract This paper introduces a new neural network based prior for real valued functions. Each weight and bias of the neural network has an independent Gaussian prior, with the key novelty that the variances decrease in the width of the network in such a way that the resulting function is well defined in the limit of an infinite width network. We show that the induced posterior over functions is amenable to Monte Carlo sampling using Hilbert space Markov chain Monte Carlo (MCMC) methods. This type of MCMC is stable under mesh refinement, i.e. the acceptance probability does not degenerate as more parameters of the function's prior are introduced, even ad infinitum. We demonstrate these advantages over other function space priors, for example in Bayesian Reinforcement Learning.
47
Sun, Zhibin, Yang Zhao, Yining Hu, Daniel Dias, and Jian Ji. "Probabilistic analysis of width‐limited 3D slope in spatially variable soils: UBLA enhanced with efficiency‐improved discretization of horn‐like failure mechanism." International Journal for Numerical and Analytical Methods in Geomechanics, September 5, 2023. http://dx.doi.org/10.1002/nag.3615.
Abstract:
AbstractReliability analysis of earth slopes considering soil spatial variability has garnered significant attention from researchers. However, previous studies have predominantly focused on long slopes with infinite or ample width, such as dams and subgrades, but rare research was dedicated to slopes with width restricted by boundary constraints. In this regard, this paper proposes an efficient reliability framework suitable for width‐limited slopes in spatially variable soils. The proposed framework employed spatial discretization‐based upper bound limit analysis (UBLA) to conduct the deterministic slope stability analysis. The established failure mechanism can well capture the width‐limited three‐dimensional slip surface characteristics and satisfies the kinematically admissible conditions in spatially variable soils. Two innovative strategies are proposed to boost the efficiency of the discretized mechanism. They reduce the computational time to determine safety factors from over 20 min to less than 2 min. For the uncertainty modeling and slope reliability computing procedure, the Sparse Polynomial Chaos Expansion (SPCE) and Monte Carlo Simulation (MCS) methods are combined to generate the distribution of safety factors and failure probabilities. This combination addresses the high‐dimensional stochastic problem arising from 3D spatial variability. Using the proposed framework, a parametric analysis of a 3D spatially variable slope is conducted to investigate the impact of various factors. The findings of this paper are beneficial to the risk evolution of width‐limited slopes, and the efficiency enhancement strategies have the potential to expedite the limit analysis of other geo‐structures, such as tunnels, retaining walls, and foundations.
48
Hu, Weidong, Xinnian Zhu, Yongqing Zeng, Xiaohong Liu, and Chucai Peng. "Active earth pressure against flexible retaining wall for finite soils under the drum deformation mode." Scientific Reports 12, no. 1 (January 11, 2022). http://dx.doi.org/10.1038/s41598-021-04411-4.
Abstract:
AbstractA reasonable method is proposed to calculate the active earth pressure of finite soils based on the drum deformation mode of the flexible retaining wall close to the basement’s outer wall. The flexible retaining wall with cohesionless sand is studied, and the ultimate failure angle of finite soils close to the basement’s outer wall is obtained using the Coulomb theory. Soil arch theory is led to get the earth pressure coefficient in the subarea using the trace line of minor principal stress of circular arc after stress deflection. The soil layers at the top and bottom part of the retaining wall are restrained when the drum deformation occurs, and the soil layers are in a non-limit state. The linear relationship between the wall movement’s magnitude and the mobilization of the internal friction angle and the wall friction anger is presented. The level layer analysis method is modified to propose the resultant force of active earth pressure, the action point’s height, and the pressure distribution. Model tests are carried out to emulate the process of drum deformation and soil rupture with limited width. Through image analysis, it is found that the failure angle of soil within the limited width is larger than that of infinite soil. With the increase of the aspect ratio, the failure angle gradually reduces and tends to be constant. Compared with the test results, it is shown that the horizontal earth pressure reduces with the reduction of the aspect ratio within critical width, and the resultant force decreases with the increase of the limit state region under the same ratio. The middle part of the distribution curve is concave. The active earth pressure strength decreases less than Coulomb’s value, the upper and lower soil layers are in the non-limit state, and the active earth pressure strength is more than Coulomb’s value.
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Pipolo de Gioia, Leonardo, and Ana Maria Raclariu. "Celestial sector in CFT: Conformally soft symmetries." SciPost Physics 17, no. 1 (July 2, 2024). http://dx.doi.org/10.21468/scipostphys.17.1.002.
Abstract:
We show that time intervals of width \Delta \tauΔτ in 3-dimensional conformal field theories (CFT_33) on the Lorentzian cylinder admit an infinite dimensional symmetry enhancement in the limit \Delta \tau → 0Δτ→0. The associated vector fields are approximate solutions to the conformal Killing equations in the strip labelled by a function and a conformal Killing vector on the sphere. An Inonu-Wigner contraction yields a set of symmetry generators obeying the extended BMS_44 algebra. We analyze the shadow stress tensor Ward identities in CFT_dd on the Lorentzian cylinder with all operator insertions in infinitesimal time intervals separated by \piπ. We demonstrate that both the leading and subleading conformally soft graviton theorems in (d-1)(d−1)-dimensional celestial CFT (CCFT_{d-1}d−1) can be recovered from the transverse traceless components of these Ward identities in the limit \Delta \tau → 0Δτ→0. A similar construction allows for the leading conformally soft gluon theorem in CCFT_{d-1}d−1 to be recovered from shadow current Ward identities in CFT_dd.
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Huang, Kan, Runing Liu, Yiwei Sun, Linyi Li, Yipeng Xie, and Xuejun Peng. "Study on the Calculation Method of Active Earth Pressure and Critical Width for Finite Soil Behind the Retaining Wall." Frontiers in Earth Science 10 (May 9, 2022). http://dx.doi.org/10.3389/feart.2022.883668.
Abstract:
The method to determine the active earth pressure and critical width for finite soil behind the retaining wall in mountainous areas is one of the concerns of geotechnical engineering. In order to study the active earth pressure distribution of the finite soil against the retaining wall and determine the critical width of the boundary between the finite soil and the semi-infinite soil, this study focuses on investigating a retaining wall with finite cohesionless backfill. The shape of the failure surface is assumed to be a cycloid passing through the heel of the wall in the limit equilibrium state. Considering the deflection of soil principal stress induced by wall–soil friction effect, a calculation method of active earth pressure for finite soil is proposed by using an arc-shaped small principal stress trajectory, and the rationality of this method is verified. On this basis, a calculation formula of the critical width for finite soil is proposed. The influence of the internal friction angle and the wall–soil friction angle on the critical width of finite soil is examined. The results indicate that the active earth pressure of finite soil presents a nonlinear drum distribution along the height of the retaining wall under the failure mode of the cycloidal surface. The maximum value of active earth pressure is close to the bottom of the wall. The critical width of finite soil decreases with the increase of the internal friction angle, and its variation rate decreases gradually. The critical width of finite soil increases with the increase of the wall–soil friction angle, and its variation rate also increases gradually. Under different internal friction angles and wall–soil friction angles, the critical width values of finite soil calculated by the assumption of the cycloidal failure surface are smaller than those calculated by the Coulomb earth pressure calculation method.