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We study the orthogonal polynomials and the Hankel determinants associated with Gaussian weight with two jump discontinuities. When the degree $n$ is finite, the orthogonal polynomials and the Hankel determinants are shown to be connected to the coupled Painlev\'e IV system. In the double scaling limit as the jump discontinuities tend to the edge of the spectrum and the degree $n$ grows to infinity, we establish the asymptotic expansions for the Hankel determinants and the orthogonal polynomials, which are expressed in terms of solutions of the coupled Painlev\'{e} II system. As applications, we re-derive the recently found Tracy-Widom type expressions for the gap probability of there being no eigenvalues in a finite interval near the the extreme eigenvalue of large Gaussian unitary ensemble and the limiting conditional distribution of the largest eigenvalue in Gaussian unitary ensemble by considering a thinned process., Comment: 34 pages, 2 figures

Ordinary differential equations of the form on the finite interval are under consideration. Here the functions and are absolutely continuous and positive and the coefficients of the differential expression are subject to the conditions where denotes the th antiderivative of the function in the sense of distributions. Our purpose is to derive analogues of the classical asymptotic Birkhoff-type representations for the fundamental system of solutions of the above equation with respect to the spectral parameter as in certain sectors of the complex plane . We reduce this equation to a system of first-order equations of the form where is a positive function, is a matrix with constant elements, the elements of the matrices and are integrable functions, and as . For systems of this kind, we obtain new results concerning the asymptotic representation of the fundamental solution matrix, which we use to make an asymptotic analysis of the above scalar equations of high order. Bibliography: 44 titles.

The last few years have seen considerable mathematical progress concerning the asymptotic structure of gravitational radiation in dynamical, astrophysical spacetimes. In this paper, we distil some of the key ideas from recent works and assemble them in a new way in order to make them more accessible to the wider general relativity community. In the process, we also discuss new physical findings. First, we introduce the conserved f(r)-modified Newman–Penrose charges on asymptotically flat spacetimes, and we show that these charges provide a dictionary that relates asymptotics of massless, general spin fields in different regions: asymptotic behaviour near i + (‘late-time tails’) can be read off from asymptotic behaviour towards I + , and, similarly, asymptotic behaviour towards I + can be read off from asymptotic behaviour near i − or I − . Using this dictionary, we then explain how: (I) the quadrupole approximation for a system of N infalling masses from i − causes the ‘peeling property towards I + ’ to be violated, and (II) this failure of peeling results in deviations from the usual predictions for tails in the late-time behaviour of gravitational radiation: instead of the Price’s law rate r Ψ [ 4 ] | I + ∼ u − 6 as u → ∞, we predict that r Ψ [ 4 ] | I + ∼ u − 4 , with the coefficient of this latter decay rate being a multiple of the monopole and quadrupole moments of the matter distribution in the infinite past.

An eigenvalue problem for Maxwell's equations with anisotropic cubic nonlinearity is studied. The problem describes propagation of transverse magnetic waves in a dielectric layer filled with (nonlinear) anisotropic Kerr medium. The nonlinearity involves two non-negative parameters a, b that are usually small. In the case a = b = 0 one arrives at a linear problem that has a finite number of solutions (eigenvalues and eigenwaves). If a > 0, b ≥ 0, then the nonlinear problem has infinitely many solutions; only a finite number of these solutions have linear counterparts. This shows that perturbation theory methods are inapplicable to study the problem in this case. For a = 0, b > 0 the nonlinear problem has a finite number of solutions; in this case each solution has a linear counterpart. Asymptotic distribution of the eigenvalues is found, periodicity of the eigenfunctions is proved and exact formula for the period is found, zeros of the eigenfunctions are determined, and a (nonlinear) eigenvalue comparison theorem is proved. Numerical experiments are presented.

In real-world networks, interactions between network elements are inherently time-delayed. These time-delays can both slow and destabilize the network, leading to poor performance. However, not all networks can be destabilized by time-delays. Previously, it has been shown that if a network is intrinsically stable, it maintains stability when constant time-delays are introduced. Here we show that intrinsically stable networks and a broad class of switched systems remain stable in the presence of any type of time-varying time-delays whether these delays are periodic, stochastic, or otherwise. We apply these results to a number of well-studied systems to demonstrate that intrinsic stability is both computationally inexpensive and improves on previous stability methods. Furthermore, we prove that the asymptotic state of an intrinsically stable switched system is independent of both the system's initial conditions and the presence of time-varying time-delays. Thus, we justify ignoring the occurance of any type of time-delays when modeling intrinsically stable real-world systems for asymptotic analysis, and provide an efficient means of engineering delay-robust systems.

Based on a formal asymptotic solution of Maxwell’s equations for a field propagating in a vacuum and of the relativistic Newton equation for an electron driven by the corresponding Lorentz force, we construct a description of the relativistic ponderomotive dynamics of an electron in the field of an intense focused electromagnetic envelope. A small parameter for the asymptotic expansion is proportional to the ratio of the radiation wavelength to the focal spot radius. Using the obtained averaged model of ponderomotive dynamics, electron beam scattering patterns are plotted with regard to an angle relative to the electromagnetic field propagation axis for laser pulses with a Gaussian transverse intensity distribution, as well as the energy spectra of scattered electrons corresponding to individual ranges of this angle. Scattered particles are absent in some ranges of the polar angle.

The loss surfaces of deep neural networks have been the subject of several studies, theoretical and experimental, over the last few years. One strand of work considers the complexity, in the sense of local optima, of high dimensional random functions with the aim of informing how local optimisation methods may perform in such complicated settings. Prior work of Choromanska et al (2015) established a direct link between the training loss surfaces of deep multi-layer perceptron networks and spherical multi-spin glass models under some very strong assumptions on the network and its data. In this work, we test the validity of this approach by removing the undesirable restriction to ReLU activation functions. In doing so, we chart a new path through the spin glass complexity calculations using supersymmetric methods in Random Matrix Theory which may prove useful in other contexts. Our results shed new light on both the strengths and the weaknesses of spin glass models in this context., 50 pages, 11 figures; references added for Kac-Rice reduction to RMT method; updates following JSTAT review and publication

We consider the one-dimensional Fisher?KPP equation with step-like initial data. Nolen et al showed that the solution u converges at long time to a travelling wave at a position , with error for any . With their methods, we find a refined shift such that in a frame moving with , the solution u satisfies for a certain profile independent of the initial data. The coefficient depends on the initial data, but is universal, and agrees with a finding of Berestycki et al. Furthermore, we predict the asymptotic forms of and u to arbitrarily high order.

In this paper, our work is based on a coupled nonlinear Schrödinger system in a two-mode nonlinear fiber. A (N,m)-generalized Darboux transformation is constructed to derive the Nth-order solutions, where the positive integers N and m denote the numbers of iterative times and of distinct spectral parameters, respectively. Based on the Nth-order solutions and the given steps to perform the asymptotic analysis, it is found that a degenerate dark-bright soliton is the nonlinear superposition of several asymptotic dark-bright solitons possessing the same profile. For those asymptotic dark-bright solitons, their velocities are z-dependent except that one of those velocities could become z-independent under certain condition, where z denotes the evolution dimension. Those asymptotic dark-bright solitons are reflected during the interaction. When a degenerate dark-bright soliton interacts with a nondegenerate/degenerate dark-bright soliton, the interaction is elastic, and the asymptotic bound-state dark-bright soliton with z-dependent or z-independent velocity could take place under certain condition. Our study extends the investigation on the degenerate solitons from the bright soliton case for the scalar equations to the dark-bright soliton case for a coupled system.

In this paper, we construct the Darboux transformation (DT) for the reverse-time integrable nonlocal nonlinear Schrodinger equation (nNLSE) by loop group method. Then we utilize the DT to derive soliton solutions with zero seed. We investigate the dynamical properties for those solutions and present a sufficient condition for the non-singularity of multi-soliton solutions. Furthermore, the asymptotic analysis of bounded multi-solutions has also been established by the determinant formula.

s The article is devoted to the study of the features of the occurrence of slip between layers of a thin two-layer plate with an initial ideal contact under the influence of critical pressure. A three-dimensional problem of the linear theory of elasticity for a thin two-layer plate with sliding conditions with friction at the interface is considered. To solve this problem, the method of asymptotic averaging is applied. An equation is established for calculating the critical pressure for a two-layer plate at which the layers begin to slip. A numerical example is considered for a two-layer fiberglass for different conditions of fixing at the ends. It is shown that the fixing conditions affect the critical pressure of the beginning of sliding.

A solution to the Boltzmann equation governing the thermal relic abundance of cold dark matter is constructed by matched asymptotic approximations. The approximation of the relic density is an asymptotic series valid when the abundance does not deviate significantly from its equilibrium value until small temperatures. Resonance and threshold effects are taken into account at leading order and found to be negligible unless the annihilation cross section is negligible at threshold. Comparisons are made to previously attempted constructions and to the freeze out approximation commonly employed in the literature. Extensions to higher order matching is outlined, and implications for solving related systems are discussed. We compare our results to a numerical determination of the relic abundance using a benchmark model and find a fantastic agreement. The method developed also serves as a solution to a wide class of problems containing an infinite order turning point., Comment: 15 pages, 6 figures

Under investigation in this paper is a relativistic Toda lattice system with one perturbation parameter α abbreviated as RTL_(α) system by Suris, which may describe the motions of particles in lattices interacting through an exponential interaction force. First of all, an integrable lattice hierarchy associated with an RTL_(α) system is constructed, from which some relevant integrable properties such as Hamiltonian structures, Liouville integrability and conservation laws are investigated. Secondly, the discrete generalized (m, 2N − m)-fold Darboux transformation is constructed to derive multi-soliton solutions, higher-order rational and semi-rational solutions, and their mixed solutions of an RTL_(α) system. The soliton elastic interactions and details of rational solutions are analyzed via the graphics and asymptotic analysis. Finally, soliton dynamical evolutions are investigated via numerical simulations, showing that a small noise has very little effect on the soliton propagation. These results may provide new insight into nonlinear lattice dynamics described by RTL_(α) system.

The mode-locking regions of a dynamical system are subsets of parameter space within which there exists an attracting periodic solution. For piecewise-linear continuous maps, these regions have a distinctive chain structure with points of zero width called shrinking points. In this paper a local analysis about an arbitrary shrinking point is performed. This is achieved by studying the symbolic itineraries of periodic solutions in nearby mode-locking regions and performing an asymptotic analysis on one-dimensional centre manifolds in order to build a comprehensive theoretical framework for the local dynamics. The main results are universal quantitative descriptions for the shape of nearby mode-locking regions, the location of nearby shrinking points, and the key properties of these shrinking points. The results are applied to the three-dimensional border-collision normal form, a model of an oscillator subject to dry friction, and a model of a DC/DC power converter.

We consider a generic biological process described by a dynamical system, subject to an input signal with a high-frequency periodic component. The rapid oscillations of the input signal induce inherently multiscale dynamics, motivating order-reduction techniques. It is intuitive that the system behaviour is well approximated by its response to the averaged input signal. However, changes to the high-frequency component that preserve the average signal are beyond the reach of such intuitive reasoning. In this study, we explore system response under the influence of such an input signal by exploiting the timescale separation between high-frequency input variations and system response time. Employing the asymptotic method of multiple scales, we establish that, in some circumstances, the intuitive approach is simply the leading-order asymptotic contribution. We focus on higher-order corrections that capture the response to the details of the high-frequency component beyond its average. This approach achieves a reduction in system complexity while providing valuable insight into the structure of the response to the oscillations. We develop the general theory for nonlinear systems, while highlighting the important case of systems affine in the state and the input signal, presenting examples of both discrete and continuum state spaces. Importantly, this class of systems encompasses biochemical reaction networks described by the chemical master equation and its continuum approximations. Finally, we apply the framework to a nonlinear system describing mRNA translation and protein expression previously studied in the literature. The analysis shines new light on several aspects of the system quantification and both extends and simplifies results previously obtained.

Using the asymptotic theory, explicit analytical expressions were obtained that allow one to calculate the stress-strain state of multilayer thin cylindrical anisotropic elastic composite shells for small deformations. Forenamed expressions allow us to obtain the distribution of all components of the stress-strain state without any assumptions about their nature of the distribution over the thickness. An algorithm was proposed that makes it possible to obtain the total stress tensor in the shell, and all components of the stress tensor are numerically calculated for the case of a cylindrical composite shell with axisymmetric bending by pressure. The character of the distribution of characteristics of the stress-strain state of the shell was analyzed and the corresponding graphs were constructed.

In this paper, we obtain the soliton solutions for the "good" Boussinesq equation on a constant background. Based on the asymptotic analysis of the solutions, we find that this equation admits both the elastic and resonant soliton interactions, as well as various partially inelastic interactions comprised of such two fundamental interactions. Via picture drawing, we present some examples of soliton interactions on nonzero backgrounds. Our results enrich the knowledge of soliton interactions in the (1+1)-dimensional integrable equation with a single field.

The paper presents an effective scheme for studying the direct and inverse problems of identifying the delamination of the elastic layer from the base. The proposed approach is based on the problem’s asymptotic analysis, based on the assumption that the relative size of the defect is small. In the framework of the asymptotic approach, the transcendental equations for reconstructing the bundle parameters are obtained. Identification is carried out using the additional information about the amplitude values of the displacement field measured at the upper boundary of the layer. The numerical results of the restoration are presented.

We review and systematize two (analytic) bootstrap techniques in two-dimensional conformal field theories using the S-modular transformation. The first one gives universal results in asymptotic regimes by relating extreme temperatures. Along with the presentation of known results, we use this technique to also derive asymptotic formulae for the Zamolodchikov recursion coefficients which match previous conjectures from numerics and from Regge asymptotic analysis. The second technique focuses on intermediate temperatures. We use it to sketch a methodology to derive a bound on off-diagonal squared OPE coefficients, as well as to improve existing bounds on the spectrum in case of non-negative diagonal OPE coefficients., 27+7 pages, 3 figures, v2 : improved the numerics for torus-one-point function bootstrap, corrected a missing factor in the torus-two-point function, conclusions unchanged, and new references added

The large N generating functions for the counting of chiral operators in , four-dimensional quiver gauge theories have previously been obtained in terms of the weighted adjacency matrix of the quiver diagram. We introduce the methods of multi-variate asymptotic analysis to study this counting in the limit of large charges. We describe a Hagedorn phase transition associated with the asymptotics, which refines and generalizes known results on the 2-matrix harmonic oscillator. Explicit results are obtained for two infinite classes of quiver theories, namely the generalized clover quivers and affine orbifold quivers.

We analyze the dynamics of an online algorithm for independent component analysis in the high-dimensional scaling limit. As the ambient dimension tends to infinity, and with proper time scaling, we show that the time-varying joint empirical measure of the target feature vector and the estimates provided by the algorithm will converge weakly to a deterministic measured-valued process that can be characterized as the unique solution of a nonlinear PDE. Numerical solutions of this PDE, which involves two spatial variables and one time variable, can be efficiently obtained. These solutions provide detailed information about the performance of the ICA algorithm, as many practical performance metrics are functionals of the joint empirical measures. Numerical simulations show that our asymptotic analysis is accurate even for moderate dimensions. In addition to providing a tool for understanding the performance of the algorithm, our PDE analysis also provides useful insight. In particular, in the high-dimensional limit, the original coupled dynamics associated with the algorithm will be asymptotically "decoupled", with each coordinate independently solving a 1-D effective minimization problem via stochastic gradient descent. Exploiting this insight to design new algorithms for achieving optimal trade-offs between computational and statistical efficiency may prove an interesting line of future research.

A theory of thin multilayer anisotropic viscoelastic cylindrical shells with allowance for quasistatic oscillations, based on the application of asymptotic analysis of small geometric parameters to the general three-dimensional equations of the viscoelasticity theory for curvilinear coordinates, is proposed. Recurrent sequences of local problems of the viscoelasticity theory for shells under the quasistatic pressure oscillations are formulated and analytical solutions are obtained. It is shown that with this approach it is possible to obtain expressions for all 6 components of the stress tensor over the thickness of the shell. An example of the calculation of a cylindrical viscoelastic shell with axisymmetric bending with allowance for quasistatic oscillations is considered. The graphs of the dependences of the components of the stress tensor on the local coordinate are given for different angles of layers in the material. An algorithm is proposed for obtaining explicit analytical equations for calculating the distribution of the total stress tensor components over a cylindrical shell under the action of quasistatic oscillations.

A mathematical model has been developed for calculating a temperature non-stationary field and thermal stresses in thin multilayer plates of fabric composite materials on an inorganic matrix based on an alum-chromophosphate binder. The model allows to take into account changes in the microstructure in the matrix and composite fibers during high- temperature heating, taking into account the physical and chemical transformations occurring in them, as well as to calculate thermal stresses in thin plates from the tissue inorganic composite taking into account the kinetics of changes in the characteristics of composites during heating. Methods of multiscale homogenization and finite element modeling of microstresses in periodic cells were used to calculate the characteristics of the inorganic composite, and the method of asymptotic averaging was used to calculate the stresses in the plates. Some results of numerical simulation of the change in stresses in the composite plate during heating are given

In this paper, we consider a three-dimensional model of cell signal transduction with delay. The deactivation of signalling proteins occurs throughout the cytosol and activation is localized to specific sites in the cell. The enzyme kinetic functions employ a constant delay to model the time lapse during reactions and also the recovery times associated with conformational changes. We use matched asymptotic expansions to construct the dynamic solutions of signalling protein concentrations. The result of the asymptotic analysis is a system of delayed differential algebraic equations. This reduced system is compared to numerical simulations of the full three-dimensional system. As well, we consider the stability of equilibrium solutions. We find that the systems under consideration may undergo Hopf bifurcations for certain delay values. In these cases sustained oscillations are observed. The Poincare–Lindstedt3 method is used to improve upon the asymptotic approximations. The simulations of the full three-dimensional system correspond well with simulations of the reduced delayed differential algebraic equations.

Problems in exponential asymptotics are typically characterized by divergence of the associated asymptotic expansion in the form of a factorial divided by a power. In this paper, we demonstrate that in certain classes of problems that involve coalescing singularities, a more general type of exponential-over-power divergence must be used. As a model example, we study the water waves produced by flow past an obstruction such as a surface-piercing ship. In the low speed or low Froude limit, the resultant water waves are exponentially small, and their formation is attributed to the singularities in the geometry of the obstruction. However, in cases where the singularities are closely spaced, the usual asymptotic theory fails. We present both a general asymptotic framework for handling such problems of coalescing singularities, and provide numerical and asymptotic results for particular examples.

With the aid of binary Bell polynomial and a general Riemann theta function, we introduce how to obtain the exact periodic wave solutions by applying the generalized Dp̄-operators in term of the Hirota direct method when the appropriate value of p̄ is determined. Furthermore, the resulting approach is applied to solve the extended (2+1)-dimensional Shallow Water Wave equation, and the periodic wave solution is obtained and reduced to soliton solution via asymptotic analysis.

We investigate the asymptotic behavior of a family of multiple orthogonal polynomials that is naturally linked with the normal matrix model with a monomial potential of arbitrary degree $d+1$. The polynomials that we investigate are multiple orthogonal with respect to a system of $d$ analytic weights defined on a symmetric $(d+1)$-star centered at the origin. In the first part we analyze in detail a vector equilibrium problem involving a system of $d$ interacting measures $(\mu_{1},\ldots,\mu_{d})$ supported on star-like sets in the plane. We show that in the subcritical regime, the first component $\mu_{1}^{*}$ of the solution to this problem is the asymptotic zero distribution of the multiple orthogonal polynomials. It also characterizes the domain where the eigenvalues in the normal matrix model accumulate, in the sense that the Schwarz function associated with the boundary of this domain can be expressed explicitly in terms of $\mu_{1}^{*}$. The second part of the paper is devoted to the asymptotic analysis of the multiple orthogonal polynomials. The asymptotic results are obtained again in the subcritical regime, and they follow from the Deift/Zhou steepest descent analysis of a Riemann-Hilbert problem of size $(d+1)\times (d+1)$. The vector equilibrium problem and the Riemann-Hilbert problem that we investigate are generalizations of those studied recently by Bleher-Kuijlaars in the case $d=2$., Comment: 65 pages, 10 figures, change in title, proof of Lemma 8.2 improved

An asymptotic expansion of the solution of an inhomogeneous matrix difference equation of general form is obtained. The case when there is no bound on the differences of the arguments is considered. The effect of the roots of the characteristic equation is taken into account. An integral estimate with a submultiplicative weight is established for the remainder in terms of the submultiplicative moment of the free term of the equation. Bibliography: 14 titles.

In this paper, a (3+1)-dimensional generalized Kadomtsev—Petviashvili (GKP) equation is investigated, which can be used to describe many nonlinear phenomena in fluid dynamics and plasma physics. Based on the generalized Bell's polynomials, we succinctly construct the Hirota's bilinear equation to the GKP equation. By virtue of multidimensional Riemann theta functions, a lucid and straightforward way is presented to explicitly construct multiperiodic Riemann theta function periodic waves (quasi-periodic waves) for the (3+1)-dimensional GKP equation. Interestingly, the one-periodic waves are well-known cnoidal waves, which are considered as one-dimensional models of periodic waves. The two-periodic waves are a direct generalization of one-periodic waves, their surface pattern is two-dimensional that they have two independent spatial periods in two independent horizontal directions. Finally, we analyze asymptotic behavior of the multiperiodic periodic waves, and rigorously present the relationships between the periodic waves and soliton solutions by a limiting procedure.

An important practical problem is the recovery of a turbulent velocity field from Lagrangian tracers that move with the fluid flow. Here, the filtering skill of L moving Lagrangian tracers in recovering random incompressible flow fields defined through a finite number of random Fourier modes is studied with full mathematical rigour. Despite the inherent nonlinearity in measuring noisy Lagrangian tracers, it is shown below that there are exact closed analytic formulas for the optimal filter for the velocity field involving Riccati equations with random coefficients for the covariance matrix. This mathematical structure allows a detailed asymptotic analysis of filter performance, both as time goes to infinity and as the number of noisy Lagrangian tracers, L, increases. In particular, the asymptotic gain of information from L-tracers grows only like ln L in a precise fashion; i.e., an exponential increase in the number of tracers is needed to reduce the uncertainty by a fixed amount; in other words, there is a practical information barrier. The proofs proceed through a rigourous mean field approximation of the random Ricatti equation. Also, as an intermediate step, geometric ergodicity with respect to the uniform measure on the period domain is proved for any fixed number L of noisy Lagrangian tracers. All of the above claims are confirmed by detailed numerical experiments presented here.