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In the field of enumeration of walks in cones, it is known how to compute asymptotically the number of excursions (finite paths in the cone with fixed length, starting and ending points, using jumps from a given step set). As it turns out, the associated critical exponent is related to the eigenvalues of a certain Dirichlet problem on a spherical domain. An important underlying question is to decide whether this asymptotic exponent is a (non-)rational number, as this has important consequences on the algebraic nature of the associated generating function. In this paper, we ask whether such an excursion sequence might admit an asymptotic expansion with a first rational exponent and a second non-rational exponent. While the current state of the art does not give any access to such many-term expansions, we look at the associated continuous problem, involving Brownian motion in cones. Our main result is to prove that in dimension three there exists a cone such that the heat kernel (the continuous analog of the excursion sequence) has the desired rational/non-rational asymptotic property. Our techniques come from spectral theory and perturbation theory. More specifically, our main tool is a new Hadamard formula, which has an independent interest and allows us to compute the derivative of eigenvalues of spherical triangles along infinitesimal variations of the angles. [ABSTRACT FROM AUTHOR]

Transient stability analysis is critical for maintaining the reliability and security of power systems. This paper provides a comprehensive review of research methods for transient stability analysis under large disturbances, detailing the modeling concepts and implementation approaches. The research methods for large disturbance transient stability analysis are categorized into five main types: simulation methods, direct methods, data-driven methods, analytical methods, and other methods. Within the analytical method category, several common analytical strategies are introduced, including the asymptotic expansion method, intrusive approximation method, and other analytical methods. The fundamental principles, characteristics, and recent research advancements of these methods are detailed, with particular attention to their performance in various aspects such as computational efficiency, accuracy, applicability to different system models, and stability region estimation. The advantages and disadvantages of each method are compared, offering insights to support further research into transient stability analysis for hybrid power grids under large disturbances. [ABSTRACT FROM AUTHOR]

Some properties of the constant in asymptotic expansion of iterates of a cubic function were investigated. This paper analyzed the monotonicity, differentiability of the constant with respect to the initial value and the functional equation that is satisfied. [ABSTRACT FROM AUTHOR]

We study an integrable connection with irregular singularities along a normally crossing divisor. The connection is obtained from an integrable connection of regular singular type by a confluence, and has irregular singularities along x = ∞ and y = ∞. Solutions are expressed by a double integral of Euler type with resonances among the exponents in the integrand. We specify twisted cycles that give main asymptotic behaviors in sectorial domains around (∞,∞). Then we obtain linear relations among the twisted cycles, and get an explicit expression of the Stokes multiplier. The methods to derive the asymptotic behaviors for double integrals and to get linear relations among twisted cycles in resonant case, which we developed in this paper, seem to be new. [ABSTRACT FROM AUTHOR]

This paper aims to provide an exact solution in the Laplace domain and related analytic approximations in short time limits for the class of boundary value problems of the one-dimensional linear parabolic equation with constant coefficients. The problem’s most general form involves a parameterized equation on a bounded interval, with unified specification of the three classical types of boundary conditions: Dirichlet, Neumann, and Robin. Under certain integrability assumptions, we have proven that a unique solution exists in the Laplace domain. This operational solution can be obtained in a closed form by using classical integral transforms. Four distinct cases have been identified based on the operational solution. Innovative formulas have been derived from these cases, which provide precise approximations within short timescales. These time-domain expressions are particularly useful for understanding the behavior of the solution at the boundaries. The formulas consist of elementary functions obtained from asymptotic expansions, and the estimation error can be minimized to the desired order of magnitude. The analytical approximations in short time limits can open up new perspectives and applications. Improved numerical efficiency in simulations of reaction-diffusion problems and of one-dimensional Stefan models are envisaged. [ABSTRACT FROM AUTHOR]

The Sturm–Liouville boundary value problem (SLBVP) stands as a fundamental cornerstone in the realm of mathematical analysis and physical modeling. Also known as the Sturm–Liouville problem (SLP), this paper explores the intricacies of this classical problem, particularly the relationship between its canonical and Liouville normal (Schrödinger) forms. While the conversion from the canonical to Schrödinger form using Liouville's transformation is well known in the literature, the inverse transformation seems to be neglected. Our study attempts to fill this gap by investigating the inverse of Liouville's transformation, that is, given any SLP in the Schrödinger form with some invariant function, we seek the SLP in its canonical form. By closely examining the second Paine–de Hoog–Anderson (PdHA) problem, we argue that retrieving the SLP in its canonical form can be notoriously difficult and can even be impossible to achieve in its exact form. Finding the inverse relationship between the two independent variables seems to be the main obstacle. We confirm this claim by considering four different scenarios, depending on the potential and density functions that appear in the corresponding invariant function. In the second PdHA problem, this invariant function takes a reciprocal quadratic binomial form. In some cases, the inverse Liouville transformation produces an exact expression for the SLP in its canonical form. In other situations, however, while an exact canonical form is not possible to obtain, we successfully derived the SLP in its canonical form asymptotically. [ABSTRACT FROM AUTHOR]

The main result of this paper presents a series expansion of the generalized Mathieu series Sm(r) with computable coefficients. This generalizes Elbert's work from m = 1 to arbitrary m ∈ Z>0, and provides a method for calculating the asymptotic formula of Sm(r) within a given error. Moreover, this paper revises an integral representation for Sm(r) given by Cerone and Lenard. [ABSTRACT FROM AUTHOR]

It is known that the constant e has the following series representations: ... The second series is extremely rapidly convergent. In this paper, we present asymptotic expansions and two-sided inequalities for the remainders ... Also, we present some inequalities and completely monotonic functions involving (1 + 1/x)x. We also consider a number of related developments on the subject of this paper. [ABSTRACT FROM AUTHOR]

Because of its flexibility and multiple meanings, the concept of information entropy in its continuous or discrete form has proven to be very relevant in numerous scientific branches. For example, it is used as a measure of disorder in thermodynamics, as a measure of uncertainty in statistical mechanics as well as in classical and quantum information science, as a measure of diversity in ecological structures and as a criterion for the classification of races and species in population dynamics. Orthogonal polynomials are a useful tool in solving and interpreting differential equations. Lately, this subject has been intensively studied in many areas. For example, in statistics, by using orthogonal polynomials to fit the desired model to the data, we are able to eliminate collinearity and to seek the same information as simple polynomials. In this paper, we consider the Tsallis, Kaniadakis and Varma entropies of Chebyshev polynomials of the first kind and obtain asymptotic expansions. In the particular case of quadratic entropies, there are given concrete computations. [ABSTRACT FROM AUTHOR]

The paper provides a precise error estimate for an asymptotic expansion of a certain stochastic control problem related to relative entropy minimization. In particular, it is shown that the expansion error depends on the regularity of functionals on path space. An efficient numerical scheme based on a weak approximation with Monte Carlo simulation is employed to implement the asymptotic expansion in multidimensional settings. Throughout numerical experiments, it is confirmed that the approximation error of the proposed scheme is consistent with the theoretical rate of convergence. [ABSTRACT FROM AUTHOR]

This paper presents a cryptanalysis attack on the RSA variant with modulus N = prq for r ≥ 2 with three public and private exponents (e1, d1), (e2, d2), (e3, d3) sharing the same modulus N where p and q are consider to be primes having the same bit size. Our attack shows that we get the private exponent σ1σ2σ3 < ( r-1/r+1)4, which makes the modulus vulnerable to Coppersmith's attacks and can lead to the factorization of N efficiently where d1 < N..., d2 < N..., and d3 < N.... The asymptotic bound of our attack is greater than the bounds for May [1], Zheng and Hu [2], and Lu et al. [3] for 2 ≤ r ≤ 10 and greater than Sarkar's [4] and [5] bounds for 5 ≤ r ≤ 10. [ABSTRACT FROM AUTHOR]

This paper aims to contribute to refining the e-values for testing precise hypotheses, especially when dealing with nuisance parameters, leveraging the effectiveness of asymptotic expansions of the posterior. The proposed approach offers the advantage of bypassing the need for elicitation of priors and reference functions for the nuisance parameters and the multidimensional integration step. For this purpose, starting from a Laplace approximation, a posterior distribution for the parameter of interest is only considered and then a suitable objective matching prior is introduced, ensuring that the posterior mode aligns with an equivariant frequentist estimator. Consequently, both Highest Probability Density credible sets and the e-value remain invariant. Some targeted and challenging examples are discussed. [ABSTRACT FROM AUTHOR]

This paper is concerned with the reconstruction of the support of the potential term for a time-fractional diffusion equation from the final measured data. The aim of this paper is to propose an accurate approach based on the topological derivative method. The idea is to formulate the reconstruction problem as a topology optimization one minimizing a given cost function. We derive a topological asymptotic expansion for the fractional model. The unknown support is reconstructed using the level-set curve of the topological gradient. We finally make some numerical examples proving the efficiency and accuracy of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we consider the application of the bifocusing method (BFM) for a fast identification of two-dimensional circle-like small inhomogeneities from measured scattered field data. Based on the asymptotic expansion formula for the scattered field in the presence of small inhomogeneities, we introduce the imaging functions of the BFM for both dielectric permittivity and magnetic permeability contrast cases. To examine the applicability and the various properties of the BFM, we show that the imaging functions can be expressed by the Bessel function of orders zero and one, as well as the characteristics (size, permittivity, and permeability) of the inhomogeneities. To support the theoretical results, various numerical results with synthetic and experimental data are presented. [ABSTRACT FROM AUTHOR]

We consider a C ∞ family of planar vector fields { X μ ˆ } μ ˆ ∈ W ˆ having a hyperbolic saddle and we study the Dulac map D (s ; μ ˆ) and the Dulac time T (s ; μ ˆ) from a transverse section at the stable separatrix to a transverse section at the unstable separatrix, both at arbitrary distance from the saddle. Since the hyperbolicity ratio λ of the saddle plays an important role, we treat it as an independent parameter, so that μ ˆ = (λ , μ) ∈ W ˆ = (0 , + ∞) × W , where W is an open subset of R N. For each μ ˆ 0 ∈ W ˆ and L > 0 , the functions D (s ; μ ˆ) and T (s ; μ ˆ) have an asymptotic expansion at s = 0 and μ ˆ ≈ μ ˆ 0 with the remainder being uniformly L -flat with respect to the parameters. The principal part of both asymptotic expansions is given in a monomial scale containing a deformation of the logarithm, the so-called Ecalle-Roussarie compensator. In this paper we are interested in the coefficients of these monomials, which are functions depending on μ ˆ that can be shown to be C ∞ in their respective domains and "universally" defined, meaning that their existence is stablished before fixing the flatness L and the unfolded parameter μ ˆ 0. Each coefficient has its own domain and it is of the form ((0 , + ∞) ∖ D) × W , where D a discrete set of rational numbers at which a resonance of the hyperbolicity ratio λ occurs. In our main result, Theorem A, we provide explicit expressions for some of these coefficients and to this end a fundamental tool is the employment of a sort of incomplete Mellin transform. With regard to these coefficients we also prove that they have poles of order at most two at D × W and we give the corresponding residue, that plays an important role when compensators appear in the principal part. Furthermore we prove a result, Corollary B, showing that in the analytic setting each coefficient given in Theorem A is meromorphic on (0 , + ∞) × W and has only poles, of order at most two, along D × W. [ABSTRACT FROM AUTHOR]

A physics-informed neural network (PINN) is a powerful tool for solving differential equations in solid and fluid mechanics. However, it suffers from singularly perturbed boundary-layer problems in which there exist sharp changes caused by a small perturbation parameter multiplying the highest-order derivatives. In this paper, we introduce Chien's composite expansion method into PINNs, and propose a novel architecture for the PINNs, namely, the Chien-PINN (C-PINN) method. This novel PINN method is validated by singularly perturbed differential equations, and successfully solves the well-known thin plate bending problems. In particular, no cumbersome matching conditions are needed for the C-PINN method, compared with the previous studies based on matched asymptotic expansions. [ABSTRACT FROM AUTHOR]

Unmanned aerial vehicle (UAV) communication and non-orthogonal multiple access (NOMA) are two promising technologies for wireless 5G networks and beyond. The UAVs can be used as flying base stations to form line-of-sight communication links to the Internet of things devices (IDs) and to enhance the performance of usual terrestrial cellular networks. Moreover, the UAVs can also be deployed as flying relay nodes for forwarding data from a base station (BS) to the IDs. On the other hand, non-orthogonal resource sharing for many concurrent users is exploited in NOMA, thus improving spectrum efficiency (SE) and supporting massive connections. The NOMA combined with energy harvesting (EH) in an amplify-and-forward (AF) with cooperative UAV systems is researched. Specifically, the UAVs act as rotary-wing relays to forward data from the BSs to two IDs. This paper focuses on the analysis of outage probabilities (OPs), system throughput, and energy efficiency (EE) for two IDs. Besides, we also do the asymptotic analysis of OPs at high signal-to-noise ratios (SNRs). Furthermore, this paper also inspects the impacts of the UAV-based relaying on the OP, system throughput, and EE of the proposed NOMA scheme. The derived asymptotic expansions show that the suggested model can enhance user fairness and the analytical results match the simulation results. [ABSTRACT FROM AUTHOR]

In this paper, we consider an approximation of the Caputo fractional derivative and its asymptotic expansion formula, whose generating function is the polylogarithm function. We prove the convergence of the approximation and derive an estimate for the error and order. The approximation is applied for the construction of finite difference schemes for the two-term ordinary fractional differential equation and the time fractional Black–Scholes equation for option pricing. The properties of the approximation are used to prove the convergence and order of the finite difference schemes and to obtain bounds for the error of the numerical methods. The theoretical results for the order and error of the methods are illustrated by the results of the numerical experiments. [ABSTRACT FROM AUTHOR]

We consider the Neumann-type problem of the heat equation in a moving thin domain around a given closed moving hypersurface. The main result of this paper is an error estimate in the sup-norm for classical solutions to the thin domain problem and a limit equation on the moving hypersurface which appears in the thin-film limit of the heat equation. To prove the error estimate, we show a uniform a priori estimate for a classical solution to the thin domain problem based on the maximum principle. Moreover, we construct a suitable approximate solution to the thin domain problem from a classical solution to the limit equation based on an asymptotic expansion of the thin domain problem and apply the uniform a priori estimate to the difference of the approximate solution and a classical solution to the thin domain problem. [ABSTRACT FROM AUTHOR]

This paper introduces a study on the horizontal and vertical deflections of the cross section of a thin-walled rotating beam. These deflections are governed by a system of two ordinary differential equations in order to describe their Cartesian directions. Based on multiple time-scales analysis, truncated asymptotic expansions are assumed to be approximate solutions to the given problem. Furthermore, an extracted autonomous system of differential equations governs the change rate of the amplitudes and phases of the beam deflections. The beam's rotation speed is adjusted to be in the neighborhood of both of the natural frequencies of the deflections such that the beam is subjected to 1 : 1 : 1 simultaneous resonance. A stability test is conducted according to the first method of Lyapunov in order to determine whether the equilibrium point is asymptotically stable or not. The beam's deflections turn unstable once its speed is in the neighborhood of its modal natural frequencies. There exists a multistable solution at some values of the beam's speed depending on the hysteresis manner of the model according to forward or backward sweeping of this speed. Furthermore, a range of centrifugal forces of the rotating hub can make the beam's deflections exhibit quasiperiodic responses which are confirmed by time response, orbital map, and amplitude spectrum. Eventually, some remarks are recommended for the external excitation frequency in order that the beam stays in the periodic behavior. [ABSTRACT FROM AUTHOR]

In this paper, a class of Stancu type Sz'asz-Mirakjan operators are introduced. The approximation properties of the operators are discussed using tools of modulus of continuity, modulus of smoothness and K-functional. The estimation of the Lipschitz function class by the operators is also studied. Later, the Voronvskaya type asymptotic expansion of the operators is established. Finally, we compare the convergence of these newly defined operators for certain functions with certain graphs. [ABSTRACT FROM AUTHOR]

In the present paper, we focus on the study of the asymptotic behaviors of solutions for the Cauchy problem of time-space fractional superdiffusion and subdiffusion equations with integral initial conditions, where the Riemann-Liouville derivative is used in the temporal direction and the integral fractional Laplacian is applied in the spatial variables. The fundamental solutions of the considered equations, which can be represented in terms of the Fox -function, are constructed and investigated by using asymptotic expansions of the Fox -function. Then, we obtain the asymptotic behaviors of solutions in the sense of and norms, where Young's inequality for convolution plays a very important role. Finally, gradient estimates and large time behaviors of solutions are also provided. In particular, we derive the optimal - decay estimate for the subdiffusion equation. [ABSTRACT FROM AUTHOR]

J.P. King constructed a sequence of positive linear operators on C[0, 1] preserving the constant function 1 and the monomial x2. After that, several papers were devoted to positive linear operators preserving prescribed functions. In 2009, J.M. Aldaz, O. Kounchev, H. Render modified the classical Bernstein operators to get new operators fixing 1 and x j for a given j 2 N. Our aim is to construct operators acting on functions defined on [0,1) and preserving 1 and x j. To this end we consider suitable modifications of the Post-Widder and Gamma operators. Using pointwise asymptotic relations for the moments we obtain convergence results for the sequences of operators. Moreover, we establish the associated Voronovskaja formulas. [ABSTRACT FROM AUTHOR]

In this paper, we generalize a sequence of positive linear operators introduced by Ismail and May and we study some of their approximation properties for different classes of continuous functions. First, we estimate the error of approximation in terms of the usual modulus of continuity and the second-order modulus of Ditzian and Totik. Then, we characterize the bounded functions that can be approximated uniformly by these new operators. In the last section, we obtain the most important results of the paper. We give the complete asymptotic expansion for the operators and we deduce a Voronovskaya-type theorem, results that hold true for smooth functions with exponential growth. [ABSTRACT FROM AUTHOR]

Theis' theory (1935), later improved by Hantush & Jacob (1955) and Moench (1971), is a technique designed to study the water level in aquifers. The key formula in this theory is a certain integral transform H [ g ] (r , t) of the pumping function g that depends on the time t and the relative position r to the pumping point as well as on other physical parameters. Several analytic approximations of H [ g ] (r , t) have been investigated in the literature that are valid and accurate in certain regions of r, t and the mentioned physical parameters. In this paper, the analysis of possible analytic approximations of H [ g ] (r , t) is completed by investigating asymptotic expansions of H [ g ] (r , t) in a region of the parameters that is of interest in practical situations, but that has not yet been investigated. Explicit and/or recursive algorithms for the computation of the coefficients of the expansions and estimates for the remainders are provided. Some numerical examples based on an actual physical experiment conducted by Layne-Western Company in 1953 illustrate the accuracy of the approximations. [ABSTRACT FROM AUTHOR]

The face gear tooth surface is a high-order variable curvature surface, and the curvature of the surface is complicated, so it is difficult to describe the characteristics of the face gear tooth surface by a specific equation. In this paper, the discrete curvature relation of the surface is used instead of an analytical equation to describe a spatial meshing surface, and the traditional meshing theory is neutralized to analyze the characteristics of the face gear tooth surface. Firstly, according to the structural characteristics of the face gear, the sampling method of the face gear tooth surface is analyzed, and the mathematical model of fitting the tooth surface contact point is established. Then, the discrete asymptotic surface development analysis method is studied, and the ellipsoid contact analysis method of the face gear pair is established by simplifying the conjugate surface and its region. Finally, the contact analysis method of the discrete tooth surface is studied, and the instantaneous contact ellipse of the face gear tooth surface is calculated, which formed a new numerical meshing method of space curved surface. [ABSTRACT FROM AUTHOR]

Over last couple of decades, various regimes of lubrication attracted lots of attention. Among these regimes of lubrication, the least understood and studied are the regimes or boundary lubrication and lubricity. Better understanding these regimes of lubrication is necessary due to the fact that many moving joints such as, for example, bearings and gears at least part of their operating time work in these regimes. It may occur due to slow motion of counter parts or their relative overload. This paper proposes a model for analysis of these regimes of lubrication. Specifically, the model takes into account roughness of one of the contact surfaces. It is shown that for sufficiently high applied load it can be expected that contact surfaces are completely separated by a lubricant layer in spite of the presence of surface asperities. The definitions of boundary lubrication regimes and super lubricity are proposed. A method based on the method of matched asymptotic expansions is applied for analyzing such cases and an approximate analytical solution of the problem away from the contact region boundary is found. In the narrow inlet and exit zones adjacent to contact boundaries equations describing the main solution terms are derived. For the case of super lubricity, a resemblance of a superposition between the lubrication and surface roughness effects is determined and the effect of surface roughness on the lubrication film thickness is considered. Specifically, certain roughness conditions are identified for which the lubrication film thickness can be higher than for smooth surfaces while in other cases it can be lower. For the super lubricity regimes, the equations for the main pressure asymptotic terms in the inlet and exit zones are reduced to the corresponding equations for the smooth (not rough) surfaces for which a series of numerical solutions is obtained earlier. A general analysis of the effect of texturing on lubrication conditions in case of super lubricity is conducted based on a simple geometric criterion. Several models of textured surfaces and their influence on the lubrication film thickness in case of super lubricity regimes are considered. [ABSTRACT FROM AUTHOR]

This paper provides asymptotic expansions for large values of n of tangent T n μ z and Apostol-tangent T n μ z ; λ polynomials of complex order. The derivation is done using contour integration with the contour avoiding branch cuts. [ABSTRACT FROM AUTHOR]

The paper develops a new deep learning based scheme for solving high-dimensional nonlinear forward-backward stochastic differential equations (FBSDE) and associated partial differential equations. Firstly, the original BSDE is split into the linear dominant BSDE part and the nonlinear residual BSDE part. Then the linear BSDE part is approximated with high accuracy using a weak approximation technique. To approximate the nonlinear BSDE part, Deep BSDE solver is applied with asymptotic expansions which work as control variates. A sharp error estimate provides how the new scheme improves the original Deep BSDE method. Numerical experiments for high-dimensional nonlinear models show the validity and the effectiveness of the new scheme in financial application. [ABSTRACT FROM AUTHOR]

Indicial operators are model operators associated to an elliptic differential operator near a corner singularity on a stratified manifold. These model operators are defined on generalized tangent cone configurations and exhibit a natural scaling invariance property with respect to dilations of the radial variable. In this paper we discuss extensions of symmetric indicial operators from a functional analytic point of view. In the first, purely abstract part of this paper, we consider a general unbounded symmetric operator that exhibits invariance with respect to an abstract scaling action on a Hilbert space, and we describe its extensions in terms of generalized eigenspaces of the infinitesimal generator of this action. Among others, we obtain a Green formula for the adjoint pairing, an algebraic formula for the signature, and in the semibounded case explicit descriptions of the Friedrichs and Krein extensions. In the second part we consider differential operators of Fuchs type on the half axis with unbounded operator coefficients that are invariant under dilation, and show that under suitable ellipticity assumptions on the indicial family these operators _t into the abstract framework of the first part, which in this case furnishes a description of extensions in terms of polyhomogeneous asymptotic expansions. We also obtain an analytic formula for the signature of the adjoint pairing in terms of the spectral flow of the indicial family for such operators. [ABSTRACT FROM AUTHOR]

In this paper we prove symmetry for solutions to the semi-linear elliptic equation Δu = ƒ(u) in B1, 0 ≤ u < M, in B1, u = M, on ∂ B1, where M > 0 is a constant, and B1 is the unit ball. Under certain assumptions on the r.h.s. ƒ(u), the C1-regularity of the free boundary ∂{u > 0} and a second order asymptotic expansion for u at free boundary points, we derive the spherical symmetry of solutions. A key tool, in addition to the classical moving plane technique, is a boundary Harnack principle (with r.h.s.) that replaces Serrin's celebrated boundary point lemma, which is not available in our case due to lack of C2-regularity of solutions. [ABSTRACT FROM AUTHOR]

Both theoretical and applied mathematics depend heavily on inequalities, which are rich in symmetries. In numerous studies, estimations of various functions based on the characteristics of their symmetry have been provided through inequalities. In this paper, we study the monotonicity of certain functions that involve Gamma functions. We were able to obtain some of the bounds of Γ (v) that are more accurate than some recently published inequalities. [ABSTRACT FROM AUTHOR]

A remarkable result which led to Ap'ery's proof of the irrationality of ζ (3) is given by the rapidly convergent series... denote the remainder of the series. In this paper, we obtain an asymptotic expansion of (-1)nRn. Based on the obtained result, we establish the upper and lower bounds of (-1)nRn. As an application of the obtained bounds, we give an approximate value of ζ (3). [ABSTRACT FROM AUTHOR]

Gigahertz acoustic streaming enables the synthesis of localized microjets reaching speeds of up to meters per second, offering tremendous potential for precision micromanipulation. However, theoretical and numerical investigations of acoustic streaming at these frequencies remain so far relatively scarce due to significant challenges including: (i) the inappropriateness of classical approaches, rooted in asymptotic development, for addressing high-speed streaming with flow velocities comparable to the acoustic velocity; and (ii) the numerical cost of direct numerical simulations generally considered as prohibitive. In this paper, we investigate high-frequency bulk streaming using high-order finite difference direct numerical simulations. First, we demonstrate that high-speed micrometric jets of several meters per second can only be obtained at high frequencies, due to diffraction limits. Second, we establish that the maximum jet streaming speed at a given actuation power scales with the frequency to the power of 3/2 in the low attenuation limit and linearly with the frequency for strongly attenuated waves. Last, our analysis of transient regimes reveals a dramatic reduction in the time required to reach the maximum velocity as the frequency increases (power law in –5/2), leading to characteristic time on the order of μs at gigahertz frequencies, and hence accelerations within the Mega-g range. [ABSTRACT FROM AUTHOR]

Partial differential equations with highly oscillatory input terms are hardly ever solvable analytically and their numerical treatment is difficult. Modulated Fourier expansion used as an ansatz is a well known and extensively investigated tool in asymptotic numerical approach for this kind of problems. Although the efficiency of this approach has been recognised, its error analysis has not been investigated rigorously for general forms of linear PDEs. In this paper, we start such kind of investigations for a general form of linear PDEs with an input term characterised by a single high frequency. More precisely we derive an analytical form of such an expansion and provide a formula for the error of its truncation. Theoretical investigations are illustrated by computational simulations. [ABSTRACT FROM AUTHOR]

Recently, Chen et al. [3] investigated the precise large deviations of aggregate claims in a renewal risk model with arbitrary dependence between claim sizes and their waiting times. In this paper, we extend their results to a nonstandard risk model in which various dependence structures are imposed on the modeling components, and obtain the asymptotic lower and upper bounds of precise large deviations for aggregate claims, which hold uniformly for all x in a t-interval. [ABSTRACT FROM AUTHOR]

In this paper, we explore the dynamic behaviour of a metamaterial lattice which is composed of elastically connected concentrated masses coupled with local resonators (mass-in-mass lattice also referred to as the Kelvin lattice). This kind of metamaterial lattice is known to be associated with interesting phenomena such as a mass reduction effect, as compared to an equivalent Lagrange lattice composed of elastically connected effective masses. The effective mass of this metamaterial lattice is exactly calculated for finite and infinite metamaterial structures. For finite systems, the effective mass is identified from the eigenfrequencies spectrum of the metamaterial lattice. These eigenfrequencies are obtained for various boundary conditions of the metamaterial lattice, by solving a linear difference eigenvalue problem. For infinite systems, the wave dispersion of the metamaterial lattice is compared to the effective Born–Karman dispersion relation. It is shown for both the finite and the infinite systems, that the effective mass related to each branch (acoustical and optical branches) is proportional to the square of frequency associated with the complementary branch. An asymptotic expansion confirms that the effective mass tends towards the sum of each mass (the primary mass and the mass of the resonator) for the long-wave approximation of the acoustical branch. The effective mass is greater than the sum of each mass for the acoustical branch and is significantly reduced for the optical branch (with eventually a negligible reduced mass). An asymptotic analysis confirms the parametric study for each branch of the lattice spectrum. The effective stiffness is also calculated for both the finite and the infinite systems. A nonlocal continuous approach is developed to approximate the metamaterial lattice response in terms of frequency spectrum or effective mass variation. This study is concluded by a design-oriented methodology for the calibration of the effective mass from the metamaterial lattice properties. [ABSTRACT FROM AUTHOR]

Thin-film piezoelectric materials are advantageous in microelectromechanical systems (MEMS), due to large motion generation, high available energy and low power requirements. In this kind of application, thin piezoelectric films are subject to mechanical and electric cyclic loading, during which damage can accumulate and eventually lead to fracture. In the present study, continuum damage mechanics and asymptotic theory are adopted to model damage evolution in piezoelectric thin films. Our purpose is to develop a new interface model for thin piezoelectric films accounting for micro-cracking damage of the material. The methods used are matched asymptotic expansions, to develop an interface law, and the classic thermodynamic framework of continuum damage mechanics combined with Kachanov and Sevostianov's theory of homogenization of micro-cracked media, to characterize the damaging behavior of the interface. The main finding of the paper is a soft imperfect interface model able to simulate the elastic and piezoelectric behavior of thin piezoelectric film in the presence of micro-cracking and damage evolution. The obtained interface model is expected to be a useful tool for damage evaluation in MEMS applications. As an example, an electromechanically active stack incorporating a damaging piezoelectric layer is studied. The numerical results indicate a non-linear evolution of the macroscopic response and a damage accumulation qualitatively consistent with experimental observations. [ABSTRACT FROM AUTHOR]

This paper characterizes those functions given by convergent q-Lidstone series expansion. We give the necessary and sufficient conditions so that the entire function f (z) has such an expansion, in which case convergence is uniform on compact sets. [ABSTRACT FROM AUTHOR]

In the paper, we apply a new approach to the comparison of the distributions of sums of random variables to the case of Poisson random sums. This approach was proposed in our previous work (Bening, Korolev, 2022) and is based on the concept of statistical deficiency. Here, we introduce a continuous analog of deficiency. In the case under consideration, by continuous deficiency, we will mean the difference between the parameter of the Poisson distribution of the number of summands in a Poisson random sum and that of the compound Poisson distribution providing the desired accuracy of the normal approximation. This approach is used for the solution of the problem of determination of the distribution of a separate term in the Poisson sum that provides the least possible value of the parameter of the Poisson distribution of the number of summands guaranteeing the prescribed value of the (1 − α) -quantile of the normalized Poisson sum for a given α ∈ (0 , 1) . This problem is solved under the condition that possible distributions of random summands possess coinciding first three moments. The approach under consideration is applied to the collective risk model in order to determine the distribution of insurance payments providing the least possible time that provides the prescribed Value-at-Risk. This approach is also used for the problem of comparison of the accuracy of approximation of the asymptotic (1 − α) -quantile of the sum of independent, identically distributed random variables with that of the accompanying infinitely divisible distribution. [ABSTRACT FROM AUTHOR]

This paper develops a two-scale asymptotic homogenization method for periodic composite Kirchhoff plates. In this method, a three-dimensional (3D) periodic plate problem is simplified as a Kirchhoff plate problem, which is governed by a fourth-order uniformly elliptic partial differential equation (PDE) with periodically oscillating coefficients. Then, a two-scale solution in an asymptotic expansion form is presented for the PDE, and it is found that the first-order perturbed displacement in the asymptotic solution is zero. Additionally, periodic boundary and normalization constraint conditions are proposed to determine the unique solution to unit cell problems. Moreover, standard finite element formulations for calculating the perturbed displacements are derived from the principle of virtual work. Physical interpretations of the influence functions are presented by analyzing the properties of self-balanced quasi-load vectors used for solving the influence functions. Numerical comparisons show that the present method is physically acceptable and highly accurate. [ABSTRACT FROM AUTHOR]

The paper deals with a coupling algorithm using shape and topological derivatives of a given cost functional and a problem governed by nonstationary Maxwell's equations in 3D. To establish the shape and topological derivatives, an adjoint method is used. For the topological asymptotic expansion, two examples of cost functionals are considered with the perturbation of the electric permittivity and magnetic permeability. We combine the shape derivative and topological one to propose an algorithm. The proposed algorithm allows to insert a small inhomogeneity (electric or magnetic) in a given shape. [ABSTRACT FROM AUTHOR]

The article contains a survey of publications studying problems characterized by the presence of fast variables with various rates of change (time scales). We consider the passage to the limit from the solution of a perturbed problem to the solution of a degenerate one, asymptotic solutions of initial and boundary value problems, stability and controllability, asymptotic solutions of optimal control problems, and problems with "hidden" multi-tempo variables. In addition, problems with control constraints, game problems, and stochastic systems are given. The last section presents practical problems with multi-tempo fast motions. [ABSTRACT FROM AUTHOR]

Droplet evaporation and autoignition are considerably influenced by transient gas-phase diffusion and variations in droplet temperature during the initial stage, especially under elevated pressure conditions and when substantial differences exist between the initial and equilibrium droplet temperatures. In this paper, we present a theoretical analysis of the combined transient effects of gas-phase diffusion (GT) and droplet temperature variation (DT) on the pure evaporation and autoignition of individual, stationary fuel droplets. Asymptotic solutions reveal that the evolution of droplet surface temperatures consists of two phases: an initial rapid response followed by a more gradual variation. By investigating the development of the evaporating boundary layer and droplet temperature, we derive the transient correction factors for both GT and DT effects, accounting for the impact of non-unity Lewis numbers. Due to the GT effect, evaporation accelerates under conditions of a reduced liquid–gas density ratio (e.g., elevated ambient pressure) and increased evaporation intensity (e.g., higher ambient temperature). Meanwhile, a larger deviation between the initial droplet temperature and the equilibrium temperature results in a more pronounced DT effect. Beyond pure evaporation, this paper also analyzes the autoignition of droplets under the combined impact of both transient effects, employing large-activation-energy asymptotics. The ignition delay time and critical conditions for droplet ignition are theoretically established. Analysis indicates that the GT effect exerts a greater influence on the critical ignition temperature than both the DT and non-unity Lewis number effects. A comprehensive comparison of the current theory with transient numerical simulations reveals its ability to accurately address both GT and DT effects on droplet evaporation and autoignition processes. Consequently, this paper successfully extends the classical theory on droplet evaporation and autoignition through an intuitive and physically meaningful analysis. Novelty and significance statement The novelty of this research stems from its detailed theoretical exploration of the concurrent transient impacts of gas-phase diffusion (GT) and droplet temperature variation (DT) on the evaporation and autoignition of individual fuel droplets. Through the development of asymptotic solutions and transient correction factors for GT and DT effects, this paper constructs a comprehensive framework that effectively refines classical theory. This work is significant for bridging a crucial gap in understanding droplet evaporation and ignition behaviors under conditions that mirror real-world scenarios, especially under elevated system pressures and notable deviations between initial and equilibrium droplet temperatures, where transient effects are prominent. This enhanced theoretical basis holds promise for advancing the design and optimization of innovative combustion systems, fostering the development of technologies that are more efficient and environmentally friendly. [ABSTRACT FROM AUTHOR]

This paper deals with the period function of the reversible quadratic centers X ν = − y (1 − x) ∂ x + (x + D x 2 + F y 2) ∂ y , where ν = (D , F) ∈ R 2. Compactifying the vector field to S 2 , the boundary of the period annulus has two connected components, the center itself and a polycycle. We call them the inner and outer boundary of the period annulus, respectively. We are interested in the bifurcation of critical periodic orbits from the polycycle Π ν at the outer boundary. A critical period is an isolated critical point of the period function. The criticality of the period function at the outer boundary is the maximal number of critical periodic orbits of X ν that tend to Π ν 0 in the Hausdorff sense as ν → ν 0. This notion is akin to the cyclicity in Hilbert's 16th Problem. Our main result (Theorem A) shows that the criticality at the outer boundary is at most 2 for all ν = (D , F) ∈ R 2 outside the segments { − 1 } × [ 0 , 1 ] and { 0 } × [ 0 , 2 ]. With regard to the bifurcation from the inner boundary, Chicone and Jacobs proved in their seminal paper on the issue that the upper bound is 2 for all ν ∈ R 2. In this paper the techniques are different because, while the period function extends analytically to the center, it has no smooth extension to the polycycle. We show that the period function has an asymptotic expansion near the polycycle with the remainder being uniformly flat with respect to ν and where the principal part is given in a monomial scale containing a deformation of the logarithm, the so-called Écalle-Roussarie compensator. More precisely, Theorem A follows by obtaining the asymptotic expansion to fourth order and computing its coefficients, which are not polynomial in ν but transcendental. Theorem A covers two of the four quadratic isochrones, which are the most delicate parameters to study because its period function is constant. The criticality at the inner boundary in the isochronous case is bounded by the number of generators of the ideal of all the period constants but there is no such approach for the criticality at the outer boundary. A crucial point to study it in the isochronous case is that the flatness of the remainder in the asymptotic expansion is preserved after the derivation with respect to parameters. We think that this constitutes a novelty that is of particular interest also in the study of similar problems for limit cycles in the context of Hilbert's 16th Problem. Theorem A also reinforces the validity of a long standing conjecture by Chicone claiming that the quadratic centers have at most two critical periodic orbits. A less ambitious goal is to prove the existence of a uniform upper bound for the number of critical periodic orbits in the family of quadratic centers. By a compactness argument this would follow if one can prove that the criticality of the period function at the outer boundary of any quadratic center is finite. Theorem A leaves us very close to this existential result. [ABSTRACT FROM AUTHOR]

The paper is concerned with development of the asymptotic formulation for surface wave field induced by vertical surface stress under the effect of gravity in the short-wave region. The approach relies on the methodology of hyperbolic-elliptic models for the Rayleigh wave and results in a regularly perturbed hyperbolic equation on the surface acting as a boundary condition for the elliptic equation governing decay over the interior. A special value of the Poisson's ratio v = 0.25 is pointed out, at which the effect of gravity disappears at leading order. [ABSTRACT FROM AUTHOR]

The paper is concerned with the asymptotically consistent theory of nanoscale plates capturing the spatial nonlocal effects. The three-dimensional (3D) elasticity equations for a thin plate are used as the governing equations. In the general case, the plate is acted upon by dynamic body forces varying in the thickness direction, and by variable surface forces. The thickness of the plate is assumed to be greater than the characteristic micro/nanoscale measure and much smaller than the in-plane characteristic dimension (e.g., the wave or deformation length). The 3D constitutive equations of gradient elasticity are used to link the fields of nonlocal stresses and strains. Using the asymptotic approach, a sequence of relations for stresses and displacements in the form of polynomials in the transverse coordinate with coefficients depending on time and in-plane coordinates was obtained. The asymptotically consistent 2D differential equation governing vibration (or static deformation) of a plate accounting for both transverse shears and the spatial nonlocal contribution of the stress and strain fields was derived. It was revealed that capturing nonlocal effects in all directions leads to an increase in the correction factor compared with the well-known 2D theories based on kinematic hypotheses and the Eringen-type gradient constitutive equations. The effect of the internal length scales parameters on free low-frequency vibrations and displacements of a plate is discoursed. [ABSTRACT FROM AUTHOR]

A well known result from the non-linear theory of elasticity applied to spherical shells is that the classical Mooney-Rivlin constitutive law may give either a monotonic or a non-monotonic pressure-inflation response for finite deformation. Specifically, this is determined by two factors: the relative shell thickness, and the relative I 1 to I 2 contribution in the M-R constitutive law. Here we consider how a residual stress field may affect this behavior. Using a constitutive framework for hyperelastic materials with residual stress that has been especially applied to tubes, this paper focuses on finite thickness spherical shells while using a similar prototypical energy response. In this context we examine different residual stress states, and show how certain of these lead to more workable analytical results than others. All of them enable Taylor and asymptotic expansions in the small and large inflation limit. On this basis it is shown how particular residual stress fields can cause a monotonic inflation graph to become non-monotonic, and vice versa. [ABSTRACT FROM AUTHOR]