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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]

This paper gives new expansions of characteristic functions of convolution of symmetric probability distributions with an explicit estimate of the remainder. [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]

We study the behavior of solutions of the Cauchy problem for a semilinear heat equation with critical nonlinearity in the sense of Joseph and Lundgren. It is known that if two solutions are initially close enough near the spatial infinity, then these solutions approach each other. In this paper, we give its optimal and sharp convergence rate of solutions with a critical exponent and two exponentially approaching initial data. This rate contains a logarithmic term which does not contain in the super critical nonlinearity case. Proofs are given by a comparison method based on matched asymptotic expansion. [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]

The Ablowitz–Ladik equations, hereafter called A L + and A L − , are distinguished integrable discretizations of respectively the focusing and defocusing nonlinear Schrödinger (NLS) equations. In this paper we first study the modulation instability of the homogeneous background solutions of A L ± in the periodic setting, showing in particular that the background solution of A L − is unstable under a monochromatic perturbation of any wave number if the amplitude of the background is greater than 1, unlike its continuous limit, the defocusing NLS. Then we use Darboux transformations to construct the exact periodic solutions of A L ± describing such instabilities, in the case of one and two unstable modes, and we show that the solutions of A L − are always singular on curves of spacetime, if they live on a background of sufficiently large amplitude, and we construct a different continuous limit describing this regime: a NLS equation with a nonlinear and weak dispersion. At last, using matched asymptotic expansion techniques, we describe in terms of elementary functions how a generic periodic perturbation of the background solution (i) evolves according to A L + into a recurrence of the above exact solutions, in the case of one and two unstable modes, and (ii) evolves according to A L − into a singularity in finite time if the amplitude of the background is greater than 1. The quantitative agreement between the analytic formulas of this paper and numerical experiments is perfect. [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]

This paper studies the monic semi‐classical Laguerre polynomials based on previous work by Boelen and Van Assche, Filipuk et al., and Clarkson and Jordaan. Filipuk et al. proved that the diagonal recurrence coefficient αn(t)$$ {\alpha}_n(t) $$ satisfies the fourth Painlevé equation. In this paper, we show that the off‐diagonal recurrence coefficient βn(t)$$ {\beta}_n(t) $$ fulfills the first member of Chazy II system. We also prove that the sub‐leading coefficient of the monic semi‐classical Laguerre polynomials satisfies both the continuous and discrete Jimbo–Miwa–Okamoto σ$$ \sigma $$‐form of Painlevé IV. By using Dyson's Coulomb fluid approach together with the discrete system for αn(t)$$ {\alpha}_n(t) $$ and βn(t)$$ {\beta}_n(t) $$, we obtain the large n$$ n $$ asymptotic expansions of the recurrence coefficients and the sub‐leading coefficient. The large n$$ n $$ asymptotics of the associated Hankel determinant (including the constant term) is derived from its integral representation in terms of the sub‐leading coefficient. [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]

In this paper, we consider effective performance transmission under generalized α-κ-μ the fading distribution. The source-user links are assumed to be nonorthogonal multiple access (NOMA) channels through a downlink power domain. Two users are selected to service in a situation with perfect channel state information (CSI) in accordance with the NOMA protocol. The closed-form expressions of outage probability (OP) and bit error rate (BER) are derived with the effect of power allocation coefficient, target rate, and channel fading parameters. In addition, we calculate numerical results to demonstrate the asymptotic expansion in the high signal-to-noise ratio (SNR) analysis. Finally, Monte Carlo simulations are provided to validate and assess the accuracy of the analytical framework proposed. [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]

This paper studies limit cycle bifurcation of an (n + 1) th order generalized Liénard differential system with an elliptic Hamiltonian H (x , y) = 1 2 x 2 + 1 2 y 2 - 1 4 x 4 , which bifurcates at least [ n 2 ] limit cycles near either a centre or a heteroclinic loop with two hyperbolic saddles for n ≤ 150 , where about two thirds of the limit cycles are large amplitude ones. To achieve this goal, we provide the precise expression of the third coefficient in the asymptotic expansion of the first order Melnikov function of an analytic near-Hamiltonian system near a polycycle with m (≥ 2) hyperbolic saddles, without any prescribed condition, whereas all the known results need. As well as we also characterize its higher order coefficients. Here [a] is the integer part function of a real number a. [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]

The Asymptotic Expansion Load Decomposition higher-order beam model is based on the classical two-scale asymptotic expansion in the linear elasticity framework.It was successively extended to eigenstrains and to plasticity in small deformations in different papers. The present paper offers a comprehensive and consistent presentation of our approach applied to civil engineering applications. [ABSTRACT FROM AUTHOR]

In this paper, we consider the classical Riemann problem for a generalized Burgers equation, ut+hα(x)uux=uxx,$$\begin{equation*} u_t + h_{\alpha }(x) u u_x = u_{xx}, \end{equation*}$$with a spatially dependent, nonlinear sound speed, hα(x)≡(1+x2)−α$h_{\alpha }(x) \equiv (1+x^2)^{-\alpha }$ with α>0$\alpha >0$, which decays algebraically with increasing distance from a fixed spatial origin. When α=0$\alpha =0$, this reduces to the classical Burgers equation. In this first part of a pair of papers, we focus attention on the large‐time structure of the associated Riemann problem, and obtain its detailed structure, as t→∞$t\rightarrow \infty$, via the method of matched asymptotic coordinate expansions (this uses the classical method of matched asymptotic expansions, with the asymptotic parameters being the independent coordinates in the evolution problem; this approach is developed in detail in the monograph of Leach and Needham, as referenced in the text), over all parameter ranges. We identify a significant bifurcation in structure at α=12$\alpha =\frac{1}{2}$. [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]

Boundary value problems for diffusion in singularly perturbed domains is a topic of considerable current interest. Applications include intracellular diffusive transport and the spread of pollutants or heat from localized sources. In a previous paper, we introduced a new method for characterizing the approach to steady state in the case of two-dimensional (2D) diffusion. This was based on a local measure of the relaxation rate known as the accumulation time T(x). The latter was calculated by solving the diffusion equation in Laplace space using a combination of matched asymptotics and Green's function methods. We thus obtained an asymptotic expansion of T(x) in powers of \nu = 1/ln \ε, where \ε specifies the relative size of the holes. In this paper, we develop the corresponding theory for three-dimensional (3D) diffusion. The analysis is a nontrivial extension of the 2D case due to differences in the singular nature of the Laplace transformed Green's function. In particular, the asymptotic expansion of the solution of the 3D diffusion equation in Laplace space involves terms of order O((\epsilon/s)n), where s is the Laplace variable. These s-singularities have to be removed by partial series resummations in order to obtain an asymptotic expansion of T(x) in powers ofε. [ABSTRACT FROM AUTHOR]

A number of recent papers have estimated ratios of the partition function $p(n-j)/p(n)$ , which appear in many applications. Here, we prove an easy-to-use effective bound on these ratios. Using this, we then study the second shifted difference of partitions, $f(\,j,n) := p(n) -2p(n-j) +p(n-2j)$ , and give another easy-to-use estimate of $f(\,j,n)$. As applications of these, we prove a shifted convexity property of $p(n)$ , as well as giving new estimates of the k -rank partition function $N_k(m,n)$ and non- k -ary partitions along with their differences. [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 constraint minimizers of the planar Schrödinger-Poisson system with a logarithmic convolution potential and a logarithmic external potential $ V(x) = \ln(1+|x|^2) $. It is known that minimizers exist if and only if the particle mass $ \rho>0 $ satisfies $ \rho<\rho^* $ for some threshold $ \rho^*\in(0,\infty) $. As a continuation of [22], this paper is devoted to analyzing the refined spike profiles of constraint minimizers as $ \rho\nearrow\rho^* $. [ABSTRACT FROM AUTHOR]

The H -function of communication theory plays an important role in the error rate analysis in digital communication with the presence of additive white Gaussian noise (AWGN) and generalized multipath fading conditions. In this paper we investigate several convergent and/or asymptotic expansions of H p (z , b , η) for some limiting values of their variables and parameters: large values of z, large values of p, small values of η, and values of b → 1. We provide explicit and/or recursive algorithms for the computation of the coefficients of the expansions. Some numerical examples illustrate the accuracy of the approximations. [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]

Many of the fastest known algorithms to compute π involve generalized hypergeometric series, such as the Ramanujan–Sato series. In this paper, we investigate the rates of convergence for several such series and we give asymptotic expansions for the error of finite approximation. For example, when using the first n terms of the Chudnovskys' series, we obtain the finite approximation π n ≈ π . It is known that the truncation error satisfies | π n - π | ≈ 53360 - 3 n. In this paper, we prove that the asymptotic expansion for the truncation error in the Chudnovskys' series is π n - π = 53360 - 3 n 106 720 10 005 π 1 672 209 n exp A 1 n + A 2 n 2 + δ n n 3 , with 0.006 907 < δ n < 0.008 429 and the exact rational values of A 1 and A 2 : A 1 = - 1 781 843 197 433 7 456 754 505 816 , A 2 = - 1 080 096 011 925 710 088 395 3 475 199 235 000 451 148 614 116. Thus we demonstrate how to establish precise error bounds for the approximations for π obtained through Ramanujan-like series for 1 / π . [ABSTRACT FROM AUTHOR]

The movement of a body subjected to a drag force proportional to the square of the velocity is of great importance in the field of mechanical engineering, since it is a realistic model in the regime of high velocity of the flow. In this paper, the trajectory of a projectile in quasi-vertical launch with quadratic drag is analyzed using the technique of matched asymptotic expansions. Through the asymptotic analysis, approximate analytical expressions for the projectile movement are found. The asymptotic solutions are compared with exact numerical results. [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]

We study in this paper a geometric inverse problem in fluid mechanics. The goal is to determine the location of an object in the fluid domain from boundary information. We consider the case of a nonstationary two‐dimensional Stokes flow. Our approach is based on the Kohn–Vogelius concept and the topological gradient method. We develop a new topological asymptotic expansion, which can be used as a basic step for developing accurate detection algorithms. [ABSTRACT FROM AUTHOR]

We study the individual behavior of the eigenvalues of the laplacian matrices of the cyclic graph of order n , where one edge has weight α ∈ C , with Re (α) < 0 , and all the others have weights 1. This paper is a sequel of a previous one where we considered Re (α) ∈ [ 0 , 1 ] (Grudsky et al., 2022 [12]). We prove that for Re (α) < 0 and n > Re (α − 1) / Re (α) , one eigenvalue is negative while the others belong to [ 0 , 4 ] and are distributed as the function x ↦ 4 sin 2 ⁡ (x / 2). Additionally, we prove that as n tends to ∞, the outlier eigenvalue converges exponentially to 4 Re (α) 2 / (2 Re (α) − 1). We give exact formulas for half of the inner eigenvalues, while for the others we justify the convergence of Newton's method and fixed-point iteration method. We find asymptotic expansions, as n tends to ∞, both for the eigenvalues belonging to [ 0 , 4 ] and the outlier. We also compute the eigenvectors and their norms. [ABSTRACT FROM AUTHOR]

In this paper, the mean velocity profiles and the multi-layer division of the turbulent boundary layer (TBL) are studied for the entire adverse pressure gradient (APG) region before separation. By asymptotic expansions of the mean velocity, the difference in the APG effect between the inner and outer regions and the problems of directly matching the mean velocity are demonstrated. The wall law of the overlapping region, i.e., the log-pressure law, is derived from the perspective of turbulent transport in combination with the results of matching the Reynolds shear stress. The results show that it is applicable in the entire APG region, thus answering the questions of the existence of the log law, the slope of the log law, and the breakdown of the log law. A new half-power law is then proposed after reconsidering the self-similarity of the outer region. This law is believed to be valid only in the outer region of larger APGs and is only an approximate self-similar law compared to the wall law in the inner region. Furthermore, a multi-layer division of APG TBLs is presented to form a complete framework for describing the mean flow of APG TBLs. In particular, the basis of the division and the specific range of each layer are specified, where the range of the overlapping region involves the definition of a critical Reynolds number. [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]

We consider a discrete-time population growth system called the Bienaymé–Galton–Watson stochastic branching system. We deal with a noncritical case, in which the per capita offspring mean $m\neq1$. The famous Kolmogorov theorem asserts that the expectation of the population size in the subcritical case $m on positive trajectories of the system asymptotically stabilizes and approaches ${1}/\mathcal{K}$ , where $\mathcal{K}$ is called the Kolmogorov constant. The paper is devoted to the search for an explicit expression of this constant depending on the structural parameters of the system. Our argumentation is essentially based on the basic lemma describing the asymptotic expansion of the probability-generating function of the number of individuals. We state this lemma for the noncritical case. Subsequently, we find an extended analogue of the Kolmogorov constant in the noncritical case. An important role in our discussion is also played by the asymptotic properties of transition probabilities of the Q-process and their convergence to invariant measures. Obtaining the explicit form of the extended Kolmogorov constant, we refine several limit theorems of the theory of noncritical branching systems, showing explicit leading terms in the asymptotic expansions. [ABSTRACT FROM AUTHOR]

In this paper, we give a new upper bound for the number NR which is defined to be the smallest positive integer such that a certain inequality due to Ramanujan involving the prime counting function π(x) holds for every x ≥ NR. [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]

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]

In this paper, we derive high‐order terms in the asymptotic expansions of the boundary perturbations for the electric field by taking into account of highly oscillating perturbations of an inhomogeneity with regular boundary. We show then that the asymptotic expansions depend on some parameter γ, that describes the depth O(δγ)$O(\delta ^{\gamma })$ of irregularity. Our study is rigorous and is founded on Field expansion method. [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]

In this paper we study a finite‐depth layer of viscous incompressible fluid in dimension n≥2, modeled by the Navier‐Stokes equations. The fluid is assumed to be bounded below by a flat rigid surface and above by a free, moving interface. A uniform gravitational field acts perpendicularly to the flat surface, and we consider the cases with and without surface tension acting on the free interface. In addition to these gravity‐capillary effects, we allow for a second force field in the bulk and an external stress tensor on the free interface, both of which are posited to be in traveling wave form, i.e., time‐independent when viewed in a coordinate system moving at a constant velocity parallel to the rigid lower boundary. We prove that, with surface tension in dimension n≥2 and without surface tension in dimension n=2, for every nontrivial traveling velocity there exists a nonempty open set of force and stress data that give rise to traveling wave solutions. While the existence of inviscid traveling waves is well‐known, to the best of our knowledge this is the first construction of viscous traveling wave solutions. Our proof involves a number of novel analytic ingredients, including: the study of an overdetermined Stokes problem and its underdetermined adjoint problem, a delicate asymptotic development of the symbol for a normal‐stress to normal‐Dirichlet map defined via the Stokes operator, a new scale of specialized anisotropic Sobolev spaces, and the study of a pseudodifferential operator that synthesizes the various operators acting on the free surface functions. © 2022 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC. [ABSTRACT FROM AUTHOR]

Designing devices composed of many small resonators is a challenging problem that can easily incur significant computational cost. Can asymptotic techniques be used to overcome this often limiting factor? Integral methods and asymptotic techniques have been used to derive concise characterizations for scattering by resonators, but can these be generalized to systems of many dispersive resonators whose material parameters have highly non‐linear frequency dependence? In this paper, we study halide perovskite resonators as a demonstrative example. We extend previous work to show how a finite number of coupled resonators can be modeled concisely in the limit of small radius. We also show how these results can be used as the basis for an inverse design strategy, to design resonator systems that resonate at specific frequencies. [ABSTRACT FROM AUTHOR]

In this paper, we study the initial layer problem of the Euler–Poisson in collisionless plasma that is rigorously proved by using the weighted energy method coupled with multiscaling asymptotic expansions. We apply a formal expansion for Debye length and derive the inner and the initial layer equations, which are used for deriving the error equations, and give out uniform estimate. [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]

This work presents an asymptotic method based on angular flux expansion in a Neumann series. The technique is aimed at effective reduction of the memory imprint of numerical methods based on collision probabilities (CPs). The asymptotic method has been implemented in the heterogeneous Cartesian cells of the integro-differential transport solver (IDT). The IDT solves the neutral-particle transport equation by discrete ordinates combined with angular-dependent CP matrices. In lattice depletion calculations, because of the change of isotopic concentration along the burnup, methods based on CP discretization, such as current-coupling CP or the one presented in this paper, would require construction and storage of a set of CP coefficients for any depleted pin cell. When the number of media grows, the performances of the solver are bounded by the memory pressure caused by the growth of coefficients. Application of the asymptotic technique, presented in this paper, transforms by two user's parameters the memory-bound solver in a compute-bound application, where the principal workload is transferred from coefficients to source iterations. In this work, a theoretical study of the method is presented together with two applications to two-dimensional assembly simulations. The effects on self-shielded and depleted materials are highlighted. Preliminary results show an encouraging reduction of memory occupation by a factor 10 without any significant loss of accuracy. [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]