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In this paper, we study the asymptotic expansion for the homogeneous model of wind‐driven oceanic circulation with the nonslip boundary condition when both of the beta‐plane parameter and the Reynolds number go to infinity. By asymptotic analysis under certain constraints on wind tensor, we derive a formal expansion including the geophysical boundary layer in the large beta‐plane parameter and high Reynolds number limit, from which one sees that the external force, wind tensor, has an important effect on the behavior of the boundary layers. Moreover, when the external force and initial data satisfy certain constraints, to exclude the appearance of strong boundary layers, we construct approximate solutions with steady boundary layers to the homogeneous model of wind‐driven oceanic circulation in the large beta‐plane parameter and high Reynolds number limit. By using an energy method, we obtain the L∞$$ {L}^{\infty } $$‐stability of small perturbation around this approximate solutions, which verifies the validity of the asymptotic expansion of weak boundary layers in the large beta‐plane parameter and high Reynolds number limit. [ABSTRACT FROM AUTHOR]

We consider the Schrödinger operator in a bounded domain $ \mathbf{R}^n (n=2,3) $ R n (n = 2 , 3) with Neumann boundary condition. We suppose that this domain contains small deformable inclusions, i.e. regions where the potentials do not have the same values as the exterior medium. Our goal is to construct an asymptotic formula for the case of multiple and simple eigenvalue problems. We find an expansion that highlights the relation between the deformation parameters and the eigenvalues. The problem is devoted to a domain containing a finite number of inhomogeneities. Also, we focus on the interface of a small inclusion. [ABSTRACT FROM AUTHOR]

In this paper we study the asymptotic behavior of the radii of starlikeness of the normalized Jackson’s second and third q-Bessel functions, focusing on their large orders. To achieve this, we use the Rayleigh sums of positive zeros of both q-Bessel functions, and determine the coefficients of the asymptotic expansions. We derive complete asymptotic expansions for these radii of starlikeness and provide recurrence relations for the coefficients of these expansions. The proofs rely on the notion of Rayleigh sums of positive zeros of q-Bessel functions, Kvitsinsky’s results on spectral zeta functions for q-Bessel functions and asymptotic inversion. Moreover, we derive bounds for the radius of starlikeness of q-Bessel functions by using potential polynomials. [ABSTRACT FROM AUTHOR]

Let N(t) be the collection of particles alive at time t in a branching Brownian motion in R d , and for u ∈ N (t) , let X u (t) be the position of particle u at time t. For θ ∈ R d , we define the additive measures of the branching Brownian motion by μ t θ (d x) : = e - (1 + ‖ θ ‖ 2 2) t ∑ u ∈ N (t) e - θ · X u (t) δ X u (t) + θ t (d x) , here ‖ θ ‖ is the Euclidean norm of θ. In this paper, under some conditions on the offspring distribution, we give asymptotic expansions of arbitrary order for μ t θ ((a , b ]) and μ t θ ((- ∞ , a ]) for θ ∈ R d with ‖ θ ‖ < 2 , where (a , b ] : = (a 1 , b 1 ] × ⋯ × (a d , b d ] and (- ∞ , a ] : = (- ∞ , a 1 ] × ⋯ × (- ∞ , a d ] for a = (a 1 , ⋯ , a d) and b = (b 1 , ⋯ , b d) . These expansions sharpen the asymptotic results of Asmussen and Kaplan (Stoch Process Appl 4(1):1–13, 1976) and Kang (J Korean Math Soc 36(1): 139–157, 1999) and are analogs of the expansions in Gao and Liu (Sci China Math 64(12):2759–2774, 2021) and Révész et al. (J Appl Probab 42(4):1081–1094, 2005) for branching Wiener processes (a particular class of branching random walks) corresponding to θ = 0 . [ABSTRACT FROM AUTHOR]

The truncation coefficient is widely utilized in non-global coverage computations of geophysics and geodesy and is always altitude dependent. As the two commonly used calculation methods for truncation coefficients, i.e., the spectral form and the recursive formula, both suffer from decreasing precision caused by high-altitude, leading to slow convergence for the former and numerical instability recursion for the latter. The asymptotic expansion mathematically converges with increasing degree and can precisely compensate for the shortcomings of the two methods. This study introduces asymptotic expansion to accurately compute the truncation coefficient for the spectral gravity forward modeling to a high degree. The evaluation at the whole altitudes and whole integral radii indicates that the proposed method has the following advantages: (i) The calculation precision increases with increasing degree and is altitude independent; (ii) the accurate calculation can be supported by a double-precision format; and (iii) the calculation can be conducted nearly without extra time cost with increasing degree. Generally, asymptotic expansion is used to calculate the high degree truncation coefficients, while the truncation coefficients at low degrees can be calculated using spectral form or recursive formulas in multiprecision format as a supplement; and the available range of asymptotic expansion is provided in the appendix. [ABSTRACT FROM AUTHOR]

This paper presents a novel generic asymptotic expansion formula of expectations of multidimensional Wiener functionals through a Malliavin calculus technique. The uniform estimate of the asymptotic expansion is shown under a weaker condition on the Malliavin covariance matrix of the target Wiener functional. In particular, the method provides a tractable expansion for the expectation of an irregular functional of the solution to a multidimensional rough differential equation driven by fractional Brownian motion with Hurst index H < 1 / 2 , without using complicated fractional integral calculus for the singular kernel. In a numerical experiment, our expansion shows a much better approximation for a probability distribution function than its normal approximation, which demonstrates the validity of the proposed method. [ABSTRACT FROM AUTHOR]

For a class of generalized holomorphic Eisenstein series, we establish complete asymptotic expansions (Theorems 1 and 2). These, together with the explicit expression of the latter remainder (Theorem 3), naturally transfer to several new variants of the celebrated formulae of Euler and of Ramanujan for specific values of the Riemann zeta-function (Theorem 4 and Corollaries 4.1–4.5), and to various modular type relations for the classical Eisenstein series of any even integer weight (Corollary 4.6) as well as for Weierstraß' elliptic and allied functions (Corollaries 4.7–4.9). Crucial roles in the proofs are played by certain Mellin-Barnes type integrals, which are manipulated with several properties Kummer's confluent hypergeometric functions. [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]

The eigenvalues of Toeplitz matrices T n (f) with a real-valued generating function f , satisfying some conditions and tracing out a simple loop over the interval [ − π , π ] , are known to admit an asymptotic expansion with the form λ j (T n (f)) = f (σ j , n) + c 1 (σ j , n) h + c 2 (σ j , n) h 2 + O (h 3) , where h = 1 / (n + 1) , σ j , n = π j h , and c k are some bounded coefficients depending only on f. The numerical results presented in the literature suggest that the effective conditions for the expansion to hold are weaker and reduce to a fixed smoothness and to having only two intervals of monotonicity over [ − π , π ]. In this article we investigate the superposition caused over this expansion, when considering the following linear combination λ j (T n (f 0) + β n , 1 T n (f 1) + β n , 2 T n (f 2)) , where β n , 1 , β n , 2 are certain constants depending on n and the generating functions f 0 , f 1 , f 2 are either simple loop or satisfy the weaker conditions mentioned before. We formally obtain an asymptotic expansion in this setting under simple-loop related assumptions, and we show numerically that there is much more to investigate, opening the door to linear in time algorithms for the computation of eigenvalues of large matrices of this type including a multilevel setting. The problem is of concrete interest, considering spectral features of matrices stemming from the numerical approximation of standard differential operators and distributed order fractional differential equations, via local methods such as Finite Differences, Finite Elements, and Isogeometric Analysis. [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]

We derive a special form of the -difference equations whose solutions exist in the form of a Dulac series. We find the coefficients of the Dulac series from some algebraic difference equations having a polynomial solution under appropriate conditions. An example is given of a -difference equation which demonstrates the lack of upper bounds for the degrees of these polynomial solutions. We provide an upper bound for the degrees of the polynomial coefficients in terms of the coefficient degrees of the initial segment of the Dulac series. We also give some examples of calculating the expansions of solutions to -difference equations in the form of Dulac series. [ABSTRACT FROM AUTHOR]

We present an optimal control problem associated to a chemical transportation phenomena in a periodic porous medium. Posing controls on the porous part of the medium (distributed control), we set up a convex minimization problem. The main objective of this article is to characterize an arbitrary control to be an optimal control. We establish a relation between the optimal control and the corresponding adjoint state. At first, we analyse the microscopic description of the controlled system, then we homogenised the system by rigorous two-scale convergence method and periodic unfolding method. [ABSTRACT FROM AUTHOR]

In this paper, as an improvement of the paper (Ishige et al. in SIAM J Math Anal 49:2167–2190, 2017), we obtain the higher order asymptotic expansions of the large time behavior of the solution to the Cauchy problem for inhomogeneous fractional diffusion equations and nonlinear fractional diffusion equations. [ABSTRACT FROM AUTHOR]

We consider a generalization of the classical coupon collector problem, where the set of available coupons consists of standard coupons (which can be part of the collection), and two coupons with special purposes: one that speeds up the collection process and one that slows it down. We obtain several asymptotic results related to the expectation and the variance of the waiting time until a portion of the collection is sampled, as the number of standard coupons tends to infinity. [ABSTRACT FROM AUTHOR]

The asymptotic behavior for a heriditary recursion$ \begin{equation*} x_1>a \, \, \text{and} \, \, x_{n+1} = \frac{1}{n^s}\sum\limits_{k = 1}^nf\left(\frac{x_k}k\right)\text{ for every }n\geq1 \end{equation*} $is studied, where $ f $ is decreasing, continuous on $ (a, \infty) $ ($ a < 0 $), and twice differentiable at $ 0 $. The result has been known for the case $ s = 1 $. This paper analyzes the case $ s > 1 $. We obtain an asymptotic sequence that is quite different from the case $ s = 1 $. Some examples and applications are provided.

This article studies an optimal control problem for the financial investment strategy with random incomes. The investment portfolio is simplified to be composed of a risk-free asset and a risky asset. The price of a risky asset is followed by a constant variance elasticity (CEV) model. We consider any correlation coefficient ρ ∈ [ − 1 , 1 ] between the income risk and the risk of risky asset. By applying the Legendre transformation, dual theory, and asymptotic expansion approach, we obtain an asymptotic strategy for the exponential utility function. Numerical examples are presented to illustrate the effects of parameters on the optimal strategy. [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.

We consider a two‐dimensional transmission problem of linear elasticity with voids in a fixed domain Ω−$$ {\Omega}_{-} $$ coated by a curved thin layer Ω+δ$$ {\Omega}_{&#x0002B;}&#x0005E;{\delta } $$. Our aim is to model the effect of the thin layer Ω+δ$$ {\Omega}_{&#x0002B;}&#x0005E;{\delta } $$ on the fixed domain Ω−$$ {\Omega}_{-} $$ by an impedance boundary condition. For that, we use the techniques of asymptotic expansion to approximate the transmission problem by an impedance problem set in the fixed domain Ω−$$ {\Omega}_{-} $$, and we prove an error estimate for the approximate impedance problem. [ABSTRACT FROM AUTHOR]

It is known that the generating function of a sequence of Toeplitz matrices may not describe the asymptotic distribution of the eigenvalues of the considered matrix sequence in the non‐Hermitian setting. In a recent work, under the assumption that the eigenvalues are real, admitting an asymptotic expansion whose first term is the distribution function, fast algorithms computing all the spectra were proposed in different settings. In the current work, we extend this idea to non‐Hermitian Toeplitz matrices with complex eigenvalues, in the case where the range of the generating function does not disconnect the complex field or the limiting set of the spectra, as the matrix‐size tends to infinity, has one nonclosed analytic arc. For a generating function having a power singularity, we prove the existence of an asymptotic expansion, that can be used as a theoretical base for the respective numerical algorithm. Different generating functions are explored, highlighting different numerical and theoretical aspects; for example, non‐Hermitian and complex symmetric matrix sequences, the reconstruction of the generating function, a consistent eigenvalue ordering, the requirements of high‐precision data types. Several numerical experiments are reported and critically discussed, and avenues of possible future research are presented. [ABSTRACT FROM AUTHOR]

Through combined applications of the transfer-matrix method and asymptotic expansion technique, we formulate a theory to predict the three-dimensional response of micropolar plates. No ad hoc assumptions regarding through-thickness assumptions of the field variables are made, and the governing equations are two-dimensional, with the displacements and microrotations of the mid-plane as the unknowns. Once the deformation of the mid-plane is solved, a three-dimensional micropolar elastic field within the plate is generated, which is exact up to the second order except in the boundary region close to the plate edge. As an illustrative example, the bending of a clamped infinitely long plate caused by a uniformly distributed transverse force is analyzed and discussed in detail. [ABSTRACT FROM AUTHOR]

We consider singularly perturbed operator differential transport equations of a special form in the case where the transport operator acts on space–time variables; a linear operator acting on an additional variable describes the interaction that "scrambles" the solution with respect to that variable. We construct a formal asymptotic expansion of the solution of the Cauchy problem for a singularly perturbed operator differential transport equation with small nonlinearity and weak diffusion in the case of several spatial variables. Under some conditions assumed for these problems, the leading term of the asymptotics is described by a quasilinear parabolic equation. The remainder term is estimated with respect to the residual under certain conditions. [ABSTRACT FROM AUTHOR]

In this paper we study how the expectations of power means behave asymptotically as some relevant parameter approaches infinity and how to approximate them accurately for general nonnegative continuous probability distributions. We derive approximation formulae for such expectations as distribution mean increases, and apply them to some commonly used distributions in statistics and financial mathematics. By numerical computations we demonstrate the accuracy of the proposed formulae which behave well even for smaller sample sizes. Furthermore, analysis of behavior depending on sample size contributes to interesting connections with the power mean of probability distribution. [ABSTRACT FROM AUTHOR]

AbstractIn this study, we consider the problems of testing for a mean vector and testing the equality of two mean vectors with monotone missing data. We propose new test statistics similar to the simplified Hotelling’s T2-type test statistic for one-sample and two-sample problems under general-step monotone missing data. Approximate upper percentiles of this new statistics are provided by asymptotic expansion along with transformed test statistics based on Bartlett adjustment. Approximate simultaneous confidence intervals for pairwise comparisons among mean vectors are also presented. Furthermore, we investigated the asymptotic behavior of the proposed statistics using a Monte Carlo simulation, and the approximate accuracy of the proposed approximations are provided and discussed. Finally, the numerical power for several statistics is given. [ABSTRACT FROM AUTHOR]

In this article we continue our research in (Yang and Estrada in Asymptot. Anal. 95(1–2) (2015) 1–19), about the asymptotic expansion of thick distributions. We compute more examples of asymptotic expansion of integral transforms using the techniques developed in (Yang and Estrada in Asymptot. Anal. 95(1–2) (2015) 1–19). Besides, we derive a new “Laplace Formula” for the situation in which a point singularity is allowed. [ABSTRACT FROM AUTHOR]

We characterize in a novel manner the physical properties of the low temperature Fermi gas in the degenerate domain as a function of temperature and chemical potential. For the first time we obtain low temperature T results in the domain where several fermions are found within a de Broglie spatial cell. In this regime, the usual high degeneracy Sommerfeld expansion fails. The other known semi-classical Boltzmann domain applies when fewer than one particle is found in the de Broglie cell. We also improve on the understanding of the Sommerfeld expansion in the regime where the chemical potential is close to the mass and also in the high temperature regime. In these calculcations we use a novel characterization of the Fermi distribution allowing the separation of the finite and zero temperature phenomena. The relative errors of the three approximate methods (Boltzmann limit, Sommerfeld expansion, and the new domain of several particles in the de Broglie cell) are quantified. [ABSTRACT FROM AUTHOR]

We consider for a small parameter > 0 a parabolic convection–diffusion problem with Péclet number of order ( − 1) in a three-dimensional graph-like junction consisting of thin curvilinear cylinders with radii of order () connected through a domain (node) of diameter (). Inhomogeneous Robin type boundary conditions, that involve convective and diffusive fluxes, are prescribed both on the lateral surfaces of the thin cylinders and the boundary of the node. The asymptotic behavior of the solution is studied as → 0 , i.e. when the diffusion coefficients are eliminated and the thin junction is shrunk into a three-part graph connected in a single vertex. A rigorous procedure for the construction of the complete asymptotic expansion of the solution is developed and the corresponding energy and uniform pointwise estimates are proved. Depending on the directions of the limit convective fluxes, the corresponding limit problems (= 0) are derived in the form of first-order hyperbolic differential equations on the one-dimensional branches with novel gluing conditions at the vertex. These generalize the classical Kirchhoff transmission conditions and might require the solution of a three-dimensional cell-like problem associated with the vertex to account for the local geometric inhomogeneity of the node and the physical processes in the node. The asymptotic ansatz consists of three parts, namely, the regular part, node-layer part, and boundary-layer one. Their coefficients are classical solutions to mixed-dimensional limit problems. The existence and other properties of those solutions are analyzed. [ABSTRACT FROM AUTHOR]

We study the two-dimensional Euler flow $ X(\cdot,x) $ for $ x $ in the torus $ {\mathbb T}^2: = {\mathbb R}^2/2\pi{\mathbb Z}^2 $, solution to the ODE: $ \partial_t X(\cdot,x) = b(X(\cdot,x)) $ for $ t\geq 0 $, with $ X(0,x) = x $, where $ b $ is the vector field defined on $ {\mathbb T}^2 $ by:$b(x) = b(x_1,x_2): = (-\,A\cos x_1-B\sin x_2\,,\,A\sin x_1+B\cos x_2),\;\;A,B\in{\mathbb R}\setminus\{0\}.$We derive for any $ x\in{\mathbb T}^2 $, the asymptotics of $ X(t,x) $ as $ t $ tends to $ \infty $, depending on whether $ |A| = |B| $ or $ |A|\neq|B| $. In the first case, the orbits of the flow are all bounded. In the second case, it turns out that one of the coordinates of $ X(t,x) $ is bounded with an explicit bound, while the other one is equivalent to $ a(x)\,t $. The function $ a $ does not vanish in $ {\mathbb T}^2 $ and satisfies uniform bounds which depend on parameters $ A,B $. When $ |A|\neq|B| $, we also prove that for any global first integral $ u $ of the flow $ X $ with a periodic gradient, $ \nabla u $ has at least a cluster point of roots in $ {\mathbb T}^2 $, which implies the non-existence of any real-analytic first-integral of the flow. [ABSTRACT FROM AUTHOR]

In the present article, we extend the results of the neural network interpolation operators activated by smooth ramp functions proposed by Yu (Acta Math. Sin.(Chin. Ed.) 59:623-638, 2016). We give different results from Yu (Acta Math. Sin.(Chin. Ed.) 59:623-638, 2016) we discuss the high-order approximation result using the smoothness of φ and a related Voronovskaya-type asymptotic expansion for the error of approximation. In addition, we showcase the related fractional estimates result and the fractional Voronovskaya type asymptotic expansion. We investigate the approximation degree for the iterated and complex extensions of the aforementioned operators. Finally, we provide numerical examples and graphs to effectively illustrate and validate our results. [ABSTRACT FROM AUTHOR]

Functional neurons built from neural circuits are capable of perceiving and processing external signals such as light illumination and magnetic radiation, by converting the physical signals into modulated bioelectric signals called action potential with diverse forms and shapes. Through modulational instability (MI), modulated wave formation and pattern transition are studied in a chain memristive network of 1 0 0 photosensitive neurons. Memristors and photocells are incorporated in a simple FitzHugh–Nagumo neuron to detect and process the external magnetic flux and light illumination. To determine regions of modulated wave formation, linear stability analysis is performed on a nonlinear envelope equation which resulted from the asymptotic expansion of the generic dynamical equations. The growth rate of MI is plotted and the distinct zones of stable/unstable MI are presented. We confirm the analytical result through numerical simulations whereby the initial plane wave solutions lead to the emergence of localized structures with traits of spiking, bursting and chaotic states. High-frequency photocurrent changes orderly localized patterns to chaotic-like patterns while high-frequency magnetic flux promotes pattern transition from bursting to 2-period spiking state and a 4-period spiking state. This could provide an adequate way to influence the behaviors of artificial neurons as well as potential mechanism of information coding in the nervous system. [ABSTRACT FROM AUTHOR]

Numerous novel integral and series representations for Ferrers functions of the first kind (associated Legendre functions on the cut) of arbitrary degree and order, various integral, series and differential relations connecting Ferrers functions of different orders and degrees as well as a uniform asymptotic expansion are derived in this article. Simple proofs of four generating functions for Ferrers functions are given. Addition theorems for P ν - μ tanh α + β are proved by basing on generation functions for three families of hypergeometric polynomials. Relations for Gegenbauer polynomials and Ferrers associated Legendre functions (associated Legendre polynomials) are obtained as special cases. [ABSTRACT FROM AUTHOR]

In this paper, we apply the steepest descent method to the Schläfli-type integral representation of the three-parameter Mittag-Leffler function (well-known as the Prabhakar function). We find the asymptotic expansions of this function for its large parameters with respect to the real and complex saddle points. For each parameter, we separately establish a relation between the variable and parameter of function to determine the leading asymptotic term. We also introduce differentiations of the three-parameter Mittag-Leffler functions with respect to parameters and modify the associated asymptotic expansions for their large parameters. As an application, we derive the leading asymptotic term of fundamental solution of the time-fractional sub-diffusion equation including the Bessel operator with large order. [ABSTRACT FROM AUTHOR]

In this paper, a modification of a renewal-reward process with dependent components is mathematically constructed and the stationary characteristics of this process are studied. Stochastic processes with dependent components have rarely been studied in the literature owing to their complex mathematical structure. We partially fill the gap by studying the effect of the dependence assumption on the stationary properties of the process . To this end, first, we obtain explicit formulas for the ergodic distribution and the stationary moments of the process. Then we analyze the asymptotic behavior of the stationary moments of the process by using the basic results of the renewal theory and the Laplace transform method. Based on the analysis, we obtain two-term asymptotic expansions of the stationary moments. Moreover, we present two-term asymptotic expansions for the expectation, variance, and standard deviation of the process . Finally, the asymptotic results obtained are examined in special cases. [ABSTRACT FROM AUTHOR]

Formulations of initial-boundary value problems for the system of Maxwell's equations in various quasi-stationary approximations in homogeneous and inhomogeneous conducting media are considered. In the case of weakly inhomogeneous media, asymptotic expansions of solutions to the considered initial-boundary value problems in a parameter characterizing the degree of inhomogeneity of the medium are formulated and substantiated. It is shown that the construction of an asymptotic expansion for the quasi-stationary electromagnetic approximation leads to successively solving independent problems for the quasi-stationary electric and quasi-stationary magnetic approximations in a homogeneous medium. Conditions for the initial data providing the convergence of the asymptotic series are given. [ABSTRACT FROM AUTHOR]

We considered the Mellin transform assigned to the convolution wave kernel associated to the Laplace-Beltrami operator on higher rank Riemannian symmetric spaces of the non-compact type. The occurrence of the analyticity strip of this transform can be deduced directly from the pointwise kernel estimates. Using the zeta function techniques, we established its meromorphic extension to the entire complex plane C with simple poles on the real line. [ABSTRACT FROM AUTHOR]

This paper is concerned with radially positive solutions of the k-Hessian equation involving a Matukuma-type source... [ABSTRACT FROM AUTHOR]

We considered the Mellin transform assigned to the convolution wave kernel associated to the Laplace-Beltrami operator on higher rank Riemannian symmetric spaces of the non-compact type. The occurrence of the analyticity strip of this transform can be deduced directly from the pointwise kernel estimates. Using the zeta function techniques, we established its meromorphic extension to the entire complex plane $ {{\Bbb C}} $ with simple poles on the real line.

In this paper, we analyze the time‐dependent flow of an incompressible micropolar fluid in a multiple‐pipe system. Motivated by the applications, we assume that the pipes have circular cross‐section and that the ratio between pipes' thickness and its length is small, denoted by the parameter ε$$ \varepsilon $$. Far from the junction, the fluid exhibits different behavior depending on the magnitude of the viscosity coefficients with respect to the small parameter ε$$ \varepsilon $$. Focusing on the critical case described by the strong coupling between velocity and microrotation, the complete asymptotic expansion of the solution (up to an arbitrary order) is built. To improve the accuracy of the asymptotic approximation, we introduce the boundary layer correctors near the pipes' ends and take into account the interior layer correction in the vicinity of the junction as well. The convergence is also proved via error estimates, providing the rigorous justification of the proposed effective model. [ABSTRACT FROM AUTHOR]

This study introduces a novel approach to analyze the stress and displacement fields around blunt notches in bi-material media, focusing on mode III loading conditions. The eigenfunction expansion method is used to derive a simplified yet accurate solution, satisfying the boundary conditions for bi-material blunt V-notches. The robustness of the proposed asymptotic solution is validated through several finite element analyses, encompassing a range of notched geometries such as blunt V-notches, VO-notches, and circular holes. Notably, it is demonstrated that when the notch tip radius approaches zero, the solution coincides with the existing sharp V-notch model. [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 Wallis power function is defined on \left (-\min \left \{ p,q\right \},\infty \right) by \begin{equation*} W_{p,q}\left (x\right) =\left (\dfrac {\Gamma \left (x+p\right) }{\Gamma \left (x+q\right) }\right) ^{1/\left (p-q\right) }\text { if }p\neq q\text { and }W_{p,p}\left (x\right) =e^{\psi \left (x+p\right) }. \end{equation*} Let p_{i},q_{i}\in \mathbb {R} with 0<\delta _{i}=p_{i}-q_{i}\leq 1, \theta _{i}=\left (1-\delta _{i}\right) /2 for i=1,2 and p_{1}+q_{1}=p_{2}+q_{2}=2\sigma +1. We prove that, if q_{1}>q_{2} then for any integer m\in \mathbb {N}, the function \begin{equation*} x\mapsto \left (-1\right) ^{m}\left [ \ln \frac {W_{p_{1},q_{1}}\left (x\right) }{W_{p_{2},q_{2}}\left (x\right) }-\sum _{k=1}^{m}\frac {a_{2k}\left (\theta _{1},\theta _{2}\right) }{\left (2k+1\right) \left (2k\right) \left (x+\sigma \right) ^{2k}}\right ] \end{equation*} is completely monotonic on \left (-\sigma,\infty \right), where \begin{equation*} a_{2k}\left (\theta _{1},\theta _{2}\right) =\frac {B_{2k+1}\left (\theta _{2}\right) }{\theta _{2}-1/2}-\frac {B_{2k+1}\left (\theta _{1}\right) }{\theta _{1}-1/2}. \end{equation*} This extends and generalizes some known results. [ABSTRACT FROM AUTHOR]

In this paper, we start from a two dimensional transmission model problem in the framework of couple stress elasticity with voids which is defined in a fixed domain Ω − juxtaposed with a planar thin layer Ω + δ . We first derive a first approximation of Dirichlet-to-Neumann operator for the thin layer Ω + δ by using the techniques of asymptotic expansion with scaling, which allows us to approximate the transmission problem by a boundary value problem doesn't take into account any more the thin layer Ω + δ , called approximate impedance problem; after that, we prove an error estimate between the solution of the transmission problem and the solution of the approximate impedance problem. [ABSTRACT FROM AUTHOR]