A complete asymptotic expansion for a sequence of certain sums

In the present note we derive a complete asymptotic expansion for a sequence of certain sums. A special case of our results solves a problem recently proposed in the Research Group in Mathematical Inequalities and Applications (RGMIA) mailing list. Moreover, the result is connected with a certain combinatorial problem.     

A generalized inverse method for asymptotic linear programming

Consider a linear program in which the entries of the coefficient matrix vary linearly with time. To study the behavior of optimal solutions as time goes to infinity, it is convenient to express the inverse of the basis matrix as a series expansion of powers of the time parameter. We show that an algorithm of Wilkinson (1982) for solving singular differential equations can be used to obtain such an expansion efficiently. The resolvent expansions of dynamic programming are a special case of this method.     

A simple approach to asymptotic expansions for Fourier integrals of singular functions

In this work, we are concerned with the derivation of full asymptotic expansions for Fourier integrals as s → ∞, where s is real positive, [a, b] is a finite interval, and the functions f(x) may have different types of algebraic and logarithmic singularities at x = a and x = b. This problem has been treated in the literature by techniques involving neutralizers and Mellin transforms. Here, we derive the relevant asymptotic expansions by a method that employs simpler and less sophisticated tools.     

A remark on the asymptotic expansion of density function of Wiener functionals

We consider the asymptotic expansion of density function of Wiener functionals as time tends to zero as in [S. Kusuoka, D.W. Stroock, Precise asymptotics of certain Wiener functionals, J. Funct. Anal. 99 (1991) 1-74], and give an explicit formula for the first coefficient.     

Some asymptotic properties of discrete means

In part 1, given n different ways of averaging n positive numbers, we iterate the resulting map in $(0,\infty {)}^n$. We prove convergence toward the diagonal, with rate estimates under smoothness assumptions. In part 2, we consider the elementary symmetric means of order p applied to the values $a_i=a(i/n),1⩽i⩽n$, of a given continuous positive function a on the normalized interval $[0,1]$ and we let $p=f(n)$. When ${\lim }_{n\to \infty }f(n)/n=0$, we prove that it admits a limit as $n\to \infty$, called the f-mean of a, which moreover coincides with ${\int }_0^{1}a(x)\mathrm{d}x$ whenever $f(n)=o(\log n)$. We record similar, quite immediate, results on the geometric side $p=n-f(n)$.     

Asymptotic expansions of Kummer hypergeometric functions with three asymptotic parameters a,b and z

In a recent paper (Temme, 2021) new asymptotic expansions are given for the Kummer functions $M(a,b,z)$ and $U(a,b+1,z)$ for large positive values of $a$ and $b$, with $z$ fixed and special attention for the case $a～b$. In this paper we extend the approach and also accept large values of $z$. The new expansions are valid when at least one of the parameters $a$, $b$, or $z$ is large. We provide numerical tables to show the performance of the expansions.     

The complete asymptotic expansion for Baskakov operators

In this paper, we derive the complete asymptotic expansion of classical Baskakov operators Vn （f;x） in the form of all coefficients of n^-k, k = 0, 1... being calculated explic- itly in terms of Stirling number of the first and second kind and another number G（i, p）. As a corollary, we also get the Voronovskaja-type result for the operators.     

The complete asymptotic expansion for Bernstein operators

In this paper we study the asymptotic behavior of the classical Bernstein operators, applied to q-times continuously differentiable functions. Our main results extend the results of S.N. Bernstein and R.G. Mamedov for all q-odd natural numbers and thus generalize the theorem of E.V. Voronovskaja. The exact degree of approximation is also proved.     

Full asymptotic expansion of the heat trace for non–self–adjoint elliptic cone operators

The operator e^–tA and its trace Tr e^–tA, for t > 0, are investigated in the case when A is an elliptic differential operator on a manifold with conical singularities. Under a certain spectral condition (parameter–ellipticity) we obtain a full asymptotic expansion in t of the heat trace as t → 0⁺. As in the smooth compact case, the problem is reduced to the investigation of the resolvent (A–λ)^–1. The main step consists in approximating this family by a parametrix of A – λ constructed within a suitable parameter–dependent calculus.     

A complete asymptotic expansion of the jacobi functions with error bounds

In this paper we give a complete asymptotic expansion of the Jacobi functions ^{(, )} (t) as + . The method we employed to get the complete expansion follows that of Olver in treating similar problems. By using a Gronwall-Bellman type inequality for an improper integral in which the integrand is an unbounded function and contains a parameter, we get an error bound of the asymptotic approximation which is different from that of Olver's.     

A note on the Landau constants

In this paper we establish new asymptotic expansions for the nth Landau constant G_n. An open problem is indicated as well.     

Uniform asymptotic methods for integrals

We give an overview of basic methods that can be used for obtaining asymptotic expansions of integrals: Watson’s lemma, Laplace’s method, the saddle point method, and the method of stationary phase. Certain developments in the field of asymptotic analysis will be compared with De Bruijn’s book Asymptotic Methods in Analysis. The classical methods can be modified for obtaining expansions that hold uniformly with respect to additional parameters. We give an overview of examples in which special functions, such as the complementary error function, Airy functions, and Bessel functions, are used as approximations in uniform asymptotic expansions.     

The asymptotic critical wave speed in a family of scalar reaction-diffusion equations

We study traveling wave solutions for the class of scalar reaction-diffusion equations

     

On functions uniquely determined by their asymptotic expansion

We present a maximal class of analytic functions. The elements of this class are uniquely determined by their asymptotic expansions. We also discuss the possibility of recovery of a function from the coefficients of its asymptotic series. In particular, we consider the problem of recovering by using Borel summation. The last published result in this direction was obtained by Alan Sokal in 1980, but his paper well known to physicists (in quantum field theory) seems to have remained unnoticed by mathematicians.     

A definitive result on asymptotic contractions

We generalize a fixed point theorem for asymptotic contractions due to Kirk [W.A. Kirk, Fixed points of asymptotic contractions, J. Math. Anal. Appl. 277 (2003) 645-650]. Our result is the final generalization in some sense.     

An asymptotic expansion for 1D steady compressible Navier-Stokes equations under nonuniform enthalpy

The solutions of the one-dimensional (1D) steady compressible Navier-Stokes equations have been thoroughly discussed before, but restrained for uniform total enthalpy, which leads to only a shock wave profile possible in an infinite domain. To date, very little progress has been made for the case with nonuniform total enthalpy. In this paper, we affirm that under nonuniform total enthalpy, there also exists steady solution for the 1D compressible Navier-Stokes equations, but the flow domain must be finite in the positive $x$-axis. The 1D steady compressible Navier-Stokes equations can be reduced to a singular perturbed nonlinear ordinary differential equation (ODE) for velocity with the assumptions of $\text{◂=▸}Pr=3/4$ and a constant viscosity coefficient. By analyzing the mathematical property of the nonlinear ODE for velocity, we propose an asymptotic expansion for the solution of it as an exponential type sequence and also prove the convergence. Unlike the case of uniform total enthalpy, where the solutions for all variables keep monotone, we show that under nonuniform total enthalpy and some specific boundary conditions, there exists extreme inside the thin boundary layer. Numerical results verify the accuracy and convergence of the asymptotic expansion. This asymptotic expansion solution can serve as an important testing to demonstrate the efficiency of numerical methods developed for compressible Navier-Stokes equations at high Reynolds number.     

An asymptotic expansion for the distribution of asymptotic maximum likelihood estimators of vector parameters

It is shown that the probability that a suitably standardized asymptotic maximum likelihood estimator of a vector parameter (i.e., an estimator which approximates the solution of the likelihood equation in a reasonably good way) lies in a measurable convex set can be approximated by an integral involving a multidimensional normal density function and a series in $\text{n}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}$ with certain polynomials as coefficients.     

Analytical solution of coupled soil erosion and consolidation equations by asymptotic expansion approach

In this work, one-dimensional approximation of internal erosion taking place in a soil made from sand and clay mixture was considered. The clay phase that is susceptible to experience erosion under water flow discharge was assumed to be small. A new erosion law fixing the initiation threshold of erosion and integrating the effect of soil consolidation on internal erosion was proposed. Conversely, the effect of erosion on elastic soil deformation was also integrated through damage mechanics concepts. Asymptotic expansion of the coupled equations in terms of a perturbation parameter linked to the total amount of internal erosion that is likely to occur has been performed. This has enabled to view the internal erosion phenomenon occurring inside the soil as a perturbation affecting the classical soil consolidation equation, and further to evaluate the critical discharge gradient for which internal erosion starts. Equations at order zero that are provided by the asymptotic expansion were exactly integrated while an adequate finite difference scheme was introduced to solve the equations at order one. A parametric study was conducted after that in order to assess effects of the main factors on internal erosion and soil deformation.