Asymptotics of the Titchmarsh-Weyl m-coefficient for integrable potentials

Solar sail trajectory analysis with asymptotic expansion method

Our aim here is to give sufficient conditions on the finite element spaces near a point so that the error in the finite element method for the function and its derivatives at the point have exact asymptotic expansions in terms of the mesh parameter h, valid for h sufficiently small. Such expansions are obtained from the so-called asymptotic expansion inequalities valid in $\mathbf{R}^N$ for $N\geq2$, studies by Schatz in [Math. Comp., 67 (1998), pp. 877–899] and [SIAM J. Numer. Anal., 38 (2000), pp. 1269–1293].

Three-dimensional sub- and supersonic flows in nozzles and channels of varying cross section: PMM vol. 36, n≗2, 1972, pp.239–247

In the paper, the authors establish explicit formulas for asymptotic and power series expansions of the exponential and the logarithm of asymptotic and power series expansions. The explicit formulas for the power series expansions of the exponential and the logarithm of a power series expansion are applied to find explicit formulas for the Bell numbers and logarithmic polynomials in combinatorics and number theory.

Modelling physical problems in mathematical form yields the governing equations that may be linear or nonlinear for known and unknown boundaries. The exact solution for those equations may or may not be obtained easily. Hence we seek an analytical approximation solution in terms of asymptotic expansion. In this study, we focus on a singular perturbation in second order ordinary differential equations. Solutions to several perturbed ordinary differential equations are obtained in terms of asymptotic expansion. The aim of this work is to find an approximate analytical solution using the classical method of matched asymptotic expansion (MMAE). The Mathematica computer algebra system is used to perform the algebraic computations. The details procedures will be discussed and the underlying concepts and principles of the MMAE will be clarified. Perturbation problem for linear equation that occurs at one boundary and two boundary layers are discussed. Approximate analytical solution obtained for both cases are illustrated by graph using selected parameter by showing the outer, inner and composite solution separately. Then, the composite solution will be compare to the exact solution to show their accuracy by graph. By comparison, MMAE is found to be one of the best methods to solve singular perturbation problems in second order ordinary differential equation since the results obtained are very close to the exact solution.

An asymptotic model of a nonlinear adaptive elastic plate

For a wide class of self-adjoint Schrödinger Hamiltonians, a detailed description of the time evolution kernel is obtained. In a setting of a d-dimensional Euclidean space without boundaries, the Schrödinger Hamiltonian H is the sum of the negative Laplacian plus a real-valued local potential v(x). The class of potentials studied is the family of bounded and continuous functions that are formed from the Fourier transforms of complex bounded measures. These potentials are suitable for the N-body problem, since they do not necessarily decrease as ‖x‖→∞. An asymptotic expansion in the complex parameter z, around z=0, is derived for the family of kernels Uz(x,y) corresponding to the analytic semigroup {e−zH:Re z>0}, which is uniform in the coordinate variables x and y. The asymptotic expansion has a simple semiclassical interpretation. Furthermore, an explicit bound for the remainder term in the asymptotic expansion is found. The expansion and the remainder term bound continue to the time axis boundary z=it/ℏ (t≠0) of the analytic semigroup domain.

Wallis ratio can be expressed as an asymptotic expansion using Stirling series and Bernoulli numbers. We prove the general inequalities for Wallis ratio for arbitrary number of terms in the asymptotic expansion. We show that the coefficients in the asymptotic expansion are the best possible.

From the point of view of physics, there are essentially two problems in the study of quantum field models. The first problem is to find the phenomena of interest to physics (e.g. particles, bound states, resonances, broken symmetries, and asymptotic power series expansions) in the two and three dimensional quantum field models.1 This problem includes obtaining the mathematical structure underlying these phenomena, and determining qualitatively which interactions and/or parameter values give rise to these phenomena and which do not. We believe this problem is feasible at the present time; indeed the initial progress is the subject of the present lectures. The second problem is the construction of nontrivial fields in four space time dimensions. Here it seems that some new ideas are needed to supplement those which have already been used in the construction of quantum fields.

Large degree asymptotics of generalized Bessel polynomials

Asymptotic expansions of two-loop Feynman diagrams in the Sudakov limit

We continue our work [F. Bornemann, Asymptotic expansions of the limit laws of Gaussian and Laguerre (Wishart) ensembles at the soft edge, preprint (2024), arXiv:2403.07628] on asymptotic expansions at the soft edge for the classical [Formula: see text]-dimensional Gaussian and Laguerre random matrix ensembles. By revisiting the construction of the associated skew-orthogonal polynomials in terms of wave functions, we obtain concise expressions for the level densities that are well suited for proving asymptotic expansions in powers of a certain parameter [Formula: see text]. In the unitary case, the expansion for the level density can be used to reconstruct the first correction term in an established asymptotic expansion of the associated generating function. In the orthogonal and symplectic cases, we can even reconstruct the conjectured first and second correction terms.

A Non-Uniform Estimate for the Convergence Speed in the Multi-Dimensional Central Theorem

We consider quasilinear and linear parabolic problems with rapidly oscillating coefficients in a domain Ω e that is e-periodically perforated by small holes of order . The holes are divided into three e-periodical sets depending on boundary conditions. The homogeneous Dirichlet boundary conditions are imposed for holes of one set, whereas, for holes in the remaining sets, different inhomogeneous Neumann and nonlinear Robin boundary conditions involving additional perturbation parameters are imposed. For a solution to the quasilinear problem we find the leading terms of the asymptotic expansion and prove asymptotic estimates that show the influence of perturbation parameters. In the linear case, we construct and justify a complete asymptotic expansion of the solution by using the two-scale asymptotic expansion method. Bibliography: 25 titles. Illustrations: 1 figure.

An interactive computer system’ service is characterized by the random waiting (or response) time perceived by users. This paper presents a novel solution to the problem of efficiently computing the second moment of the waiting time of a class of large interactive systems. The physical system consists of a bank of terminals, each of which asynchronously alternates between “thinking” and waiting for service from a CPU which operates under the processor-sharing discipline. The problem of obtaining higher order waiting time moments is quite different from that of obtaining CPU queue statistics. Only the first moment of the waiting time is obtainable from the CPU queue statistics. The technique for arriving at the second moment of the waiting time consists of developing an asymptotic expansion in inverse powers of the number of terminals. Hence, as the system grows, quite fortuitously, fewer terms of the series require computation to achieve the desired accuracy. Beside its numerical advantages, the results give new insight since the leading terms of the series, which contain most of the information, are obtained explicitly. The novelty also rests on the fact that instead of solving matrix equations, the problem is turned into one of solving a second order differential equation. A simple two-dimensional recursion yields all the terms of the asymptotic expansion.