Two-term Asymptotic Expansion Research Articles

Two-Term Asymptotics of the Exchange Energy of the Electron Gas on Symmetric Polytopes in the High-Density Limit

We derive a two-term asymptotic expansion for the exchange energy of the free electron gas on strictly tessellating polytopes and fundamental domains of lattices in the thermodynamic limit. This expansion comprises a bulk (volume-dependent) term, the celebrated Dirac exchange, and a novel surface correction stemming from a boundary layer and finite-size effects. Furthermore, we derive analogous two-term asymptotic expansions for semi-local density functionals. By matching the coefficients of these asymptotic expansions, we obtain an integral constraint for semi-local approximations of the exchange energy used in density functional theory.

Two-term large-time asymptotic expansion of the value function for dissipative nonlinear optimal control problems

Two-term large-time asymptotic expansion of the value function for dissipative nonlinear optimal control problems

Entanglement Entropy of Ground States of the Three-Dimensional Ideal Fermi Gas in a Magnetic Field

We study the asymptotic growth of the entanglement entropy of ground states of non-interacting (spinless) fermions in R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathbb {R}}}^3$$\\end{document} subject to a constant magnetic field perpendicular to a plane. As for the case with no magnetic field we find, to leading order L2ln(L)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2\\ln (L)$$\\end{document}, a logarithmically enhanced area law of this entropy for a bounded, piecewise Lipschitz region LΛ⊂R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L\\Lambda \\subset {{\\mathbb {R}}}^3$$\\end{document} as the scaling parameter L tends to infinity. This is in contrast to the two-dimensional case since particles can now move freely in the direction of the magnetic field, which causes the extra ln(L)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ln (L)$$\\end{document} factor. The explicit expression for the coefficient of the leading order contains a surface integral similar to the Widom–Sobolev formula in the non-magnetic case. It differs, however, in the sense that the dependence on the boundary, ∂Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\partial \\Lambda $$\\end{document}, is not solely on its area but on the “surface perpendicular to the direction of the magnetic field”. We utilize a two-term asymptotic expansion by Widom (up to an error term of order one) of certain traces of one-dimensional Wiener–Hopf operators with a discontinuous symbol. This leads to an improved error term of the order L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document} of the relevant trace for piecewise C1,α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf{C}^{1,\\alpha }$$\\end{document} smooth surfaces ∂Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\partial \\Lambda $$\\end{document}.

Asymptotics of sloshing eigenvalues for a triangular prism

AbstractWe consider the three-dimensional sloshing problem on a triangular prism whose angles with the sloshing surface are of the form ${\pi}/{2q}$ , where q is an integer. We are interested in finding a two-term asymptotic expansion of the eigenvalue counting function. When both angles are ${\pi}/{4}$ , we compute the exact value of the second term. As for the general case, we conjecture an asymptotic expansion by constructing quasimodes for the problem and computing the counting function of the related quasi-eigenvalues. These quasimodes come from solutions of the sloping beach problem and correspond to two kinds of waves, edge waves and surface waves. We show that the quasi-eigenvalues are exponentially close to real eigenvalues of the sloshing problem. The asymptotic expansion of their counting function is closely related to a lattice counting problem inside a perturbed ellipse where the perturbation is in a sense random. The contribution of the angles can then be detected through that perturbation.

Asymptotic Expansions and Strategies in the Online Increasing Subsequence Problem

We study two closely related problems in the online selection of increasing subsequence. In the first problem, introduced by Samuels and Steele (Ann. Probab. 9(6):937–947, 1981), the objective is to maximise the length of a subsequence selected by a nonanticipating strategy from a random sample of given size n. In the dual problem, recently studied by Arlotto et al. (Random Struct. Algorithms 49:235–252, 2016), the objective is to minimise the expected time needed to choose an increasing subsequence of given length k from a sequence of infinite length. Developing a method based on the monotonicity of the dynamic programming equation, we derive the two-term asymptotic expansions for the optimal values, with O(1) remainder in the first problem and O(k) in the second. Settling a conjecture in Arlotto et al. (Random Struct. Algorithms 52:41–53, 2018), we also design selection strategies to achieve optimality within these bounds, that are, in a sense, best possible.

Estimation of the creep crack-tip stress field considering the effect of threshold stress

Because of the presence of second-phase particles or nanoscale particles, a term of threshold stress incorporated into the power-law creep model of some materials, such as magnesium, aluminum, and chromium alloys. The estimation of the creep crack-tip stress field for these materials is notably important to facilitate their exploitation in high-temperature structures. In the present work, a two-term asymptotic expansion of the stress field is developed to predict the creep crack-tip stress field for power-law creep materials with threshold stress. The finite element method is used to verify the accuracy of the two-term asymptotic expansion of the stress field. The simulation results reveal that the two-term asymptotic expansion of the stress field is accurate enough as an engineering method in estimating the creep crack-tip stress field. In addition, the existing estimation method of the time-dependent C-integral of C(t) can be directly implemented in power-law creep materials with threshold stress.

Effective estimation of some oscillatory integrals related to infinitely divisible distributions

We present a practical framework to prove, in a simple way, two-term asymptotic expansions for Fourier integrals $$\begin{aligned} {{\mathcal {I}}}(t) = \int _{{\mathbb {R}}}(\mathrm{e}^{it\phi (x)}-1) \mathop {}\!\mathrm {d}\mu (x), \end{aligned}$$where \(\mu \) is a probability measure on \({{\mathbb {R}}}\) and \(\phi \) is measurable. This applies to many basic cases, in link with Levy’s continuity theorem. We present applications to limit laws related to rational continued fraction coefficients.

Uniform stress field inside a non-parabolic open inhomogeneity interacting with a mode III crack

Using conformal mapping techniques, analytic continuation and the theory of Cauchy singular integral equations, we prove that a non-parabolic open inhomogeneity embedded in an elastic matrix subjected to a uniform remote anti-plane stress nevertheless admits an internal uniform stress field despite the presence of a finite mode III crack in its vicinity. Our analysis indicates that: (i) the internal uniform stress field is independent of the specific shape of the inhomogeneity and the presence of the finite crack; (ii) the existence of the finite crack plays a key role in the non-parabolic open shape of the inhomogeneity and in the non-uniform stresses in the surrounding matrix; (iii) the two-term asymptotic expansion at infinity of the stress field in the matrix is independent of the presence of the finite crack. Detailed numerical results are presented to demonstrate the proposed theory.

Boundary Element Solution of Electromagnetic Fields for Non-Perfect Conductors at Low Frequencies and Thin Skin Depths

A novel boundary element formulation for solving problems involving eddy currents in the thin skin depth approximation is developed. It is assumed that the time-harmonic magnetic field outside the scatterers can be described using the quasistatic approximation. A two-term asymptotic expansion with respect to a small parameter characterizing the skin depth is derived for the magnetic and electric fields outside and inside the scatterer, which can be extended to higher order terms if needed. The introduction of a special surface operator (the inverse surface gradient) allows the reduction of the problem complexity. A method to compute this operator is developed. The obtained formulation operates only with scalar quantities and requires computation of surface operators that are usual for boundary element (method of moments) solutions to the Laplace equation. The formulation can be accelerated using the fast multipole method. The method is much faster than solving the vector Maxwell equations. The obtained solutions are compared with the Mie solution for scattering from a sphere and the error of the solution is studied. Computations for much more complex shapes of different topologies, including for magnetic and electric field cages used in testing are also performed and discussed.

Limit Theorem for a Semi-Markovian Random Walk with General Interference of Chance

A semi-Markovian random-walk process with general interference of chance was constructed and investigated. The key point of this study is the assumption that the discrete interference of chance has a general form. Under some conditions, it is proved that the process is ergodic, and the exact forms of the ergodic distribution and characteristic function of the process are obtained. By using basic identity for random walks, the characteristic function of the process is expressed by the characteristic function of a boundary functional. Then, two-term asymptotic expansion for the characteristic function of the standardized process is found. Using this asymptotic expansion, a weak convergence theorem for the ergodic distribution of the standardized process is proved, and the limiting form for the ergodic distribution is obtained. The obtained limit distribution coincides with the limit distribution of the residual waiting time of the renewal process generated by a sequence of random variables expressing the discrete interference of chance.

Poisson–Voronoi Tessellation on a Riemannian Manifold

Abstract In this paper, we consider a Riemannian manifold $M$ and the Poisson–Voronoi tessellation generated by the union of a fixed point $x_0$ and a Poisson point process of intensity $\lambda $ on $M$. We obtain a two-term asymptotic expansion, when $\lambda $ goes to infinity, of the mean number of vertices of the Voronoi cell associated with $x_0$. The 1st term of the estimate is equal to the mean number of vertices in the Euclidean setting, while the 2nd term involves the scalar curvature of $M$ at $x_0$. This settles with the proper and rigorous frame the former 2D statement from [ 19] and extends it to higher dimension. The key tool for proving this result is a new change of variables formula of Blaschke–Petkantschin type in the Riemannian setting, which brings out the Ricci curvatures of the manifold.

Limit Theorem for a Semi - Markovian Stochastic Model of Type (s,S)

In this study, a semi-Markovian inventory model of type $(s,S)$ is considered and the model is expressed by means of renewal-reward process $(X(t))$ with an asymmetric triangular distributed interference of chance and delay. The ergodicity of the process $X(t)$ is proved and the exact expression for the ergodic distribution is obtained. Then, two-term asymptotic expansion for the ergodic distribution is found for standardized process $W(t)\equiv (2X(t)) / (S-s)$. Finally, using this asymptotic expansion, the weak convergence theorem for the ergodic distribution of the process $W(t)$ is proved and the explicit form of the limit distribution is found.

Asymptotic approach for a renewal-reward process with a general interference of chance

ABSTRACTIn this study, a renewal-reward process with a discrete interference of chance is constructed and considered. Under weak conditions, the ergodicity of the process X(t) is proved and exact formulas for the ergodic distribution and its moments are found. Within some assumptions for the discrete interference of chance in general form, two-term asymptotic expansions for all moments of the ergodic distribution are obtained. Additionally, kurtosis coefficient, skewness coefficient, and coefficient of variation of the ergodic distribution are computed. As a special case, a semi-Markovian inventory model of type (s, S) is investigated.

Direct and inverse asymptotic problems with high-frequency terms

This paper is devoted to the first boundary value problem for the heat equation with a fast oscillating source. Direct and inverse problems are solved. The direct problem is to construct and justify an asymptotic expansion for the solution under appropriate assumptions. The inverse problem is to find the source if the value of two-term asymptotic expansion for solution at some point of space is given.

SPECTRAL THEORETIC CHARACTERIZATION OF THE MASSLESS DIRAC ACTION

We consider an elliptic self-adjoint first order differential operator L acting on pairs (2-columns) of complex-valued half-densities over a connected compact 3-dimensional manifold without boundary. The principal symbol of the operator L is assumed to be trace-free and the subprincipal symbol is assumed to be zero. Given a positive scalar weight function, we study the weighted eigenvalue problem for the operator L. The corresponding counting function (number of eigenvalues between zero and a positive lambda) is known to admit, under appropriate assumptions on periodic trajectories, a two-term asymptotic expansion as lambda tends to plus infinity and we have recently derived an explicit formula for the second asymptotic coefficient. The purpose of this paper is to establish the geometric meaning of the second asymptotic coefficient. To this end, we identify the geometric objects encoded within our eigenvalue problem - metric, nonvanishing spinor field and topological charge - and express our asymptotic coefficients in terms of these geometric objects. We prove that the second asymptotic coefficient of the counting function has the geometric meaning of the massless Dirac action.

Spectral problem with Steklov condition on a thin perforated interface

A two-dimensional Steklov-type spectral problem for the Laplacian in a domain divided into two parts by a perforated interface with a periodic microstructure is considered. The Steklov boundary condition is set on the lateral sides of the channels, a Neumann condition is specified on the rest of the interface, and a Dirichlet and Neumann condition is set on the outer boundary of the domain. Two-term asymptotic expansions of the eigenvalues and the corresponding eigenfunctions of this spectral problem are constructed.

On radiation of omnidirectional axisymmetric antennas with circular ground planes

Two-term asymptotic expansions of the solution to the key scattering problem for the vector toroidal wave with general form of the azimuthal dependence scattered by a semitransparent disk are obtained. The problem is solved in the Kirchhoff approximation. The obtained asymptotic expansions of the scattering pattern can be used for simulation of radiation of omnidirectional axisymmetric antennas with circular semitransparent ground planes. For a metal disk ground plane, corrections to the edge waves in the approximation of the physical theory of diffraction are obtained. The developed theory is used to obtain asymptotic formulas for the radiation pattern of a loop antenna with ground plane and a monopole. The results of simulation with the use of these formulas are compared to the corresponding numerical results obtained with the finite-element method.

Asymptotic analysis of narrow escape problems in nonspherical three-dimensional domains.

Narrow escape problems consider the calculation of the mean first passage time (MFPT) for a particle undergoing Brownian motion in a domain with a boundary that is everywhere reflecting except for at finitely many small holes. Asymptotic methods for solving these problems involve finding approximations for the MFPT and average MFPT that increase in accuracy with decreasing hole sizes. While relatively much is known for the two-dimensional case, the results available for general three-dimensional domains are rather limited. This paper addresses the problem of finding the average MFPT for a class of three-dimensional domains bounded by the level surface of an orthogonal coordinate system. In particular, this class includes spheroids and other solids of revolution. The primary result presented is a two-term asymptotic expansion for the average MFPT of such domains containing an arbitrary number of holes. Steps are taken towards finding higher-order asymptotic expansions for both the average MFPT and the MFPT in these domains. The results for the average MFPT are compared to full numerical calculations performed with the comsol Multiphysics finite element solver for three distinct domains: prolate and oblate spheroids and biconcave disks. This comparison shows good agreement with the proposed two-term expansion of the average MFPT in the three domains.

Semi-classical analysis of the Laplace operator with Robin boundary conditions

We prove a two-term asymptotic expansion of eigenvalue sums of the Laplacian on a bounded domain with Neumann, or more generally, Robin boundary conditions. We formulate and prove the asymptotics in terms of semi-classical analysis. In this reformulation it is natural to allow the function describing the boundary conditions to depend on the semi-classical parameter and we identify and analyze three different regimes for this dependence.

Correlating data: examples from turbulent wall layers

Flow fields in fluid mechanics often contain regions where different physical events occur. The relative size of the regions changes as a parameter is varied. Correlating field data, either physical or DNS calculation experiments, in these situations can be aided by using ideas from matched asymptotic expansions from applied mathematics. A second situation is when two slightly different processes occur in the same spatial region. For this case, a two-term asymptotic expansion is needed. This article discusses how composite expansions and the common part matching behavior are useful in correlating data.