Uniform Asymptotics Research Articles

Uniform Far-Field Asymptotics of the Two-Layered Green Function in Two Dimensions and Application to Wave Scattering in a Two-Layered Medium

Uniform Far-Field Asymptotics of the Two-Layered Green Function in Two Dimensions and Application to Wave Scattering in a Two-Layered Medium

Uniform Spectral Asymptotics for a Schrödinger Operator on a Segment with Delta-Interaction

Uniform Spectral Asymptotics for a Schrödinger Operator on a Segment with Delta-Interaction

Uniform asymptotics for a risk model with constant force of interest and a random number of delayed claims

Consider an insurance risk model with constant force of interest under the assumption that an immediate claim is accompanied by a random number of delayed claims. In the presence of heavy tails on claim distributions and dependence structures among modelling components, we study the uniform asymptotics for an insurer's ruin-related quantities, where the uniformity holds for all time horizons varying in a relevant interval.

Uniform Asymptotics for Finite-Time Ruin Probabilities of Risk Models with Non-Stationary Arrivals and Strongly Subexponential Claim Sizes

This paper considers the one- and two-dimensional risk models with a non-stationary claim-number process. Under the assumption that the claim-number process satisfies the large deviations principle, the uniform asymptotics for the finite-time ruin probability of a one-dimensional risk model are obtained for the strongly subexponential claim sizes. Further, as an application of the result of one-dimensional risk model, we derive the uniform asymptotics for a kind of finite-time ruin probability in a two dimensional risk model sharing a common claim-number process which satisfies the large deviations principle.

Uniform asymptotics for a renewal risk model with a random number of delayed claims

Uniform asymptotics for a renewal risk model with a random number of delayed claims

FEATURES OF THE CONSTRUCTION OF UNIFORM ASYMPTOTIC SOLUTION OF A SYSTEM OF DIFFERENTIAL EQUATIONS WITH A TURNING POINT WITH POSITIVE COEFFICIENTS OF THE MATRIX

The work is devoted to the analysis of the coefficients of the singular operator of the Orr-Sommerfeld type in vector form including the turning point. The system of singularly perturbed differential equations with a small parameter at the highest derivative is investigated. We consider the case when the spectrum of the limit operator contains multiple and identically equal zero elements. Using the method of essentially singular functions, the uniform asymptotic solution of the system is constructed. For the case of a stable turning point, the asymptotics of the solutions of the system are constructed in the sector that contains the turning point. The asymptotics of the first two solutions for the homogeneous problem are constructed using the Airy functions and their derivatives. The third formal solution of a homogeneous system for this case presents certain difficulties. Therefore, taking into account the specified conditions, in order to construct the uniform asymptotics of the solution for the given system, we used the partial solution of the heterogeneous system as the third solution of the homogeneous system.

Uniform asymptotics for finite-time ruin probabilities of a bidimensional compound risk model with stochastic returns

Consider a continuous-time bidimensional compound risk model with stochastic returns, where an insurance company operates two lines of business at the same time and is allowed to invest its wealth into financial assets. In this model, each accident may cause a random number of heavy-tailed claims and the claim sizes from the same line of business are upper tail asymptotically independent, while the accident arrival processes from different lines of business are arbitrarily dependent. Under some moment conditions on the accident arrival processes, some uniform asymptotic formulae for finite-time ruin probabilities are established.

Eigenvalues of truncated unitary matrices: disk counting statistics

Let T be an n×n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n\ imes n$$\\end{document} truncation of an (n+α)×(n+α)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(n+\\alpha )\ imes (n+\\alpha )$$\\end{document} Haar distributed unitary matrix. We consider the disk counting statistics of the eigenvalues of T. We prove that as n→+∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n\\rightarrow + \\infty $$\\end{document} with α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document} fixed, the associated moment generating function enjoys asymptotics of the form exp(C1n+C2+o(1)),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\exp \\big ( C_{1} n + C_{2} + o(1) \\big ), \\end{aligned}$$\\end{document}where the constants C1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_{1}$$\\end{document} and C2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_{2}$$\\end{document} are given in terms of the incomplete Gamma function. Our proof uses the uniform asymptotics of the incomplete Beta function.

Surface measure on, and the local geometry of, sub-Riemannian manifolds

We prove an integral formula for the spherical measure of hypersurfaces in equiregular sub-Riemannian manifolds. Among various technical tools, we establish a general criterion for the uniform convergence of parametrized sub-Riemannian distances, and local uniform asymptotics for the diameter of small metric balls.

CONSTRUCTION OF THE SOLUTION OF THE SYSTEM OF SINGULARLY PERTURBATED DIFFERENTIAL EQUATIONS WITH A DIFFERENTIAL TURNING POINT

In this work, a system of singularly perturbed differential equations of the 4th order with a small parameter at the highest derivative and a turning point is considered. The turning point x=0 is located in the end of the segment [-l;0] under consideration. The coefficients of a given matrix are infinitely differentiable functions on a given interval. The goal is to construct a uniform solution asymptotics for a system of singularly perturbed differential equations with a differential turning point on the segment [-l;0]. In this case, the spectrum of the limit operator contains multiple and identically zero elements. The uniform asymptotic solution is constructed by the method of essential-singular functions using the Airy functions [0;l] and their derivatives. Studies have shown that the Apparatus of Airy functions and model equations is quite effective for constructing a uniform asymptotic solution of problems with turning points. It was also established that in order to select these functions along with the independent variable it is necessary to introduce a new variable t of the segment [0;l]. As a result, an extended problem will be obtained, which will make it possible to formally find the solution in the form of series with a small parameter. The constructions of the solutions are obtained by sequentially solving the systems of iterative equations, which at a certain step of the iterations make it possible to determine all the components of the sought-after vector functions with an accuracy of two scalar factors that form an arbitrary vector. The regularizing function Bi(t) is chosen in such a way that the function increases indefinitely when t approaches the infinity. The conducted studies showed that for any ratio of the signs of the matrix coefficient near which the inflection point is located, it is not possible to write down a smooth solution in the form of a single analytical expression. The solution of a degenerate vector equation in the general case has a discontinuity of the second kind at the inflection point, so it cannot be used explicitly to construct the asymptotic solution of this problem. For this purpose, a technique from the classical theory of linear differential equations was applied and solutions for a degenerate equation of the 2nd order were obtained.

Mutually inverse series relating Ferrers and associated Legendre functions and generating functions pertaining to them

This article deals with three types of mutually inverse series relating Ferrers and associated Legendre functions of arbitrary complex indexes and orders established on the base of integral representations by using a number of generating functions (some of them are novel) for polynomials expressed in terms of hypergeometric or generalized hypergeometric polynomials. High-order asymptotics of above-mentioned polynomials are indicated. In particular, as n→∞, a uniform asymptotics for Gegenbauer polynomials Cnλ−n(x), λ∈C, is obtained. In special cases, the inverse series turn into novel mutually inverse finite sums relating Gegenbauer or associated Legendre polynomials of different arguments as well as into novel connection formulas for Gegenbauer and associated Legendre polynomials.

Uniform Asymptotics of Solutions to Linear Differential Equations with Holomorphic Coefficients in the Neighborhood of an Infinitely

Uniform Asymptotics of Solutions to Linear Differential Equations with Holomorphic Coefficients in the Neighborhood of an Infinitely

Connection problem of the first Painlevé transcendents with large initial data

In previous work, Bender and Komijani (2015 J. Phys. A: Math. Theor. 48 475202) studied the first Painlevé (PI) equation and showed that the sequence of initial conditions giving rise to separatrix solutions could be asymptotically determined using a -symmetric Hamiltonian. In the present work, we consider the initial value problem of the PI equation in a more general setting. We show that the initial conditions located on a sequence of curves , , will give rise to separatrix solutions. These curves separate the singular and the oscillating solutions of PI. The limiting form equation for the curves as is derived, where A is a positive constant. The discrete set could be regarded as the nonlinear eigenvalues. Our analytical asymptotic formula of matches the numerical results remarkably well, even for small n. The main tool is the method of uniform asymptotics introduced by Bassom et al (1998 Arch. Rational Mech. Anal. 143 241–71) in the studies of the second Painlevé (PII) equation.

Dimer–Dimer Correlations at the Rough–Smooth Boundary

Three phases of macroscopic domains have been seen for large but finite periodic dimer models; these are known as the frozen, rough and smooth phases. The transition region between the frozen and rough region has received a lot of attention for the last twenty years and recently work has been underway to understand the rough–smooth transition region in the case of the two-periodic Aztec diamond. We compute uniform asymptotics for dimer–dimer correlations of the two-periodic Aztec diamond when the dimers lie in the rough–smooth transition region. These asymptotics rely on a formula found in Chhita and Johansson (Adv Math 294:37–149, 2016) for the inverse Kasteleyn matrix, they also apply to the infinite square grid dimer model with a variable weighting which interpolates between the rough and smooth phase (Kenyon et al. Ann Math (2) 163(3):1019–1056, 2006). In particular, we find that distant dimers initially decay exponentially when the magnetic coordinates are very close to the bounded complementary component of the associated amoebae, they then transition to a power law decay once far enough apart.

Uniform asymptotics for ruin probabilities of a time-dependent bidimensional renewal risk model with dependent subexponential claims

This paper considers a continuous-time bidimensional renewal risk model with subexponential claims. In this model, the claim size vectors and their inter-arrival times form a sequence of independent and identically distributed random vectors following some general dependence structures introduced by [T. Jiang, Y. Wang, Y. Chen and H. Xu, Uniform asymptotic estimate for finite-time ruin probabilities of a time-dependent bidimensional renewal model, Insur. Math. Econom. 64 (2015), pp. 45–53.]. We not only extend the results of [T. Jiang, Y. Wang, Y. Chen and H. Xu, Uniform asymptotic estimate for finite-time ruin probabilities of a time-dependent bidimensional renewal model, Insur. Math. Econom. 64 (2015), pp. 45–53.] under some weak conditions, but we also obtain an asymptotic estimate of the finite-time sum-ruin probability. Furthermore, when the distributions of claim sizes have nonzero lower Karamata indices, some explicit asymptotic formulas are established for the infinite-time ruin probabilities.

Parity violation in spin-precessing binaries: Gravitational waves from the inspiral of black holes in dynamical Chern-Simons gravity

Spin precession in compact binaries is intricately tuned to the multipole structure of the underlying bodies. For black holes, violations of the no-hair theorems induced by modifications to general relativity correct the precession dynamics, which in turn imprints onto the amplitude and phase modulations of the gravitational waves emitted by the binary. Recently, the spin precession equations were derived up to second order in spin for dynamical Chern-Simons (dCS) gravity, a parity-violating modified theory of gravity. We here solve these equations and construct, for the first time, analytic expressions for the time- and frequency-domain gravitational waves emitted in the quasicircular inspiral of spin-precessing black hole binaries in a modified theory of gravity using the post-Newtonian approximation. Working within the small coupling approximation and using multiple scale analysis, we show that the corrections to the nutation phase enter at relative first post-Newtonian (1PN) order, and the corrections to the precession angle and Thomas phase enter at relative 0PN order. Making use of the stationary phase approximation and shifted uniform asymptotics, we find that the Fourier phase of the waveform is characterized by three modifications: two due to the backreaction of the precession dynamics onto the spin-orbit and spin-spin couplings that enter at 1.5PN and 2PN orders, and a 2PN modification due to the emission of dipole radiation. We also find that backreaction of the precession dynamics forces the dCS corrections to the Fourier amplitude to enter at 0PN order, as opposed to 2PN order, as expected for spin-aligned binaries. Our work lays the first foundational stones to build an inspiral-merger-ringdown phenomenological model for spin-precessing binaries in a modified theory of gravity.

Uniform asymptotics for ruin probabilities of a delayed renewal risk model with one-sided linear dependence and stochastic returns

In this article, we consider a renewal risk model with by-claims, where the price process of the investment portfolio follows an exponential Lévy process. We further assume that the main claim is a one-sided linear process and there exists a certain dependence structure between the innovations and by-claims. In the presence of heavy tails, we obtain a series of uniform formulas in finite and infinite intervals. In order to better describe the obtained results, we carry on the numerical simulations.

Asymptotics of determinants with a rotation-invariant weight and discontinuities along circles

We study the moment generating function of the disk counting statistics of a two-dimensional determinantal point process which generalizes the complex Ginibre point process. This moment generating function involves an n×n determinant whose weight is supported on the whole complex plane, is rotation-invariant, and has discontinuities along circles centered at 0. These discontinuities can be thought of as a two-dimensional analogue of jump-type Fisher-Hartwig singularities. In this paper, we obtain large n asymptotics for this determinant, up to and including the term of order n−12. We allow for any finite number of discontinuities in the bulk, one discontinuity at the edge, and any finite number of discontinuities bounded away from the bulk. As an application, we obtain the large n asymptotics of all the cumulants of the disk counting function up to and including the term of order n−12, both in the bulk and at the edge. This improves on the best known results for the complex Ginibre point process, and for general values of our parameters these results are completely new. Our proof makes a novel use of the uniform asymptotics of the incomplete gamma function.

Uniform Asymptotics of Solutions of Second-Order Differential Equations with Meromorphic Coefficients in a Neighborhood of Singular Points and Their Applications

In this paper, we consider the problem of obtaining the asymptotics of solutions of differential operators in a neighborhood of an irregular singular point. More precisely, we construct uniform asymptotics for solutions of linear differential equations with second-order meromorphic coefficients in a neighborhood of a singular point and apply the results obtained to the equations of mathematical physics. The main results related to the construction of uniform asymptotics are obtained using resurgent analysis methods applied to differential equations with irregular singularities. These results allow us to construct asymptotics for any second-order equations with meromorphic coefficients—that is, with an arbitrary order of degeneracy. This also allows one to determine the type of a singular point and highlight the cases where the point is non-singular or regular.

Uniform tail asymptotics for Airy kernel determinant solutions to KdV and for the narrow wedge solution to KPZ

We obtain uniform asymptotics for deformed Airy kernel determinants, which arise in models of finite temperature free fermions and which characterize the narrow wedge solution of the Kardar–Parisi–Zhang equation. The asymptotics for the determinants yield uniform initial data for an associated family of solutions to the Korteweg–de Vries equation, and uniform lower tail asymptotics for the narrow wedge solution of the Kardar–Parisi–Zhang equation.