Asymptotic Research Articles

Precise large deviations of the net loss process in a non-standard two-dimensional risk model

Abstract This paper investigates the precise large deviations of the net loss process in a two-dimensional risk model with consistently varying tails and dependence structures, and gives some asymptotic formulas which hold uniformly for all x varying in t-intervals. The study is among the initial efforts to analyze potential risk via large deviation results for the net loss process of the two-dimensional risk model, and can provide a novel insight to assess the operation risk in a long run by fully considering the premium income factors of the insurance company.

One loop effective actions in Kerr-(A)dS black holes

We compute new exact analytic expressions for one-loop scalar effective actions in Kerr (A)dS black hole (BH) backgrounds in four and five dimensions. These are computed by the connection coefficients of the Heun equation via a generalization of the Gelfand-Yaglom formalism to second-order linear ordinary differential equations with regular singularities. The expressions we find are in terms of Nekrasov-Shatashvili special functions, making explicit the analytic properties of the one-loop effective actions with respect to the gravitational parameters and the precise contributions of the quasinormal modes. The latter arise via an associated integrable system. In particular, we prove asymptotic formulas for large angular momenta in terms of hypergeometric functions and give a precise mathematical meaning to Rindler-like region contributions. Moreover, we identify the leading terms in the large distance expansion as the point particle approximation of the BH and their finite size corrections as encoding the BH tidal response. We also discuss the exact properties of the thermal version of the BH effective actions by providing a proof of the Denef-Hartnoll-Sachdev formula and explicitly computing it for new relevant cases. Although we focus on the real scalar field in dS-Kerr and (A)dS-Schwarzschild in four and five dimensions, similar formulas can be given for higher spin matter and radiation fields in more general gravitational backgrounds. Published by the American Physical Society 2024

A further generalization of sums of higher derivatives of the Riemann zeta function

We obtain a full asymptotic for the sum of [Formula: see text], where [Formula: see text] denotes the [Formula: see text]th derivative of the Riemann zeta function, [Formula: see text] is a positive real number, and [Formula: see text] denotes a nontrivial zero of the Riemann zeta function. The sum is over the zeros with imaginary parts up to a height [Formula: see text], as [Formula: see text]. We also specify what the asymptotic formula becomes when [Formula: see text] is a positive integer, highlighting the differences in the asymptotic expansions as [Formula: see text] changes its arithmetic nature.

Effective count of square-tiled surfaces with prescribed real and imaginary foliations in connected components of strata

Abstract We prove an effective estimate with a power saving error term for the number of square-tiled surfaces in a connected component of a stratum of quadratic differentials whose vertical and horizontal foliations belong to prescribed mapping class group orbits and which have at most L squares. This result strengthens asymptotic counting formulas in the work of Delecroix, Goujard, Zograf, Zorich, and the author.

The mean number of $2$-torsion elements in the class groups of cubic orders

We determine the mean number of 2-torsion elements in class groups of cubic orders, when such orders are enumerated by discriminant. Specifically, we prove that when isomorphism classes of totally real (resp., complex) cubic orders are enumerated by discriminant, the average 2 -torsion in the class group is 1 + \frac{1}{4}\times \frac{\zeta(2)}{\zeta(4)} (resp., 1 + \frac{1}{2}\times \frac{\zeta(2)}{\zeta(4)} ). In particular, we find that the average 2 -torsion in the class group increases when one ranges over all orders in cubic fields instead of restricting to the subfamily of rings of integers of cubic fields, where the average 2 -torsion in the class group was first determined in work of Bhargava to be \frac{5}{4} (resp., \frac{3}{2} ).By work of Bhargava–Varma, proving this result amounts to obtaining an asymptotic count of the number of “reducible” \operatorname{SL}_{3}(\mathbb{Z}) -orbits on the space \mathbb{Z}^{2} \otimes_{\mathbb{Z}}\operatorname{Sym}^{2} \mathbb{Z}^{3} of 3 \times 3 symmetric integer matrices having bounded invariants and satisfying local conditions. In this paper, we resolve the generalization of this orbit-counting problem where the dimension 3 is replaced by any fixed odd integer N \geq 3 . More precisely, we determine asymptotic formulas for the number of reducible \operatorname{SL}_{N}(\mathbb{Z}) -orbits on \mathbb{Z}^{2} \otimes_{\mathbb{Z}}\operatorname{Sym}^{2} \mathbb{Z}^{N} satisfying general infinite sets of congruence conditions.

Interatomic Coulombic electron capture beyond the virtual photon approximation.

Via the interatomic Coulombic electron capture (ICEC) process, an electron can be captured by an atom or a molecule, while the binding and excess energy is transferred, via a long-range Coulomb interaction, to a neighboring atom or molecule. The transferred energy can be used to ionize or electronically excite the neighboring species. When the two species are asymptotically far apart, an analytical formula for the ICEC cross sections can be derived. The latter can then be estimated using only the energies and the photoionization cross sections of each species. In this work, we develop an analytical model that allows us to predict the ICEC cross sections when the size of the involved species is comparable to the distance between the two entities. Using abinitio R-matrix results for various systems, we show that the new model reduces the error of the asymptotic formula by two orders of magnitude on average while only using parameters that can be taken from the properties of each species.

Twisted mixed moments of the Riemann zeta function

AbstractWe analyse a collection of twisted mixed moments of the Riemann zeta function and establish the validity of asymptotic formulae comprising on some instances secondary terms of the shape for a suitable constant and a polynomial . Such examinations are performed both unconditionally and under the assumption of a weaker version of the ‐conjecture.

An asymptotic formula for perturbations in an electromagnetic eigenvalue problem due to the presence of small deformable dielectric inclusion

AbstractWe consider a homogeneous electromagnetic material enclosing a bounded ‐domain with a small dielectric inclusion. We derive a rigorous asymptotic expansion of the eigenvalues of the Maxwell operator as the diameter of the inclusion goes to zero.

Interference of echo-signals from spherical scatterers located near the seabed

The paper investigates the impact of the seabed on the echo signal from spherical scatterers. The seabed is modeled as a liquid absorbing half-space. The transmitter/receiver is located in the water half-space. The distance between the transmitter/receiver and the scatterer is assumed to be large compared to the wavelengths of acoustic waves in water and the seafloor. Numerical results are obtained for acoustically rigid spherical scatterers of the same radius. Interaction between the scatterers is not taken into account. The echo signal from a single sphere over a wide frequency range is computed using a method proposed by R.H. Hackman and G.S. Sammelmann, with a crucial step being the computation of the scattering coefficients of the sphere. Asymptotic formulae obtained using the saddle-point method are used in the paper to compute these coefficients. The obtained asymptotic expressions for the scattering coefficients of the sphere significantly reduce the number of summands in the formula for the form function of the echo signal.

Generalized quadratic Gauss sums and their 2mth power mean

Abstract The main purpose of this article is to study the problem of calculating the 2mth power mean of the generalized quadratic Gauss sums, and using the analytic method and an interesting combinatorial identity to give a sharp asymptotic formula for the 2mth power mean. Thus, a new simple proof of this existing result [N. Bag, A. Rojas-León, and W. P. Zhang, On some conjectures on generalized quadratic Gauss sums and related problems, Finite Fields Appl. 86 (2023), 10213] is provided.

Asymptotic behaviors of Hausdorff dimensions of radial Julia sets of exponential and cosine functions

In this paper, we obtained a four terms asymptotic formula of the Hausdorff dimension of the radial Julia set Jr(fλ) for the exponential function fλ(z) = λez as λ → 0. Moreover, we also discussed the asymptotic behavior of the Hausdorff dimension of the radial Julia set of cosine functions fa,b(z) = aez + be−z as a, b → 0 in various ways.

On the Large-x Asymptotic of the Classical Solutions to the Non-Linear Benjamin Equation in Fractional Sobolev Spaces

In this work, we study the large-x asymptotic of classical solutions to the non-linear Benjamin equation modeling propagation of small amplitude internal waves in a two fluid system. In our analysis, we extend known Hs-well-posedness results to the case of the variable-weight Sobolev spaces. The spaces provide a direct control over the asymptotics of classical solutions and their weak derivatives, and permit us to compute the bulk large-x asymptotic of classical solutions explicitly in terms of input data. The asymptotic formula provides a precise description of the qualitative behaviour of classical solutions in weighted spaces and yields a number of weighted persistence and continuation results automatically.

AVERAGE GEODESIC DISTANCE OF SIERPIŃSKI NETWORKS ON TORUS

In this paper, we investigate the average geodesic distance on Sierpiński torus networks. We construct Sierpiński torus networks based on the classic Sierpiński carpet in an iterative way. By applying finite patterns on integrals, we deduce the exact value of the average geodesic distance of the Sierpiński torus. Furthermore, the asymptotic formula for the average geodesic distance of the corresponding networks can be obtained.

Analysis of the three‐dimensional elasticity theory problem for a radially inhomogeneous cylinder

AbstractIn this paper an axisymmetric problem of elasticity theory for a radially inhomogeneous cylinder of small thickness is studied. It is assumed that homogeneous mixed boundary conditions are specified on lateral surfaces of the cylinder, and boundary conditions that leave the cylinder in equilibrium are specified on the ends of the cylinder. It is considered that elastic moduli are arbitrary continuous functions of a variable along the radius of the cylinder. The formulated boundary value problem is reduced to a spectral problem containing a small parameter characterizing the thinness of the cylinder walls. To determine its solution, the method of asymptotic integration is used, based on three iteration processes. All solutions of the equilibrium equation are constructed, satisfying the specified homogeneous boundary conditions on lateral surfaces. It is obtained that the solution of the problem consists of a penetrating solution and a solution of the nature of the boundary layer. An analysis of stress‐strain states corresponding to different types of solutions is carried out. Based on the constructed asymptotic solutions, a connection is found between the solutions obtained by the equation of elasticity theory and by the applied theory of shells. It is shown that the penetrating solution determines the internal stress‐strain state of a radially inhomogeneous cylinder. The stress state determined by the penetrating solution is equivalent to the principal vector of stress acting in a cross section perpendicular to the axis of the cylinder.The first terms of asymptotic expansions of the penetrating solution in a small parameter coincide with solutions in the applied theory of shells. New classes of solutions are defined as the character of a boundary layer, which are localized at the ends of the cylinder and decrease exponentially with distance from the ends. These solutions are absent in the applied theories of shells. The first terms of the asymptotic expansion of the solution, having the nature of a boundary layer, are equivalent to the Saint‐Venant edge effect in the theory of heterogeneous plates. Asymptotic formulas for displacement and stresses are constructed, allowing one to calculate the three‐dimensional stress‐strain state of a radially heterogeneous cylinder of small thickness with any predetermined accuracy. Based on the obtained asymptotic expansions, one can estimate the applicability areas of applied theories and construct a refined applied theory for radially heterogeneous cylindrical shells.

Trace regularization problem for a fourth‐order differential operator on separable Banach space

In this study, we employ new results on the semi‐inner product, the adjoint operator, and the Schatten class of operators on a separable Banach space, which allow us to investigate a differential operator with unbounded operator‐valued coefficients. Specifically, we derive an asymptotic formula for the second regularized trace based on the asymptotic and spectral properties of the extended operator. For this purpose, we use the theory of continuous dense embeddings and known results regarding the regularized trace in Hilbert space. We conclude our paper by providing examples that support our findings.

Deterministic equivalent and error universality of deep random features learning*

Abstract This manuscript considers the problem of learning a random Gaussian network function using a fully connected network with frozen intermediate layers and trainable readout layer. This problem can be seen as a natural generalization of the widely studied random features model to deeper architectures. First, we prove Gaussian universality of the test error in a ridge regression setting where the learner and target networks share the same intermediate layers, and provide a sharp asymptotic formula for it. Establishing this result requires proving a deterministic equivalent for traces of the deep random features sample covariance matrices which can be of independent interest. Second, we conjecture the asymptotic Gaussian universality of the test error in the more general setting of arbitrary convex losses and generic learner/target architectures. We provide extensive numerical evidence for this conjecture. In light of our results, we investigate the interplay between architecture design and implicit regularization.

Three-Term Asymptotic Formula for Large Eigenvalues of the Quantum Rabi Model with a Resonant Bias

Three-Term Asymptotic Formula for Large Eigenvalues of the Quantum Rabi Model with a Resonant Bias

Three-term asymptotic formula for large eigenvalues of the two-photon quantum Rabi model

We prove that the spectrum of the two-photon quantum Rabi Hamiltonian consists of two eigenvalue sequences (E_{m}^{+})_{m=0}^{\infty} , (E_{m}^{-})_{m=0}^{\infty} satisfying a three-term asymptotic formula with the remainder estimate O(m^{-1}\ln m) when m tends to infinity. Our asymptotic formula can be written so that the third term is given by an explicit oscillatory integral and an explicit remainder estimate.

A note on the satisficing policy of the secretary problem

Wu and Li (2022) presented a satisficing policy for the secretary problem (SP). Moreover, Wu and Li (2022) derived asymptotic formulas of the maximum probability of choosing one of the k satisfactory applicants with special cases k = 2 and 3, respectively, in the satisficing policy. “Based on the work of Wu and Li (2022), this paper derives the asymptotic formulas for the probability of selecting one of the k satisfactory applicants and provides the equation for the approximate optimal cutoff value to achieve the asymptotic maximum probability in the general case of k satisfactory applicants under the satisficing policy.” Moreover, a series of computer simulations are conducted to test the theoretical results for k = 4 and 5.

Existence, stability and the number of two-dimensional invariant manifolds for the convective Cahn–Hilliard equation

We study the well-known generalised version of the nonlinear Cahn–Hilliard evolution equation, supplemented with periodic boundary conditions. We study local bifurcations in the vicinity of spatially homogeneous equilibrium states. We show the possibility of the existence of a finite or countable set of equilibrium states of the boundary value problem under study, in the vicinity of which, if appropriate conditions are met, there exist two-dimensional invariant manifolds filled with solutions that are periodic in the evolutionary variable. Moreover, we derive asymptotic formulas for these periodic solutions. Finally, we study the stability of invariant manifolds and the solutions belonging to them.In order to analyse the bifurcation problem, we used methods from the theory of dynamical systems with infinite-dimensional phase, namely the method of invariant manifolds and the method of normal forms.