Asymptotic Research Articles

Square-full numbers multiple of a certain set of primes and hybrid numbers

Let h ≥ 1 be an arbitrary but fixed positive integer. Let us consider the s distinct primes q1, . . . , qs. Let Bq1···qs(x) be the number of h-full numbers not exceeding x multiple of q1 · · · qs and let Aq1···qs (x) be the number of h-full numbers not exceeding x relatively prime to q1 · · · qs. In this note we obtain asymptotic formulas for Aq1···qs (x) and Bq1···qs (x). Then we apply the results obtained in the study of hybrid h-full numbers. That is, the h-full numbers of the form qhQ, where q > 1 is square-free, Q > 1 is a (h + 1)-full number, and q, Q are relatively prime.

Asymptotic behaviors for the eigenvalues of the Schrödinger equation

We consider the Schrödinger operator in a bounded domain R n ( n = 2 , 3 ) with Neumann boundary condition. We suppose that this domain contains small deformable inclusions, i.e. regions where the potentials do not have the same values as the exterior medium. Our goal is to construct an asymptotic formula for the case of multiple and simple eigenvalue problems. We find an expansion that highlights the relation between the deformation parameters and the eigenvalues. The problem is devoted to a domain containing a finite number of inhomogeneities. Also, we focus on the interface of a small inclusion.

Reversible primes

AbstractFor an ‐bit positive integer written in binary as where , , , let us define the digital reversal of . Also let . With a sieve argument, we obtain an upper bound of the expected order of magnitude for the number of such that and are prime. We also prove that for sufficiently large , where denotes the number of prime factors counted with multiplicity of and is an absolute constant. Finally, we provide an asymptotic formula for the number of ‐bit integers such that and are both squarefree. Our method leads us to provide various estimates for the exponential sum

ON THE ASYMPTOTICS OF EIGENVALUES OF THE NEUMANN PROBLEM FOR THE SCHRODINGER ¨ OPERATOR

. In this paper, we study the change in the eigenvalues of the Neumann problem for the Schr¨odinger equation with respect to the radius of the ball. We prove the self-adjointness of the Schr¨odinger operator with a spherically symmetric homogeneous potential and obtain asymptotic formulas for the eigenvalues of the Neumann problem as the radius of the ball tends to zero.

Construction of homogeneous solutions of the torsion problem for a radially inhomogeneous transversely isotropic cylinder

The torsion problem for a radially inhomogeneous transversely isotropic cylinder of small thickness was investigated by the method of asymptotic integration of elasticity theory equations. It is assumed that the side part of the cylinder is stress-free, and boundary conditions are set at the ends of the cylinder, leaving the cylinder in equilibrium. The elastic moduli are thought to be arbitrary continuous functions of the variable along the cylinder radius. The formulated boundary value problem is reduced to a spectral problem containing a small parameter characterizing the thin-walledness of the cylinder. Homogeneous solutions are built, i.e. any solutions of the equilibrium equation satisfying the condition of no stresses on the side surfaces. It is shown that the solution of the torsion problem consists of a penetrating solution and a boundary layer character solution similar to Saint-Venant's edge effect in the theory of inhomogeneous plates. 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 torsional moments of stresses acting in the cross-section perpendicular to the cylinder axis. Solutions having the boundary layer character are localized at the ends of the cylinder and decrease exponentially with distance from the ends. These solutions are absent in applied shell theories. Asymptotic formulas for displacement and stresses are built, which make it possible to calculate the three-dimensional stress-strain state of a radially inhomogeneous transversely isotropic cylinder of small thickness. Based on the obtained asymptotic expansions, it is possible to assess the applicability of applied theories and build a refined applied theory for radially inhomogeneous cylindrical shells

Asymptotics for the ruin probability in a proportional reinsurance risk model with dependent insurance and financial risks

. This article studies the joint ruin problem for two insurance companies that divide claims in positive proportions (modeling an insurance and re-insurance company). The arrival times of claims are delayed by a common random time. Suppose that the two insurance companies are allowed to make risk-free and risky investments, and the price processes of the corresponding investment portfolios are exponentials of jump-diffusion processes with common jumps. Furthermore, assuming that the claim sizes and their corresponding investment return jump possess a dependence structure, this article establishes an asymptotic formula for the ruin probability.

Integral points of bounded height on a certain toric variety

We determine an asymptotic formula for the number of integral points of bounded height on a certain toric variety, which is incompatible with part of a preprint by Chambert-Loir and Tschinkel. We provide an alternative interpretation of the asymptotic formula we get. To do so, we construct an analogue of Peyre’s constant α \alpha and describe its relation to a new obstruction to the Zariski density of integral points in certain regions of varieties.

Asymptotic expansions for partitions generated by infinite products

Recently, Debruyne and Tenenbaum proved asymptotic formulas for the number of partitions with parts in Λ⊂N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda \\subset {\\mathbb {N}}$$\\end{document} (gcd(Λ)=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gcd (\\Lambda )=1$$\\end{document}) and good analytic properties of the corresponding zeta function, generalizing work of Meinardus. In this paper, we extend their work to prove asymptotic formulas if Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda $$\\end{document} is a multiset of integers and the zeta function has multiple poles. In particular, our results imply an asymptotic formula for the number of irreducible representations of degree n of so(5)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathfrak {so}{(5)}}$$\\end{document}. We also study the Witten zeta function ζso(5)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\zeta _{{\\mathfrak {so}{(5)}}}$$\\end{document}, which is of independent interest.

Long-time asymptotics for the integrable nonlocal Lakshmanan–Porsezian–Daniel equation with decaying initial value data

In this work, we study the Cauchy problem of integrable nonlocal Lakshmanan-Porsezian-Daniel equation with rapid attenuation of initial data. The basic Riemann–Hilbert problem of integrable nonlocal Lakshmanan-Porsezian-Daniel equation is constructed from Lax pair. Using Deift-Zhou nonlinear steepest descent method, the explicit long-time asymptotic formula of integrable nonlocal Lakshmanan-Porsezian-Daniel equation is derived, which is different from the local model. Besides, compared to the nonlocal nonlinear Schrödinger equation, since the increase of real stationary phase points, the long-time asymptotic formula for nonlocal Lakshmanan-Porsezian-Daniel equation becomes more complex.

Higher order Turán inequalities for the distinct partition function

We prove that the number q(n) of partitions of n with distinct parts is log-concave for n≥33 and satisfies the third-order Turán inequalities for n≥121 conjectured by Craig and Pun. In doing so, we establish explicit error terms for q(n) and for q(n−1)q(n+1)/q(n)2 based on Chern's asymptotic formulas for η-quotients.

A regularized (XP)2 model

We investigate a dynamic model described by the classical Hamiltonian H(x, p) = (x 2 + a 2)(p 2 + a 2), where a 2 > 0, in classical, semi-classical, and quantum mechanics, respectively. In the high-energy E limit, the phase path resembles that of the XP2 model. However, the non-zero value of a acts as a regulator, removing the singularities that appear in the region where x, p ∼ 0, resulting in a discrete spectrum characterized by a logarithmic increase in state density. Classical solutions are described by elliptic functions, with the period being determined by elliptic integrals. In semi-classical approximation, we speculate that the asymptotic Riemann-Siegel formula may be interpreted as summing over contributions from multiply phase paths. We present three different forms of quantized Hamiltonians, and reformulate them into the standard Schrödinger equation with cosh2x -like potentials. Numerical evaluations of the spectra for these forms are carried out and reveal minor differences in energy levels. Among them, one interesting form possesses Hamiltonian in the Schrödinger equation that is identical to its classical version. In such scenarios, the eigenvalue equations can be expressed as the vanishing of the Mathieu functions’ value at i ∞ points, and furthermore, the Mathieu functions can be represented as the wave functions.

Steklov Type Operators and Functional Equations

Abstract We consider sequences of Steklov type operators and an associated functional equation. For a suitable sequence, we establish asymptotic formulas.

Tunneling effect between radial electric wells in a homogeneous magnetic field

We establish a tunneling formula for a Schrödinger operator with symmetric double-well potential and homogeneous magnetic field, in dimension two. Each well is assumed to be radially symmetric and compactly supported. We obtain an asymptotic formula for the difference between the two first eigenvalues of this operator, that is exponentially small in the semiclassical limit.

Behaviour of large eigenvalues for the asymmetric quantum Rabi model

We prove that the spectrum of the asymmetric quantum Rabi model consists of two eigenvalue sequences ( E m + ) m = 0 ∞ , ( E m − ) m = 0 ∞ , satisfying a two-term asymptotic formula with error estimate of the form O ( m − 1 / 4 ), when m tends to infinity.

Schatten Index of the Sectorial Operator via the Real Component of Its Inverse

In this paper, we study spectral properties of non-self-adjoint operators with the discrete spectrum. The main challenge is to represent a complete description of belonging to the Schatten class through the properties of the Hermitian real component. The method of estimating the singular values is elaborated by virtue of the established asymptotic formulas. The latter fundamental result is advantageous since, of many theoretical statements based upon it, one of them is a concept on the root vectors series expansion, which leads to a wide spectrum of applications in the theory of evolution equations. In this regard, the evolution equations of fractional order with the sectorial operator in the term not containing the time variable are involved. The concrete well-known operators are considered and the advantage of the represented method is convexly shown.

Isotropically active particle closely fitting in a cylindrical channel: spontaneous motion at small Péclet numbers

Spontaneous motion due to symmetry breaking has been predicted theoretically for both active droplets and isotropically active particles in an unbounded fluid domain, provided that their intrinsic Péclet number $Pe$ exceeds a critical value. However, due to their inherently small $Pe$ , this phenomenon has yet to be observed experimentally for active particles. In this paper, we demonstrate theoretically that spontaneous motion for an active spherical particle closely fitting in a cylindrical channel is possible at arbitrarily small $Pe$ . Scaling arguments in the limit where the dimensionless clearance is $\epsilon \ll 1$ reveal that when $Pe=O(\epsilon ^{1/2})$ , the confined particle reaches speeds comparable to those achieved in an unbounded fluid at moderate (supercritical) $Pe$ values. We use matched asymptotic expansions in that distinguished limit, where the fluid domain decomposes into several asymptotic regions: a gap region, where the lubrication approximation applies; particle-scale regions, where the concentration is uniform; and far-field regions, where solute transport is one-dimensional. We derive an asymptotic formula for the particle speed, which is a monotonically decreasing function of $\overline {Pe}=Pe/\epsilon ^{1/2}$ and approaches a finite limit as $\overline {Pe}\searrow 0$ . Our results could pave the way for experimental realisations of symmetry-breaking spontaneous motion in active particles.

Sparse sets that satisfy the prime number theorem

For arbitrary real t>1 we examine the set {⌊x/nt⌋:n≤x}. Asymptotic formulas for the cardinality of this set and the number of primes in this set are given. The prime counting result uses an alternate Vaughan's decomposition for the von Mangoldt function, with triple exponential sums instead of double exponential sums. These sets are the sparsest known sets that satisfy the prime number theorem, in the sense that the number of primes is asymptotically given by the cardinality of the set divided by the natural logarithm of the cardinality of the set.

Asymptotic Representation of Vorticity and Dissipation Energy in the Flux Problem for the Navier–Stokes Equations in Curved Pipes

This study explores the problem of describing viscous fluid motion for Navier–Stokes equations in curved channels, which is important in applications like hemodynamics and pipeline transport. Channel curvature leads to vortex flows and closed vortex zones. Asymptotic models of the flux problem are useful for describing viscous fluid motion in long pipes, thus considering geometric parameters like pipe diameter and characteristic length. This study provides a representation for the vorticity vector and energy dissipation in the flow problem for a curved channel, thereby determining the magnitude of vorticity and energy dissipation depending on the channel’s central line curvature and torsion. The accuracy of the asymptotic formulas are estimated in terms of small parameter powers. Numerical calculations for helical tubes demonstrate the effectiveness of the asymptotic formulas.

Numerical validation of analytical formulas for channel flows over liquid-infused surfaces

This paper provides numerical validation of some new explicit, asymptotically exact, analytical formulas describing channel flows over liquid-infused surfaces, an important class of surfaces of current interest in surface engineering. The new asymptotic formulas, reproduced here, were derived in a recent companion paper by the authors. The numerical validation is done by presenting a novel computational method for calculating longitudinal flow in a periodic channel involving finite-length closed liquid-filled grooves with a flat two-fluid interface, a challenging problem given the two-fluid nature of the flow. The formulas are asymptotically exact for wide channels where the grooves on the lower wall of the channel are well separated; the numerical method devised here, however, is subject to no such restrictions. Significantly, it is shown here that the asymptotic formulas remain good global approximants for the flow over a wide range of flow geometries, including those well outside the asymptotic parameter range for which they were derived. It is found that the formulas are more reliable for liquid-infused surfaces than for superhydrophobic surfaces.

Calculation of Sommerfeld Integrals in Dipole Radiation Problems

This article proposes asymptotic methods for calculating Sommerfeld integrals, which enable us to calculate the integral using the expansion of a function into an infinite power series at the saddle point, where the role of a rapidly oscillating function under the integral can be fulfilled either by an exponential or by its product by the Hankel function. The proposed types of Sommerfeld integrals are generalized on the basis of integral representations of the Hertz radiator fields in the form of the inverse Hankel transform with the subsequent replacement of the Bessel function by the Hankel function. It is shown that the numerical values of the saddle point are complex. During integration, reference or so-called standard integrals, which contain the main features of the integrand function, were used. As a demonstration of the accuracy of the technique, a previously known asymptotic formula for the Hankel functions was obtained in the form of an infinite series. The proposed method for calculating Sommerfeld integrals can be useful in solving the half-space Sommerfeld problem. The authors present an example in the form of an infinite series for the magnetic field of reflected waves, obtained directly through the Sommerfeld integral (SI).