Asymptotics of Information Entropy for the Two-Dimensional Analog of the Relativistic Hydrogen Atom in the Kozlov-Nikishin Model

For the two-dimensional Lorentz-invariant model of the hydrogen atom, we obtain wave functions of bound states in coordinate representation and, for nonexcited (in time) states, also in momentum representation. For such states, the short-wave asymptotics of the information entropy is studied.     

The Asymptotics of the Two-Dimensional Wave Equation for a General Multi-Connected Vibrating Membrane with Piecewise Smooth Robin Boundary Conditions

The asymptotic expansion for small |t| of the trace of the wave kernel ∧↑μ(t) =∑v=1^∞exp(-it μv^1/2), where i= √-1 and {μv}v=1^∞ are the eigenvalues of the negative Laplacian -△=-∑β=1^2(δ/δx^β)^2 in the (x^1, x^2)-plane, is studied for a multi-connected vibrating membrane Ω in R^2 surrounded by simply connected bounded domains Ωj with smooth boundaries δΩj(j=1,...,n), where a finite number of piecewise smooth Robin boundary conditions on the piecewise smooth components Гi(i=1 κj-1,...,κj) of the boundaries δΩj are considered, such that δΩj=∪i=1 κj-1^κj Гi and κ0=0. The basic problem is to extract information on the geometry of Ω using the wave equation approach. Some geometric quantities of Ω (e.g. the area of Ω, the total lengths of its boundary, the curvature of its boundary, the number of the holes of Ω, etc.) are determined from the asymptotic expansion of the trace of the wave kernel ∧↑μ(t) for small |t|.     

Near-Boundary Asymptotics for Correlation Kernels

We discuss asymptotic formulas for the correlation kernel corresponding to a (more or less) general potential Q in the plane. If K _n is such a kernel, there is a known asymptotic formula K _n (z,z)=nΔQ(z)+B ₁(z)+n ^?1 B ₂(z)+? valid for z in the interior of a certain compact set known as the “droplet” corresponding to Q (on which ΔQ>0). On the other hand, if z is in the exterior of the droplet, K _n (z,z) converges to zero exponentially in n. Results of this type are useful in random matrix theory and conformal field theory; they have recently been used to prove the Gaussian field convergence in the interior of the droplet for fluctuations of eigenvalues of random normal matrices. To be able to extend such results beyond the interior, it becomes necessary to have a certain uniformity of estimates as z approaches the boundary either from the interior or from the exterior. Such estimates have to our knowledge hitherto not been known on a rigorous level. This note intends to fill this gap. We will consider applications in later publications. Our treatment of the (interior) asymptotics relies in an essential way on previous work due to Berman, Berndtsson, and Sjöstrand (Ark. Mat. 46, 2008), and Berman (Indiana Univ. Math. J. 58, 2009). We hope that this note can to some extent be regarded as a contribution to that work.     

Asymptotics for the Tukey Median

The asymptotic distribution of the Tukey median has recently been obtained by Nolan in a bivariate setting and by Bai and He in the general multivariate case. To establish their theorem, these authors made a strong symmetry hypothesis on the distribution and, in the case of Bai and He, assumed the existence of a density function with a gradient satisfying a finite-moment condition. This paper extends the above result by deriving the asymptotic distribution of the above location estimate without making any symmetry or differentiability assumption on the distribution.     

Asymptotics for multiple Meixner polynomials

We study the asymptotic behavior of multiple Meixner polynomials of first and second kind, respectively [6]. We use an algebraic function formulation for the solution of the equilibrium problem with constraint to describe their zero distribution. Moreover, analyzing the limiting behavior of the coefficients of the recurrence relations for Multiple Meixner polynomials we obtain the main term of their asymptotics.     

Asymptotics of the Distribution of Eigenvalues of a Selfadjoint Second Order Hyperbolic Differential Operator on the Two-Dimensional Torus

We study the asymptotic properties of the discrete spectrum for general selfadjoint second order hyperbolic operators on the two-dimensional torus. For a broad class of operators with sufficiently smooth coefficients and the principal part coinciding with the wave operator in the light cone coordinates we prove the discreteness of the spectrum and obtain an asymptotic formula for the distribution of eigenvalues. In some cases we can indicate the first two asymptotic terms. We discuss the relations of these questions to analytic number theory and mathematical physics.     

Asymptotics for the late arrivals problem

We study a discrete time queueing system where deterministic arrivals have i.i.d. exponential delays \(\xi _{i}\). We describe the model as a bivariate Markov chain, prove its ergodicity and study the joint equilibrium distribution. We write a functional equation for the bivariate generating function, finding the solution on a subset of its domain. This solution allows us to prove that the equilibrium distribution of the chain decays super-exponentially fast in the quarter plane. We exploit the latter result and discuss the numerical computation of the solution through a simple yet effective approximation scheme in a wide region of the parameters. Finally, we compare the features of this queueing model with the standard M / D / 1 system, showing that the congestion turns out to be very different when the traffic intensity is close to 1.     

Ratio Asymptotics for Orthogonal Matrix Polynomials

Ratio asymptotic results give the asymptotic behaviour of the ratio between two consecutive orthogonal polynomials with respect to a positive measure. In this paper, we obtain ratio asymptotic results for orthogonal matrix polynomials and introduce the matrix analogs of the scalar Chebyshev polynomials of the second kind.     

Asymptotics for Fractional Nonlinear Heat Equations

The Cauchy problem is studied for the nonlinear equations withfractional power of the negative Laplacian where (0,2), with critical = /n and sub-critical (0,/n)powers of the nonlinearity. Let u₀ L^1,a L C, u₀(x) 0 in Rⁿ, = . The case of not small initial data is of interest. It is proved that the Cauchy problemhas a unique global solution u C([0,); L L^1,a C) and the largetime asymptotics are obtained.     

Strong Asymptotics for Multiple Laguerre Polynomials

We consider multiple Laguerre polynomials l_n of degree 2n orthogonal on (0,∞) with respect to the weights and , where -1 < α, 0 < β₁ < β₂, and we study their behavior in the large n limit. The analysis differs among three different cases which correspond to the ratio β₂/β₁ being larger, smaller, or equal to some specific critical value κ. In this paper, the first two cases are investigated and strong uniform asymptotics for the scaled polynomials l_n(nz) are obtained in the entire complex plane by using the Deift-Zhou steepest descent method for a (3 × 3)-matrix Riemann-Hilbert problem.     

Asymptotics for Bailey-type mock theta functions

The Ramanujan Journal - We compute asymptotic estimates for the Fourier coefficients of two mock theta functions, which come from Bailey pairs derived by Lovejoy and Osburn. To do so, we employ the...