Two Parameter Asymptotic Spectra in the Uniformly Elliptic Case

In this article we study the abstract two parameter eigenvalue problem $$\begin{gathered} T_1 u_1 = \left( {\lambda _1 V_{11} + \lambda _2 V_{12} } \right)u_1 , \left\| {u_1 } \right\| = 1 \hfill \\ T_2 u_2 = \left( {\lambda _1 V_{21} + \lambda _2 V_{22} } \right)u_2 , \left\| {u_2 } \right\| = 1 \hfill \\ \end{gathered}$$ where, in the Hilbert spaces H_j, T_j is self-adjoint, bounded below and has compact resolvent, and V_jk are self-adjoint bounded operators, (?1)^j+kV_jk >> 0, j, k = 1, 2. An eigenvalue λ for this problem is a point in R² satisfying both equations. Under appropriate conditions, the eigenvalues λⁿ = (λ₁ ⁿ, λ₂ ⁿ) are countable and in R². We aim to describe the set of limit points of λⁿ/∥λⁿ∥, as ∥λⁿ∥ → ∞, in terms of the V_jk.     

Asymptotic Spectra of Dense Toeplitz Matrices Are Unstable

The paper is concerned with the limiting set of the eigenvalues of the truncations of an infinite Toeplitz matrix whose symbol is continuous but not rational. This limiting set is shown to be unstable with respect to small perturbations of the symbol in the uniform norm, which reveals that the numerical computation of the asymptotic spectra of dense Toeplitz matrices is a genuine mathematical challenge.     

指数族刻度参数EB估计的渐近最优性

依据经验Bayes(EB)估计的思想方法,研究在LINEX损失函数下指数族刻度参数的EB估计问题.在这种损失函数下,求得参数的Bayes估计,利用密度函数的核估计方法,构造了总体X的密度函数估计,从而得到参数的EB估计,证明了这种EB估计是渐近最优的,并获得了它的收敛速度,最后将这种方法推广到多参数情形,并举例、模拟说明了它的应用.     

Asymptotic Zero Behavior of Laguerre Polynomials with Negative Parameter

We consider Laguerre polynomials $L_n^{(\alpha_n)}(nz)$ with varying negative parameters $\alpha_n$, such that the limit $A = -\!\lim_n \alpha_n/n$ exists and belongs to $(0,1)$. For $A > 1$, it is known that the zeros accumulate along an open contour in the complex plane. For every $A \in (0,1)$, we describe a one-parameter family of possible limit sets of the zeros. Under the condition that the limit $r= - \!\lim_n {1}/{n} \log [\dist\!(\alpha_n, \mbox{\footnotesize{\bf Z}})]$ exists, we show that the zeros accumulate on $\Gamma_r \cup [\beta_1,\beta_2]$ with $\beta_1$ and $\beta_2$ only depending on $A$. For $r \in [0,\infty)$, $\Gamma_r$ is a closed loop encircling the origin, which for $r = +\infty$, reduces to the origin. This shows a great sensitivity of the zeros to $\alpha_n$s proximity to the integers. We use a Riemann–Hilbert formulation for the Laguerre polynomials, together with the steepest descent method of Deift and Zhou to obtain asymptotics for the polynomials, from which the zero behavior follows.     

Inverse Spectra with Two and Three Maps

It is shown that, for any 1 n < , there exist four maps of the n-dimensional cube to itself such that the limit of any inverse sequence of n-cubes is the limit of some sequence with only these four bonding maps. A universal continuum in the class of all limits of sequences of n-cubes is constructed as the limit of an inverse sequence of n-cubes with one bonding map. All compact sets of trivial shape are represented by using only three maps of the Hilbert cube to itself. Two maps of the closed interval to itself such that any Knaster continuum can be obtained as the limit of an inverse sequence with only these two bonding maps are constructed.     

Asymptotic Approximations for Radial Spheroidal Wavefunctions with Complex Size Parameter

Radial spheroidal wavefunctions are functions of four variables, usually denoted by m , n , x , and γ, the last of which is known as the size parameter. This parameter becomes complex when the problem of scattering of a sound pulse by a spheroid is treated using a Laplace transform with respect to time together with the method of separation of variables. Several asymptotic approximations, involving modified Bessel functions, are developed and analyzed.     

Modified Unit Root Tests with Nuisance Parameter Free Asymptotic Distributions

It is well-known that the asymptotic distributions of the Dickey-Fuller (DF) tests for a unit root with linear process errors are not free of nuisance parameters. In this paper, we introduce a consistent estimator for the nuisance parameters and then use it to modify the DF tests, denoted as R-tests. Under fairly mild moment and summability conditions on the errors, we show that the asymptotic distributions of the R-tests are of the same as the Dickey-Fuller distributions. In Monte Carlo experiments, the R-tests are shown to have improved size properties.     

The Asymptotic Workload Behavior of Two Coupled Queues

We consider a system of two coupled queues Q ₁ and Q ₂. When both queues are backlogged, they are each served at unit rate. However, when one queue empties, the service rate at the other queue increases. Thus, the two queues are coupled through the mechanism for dynamically sharing surplus service capacity. We derive the asymptotic workload behavior at Q ₁ for various scenarios where at least one of the two queues has a heavy-tailed service time distribution. First of all, we consider a situation where the traffic load at Q ₁ is below the nominal unit service rate. We show that if the service time distribution at Q ₁ is heavy-tailed, then the workload behaves exactly as if Q ₁ is served in isolation at a constant rate, which only depends on the service time distribution at Q ₂ through its mean. In addition, we establish that if the service time distribution at Q ₁ is exponential, then the workload distribution is either exponential or semi-exponential, depending on whether the traffic load at Q ₂ exceeds the nominal service rate or not. Next, we focus on a regime where the traffic load at Q ₁ exceeds the nominal service rate, so that Q ₁ relies on the surplus capacity from Q ₂ to maintain stability. In that case, the workload distribution at Q ₁ is determined by the heaviest of the two service time distributions, so that Q ₁ may inherit potentially heavier-tailed characteristics from Q ₂.     

Asymptotic Expansion of Solutions of Parabolic Equations with a Small Parameter

The heat equation with a small parameter, $\left( {1 + \varepsilon ^{ - m} \chi \left( {\frac{x}{\varepsilon }} \right)} \right)ut = u_{xx} $ , is considered, where ε ∈ (0, 1), m < 1 and χ is a finite function. A complete asymptotic expansion of the solution in powers ε is constructed.     

Asymptotic positivity of Hurwitz product traces: Two proofs

Consider the polynomial tr(A+tB)^m in t for positive hermitian matrices A and B with m∈N. The Bessis-Moussa-Villani conjecture (in the equivalent form of Lieb and Seiringer) states that this polynomial has nonnegative coefficients only. We prove that they are at least asymptotically positive, for the nontrivial case of AB≠0. More precisely, we show—once complex-analytically, once combinatorially—that the k-th coefficient is positive for all integer m?m₀, where m₀ depends on A, B and k.     

Convergent and Asymptotic Methods for Second-order Difference Equations with a Large Parameter

We consider the second-order linear difference equation \(y(n+2)-2a y(n+1)-\Lambda ^2 y(n)=g(n)y(n)+f(n)y(n+1)\), where \(\Lambda \) is a large complex parameter, \(a\ge 0\) and g and f are sequences of complex numbers. Two methods are proposed to find the asymptotic behavior for large \(\vert \Lambda \vert \) of the solutions of this equation: (i) an iterative method based on a fixed point method and (ii) a discrete version of Olver’s method for second-order linear differential equations. Both methods provide an asymptotic expansion of every solution of this equation. The expansion given by the first method is also convergent and may be applied to nonlinear problems. Bounds for the remainders are also given. We illustrate the accuracy of both methods for the modified Bessel functions and the associated Legendre functions of the first kind.     

泊松分布中一个参数的两个最大似然估计的比较

给出泊松分布中参数e-λ的两个最大似然估计,讨论了不同标准条件下各自的性质,指出在大样本条件下,两者具有相当的性质.     

Strong Convergence Theorems for Two Parameter Vilenkin-Fourier Series

As main result we prove that certain means of the partial sums of two-parameter Vilenkin-Fourier series are uniformly bounded operators from H ^P to L ^p (0 < p ≦ 1). The Hardy space H ^p (0 < p ≦ 1) will be defined by means of a diagonal maximal function. As a consequence we obtain a so-called strong convergence theorem for the Vilenkin-Fourier partial sums. Some dual inequalities are also verified for BMO spaces.     

一类二元离散神经网络模型的渐近性

本文考虑的是一类具有分段常数非线性时滞差分系统,该系统可作为二元人工神经网络模型的离散形式,本文得到了系统解的渐近性的一些结果。     

Asymptotic Formulas for Two Continued Fractions in Ramanujan's Lost Notebook

On page 45 of his lost notebook, Ramanujan recorded two asymptoticformulas for two continued fractions involving the Riemann zeta-functionand Dirichlet L-functions. The paper proves a more general theoremand derives Ramanujan's claims as a corollary of the theorem.     

On the Restricted Order of Asymptotic Bases of Order Two

The restricted order of an asymptotic basis A is the least integer h, if it exists, such that every sufficiently large integer is the sum of h or fewer distinct elements of A. We show that any asymptotic basis of order 2 has a restricted order at most equal to 4. We also provide an example of an additive basis of order 2 whose restricted order is 4.To Jean-Louis Nicolas2000 Mathematics Subject Classification: Primary—11B13     

A Relation between formal and Asymptotic Equivalence for linear Systems Containing A Parameter

We use an elementary method to draw analytic conclusions from divergent formal power series solutions of systems of differential equations containing a parameter and give some applications to the theory of turning points. Our main result shows that a divergent formal series transformation of one system into another in which the coefficients satisfy certain estimates is necessarily the asymptotic expansion of an actual transformation. We use it to show the following. Given a two dimensional system ε^Pdy/dx = A(x,ε)y with A holomorphic at (x₀,0), suppose that x₀ is formally not a turning point in the sense that no singularities appear at x₀ during the standard formal solution procedure with formal fractional power series in ε. Then the formal solution is necessarily a uniform asymptotic representation of a fundamental matrix of the system on a full neighborhood of x₀. (This conclusion is known to fail under weaker hypotheses on A). We also obtain similar but less complete results for higher order systems under more specialized hypotheses.     

Momentum Operators in Two Intervals: Spectra and Phase Transition

We study the momentum operator defined on the disjoint union of two intervals. Even in one dimension, the question of two non-empty open and non-overlapping intervals has not been worked out in a way that extends the cases of a single interval and gives a list of the selfadjoint extensions. Starting with zero boundary conditions at the four endpoints, we characterize the selfadjoint extensions and undertake a systematic and complete study of the spectral theory of the selfadjoint extensions. In an application of our extension theory to harmonic analysis, we offer a new family of spectral pairs. Compared to earlier studies, it yields a more direct link between spectrum and geometry.