Discrete Spectrum of Differential Operators in Spectral Gaps under Nonnegative Perturbations of High Order

Let A be a self-adjoint elliptic second-order differential operator, let (, ) be an inner gap in the spectrum of A, and let B(t) = A + tW ^* W, where W is a differential operator of higher order. Conditions are obtained under which the spectrum of the operator B(t) in the gap (, ) is either discrete, or does not accumulate to the right-hand boundary of the spectral gap, or is finite. The quantity N(, A, W, ), (, ), > 0 (the number of eigenvalues of the operator B(t) passing the point (, ) as t increases from 0 to ) is considered. Estimates of N(, A, W, ) are obtained. For the perturbation W ^* W of a special form, the asymptotics of N(, A, W, ) as + is given. Bibliography: 5 titles.     

Nonnegative Linearization for Polynomials Orthogonal with Respect to Discrete Measures

We give conditions for the coefficients in three term recurrence relations implying nonnegative linearization for polynomials orthogonal with respect to measures supported on convergent sequences of points. The previous methods were unable to cover this case. January 14, 1999. Date revised: January 31, 2000. Date accepted: October 24, 2000.     

Strong Asymptotics of the Orthogonal Polynomials with Respect to a Measure Supported on the Plane

We consider the orthogonal polynomials with respect to the measure over the whole complex plane. We obtain the strong asymptotic of the orthogonal polynomials in the complex plane and the location of their zeros in a scaling limit where n grows to infinity with N . The asymptotics are described in terms of three (probability) measures associated with the problem. The first measure is the limit of the counting measure of zeros of the polynomials, which is captured by the g‐function much in the spirit of ordinary orthogonal polynomials on the real line. The second measure is the equilibrium measure that minimizes a certain logarithmic potential energy, supported on a region K of the complex plane. The third measure is the harmonic measure of K ^c with a pole at ∞ . This appears as the limit of the probability measure given (up to the normalization constant) by the squared modulus of the n^th orthogonal polynomial times the orthogonality measure, i.e., The compact region K that is the support of the second measure undergoes a topological transition under the variation of the parameter in a double scaling limit near the critical point given by we observe the Hastings‐McLeod solution to Painlevé II in the asymptotics of the orthogonal polynomials. © 2014 Wiley Periodicals, Inc.     

Szego Orthogonal Polynomials with Respect to an Analytic Weight: Canonical Representation and Strong Asymptotics

We provide a representation in terms of certain canonical functions for a sequence of polynomials orthogonal with respect to a weight that is strictly positive and analytic on the unit circle. These formulas yield a complete asymptotic expansion for these polynomials, valid uniformly in the whole complex plane. As a consequence, we obtain some results about the distribution of zeros of these polynomials. The main technique is the steepest descent analysis of Deift and Zhou, based on the matrix Riemann-Hilbert characterization proposed by Fokas, Its, and Kitaev.     

The Length Spectrum of a Compact Constant Curvature Complex is Discrete

In this short note we prove that the length spectrum of a compact constant curvature complex is discrete. After recalling the relevant definitions and reducing to a relatively simple situation, the result follows easily from a foundational result about real semi-algebraic sets and the Morse–Sard theorem. We conclude with a conjecture which remains open, a few remarks and an easy application.Partially supported by NSF grant no. DMS-0123344 (N. Brady).Partially supported by NSF grant no. DMS-0101506 (J. McCammond)     

The Discrete Spectrum in the Spectral Gaps of Semibounded Operators with Non-sign-definite Perturbations

Given two self-adjoint operators A and V = V₊  − V₋ , we study the motion of the eigenvalues of the operator A(t) = A − tV as t increases. Let α > 0 and let λ be a regular point for A. We consider the quantities N₊ (V; λ, α), N₋ (V; λ, α), and N₀(V; λ, α) defined as the number of eigenvalues of the operator A(t) that pass point λ from the right to the left, from the left to the right, or change the direction of their motion exactly at point λ, respectively, as t increases from 0 to α > 0. We study asymptotic characteristics of these quantities as α → ∞. In the present paper, the results obtained previously [O. L. Safronov, Comm. Math. Phys.193 (1998), 233–243] are extended and given new applications to differential operators.     

离散大系统关于部分变元的关联集合稳定性

本文利用Liapunov函数方法，研究了离散大系统关于部分变元的关联集合稳定性，给出了关于部分变元关联集合稳定性的几个基本定理的一些充分条件.     

Asymptotics of the Spectrum and Eigenharmonics of a Discrete String Containing a Bead of Vanishing Mass

A discrete string with fixed endpoints containing a finite number of pointlike masses (beads) is considered. The behavior of the eigenfrequences and eigenharmonics of the string is studied as one of the masses tends to zero. Bibliography: 1 title.     

Painlevé IV Asymptotics for Orthogonal Polynomials with Respect to a Modified Laguerre Weight

We study polynomials that are orthogonal with respect to the modified Laguerre weight   z ^{− n +ν} e ^{− Nz} ( z − 1)^{2 b}   , in the limit where   n , N →∞  with   N / n → 1  and ν is a fixed number in     . With the effect of the factor (   z − 1)^{2 b}   , the local parametrix near the critical point z = 1 can be constructed in terms of Ψ functions associated with the Painlevé IV equation. We show that the asymptotics of the recurrence coefficients of orthogonal polynomials can be described in terms of specified solution of the Painlevé IV equation in the double scaling limit. Our method is based on the Deift/Zhou steepest decent analysis of the Riemann–Hilbert problem associated with orthogonal polynomials.     

离散大系统在结构扰动下周期解的存在性

对于离散系统稳定性的研究,近年来受到人们的重视,但对于周期解的研究,在文献中还很少看到。本文首先讨论了离散系统解的有界性,并且得到了若一个具有周期系数的差分方程的解为最终有界的,则存在周期解的结果。然后利用李雅普诺夫函数方法研究了离散大系统在结构扰动之下周期解的存在性和离散大系统的平稳振荡。     

Logarithmic L 2-Small Ball Asymptotics with Respect to a Self-Similar Measure for Some Gaussian Processes

We find logarithmic small ball asymptotics for the L₂-norm with respect to self-similar measures for a certain class of Gaussian processes including Brownian motion, Ornstein-Uhlenbeck process, and their integrated counterparts. Bibliography: 46 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 311, 2004, pp. 190–213.     

Periodic Solutions of Chebyshev Polynomials with Respect to the Discrete Variable

In this paper, we study a very simple three term recurrence relation involving the discontinuous Heaviside step function. One reason for studying such an relation is that solutions of our recurrence relation are steady state distributions in some basic neural network models. Since analytic tools cannot be used to handle discontinuous models such as ours, existence of periodic solutions is investigated by combining combinatorial elimination technique as well as existence arguments for linear systems. By such means, we are able to obtain all periodic solutions with least periods 1 through 8. Some periodic solutions with periods 12, 20 and 36 can also be found, but exhaustive results are not yet available.     

Asymptotics of Laurent Polynomials of Odd Degree Orthogonal with Respect to Varying Exponential Weights

Let denote the linear space over spanned by . Define the (real) inner product , where V satisfies: (i) V is real analytic on ; (ii) ; and (iii) . Orthogonalisation of the (ordered) base with respect to yields the even degree and odd degree orthonormal Laurent polynomials , and . Define the even degree and odd degree monic orthogonal Laurent polynomials: and . Asymptotics in the double-scaling limit such that of (in the entire complex plane), , and (in the entire complex plane) are obtained by formulating the odd degree monic orthogonal Laurent polynomial problem as a matrix Riemann-Hilbert problem on , and then extracting the large-n behaviour by applying the non-linear steepest-descent method introduced in [1] and further developed in [2],[3].     

椭球约束下误差方差非负二次估计的可容许性

本文给出了线性模型中椭球约束下，误差方差非负二次估计可容许的一个必要条件，并且，对于线性模型中设计阵X＝1nj=(1,1,…，1）′的特殊情形，本文给出了一个充要条件。     

Large Time Asymptotics with Error Estimates to Solutions of a Forced Burgers Equation

This article deals with a forced Burgers equation (FBE) subject to the initial function, which is continuous and summable on . Large time asymptotic behavior of solutions to the FBE is determined with precise error estimates. To achieve this, we construct solutions for the FBE with a different initial class of functions using the method of separation of variables and Cole–Hopf like transformation. These solutions are constructed in terms of Hermite polynomials with the help of similarity variables. The constructed solutions would help us to pick up an asymptotic approximation and to show that the magnitude of the difference function of the true and approximate solutions decays algebraically to 0 for large time.     

一类带扰动项的拟线性抛物型方程解的存在性

主要利用算子的性质证明了一类带扰动项的拟线性方程的L~2(Ω)初值和狄立克莱边值问题解的存在性和唯一性.     

Large Deviation Principle for Fractional Brownian Motion with Respect to Capacity

Potential Analysis - We show that the fractional Brownian motion (fBM) defined via the Volterra integral representation with Hurst parameter $Hgeq frac {1}{2}$ is a quasi-surely defined Wiener...     

Asymptotics of Eigenvalues for a Class of Jacobi Matrices with Limiting Point Spectrum

In this paper, we study a class of Jacobi matrices with very rapidly decreasing weights. It is shown that the Weyl function (the matrix element of the resolvent of the operator) for the class under study can be expressed as the ratio of two entire transcendental functions of order zero. It is shown that the coefficients in the expansion of these functions in Taylor series are proportional to the generating functions of the number of integral solutions defined by certain Diophantine equations. An asymptotic estimate for the eigenvalues is obtained.