Inverse Problem for Polynomials Orthogonal on the Unit Circle

Polynomials orthogonal on the unit circle with random recurrence coefficients and finite band spectrum are investigated. It is shown that the coefficients are in fact quasi-periodic. The measures associated with these quasi-periodic coefficients are exhibited and necessary and sufficient conditions relating quasi-periodicity and spectral measures of this type are given. Analogs for polynomials orthogonal on subsets of the real line are also presented. Received: 21 May 1997 / Accepted: 28 July 1997     

A Maximum Entropy Method Based on Orthogonal Polynomials for Frobenius-Perron Operators

Let $S$: [0, 1]→[0, 1] be a chaotic map and let $f^∗$ be a stationary density of the Frobenius-Perron operator $P_S$: $L^1$→$L^1$ associated with $S$. We develop a numerical algorithm for approximating $f^∗$, using the maximum entropy approach to an under-determined moment problem and the Chebyshev polynomials for the stability consideration. Numerical experiments show considerable improvements to both the original maximum entropy method and the discrete maximum entropy method.     

Non-Intersecting Squared Bessel Paths and Multiple Orthogonal Polynomials for Modified Bessel Weights

We study a model of n non-intersecting squared Bessel processes in the confluent case: all paths start at time t = 0 at the same positive value x = a, remain positive, and are conditioned to end at time t = T at x = 0. In the limit n → ∞, after appropriate rescaling, the paths fill out a region in the tx-plane that we describe explicitly. In particular, the paths initially stay away from the hard edge at x = 0, but at a certain critical time t* the smallest paths hit the hard edge and from then on are stuck to it. For t ≠ t* we obtain the usual scaling limits from random matrix theory, namely the sine, Airy, and Bessel kernels. A key fact is that the positions of the paths at any time t constitute a multiple orthogonal polynomial ensemble, corresponding to a system of two modified Bessel-type weights. As a consequence, there is a 3 × 3 matrix valued Riemann-Hilbert problem characterizing this model, that we analyze in the large n limit using the Deift-Zhou steepest descent method. There are some novel ingredients in the Riemann-Hilbert analysis that are of independent interest.     

Analytic Expansion for Ground-State Wavefunction of Time-Dependent Strong-CouplingSchrödinger Equation

The new method proposed recently by Friedberg, Lee,and Zhao is extended to obtain an analytic expansion for the ground-state wavefunction of a time-dependent strong-coupling Schrödinger equation. Two different types of the time-dependent harmonic oscillators are considered as examples for application of the time-dependent expansion. It is shown that the time-dependent strong-coupling expansion is applicable to the time-dependent harmonic oscillators with a slowly varying time-dependent parameter.     

Hankel Determinant and Orthogonal Polynomials for a Gaussian Weight with a Discontinuity at the Edge

We compute asymptotics for Hankel determinants and orthogonal polynomials with respect to a discontinuous Gaussian weight, in a critical regime where the discontinuity is close to the edge of the associated equilibrium measure support. Their behavior is described in terms of the Ablowitz–Segur family of solutions to the Painlevé II equation. Our results complement the ones in [33]. As consequences of our results, we conjecture asymptotics for an Airy kernel Fredholm determinant and total integral identities for Painlevé II transcendents, and we also prove a new result on the poles of the Ablowitz–Segur solutions to the Painlevé II equation. We also highlight applications of our results in random matrix theory.     

The Convergence Estimation of the Parallel Algorithm of the Linear Cauchy Problem Solution for Large Systems of First-Order Ordinary Differential Equations Using the Solution as Expansion over Orthogonal Polynomials

This paper is devoted to the algorithm of the linear Cauchy problem solution for large systems of first-order ordinary differential equations using parallel calculations. The proof of the convergence of the iteration process using the solution as expansion over orthogonal polynomials for the interval [0,1] is presented. The features of this algorithm are its simplicity, the opportunity to get a solution by parallel calculations, and also the possibility to get a solution for nonlinear problems by changing the operator using the solution from the iteration process.     

单模激光Haken-Lorenz系统的振荡解析解

研究了单模激光Haken-Lorenz系统在Hopf 分歧点处的动力学行为.求出了Haken-Lorenz系统的定态解,采用线性稳定性原理对定态解进行了稳定性分析,获得了本征值方程,进而确定了系统的Hopf 分歧点μc.利用级数法求出了系统在分歧点处的时间周期振荡解的解析表达式.通过计算机对系统分歧点处的动力学行为进行了数值模拟,结果表明,系统在分歧点处存在一个极限环,即时间周期振荡解.与理论分析的解析结果相一致.     

Formation for Bass-Relief Microprofiles Based on an Analytic Formulation

A method of fast design and fabrication for bass-relief micro-profiles is developed by using an analytic formulation to determine the exposure distribution. Based on an equivalent exposure threshold model, the formulation is simplified for the case of bass-relief profile corresponding to the smaller exposure dose. The mask function for a microlens array is designed without iteration involved by the analytic formulation. The experiment is performed to validate the method, and the fabrication result is obtained with the profile error less than 30nm （rms）.     

Fundamentals of a Technique for Determining Electron Distribution Functions by Multi-Term Even-Order Expansion in Legendre Polynomials 1. Theory

On the basis of our recent investigations concerning the mathematical structure of the hierarchy which results from the Legendre polynomial expansion of the electron velocity distribution function in Boltzmann's equation a new technique for solving this equation in multi-term even-order approximation is presented. This method is, even if more complex, the logical generalization of the well known technique for solving Boltzmann's equation by backward integration in the conventional two-term approximation. A weakly ionized, spatially homogeneous and stationary plasma with elastic and exciting electron-atom collisions is considered acted upon by a dc electric field. The technique, presented in detail, determines the distribution function in even order 2l of the expansion at the end by l-fold backward and 2l-fold forward integration of the hierarchy and by continuous connection of the resulting non-singular parts of the general solutions at low and high energies at an appropriate connection point. A first application of this method is made on a model gas for the even orders from 2 to 10 and under conditions with distinct anisotropy in the velocity space due to intensive exciting collisions. The converged macroscopic quantities and the corresponding first coefficients of the distribution expansion itself are compared with very accurate Monte Carlo simulations under the same conditions where a perfect agreement between the results obtained with both techniques was found confirming the high accuracy of the new technique to be presented.     

An Asymptotic Expansion and Recursive Inequalities for the Monomer-Dimer Problem

Let λ _d (p) be the p monomer-dimer entropy on the d-dimensional integer lattice ℤ ^d , where p∈[0,1] is the dimer density. We give upper and lower bounds for λ _d (p) in terms of expressions involving λ _d−1(q). The upper bound is based on a conjecture claiming that the p monomer-dimer entropy of an infinite subset of ℤ ^d is bounded above by λ _d (p). We compute the first three terms in the formal asymptotic expansion of λ _d (p) in powers of  \frac1d\frac{1}{d}. We prove that the lower asymptotic matching conjecture is satisfied for λ _d (p). Converted to a power series in p, our “formal” expansion shows remarkable validity in low dimensions, d=1,2,3, in which dimensions we give some numerical studies.     

Optimizing Expected Shortfall under an ℓ1 Constraint—An Analytic Approach

Expected Shortfall (ES), the average loss above a high quantile, is the current financial regulatory market risk measure. Its estimation and optimization are highly unstable against sample fluctuations and become impossible above a critical ratio r=N/T, where N is the number of different assets in the portfolio, and T is the length of the available time series. The critical ratio depends on the confidence level α, which means we have a line of critical points on the α−r plane. The large fluctuations in the estimation of ES can be attenuated by the application of regularizers. In this paper, we calculate ES analytically under an ℓ1 regularizer by the method of replicas borrowed from the statistical physics of random systems. The ban on short selling, i.e., a constraint rendering all the portfolio weights non-negative, is a special case of an asymmetric ℓ1 regularizer. Results are presented for the out-of-sample and the in-sample estimator of the regularized ES, the estimation error, the distribution of the optimal portfolio weights, and the density of the assets eliminated from the portfolio by the regularizer. It is shown that the no-short constraint acts as a high volatility cutoff, in the sense that it sets the weights of the high volatility elements to zero with higher probability than those of the low volatility items. This cutoff renormalizes the aspect ratio r=N/T, thereby extending the range of the feasibility of optimization. We find that there is a nontrivial mapping between the regularized and unregularized problems, corresponding to a renormalization of the order parameters.     

An Analytic Approximation for Researching Tunneling Rate from Black Hole in Proca Field

The analytic solution of a static spherical symmetrical Proca black hole is discussed in this paper. As in the massive vector field, Proca black hole can be considered as the analogy of RN background plus a perturbation with the same order as μ ² due to the mass of vector particle μ satisfying μ ² ? 1. Through the action of Proca field, we find the analytic form with the first and arbitrary order approximation. Furthermore, we divide the results into 3 groups according to the real zero solutions of the background (i.e., spacetime in massless vector field). Finally we analyze the Hawking radiation of such black hole, which is significant for constructing black hole thermodynamic.     

An analytic method for S‐Expansion involving resonance and reduction

In this paper we describe an analytic method able to give the multiplication table(s) of the set(s) involved in an S‐expansion process (with either resonance or 0_S‐resonant‐reduction) for reaching a target Lie (super)algebra from a starting one, after having properly chosen the partitions over subspaces of the considered (super)algebras. This analytic method gives us a simple set of expressions to find the subset decomposition of the set(s) involved in the process. Then, we use the information coming from both the initial (super)algebra and the target one for reaching the multiplication table(s) of the mentioned set(s). Finally, we check associativity with an auxiliary computational algorithm, in order to understand whether the obtained set(s) can describe semigroup(s) or just abelian set(s) connecting two (super)algebras. We also give some interesting examples of application, which check and corroborate our analytic procedure and also generalize some result already presented in the literature.     

一种基于简化正交距离的近红外异常光谱判断方法

及时发现并剔除异常光谱对于提高近红外光谱分析的可靠性具有重要意义。首先介绍了基于正交距离的稳健主成分分析法判断异常光谱的原理,分析了该方法的局限性,在此基础上提出了一种基于简化正交距离的异常光谱判断方法,即以高浓度样品的光谱直接估计第一稳健主成分,再通过重复测量估计正交距离的统计参数来判断异常光谱。最后,采用上述两种方法对葡萄糖水溶液和2% Intralipid仿体溶液透射实验中受温度影响的异常光谱进行了判别。结果表明,基于正交距离的稳健主成分分析法,当崩溃值设定为40%时,葡萄糖水溶液和仿体溶液实验中受温度影响的异常光谱均能被正确识别;当崩溃值设定为25%时,对葡萄糖水溶液和仿体溶液实验中受温度影响的异常光谱漏判率分别为54.5%和72.7%。而简化正交距离法可以准确识别出葡萄糖水溶液和仿体溶液实验中受温度干扰的异常光谱,且具有无需预先设定崩溃值和测量持续时间较短等优点。因此,简化正交距离法在实际应用中相比基于正交距离的稳健主成分分析法有着显著的优势。     

基于点和直线段对应的扩展正交迭代位姿估计算法

正交迭代算法是基于点特征,全局且快速收敛的位姿估计算法,是目前性能最优的实时位姿估计算法之一.对摄像机观测到的直线段提出了其不确定性描述的新方法:目标空间直线段误差,并把该误差融入到正交迭代算法的位姿计算过程中,形成扩展正交迭代算法.该算法可以同时利用点和直线段特征.算法先把点共面性方程表达为与正交迭代算法中点共线性方程一致的数学形式,再根据目标空间点共面性误差定义了目标空间直线段共面性误差.接着把目标空间点共线性误差和直线段共面性误差二者之和作为误差函数.最后推导出使该函数最小化的迭代求解过程.实验结果表明,本文算法是有效的,精确的.与正交迭代算法的计算结果对比,由于可以同时利用直线段特征,扩展正交迭代算法的位姿估计误差更低,抗噪声性能更强,算法更稳定.     

Fundamentals of a Technique for Determining Electron Distribution Functions by Multi-Term Even-Order Expansion in Legendre Polynomials. 2. Comprehensive Investigation of a Model Plasma

Applying the new technique for finding the converged solution of the Boltzmann equation in a weakly ionized plasma, which was developed in the first part of this paper, a comprehensive study of the electron velocity distribution function for a model plasma with elastic and exciting collisions is performed by solving the Boltzmann equation with increasing order of approximation. The purpose of this investigation is that of calculating the isotropic distribution f₀, the first contribution f₁ to the anisotropy of the velocity distribution, the important macroscopic quantities and, more generally, that of studying the total anisotropy as well as the changes of all these quantities when the approximation degree is enlarged beyond the 2 terms of the conventional Lorentz approximation. By varying some parameters of the model plasma, that is the electric field strength, the magnitude of the excitation cross section and the excitation threshold, the main features of plasmas in inert as well as molecular gases are modelled and the impact of these parameters on the mentioned quantities is analysed. Some of the converged results are compared with results of corresponding Monte Carlo simulations. The approximation degree required to find the converged values of isotropic distribution, main macroscopic quantities and electron distribution in the velocity space (and thus its real anisotropy) is estimated by solving the Boltzmann equation over wide parameter ranges.     

Travelling Wave Solutions for Konopelchenko-Dubrovsky Equation Using an Extended sinh-Gordon Equation Expansion Method

The sinh-Gordon equation expansion method is further extended by generMizing the sinh-Gordon equation and constructing new ansatz solution of the considered equation. As its application, the （2＋1）-dimensional Konopelchenko-Dubrovsky equation is investigated and abundant exact travelling wave solutions are explicitly obtained including solitary wave solutions, trigonometric function solutions and Jacobi elliptic doubly periodic function solutions, some of which are new exact solutions that we have never seen before within our knowledge. The method can be applied to other nonlinear evolution equations in mathematical physics.     

Travelling Wave Solutions for Konopelchenko-Dubrovsky Equation Using an Extended sinh-Gordon Equation Expansion Method

The sinh-Gordon equation expansion method is further extended by generalizing the sinh-Gordon equation and constructing new ansatz solution of the considered equation. As its application, the (2+1)-dimensional Konopelchenko-Dubrovsky equation is investigated and abundant exact travelling wave solutions are explicitly obtained including solitary wave solutions, trigonometric function solutions and Jacobi elliptic doubly periodic function solutions, some of which are new exact solutions that we have never seen before within our knowledge. The method can be applied to other nonlinear evolution equations in mathematical physics.     

Cluster Expansion for Abstract Polymer Models. New Bounds from an Old Approach

We revisit the classical approach to cluster expansions, based on tree graphs, and establish a new convergence condition that improves those by Kotecky-Preiss and Dobrushin, as we show in some examples. The two ingredients of our approach are: (i) a careful consideration of the Penrose identity for truncated functions, and (ii) the use of iterated transformations to bound tree-graph expansions.