Note on Fractional Green's Function

In this paper, we modify some errors on the definition of fractional Green''s function in monograph [5], and give the solution of the inhomogenous equation which satisfies the given inhomogenous initial conditions by fractional Green''s function.     

Wandering vector multipliers for unitary groups

A wandering vector multiplier is a unitary operator which maps the set of wandering vectors for a unitary system into itself. A special case of unitary system is a discrete unitary group. We prove that for many (and perhaps all) discrete unitary groups, the set of wandering vector multipliers is itself a group. We completely characterize the wandering vector multipliers for abelian and ICC unitary groups. Some characterizations of special wandering vector multipliers are obtained for other cases. In particular, there are simple characterizations for diagonal and permutation wandering vector multipliers. Similar results remain valid for irrational rotation unitary systems. We also obtain some results concerning the wandering vector multipliers for those unitary systems which are the ordered products of two unitary groups. There are applications to wavelet systems.

     

SINE TRANSFORM MATRIX FOR SOLVING TOEPLITZ MATRIX PROBLEMS

1. IntroductionStrang[1] first studied the use of circulallt matrices C for solving systems of linear eqllationsTi x = b witha symmetric positive definite Toeplitz matrix.Numerous authors such as T.Chan[2],R.Chan,etc.[3],[4],[5], Tyrtyshnikov[6], Huckle[7] and T.Ku and C.Kuo[8] proposed differentfamilies of circulallt / skew- circulant precondit ioners.Appling the preconditioned conjugate gradient algorithm(PCGA) to solve the systems Ti x -b, we must find a preconditioner P such that P…     

High-Resolution Color Image Reconstruction with Neumann Boundary Conditions

This paper studies the application of preconditioned conjugate gradient methods in high-resolution color image reconstruction problems. The high-resolution color images are reconstructed from multiple undersampled, shifted, degraded color frames with subpixel displacements. The resulting degradation matrices are spatially variant. To capture the changes of reflectivity across color channels, the weighted H ₁ regularization functional is used in the Tikhonov regularization. The Neumann boundary condition is also employed to reduce the boundary artifacts. The preconditioners are derived by taking the cosine transform approximation of the degradation matrices. Numerical examples are given to illustrate the fast convergence of the preconditioned conjugate gradient method.     

When the “Bull” Meets the “Bear”—A First Passage Time Problem for a Hidden Markov Process

Let _t be a continuous Markov chain on N states. Consider adjoining a Brownian motion with this Markov chain so that the drift and the variance take different values when _t is in different states. This new process Z_t is a hidden Markov process. We study the probability distribution of the first passage time for Z_t.Our result, when applied to the stock market, provides an explicit mathematical interpretation of the fact that in finite time, there is positive probability for the bull (bear) market to become bear (bull).     

Second-Order Subexponential Sequences and the Asymptotic Behavior of Their De Pril Transform

Suppose that { f(n), n N ₀ } is a sequence of positive real numbers and suppose that the sequence { a(n), n N ₀ } is given by a(0) = 0, and, for n 1, by the convolution equation nf(n) = a* f(n). The resulting sequence is denoted by a(n) = _f (n) and is called the De Pril transform of { f(n), n N ₀ } . In this paper, we consider first- and second-order asymptotic behavior of { _{f (n), n N 0} } for a large class of subexponential sequences { f(n), n N ₀ } . We also discuss some applications.     

OCCUPATION TIMES OF BALLS BY BROWNIAN MOTION WITH DRIFT

51. Introduction and Statement of ResultsLet X ~ {Xt, t 2 0} be a standajrd d-dimensional Brownian motion with drift c startedat fiXed XO ~ x:Xo ~ Wb ct, t 2 0,where Wt is the standard d--dimensional Brownian motion, c E R'(d 2 2) is a fixed vector.Denote by P:(.) the probability meajsure on the path space of X corresponding to initialstate XO = x and drift vector c, with E;(.) the corresponding expection operator. Forsimplity, we shall write Pz(.) and Ex(.) to refer to the case c ~ …     

Orthogonal polynomials on the unit circle associated with the Laguerre polynomials

Using the well-known fact that the Fourier transform is unitary, we obtain a class of orthogonal polynomials on the unit circle from the Fourier transform of the Laguerre polynomials (with suitable weights attached). Some related extremal problems which arise naturally in this setting are investigated.

     

基于双尺度方程近似解的适合任何连续信号的近似采样定理

提出了一个对任何连续信号,利用双尺度方程的近似解的近似采样定理.采样函数是一个双尺度方程的近似解,是一个逐段线性函数,有明确的计算公式,且精度可以控制.采样函数由双尺度方程的系数唯一决定,可以通过选择双尺度方程的系数而使采样函数具有良好的性质.     

非线性动力学系统的辨识：时频滤波与骨架曲线

利用非平稳信号的时频分析方法研究了一类非线性系统的频率特性和阻尼特性随运动形态的变化规律,得到了能简洁、直观地反映系统基本非线性动力学特性的广义骨架线性系统（简称GSLS）和骨架曲线。在此基础上,利用时频滤波方法根据系统非平稳响应信号对非线性系统进行辨识。该项工作为非线性系统反问题的研究提供了一条新的途径。