Correlation Functions, Cluster Functions, and Spacing Distributions for Random Matrices

The usual formulas for the correlation functions in orthogonal and symplectic matrix models express them as quaternion determinants. From this representation one can deduce formulas for spacing probabilities in terms of Fredholm determinants of matrix-valued kernels. The derivations of the various formulas are somewhat involved. In this article we present a direct approach which leads immediately to scalar kernels for the unitary ensembles and matrix kernels for the orthogonal and symplectic ensembles, and the representations of the correlation functions, cluster functions, and spacing distributions in terms of them.     

The fractal interpretation of the weak scattering of elastic waves

Nondestructive methods, in particular the measurement of elastic waves, have become increasingly important in determining the microstructure of many materials in recent years. One of these methods is observing the attenuation of ultrasonic waves of known amplitude and direction, e.g., in granular metals. The waves are exponentially attenuated with distance with a frequency-dependent attenuation fractor. The attenuation factor can be decomposed into two parts: absorption and scattering. Experimentally, the absorption part varies linearly with frequency, while the scattering part has a noninteger power law behavior, the exponent of which is related to the strength of the material. Theoretically, at long wavelengths the exponent is 4 (Rayleigh scattering) while for grain-sized wavelengths it is 2 (diffusive scattering). We relate the attenuation factor to the forward scattering amplitude which is related to the frequency dependence of the scatterers and their cross sections. We attribute the noninteger attenuation exponent to a fractal distribution of grain shapes and sizes.Supported by the Defense Advanced Research Projects Agency.     

Convergence in the pth-mean and some Weak Laws of Large Numbers for weighted sums of random elements in separable normed linear spaces

Theorem. Let X_n, n ≥ 1, be a sequence of tight random elements taking values in a separable Banach space B such that |X_n|, n ≥ 1, is uniformly integrable. Let a_nk, n ≥ 1, k ≥ 1, be a double array of real numbers satisfying Σ_{k ≥ 1} |a_nk| ≤ Γ for every n ≥ 1 for some positive constant Γ. Then Σ_{k ≥ 1}a_nkX_k, n ≥ 1, converges to 0 in probability if and only if Σ_{k ≥ 1}a_nkf(X_k), n ≥ 1, converges to 0 in probability for every f in the dual space B¹.     

随机集的独立性与同分布性的刻划

本文绘出了随机集的独立性与同分布性的刻化，所给出的准则比Hess[4]，[5]更易于验证．     

Self-organization driven by broad band light

We show that strong incoherent broad band light transports electrons in random media in opposite to the electric force eE direction. Current flows against the voltage drop and static polarization is generated reversed to the applied electric field. This unusual response amplifies the internal electric fields and drives self-organization phenomena in wide class of disordered materials. The model proposed gives an idea on how the life on Earth could have emerged.     

Almost periodic random sequences in probability

In this paper almost periodic random sequence in probability is defined and investigated. It is also applied to random difference equations by means of exponential dichotomy. The existence of such kind of solutions of random difference equations is discussed.     

The expected number of extreme points of a random linear program

There has been increasing attention recently on average case algorithmic performance measures since worst case measures can be qualitatively quite different. An important characteristic of a linear program, relating to Simplex Method performance, is the number of vertices of the feasible region. We show 2 ⁿ to be an upper bound on the mean number of extreme points of a randomly generated feasible region with arbitrary probability distributions on the constraint matrix and right hand side vector. The only assumption made is that inequality directions are chosen independently in accordance with a series of independent fair coin tosses.We would like to thank the Institute of Pure and Applied Mathematics in Rio de Janeiro for supporting the authors' collaboration that led to this paper.     

有限级Dirchlet级数及随机Dirchlet级数

本文研究了全平面上有限级Dirichlet级数的增长性和正规增长性,得到了两个充要条 件;证明了有限级随机Dirichlet级数的增长性几乎必然与其在每条水平直线上的增长性相同．     

Agent-based randomized broadcasting in large networks

Mobile agents are software abstractions that can migrate across the links of a network. They naturally extend the object oriented program style and nicely correspond to agents as examined in game theory. In this paper, we introduce a simple, robust, and efficient randomized broadcast protocol within this mobile agent programming paradigm. We show that by using this scheme, broadcasting enquiries in a random graph of certain density O(lnn) steps, where n denotes the number of nodes in the graph. Then, we consider bounded degree graphs and prove that we are able to distribute an information among all nodes in O(D) steps, where D denotes the diameter of the graph. We also show that, in contrast to traditional randomized broadcasting (TRB), graphs exist in which agent-based randomized broadcasting requires Ω(n²) steps. On the other hand, some graphs which require Ω(nlnn) steps to spread the information in the traditional broadcast model, allow very fast agent-based broadcasting. It should be noted that the previously mentioned results are guaranteed with probability 1-o(1/n).     

On Consistent Statistical Procedures in Regression

Necessary and sufficient conditions are provided for the existence of consistent statistical procedures in regression models with random predictors under various error assumptions