Fractional integral operators in the complex matrix variate case

A formal definition of fractional integrals in the complex matrix variate case is given here. This definition will encompass all the various fractional integral operators introduced by various authors in the real scalar and matrix cases. The new definition is introduced in terms of M-convolutions of products and ratios of matrices in the complex domain. Their connections to statistical distribution theory, Mellin convolutions, M-transforms and Mellin transform are pointed out. Some basic properties are given and a pathway extension of the new definition is also given. The pathway extension will provide a switching mechanism to move among three different families of functions.     

Asymptotics of Brownian integrals and pressure: Bose-Einstein statistics

The paper studies the asymptotics of the Brownian integrals with paths restricted to a bounded domain of ? ^v , when the domain is dilated to infinity. The framework is that of the Bose-Einstein statistics with paths observed within random time intervals which are integer multiplies of some fixed β > 0. The three first terms of the asymptotics are found explicitly via the functional integrals. In the case of a gas of interacting Brownian loops an expression for the volume term of the asymptotics of the log-partition function is found and the correction term is proved to by order be the boundary area of the domain.     

Exact distribution of the generalized Wilks’s statistic and applications

We establish the exact expression of the density of Wilks’s statistic Λ(n,p,q), and also those of the densities of the product and ratio of two independent such statistics, in terms of Meijer functions, and provide applications with numerical illustrations in various domains of Multivariate Analysis.     

SLE Boundary Visits

We study the probabilities with which chordal Schramm–Loewner evolutions (SLE) visit small neighborhoods of boundary points. We find formulas for general chordal SLE boundary visiting probability amplitudes, also known as SLE boundary zig-zags or order refined SLE multi-point Green’s functions on the boundary. Remarkably, an exact answer can be found to this important SLE question for an arbitrarily large number of marked points. The main technique employed is a spin chain–Coulomb gas correspondence between tensor product representations of a quantum group and functions given by Dotsenko–Fateev type integrals. We show how to express these integral formulas in terms of regularized real integrals, and we discuss their numerical evaluation. The results are universal in the sense that apart from an overall multiplicative constant the same formula gives the amplitude for many different formulations of the SLE boundary visit problem. The formula also applies to renormalized boundary visit probabilities for interfaces in critical lattice models of statistical mechanics: we compare the results with numerical simulations of percolation, loop-erased random walk, and Fortuin–Kasteleyn random cluster models at Q = 2 and Q = 3, and find good agreement.     

Poisson fields and distribution functions in the statistical mechanics of charged particles

A brief survey of the Poisson analysis on the spaces of tempered distributions is given and the generalized Wick theorem for Poisson fields is formulated. For systems of charged particles, new representations in terms of integrals with respect to the Poisson measure are obtained for distribution functions and diagonal elements of a reduced density matrix; these representations are convenient for investigation of model systems of statistical mechanics by the cluster expansion method. In the quantum case, the Boltzmann, Fermi-Dirac, and Bose-Einstein statistics are studied.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 5, pp. 710–719, May, 1995.The author expresses his deep gratitude to A. L. Rebenko for posing the problem and for his attention to the work and to G. F. Us for helpful discussions.This research was partially supported by the International Science Foundation, grant No. UB3000.     

Loop quantum mechanics and the fractal structure of quantum spacetime

We discuss the relation between string quantization based on the Schild path integral and the Nambu-Goto path integral. The equivalence between the two approaches at the classical level is extended to the quantum level by a saddle-point evaluation of the corresponding path integrals. A possible relationship between M-Theory and the quantum mechanics of string loops is pointed out. Then, within the framework of “loop quantum mechanics”, we confront the difficult question as to what exactly gives rise to the structure of spacetime. We argue that the large scale properties of the string condensate are responsible for the effective Riemannian geometry of classical spacetime. On the other hand, near the Planck scale the condensate “evaporates”, and what is left behind is a “vacuum” characterized by an effective fractal geometry.     

Vector Hamiltonians in Nambu Mechanics

We give a generalization of the Nambu mechanics based on vector Hamiltonians theory. It is shown that any divergence-free phase flow in ? ⁿ can be represented as a generalized Nambu mechanics with n ? 1 integral invariants. For the case when the phase flow in ? ⁿ has n ? 3 or less first integrals, we introduce the Cartan concept of mechanics. As an example we give the fifth integral invariant of Euler top.     

Solution of Poisson’s equation by analytical boundary element integration

The solution of Poisson’s equation is essential for many branches of science and engineering such as fluid-mechanics, geosciences, and electrostatics. Solution of two-dimensional Poisson’s equations is carried out by BEM based on Galerkin Vector Method in which the integrals that appear in the boundary element method are expressed by analytical integration. In this paper, the Galerkin vector method is developed for more general case than presented in the previous works. The integrals are computed for constant and linear elements in BEM. By employing analytical integration in BEM computation, the numerical schemes and coordinate transformations can be avoided. The presented method can also be used for the multiple domain case. The results of the analytical integration are employed in BEM code and the obtained analytical expression will be applied to several examples where the exact solution exists. The produced results are in good agreement with the exact solution.     

Comparison of three semiparametric methods for estimating dependence parameters in copula models

Three semiparametric methods for estimating dependence parameters in copula models are compared, namely maximum pseudo-likelihood estimation and the two method-of-moment approaches based on the inversion of Spearman’s rho and Kendall’s tau. For each of these three asymptotically normal estimators, an estimator of their asymptotic (co)variance is stated in three different situations, namely the bivariate one-parameter case, the multivariate one-parameter case and the multivariate multiparameter case. An extensive Monte Carlo study is carried out to compare the finite-sample performance of the three estimators under consideration in these three situations. In the one-parameter case, it involves up to six bivariate and four-variate copula families, and up to five levels of dependence. In the multiparameter case, attention is restricted to trivariate and four-variate normal and t copulas. The maximum pseudo-likelihood estimator appears as the best choice in terms of mean square error in all situations except for small and weakly dependent samples. It is followed by the method-of-moment estimator based on Kendall’s tau, which overall appears to be significantly better than its analogue based on Spearman’s rho. The simulation results are complemented by asymptotic relative efficiency calculations. The numerical computation of Spearman’s rho, Kendall’s tau and their derivatives in the case of copula families for which explicit expressions are not available is also investigated.     

Tsallis statistics and turbulence

In order to reveal the underlying statistics describing properly the fully developed turbulence, the probability density function of the local dissipation is derived by taking extremal of a generalized entropy (Tsallis entropy) under the two constraints, i.e., one is the normalization of probability and the other is to fix the intermittency exponent being constant. The generalized entropy includes the Boltzmann–Gibbs entropy as a special case where the Tsallis index q is equal to 1. The multifractal spectrum f(α) corresponding to the probability density function is determined self-consistently in the sense that all quantities can be determined by the observed value of the intermittency exponent. It is shown that the scaling exponents ζ_m of velocity structure function derived by making use of f(α) explains experimental data very well. It is also revealed that the asymptotic expression of ζ_m for m≫1 has a log term. The Tsallis index q turns out to be 0.380 which manifests itself that the system of fully developed turbulence has a nonextensive character.     

A note on Chrystal’s equation

The exact, explicit form of the transcendental solution of Chrystal’s equation, a first order nonlinear ordinary differential equation (ODE) of degree two, is derived in terms of the Lambert W-function. It is shown that this case of the general solution is dual-valued over a finite interval and that, for a special case of the coefficients, its zeros involve the Golden ratio. Additionally, a number of applications involving special cases of this ODE are noted and the main properties of the Lambert W-function are briefly reviewed.     

Some properties of complex matrix-variate generalized Dirichlet integrals

Dirichlet integrals and the associated Dirichlet statistical densities are widely used in various areas. Generalizations of Dirichlet integrals and Dirichlet models to matrix-variate cases, when the matrices are real symmetric positive definite or hermitian positive definite, are available [4]. Real scalar variables case of the Dirichlet models are generalized in various directions. One such generalization of the type-2 or inverted Dirichlet is looked into in this article. Matrix-variate analogue, when the matrices are hermitian positive definite, are worked out along with some properties which are mathematically and statistically interesting.     

Classical and quantum non-extensive statistics effects in nuclear many-body problems

Recent progress in statistical mechanics indicates the Tsallis non-extensive thermostatistics as the natural generalization of the standard classical and quantum statistics, when memory effects and long-range forces are not negligible. In this framework, we show that in weakly non-ideal plasma non-extensive and quantum uncertainty effects should be taken into account to derive the equilibrium distribution functions, quantum fluctuations and correlations between the particles. The strong influence of these effects is discussed in the context of the solar neutrino problem and in the high-energy nuclear–nuclear collision experiments. Although the deviation from the Boltzmann–Gibbs statistics, in both cases, is very small, the nuclear reaction rates and quantum fluctuations are strongly influenced by the non-extensive feature of the solar plasma and the hadronic gas produced by a quark–gluon plasma. The discrepancies between the experimental data and theoretical previsions are sensibly reduced.     

Decision-making in testing process performance with fuzzy data

Over the years, numerous process capability indices (PCIs) have been proposed to the manufacturing industry to provide numerical measures of process performance. Most research efforts have focused on developing and investigating PCIs that assess process capability by precise measurements of output quality. However, real observations of continuous quantities are not precise numbers; in practice, they are more or less imprecise. Since observations of continuous random variables are imprecise the values of related test statistics become imprecise. Therefore, decision rules for statistical tests have to be adapted to this situation. This article presents a set of confidence intervals that produces triangular fuzzy numbers for the estimation of C_pk index using Buckley’s approach with some modification. Additionally, a three-decision testing rule and step-by-step procedure are developed to assess process performance based on fuzzy critical values and fuzzy p-values. This concept is also illustrated with an example for testing process performance.     

Asymptotic expansions of Mellin convolution integrals: An oscillatory case

In a recent paper [J.L. López, Asymptotic expansions of Mellin convolution integrals, SIAM Rev. 50 (2) (2008) 275-293], we have presented a new, very general and simple method for deriving asymptotic expansions of for small x. It contains Watson’s Lemma and other classical methods, Mellin transform techniques, McClure and Wong’s distributional approach and the method of analytic continuation used in this approach as particular cases. In this paper we generalize that idea to the case of oscillatory kernels, that is, to integrals of the form , with c∈R, and we give a method as simple as the one given in the above cited reference for the case c=0. We show that McClure and Wong’s distributional approach for oscillatory kernels and the summability method for oscillatory integrals are particular cases of this method. Some examples are given as illustration.     

Characterization of integrals with respect to arbitrary radon measures by the boundedness indices

The problem of characterization of integrals as linear functionals is considered in the paper. It starts from the familiar results of F. Riesz (1909) and J. Radon (1913) on integral representation of bounded linear functionals by Riemann?CStieltjes integrals on a segment and by Lebesgue integrals on a compact in $ {\mathbb{R}^n} $ , respectively. After works of J. Radon, M. Fréchet, and F. Hausdorff the problem of characterization of integrals as linear functionals took the particular form of the problem of extension of Radon??s theorem from $ {\mathbb{R}^n} $ to more general topological spaces with Radon measures. This problem has turned out difficult and its solution has a long and rich history. Therefore, it may be naturally called the Riesz?CRadon?CFréchet problem of characterization of integrals. The important stages of its solution are connected with such mathematicians as S. Banach, S. Saks, S. Kakutani, P. Halmos, E. Hewitt, R. E. Edwards, N. Bourbaki, V. K. Zakharov, A. V. Mikhalev, et al. In this paper, the Riesz?CRadon?CFr??echet problem is solved for the general case of arbitrary Radon measures on Hausdorff spaces. The solution is given in the form of a general parametric theorem in terms of a new notion of the boundedness index of a functional. The theorem implies as particular cases well-known results of the indicated authors characterizing Radon integrals for various classes of Radon measures and topological spaces.     

Weak and strong convergence theorems for nonspreading-type mappings in Hilbert spaces

Weak and strong convergence theorems are proved in real Hilbert spaces for a new class of nonspreading-type mappings more general than the class studied recently in Kurokawa and Takahashi [Y. Kurokawa, W. Takahashi, Weak and strong convergence theorems for nonspreading mappings in Hilbert spaces, Nonlinear Anal. 73 (2010) 1562-1568]. We explored an auxiliary mapping in our theorems and proofs and this also yielded a strong convergence theorem of Halpern’s type for our class of mappings and hence resolved in the affirmative an open problem posed by Kurokawa and Takahashi in their final remark for the case where the mapping T is averaged.     

Nonlinear integration and Weierstrass integral over a manifold: Connections with theorems on martingales

This paper is a survey on the integrals of the calculus of variations as Weierstrass integrals. While the classic integrals in terms of Lebesgue processes are valid only overAC curves, surfaces, or variety, the corresponding Weierstrass or Burkill integrals are valid in theBV case. This is the immediate advantage of the Weierstrass-Burkill form. Moreover, Cesari defined a very general integration process for set functions, in an abstract setting, which extends the Weierstrass and Burkill integrals. In more detail, we examine here some existence, approximation, representation, and semicontinuity results both in the parametric case and in the nonparametric case.     

Exact nonnull distribution of Wilks’ statistic: The ratio and product of independent components

The study of the noncentral matrix variate beta type distributions has been sidelined because the final expressions for the densities depend on an integral that has not been resolved in an explicit way. We derive an exact expression for the nonnull distribution of Wilks’ statistic and precise expressions for the densities of the ratio and product of two independent components of matrix variates where one matrix variate has the noncentral matrix variate beta type I distribution and the other has the matrix variate beta type I distribution. We provide the expressions for the densities of the determinant of the ratio and the product of these two components. These distributions play a fundamental role in various areas of statistics, for example in the criteria proposed by Wilks.     

基于Tsallis理论的中国股市收益分布研究

应用Tsallis提出的非广延统计力学理论以及与之密切相关的非线性Fokker-Planck方程所描述的动力系统,根据我国上证指数和深证指数2004年1月1日～2008年11月13日的高频数据,分析了在三种不同的时间标度下股指收益的概率分布,发现Tsallis分布可以很好地描述两市收益分布的尖峰厚尾有限方差等特征,同时也给出了市场微观动力学层面的解释。揭示出我国上海和深圳股市的价格过程并不符合随机游走,而是反常扩散过程,两市具有十分接近的非线性动力系统特征。所得结论对于研究我国金融市场的资产配置和定价、风险管理和制度建设都具有重要的意义。