a pathway to matrix variate gamma and normal densities a general real matrix variate probability model is introduced here which covers almost all real matrix variate densities used in multivariate statistical analysis through the new density introduced here a pathway is created to go from matrix variate type 1 beta to matrix variate type 2 beta to matrix variate gamma to matrix variate gaussian or normal densities other densities such as extended matrix variate student t f cauchy density will also come in as particular cases connections to the distributions of quadratic forms and generalized quadratic forms in the new matrix are established the present day analysis of these problems is mainly confined to gaussian random variables thus through the new distribution all these theories are extended connections to certain geometrical probability problems such as the distribution of the volume of a random parallelotope in euclidean space is also established

fractional differential operators in the complex matrix variate case abstract there are many applications for fractional integrals fractional derivatives and fractional differential equations in the real scalar variable cases however when it comes to fractional integrals in the real and complex matrix variate cases there are not many papers the author has given some definitions and properties for fractional integrals and fractional derivatives in the real matrix variate cases recently some situations of fractional derivatives in the complex matrix variate case are discussed in the present article along with the definitions for fractional derivatives in the complex matrix variate case the definition introduced here enables us to handle certain types of derivatives of real valued scalar functions of matrix argument in the complex domain it is not universally applicable fractional derivatives in the riemann liouville and caputo senses are evaluated when the arbitrary function is compatible with right and left sided fractional integrals in the complex matrix variate cases kober operators of the first and second kind in the complex matrix variate cases are also examined here

modelling temporal decay of aftershocks by a solution of the fractional reactive equation abstract we propose a new analytical perspective to explain the behavior of the number of seismic events observed post an intense earthquake as time elapses through the application of a fractional solution of the reactive equation according to the results obtained a double power law model shows the number density decay in several possible ways among which is a particular case the modified version of omori law proposed by utsu in 1961

gamma function to beck cohen superstatistics abstract in this paper we show a mathematical construction of beck cohen superstatistics in the bayesian point of view with the help of the two representations of a gamma function furthermore it is shown how some results for superstatistics are related to each other

role of short periodic orbits in quantum maps with continuous openings we apply a recently developed semiclassical theory of short periodic orbits to the continuously open quantum tribaker map in this paradigmatic system the trajectories are partially bounced back according to continuous reflectivity functions this is relevant in many situations that include optical microresonators and more complicated boundary conditions in a perturbative regime the shortest periodic orbits belonging to the classical repeller of the open map a cantor set given by a region of exactly zero reflectivity prove to be extremely robust in supporting a set of long lived resonances of the continuously open quantum maps moreover for steplike functions a significant reduction in the number needed is obtained similarly to the completely open situation this happens despite a strong change in the spectral properties when compared to the discontinuous reflectivity case in order to give a more realistic interpretation of these results we compare with a fresnel type reflectivity function

some complex matrix variate statistical distributions on rectangular matrices abstract in a recent survey olkin the 70th anniversary of the distribution of random matrices a survey linear algebra appl 354 2002 231 243 has looked into 70 years of development on random matrices giving special attention to rectangular matrices gupta and richards multivariate liouville distributions j multivariate anal 23 1987 233 256 have looked into generalizations of the dirichlet family of distributions for the multivariate case mathai an introduction to geometrical probability distributional aspects with applications gordon and breach scientific publishers new york 1999 developed a gamma type distribution on rectangular matrices in connection with the distributional aspects of random volumes this idea is developed further in the present article to define rectangular matrix variate gamma type beta type and dirichlet type distributions with location and scale matrices connections among these distributions and various properties are pointed out this paper discusses the complex matrix variate cases and the corresponding real cases are also listed along with each result the complex rectangular matrix variate t and cauchy distributions are obtained as particular cases

pathway model superstatistics tsallis statistics and a generalized measure of entropy abstract the pathway model of mathai a pathway to matrix variate gamma and normal densities linear algebra appl 396 2005 317 328 is shown to be inferable from the maximization of a certain generalized entropy measure this entropy is a variant of the generalized entropy of order considered in mathai and rathie basic concepts in information theory and statistics axiomatic foundations and applications wiley halsted new york and wiley eastern new delhi 1975 and it is also associated with shannon boltzmann gibbs renyi tsallis and havrda charvat entropies the generalized entropy measure introduced here is also shown to have interesting statistical properties and it can be given probabilistic interpretations in terms of inaccuracy measure expected value and information content in a scheme particular cases of the pathway model are shown to be tsallis statistics c tsallis possible generalization of boltzmann gibbs statistics j stat phys 52 1988 479 487 and superstatistics introduced by beck and cohen superstatistics physica a 322 2003 267 275 the pathway model s connection to fractional calculus is illustrated by considering a fractional reaction equation

multivariate and matrix variate analogues of maxwell boltzmann and raleigh densities abstract the maxwell boltzmann and raleigh densities are basic densities in many problems in physics a multivariate analogue and a rectangular matrix variate analogue of these densities are explored in this article the results may become useful in extending the usual theories where these densities for the real scalar variable case occur to multivariate and matrix variable situations various properties are studied and connection to the volumes of parallelotopes determined by p linearly independent random points in euclidean n space n p is also established structural decompositions of these random determinants and pathway extensions of maxwell boltzmann and raleigh densities are also considered

direct scheme in optimal control problems with fast and slow motions abstract the nonlinear optimal control problems described by singularly perturbed models are studied in order to obtain the ways of decomposition and to construct the suboptimal control sequences a new scheme the so called direct scheme applying the vasil eva boundary layer functions method in control theory is introduced it consists in directly expanding the problem s conditions into postulated asymptotic series and then in successively solving lower dimensional approximate decomposition problems of optimal control by means of the direct scheme a new property of an asymptotic expansion the relaxation is obtained i e the decrease of the performance index with each new control approximation illustrating examples are given

fractal weyl law for quantum fractal eigenstates the properties of the resonant gamow states are studied numerically in the semiclassical limit for the quantum chirikov standard map with absorption it is shown that the number of such states is described by the fractal weyl law and their husimi distributions closely follow the strange repeller set formed by classical orbits nonescaping in future times for large matrices the distribution of escape rates converges to a fixed shape profile characterized by a spectral gap related to the classical escape rate

stretched exponentials from superstatistics abstract distributions exhibiting fat tails occur frequently in many different areas of science a dynamical reason for fat tails can be a so called superstatistics where one has a superposition of local gaussians whose variance fluctuates on a rather large spatio temporal scale after briefly reviewing this concept we explore in more detail a class of superstatistics that hasn t been subject of many investigations so far namely superstatistics for which a suitable power of the local inverse temperature is 2 distributed we show that 0 leads to power law distributions while 0 leads to stretched exponentials the special case 1 corresponds to tsallis statistics and the special case 1 to exponential statistics of the square root of energy possible applications for granular media and hydrodynamic turbulence are discussed

some physical features of the burr type xii distribution aspects related to driven nonequilibrium systems considered in beck cohen superstatistics bcs formalism and in addition the unbiased maximum entropy density estimation show the pertinence of considering the three parameter burr type xii distribution as a suitable way to describe stationary states of complex and nonequilibrium systems the above is shown following a variant to the procedure presented in the formulation of bcs which consists of the incorporation of an expanded boltzmann factor through the mittag leffler function on the other hand maximization of shannon boltzmann gibbs entropy and other generalized forms of entropy show that the burr type xii distribution may emerge as the least informative distribution for the mentioned systems

what should a statistical mechanics satisfy to reflect nature there is no compelling reason imposing that the methods of statistical mechanics should be restricted to the dynamical systems which follow the usual boltzmann gibbs prescriptions more specifically ubiquitous natural and artificial systems exhibit generic stationary states which are not ergodic nor close to it in any geometrically simple subset of the a priori allowed phase space in any even extended trivial sense a vast class of such systems appears nevertheless to be tractable within thermostatistical methods completely analogous to the usual ones the question posed in the title arises then naturally some answer to this complex question is advanced in the present review of nonextensive statistical mechanics and its recent connections

fractal weyl law for chaotic microcavities fresnel s laws imply multifractal scattering we demonstrate that the harmonic inversion technique is a powerful tool to analyze the spectral properties of optical microcavities as an interesting example we study the statistical properties of complex frequencies of the fully chaotic microstadium we show that the conjectured fractal weyl law for open chaotic systems lu et al phys rev lett 91 154101 2003 is valid for dielectric microcavities only if the concept of the chaotic repeller is extended to a multifractal by incorporating fresnel s laws

generalized boltzmann factors induced by weibull type distributions abstract the inverse mellin transform technique is utilized to obtain closed form representations of the generalized boltzmann factors associated with several weibull type models such as the generalized gamma maxwell rayleigh and half normal distributions the results complement those already available in the physics literature in connection with the distribution of certain variables affecting the behavior of nonequilibrium systems subject to complex dynamics which include for instance computable expressions for the generalized boltzmann factors induced by the gamma f uniform and lognormal statistical models

on generalized entropy measures and pathways abstract product probability property known in the literature as statistical independence is examined first then generalized entropies are introduced all of which give generalizations to shannon entropy it is shown that the nature of the recursivity postulate automatically determines the logarithmic functional form for shannon entropy due to the logarithmic nature shannon entropy naturally gives rise to additivity when applied to situations having product probability property it is argued that the natural process is non additivity important for example in statistical mechanics c tsallis possible generalization of boltzmann gibbs statistics j stat phys 52 1988 479 487 e g d cohen boltzmann and einstein statistics and dynamics an unsolved problem pramana 64 2005 635 643 even in product probability property situations and additivity can hold due to the involvement of a recursivity postulate leading to a logarithmic function generalized entropies are introduced and some of their properties are examined situations are examined where a generalized entropy of order leads to pathway models exponential and power law behavior and related differential equations connection of this entropy to kerridge s measure of inaccuracy is also explored

fractal weyl law behavior in an open hamiltonian system we numerically show fractal weyl law behavior in an open hamiltonian system that is described by a smooth potential and which supports numerous above barrier resonances this behavior holds even relatively far away from the classical limit the complex resonance wave functions are found to be localized on the fractal classical repeller

possible generalization of boltzmann gibbs statistics with the use of a quantity normally scaled in multifractals a generalized form is postulated for entropy namelys q k 1 i 1 w p i q q 1 whereq characterizes the generalization andp i are the probabilities associated withw microscopic configurations w the main properties associated with this entropy are established particularly those corresponding to the microcanonical and canonical ensembles the boltzmann gibbs statistics is recovered as theq 1 limit

some properties of q logarithm and q exponential functions in tsallis statistics we present some formulae which q logarithmic and q exponential functions introduced in tsallis statistics satisfy exact expression of the mellin transform of q exponential function is given moreover the q generalization of the digamma function is presented

evaluation of matrix variate gamma and beta integrals as multiple integrals and koberfractional integral operators in the complex matrix variate case explicit evaluations of matrix variate gamma and beta integrals in the complex domain by using conventional procedures is extremely difficult such an evaluation will reveal the structure of these matrix variate integrals in this article explicit evaluations of matrix variate gamma and beta integrals in the complex domain for the order of the matrix p 1 2 are given then fractional integral operators of the kober type are given for some specific cases of the arbitrary function a formal definition of fractional integrals in the complex matrix variate case was given by the author earlier as the m convolution of products and ratios where kober operators become a special class of fractional integral operators

fractional integral operators involving many matrix variables abstract fractional integral operators of the first and second kind in many and complex matrix variate cases are the main topic of discussion here after discussing fractional integral operators in the real scalar variable case the ideas are extended to fractional integral operators involving one matrix variable in the real and complex cases then these ideas are extended to define fractional integrals of the first and second kinds in many matrix variables in the real as well as complex cases

first asymptotic approximations to a solution of singularly perturbed optimal control problem with intersecting solutions of degenerate problem using the direct scheme method for constructing asymptotic solution of optimal control problems with small parameters consisting of immediate substituting a postulated asymptotic expansion of a solution into the problem condition and receiving problems for finding asymptotic terms we construct formally the zero order asymptotic approximation to an optimal control and the first order approximation to an optimal trajectory of a singularly perturbed optimal control problem with a weakly controllable state equation a cheap control a not uniquely solvable degenerate state equation fixed endpoints and with intersecting trajectories of the degenerate state equation corresponding to slow trajectories of the perturbed problem together with boundary layer functions in the vicinities of both ends of the considered interval the constructed asymptotics contains inner boundary layer functions an illustrative example is given

proportional reversed hazard rates weighted frailty model abstract in this paper a proportional reversed hazards weighted frailty model is considered the model is treated by the concept of weighted distributions where a natural alteration in the baseline distribution in the ordinarily proportional reversed hazards model illustrates the formation of the model stochastic variation of the model based on some stochastic orders is discussed finally the model is motivated with several expository examples by which the model is potentially applied to analyze parallel systems with different formations of lifetime components data analysis of the model for a physical distribution is carried out

random volumes under a general matrix variate model abstract the convex hull generated by p linearly independent points in euclidean n space n p will almost surely determine a p simplex and the corresponding p parallelotope the volume of this p parallelotope is v xx 1 2 where the rows of the p n n p matrix of rank p represent the p linearly independent points if the points are random points in some sense then v becomes a random volume the distribution of this random volume v when the matrix x has a very general real rectangular matrix variate density is the topic of this paper the complicated classical procedures based on integral geometry techniques for dealing with such problems are replaced by a simpler procedure based on jacobians of matrix transformations and functions of matrix argument apart from the distribution of v under this general model arbitrary moments of v connection to the likelihood ratio statistic or criterion for testing hypotheses on the parameters of multivariate normal distributions connections to mellin barnes integrals and meijer s g function connection to the concept of generalized variance various structural decompositions of v and special cases are also examined here

analogues of reliability analysis for matrix variate cases abstract an attempt is made in this paper to extend various aspects of reliability analysis in the real scalar variable case to the corresponding real matrix variate cases probability of a matrix random variable being bigger than a given matrix is defined in terms of norms and positive definite matrices then by using these definitions various aspects of reliability analysis are extended to the corresponding matrix variate cases a multivariate in the sense of many real scalar variables pathway model is also defined and some probabilities connected with reliability are also evaluated here

solution of fractional kinetic equation by a class of integral transform of pathway type solutions of fractional kinetic equations are obtained through an integral transform named p transform introduced in this paper the p transform is a binomial type transform containing many class of transforms including the well known laplace transform the paper is motivated by the idea of pathway model introduced by mathai linear algebra appl 396 317 328 2005 10 1016 j laa 2004 09 022 the composition of the transform with differential and integral operators are proved along with convolution theorem as an illustration of applications to the general theory of differential equations a simple differential equation is solved by the new transform being a new transform the p transform of some elementary functions as well as some generalized special functions such as h function g function wright generalized hypergeometric function generalized hypergeometric function and mittag leffler function are also obtained the results for the classical laplace transform is retrieved by letting 1

asymptotic solution of a linear quadratic problem with discontinuous coefficients and cheap control abstract this paper is devoted to a linear quadratic problem with cheap control and discontinuous coefficients using the direct scheme consisting of an immediate substitution of a postulated asymptotic expansion of a solution into the problem condition and finding a family of control problems to define the terms of the asymptotic expansion the asymptotic solution of the considered problem is obtained the proximity of the exact solution to the approximate asymptotic one is estimated for the control trajectory and functional it is shown that the minimized functional does not increase when a higher order asymptotic approximation to the optimal control is used an illustrative example is presented

a class of asymmetric pathway distributions and an entropy interpretation abstract asymmetric distributions are widely used in probability modeling and statistical analysis recently various asymmetric distributions are being developed by many researchers for modeling various data sets in real life contexts in the present paper we introduce a new class of q type asymmetric distributions which include q analogues of asymmetric laplace exponential power weibull etc and corresponding standard distributions as special cases also we show that this pathway model can be obtained by optimizing mathai s generalized entropy with more general setup which is a generalization of various entropy measures due to shannon and others

fractional integral operators in the complex matrix variate case abstract 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

on generalized fractional kinetic equations abstract in a recent paper saxena et al astro phys space sci 282 2002 281 developed solutions of generalized fractional kinetic equations in terms of mittag leffler functions the object of the present paper is to derive the solution of further generalized fractional kinetic equations their relation to fundamental laws of physics is briefly discussed results are obtained in a compact form in terms of generalized mittag leffler functions and a number of representations of these functions which are widely distributed in the literature are compiled for the first time

quadratic forms in random variables theory and applications textbook for a one semester graduate course for students specializing in mathematical statistics or in multivariate analysis or reference for theoretical as well as applied statisticians confines its discussion to quadratic forms and second degree polynomials in real normal random vectors and matr

a pathway from bayesian statistical analysis to superstatistics abstract superstatistics and tsallis statistics in statistical mechanics are given an interpretation in terms of bayesian statistical analysis subsequently superstatistics is extended by replacing each component of the conditional and marginal densities by mathai s pathway model and further both components are replaced by mathai s pathway models this produces a wide class of mathematically and statistically interesting functions for prospective applications in statistical physics it is pointed out that the final integral is a particular case of a general class of integrals introduced by the authors earlier those integrals are also connected to kratzel integrals in applied analysis inverse gaussian densities in stochastic processes reaction rate integrals in the theory of nuclear astrophysics and tsallis statistics in nonextensive statistical mechanics the final results are obtained in terms of fox s h function matrix variate analogue of one significant specific case is also pointed out

fractal weyl law for three dimensional chaotic hard sphere scattering systems the fractal weyl law connects the asymptotic level number with the fractal dimension of the chaotic repeller we provide the first test for the fractal weyl law for a three dimensional open scattering system for the four sphere billiard we investigate the chaotic repeller and discuss the semiclassical quantization of the system by the method of cycle expansion with symmetry decomposition we test the fractal weyl law for various symmetry subspaces and sphere to sphere separations

singular perturbations and time scale methods in control theory survey 1976 1983 recent progress in the use of singular perturbation and two time scale methods of modeling and design for control systems is reviewed over 350 references are organized into major problem areas representative issues and results are discussed with a view to outlining research directions and indicating potential areas of application the survey is aimed at engineers and applied mathematicians interested in model order reduction separation of time scales and allied simplified methods of control system analysis and design the exposition does not assume prior knowledge of singular perturbation methods

the quantum perturbed cat map and symmetry we investigate a class of quantum symmetries of the perturbed cat map which exist only for a subset of possible values of planck s constant the effect of these symmetries is to change the spectral statistics along this positive density subset the symmetries are shown to be related to some simple classical symmetries of the map