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

ergodic averages of commuting transformations with distinct degree polynomial iterates we prove mean convergence as n to infty for the multiple ergodic averages frac 1 n sum n 1 n f 1 t 1 p 1 n x cdot ldots cdot f ell t ell p ell n x where p 1 ldots p ell are integer polynomials with distinct degrees and t 1 ldots t ell are invertible commuting measure preserving transformations acting on the same probability space this establishes several cases of a conjecture of bergelson and leibman that complement the case of linear polynomials recently established by tao furthermore we show that unlike the case of linear polynomials for polynomials of distinct degrees the corresponding characteristic factors are mixtures of inverse limits of nilsystems we use this particular structure together with some equidistribution results on nilmanifolds to give an application to multiple recurrence and a corresponding one to combinatorics

nonconventional ergodic averages and nilmanifolds we study the l2 convergence of two types of ergodic averages the first is the average of a product of functions evaluated at return times along arithmetic progressions such as the expressions appearing in furstenberg fs proof of szemer ledi fs theorem the second average is taken along cubes whose sizes tend to for each average we show that it is sufficient to prove the convergence for special systems the characteristic factors we build these factors in a general way independent of the type of the average to each of these factors we associate a natural group of transformations and give them the structure of a nilmanifold from the second convergence result we derive a combinatorial interpretation for the arithmetic structure inside a set of integers of positive upper density

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

a multidimensional szemeredi theorem for hardy sequences of different growth abstract we prove a variant of the multidimensional polynomial szemeredi theorem of bergelson and leibman where one replaces polynomial sequences with other sparse sequences defined by functions that belong to some hardy field and satisfy certain growth conditions we do this by studying the limiting behavior of the corresponding multiple ergodic averages and obtaining a simple limit formula a consequence of this formula in topological dynamics shows denseness of certain orbits when the iterates are restricted to suitably chosen sparse subsequences another consequence is that every syndetic set of integers contains certain nonshift invariant patterns and every finite coloring of n with each color class a syndetic set contains certain polychromatic patterns results very particular to our non polynomial setup

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

simple approximations to the cramer rao lower bound on direction of arrival for closely spaced sources it is demonstrated that the cramer rao lower bound crlb on the directions of arrival doas of two closely spaced uncorrelated plane wave sources impinging on a uniform line array can be approximated by simple algebraic formulas which provide insight into the behavior of the bound with respect to signal to noise ratios snrs source separation etc these formulas are sufficiently accurate for general use with the number of sensors between about 6 and 60 for source separations down to about 0 1 of a classical beamwidth and for snrs from 30 to 30 db

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

a non singular dynamical system without maximal ergodic inequality abstract it is shown that there is a non singular dynamical system for which the maximal ergodic inequality does not hold we further discuss the connection between a non singular dynamical systems and the pointwise convergence of the furstenberg ergodic averages

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

polynomial sequences in groups abstract given a groupgwith lower central seriesg g1 g2 g3 we say that a sequenceg z gispolynomialif for anykthere isdsuch that the sequence obtained fromgby applying the difference operatordg n g n 1g n 1 dtimes takes its values ingk we introduce the notion ofthe degree of a polynomial sequenceand we prove that polynomial sequences of degrees not exceeding a given one form a group as an application we obtain the following extension of the hall petresco theorem theorem letg g1 g2 g3 be the lower central series of a group g let x gk y gland letp qbe polynomials z z of degrees k and l respectively then there is a sequencez0 g zi gifori n such thatfor all n n

generalized superstatistics of nonequilibrium markovian systems abstract the paper is devoted to the construction of the superstatistical description for nonequilibrium markovian systems it is based on kirchhoff s diagram technique and the assumption on the system under consideration to possess a wide variety of cycles with vanishing probability fluxes the latter feature enables us to introduce equivalence classes called channels within which detailed balance holds individually then stationary probability as well as flux distributions are represented as some sums over the channels the latter construction actually forms the superstatistical description which however deals with a certain superposition of equilibrium subsystems rather than a formal expansion of the nonequilibrium steady state distribution into terms of the boltzmann type

pointwise convergence of some multiple ergodic averages abstract we show that for every ergodic system x t 1 t d with commuting transformations the average 1 n d 1 0 n 1 n d n 1 0 n n 1 f 1 t 1 n j 1 d t j n j x f 2 t 2 n j 1 d t j n j x f d t d n j 1 d t j n j x converges for a e x x as n if x is distal we prove that the average 1 n n 0 n 1 f 1 t 1 n x f 2 t 2 n x f d t d n x converges for a e x x as n we also establish the pointwise convergence of averages along cubical configurations arising from a system with commuting transformations our methods combine the existence of sated and magic extensions introduced by austin and host respectively with ideas on topological models by huang shao and ye

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

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

norm convergence of nilpotent ergodic averages we show that multiple polynomial ergodic averages arising from nilpotent groups of measure preserving transformations of a probability space always converge in the l 2 norm

multiple correlation sequences and nilsequences we study the structure of multiple correlation sequences defined by measure preserving actions of commuting transformations when the iterates of the transformations are integer polynomials we prove that any such correlation sequence is the sum of a nilsequence and an error term that is small in uniform density this was previously known only for measure preserving actions of a single transformation we then use this decomposition result to give convergence criteria for multiple ergodic averages and deduce some rather surprising results for instance we infer convergence for actions of commuting transformations from the special case of actions of a single transformation our proof of the decomposition result differs from previous works of bergelson host kra and leibman as it does not rely on the theory of characteristic factors it consists of a simple orthogonality argument and the main tool is an inverse theorem of host and kra for general bounded sequences

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

a nilpotent roth theorem let t and s be invertible measure preserving transformations of a probability measure space x we prove that if the group generated by t and s is nilpotent then lim n ras infty frac 1 n sum n 1 n u t n x v s n x exists in l 2 norm for any u v l x we also show that for a with a 0 one has lim n ras infty frac 1 n sum n 1 n mu a cap t n a cap s n a 0 by the way of contrast we bring examples showing that if measure preserving transformations t s generate a solvable group then i the above limits do not have to exist ii the double recurrence property fails that is for some a a 0 one may have a t n a s n a 0 for all n finally we show that when t and s generate a nilpotent group of class c lim n ras infty frac 1 n sum n 1 n u t n x v s n x int ud mu int vd mu in l 2 x for all u v l x if and only if t s is ergodic on x x and the group generated by t 1 s t 2 s 2 t c s c acts ergodically on x

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

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

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

multiple ergodic theorems for arithmetic sets we establish results with an arithmetic flavor that generalize the polyno mial multidimensional szemeredi theorem and related multiple recurrence and conver gence results in ergodic theory for instance we show that in all these statements we can restrict the implicit parameter n to those integers that have an even number of dis tinct prime factors or satisfy any other congruence condition in order to obtain these refinements we study the limiting behavior of some closely related multiple ergodic averages with weights given by appropriately chosen multiplicative functions these averages are then analysed using a recent structural result for bounded multiplicative functions proved by the authors

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

local stability of ergodic averages we consider the extent to which one can compute bounds on the rate of convergence of a sequence of ergodic averages it is not difficult to construct an example of a computable lebesgue measure preserving transformation of 0 1 and a characteristic function f xa such that the ergodic averages a n f do not converge to a computable element of l 2 0 1 in particular there is no computable bound on the rate of convergence for that sequence on the other hand we show that for any nonexpansive linear operator t on a separable hilbert space and any element f it is possible to compute a bound on the rate of convergence of a n f from t f and the norm f of the limit in particular if t is the koopman operator arising from a computable ergodic measure preserving transformation of a probability space x and f is any computable element of l 2 x then there is a computable bound on the rate of convergence of the sequence anf the mean ergodic theorem is equivalent to the assertion that for every function k n and every e 0 there is an n with the property that the ergodic averages a m f are stable to within e on the interval n k n even in situations where the sequence a n f does not have a computable limit one can give explicit bounds on such n in terms of k and f e this tells us how far one has to search to find an n so that the ergodic averages are locally stable on a large interval we use these bounds to obtain a similarly explicit version of the pointwise ergodic theorem and we show that our bounds are qualitatively different from ones that can be obtained using upcrossing inequalities due to bishop and ivanov finally we explain how our positive results can be viewed as an application of a body of general proof theoretic methods falling under the heading of proof mining

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