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

fast high order ader schemes for linear hyperbolic equations a reformulation of the ader approach arbitrary high order schemes using derivatives for linear hyperbolic pde s is presented this reformulation leads to a drastic decrease of the computational effort a formula for the construction of ader schemes that are arbitrary high order accurate in space and time is given the accuracy for some selected schemes is shown numerically for the two dimensional linearized euler equations as a mathematical model for noise propagation in the time domain in aeroacoustics

numerical treatment of two dimensional interfaces for acoustic and elastic waves we present a numerical method to take into account 2d arbitrary shaped interfaces in classical finite difference schemes on a uniform cartesian grid this work extends the explicit simplified interface method esim previously proposed in 1d j comput phys 168 2001 227 248 the physical problem under study concerns the linear hyperbolic systems of acoustics and elastodynamics with stationary interfaces our method maintains near the interfaces properties of the schemes in homogeneous medium such as the order of accuracy and the stability limit moreover it enforces the numerical solution to satisfy the exact interface conditions lastly it provides subcell geometrical features of the interface inside the meshing the esim can be coupled automatically with a wide class of numerical schemes lax wendroff flux limiter schemes etc for a negligible additional computational cost throughout the paper we focus on the challenging case of an interface between a fluid and an elastic solid in numerical experiments we provide comparisons between numerical solutions and original analytic solutions showing the efficiency of the method

the mean field traveling salesman and related problems the edges of a complete graph on n vertices are assigned i i d random costs from a distribution for which the interval 0 t has probability asymptotic to t as t 0 through positive values in this so called pseudo dimension 1 mean field model we study several optimization problems of which the traveling salesman is the best known we prove that as n the cost of the minimum traveling salesman tour converges in probability to a certain number approximately 2 0415 which is characterized analytically

a time domain numerical modeling of two dimensional wave propagation in porous media with frequency dependent dynamic permeability an explicit finite difference scheme is presented for solving the two dimensional biot equations of poroelasticity across the full range of frequencies the key difficulty is to discretize the johnson koplik dashen jkd model which describes the viscous dissipations in the pores indeed the time domain version of biot jkd model involves order 1 2 fractional derivatives which amount to a time convolution product to avoid storing the past values of the solution a diffusive representation of fractional derivatives is used the convolution kernel is replaced by a finite number of memory variables that satisfy local in time ordinary differential equations the coefficients of the diffusive representation follow from an optimization procedure of the dispersion relation then various methods of scientific computing are applied the propagative part of the equations is discretized using a fourth order finite difference scheme whereas the diffusive part is solved exactly an immersed interface method is implemented to discretize the geometry on a cartesian grid and also to discretize the jump conditions at interfaces numerical experiments are proposed in various realistic configurations

near optimal configurations in mean field disordered systems we present a general technique to compute how the energy of a configuration varies as a function of its overlap with the ground state in the case of optimization problems our approach is based on a generalization of the cavity method to a system interacting with its ground state with this technique we study the random matching problem as well as the mean field diluted spin glass as a by product of this approach we calculate the de almeida thouless transition line of the spin glass on a fixed connectivity random graph

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

time domain numerical modeling of wave propagation in 2d heterogeneous porous media this paper deals with the numerical modeling of wave propagation in porous media described by biot s theory the viscous efforts between the fluid and the elastic skeleton are assumed to be a linear function of the relative velocity which is valid in the low frequency range the coexistence of propagating fast compressional wave and shear wave and of a diffusive slow compressional wave makes numerical modeling tricky to avoid restrictions on the time step the biot s system is splitted into two parts the propagative part is discretized by a fourth order ader scheme while the diffusive part is solved analytically near the material interfaces a space time mesh refinement is implemented to capture the small spatial scales related to the slow compressional wave the jump conditions along the interfaces are discretized by an immersed interface method numerical experiments and comparisons with exact solutions confirm the accuracy of the numerical modeling the efficiency of the approach is illustrated by simulations of multiple scattering

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

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

stochastic theory of dynamic permeability in poroelastic media a theory for the dynamic permeability in deformable porous media is developed the analysis is based on the momentum flux transfer from the slow compressional into the slow shear wave a proxy for the viscous wave in a newtonian fluid in the presence of random pore scale heterogeneities a first order statistical smoothing approximation is used to infer a dynamic permeability in the form of an integral over the covariance function modulated by the slow shear wave in a smooth pore throat limit the results reproduce the model proposed by johnson et al j fluid mech 176 379 1987

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

the parisi formula in this chapter we prove the parisi formula which gives the limiting value of the free energy per site for the sherrington kirkpatrick model at each temperature starting with the famous result of guerra that the parisi formula provided a bound for this free energy we make full use of poisson dirichlet cascades we then use the same techniques to prove in a strong sense that in presence of an external field the overlaps are positive

asymptotics in the random assignment problem we show that in the usual probabilistic model for the random assignment problem the optimal cost tends to a limit constant in probability and in expectation the method involves construction of an infinite limit structure in terms of which the limit constant is defined but we cannot improve on the known numerical bounds for the limit

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

replica symmetry of the minimum matching we establish the soundness of the replica symmetric ansatz introduced by m m ezard and g parisi for the minimum matching problem in the pseudo dimensiond mean eld model for d 1 the cased 1 corresponds to the 2 6 limit for the assignment problem proved by d aldous in 2001 we introduce a game theoretical framework by which we establish the analogous limit also for d 1

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

on random symmetric travelling salesman problems let the edges of the complete graphk nbe assigned independent uniform 0 1 random edge weights letztsp andz2fac be the weights of the minimum length travelling salesman tour and minimum weight 2 factor respectively we show thatwhp ztsp z2fac o 1 the proof is obtained by the analysis of a polynomial time algorithm that finds a tour only a little longer thanz2fac

wave field simulation for heterogeneous porous media with singular memory drag force the objective of this paper is to use biot s theory and the johnson koplik dashen dynamic permeability model in wave field simulation of a heterogeneous porous medium the johnson koplik dashen dynamic permeability model was originally formulated in the frequency domain in this paper the time domain drag force expression of the model is expressed in terms of the shifted fractional derivative of the relative fluid velocity in contrast to the exponential type viscous relaxation models the convolution operator in the johnson koplik dashen dynamic permeability model cannot be replaced by memory variables satisfying first order relaxation differential equations a new method for calculating the shifted fractional derivative without storing and integrating the entire velocity histories is developed using the new method to calculate the fractional derivative the governing equations for the two dimensional porous medium are reduced to a system of first order differential equations for velocities stresses pore pressure and the quadrature variables associated with the drag forces spatial derivatives involved in the first order differential equations system are calculated by fourier pseudospectral method while the time derivative of the system is discretized by a predictor corrector method for the demonstration of our method some numerical results are presented

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

the stochastic traveling salesman problem finite size scaling and the cavity prediction we study the random link traveling salesman problem where lengths lij between city i and city j are taken to be independent identically distributed random variables we discuss a theoretical approach the cavity method that has been proposed for finding the optimum tour length over this random ensemble given the assumption of replica symmetry using finite size scaling and a renormalized model we test the cavity predictions against the results of simulations and find excellent agreement over a range of distributions we thus provide numerical evidence that the replica symmetric solution to this problem is the correct one finally we note a surprising result concerning the distribution of k th nearest neighbor links in optimal tours and invite a theoretical understanding of this phenomenon

numerical modeling of elastic waves across imperfect contacts a numerical method is described for studying how elastic waves interact with imperfect contacts such as fractures or glue layers existing between elastic solids these contacts have been classicaly modeled by interfaces using a simple rheological model consisting of a combination of normal and tangential linear springs and masses the jump conditions satisfied by the elastic fields along the interfaces are called the spring mass conditions by tuning the stiffness and mass values it is possible to model various degrees of contact from perfect bonding to stress free surfaces the conservation laws satisfied outside the interfaces are integrated using classical finite difference schemes the key problem arising here is how to discretize the spring mass conditions and how to insert them into a finite difference scheme this was the aim of the present paper for this purpose we adapted an interface method previously developed for use with perfect contacts j comput phys 195 2004 pp 90 116 this numerical method also describes closely the geometry of arbitrarily shaped interfaces on a uniform cartesian grid at negligible extra computational cost comparisons with original analytical solutions show the efficiency of this approach

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

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

asymptotic stability for the lame system with fractional boundary damping abstract we consider a lame system damped by fractional boundary damping of neumann type we prove well posedness by semigroup theory moreover we prove stabilization by using multipliers method and rellich type relation combined with the frequency domain method

finite difference modeling of viscoelastic materials with quality factors of arbitrary magnitude to minimize acoustic noise designers of sonic logging tools often consider coatings of viscoelastic materials with very high attenuation properties efficient finite difference modeling of viscoelastic materials is a topic of current research to model viscoelastic materials in the time domain through finite differences efficiently one needs to replace the time convolution which enters in the stress strain relations by a set of first order differential equations this procedure is equivalent to computing a rational approximation of a certain form to the frequency dependent complex modulus of viscoelasticity known schemes for computing such approximations are designed to treat materials with low attenuation such as underground formations but fail to produce accurate or even physically meaningful results for highly attenuative materials we propose a novel scheme that allows one to construct for a given frequency range a uniformly optimal rational approximation for the most widely used model of mater

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

fourth order symplectic integration abstract in this paper we present an explicit fourth order method for the integration of hamilton s equations this method preserves the property that the time evolution of such a system yields a canonical transformation from the initial conditions to the final state that is the integration step is an explicit symplectic map although the result is first derived for a specific type of hamiltonian it is shown to be quite general in particular the results can be applied to any lie group

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

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

nature s way of optimizing we propose a general purpose method for finding high quality solutions to hard optimization problems inspired by self organizing processes often found in nature the method called extremal optimization successively eliminates extremely undesirable components of sub optimal solutions drawing upon models used to simulate far from equilibrium dynamics it complements approximation methods inspired by equilibrium statistical physics such as simulated annealing with only one adjustable parameter its performance proves competitive with and often superior to more elaborate stochastic optimization procedures we demonstrate it here on two classic hard optimization problems graph partitioning and the traveling salesman problem 2000 elsevier science b v all rights reserved

numerical modeling of 1d transient poroelastic waves in the low frequency range propagation of transient mechanical waves in porous media is numerically investigated in 1d the framework is the linear biot model with frequency independent coefficients the coexistence of a propagating fast wave and a diffusive slow wave makes numerical modeling tricky a method combining three numerical tools is proposed a fourth order ader scheme with time splitting to deal with the time marching a space time mesh refinement to account for the small scale evolution of the slow wave and an interface method to enforce the jump conditions at interfaces comparisons with analytical solutions confirm the validity of this approach

modeling of a constant q methodology and algorithm for an efficient and optimally inexpensive viscoelastic technique linear anelastic phenomena in wave propagation problems can be well modeled through a viscoelastic mechanical model consisting of standard linear solids in this paper we present a method for modeling of constant q as a function of frequency based on an explicit closed formula for calculation of the parameter fields several standard linear solids connected in parallel can be tuned through a single parameter to yield an excellent constant q approximation the proposed method enables substantial savings in computations and memory requirements experiments show that the new method also yields higher accuracy in the modeling of q than e g the pade approximant method

modeling power law absorption and dispersion in viscoelastic solids using a split field and the fractional laplacian the absorption of compressional and shear waves in many viscoelastic solids has been experimentally shown to follow a frequency power law it is now well established that this type of loss behavior can be modeled using fractional derivatives however previous fractional constitutive equations for viscoelastic media are based on temporal fractional derivatives these operators are non local in time which makes them difficult to compute in a memory efficient manner here a fractional kelvin voigt model is derived based on the fractional laplacian this is obtained by splitting the particle velocity into compressional and shear components using a dyadic wavenumber tensor this allows the temporal fractional derivatives in the kelvin voigt model to be replaced with spatial fractional derivatives using a lossless dispersion relation with the appropriate compressional or shear wave speed the model is discretized using the fourier collocation spectral method which allows the fractional operators to be efficiently computed the field splitting also allows the use of a k space corrected finite difference scheme for time integration to minimize numerical dispersion the absorption and dispersion behavior of the fractional laplacian model is analyzed for both high and low loss materials the accuracy and utility of the model is then demonstrated through several numerical experiments including the transmission of focused ultrasound waves through the skull

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

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

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

ultrasonic characterization of human cancellous bone using the biot theory inverse problem this paper concerns the ultrasonic characterization of human cancellous bone samples by solving the inverse problem using experimental transmitted signals the ultrasonic propagation in cancellous bone is modeled using the biot theory modified by the johnson et al model for viscous exchange between fluid and structure the sensitivity of the young modulus and the poisson ratio of the skeletal frame is studied showing their effect on the fast and slow wave forms the inverse problem is solved numerically by the least squares method five parameters are inverted the porosity tortuosity viscous characteristic length young modulus and poisson ratio of the skeletal frame the minimization of the discrepancy between experiment and theory is made in the time domain the inverse problem is shown to be well posed and its solution to be unique experimental results for slow and fast waves transmitted through human cancellous bone samples are given and compared with theoretical predictions

nineteen dubious ways to compute the exponential of a matrix twenty five years later in principle the exponential of a matrix could be computed in many ways methods involving approximation theory differential equations the matrix eigenvalues and the matrix characteristic polyn

3 d wave simulation in anelastic media using the kelvin voigt constitutive equation we design a numerical algorithm for wave simulation in anelastic media in the presence of free surface which can be used to model seismic waves at the earth s surface and ultrasonic waves in heterogeneous materials the stress strain relation is based on the kelvin voigt mechanical model which has the advantage of not requiring additional field variables the model requires two anelastic parameters and twice the spatial derivatives of the lossless case the high frequency components of the wave field are more attenuated than the low frequency components with the attenuation factors being approximately proportional to the square of the frequency the modeling simulates 3 d waves by using the fourier and chebyshev methods to compute the spatial derivatives along the horizontal and vertical directions respectively we stretch the mesh in the vertical direction to increase the minimum grid spacing and reduce the computational cost instabilities of the chebyshev differential operator due to the implementation of the free surface boundary conditions are solved with a characteristic approach where the characteristic variables are evaluated at the source central frequency the results of the modeling are verified by comparisons to the analytical solutions for lamb s problem and propagation in unbounded homogeneous media examples illustrating the propagation of rayleigh and love waves are presented

incorporation of attenuation into time domain computations of seismic wave fields the only numerically tractable way yet found to incorporate attenuation into numerical time domain computations of seismic wave fields is to approximate the viscoelastic modulus by a low order rational function of frequency the coefficients of this function can be determined by the pade approximation our test computations show however that this approximation generally is of poor quality therefore we suggest a new approach which is based on the rheological model of the generalized maxwell body which has a modulus of the desired rational form we choose the relaxation frequencies logarithmically equidistant in the frequency band of interest and determine the weight factors by simple numerical curve fitting to an arbitrary q law this approach is superior to the method above both in accuracy and in computational efficiency for most practical applications approximations of orders 2 or 3 are sufficient the computing time and memory requirements for a finite difference calculation are then approximately twice those of a purely elastic calculation as a first application of the method we compute sh channel waves in discontinuous coal seams with q 50 within the coal the results show that the high frequency airy phase is strongly attenuated this indicates that care has to be taken in comparing the results of purely elastic model calculations of the propagation of seam waves with experimental data

numerical modeling of transient two dimensional viscoelastic waves this paper deals with the numerical modeling of transient mechanical waves in linear viscoelastic solids dissipation mechanisms are described using the generalized zener model no time convolutions are required thanks to the introduction of memory variables that satisfy local in time differential equations by appropriately choosing the relaxation parameters it is possible to accurately describe a large range of materials such as solids with constant quality factors the evolution equations satisfied by the velocity the stress and the memory variables are written in the form of a first order system of pdes with a source term this system is solved by splitting it into two parts the propagative part is discretized explicitly using a fourth order ader scheme on a cartesian grid and the diffusive part is then solved exactly jump conditions along the interfaces are discretized by applying an immersed interface method numerical experiments of wave propagation in viscoelastic and fluid media show the efficiency of this numerical modeling for dealing with challenging problems such as multiple scattering configurations

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

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

diffusive approximation of a time fractional burger s equation in nonlinear acoustics a fractional time derivative is introduced into burger s equation to model losses of nonlinear waves this term amounts to a time convolution product which greatly penalizes the numerical modeling a diffusive representation of the fractional derivative is adopted here replacing this nonlocal operator by a continuum of memory variables that satisfy local in time ordinary differential equations then a quadrature formula yields a system of local partial differential equations well suited to numerical integration the determination of the quadrature coefficients is crucial to ensure both the well posedness of the system and the computational efficiency of the diffusive approximation for this purpose optimization with constraint is shown to be a very efficient strategy strang splitting is used to solve successively the hyperbolic part by a shock capturing scheme and the diffusive part exactly numerical experiments are proposed to assess the efficiency of the numerical modeling and to illustrate the effect of the fractional attenuation on the wave propagation

finite difference modeling of biot s poroelastic equations across all frequencies an explicit time stepping finite difference scheme is presented for solving biot s equations of poroelasticity across the entire band of frequencies in the general case for which viscous boundary layers in the pores must be accounted for the time domain version of darcy s law contains a convolution integral it is shown how to efficiently and directly perform the convolution so that the darcy velocity can be properly updated at each time step at frequencies that are low enough compared to the onset of viscous boundary layers no memory terms are required at higher frequencies the number of memory terms required is the same as the number of time points it takes to sample accurately the wavelet being used in practice we never use more than 20 memory terms and often considerably fewer allowing for the convolution makes the scheme even more stable even larger time steps might be used than it is when the convolution is entirely neglected the accuracy of the scheme is confirmed by comparing numerical examples to exact analytic results

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

a proof of parisi s conjecture on the random assignment problem an assignment problem is the optimization problem of finding in an m by n matrix of nonnegative real numbers k entries no two in the same row or column such that their sum is minimal such an optimization problem is called a random assignment problem if the matrix entries are random variables we give a formula for the expected value of the optimal k assignment in a matrix where some of the entries are zero and all other entries are independent exponentially distributed random variables with mean 1 thereby we prove the formula 1 1 4 1 9 1 k2 conjectured by g parisi for the case k m n and the generalized conjecture of d coppersmith and g b sorkin for arbitrary k m and n

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

belief propagation an asymptotically optimal algorithm for the random assignment problem the random assignment problem asks for the minimum cost perfect matching in the complete n n bipartite graph knn with i i d edge weights say uniform on 0 1 in a remarkable work by aldous aldous d 2001 the 2 limit in the random assignment problem rsa18 381 418 the optimal cost was shown to converge to 2 as n as conjectured by mezard and parisi mezard m g parisi 1987 on the solution of the random link matching problem j phys 48 1451 1459 through the so called cavity method the latter also suggested a nonrigorous decentralized strategy for finding the optimum which turned out to be an instance of the belief propagation bp heuristic discussed by pearl pearl j 1988 probabilistic reasoning in intelligent systems networks of plausible inference morgan kaufmann san francisco in this paper we use the objective method to analyze the performance of bp as the size of the underlying graph becomes large specifically we establish that the dynamic of bp on knn converges in distribution as n to an appropriately defined dynamic on the poisson weighted infinite tree and we then prove correlation decay for this limiting dynamic as a consequence we obtain that bp finds an asymptotically correct assignment in o n2 time only this contrasts with both the worst case upper bound for convergence of bp derived by bayati et al bayati m d shah m sharma 2008 max product for maximum weight matching convergence correctness and lp duality ieee trans inform theory54 3 1241 1251 and the best known computational cost of n3 achieved by edmonds and karp s algorithm edmonds j r karp 1972 theoretical improvements in algorithmic efficiency for network flow problems j acm19 248 264

biot jkd model simulation of 1d transient poroelastic waves with fractional derivatives a time domain numerical modeling of biot poroelastic waves is presented the viscous dissipation occurring in the pores is described using the dynamic permeability model developed by johnson koplik dashen jkd some of the coefficients in the biot jkd model are proportional to the square root of the frequency in the time domain these coefficients introduce order 1 2 shifted fractional derivatives involving a convolution product based on a diffusive representation the convolution kernel is replaced by a finite number of memory variables that satisfy local in time ordinary differential equations thanks to the dispersion relation the coefficients in the diffusive representation are obtained by performing an optimization procedure in the frequency range of interest a splitting strategy is then applied numerically the propagative part of biot jkd equations is discretized using a fourth order ader scheme on a cartesian grid whereas the diffusive part is solved exactly comparisons with analytical solutions show the efficiency and the accuracy of this approach

efficient solution of a wave equation with fractional order dissipative terms we consider a wave equation with fractional order dissipative terms modeling visco thermal losses on the lateral walls of a duct namely the webster lokshin model diffusive representations of fractional derivatives are used first to prove existence and uniqueness results then to design a numerical scheme which avoids the storage of the entire history of past data two schemes are proposed depending on the choice of a quadrature rule in the laplace domain the first one mimics the continuous energy balance but suffers from a loss of accuracy in long time simulation the second one provides uniform control of the accuracy however even though the latter is more efficient and numerically stable under the standard cfl condition no discrete energy balance has been yet found for it numerical results of comparisons with a closed form solution are provided

wave simulation in 2d heterogeneous transversely isotropic porous media with fractional attenuation a cartesian grid approach a time domain numerical modeling of transversely isotropic biot poroelastic waves is proposed in two dimensions the viscous dissipation occurring in the pores is described using the dynamic permeability model developed by johnson koplik dashen jkd some of the coefficients in the biot jkd model are proportional to the square root of the frequency in the time domain these coefficients introduce shifted fractional derivatives of order 1 2 involving a convolution product based on a diffusive representation the convolution kernel is replaced by a finite number of memory variables that satisfy local in time ordinary differential equations resulting in the biot da diffusive approximation model the properties of both the biot jkd and the biot da models are analyzed hyperbolicity decrease of energy dispersion to determine the coefficients of the diffusive approximation two approaches are analyzed gaussian quadratures and optimization methods in the frequency range of interest the nonlinear optimization is shown to be the better way of determination a splitting strategy is then applied to approximate numerically the biot da equations the propagative part is discretized using a fourth order ader scheme on a cartesian grid whereas the diffusive part is solved exactly an immersed interface method is implemented to take into account heterogeneous media on a cartesian grid and to discretize the jump conditions at interfaces numerical experiments are presented comparisons with analytical solutions show the efficiency and the accuracy of the approach and some numerical experiments are performed to investigate wave phenomena in complex media such as multiple scattering across a set of random scatterers

numerical modeling of nonlinear acoustic waves in a tube connected with helmholtz resonators acoustic wave propagation in a one dimensional waveguide connected with helmholtz resonators is studied numerically finite amplitude waves and viscous boundary layers are considered the model consists of two coupled evolution equations a nonlinear pde describing nonlinear acoustic waves and a linear ode describing the oscillations in the helmholtz resonators the thermal and viscous losses in the tube and in the necks of the resonators are modeled by fractional derivatives a diffusive representation is followed the convolution kernels are replaced by a finite number of memory variables that satisfy local ordinary differential equations a splitting method is then applied to the evolution equations their propagative part is solved using a standard tvd scheme for hyperbolic equations whereas their diffusive part is solved exactly various strategies are examined to compute the coefficients of the diffusive representation finally an optimization method is preferred to the usual quadrature rules the numerical model is validated by comparisons with exact solutions the properties of the full nonlinear solutions are investigated numerically in particular the existence of acoustic solitary waves is confirmed

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

viscoelastic finite difference modeling real earth media disperse and attenuate propagating mechanical waves this anelastic behavior can be described well by a viscoelastic model we have developed a finite difference simulator to model wave propagation in viscoelastic media the finite difference method was chosen in favor of other methods for several reasons finite difference codes are more portable than for example pseudospectral codes moreover finite difference schemes provide a convenient environment in which to define complicated boundaries a staggered scheme of second order accuracy in time and fourth order accuracy in space appears to be optimally efficient because of intrinsic dispersion no fixed grid points per wavelength rule can be given instead we present tables which enable a choice of grid parameters for a given level of accuracy since the scheme models energy absorption natural and efficient absorbing boundaries may be implemented merely by changing the parameters near the grid boundary the viscoelastic scheme is

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