domain of definition of levermore s five moment system the simplest system in levermore s moment hierarchy involving moments higher than second order is the five moment closure it is obtained by taking velocity moments of the one dimensional boltzmann equation under the assumption that the velocity distribution is a maximum entropy function the moment vectors for which a maximum entropy function exists consequently make up the domain of definition of the system the aim of this article is a complete characterization of the structure of the domain of definition and the connected maximum entropy problem the space homogeneous case of the equation and numerical aspects are also addressed

perturbation theory for the eckart young mirsky theorem and the constrained total least squares problem abstract golub et al linear algebra appl 88 89 1987 317 327 j demmel siam j numer anal 24 1987 199 206 generalized the eckart young mirsky eym theorem which solves the problem of approximating a matrix by one of lower rank with only a specific rectangular subset of the matrix allowed to be changed based on their results this paper presents perturbation analysis for the eym theorem and the constrained total least squares problem ctls

perturbation bounds for least squares problems with equality constraints we present some new results on the perturbation analysis for least squares problems with equality constraints based on an equivalent formation of the problem and the optimal perturbation result for general least squares problems

rank revealing qr factorizations abstract an algorithm is presented for computing a column permutation pi and a qr factorization a pi qr of an m by n m or n matrix a such that a possible rank deficiency of a will be revealed in the triangular factor r having a small lower right block for low rank deficient matrices the algorithm is guaranteed to reveal the rank of a and the cost is only slightly more than the cost of one regular qr factorization a posteriori upper and lower bounds on the singular values of a are derived and can be used to infer the numerical rank of a keywords qr factorization rank deficient matrices least squares computation subset selection rank singular values

lagrangians of physics and the game of fisher information transfer the lagrangians of physics arise out of a mathematical game between a smart measurer and nature personified by a demon each contestant wants to maximize his level of fisher information i the game is zero sum by conservation of information in the closed system the payoff of the game introduces a variational principle extreme physical information epi which fixes both the lagrangian and the physical constant of each scenario the epi approach provides an understanding of the relationship between measurement and physical law epi also defines a prescription for constructing lagrangians the prior knowledge required for this purpose is a rule of symmetry or conservation that implies a unitary transformation for which i remains invariant as an example when applied to the smart measurement of the space time coordinate of a particle the symmetry used is that between position time space and momentum energy space then the unitary transformation is the fourier one and epi derives the following the equivalence of energy momentum and mass the constancy of planck s parameter h and the lagrangian that implies both the klein gordon equation and the dirac equation of quantum mechanics

hausdorff moment problem and maximum entropy a unified approach hausdorff moment problem in the approach of maximum entropy is reconsidered some aspects as existence condition entropy convergence entropy decreasing stability are studied through a unified approach results known in literature and some new ones are illustrated

rigorous properties and uncertainty like relationships on product complexity measures application to atomic systems most of research on complexities and the corresponding conclusions have been obtained by numerically quantifying their values and little attention has been paid to their theoretical properties and the exact meaning within an statistical framework valid for any arbitrary n dimensional distribution in the present work different product complexities constructed as a product of two relevant information quantities associated to the concepts of uncertainty and localization are investigated different results on product complexities such as rigorous bounds uncertainty like inequalities relationships among themselves and also statistical interpretations are given for the sake of completeness some of these analytical results on product complexities are numerically checked for the one particle densities of atomic systems in both conjugated spaces

on the calculations of maximum entropy distributions having prescribed the moments it is shown that maximum entropy distributions having prescribed the finite number of moments can be determined from a suitable system of differential equations

a note on a result of sun ji guang sensitivity of the cs and gsv decompositions in siam j numer anal 20 1983 pp 611 625 sun ji guang gives a perturbation analysis of a generalized singular value decomposition this note rephrases some of his ideas and derives an improved version of one of his results using straightforward matrix ideas some elaboration is given on the meaning of the result and the analysis is extended to cover the case of most usual interest to the numerical analyst

towards a generalized singular value decomposition we suggest a form for and give a constructive derivation of the generalized singular value decomposition of any two matrices having the same number of columns we outline its desirable characteristics and compare it to an earlier suggestion by van loan siam j numer anal 13 1976 pp 76 83 the present form largely follows from the work of van loan but is slightly more general and computationally more amenable than that in the paper cited we also prove a useful extension of a theorem of stewart siam rev 19 1977 pp 634 662 on unitary decompositions of submatrices of a unitary matrix

maximum entropy and lacunary stieltjes moment problem the lacunary stieltjes moment problem in the framework of the maximum entropy approach is considered a proof for the existence conditions of the solution is provided

high order entropy based closures for linear transport in slab geometry ii a computational study of the optimization problem we present a numerical algorithm to implement entropy based m n moment models in the context of a simple linear kinetic equation for particles moving through a material slab the closure for these models as is the case for all entropy based models is derived through the solution of a constrained convex optimization problem the algorithm has two components the first component is a discretization of the moment equations which preserves the set of realizable moments thereby ensuring that the optimization problem has a solution in exact arithmetic the discretization is a second order kinetic scheme which uses muscl type limiting in space and a strong stability preserving runge kutta time integrator the second component of the algorithm is a newton based solver for the dual optimization problem which uses an adaptive quadrature to evaluate integrals in the dual objective and its derivatives the accuracy of the numerical solution to the dual problem plays a key role in the time step restric

on the fuzzy initial value problem abstract the initial value problem x t f t x t x 0 x 0 with fuzzy initial value and with deterministic or fuzzy function f is considered two different approaches viz the extension principle and the use of extremal solutions of deterministic initial value problems are applied generalizations to fuzzy integral equations and fuzzy functional differential equations are indicated

the existence conditions for maximum entropy distributions having prescribed the first three moments the criteria for existence of maximum entropy distributions having prescribed the first three moments in a semi infinite range are given some relationships between moments and lagrange multipliers are also presented

fisher based thermodynamics its legendre transform and concavity properties it is shown that the legendre transform structure of thermodynamics can be replicated without any change if one replaces the entropy s by fisher s information measure i also the important thermodynamic property of concavity is shown to be obeyed by i by this use of the fisher information measure we develop a thermodynamics that seems to be able to treat equilibrium and nonequilibrium situations in a manner entirely similar to the conventional one

perturbation analysis for the generalized singular value problem this paper discusses perturbation bounds for generalized singular values and for subspaces associated with the generalized singular value decomposition of two matrices having the same number of columns by a limiting procedure from our results we can derive those for the ordinary singular value problem which were proved by l mirsky quart j math oxford 11 1960 pp 50 59 and p a wedin bit 12 1972 pp 99 111

conditional hyperbolic quadrature method of moments for kinetic equations the conditional quadrature method of moments cqmom was introduced by yuan and fox j comput phys 230 22 8216 8246 2011 to reconstruct a velocity distribution function vdf from a finite set of its integer moments the reconstructed vdf takes the form of a sum of weighted dirac delta functions in velocity phase space and provides a closure for the spatial flux term in the corresponding kinetic equation the cqmom closure for the flux leads to a weakly hyperbolic system of moment equations in subsequent work chalons et al proceed ctr sum prog 2010 347 358 2010 the dirac delta functions were replaced by gaussian distributions which make the moment system hyperbolic but at the added cost of dealing with continuous distributions here a hyperbolic version of cqmom is proposed that uses weighted dirac delta functions while the moment set employed for multi gaussian and conditional hyqmom chyqmom are equivalent the latter is able to access all of moment space whereas the former cannot e g arbitrary values of the fourth order velocity moment in 1 d phase space with two nodes by making use of the properties of chyqmom in 2 d phase space it is possible to control a symmetrical subset of the optimal moments fox indust engng chem res 48 21 9686 9696 2009 furthermore the moment sets for 2 d problems are smaller for chyqmom than in the original cqmom thanks to a judicious choice of the velocity abscissas in phase space

maximum entropy in the hamburger moments problem the maximum entropy approach to the solution of the hamburger inverse problem of moments in which one seeks to recover a positive density function p x where x from the values of a finite n 1 of its moments is considered the obtained results show that unexpected upper bounds for the moments do not exist in the general hamburger finite moment problem unlike in the symmetric case previously considered some physical examples illustrating the use of partial information to determine the approximate function are presented

the cauchy problem for continuous fuzzy differential equations abstract we prove a version of the classical peano s theorem for the initial value problem for a fuzzy differential equation in the metric space of normal fuzzy convex sets with distance given by the maximum of the hausdorff distances between level sets

condition numbers for a linear function of the solution of the linear least squares problem with equality constraints abstract in this paper we consider the normwise mixed and componentwise condition numbers for a linear function l x of the solution x to the linear least squares problem with equality constraints lse the explicit expressions of the normwise mixed and componentwise condition numbers are derived also we revisit some previous results on the condition numbers of linear least squares problem ls and lse it is shown that some previous explicit condition number expressions on ls and lse can be recovered from our new derived condition numbers formulas the sharp upper bounds for the derived normwise mixed and componentwise condition numbers are obtained which can be estimated efficiently by means of the classical hager higham algorithm for estimating matrix one norm moreover the proposed condition estimation methods can be incorporated into the generalized qr factorization method for solving lse the numerical examples show that when the coefficient matrices of lse are sparse and badly scaled the mixed and componentwise condition numbers can give sharp perturbation bounds on the other hand normwise condition numbers can severely overestimate the exact relative errors because normwise condition numbers ignore the data sparsity and scaling however from the numerical experiments for random lse problems if the data is not either sparse or badly scaled it is more suitable to adopt the normwise condition number to measure the conditioning of lse since the explicit formula of the normwise condition number is more compact

differential escort transformations and the monotonicity of the lmc renyi complexity measure abstract escort distributions have been shown to be very useful in a great variety of fields ranging from information theory nonextensive statistical mechanics till coding theory chaos and multifractals in this work we give the notion and the properties of a novel type of escort density the differential escort densities which have various advantages with respect to the standard ones we highlight the behavior of the differential shannon renyi and tsallis entropies of these distributions then we illustrate their utility to prove the monotonicity property of the lmc renyi complexity measure and to study the behavior of general distributions in the two extreme cases of minimal and very high lmc renyi complexity finally this transformation allows us to obtain the tsallis q exponential densities as the differential escort transformation of the exponential density

general entropic approximations for canonical systems described by kinetic equations in this paper we extend the general construction of entropic approximation for kinetic operators modelling canonical systems more precisely this paper aims at pursuing to thermalized systems the works of levermore schneider and junk on moments problems relying on entropy minimization in order to construct bgk approximations and moments based equations

exact maximum entropy closure of the hydrodynamical model for si semiconductors the 8 moment case an exact closure is obtained of the 8 moment model for semiconductors based on the maximum entropy principle in the case of silicon semiconductors the validity of the model is assessed and comparisons with an approximate closure are presented and discussed

error analysis and implementation aspects of deferred correction for equality constrained least squares problems the deferred correction procedure recently proposed by van loan on the method of weighting for equality constrained least squares problems siam j numer anal 22 1985 pp 851 864 solves the problem of minimizing f ex 2 subject to cx g by repeatedly solving the related least squares problem of minimizing left left begin array 20 c tau delta g delta f end array right left begin array 20 c tau c e end array right delta x right 2 for some large weight tau the convergence behavior of the procedure is shown to be greatly dependent upon error analysis considerations an analysis of its convergence with an approximation factorization of left begin array 20 c tau c e end array right is given an implementation improvement to the procedure that makes it more robust is also given van loan s conjecture that the method converges for all but ill conditioned problems is shown to be true

new perturbation results for equality constrained least squares problems abstract we present some perturbation results for least squares problems with equality constraints relative errors are obtained on perturbed solutions and lagrange multipliers of the problem based on the equivalence of the problem to a consistent system of linear equations

on the application of maximum entropy to the moments problem the maximum entropy approach to the solution of classical inverse problem of moments in which one seeks to reconstruct a function p x where x 0 from the values of a finite set n 1 of its moments is studied it is shown that for n 4 such a function always exists while for n 2 and n 3 the acceptable values of the moments are singled out analytically the paper extends to the general case where the results were previously bounded to the case n 2

global existence of solutions to fuzzy differential equations abstract the local existence and uniqueness theorems are investigated by wu et al j math anal appl 202 1996 629 644 for the cauchy problem of the fuzzy valued mappings of a real variable whose values are in the fuzzy number space e n d in this paper we point out a variety of results which assure global existence of solutions to fuzzy differential equations

perturbation theory for the rank deficient equality constrained least squares problem this paper presents perturbation theory for the equality constrained least squares problem min f in s kf g 2 with s f lf h 2 min y in c n ly h 2 lse in which l and begin array 20 c l k end array may not be of full rank the analysis generalizes the results of elden 4 perturbation theorems for the lse problem are proved numerical experiments are also given to verify the error bounds

grad s 13 moment equations in a modified form we describe a hierarchy of formal expansions that represent the fourier transform of a solution of the boltzmann equation the constructed approximations are based on the family of weighted taylor expansions the first two representations correspond to the maxwellian and to the gaussian expansions the third representation has a weight that generalizes the gaussian and it depends on the first 13 moments of the boltzmann density f it can be shown that this weight is galilean invariant and it is close to the gaussian providing that the heat fluxes are not too large the 13 moment weight yields a revised form of grad s 13 moment expansion for the boltzmann equation in search for the entropy dissipation inequality we also examine the relation between levermore s 14 moment and grad s 13 moment expansion first we show that the coefficients of the godunov potential are described by a system of partial differential equations with coefficients that depend on the fourier transform of the levermore s density f then we argue that the same taylor expansion exploited in the grad s scheme can be used to approximate levermore s 14 moment density we also show that the weighted taylor expansions are related to a formal solution of the hamburger problem

tsallis entropy maximization procedure revisited abstract the proper way of averaging is an important question with regards to tsallis thermostatistics three different procedures have been thus far employed in the pertinent literature the third one i e the tsallis mendes plastino tmp physica a 261 1998 534 normalization procedure exhibits clear advantages with respect to earlier ones in this work we advance a distinct from the tmp one way of handling the lagrange multipliers involved in the extremization process that leads to tsallis statistical operator it is seen that the new approach considerably simplifies the pertinent analysis without losing the beautiful properties of the tsallis mendes plastino formalism

stieltjes moment problem via fractional moments stieltjes moment problem is considered to recover a probability density function from the knowledge of its infinite sequence of ordinary moments the approximate density is obtained through maximum entropy technique under the constraint of few fractional moments the latter are numerically obtained from the infinite sequence of ordinary moments and are chosen in such a way as to convey the maximum information content carried by the ordinary moments as a consequence a model with few parameters is obtained and intrinsic numerical instability is avoided it is proved that the approximate density is useful for calculating expected values and some other interesting probabilistic quantities

equivalent formulae for the supremum and stability of weighted pseudoinverses during recent decades there have been a great number of research articles studying interior point methods for solving problems in mathematical programming and constrained optimization stewart and o leary obtained an upper bound for scaled pseudoinverses sup w1 2x w1 2 2 of a matrix x w p where p is a set of diagonal positive definite matrices we improved their results to obtain the supremum of scaled pseudoinverses and derived the stability property of scaled pseudoinverses forsgren further generalized these results to derive the supremum of weighted pseudoinverses sup w1 2x w1 2 2 where w p p is a set of diagonally dominant positive semidefinite matrices by using a signature decomposition of weighting matrices w and by applying the binet cauchy formula and cramer s rule for determinants the results are also extended to equality constrained linear least squares problems in this paper we extend forsgren s results to a general complex matrix x to establish several equivalent formulae for sup w1 2x w1 2 2 where p is a w p set of diagonally dominant positive semidefinite matrices or a set of weighting matrices arising from solving equality constrained least squares problems we also discuss the stability property of these weighted pseudoinverses

fuzzy and set valued stochastic differential equations with local lipschitz condition we are concerned with the fuzzy stochastic differential equations driven by multidimensional brownian motion viewed as a tool used to describe the behavior of dynamic systems operating in fuzzy environments with stochastic noises under the uniform lipschitz condition we prove the local uniqueness theorem for the solutions of fuzzy stochastic differential equations next we show assuming the lipschitz condition is satisfied only locally that these equations have a unique solution the fact that the solution is bounded is also proved we conclude the paper with a number of corresponding results holding for the deterministic fuzzy differential equations and set valued stochastic differential equations with local lipschitz condition

a new technique of quaternion equality constrained least squares problem by means of complex representation of a quaternion matrix we study the relationship between the solutions of the quaternion equality constrained least squares problem and that of complex equality constrained least squares problem and obtain a new technique of finding a solution of the quaternion equality constrained least squares problem

a realizability preserving discontinuous galerkin method for the m1 model of radiative transfer the m 1 model for radiative transfer coupled to a material energy equation in planar geometry is studied in this paper for this model to be well posed its moment variables must fulfill certain realizability conditions our main focus is the design and implementation of an explicit runge kutta discontinuous galerkin method which under a more restrictive cfl condition guarantees the realizability of the moment variables and the positivity of the material temperature an analytical proof for our realizability preserving scheme which also includes a slope limiting technique is provided and confirmed by various numerical examples among other things we present accuracy tests showing convergence up to fourth order compare our results with an analytical solution in a riemann problem and consider a marshak wave problem

size velocity correlations in hybrid high order moment multi fluid methods for polydisperse evaporating sprays modeling and numerical issues kah et al 2010 30 33 recently developed the eulerian multi size moment model emsm which tackles the modeling and numerical simulation of polydisperse multiphase flows using a high order moment method in a compact interval they suggested to reconstruct the number density function ndf by entropy maximization which leads to a unique and realizable ndf potentially in several size intervals thus leading to an hybrid method between multifluid and high order this reconstruction is used to simulate the evaporation process by an evaluation of the flux of droplet disappearance at zero size the fluxes of droplets between size intervals and an accurate description of the size shift induced by evaporation massot et al 2010 15 although this method demonstrated its potential for evaporating polydisperse flows two issues remain to be addressed first the emsm only considers one velocity for all droplets thus decoupling size from velocity which is too restrictive for distributions with a large size spectrum in most applications size conditioned dynamics have to be accounted for second the possibility to have separated dynamics for each size can lead to quasi monodisperse distributions which corresponds to a hard limiting case for the em algorithm so the behavior of the algorithm needs to be investigated in order to reproduce the entire moment space with a reasonable accuracy the aim of this paper is thus twofold the em and its related algorithm are enhanced by using a more accurate integration method in order to handle ndf close to the frontier of the moment space associated with an adaptive number of parameters to reconstruct the ndf accurately and efficiently as well as tabulated initial guess to optimize the computational time then a new model called csvm coupled size velocity moments model is introduced size velocity correlations are addressed either in the evaporation and drag processes or in the convective transport to reach this goal a velocity reconstruction for each size is suggested using only one additional moment per dimension and which can be directly applied to several size intervals thus this method is a direct generalization of emsm to handle the convective transport a flux splitting scheme is proposed based on the underlying kinetic description of the disperse phase comparing to existing approaches a main novelty of the csvm is that our kinetic approach ensures built in realizability conditions no additional corrections of the moments being needed at each time step the full strategy is first evaluated in 0d and 1d cases which either demonstrates the ability to reproduce both evaporation drag force and convection with size velocity correlations or the possible extension to several size intervals finally the method is applied on 2d cases with only one section showing the ability of the csvm and its related algorithms to capture the main physics of polydisperse evaporating sprays with a minimal number of moments

perturbation bounds for constrained and weighted least squares problems we derive perturbation bounds for the constrained and weighted linear least squares ls problems both the full rank and rank deficient cases are considered the analysis generalizes some results of earlier works

estimation of distribution laws and physical laws by a principle of extremized physical information abstract a new method of estimating physical probability laws is given this arises from a new form of information which in turn follows from four physically based axioms extremization of this physical information gives rise to the required probability laws as verifications of the approach the following statistical and non statistical laws of physics are derived the complex schrodinger and helmholtz wave equations relativistic quantum mechanics in the klein gordon dirac and weyl pauli formulations the boltzmann energy and maxwell boltzmann velocity distributions the lorentz transformation group of special relativity and maxwell s equations also derived are inequalities defining a class of uncertainty principles e g heisenberg s finally the einstein equations of motion but not the fields of general relativity are derived along with an equivalence d i d mc 2 between information flow rate and matter energy

maximum entropy in the generalized moment problem the generalized moment problem in the framework of the maximum entropy approach is considered a proof for the existence conditions of the solution is provided

affordable robust moment closures for cfd based on the maximum entropy hierarchy the use of moment closures for the prediction of continuum and moderately non equilibrium flows offers modelling and numerical advantages over other methods the maximum entropy hierarchy of moment closures holds the promise of robustly hyperbolic stable moment equations however their are two issues that limit their practical implementation firstly for closures that have a treatment for heat transfer fluxes cannot be written in closed form and a very expensive iterative procedure is required at every flux evaluation secondly for these same closures there are physically possible moment states for which the entropy maximization problem has no solution and the entire framework breaks down this paper demonstrates that affordable closed form moment closures that are inspired by the maximum entropy framework can be proposed it is known that closing fluxes in the maximum entropy hierarchy approach a singularity as the region of non solvability is approached this paper shows that far from a disadvantage this singularity allows for smooth and accurate prediction of shock wave structure even for high mach numbers the presence of unphysical sub shocks within shock profile predictions of traditional closures has long been regarded as an unfortunate limitation of the entire moment closure technique the realization that smooth shock profiles are in fact possible for moment methods with a moderate number of moments greatly increases the method s applicability to high speed flows in this paper a 5 moment system for a simple one dimensional gas and a 14 moment system for realistic gases are developed and examined numerical solution for shock waves at a variety of incoming flow mach numbers demonstrate both the robustness and the accuracy of the closures

on the existence and characterization of the maxent distribution under general moment inequality constraints a broad set of sufficient conditions that guarantees the existence of the maximum entropy maxent distribution consistent with specified bounds on certain generalized moments is derived most results in the literature are either focused on the minimum cross entropy distribution or apply only to distributions with a bounded volume support or address only equality constraints the results of this work hold for general moment inequality constraints for probability distributions with possibly unbounded support and the technical conditions are explicitly on the underlying generalized moment functions an analytical characterization of the maxent distribution is also derived using results from the theory of constrained optimization in infinite dimensional normed linear spaces several auxiliary results of independent interest pertaining to certain properties of convex coercive functions are also presented

special features of the relation between fisher information and schrodinger eigenvalue equation it is well known that a suggestive relation exists that links schrodinger s equation se to the information optimizing principle based on fisher s information measure the connection entails the existence of a legendre transform structure underlying the se here we show that appeal to this structure leads to a first order differential equation for the se s eigenvalues that in certain cases can be used to obtain the eigenvalues without explicitly solving the se complying with the above mentioned equation constitutes a necessary condition to be satisfied by an energy eigenvalue we show that the general solution is unique

a maximum entropy method for particle filtering standard ensemble or particle filtering schemes do not properly represent states of low priori probability when the number of available samples is too small as is often the case in practical applications we introduce here a set of parametric resampling methods to solve this problem motivated by a general h theorem for relative entropy we construct parametric models for the filter distributions as maximum entropy minimum information models consistent with moments of the particle ensemble when the prior distributions are modeled as mixtures of gaussians our method naturally generalizes the ensemble kalman filter to systems with highly non gaussian statistics we apply the new particle filters presented here to two simple test cases a one dimensional diffusion process in a double well potential and the three dimensional chaotic dynamical system of lorenz

adaptive change of basis in entropy based moment closures for linear kinetic equations entropy based m n moment closures for kinetic equations are defined by a constrained optimization problem that must be solved at every point in a space time mesh making it important to solve these optimization problems accurately and efficiently we present a complete and practical numerical algorithm for solving the dual problem in one dimensional slab geometries the closure is only well defined on the set of moments that are realizable from a positive underlying distribution and as the boundary of the realizable set is approached the dual problem becomes increasingly difficult to solve due to ill conditioning of the hessian matrix to improve the condition number of the hessian we advocate the use of a change of polynomial basis defined using a cholesky factorization of the hessian that permits solution of problems nearer to the boundary of the realizable set we also advocate a fixed quadrature scheme rather than adaptive quadrature since the latter introduces unnecessary expense and changes the computationally realizable set as the quadrature changes for very ill conditioned problems we use regularization to make the optimization algorithm robust we design a manufactured solution and demonstrate that the adaptive basis optimization algorithm reduces the need for regularization this is important since we also show that regularization slows and even stalls convergence of the numerical simulation when refining the space time mesh we also simulate two well known benchmark problems there we find that our adaptive basis fixed quadrature algorithm uses less regularization than alternatives although differences in the resulting numerical simulations are more sensitive to the regularization strategy than to the choice of basis

on the generalized problem of the boltzmann equation and the moment method in kinetic theory of gases in this paper we formulate the generalized problem of the boltzmann equation based on the kinetic entropy balance equation in conjunction with the maximum entropy principle first we prove that the solution of this generalized problem is unique we then prove that the entropy balance equation obtained by eu in extended irreversible thermodynamics is valid if and only if the one particle distribution function fa is the solution of this generalized problem as a by product of this result we also obtain a statistical expression of the thermodynamic entropy balance equation that shares the same formula as the kinetic entropy balance equation

nine moment phonon hydrodynamics based on the modified grad type approach hyperbolicity of the one dimensional flow after expanding the distribution function about an anisotropic planck function the new moment closure method of banach and larecki applied to the boltzmann peierls equation for the phonon gas dynamics leads to a whole hierarchy of closed systems of moment equations the system of equations for the energy density and the heat flux is the first non perturbative member of this hierarchy of closures in our previous paper 2004 j phys a math gen 37 9805 emphasis was placed on deriving the next member the 9 moment anisotropic closure that involves the flux of the heat flux as an extra gas state variable here as a first step in effectively analysing this system we present a study of the one dimensional rotationally symmetric reduction of these equations under the assumption of callaway s model a systematic procedure is derived which shows that the obtained system of three evolution equations for three nonvanishing gas state variables can be cast into a symmetric hyperbolic form for the sake of completeness we describe explicitly the region of symmetric hyperbolicity in parameter space the space defined by the gas state variables the evolution system is symmetric hyperbolic for significant ranges of physical conditions i e there are effectively no unphysical limitations on the magnitude of the energy density and the heat flux this paper also deals with the eigenvalue problem and calculates approximately the characteristic speeds

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

on perturbations of some constrained subspaces perturbation bounds of subspaces such as eigen spaces singular subspaces and canonical subspaces have been extensively studied in the literature in this paper we study perturbations of some constrained subspaces of 1x2 2x1 and 2x2 block matrices in which only one of the sub matrices can be changed such problems rise from the least squares total least squares problem the constrained least squares problem and the constrained total least squares problem

maximum entropy autoregressive conditional heteroskedasticity model abstract in many applications it has been found that the autoregressive conditional heteroskedasticity arch model under the conditional normal or student s t distributions are not general enough to account for the excess kurtosis in the data moreover asymmetry in the financial data is rarely modeled in a systematic way in this paper we suggest a general density function based on the maximum entropy me approach that takes account of asymmetry excess kurtosis and also of high peakedness the me principle is based on the efficient use of available information and as is well known many of the standard family of distributions can be derived from the me approach we demonstrate how we can extract information functional from the data in the form of moment functions we also propose a test procedure for selecting appropriate moment functions our procedure is illustrated with an application to the nyse stock returns the empirical results reveal that the me approach with a fewer moment functions leads to a model that captures the stylized facts quite effectively

on the error estimate for the projection of a point onto a linear manifold abstract suppose that the linear system a h x b is consistent the problem of finding the solution x to min a h x b 1 2 x p 2 2 is studied where a c n m b c m x p c n and a b p are given when a b and p are perturbed an error bound for the solution x is obtained roundoff error bounds for the singular value decomposition method the column pivoting method and the rank revealing method are obtained the results in this paper extend those of arioli laratta and menchi to the case when a is not of full rank and either m n or m n

on the q 1 2 non extensive maximum entropy distribution abstract a detailed mathematical analysis on the q 1 2 non extensive maximum entropy distribution of tsallis is undertaken the analysis is based upon the splitting of such a distribution into two orthogonal components one of the components corresponds to the minimum norm solution of the problem posed by the fulfillment of the a priori conditions on the given expectation values the remaining component takes care of the normalization constraint and is the projection of a constant onto the null space of the expectation values transformation

existence conditions and spreading properties of extreme entropy d dimensional distributions abstract the extremization of the information theoretic measures fisher information shannon entropy tsallis entropy which complementary describe the spreading of the physical states of natural systems gives rise to fundamental equations of motion and or conservation laws at times the associated extreme entropy distributions are known for some given constraints usually moments or radial expectation values in this work first we give the existence conditions of the maxent probability distributions in a d dimensional scenario where two moments not necessarily of consecutive order are known then we find general relations which involve four elements the extremized entropy the other two information theoretic measures and the variance of the extremum density in scenarios with different dimensionalities and moment constraints

extremum entrop y based heisenberg like uncertainty relations in this work we use the extremization method of various information theoretic measures fisher information shannon entropy tsallis entropy for d dimensional quantum systems which complementarily describe the spreading of the quantum states of natural systems under some given constraints usually one or two radial expectation values this variational method allows us to determine an extremum entropy distribution which is the least biased one to characterize the state among all those compatible with the known data then we use it together with the spin dependent uncertainty like relations of daubechies thakkar type as a tool to obtain relationships between the position and momentum radial expectation values of the type for d dimensional systems of n fermions with spin s the resulting uncertainty like products which take into account both spatial and spin degrees of freedom of the fermionic constituents of the system are shown to often improve the best corresponding relationships existing in the literature

existence and uniqueness of fuzzy solution for semilinear fuzzy integrodifferential equations with nonlocal conditions abstract in this paper the existence and uniqueness of a fuzzy solution for the semilinear fuzzy integrodifferential equation is established via the banach fixed point analysis approach and using the fuzzy number whose values are normal convex upper semicontinuous and compactly supported interval in e n

entropy convergence in stieltjes and hamburger moment problem in the classical stieltjes and hamburger moment problem the sequence of maximum entropy approximants whose first m moments are equal to given ones is considered it is proved that whenever an infinite moment problem is determined then maximum entropy approximants converge in entropy to the function characterized by given moments entropy convergence is proved by using exclusively existence and uniqueness conditions

some new properties of the equality constrained and weighted least squares problem abstract in this paper some new properties of the equality constrained and weighted least squares problem wlse min w 1 2 kx g 2 subject to lx h are obtained we derive a perturbation bound based on an unconstrained least squares problem and deduce some equivalent formulae for the projectors of this unconstrained ls problem we also present a new way to compute the minimum norm solution x wlse of the wlse problem by using the qr decomposition of the corresponding matrices and propose an algorithm to compute x wlse using the qr factorizations some numerical examples are provided to compare different methods for solving the wlse problem

moment closure hierarchies for kinetic theories this paper presents a systematicnonperturbative derivation of a hierarchy of closed systems of moment equations corresponding to any classical kinetic theory the first member of the hierarchy is the euler system which is based on maxwellian velocity distributions while the second member is based on nonisotropic gaussian velocity distributions the closure proceeds in two steps the first ensures that every member of the hierarchy is hyperbolic has an entropy and formally recovers the euler limit the second involves modifying the collisional terms so that members of the hierarchy beyound the second also recover the correct navier stokes behavior this is achieved through the introduction of a generalization of the bgk collision operator the simplest such system in three spatial dimensions is a 14 moment closure which also recovers the behavior of the grad 13 moment system when the velocity distributions lie near local maxwellians the closure procedure can be applied to a general class of kinetic theories

maximum entropy in the problem of moments the maximum entropy approach to the solution of underdetermined inverse problems is studied in detail in the context of the classical moment problem in important special cases such as the hausdorff moment problem we establish necessary and sufficient conditions for the existence of a maximum entropy solution and examine the convergence of the resulting sequence of approximations a number of explicit illustrations are presented in addition to some elementary examples we analyze the maximum entropy reconstruction of the density of states in harmonic solids and of dynamic correlation functions in quantum spin systems we also briefly indicate possible applications to the lee yang theory of ising models to the summation of divergent series and so on the general conclusion is that maximum entropy provides a valuable approximation scheme a serious competitor of traditional pade like procedures

maxentropic solution of fractional moment problems the problem of reconstructing a function from the knowledge of a few of its moments is a special case of the problem inverting an integral transform when there are finitely many data points the classical method of maximum entropy and the method of maximum entropy in the mean are well suited to attack these problems what is interesting is that when we are free to choose the data points the choice can be made to improve the quality of the reconstruction when the method of maximum entropy is used too in this note we review how to apply these methods and compare them in a couple of examples

fuzzy differential equations abstract this paper deals with fuzzy set valued mappings of a real variable whose values are normal convex upper semicontinuous and compactly supported fuzzy sets in rn we study differentiability and integrability properties of such functions and give an existence and uniqueness theorem for a solution to a fuzzy differential equation

a new approach for the optimal fuzzy linear time invariant controlled system with fuzzy coefficients in this article a new technique for solving the linear time invariant dynamical system with fuzzy parameters is proposed this approach is used to solve a fuzzy controlled system by using a cuts and presentation of numbers in a more compact form by moving to the field of complex numbers by considering a new feature of the hamiltonian function and using the pontryagin maximum principle the solution of the fuzzy optimal controlled system is considered

moment closure approximations of the boltzmann equation based on varphi divergences this paper is concerned with approximations of the boltzmann equation based on the method of moments we propose a generalization of the setting of the moment closure problem from relative entropy to varphi divergences and a corresponding closure procedure based on minimization of varphi divergences the proposed description encapsulates as special cases grad s classical closure based on expansion in hermite polynomials and levermore s entropy based closure we establish that the generalization to divergence based closures enables the construction of extended thermodynamic theories that avoid essential limitations of the standard moment closure formulations such as inadmissibility of the approximate phase space distribution potential loss of hyperbolicity and singularity of flux functions at local equilibrium the divergence based closure leads to a hierarchy of tractable symmetric hyperbolic systems that retain the fundamental structural properties of the boltzmann equation

legendre transform structure derived from quantum theorems abstract by recourse to i the hellmann feynman theorem and ii the virial one the information optimizing principle based on fisher s information measure uncovers a legendre transform structure associated with schrodinger s equation in close analogy with the structure that lies behind the standard thermodynamical formalism the present developments provide new evidence for the information theoretical links based on fisher s measure that exist between schrodinger s equation on the one hand and thermodynamics thermostatistics on the other hand

entropic quadrature for moment approximations of the boltzmann bgk equation abstract using the method of moments to approximate the solution of kinetic equations has become a standard technique especially in the context of solving the boltzmann equation to model rarefied gases one attractive example is the maximum entropy closure which however is computationally barely affordable in this paper we will combine the maximum entropy approach with the quadrature method of moments qmom two methods which have in some sense diametric properties thereby introducing the entropic quadrature eq closure eq calculates a sparse quadrature based reconstruction of the unknown velocity distribution like the qmom where the physical meaningful maximum entropy principle is employed as a criterion for selecting a quadrature among different quadratures fitting the given moments we will discuss the construction of the closure and give several numerical examples of its performance in one and two dimensions

quantum moment hydrodynamics and the entropy principle this paper presents how a non commutative version of the entropy extremalization principle allows to construct new quantum hydrodynamic models our starting point is the moment method which consists in integrating the quantum liouville equation with respect to momentum p against a given vector of monomials of p like in the classical case the so obtained moment system is not closed inspired from levermore s procedure in the classical case 26 we propose to close the moment system by a quantum wigner distribution function which minimizes the entropy subject to the constraint that its moments are given in contrast to the classical case the quantum entropy is defined globally and not locally as the trace of an operator therefore the relation between the moments and the lagrange multipliers of the constrained entropy minimization problem becomes nonlocal and the resulting moment system involves nonlocal operators instead of purely local ones in the classical case in the present paper we discuss some practical aspects and consequences of this nonlocal feature

inferring an optimal fisher measure abstract it is well known that a suggestive relation exists that links the schrodinger equation se to the information optimizing principle based on the fisher information measure fim we explore here an approach that will allow one to infer the optimal fim compatible with a given amount of prior information without explicitly solving first the associated se this technique is based on the virial theorem and it provides analytic solutions for the physically relevant fim that which is minimal subject to the constraints posed by the prior information

maximisation of the entropy in non equilibrium this author proves that the maximisation of the entropy in non equilibrium is equivalent to the exploitation of the entropy inequality in extended thermodynamics

on discontinuous galerkin approximations of boltzmann moment systems with levermore closure abstract this work considers the discontinuous galerkin dg finite element discretization of first order systems of conservation laws derivable as moments of the kinetic boltzmann equation with levermore c d levermore moment closure hierarchies for kinetic theories j statist phys 83 5 6 1996 1021 1065 closure using standard energy analysis techniques a new class of energy stable numerical flux functions are devised for the dg discretization of boltzmann moment systems simplified energy stable numerical fluxes are then constructed which replace exact state space integration in the numerical flux with gauss lobatto quadrature numerical results for supersonic flow over a cylinder geometry in the continuum and transitional regimes using 5 and 10 moment approximations are presented using the newly devised dg discretizations

maximum entropy in the finite stieltjes and hamburger moment problem a necessary and sufficient condition for the existence of the maximum entropy me function defined in an infinite or semiinfinite interval is provided the conclusions reached show that except in a few particular cases the necessary and sufficient conditions for the existence of maximum entropy function are identical to the conditions for the solution of the moment problem when the first m 1 moments are assigned even if the conclusions reached are very similar to the hausdorff case the specificity of the hamburger and stieltjes cases demands a different handling a sufficient condition for the entropy convergence of the resulting sequence of maximum entropy estimators to the entropy of the recovering function is also provided

a hierarchy of diffusive higher order moment equations for semiconductors a hierarchy of diffusive partial differential equations is derived by a moment method and a chapman enskog expansion from the semiconductor boltzmann equation assuming dominant collisions the moment equations are closed by employing the entropy maximization principle of levermore the new hierarchy contains the well known drift diffusion model the energy transport equations and the six moments model of grasser et al it is shown that the diffusive models are of parabolic type two different formulations of the models are derived a drift diffusion formulation allowing for a numerical decoupling and a symmetric formulation in generalized dual entropy variables inspired by nonequilibrium thermodynamics an entropy inequality or h theorem follows from the latter formulation

the role of constraints within generalized nonextensive statistics the gibbs jaynes path for introducing statistical mechanics is based on the adoption of a specific entropic form sand of physically appropriate constraints for instance for the usual canonical ensemble one adopts i ii ipi 1 and iii ei eigenvalues of the hamiltonian u1 internal energy equilibrium consists in optimizing s1 with regard to pi in the presence of constraints ii and iii within the recently introduced nonextensive statistics i is generalized into sq k 1 ipiq q 1 q 1 reproduces s1 ii is maintained and iii is generalized in a manner which might involve piq in the present effort we analyze the consequences of some special choices for iii and their formal and practical implications for the various physical systems that have been studied in the literature to illustrate some mathematically relevant points we calculate the specific heat respectively associated with a nondegenerate two level system as well as with the classical and quantum harmonic oscillators