Integral transforms and probability distributions involving a generalized hypergeometric function

Various extensions of the beta function together with their associated extended hypergeometric and confluent hypergeometric functions have been introduced and investigated. In this paper, using the very recently contrived extended beta function, we aim to introduce an extension F v p , q ; λ ; σ , τ u {{}_{u}F_{v}^{p,q;\lambda;\sigma,\tau}} of the generalized hypergeometric function F v u {{}_{u}F_{v}} and investigate certain classes of transforms and several identities of a generalized probability distribution involving this extension. In fact, we present some interesting formulas of Jacobi, Gegenbauer, pathway, Laplace, and Legendre transforms of this extension multiplied by a polynomial. We also introduce a generalized probability distribution to investigate its several related properties. Further, we consider some special cases of our main results with an argument about the derived process of a known result.

Model reduction by mean-field homogenization in viscoelastic composites. I. Primal theory

A homogenization scheme for viscoelastic composites proposed by Lahellec & Suquet (2007 Int. J. Solids Struct. 44 , 507–529 ( doi:10.1016/j.ijsolstr.2006.04.038 )) is revisited. The scheme relies upon an incremental variational formulation providing the inelastic strain field at a given time step in terms of the inelastic strain field from the previous time step, along with a judicious use of Legendre transforms to approximate the relevant functional by an alternative functional depending on the inelastic strain fields only through their first and second moments over each constituent phase. As a result, the approximation generates a reduced description of the microscopic state of the composite in terms of a finite set of internal variables that incorporates information on the intraphase fluctuations of the inelastic strain and that can be evaluated by mean-field homogenization techniques. In this work we provide an alternative derivation of the scheme, relying on the Cauchy–Schwarz inequality rather than the Legendre transform, and in so doing we expose the mathematical structure of the resulting approximation and generalize the exposition to fully anisotropic material systems.

Thermodynamic Potentials

Thermodynamics specifies the relation between an independent, predictor variable, and what is predicted. It is often the case that changing the variables regarded as independent can greatly simplify problem solving. The chapter shows how using an intensive variable (like temperature or pressure) as the predictor loses information that can be retained if it is expressed by a different function. It shows the importance of Legendre transforms, which contain the same information about the system as is available by using extensive variables. Legendre transforms exploiting the fundamental relation are shown to yield the Helmholtz free energy, the enthalpy, and the Gibbs free energy. Massieu functions are introduced as an alternative that is particularly important for models exhibiting negative temperatures.

An Introduction to Statistical Mechanics and Thermodynamics

This is a textbook on statistical mechanics and thermodynamics. It begins with the molecular nature of matter and the fact that we want to describe systems containing many (1020) particles. The first part of the book derives the entropy of the classical ideal gas using only classical statistical mechanics and Boltzmann’s analysis of multiple systems. The properties of this entropy are then expressed as postulates of thermodynamics in the second part of the book. From these postulates, the structure of thermodynamics is developed. Special features are systematic methods for deriving thermodynamic identities using Jacobians, the use of Legendre transforms as a basis for thermodynamic potentials, the introduction of Massieu functions to investigate negative temperatures, and an analysis of the consequences of the Nernst postulate. The third part of the book introduces the canonical and grand canonical ensembles, which are shown to facilitate calculations for many models. An explanation of irreversible phenomena that is consistent with time-reversal invariance in a closed system is presented. The fourth part of the book is devoted to quantum statistical mechanics, including black-body radiation, the harmonic solid, Bose–Einstein and Fermi–Dirac statistics, and an introduction to band theory, including metals, insulators, and semiconductors. The final chapter gives a brief introduction to the theory of phase transitions. Throughout the book, there is a strong emphasis on computational methods to make abstract concepts more concrete.

Some trace inequalities for exponential and logarithmic functions

Consider a function [Formula: see text] of pairs of positive matrices with values in the positive matrices such that whenever [Formula: see text] and [Formula: see text] commute [Formula: see text] Our first main result gives conditions on [Formula: see text] such that [Formula: see text] for all [Formula: see text] such that [Formula: see text]. (Note that [Formula: see text] is absent from the right side of the inequality.) We give several examples of functions [Formula: see text] to which the theorem applies. Our theorem allows us to give simple proofs of the well-known logarithmic inequalities of Hiai and Petz and several new generalizations of them which involve three variables [Formula: see text] instead of just [Formula: see text] alone. The investigation of these logarithmic inequalities is closely connected with three quantum relative entropy functionals: The standard Umegaki quantum relative entropy [Formula: see text], and two others, the Donald relative entropy [Formula: see text], and the Belavkin–Stasewski relative entropy [Formula: see text]. They are known to satisfy [Formula: see text]. We prove that the Donald relative entropy provides the sharp upper bound, independent of [Formula: see text] on [Formula: see text] in a number of cases in which [Formula: see text] is homogeneous of degree [Formula: see text] in [Formula: see text] and [Formula: see text] in [Formula: see text]. We also investigate the Legendre transforms in [Formula: see text] of [Formula: see text] and [Formula: see text], and show how our results for these lead to new refinements of the Golden–Thompson inequality.

Directional Thermodynamic Formalism

The usual thermodynamic formalism is uniform in all directions and, therefore, it is not adapted to study multi-dimensional functions with various directional behaviors. It is based on a scaling function characterized in terms of isotropic Sobolev or Besov-type norms. The purpose of the present paper was twofold. Firstly, we proved wavelet criteria for a natural extended directional scaling function expressed in terms of directional Sobolev or Besov spaces. Secondly, we performed the directional multifractal formalism, i.e., we computed or estimated directional Hölder spectra, either directly or via some Legendre transforms on either directional scaling function or anisotropic scaling functions. We obtained general upper bounds for directional Hölder spectra. We also showed optimal results for two large classes of examples of deterministic and random anisotropic self-similar tools for possible modeling turbulence (or cascades) and textures in images: Sierpinski cascade functions and fractional Brownian sheets.

Legendre Transforms

This chapter introduces vector calculus to the reader from the very basics to a level appropriate for studying classical mechanics. However, it provides only the necessary vector calculus required to understand some of the operations perform in the text and perhaps support self-learning in more advanced topics, so the analysis is not be definitive. The chapter begins by examining the axioms of vector algebra, vector multiplication and vector differentiation, and then tackles the gradient, divergence and curl and other elements of vector integration. Topics discussed include contour integrals, the continuity equation, the Kronecker delta and the Levi-Civita symbol. Particular care is taken to explain every mathematical relation used in the main text, leaving no stone unturned!