Narrow Results

The present paper studies continuity of generalized entropy functions and relative entropies defined using the notion of a deformed logarithmic function. In particular, two distinct definitions of relative entropy are discussed. As an application, all considered entropies are shown to satisfy Lesche's stability condition. The entropies of Tsallis' non-extensive thermostatistics are taken as examples.

The effective mass m_eff of the Pauli–Fierz Hamiltonian with ultraviolet cutoff Λ and the bare mass m in non-relativistic quantum electrodynamics (QED) with spin ½ is investigated. Analytic properties of m_eff in coupling constant e are shown. Let us set . The explicit form of constant a₂(Λ/m) depending on Λ/m is given. It is shown that the spin interaction enhances the effective mass and that there exist strictly positive constants c₁ and c₂ such that

Let $A$ be a reduced ring containing $\mathbb{Q}$ and let ${\xi }_1,{\xi }_2$ be commuting locally nilpotent derivations of $A$. In this paper, we give an algorithm to decide the local nilpotency of derivations of the form $\xi =a_1{\xi }_1+a_2{\xi }_2$, where $a_1,a_2$ are elements in $A$.

A method of constructing (finitely generated and projective) right module structure on a finitely generated projective left module over an algebra is presented. This leads to a construction of a first-order differential calculus on such a module which admits a hom-connection or a divergence. Properties of integrals associated to this divergence are studied, in particular the formula of integration by parts is derived. Specific examples include inner calculi on a non-commutative algebra, the Berezin integral on the supercircle and integrals on Hopf algebras.

In this paper, flutter and divergence instabilities of functionally graded porous plate strip reinforced with graphene nanoplatelets in supersonic flow and subjected to an axial loading are studied. The graphene nanoplatelets are distributed in the matrix either uniformly or non-uniformly along the thickness direction. Four graphene nanoplatelets distribution patterns namely, Patterns A through D are considered. Based on the modified Halpin–Tsai micromechanics model and the rule of mixture, the effective material properties of functionally graded plate strip reinforced with graphene nanoplatelets are obtained. The aerodynamic pressure is considered in accordance with the quasi-steady supersonic piston theory. To transform the governing equations of motion to a general eigenvalue problem, the Galerkin method is employed. The flutter aerodynamic pressure and stability boundaries are determined by solving standard complex eigenvalue problem. The effects of graphene nanoplatelets distributions, graphene nanoplatelets weight fraction, geometry of graphene nanoplatelets, porosity coefficient and porosity distributions on the flutter and divergence instabilities of the system are studied. The results show that the plate strip with symmetric distribution pattern (stiffness in the surface areas) and GPLs pattern A predict the highest stable area. The flutter and divergence regions decrease as the porosity coefficient increases. Besides, the critical aerodynamic loads increase by adding a small amount of GPL to the matrix.

In this paper we provide a framework for the study of isoperimetric problems in finitely generated groups, through a combinatorial study of universal covers of compact simplicial complexes. We show that, when estimating filling functions, one can restrict to simplicial spheres of particular shapes, called “round” and “unfolded”, provided that a bounded quasi-geodesic combing exists. We prove that the problem of estimating higher dimensional divergence as well can be restricted to round spheres. Applications of these results include a combinatorial analogy of the Federer–Fleming inequality for finitely generated groups.

Geometry for q-exponential families is studied in this paper. A q-exponential family is a set of probability distributions, which is a natural generalization of the standard exponential family. A q-exponential family has information geometric structure and a dually flat structure. To describe these relations, generalized conformal structures for statistical manifolds are studied in this paper. As an application of geometry for q-exponential families, a geometric generalization of statistical inference is also studied.

We give a survey on geometry of statistical manifolds in terms of estimating functions. A statistical model naturally has a statistical manifold structure. In particular, a q-exponential family which is a generalization of an exponential family admits several statistical manifold structures. An estimating function can be regarded as a tangent vector of a statistical model, and it gives rise to dualistic structures on a statistical manifold. In this paper, we construct statistical manifold structures on statistical models and divergence functions from the viewpoint of estimating functions. We also study geometry of nonintegrable estimating functions.