Weighted Relative Group Entropies and Associated Fisher Metrics

A large family of new α-weighted group entropy functionals is defined and associated Fisher-like metrics are considered. All these notions are well-suited semi-Riemannian tools for the geometrization of entropy-related statistical models, where they may act as sensitive controlling invariants. The main result of the paper establishes a link between such a metric and a canonical one. A sufficient condition is found, in order that the two metrics be conformal (or homothetic). In particular, we recover a recent result, established for α=1 and for non-weighted relative group entropies. Our conformality condition is “universal”, in the sense that it does not depend on the group exponential.

Identification of Structure Ordering of Melt-Spun Fe70Cr15B15 Alloy by the Entropy Functionals

New techniques for the analysis of complex images were advanced by the example of asurface microrelief of the Fe70Cr15B15 alloys, obtained by the melt-spinning at different linearvelocity of a quenching roller. Proposed techniques are based on the fast Fourier transform of theimage using digital signal processors. The degree of morphological ordering/disordering of ribbonswas determined due to the parameterization of mode spectra by entropy functionals. The best glassforming ability for alloys of the studied composition was obtained at 30 m/s.

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.