The purpose of this study is to define the concepts of Tsallis entropy and conditional Tsallis entropy of fuzzy partitions and to obtain some results concerning this kind entropy. We show that the Tsallis entropy of fuzzy partitions has the subadditivity and concavity properties. We study this information measure under the refinement and zero mode subset relations. We check the chain rules for ...

Mathematics is about ”interesting structures”. What make a structure interesting is an abundance of interesting problems; we study a structure by solving these problems. The worlds of science, as well as of mathematics itself, is abundant with gems (germs?) of simple beautiful ideas. When and how may such an idea direct you toward beautiful mathematics? I try to present in this talk a few sugge...

A 1993 result of J. Llibre, and M. Misiurewicz, (Theorem A [5]), states that if a continuous map f of a graph into itself has an s-horseshoe, then the topological entropy of f is greater than or equal to logs, that is h( f ) ? logs. Also a 1980 result of L.S. Block, J. Guckenheimer, M. Misiurewicz and L.S. Young (Lemma 1.5 [3]) states that if G is an A-graph of f then h(G) ? h( f ). In this pap...

The notion of entropy was first introduced in the 1850‘s by Rudolf Clausius in the context of thermodynamics. Focusing on the meaning that an increase of entropy means an increase of disorder, dissipation and decay, subsequent generations of authors have imported the notion into practically every area of intellectual discussion. This paper will mention numerous definitions of entropy and consid...

Many people are familiar with the geometrical shape called the circle. Based on this figure, the circle space S connects the endpoints of the interval [0, 1] together so that 0 ≡ 1 (mod 1). On this space, the expanding map f : S → S stretches an initial distribution of points along the circle and then rewraps the lengthened distribution tightly about the circle; overlapping regions are compress...

Extremal black holes in M-theory compactification on M × S 1 are microscopically represented by fivebranes wrapping P × S 1 , where M is a Calabi-Yau threefold and P is a four-cycle in M. Additional spacetime charges arise from momentum around the S 1 and expectation values for the self-dual three-form field strength in the fivebrane. The microscopic entropy of the fivebrane as a function of al...

The string-black hole correspondence is considered in the context of the correspondence principle proposed recently by Horowitz and Polchinski [7]. We demonstrate that the entropy of string states and the entropy of a Schwarzschild black hole can be matched including the subleading terms which depend on mass logarithmically. We argue the necessity to include the string interaction (with couplin...

Privacy ampliication allows two parties Alice and Bob knowing a partially secret string S to extract, by communication over a public channel, a shorter, highly secret string S 0. Bennett, Brassard, Cr epeau, and Maurer showed that the length of S 0 can be almost equal to the conditional R enyi entropy of S given an opponent Eve's knowledge. All previous results on privacy ampliication assumed t...