Generalized algebra within a nonextensive statistics

Abstract

By considering generalized logarithm and exponential functions used in nonextensive statistics, the four usual algebraic operators : addition, subtraction, product and division, are generalized. The properties of the generalized operators are investigated. Some standard properties are preserved, e.g., associativity, commutativity and existence of neutral elements. On the contrary, the distributivity law and the opposite element is no more universal within the generalized algebra.Comment: 11 pages, no figure, TeX. Reports on Mathematical Physics (2003), in pres

Multiplicative duality, q-triplet and (μ,ν,q)-relation derived from the one-to-one correspondence between the (μ

“…For this purpose, the new multiplication operation ⊗ q is introduced in [11] [12]. The concrete forms of the q-logarithm and q-exponential are given in (11) and (12), so that the above requirement (13) or (14) as the q-exponential law leads to the definition of ⊗ q between two positive numbers.…”

“…By means of the qproduct uniquely determined by the q-exponential function [11][12] as the q-exponential law, the one-to-one correspondence between the q-multinomial coefficient and Tsallis entropy is obtained as follows [8]: for n = k i=1 n i and n i ∈ N if q = 2,…”

Group Invariance of Information Geometry on q-Gaussian Distributions Induced by Beta-Divergence

“…In statistics, it is reported to provide a reasonable statistical model in robust inference from data losing normality [1,[12][13][14]. In addition, quite interesting and abundant mathematical structures have been developed [15][16][17][18][19] for the q-exponential function itself.…”

Generalised cascades

“…One of the most exciting applications which have emerged within the non-extensive scope is the definition of a whole new set of mathematical operations/functions that goes from the generalised algebra independently defined by Borges [3] and Nivanen et al [4] and the integro-differential operators by Borges to the q-trigonometric functions [5]. Besides its inherent beauty, these generalisations have found its own field of applicability.…”