Makoto Tsukada

Mathematical structures derived from the q-product uniquely determined by tsallis entropy

For a unified description of power-law behaviors such as chaos, fractal and scale-free network, Tsallis entropy has been applied to the generalization of the traditional Boltzmann-Gibbs statistics as a fundamental information measure. Tsallis entropy Sq is an one-parameter generalization of Shannon entropy S1 in the sense that limqrarr1 Sq = Si. The generalized Boltzmann-Gibbs statistics by means of Tsallis entropy is nowadays called Tsallis statistics. The main approach in Tsallis statistics has been the maximum entropy principle, but there have been missing some fundamental mathematical formulae such as law of error, q-Stirlings formula and q-multinomial coefficient. Recently, we have succeeded in proving law of error in Tsallis statistics using the q-product uniquely determined by Tsallis entropy. Along the same lines as the proof, we present q-Stirlings formula, q-multinomial coefficient and a conjecture on the q-central limit theorem in Tsallis statistics

Tsallis differential entropy and divergences derived from the generalized Shannon-Khinchin axioms

In discrete systems, Shannon entropy is well known to be characterized by the Shannon-Khinchin axioms. Recently, this set of axioms was generalized for Tsallis entropy, one-parameter generalization of Shannon entropy. In continuos systems, Shannon differential entropy has been introduced as a natural extension of the above Shannon entropy without using an axiomatic approach. We derive the generalized entropy function as a solution of the functional equation determined by the generalized Shannon additivity, one of the most important axiom of the generalized Shannon-Khinchin axioms for Tsallis entropy. This generalized entropy function naturally introduces Tsallis differential entropy and two Tsallis divergences. In particular, one (Csiszár type) of the divergences has almost the same form as the α-divergence in information geometry and the other the Bregman type divergence. Our results reveal that the generalized Shannon additivity representing a branch structure of a rooted tree plays an essential role in the determination of these entropies.

Classification of three-dimensional zeropotent algebras over an algebraically closed field

ABSTRACT A nonassociative algebra is defined to be zeropotent if the square of any element is zero. Zeropotent algebras are exactly the same as anticommutative algebras when the characteristic of the ground field is not two. The class of zeropotent algebras properly contains that of Lie algebras. In this paper, we give a complete classification of three-dimensional zeropotent algebras over an algebraically closed field of characteristic not equal to two. By restricting the result to the subclass of Lie algebras, we can obtain a classification of three-dimensional complex Lie algebras, which is in accordance with the conventional one.

Ulam type stability problems for alternative homomorphisms

We introduce an alternative homomorphism with respect to binary operations and investigate the Ulam type stability problem for such a mapping. The obtained results apply to Ulam type stability problems for several important functional equations.MSC: Primary 39B82; secondary 47H10.

Classification of three-dimensional zeropotent algebras over the real number field

ABSTRACT A nonassociative algebra is defined to be zeropotent if the square of any element is zero. In this paper, we give a complete classification of three-dimensional zeropotent algebras over the real number field up to isomorphism. By restricting the result to the subclass of Lie algebras, we can obtain a classification of three-dimensional real Lie algebras, which is in accordance with the Bianchi classification. Moreover, three-dimensional zeropotent algebras over a real closed field are classified in the same manner as those over the real number field.

A complete classification of continuous fractional operations on \(\mathbb {C}\)

We completely classify continuous fractional operations on the complex number field

Superstability of Generalized Multiplicative Functionals

\mathbb {C}

Classification of Continuous Fractional Binary Operations on the Real and Complex Fields

C modulo equivalence. A continuous fraction is described by a pair of complex numbers. We prove that a continuous fraction is completely characterized by the (conjugate) ratio of two numbers describing the fraction. Furthermore, we show that the set of all the equivalence classes of continuous fractions is equipped with a natural topology and it is homeomorphic to the unit disk