The classical Poincaré weak recurrence theorem states that for any probability space (Ω,S, P ), any P -measure preserving transformation T , and any A ∈ S, almost all points of A return to A. In the present paper the Poincaré theorem is proved when the σ-algebra S is replaced by any σ-complete MV-algebra. Keywords— Measure preserving transformation, Poincaré recurrence theorem, σ-complete MV-al...

We shall start by giving the definition of the entropy of dynamical system. Consider dynamical systems with discrete time. The phase space of dynamical system is denoted by M . It is equipped with σ-algebra M and a probability measure μ defined on M. In the general ergodic theory dynamics is given by a measurable transformation T of M onto itself preserving the measure μ. It is enough for many ...

Let R be an n-dimensional Euclidean space and D be an n-dimensional hyperbolic space with the Poincaré metric for n > 1. In this paper, we shall prove the following results. (i) A bijection f : D → D n is an isometry (Möbius transformation) if and only if f is triangle preserving. (ii) A bijection f : R → R is an affine transformation if and only if f is triangle preserving.

This paper presents a general approach to automatically validate code-improving transformations on low-level program representations. The approach ensures the correctness of compiler and hand-specified optimizations at the machine instruction level. The method verifies the semantic equivalence of the program representation before and after a transformation to determine the validity of the trans...

This paper studies the creation of schematic maps from traditional vector-based, carto-graphic information. An algorithm is proposed to modify positions of lines in the original input map with the goal of producing as output a schematic map that meets certain geometric and aesthetic criteria. Special emphasis is placed here on preserving topological structure of features during this transformat...

Cryptographic solutions to privacy-preserving multiparty linear programming are slow. This makes them unsuitable for many economically important applications, such as supply chain optimization, whose size exceeds their practically feasible input range. In this paper we present a privacy-preserving transformation that allows secure outsourcing of the linear program computation in an efficient ma...

In this paper, a conflict problem caused by applying traditional operational transformation algorithms on Replace operation is firstly described. Then a new conception of operation group is introduced into the collaborative editing area. T-Group is an important type of operation group, when a Replace operation can be considered as a T-Group. To preserve T-Groups in operational transformation ne...