In this paper, Kolmogorov-Sinai entropy is studied using mathematical modeling of an observer $ Theta $. The relative entropy of a sub-$ sigma_Theta $-algebra having finite atoms is defined and then   the ergodic properties of relative  semi-dynamical systems are investigated.  Also,  a relative version of Kolmogorov-Sinai theorem  is given. Finally, it is proved  that the relative entropy of a...

Let A and B be two factors. In this paper, it is proved that a not necessarily linear bijective map ? : satisfies ?([A, B]* C) = [?(A), ?(B)]* ?(C) for all A, B,C if only *-isomorphism, conjugate the negative of or *-isomorphism.

We prove that if the squaring map in the factor loop of a Moufang loop Q over its nucleus is surjective, then every half-isomorphism of Q onto a Moufang loop is either an isomorphism or an anti-isomorphism. This generalizes all earlier results in this vein.

Almost isomorphism is an equivalence relation on countable state Markov shifts which provides a strong version of Borel conjugacy; still, for mixing SPR shifts, entropy is a complete invariant of almost isomorphism [2]. In this paper, we establish a class of potentials on countable state Markov shifts whose thermodynamic formalism is respected by almost isomorphism.

Whittlesey 12,13,14] gave a criterion which decides when two nite 2-dimensional complexes are homeomorphic. We show that graph isomorphism can be reduced eeciently to 2-complex homeomorphism, and that Whittlesey's criterion can be reduced eeciently to graph isomorphism. Therefore graph isomorphism and 2-complex homeomorphism are polynomial-time equivalent.

In this paper, the order, size and degree of the nodes of the isomorphic fuzzy graphs are discussed. Isomorphism between fuzzy graphs is proved to be an equivalence relation. Some properties of self complementary and self weak complementary fuzzy graphs are discussed. Keywords—complementary fuzzy graphs, co-weak isomorphism, equivalence relation, fuzzy relation, weak isomorphism.

The questions of isomorphism degree estimation of fuzzy graphs are discussed in this paper. The definition of independent fuzzy set of fuzzy graph is presented. The estimation method of isomorphism degree of fuzzy graphs is suggested and proved. The example of isomorphism degree estimation is considered.

We show how the formal Wirthmüller isomorphism theorem proven in [2] simplifies the proof of the Wirthmüller isomorphism in equivariant stable homotopy theory. Other examples from equivariant stable homotopy theory show that the hypotheses of the formal Wirthmüller and formal Grothendieck isomorphism theorems in [2] cannot be weakened.