Narrow Results

Let $D=(V,A)$ be a simple strongly connected digraph and let Aut$(D)$ be an automorphism of $V(D)$. For $x\in V(D)$, the set $\{x^g:g\in \mathrm{\text{Aut}}(D)\}$ is called an orbit of Aut$(D)$. In this paper, first, we show that if $D$ is a 2-regular strongly connected digraph with two orbits then $\lambda (D)=2$, if $D$ is a $k$-regular strongly connected digraph with two orbits and $\lambda (G)<k$$(k\ge 3)$, then $\lambda (G)\ge 2$. Second, we prove that if $g(D)\ge 3$, then $\lambda (D)=3$. Last, we characterize the arc atoms of 3-regular strongly connected digraphs with two orbits and $\lambda (D)=2$.

Let M be a closed n(≥2)-dimensional smooth Riemannian manifold and let X be a vector field on M. In this paper, we show that the robust chain transitive set is hyperbolic if and only if there are a C¹-neighborhood of X and a compact neighborhood U of the chain transitive set such that for any , the index of the continuation on Λ_Y(U) = ⋂_t∈ℝY^t(U) of every critical point does not change.

Let $X$ be a compact metric space, and let $f:X\to X$ be a homeomorphism. We show that if $f$ has the limit shadowing property then $f$ is chaotic in the sense of Li–Yorke. Moreover, $f$ is dense Li–Yorke chaos.

Functional equations are equations in which the unknown (or unknowns) are functions. We consider equations of generalized associativity, mediality (bisymmetry, entropy), paramediality, transitivity as well as the generalized Kolmogoroff equation. Their usefulness was proved in applications both in mathematics and in other disciplines, particularly in economics and social sciences (see J. Aczél, On mean values, Bull. Amer. Math. Soc.54 (1948) 392–400; J. Aczél, Remarques algebriques sur la solution donner par M. Frechet a l’equation de Kolmogoroff, Pupl. Math. Debrecen4 (1955) 33–42; J. Aczél, A Short Course on Functional Equations Based Upon Recent Applications to the Social and Behavioral Sciences, Theory and decision library, Series B: Mathematical and statistical methods (D. Reidel Publishing Company, Dordrecht, Boston, Lancaster, Tokyo, 1987); J. Aczél, Lectures on Functional Equations and Their Applications (Supplemented by the author, ed. H. Oser) (Dover Publications, Mineola, New York, 2006); J. Aczél, V. D. Belousov and M. Hosszu, Generalized associativity and bisymmetry on quasigroups, Acta Math. Acad. Sci. Hungar.11 (1960) 127–136; J. Aczél and J. Dhombres, Functional Equations in Several Variables (Cambridge University Press, New York, 1991); J. Aczél and T. L. Saaty, Procedures for synthesizing ratio judgements, J. Math. Psych.27(1) (1983) 93–102). We use unifying approach to solve these equations for division and regular operations generalizing the classical quasigroup case.

A finite group is called a CN – group if every subgroup of it is c–normal. A finite group G is called a CT – group if c–normality is transitive in G. In this paper, we determine the structure of CN – groups and CT-groups and prove that solvable CT – groups are exactly the same as CN – groups.