For a ring endomorphism $alpha$ and an $alpha$-derivation $delta$, we introduce a concept, so called skew $pi$-Armendariz ring, that is a generalization of both $pi$-Armendariz rings, and $(alpha,delta)$-compatible skew Armendariz rings. We first observe the basic properties of skew $pi$-Armendariz rings, and extend the class of skew $pi$-Armendariz rings through various ring extensions. We nex...

<p style='text-indent:20px;'>We investigate some asymptotic properties of trigonometric skew-product maps over irrational rotations the circle. The limits are controlled using renormalization. considered here arise in connection with self-dual Hofstadter Hamiltonian at energy zero. They analogous to almost Mathieu maps, but factors commute. This allows us construct periodic orbits under r...

A k-orbit map is a map with its automorphism group partitioning the set of flags into k orbits. Recently k-orbit maps were studied by Orbanić, Pellicer and Weiss, for k 6 4. In this paper we use symmetry type graphs to extend such study and classify all the types of 5-orbit maps, as well as all self-dual, properly and improperly, symmetry type of k-orbit maps with k 6 7. Moreover, we determine,...

Denote by Σn and Qn the set of all n × symmetric skew-symmetric matrices over a field $\mathbb {F}$ , respectively, where $\text {char}(\mathbb {F})\neq 2$ $|\mathbb {F}| \geq n^{2}+1$ . A characterization $\phi ,\psi :{\varSigma }_{n} \rightarrow {\varSigma }_{n}$ for which at least one them is surjective, satisfying $ \det (\phi (x)+\psi (y))=\det (x+y)\qquad (x,y\in }_{n}) given. Furthermore...

Let $S(G^{sigma})$ be the skew-adjacency matrix of the oriented graph $G^{sigma}$, which is obtained from a simple undirected graph $G$ by assigning an orientation $sigma$ to each of its edges. The skew energy of an oriented graph $G^{sigma}$ is defined as the sum of absolute values of all eigenvalues of $S(G^{sigma})$. Two oriented graphs are said to be skew equienergetic iftheir skew energies...