let $g$ be a connected graph of order $3$ or more and $c:e(g)rightarrowmathbb{z}_k$‎ ‎($kge 2$) a $k$-edge coloring of $g$ where adjacent edges may be colored the same‎. ‎the color sum $s(v)$ of a vertex $v$ of $g$ is the sum in $mathbb{z}_k$ of the colors of the edges incident with $v.$ the $k$-edge coloring $c$ is a modular $k$-edge coloring of $g$ if $s(u)ne s(v)$ in $mathbb{z}_k$ for all pa...

Let $K$ be an algebraically closed field of characteristic zero and let $G$ a finitely generated subgroup the multiplicative group $K$. We consider $K$-valued sequences form $a_n:=f(\varphi^n(x_0))$, where $\varphi\colon X\to X$ $f\colon X\to\mathbb{P}^1$ are rational maps defined over $x_0\in is point whose forward orbit avoids indeterminacy loci $\varphi$ $f$. Many classical from number theor...

We explore an elementary construction that produces finitely presented groups with diverse homological finiteness properties – the binary subgroups, $B(\Sigma,\mu)\<G\_1\times\dots\times G\_m$. These full subdirect products require strikingly few generators. If each $G\_i$ is presented, $B(\Sigma,\mu)$ presented. When are non-abelian limit (e.g. free or surface groups), provide new examples of ...

Let $G$ be a $t$-uniform hypergraph, and let $c(G)$ denote the cyclic index of adjacency tensor $G$. $m,s,t$ positive integers such that $t \ge 2$, $s 2$ $m=st$. The generalized power $G^{m,s}$ is obtained from by blowing up each vertex into an $s$-set preserving relation. It was conjectured $c(G^{m,s})=s \cdot c(G)$. In this paper we show conjecture false giving counterexample, give some suffi...

If $\psi\colon M^n\to \mathbb{R}^{n+1}$ is a smooth immersed closed hypersurface, we consider the functional $$ \mathcal{F}\_m(\psi) = \int\_M 1 + |\nabla^m \nu |^2 , d\mu, where $\nu$ local unit normal vector along $\psi$, $\nabla$ Levi-Civita connection of Riemannian manifold $(M,g)$, with $g$ pull-back metric induced by immersion and $\mu$ associated volume measure. We prove that if $m>\lflo...