Let $R$ be a strongly $\mathbb {Z}^2$ -graded ring, and let $C$ bounded chain complex of finitely generated free -modules. The is $R_{(0,0)}$ -finitely dominated, or type $FP$ over , if it homotopy equivalent to projective We show that this happens only becomes acyclic after taking tensor product with certain eight rings formal power series, the graded analogues classical Novikov rings. This ex...

‎in this paper‎, ‎we investigate a damped korteweg-de‎ ‎vries equation with forcing on a periodic domain‎ ‎$mathbb{t}=mathbb{r}/(2pimathbb{z})$‎. ‎we can obtain that if the‎ ‎forcing is periodic with small amplitude‎, ‎then the solution becomes‎ ‎eventually time-periodic.

Let r, s, t be three positive integers and C be a binary linear code of lenght r + s + t. We say that C is a triple cyclic code of lenght (r, s, t) over Z 2 if the set of coordinates can be partitioned into three parts that any cyclic shift of the coordinates of the parts leaves invariant the code. These codes can be considered as Z 2 [x]-submodules of Z2[x] x r −1 × Z2[x] x s −1 × Z2[x] x t −1...

It is known that a Prüfer domain either with dimension 1 or with finite character has the stacked bases property. Following Brewer and Klinger, some rings of integer-valued polynomials provide, for every n ≥ 2, examples of n-dimensional Prüfer domains without finite character which have the stacked bases property. But, the following question is still open: does the two-dimensional Prüfer domain...

The random-cluster model has been widely studied as a unifying framework for random graphs, spin systems and electrical networks, but its dynamics have so far largely resisted analysis. In this paper we analyze the Glauber dynamics of the random-cluster model in the canonical case where the underlying graph is an n × n box in the Cartesian lattice Z. Our main result is a O(n log n) upper bound ...

a modular $k$-coloring, $kge 2,$ of a graph $g$ without isolated vertices is a coloring of the vertices of $g$ with the elements in $mathbb{z}_k$ having the property that for every two adjacent vertices of $g,$ the sums of the colors of the neighbors are different in $mathbb{z}_k.$ the minimum $k$ for which $g$ has a modular $k-$coloring is the modular chromatic number of $g.$ except for some s...