The ‘dollar game’ represents a kind of diffusion process on a graph. Under the rules of the game some configurations are both stable and recurrent, and these are known as critical configurations. The set of critical configurations can be given the structure of an abelian group, and it turns out that the order of the group is the tree-number of the graph. Each critical configuration can be assig...

we find an explicit expression of the tutte polynomial of an $n$-fan. we also find a formula of the tutte polynomial of an $n$-wheel in terms of the tutte polynomial of $n$-fans. finally, we give an alternative expression of the tutte polynomial of an $n$-wheel and then prove the explicit formula for the tutte polynomial of an $n$-wheel.

For a positive real α, we can consider the additive submonoid M of line that is generated by nonnegative powers α. When α transcendental, unique factorization monoid. However, when algebraic, may not be atomic, and even it contain elements having more than one (i.e., decomposition as sum irreducibles). The main purpose this paper to study phenomenon multiple factorizations inside M. algebraic b...

In this paper we show some properties of the Newton polygon of a polynomial ordinary differential equation. We give the relation between the Newton polygons of a differential polynomial and its partial derivatives. Newton polygons of evaluations of differential polynomials are also described.

Two essential problems in computer algebra, namely polynomial factorization and greatest common divisor computation, can be efficiently solved thanks to multiple evaluations two variables using modular arithmetic. In this article, we focus on the efficient computation of such one single CPU core. We first show how leverage SIMD (single instruction, data) computing for arithmetic AVX2 AVX-512 un...