We describe the local structure of an irreducible algebraic monoid M at an idempotent element e. When e is minimal, we show that M is an induced variety over the kernel MeM (a homogeneous space) with fibre the two-sided stabilizer Me (a connected affine monoid having a zero element and a dense unit group). This yields the irreducibility of stabilizers and centralizers of idempotents when M is n...

In this paper, we consider the long time behaviour of collisionless kinetic equation with stochastic diffuse boundary operators for velocities bounded away from zero. We show that under suitable reasonable conditions, semigroup is eventually compact. particular, without any irreducibility assumption, converges exponentially to spectral projection associated zero eigenvalue as $$t \rightarrow \i...

Let Fq be a Galois field with q = p elements of characteristic p, and Fq is multiplicative group of Fq. The problem of irreducibility of polynomials and determining the reducibility of a given polynomial stems both from mathematical theory and applications. At mathematical aspects determining the reducibility of a polynomial often appears in number theory, combinatorics and algebraic geometries...

The program of Langlands is studied here on the basis of: • new concepts of global class field theory related to the explicit construction of global class fields and of reciprocity laws; • the representations of the reductive algebraic groups GL(n) constituting the n-dimensional representations of the associated global Weil-Deligne groups; • a toroidal compactification of the conjugacy classes ...

We study applications of discrete valuations to ideals in analytically irreducible domains, in particular applications to zero divisors modulo powers of ideals. We prove a uniform version of Izumi's theorem and calculate several examples illustrating it, such as for rational singularities. The paper contains a new criterion of analytic irreducibility, a new criterion of one-beredness, and a val...

For readers already familiar with Partial Information Decomposition (PID), we show that PID’s definition of synergy enables quantifying at least four different notions of irreducibility. First, we show four common notions of “parts” give rise to a spectrum of four distinct measures of irreducibility. Second, we introduce a nonnegative expression based on PID for each notion of irreducibility. T...

The purpose of this paper is to give an algorithm for deciding the irreducibility of reduced algebroid curves over an algebraically closed field of arbitrary characteristic. To do this, we introduce a new notion of local tropical variety which is a straightforward extension of tropism introduced by Maurer, and then give irreducibility criterion for algebroid curves in terms local tropical varie...

‎We consider Albeverio's linear representations of the braid groups $B_3$ and $B_4$‎. ‎We specialize the indeterminates used in defining these representations to non zero complex numbers‎. ‎We then consider the tensor products of the representations of $B_3$ and the tensor products of those of $B_4$‎. ‎We then determine necessary and sufficient conditions that guarantee the irreducibility of th...

We present a number of classical proofs of the irreducibility of the n-th cyclotomic polynomial Φn(x). For n prime we present proofs due to Gauss (1801), in both the original and a simplified form, Kronecker (1845), and Schönemann/Eisenstein (1846/1850), and for general n proofs due to Dedekind (1857), Landau (1929), and Schur (1929). Let Φn(x) denote the n-th cyclotomic polynomial, defined by ...

Reaction networks are commonly used to model the dynamics of populations subject to transformations that follow an imposed stoichiometry. This paper focuses on the efficient characterisation of dynamical properties of Discrete Reaction Networks (DRNs). DRNs can be seen as modeling the underlying discrete nondeterministic transitions of stochastic models of reaction networks. In that sense, a pr...