Related papers: Products of Factorials Modulo p

Properties of partial integrals such as real and complex-valued polynomial, multiple polynomial, exponential, and conditional for ordinary differential systems are studied. The possibilities of constructing first integrals and last…

The concept of a composed product for univariate polynomials has been explored extensively by Brawley, Brown, Carlitz, Gao, Mills, et al. Starting with these fundamental ideas and utilizing fractional power series representation (in…

In this paper, we establish continuous bilinear decompositions that arise in the study of products between elements in martingale Hardy spaces $ H^p\ (0<p\leqslant 1) $ and functions in their dual spaces. Our decompositions are based on…

Let $p$ be a prime. We prove that if a modular unit has a $p^{th}$ root that is again a modular unit then the level of that root is at most $p$ times the level of the original unit.

This note deals with the computation of the factorization number $F_2(G)$ of a finite group $G$. By using the M\"{o}bius inversion formula, explicit expressions of $F_2(G)$ are obtained for two classes of finite abelian groups, improving…

Part I. Some Facts From p-Adic Analysis. Part II. Tables of Integrals.

An \emph{indexing} of a finite set $S$ is a bijection $D : \{1,...,|S|\} \rightarrow S$. We present an indexing for the set of quadratic residues modulo $N$ that is decodable in polynomial time on the size of $N$, given the factorization of…

Let $p$ be a prime number. In this paper, we estimate the variation of the sizes of quotients of certain finitely generated $p$-torsion Iwasawa modules, which are closely related to class numbers. We also construct some…

We investigate the group of normalized units of the group algebra $\mathbb{Z}_{p^e}G$ of a finite abelian $p$-group $G$ over the ring $\mathbb{Z}_{p^e}$ of residues modulo $p^e$ with $e\geq 1$.

Let H be a finite-dimensional Hopf algebra. We give a description of the tensor product of bimodule categories over Rep(H). When the bimodule categories are invertible this description can be given explicitly. We present some consequences…

The paper proposes a polynomial formula for solution quadratic congruences in $\mathbb{Z}_p$. This formula gives the correct answer for quadratic residue and zeroes for quadratic nonresidue. The general form of the formula for $p=3…

A series of physically motivated operations appearing in the study of composite materials are interpreted in terms of elementary continued fraction transforms of matrix valued, rational Stieltjes functions.

The Mellin transform of quartic products of shifted Airy functions is evaluated in a closed form. Some particular cases expressed in terms of the logarithm function and complete elliptic integrals special values are presented.

We investigate the continued fraction expansion of the infinite products $g(x) = x^{-1}\prod_{t=0}^\infty P(x^{-d^t})$ where polynomials $P(x)$ satisfy $P(0)=1$ and $\deg(P)<d$. We construct relations between partial quotients of $g(x)$…

We consider polynomials with integer coefficients and discuss their factorization properties in Z[[x]], the ring of formal power series over Z. We treat polynomials of arbitrary degree and give sufficient conditions for their reducibility…

Given n quaternions we investigate the extent of non-commutativity of their multiple products, commutators and exponential products.

Let k be a perfect field of characteristic p>0. When p>2, Fontaine and Laffaille have classified p-divisibles groups and finite flat p-groups over the Witt vectors W(k) in terms of filtered modules. Still assuming p>2, we extend these…

The number of maximal abelian subgroups of a finite p-group is shown to be congruent to 1 modulo p.

In this paper we study the relative tensor product of module categories over braided fusion categories using, in part, the notion of the relative center of a module category. In particular we investigate the canonical tensor category…

We present a deterministic polynomial-time algorithm that determines whether a finite module over a finite commutative ring is cyclic, and if it is, outputs a generator.