Numerical semigroups: Apéry sets and Hilbert series

Let a ₁,…,a _n be relatively prime positive integers, and let S be the semigroup consisting of all non-negative integer linear combinations of a ₁,…,a _n . In this paper, we focus our attention on AA-semigroups, that is semigroups being generated by almost arithmetic progressions. After some general considerations, we give a characterization of the symmetric AA-semigroups. We also present an efficient method to determine an Apéry set and the Hilbert series of an AA-semigroup. Dedicated to the memory of Ernst S. Selmer (1920–2006), whose calculations revealed the “Selmer group”.     

Quadratic transformations and Guillera’s formulas for 1/π2

We prove two new series of Ramanujan type for 1/π².     

Type and conductor of simplicial affine semigroups

We provide a generalization of pseudo-Frobenius numbers of numerical semigroups to the context of the simplicial affine semigroups. In this way, we characterize the Cohen-Macaulay type of the simplicial affine semigroup ring $\mathbb{K}[S]$. We define the type of S, $\mathrm{type}(S)$, in terms of some Apéry sets of S and show that it coincides with the Cohen-Macaulay type of the semigroup ring, when $\mathbb{K}[S]$ is Cohen-Macaulay. If $\mathbb{K}[S]$ is a d-dimensional Cohen-Macaulay ring of embedding dimension at most $d+2$, then $\mathrm{type}(S)\le 2$. Otherwise, $\mathrm{type}(S)$ might be arbitrary large and it has no upper bound in terms of the embedding dimension. Finally, we present a generating set for the conductor of S as an ideal of its normalization.     

Factorizations in numerical semigroup algebras

We study a numerical semigroup ring as an algebra over another numerical semigroup ring. The complete intersection property of numerical semigroup algebras is investigated using factorizations of monomials into minimal ones. The goal is to study whether a flat rectangular algebra is a complete intersection. Along this direction, special types of algebras generated by few monomials are worked out in detail.     

Rhin integrals

We study a generalization of the integrals examined by G. Rhin, in the form of multiple integrals. These integrals yield rational approximations to the values of the Riemann zeta function. In a particular case, we obtain Apéry approximations used to prove the irrationality of the number ζ(3).     

广义差集与几乎完美序列

提出了广义差集的概念,并且给出了广义差集的一些初等性质.从应用的角度讲,广义差集就是使得其±1特征序列的自相关函数是(最多)三值的一种组合结构.因此,广义差集不仅仅是在概念(理论)上的推广,它还具有深层次的应用背景.事实上,给出了一些广义差集,它不是可分差集,也不是相对差集.同时也给出了一类广义差集存在的一些必要条件,使得这些广义差集对应的±1特征序列成为几乎完美序列.并举例说明本文中的方法是有效的.     

Transformational antiatomic characteristic of ordered sets

To any ordered set with a universally maximal element, a semigroup of its transformations with some natural properties that defines the ordered set up to an isomorphism is assigned. The system of such transformation semigroups is proved to be the minimal element in the set of all defining systems of transformation semigroups with respect to the following ordering: one system precedes another if for each ordered set from the class in question, the semigroup of its transformation belonging to the first system is contained in the semigroup of its transformation from the second system. Translated fromMatematicheskie Zametki, Vol. 66, No. 1, pp. 112–119, July, 1999.     

Numerical Semigroups with a Monotonic Apery Set

We study numerical semigroups S with the property that if m is the multiplicity of S and w(i) is the least element of S congruent with i modulo m, then 0 < w(1) < ... < w(m − 1). The set of numerical semigroups with this property and fixed multiplicity is bijective with an affine semigroup and consequently it can be described by a finite set of parameters. Invariants like the gender, type, embedding dimension and Frobenius number are computed for several families of this kind of numerical semigroups. This paper was supported by the project BFM2000-1469. The fourth author wishes to acknowledge support from the Universidade de Evora and the CIMA-UE.     

0-Minimal Quasi-ideals of Generalized Linear Transformation Semigroups

Abstract

For vector spaces V and W over a field F, L _F (V, W) denotes the set of all linear transformations α : V → W, and for a cardinal number k > 0, let L _F (V, W, k) be the set of all α ∈ L _F (V, W) of rank less than k. For θ ∈ L _F (W, V), let (L _F (V, W, k), θ) denote the semigroup L _F (V, W, k) under the operation ? defined by α ? β = αθβ for all α, β ∈ L _F (V, W, k). In this paper, all 0-minimal quasi-ideals of the semigroup (L _F (V, W, k), θ) are completely characterized. It is also shown from this characterization that every nonzero semigroup (L _F (V, W, k), θ) always has a 0-minimal quasi-ideal.     

广义算子半群的遍历及概率逼近定理

首先给出了广义算子半群的Abel-遍历和Cesaro-遍历的定义,对两种遍历的性质进行了刻画,研究了两种遍历的等价条件.其次,利用Pettis积分、算子值数学期望及广义连续修正模等工具给出广义算子半群的概率逼近表达式.     

A short proof of a theorem on degree sets of graphs

For a finite simple graph G, we denote the set of degrees of its vertices, known as its degree set, by D(G). Kapoor, Polimeni and Wall [Degree sets for graphs, Fund. Math. 95 (1977) 189-194] have determined the least number of vertices among graphs with a given degree set. We give a very short proof of this result.     

Semigroups associated to generalized polynomials and some classical formulas

We study operator semigroups associated with a family of generalized orthogonal polynomials with Hermitian matrix entries. For this we consider a Markov generator sequence, and therefore a Markov semigroup, for the family of orthogonal polynomials on related to the generalized polynomials. We give an expression of the infinitesimal generator of this semigroup and under the hypothesis of diffusion we prove that this semigroup is also Markov. We also give expressions for the kernel of this semigroup in terms of the one-dimensional kernels and obtain some classical formulas for the generalized orthogonal polynomials from the correspondent formulas for orthogonal polynomials on .     

Proportionally modular diophantine inequalities

We study the sets of nonnegative solutions of Diophantine inequalities of the form with a,b and c positive integers. These sets are numerical semigroups, which we study and characterize.