A characterization of some families of Cohen--Macaulay, Gorenstein and/or Buchsbaum rings

We provide algorithmic methods to check the Cohen--Macaulayness, Buchsbaumness and/or Gorensteiness of some families of semigroup rings that are constructed from the dilation of bounded convex polyhedrons of $\R^3_{\geq}$. Some families of semigroup rings are given satifying these properties

Union of Sets of Lengths of Numerical Semigroups

Let S = be a numerical semigroup, let s is an element of S and let Z(s) be its set of factorizations. The set of lengths is denoted by L(s) = {L(x(1), ... , x(p)) vertical bar (x(1), ... , x(p)) is an element of Z(s)}, where L(x(1), ... , x(p)) = x(1) + ... + x(p). The following sets can then be defined: W(n) = {s is an element of S vertical bar there exists x is an element of Z(s) such that L(x) = n}, nu(n) = boolean OR(s is an element of W(n)) L(s) = {l(1) P(N) is almost periodic with period lcm(a(1), a(p))

An extension of Wilf's conjecture to affine semigroups

Producción CientíficaLet C ⊂ Qp be a rational cone. An affine semigroup S ⊂ C is a C-semigroup whenever (C \ S) ∩ Np has only a finite number of elements. In this work, we study the tree of C-semigroups, give a method to generate it and study their subsemigroups with minimal embedding dimension. We extend Wilf’s conjecture for numerical semigroups to C- semigroups and give some families of C-semigroups fulfilling the extended conjecture. We also check that other conjectures on numerical semigroups seem to be also satisfied by C-semigroups

Combining microsatellites, growth, and adaptive traits for managing in situ genetic resources of Eucalyptus urophylla

International audienceGenetic diversity and structure of 17 populations representative of the natural range of Eucalyptus urophylla S.T. Blake in Indonesia were analysed with 10 microsatellite markers. Two provenance and progeny trials, using the same populations, were established in the Republic of the Congo and analysed for growth and survival at 37months. Observed microsatellite heterozygosity (Ho) was moderate to high within populations (Ho = 0.51-0.72). The index of fixation (FIS) was significantly different from zero for all populations (FIS = 0.13-0.31) and possibly resulted from a Wahlund effect. The differentiation parameter was low (FST = 0.04) and not significantly different from zero, which can be explained by effective gene flow via pollen. The genetic variances within and among provenances for survival and growth traits were significantly different from zero, representing 13%-23% and 14%-50% of the total variation, respectively. The differentiation between populations based on quantitative traits was marked (QST = 0.07-0.33). A negative relationship between altitude of the seed source and its performance in the Congo was observed (R2 = 0.59-0.67) and could be explained by the effect of natural selection along the altitudinal gradient. This marked differentiation for quantitative traits despite high apparent gene flow results in a clinal variation, which suggests the use of altitude of seed source as an important principle for in situ or ex situ management of E. urophylla genetic resources. À l'aide de 10 marqueurs microsatellites, les auteurs ont analysé la diversité et la structure génétiques de 17 populations représentatives de l'aire naturelle d'Eucalyptus urophylla S.T. Blake en Indonésie. La croissance et la survie après 37 mois ont été analysées dans deux essais établis en République du Congo et contenant les mêmes provenances et descendances. L'hétérozygotie observée (Ho) chez les marqueurs microsatellites variait de modérée à élevée au sein des populations (Ho = 0,51-0,72). L'indice FIS était significativement différent de zéro pour toutes les populations (FIS = 0,13-0,31) et était possiblement le résultat d'un effet de Wahlund. La différenciation de population était faible (FST = 0,04) et n'était pas significativement différente de zéro, ce qui peut s'expliquer par un flux génique efficace attribuable au pollen. Les variances génétiques au sein et parmi les provenances pour les caractères de survie et de croissance étaient significativement différentes de zéro, représentant respectivement 13% à 23% et 14% à 50% de la variation totale. La différenciation de population estimée à partir des caractères quantitatifs était élevée, avec une valeur de QST = 0,07-0,33. Une relation négative entre l'altitude des sources de semences et leur performance au Congo a été remarquée (R2 = 0,59-0,67). Cette relation pourrait s'expliquer par l'effet de la sélection naturelle le long du gradient altitudinal. Cette différenciation marquée chez les caractères quantitatifs, en dépit d'un flux génique apparent élevé, se manifeste par un patron de variation clinale, ce qui indique que l'altitude des sources de semences est un critère important qui devrait être utilisé pour la gestion in situ et ex situ des ressources génétiques d'E. urophyll

An Elementary Algorithm for Reporting Intersections of Red/Blue Curve Segments

Let E_r and E_b be two sets of x-monotone and non-intersecting curve segments, E=E_r \cup E_b and |E|=n. We give a new sweep-line algorithm that reports the $k$ intersecting pairs of segments of E. Our algorithm uses only three simple predicates that allow to decide if two segments intersect, if a point is left or right to another point, and if a point is above, below or on a segment. These three predicates seem to be the simplest predicates that lead to subquadratic algorithms. Our algorithm is almost optimal in this restricted model of computation. Its time complexity is $O((n+k)\logn) and it requires O(n) space. The same algoritm has been described in our previous report [5]. That report presented also an algoritm for the general case but its analysis was not not correct

Elementary Algorithms for Reporting Intersections of Curve Segments

We propose several algorithms to report the k intersecting pairs among a set of n curve segments. Apart from the intersection predicate, our algorithms only use two simple predicates : the predicate that compares the coordinates of two points and the predicate that says if a point is below, on, or above a segment. In particular, the predicates we use do not allow to count the number of intersection points nor to sort them, and the time complexity of our algorithms depends on the number of intersectin- g pairs, not on the number of intersection points (differently from the other non trivial algorithms). We present an algorithm for the red-blue variant of the problem where we have a set of blue segments and a set of red segments so that no two segments of the same set intersect. The time complexity is O((n+k)\log n). This algorithm is then used to solve the general case in O(n\sqrtk\log n) time. In the case of pseudo-segments (i.e. segments that intersect in at most one point) we propose a better algorithm whose time complexity is O((k+n)\log n+ n\sqrt k). All our time complexity results are a log factor from optimal

On reducible non-Weierstrass semigroups

Weierstrass semigroups are well known along the literature. We present a new family of non- Weierstrass semigroups which can be written as an intersection of Weierstrass semigroups. In addition, we provide methods for computing non-Weierstrass semigroups with genus as large as desired.Funding information: Part of this paper was written during a visit of Fernando Torres to the Universidad de Cadiz (Spain) ; his visit was partially supported by Ayudas para Estancias Cortas de Investigadores (EST2018-R0, Programa de Fomento e Impulso de la Investigacion y la Transferencia en la Universidad de Cadiz) . Fernando Torres was partially supported by CNPq/Brazil (Grant 310623/2017-0) . Juan Ignacio Garcia-Garcia, Daniel Marin-Aragon, and Alberto Vigneron-Tenorio were partially supported by Junta de Andalucia research groups FQM-343 and FQM-366, and by the project MTM2017-84890-P (MINECO/FEDER, UE)

The Buchweitz Set of a Numerical Semigroup

Let A subset of Z be a finite subset. We denote by B(A) the set of all integers n >= 2 such that |nA|>(2n-1)(|A|-1), where nA=A+middotmiddotmiddot+A denotes the n-fold sumset of A. The motivation to consider B(A)stems from Buchweitz's discovery in 1980 that if a numerical semigroup S subset of N is a Weierstrass semigroup, then B(N\S)= empty set . By constructing instances where this condition fails, Buchweitz disproved a longstanding conjecture by Hurwitz (Math Ann 41:403-442, 1893). In this paper, we prove that for any numerical semigroup S subset of N of genus g >= 2, the set B(N\S) is finite, of unbounded cardinality as S varies