Perfect numerical semigroups with embedding dimension three

A numerical semigroup is perfect if it has no isolated gaps. In this paper, we will characterize the perfect numerical semigroups with embedding dimension three, and we show how to obtain them all. Also, we obtain formulas for each of the genus and the pseudo-Frobenius numbers of these semigroups

Perfect numerical semigroups

A numerical semigroup is perfect if it does not have isolated gaps. In this paper we will order the perfect numerical semigroups with a fixed multiplicity. This ordering allows us to give an algorithm procedure to obtain them. We also study the perfect monoid, which is a subset of N that can be expressed as an intersection of perfect numerical semigroups, and we present the perfect monoid generated by a subset of N. We give an algorithm to calculate it. We study the perfect closure of a numerical semigroup, as well as the perfect numerical semigroup with maximal embedding dimension, in particular Arf and saturated numerical semigroups

Parity numerical semigroups

A numerical semigroup is parity (P-semigroup) if the sum of two consecutive elements of S is odd. In this paper we will order the P-semigroups creating a tree with root. This ordering allows us to give some algorithms to build all the P-semigroups with a fixed genus and Frobenius number. Also, we will study the maximal P-semigroups with a given Frobenius number as well as the P-semigroups with maximum embedding dimension

Numerical semigroups bounded by the translation of a plane monoid

Let N be the set of nonnegative integer numbers. A plane monoid is a submonoid of (N-2, +). Let M be a plane monoid and p, q is an element of N. We will say that an integer number n is M(p, q)- bounded if there is (a, b) is an element of M such that a + p <= n <= b - q. We will denote by A(M(p, q)) = {n is an element of N | n is M(p, q)-bounded}. An A( p, q)-semigroup is a numerical semigroup S such that S = A(M(p, q)) boolean OR {0} for some plane monoid M. In this work we will study these kinds of numerical semigroups.M. A. Moreno-Frias: Partially supported by MTM2017-84890-P and by Junta de Andalucia group FQM-298. J. C. Rosales: Partially supported by MTM2017-84890-P and by Junta de Andalucia group FQM-343

Proportionally modular affine semigroups

Producción CientíficaThis work introduces a new kind of semigroup of Np called proportionally modular affine semigroup. These semigroups are defined by modular Diophantine inequalities and they are a generalization of proportionally modular numerical semigroups. We give an algorithm to compute their minimal generating sets. We also specialize on the case p = 2. For this case, we provide a faster algorithm to compute their minimal system of generators, prove they are Cohen-Macaulay and Buchsbaum, and determinate their (minimal) Frobenius vectors. Besides, Gorenstein proportionally modular affine semigroups are characterized.Ministerio de Economía, Industria y Competitividad ( grants MTM2014-55367-P / MTM2015-65764- C3-1-P)Junta de Andalucía (FQM-366 / FQM-298

Semigroups with fixed multiplicity and embedding dimension

Given m is an element of N, a numerical semigroup with multiplicity m is called a packed numerical semigroup if its minimal generating set is included in {m, m + 1 , ..., 2m - 1}. In this work, packed numerical semigroups are used to build the set of numerical semigroups with a given multiplicity and embedding dimension, and to create a partition of this set. Wilf's conjecture is verified in the tree associated to some packed numerical semigroups. Furthermore, given two positive integers m and e, some algorithms for computing the minimal Frobenius number and minimal genus of the set of numerical semigroups with multiplicity m and embedding dimension e are provided. We also compute the semigroups where these minimal values are achieved

The Clinical Impact of Neoadjuvant Endocrine Treatment on Luminal-like Breast Cancers and Its Prognostic Significance: Results from a Single-Institution Prospective Cohort Study

Purpose: Neoadjuvant endocrine treatment (NET) has become a useful tool for the down-staging of luminal-like breast cancers in postmenopausal patients. It enables us to increase breast-conserving surgery (BCS) rates, provides an opportunity for us to assess in vivo NET effectiveness, and allows us to study any biological changes that may act as valid biomarkers. The purpose of this study was to evaluate the safety and effectiveness of NET, and to assess the role of Ki67 proliferation rate changes as an indicator of endocrine responsiveness. Methods: From 2016 to 2020, a single-institution cohort of patients, treated with NET and further surgery, was evaluated. In patients with Ki67 ≥ 10%, a second core biopsy was performed after four weeks. Information regarding histopathological and clinical changes was gathered. Results: A total of 115 estrogen receptor-positive (ER+)/HER2-negative patients were included. The median treatment duration was 5.0 months (IQR: 2.0–6.0). The median maximum size in the surgical sample was 40% smaller than the pretreatment size measured by ultrasound (p < 0.0001). The median pretreatment Ki67 expression was 20.0% (IQR: 12.0–30.0), and was reduced to 5.0% (IQR: 1.8–10.0) after four weeks, and to 2.0% (IQR: 1.0–8.0) in the surgical sample (p < 0.0001). BCS was performed on 98 patients (85.2%). No pathological complete responses were recorded. A larger Ki67 fold change after four weeks was significantly related to a PEPI score of zero (p < 0.002). No differences were observed between luminal A-and B-like tumors, with regard to fold change and PEPI score. Conclusions: In our cohort, NET was proven to be effective for tumor size and Ki67 downstaging. This resulted in a higher rate of conservative surgery, aided in therapeutic decision making, provided prognostic information, and constituted a safe and well-tolerated approachThis research received no external fundin

Numerical semigroups bounded by a cyclic monoid

We will say that a numerical semigroup S is bounded by a cyclic monoid if there exist integer numbers 0 ≤ α < β such that S = {x ∈ ℕ | kα < x < kβ for some k ∈ ℕ}∪{0}. The goal of this work is to study this kind of numerical semigroups. In particular, we will determine important invariants of them such as multiplicity, embedding dimension, Frobenius number and genus.Partially supported by MTM2017-84890-PJunta de Andalucía group FQM-298Partially supported by MTM2017-84890-PJunta de AndalucÍa group FQM-34

La formación de profesores : aportaciones para la formación inicial de profesores de Matemáticas de Secundaria

En este trabajo se expone la situación de la asignatura Didáctica de la Matemática en la Educación Secundaria del plan de estudios vigente en la Universidad de Almería para la Licenciatura en Matemáticas. Dado que esta asignatura junto con las otras dos optativas que ofrece el área de conocimiento de este plan, constituye una primera oportunidad de contribuir a la formación inicial del profesor de Matemáticas de Secundaria, se aportan algunas consideraciones sobre formación de profesores que permiten ubicar objetivos y actuaciones dentro de un marco más general.Castilla y LeónES