The numerical semigroup of phrases' lengths in a simple alphabet

Let A be an alphabet with two elements. Considering a particular class of words (the phrases) over such an alphabet, we connect with the theory of numerical semigroups. We study the properties of the family of numerical semigroups which arise from this starting point.Both of the authors are supported by FQM-343 (Junta de Andalucía), MTM2010-15595 (MICINN, Spain), and FEDER funds. The second author is also partially supported by Junta de Andalucía/Feder Grant no. FQM-5849

Numerical semigroups in a problem about cost-effective transport

Let N be the set of nonnegative integers. A problem about how to transport profitably an organized group of persons leads us to study the set T formed by the integers n such that the system of inequalities, with nonnegative integer coefficients, a_1x_1+⋯+a_px_p<n<b_1x_1+⋯+b_px_p has at least one solution in N^p. We will see that T∪{0} is a numerical semigroup. Moreover, we will show that a numerical semigroup S can be obtained in this way if and only if {a+b−1,a+b+1}⊆S, for all a,b∈S∖{0}. In addition, we will demonstrate that such numerical semigroups form a Frobenius variety and we will study this variety. Finally, we show an algorithmic process in order to compute T.Both authors are supported by the project MTM2014-55367-P, which is funded by Ministerio de Economía y Competitividad and Fondo Europeo de Desarrollo Regional FEDER, and by the Junta de Andalucía Grant Number FQM-343. The second author is also partially supported by Junta de Andalucía/Feder Grant Number FQM-5849

Collected results on semigroups, graphs and codes

In this thesis we present a compendium of _ve works where discrete mathematics play a key role. The _rst three works describe di_erent developments and applications of the semigroup theory while the other two have more independent topics. First we present a result on semigroups and code e_ciency, where we introduce our results on the so-called Geil-Matsumoto bound and Lewittes' bound for algebraic geometry codes. Following that, we work on semigroup ideals and their relation with the Feng-Rao numbers; those numbers, in turn, are used to describe the Hamming weights which are used in a broad spectrum of applications, i.e. the wire-tap channel of type II or in the t-resilient functions used in cryptography. The third work presented describes the non-homogeneous patterns for semigroups, explains three di_erent scenarios where these patterns arise and gives some results on their admissibility. The last two works are not as related as the _rst three but still use discrete mathematics. One of them is a work on the applications of coding theory to _ngerprinting, where we give results on the traitor tracing problem and we bound the number of colluders in a colluder set trying to hack a _ngerprinting mark made with a Reed-Solomon code. And _nally in the last work we present our results on scientometrics and graphs, modeling the scienti_c community as a cocitation graph, where nodes represent authors and two nodes are connected if there is a paper citing both authors simultaneously. We use it to present three new indices to evaluate an author's impact in the community

Contagem de semigrupos numéricos pelo gênero e lacunas pares e generalizações. Patterns em semigrupos numéricos

Orientador: Fernando Eduardo Torres OrihuelaTese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação CientíficaResumo: Neste trabalho, apresentamos uma abordagem para o problema de contagem de semigrupos numéricos pelo gênero, usando o fato de que cada semigrupo numérico de gênero g possui uma quantidade de lacunas pares \gamma e o número n_g dos semigrupos de gênero g pode ser calculado como a soma dos números N_{\gamma}(g), que denota a quantidade de semigrupos numéricos de gênero g e \gamma lacunas pares. Um dos principais resultados do trabalho é o fato de N_{\gamma}(g) é constante para \gamma fixado e g \geq 3\gamma. De forma natural estudamos o comportamento da sequência N_{\gamma}(3\gamma). Motivados pela similaridade entre as sequências de Fibonacci e (n_g), estudamos o comportamento assintótico de sequências envolvendo os números n_g. Usando as ideias do Capítulo 2 deste trabalho, estudamos uma generalização natural de semigrupo \gamma-hiperelíptico. Ao final do trabalho, introduzimos o conceito de patterns e tentamos entender como eles podem ser aplicados a problemas envolvendo semigrupos numéricosAbstract: In this work, we present an approach to the problem of counting numerical semigroups by genus, using the fact that each numerical semigroup with genus g has a number of even gaps \gamma and the number n_g, that denotes the number of numerical semigroups of genus g, can be computed as a sum of the numbers N_{\gamma}(g), which denotes the number of numerical semigroups with genus g and \gamma even gaps. One of the results of this work is the fact that N_{\gamma}(g) is constant for a fixed \gamma and g \geq 3\gamma. Naturally, we study the behaviour of the sequence N_{\gamma}(3\gamma). Motivated by similarity between the Fibonacci and (n_g) sequences, we study the asymptotic behaviour of sequences involving the numbers n_g. By using some ideas of Chapter 2 of this work, we study a natural generalization of \gamma-hyperelliptic semigroup. At the end, we introduce the concept of patterns and we try to understand how they can be applied to the problems involving numerical semigroupsDoutoradoMatematicaDoutor em Matemática140292/2015CNPQCAPE