Implicit operations on DS

In this work we illustrate how the study of the topological semigroups of implicit operations of a pseudovariety V can be useful to the knowledge of V. In particular, for subpseudovarieties of DS, we find factorizations of the implicit operations in terms of regular and explicit ("words'') ones. For a better knowledge of V, we try to solve some kind of "word problem'' on V i.e., decide when two implicit operations are the same in V. Such study has already been done for some pseudovarieties, namely J, R and L. As an application of this study we calculate some joins of pseudovarieties

Characteristic functions and averages

Let $\Omega$ be a set and $\Omega_1,\ldots,\Omega_{m-1}$ subsets of $\Omega$, being $m$ an integer greater than one. For a given function $f=(f_1,\ldots f_m):\Omega\rightarrow\mathbb{R}^m$, we prove the existence of a unique function $\alpha=(\alpha_1,\ldots,\alpha_m):\Omega\rightarrow\mathbb{R}^m$ such that \begin{eqnarray*} \left\{ \begin{array}{lcl} \alpha_i & = & \alpha_{i+1} \quad\mbox{on $\Omega_i$}\\ \alpha_1+\cdots+\alpha_i & = & f_1+\cdots+f_i\quad\mbox{on $\Omega\setminus\Omega_i$, for all $i< m$}\\ \alpha_1+\cdots+\alpha_m & = & f_1+\cdots+f_m, \end{array} \right. \end{eqnarray*} called the average function of $f:\Omega\rightarrow\mathbb{R}^m$ relatively to $\left(\Omega,\Omega_1,\ldots,\Omega_{m-1}\right)$. When $\Omega$ is a topological space and $f$ is a continuous function, we find necessary and sufficient conditions for the continuity of the average function of $f$. We write $\alpha_i$ as a linear combination of characteristic functions of the (coincidence) sets $\cap_{j=r}^s\Omega_j$, $1\leq r\leq s\leq m-1$, belonging the coefficients to $\mathbb{Q}[f_1,\ldots,f_m]$.Fundação para a Ciência e a Tecnologia (FCT

Asymptotics for generating functions of the Fuss-Catalan numbers

We consider a certain class of polynomials with coefficients in Z_M, all of which admit a unique zero. We prove that the zero of each of those can be given by a (multiple) sum involving the coefficients and a vectorial generalization of the Fuss-Catalan numbers. We also consider the sequence of the partial sums of the generating function of the d -Fuss- Catalan numbers. Using the holonomy of this sequence, we study its asymptotic behaviour. The main difference from the known case d = 2 is, in that one, we have a “closed” expression for the generating function.The research of the authors were partially financed by Portuguese Funds through Fundação para a Ciência e a Tecnologia within the Projects UIDB/00013/2020 and UIDP/00013/2020

A two obstacles coupled problem

We consider a system of an evolutionary variational inequality of two obstacles type, depending on the temperature, coupled with the heat equation. We prove existence of solution of this system and we present examples that motivated this work. In particular, with additional assumptions on the data, we prove that solutions of this problem are also solutions of a similar problem where the convex set is of gradient constraint type (that depends on the temperature), improving a previous result.This research was partially supported by CMAT - Centro de Matemática da Universidade do Minho, financed by the Strategic Project PEst-OE/MAT/UI0013/2014

A diffusion problem with gradient constraint depending on the temperature

We consider a system of a variational inequality with gradient constraint depending on the temperature, coupled with the heat equation. We prove existence of solution of this system by approximating it by a system of equations and using a fixed point argument.Fundação para a Ciência e a Tecnologia (FCT) - Pluriannual Funding Program, projecto UT-Austin/MAT/0035/2008.Centro de Matemática da Universidade do Minh

Convergence of convex sets with gradient constraint

Given a bounded open subset of R^N, we study the convergence of a sequence (K_n)_{n\in\N} of closed convex subsets of W_0^{1,p}(\Omega) (p\in]1,\infty[) with gradient constraint, to a convex set K, in the Mosco sense. A particular case of the problem studied is when K_n={v\in W_0^{1,p}(\Omega):: F_n(x,\nabla v(x))<= g_n(x) for a.e. x in \Omega}. Some examples of non-convergence are presented. We also present an improvement of a result of existence of a solution of a quasivariational inequality, as an application of this Mosco convergence result.Fundação para a Ciência e a Tecnologia (FCT) - Programa Operacional "Ciência, Tecnologia, Inovação (POCTI), União Europeia (UE). Fundo Europeu de Desenvolvimento Regional - (Portugal/FEDER-EU)

Three examples of join computations

This article answers three questions of J. Almeida. Using combinatorial, algebraic and topological methods, we compute joins involving the pseudovariety of finite groups, the pseudovariety of semigroups in which each idempotent is a right zero and the pseudovariety generated by monoids M such that each idempotent of M\{1} is a left zero.ESPRIT - BRA Working Group 6317 Asmics-2.Project de Recherche Coordonnée "Mathématique et Informatique".Junta Nacional de Investigação Científica e Tecnológica (JNICT) - Projecto SAL (PBIC/C/CEN/1021/92)

Multiplicador de Lagrange num problema com restrição não constante no gradiente

Prova-se a existência de solução, num sentido generalizado, de um problema com um multiplicador de Lagrange, para uma restrição arbitrária no gradiente e condição de Dirichlet homogénea na fronteira. Prova-se ainda a equivalência deste problema com a correspondente inequação variacional elíptica. A abordagem utilizada para provar o resultado de existência baseia-se na utilização de soluções de uma família aproximante de equações quasilineares elípticas

Um problema de grandes denominadores

Fixado $M\in \N$, escolhamos aleatoriamente $a_1\in \N$ e consideremos $M_1=\frac{M}{(M,a_1)}$. Repita-se este procedimento, seleccionando ao acaso $a_2$ e definindo $M_2=\frac{M_1}{(M_1,a_2)}$, e assim sucessivamente. Dados $M, n\in\N$, qual é a probabilidade, digamos $\mathcal{P}(n,M)$, de ser $M_n=1$? Tem-se $\mathcal{P}(1,M)=\frac{1}{M}$ e a relação de recorrência $\mathcal{P}(n+1,M)=\sum_{d|M}\frac{\varphi(d)}{M} \mathcal{P}(n,d)$, onde $\varphi$ é a função de Euler. O que implica que, com probabilidade um, $M_n=1$ para algum $n\in \N$. O sistema dinâmico discreto associado à aplicação \cchi:\Q\to \Q dada por \cchi(x)= x\ceil{x} simula o comportamento deste processo aleatório, tratando-se agora de saber se a órbita por \cchi de qualquer racional de $[1, +\infty[$ entra em $\Z$. Em \cite{A}, provou-se que o conjunto das fracções irredutíveis com denominador $M$ cujas órbitas entram em $\Z$ no $n$-ésimo iterado é uma união disjunta de classes de congruência módulo $M^{n+1}$. Este resultado sugeriu um algoritmo eficiente para decidir se um racional está nesta união e, do número daquelas classes, deduzimos que, com probabilidade $1$, a órbita de um racional de $[1, +\infty[$ entra em $\Z$.Given M 2 N, suppose one randomly chooses a1 2 N, sets M1 = M (M,a1) , then repeats the process by randomly sorting out a2 and letting M2 = M1 (M1,a2) , and so on. Given M, n 2 N, what is the probability, say P(n,M), that Mn = 1? Clearly P(1,M) = 1 M and the numbers P(n,M) satisfy the recurrence relation P(n+1,M) = P d|M '(d) M P(n, d), where ' is the Euler function. This implies that, with probability one, Mn = 1 for some n. The map : Q ! Q given by (x) = xdxe induces a deterministic dynamical system modeling this random behavior. The question we address now is whether the orbit by of any rational bigger than 1 enters Z. In [1] we proved that the set of irreducible fractions with denominator M whose orbits by reach an integer in exactly n iterations is a disjoint union of congruence classes modulo Mn+1. The proof of this result suggested how to build an efficient algorithm to decide if an orbit fails to hit an integer before a prescribed number of iterations have elapsed. Besides, from the number of those classes, we deduced that, with probability 1, the orbit of a rational in [1,+1[ enters Z. palavras-chave: sistemas dinâmicos discretos; função tecto; densidade; cobertur

43 Miniaturas Matemáticas

[Excerto]A divulgação da matemática é uma tarefa importante, diria mesmo, uma obrigação de todos quantos gostam de matemática e a ensinam a um nível superior. Mostrar, pelo menos a algum público, que a matemática não é aquele edifício monstruoso que lhe foi dado a conhecer quando frequentava os bancos da escola, apresentando-a vista de um prisma lúdico e desmontando, com textos simples, as barreiras que se colocam à sua compreensão, é um serviço louvável que se presta a esse público e à matemática. Esta parece ter sido a corajosa intenção dos autores