In this thesis, we introduce the notion of quadratic forms and provide motivation for their study. We begin by discussing Diophantine equations, the field of p-adic numbers, and the Hasse-Minkowski Theorem that allows us to use p-adic analysis determine whether a quadratic form has a rational root. We then discuss search bounds and state Cassels' Theorem for small-height zeros of rational quadratic forms. We end with a proof of Cassels' Theorem and suggestions for further reading.

no / Inverse limits and profinite groups are used in a quantum mechanical context. Two cases are considered: a quantum system with positions in the profinite group Z(p) and momenta in the group Q(p)/Z(p), and a quantum system with positions in the profinite group (Z) over cap and momenta in the group Q/Z. The corresponding Schwatz-Bruhat spaces of wavefunctions and the Heisenberg-Weyl groups are discussed. The sets of subsystems of these systems are studied from the point of view of partial order theory. It is shown that they are directed-complete partial orders. It is also shown that they are topological spaces with T-0-topologies, and this is used to define continuity of various physical quantities. The physical meaning of profinite groups, non-Archimedean metrics, partial orders and T-0-topologies, in a quantum mechanical context, is discussed.

Submitted by Cláudia Bueno (claudiamoura18@gmail.com) on 2016-03-10T17:35:32Z No. of bitstreams: 2 Dissertação - Adriana Marcela Fonce Camacho - 2014.pdf: 981401 bytes, checksum: a14522ebe9ae77cf599946d25752f8b4 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2016-03-14T14:08:40Z (GMT) No. of bitstreams: 2 Dissertação - Adriana Marcela Fonce Camacho - 2014.pdf: 981401 bytes, checksum: a14522ebe9ae77cf599946d25752f8b4 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Made available in DSpace on 2016-03-14T14:08:40Z (GMT). No. of bitstreams: 2 Dissertação - Adriana Marcela Fonce Camacho - 2014.pdf: 981401 bytes, checksum: a14522ebe9ae77cf599946d25752f8b4 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Previous issue date: 2014-08-22 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / This work is based mainly on the Brunder and Godinho article [2] which shows proof of the conjecture of Artin methods using p-adic, although the conjecture is stated on the real numbers which makes the proof is show an equivalence on the field of the number p-adic method with the help of colored variables ya contraction of variables so as to prove the statement, taking the first level and ensuring a nontrivial solution in the following levels. / Este trabalho é baseado principalmente no artigo de Brunder e Godinho [2] o qual mostra a prova da conjetura de Artin usando métodos p-ádicos, ainda que a conjetura se afirma sobre o números reais o que faz a prova é mostrar uma equivalência sobre o corpo dos número p-ádicos com ajuda do método de variáveis coloridas e a contração de variáveis para assim provar a afirmação, tomando o primeiro nível e assim garantindo uma solução não trivial nos níveis seguintes.

We study bounds on the distances of roots of integer polynomials and applications of such results. The separation of complex roots for reducible monic integer polynomials of fourth degree is thoroughly explained. Lemmas on roots of polynomials in the p-adic setting are proved. Explicit families of polynomials of general degree as well as families in some classes of quadratic and cubic polynomials with very good separation of roots in the same setting are exhibited. The second part of the thesis is concerned with results on p-adic versions of Mahler's and Koksma's functions wn and w*n and the related classifications of transcendental numbers in Cp. The main result is a construction of numbers such that the two functions wn and w*n differ on them for every n and later on expanding the interval of possible values for wn-w*n. The inequalities linking values of Koksma's functions for algebraically dependent numbers are proved.

In this thesis we study discrete dynamical system, given by a polynomial, over both finite extension of the fields of p-adic numbers and over finite fields. Especially in the p-adic case, we study fixed points of dynamical systems, and which elements that are attracted to them. We show with different examples how complex these dynamics are. For certain polynomial dynamical systems over finite fields we prove that the normalized average of the numbers of linear factors modulo prime numbers exists. We also show how to calculate the average, by using Chebotarev's Density Theorem. The non-normalized version of the average of the number of linear factors of linearized polynomials modulo prime numbers, tends to infinity, so in that case we find an asymptotic formula instead. We have also used a computer to study different behaviors, such as iterations of polynomials over the p-adic fields and the asymptotic relation mention above. In the last chapter we present the computer programs used in different part of the thesis.

No description available.

This thesis concerns decision procedures for fragments of linear arithmetic and their application to model-checking one-counter automata. The first part of this thesis covers the complexity of decision problems for different types of linear arithmetic, namely the existential subset of the first-order linear theory over the p-adic numbers and the existential subset of Presburger arithmetic with divisibility, with all integer constants and coefficients represented in binary. The most important result of this part is a new upper complexity bound of <b>NEXPTIME</b> for existential Presburger arithmetic with divisibility. The best bound that was known previously was <b>2NEXPTIME</b>, which followed directly from the original proof of decidability of this theory by Lipshitz in 1976. Lipshitz also gave a proof of <b>NP</b>-hardness of the problem in 1981. Our result is the first improvement of the bound since this original description of a decision procedure. Another new result, which is both an important building block in establishing our new upper complexity bound for existential Presburger arithmetic with divisibility and an interesting result in its own right, is that the decision problem for the existential linear theory of the p-adic numbers is in the Counting Hierarchy <b>CH</b>, and thus within <b>PSPACE</b>. The precise complexity of this problem was stated as open by Weispfenning in 1988, who showed that it is in <b>EXPTIME</b> and <b>NP</b>-hard. The second part of this thesis covers two problems concerning one-counter automata. The first problem is an LTL synthesis problem on one-counter automata with integer-valued and parameterised updates and with equality tests. The decidability of this problem was stated as open by G&ouml;ller et al. in 2010. We give a reduction of this problem to the decision problem of a subset of Presburger arithmetic with divisibility with one quantifier alternation and a restriction on existentially quantified variables. A proof of decidability of this theory is currently under review. The final result of this thesis concerns a type of one-counter automata that differs from the previous one in that it allows parameters only on tests, not actions, and it includes both equality and disequality tests on counter values. The decidability of the basic reachability problem on such one-counter automata was stated as an open problem by Demri and Sangnier in 2010. We show that this problem is decidable by a reduction to the decision problem for Presburger arithmetic.

Submitted by ANA KARLA PEREIRA RODRIGUES (anakarla_@hotmail.com) on 2017-08-29T16:07:28Z No. of bitstreams: 1 arquivototal.pdf: 945033 bytes, checksum: cab32f76c5a55b0c6400063ed6b40ff6 (MD5) / Approved for entry into archive by Fernando Souza (fernandoafsou@gmail.com) on 2017-08-29T16:11:36Z (GMT) No. of bitstreams: 1 arquivototal.pdf: 945033 bytes, checksum: cab32f76c5a55b0c6400063ed6b40ff6 (MD5) / Made available in DSpace on 2017-08-29T16:11:36Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 945033 bytes, checksum: cab32f76c5a55b0c6400063ed6b40ff6 (MD5) Previous issue date: 2015-08-25 / We introduce and de ne the p-adics integer numbers as a result of a search for solutions, for a congruences system that derives from a variable polynomial equation with rational coe cients. We evidence that the p-adic integers set is strictly larger than the integers. We present a criterion so that a rational that holds a correspondent in a p-adic integers set. We search for the possibility to represent irrational and complex numbers as p-adics integers. Algebraically, the p-adic integers set will be an integral domain and, from this, we search for the construction of p-adic integers quotient eld so that shall form the p-adic rationals eld, from a purely algebraically point of view. In the second part, we will expose the bases for the construction of a norm that's di erent from the usual, establishing so a new metric in the rational numbers set and the construction of a non-archimedian eld. / Apresentamos e de nimos os números inteiros p-ádicos como o resultado de uma busca por soluções, para um sistema de congruências, que parte de uma equação polinomial de uma variável, com coe cientes racionais. Constatamos que o conjunto dos inteiros p-ádicos é estritamente maior que os inteiros. Mostramos um critério para que um racional possua um correspondente num conjunto de inteiros p-ádicos. Buscamos a possibilidade de representarmos números irracionais e números complexos como inteiros p-ádicos. Algebricamente, o conjunto dos inteiros p-ádicos será um domínio de integridade e, partindo disto, buscamos a construção de um corpo de frações dos inteiros p-ádicos, que formarão, assim, o corpo dos racionais p-ádicos, de um ponto de vista puramente algébrico. Na segunda parte, vamos expor os fundamentos para a construção de uma norma diferente da habitual, estabelecendo assim uma nova métrica, no conjunto dos números racionais, e a construção de um corpo não-arquimediano.

We study bounds on the distances of roots of integer polynomials and applications of such results. The separation of complex roots for reducible monic integer polynomials of fourth degree is thoroughly explained. Lemmas on roots of polynomials in the p-adic setting are proved. Explicit families of polynomials of general degree as well as families in some classes of quadratic and cubic polynomials with very good separation of roots in the same setting are exhibited. The second part of the thesis is concerned with results on p-adic versions of Mahler's and Koksma's functions wn and w*n and the related classifications of transcendental numbers in Cp. The main result is a construction of numbers such that the two functions wn and w*n differ on them for every n and later on expanding the interval of possible values for wn-w*n. The inequalities linking values of Koksma's functions for algebraically dependent numbers are proved.

Nous étudions les bornes sur les distances des racines des polynômes entiers et les applications de ces résultats. La séparation des racines complexes pour les polynômes réductibles normalisés de quatrième degré à coefficients entiers est examinée plus à fond. Différents lemmes sur les racines des polynômes en nombres p-adiques sont prouvés. Sont fournies les familles explicites de polynômes de degré général, ainsi que les familles dans certaines classes de polynômes quadratiques et cubiques avec une très bon separation des racins dans le cadre p-adique. Le reste de la thèse est dédié aux résultats liés aux versions p-adiques des fonctions de Mahler et de Koksma wn et w*n , ainsi qu'aux classifications correspondantes des nombres transcendants dans Cp. Le résultat principal est une construction des nombres pour lesquelles les deux fonctions wn et w*n sont différentes pour tous les n et puis l'intervalle de valeurs possibles pour wn-w*n est élargi. Les inégalités reliant les valeurs des fonctions de Koksma en nombres algébriquement dépendants sont prouvées. / We study bounds on the distances of roots of integer polynomials and applications of such results. The separation of complex roots for reducible monic integer polynomials of fourth degree is thoroughly explained. Lemmas on roots of polynomials in the p-adic setting are proved. Explicit families of polynomials of general degree as well as families in some classes of quadratic and cubic polynomials with very good separation of roots in the same setting are exhibited. The second part of the thesis is concerned with results on p-adic versions of Mahler's and Koksma's functions wn and w*n and the related classifications of transcendental numbers in Cp. The main result is a construction of numbers such that the two functions wn and w*n differ on them for every n and later on expanding the interval of possible values for wn-w*n. The inequalities linking values of Koksma's functions for algebraically dependent numbers are proved.