Abstract The focus of this thesis is on two functions: the exponential function and Euler’s factorial series. By constructing explicit Padé approximations, we are able to improve lower bounds for linear forms in the values of these functions. In particular, the dependence on the height of the coefficients of the linear form will be sharpened in the lower bound. The first chapter contains some necessary definitions and auxiliary results needed in later chapters.We give precise definitions for a transcendence measure and Padé approximations of the second type. Siegel’s lemma will be introduced as a fundamental tool in Diophantine approximation. A brief excursion to exterior algebras shows how they can be used to prove determinant expansion formulas. The reader will also be familiarised with valuations of number fields. In Chapter 2, a new transcendence measure for e is proved using type II Hermite-Padé approximations to the exponential function. An improvement to the previous transcendence measures is achieved by estimating the common factors of the coefficients of the auxiliary polynomials. The exponential function is the underlying topic of the third chapter as well. Now we study the common factors of the maximal minors of some large block matrices that appear when constructing Padé-type approximations to the exponential function. The factorisation of these minors is of interest both because of Bombieri and Vaaler’s improved version of Siegel’s lemma and because they are connected to finding explicit expressions for the approximation polynomials. In the beginning of Chapter 3, two general theorems concerning factors of Vandermonde-type block determinants are proved. In the final chapter, we concentrate on Euler’s factorial series which has a positive radius of convergence in p-adic fields. We establish some non-vanishing results for a linear form in the values of Euler’s series at algebraic integer points. A lower bound for this linear form is derived as well.

Submitted by Liliane Ferreira (ljuvencia30@gmail.com) on 2018-04-11T12:39:47Z No. of bitstreams: 2 Dissertação - Carlos Alirio Rico Acevedo - 2018.pdf: 1289579 bytes, checksum: 0b60c803c3d9f6f61772e58e7d624086 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2018-04-12T11:29:32Z (GMT) No. of bitstreams: 2 Dissertação - Carlos Alirio Rico Acevedo - 2018.pdf: 1289579 bytes, checksum: 0b60c803c3d9f6f61772e58e7d624086 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2018-04-12T11:29:32Z (GMT). No. of bitstreams: 2 Dissertação - Carlos Alirio Rico Acevedo - 2018.pdf: 1289579 bytes, checksum: 0b60c803c3d9f6f61772e58e7d624086 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2018-03-16 / Conselho Nacional de Pesquisa e Desenvolvimento Científico e Tecnológico - CNPq / Let $ k \geq 2.$ an integer. The recurrence $ \fk{n} = \sum_ {i = 0}^k \fk{n-i} $ for $ n> k $, with initial conditions $F_{-(k-2)}^{(k)}=F_{-(k-3)}^{(k)}=\cdots=F_{0}^{(k)}=0$ and $F_1^{ (k)} = 1$, which is called the $k$-generalized Fibonacci sequence. When $ k = 2 ,$ we have the Fibonacci sequence $ \{ F_n \}_{n\geq 0}.$ We will show that the equation $F_{n}^{x}+F_{n+1}^x=F_{m}$ does not have no non-trivial integer solutions $ (n, m, x) $ to $ x> 2 $. On the other hand, for $ k \geq 3,$ we will show that the diophantine equation $\epi$ does not have integer solutions $ (n, m, k, x) $ with $ x \geq 2 $. In both cases, we will use initially Matveev's Theorem, for linear forms in logarithms and the reduction method due to Dujella and Pethö, to limit the variables $ n, \; m $ and $ x $ at intervals where the problem is computable. In addition, in the case for $ k\geq 3 $, we will use the fact that the dominant root the $k$-generalized Fibonacci sequence is exponentially close to 2 to bound $k$, a method developed by Bravo and Luca. / Seja $k\geq 2$ inteiro, considere-se a recorrência $\fk{n}=\sum_{i=0}^{k}\fk{n-i}$ para $n>k$, com condições iniciais $F_{-(k-2)}^{(k)}=F_{-(k-3)}^{(k)}=\cdots=F_{0}^{(k)}=0$ e $F_{1}^{(k)}=1$, que é a sequência de Fibonacci $k$-generalizada. No caso quando $k=2$, é dizer, para a sequência de Fibonacci $\{F_n\}_{n\geq 0}$, vai-se mostrar que a equação $F_{n}^{x}+F_{n+1}^x=F_{m}$ não possui soluções inteiras não triviais $(n,m,x)$ para $x>2$. Por outro lado para, $k\geq 3$ se mostrar que a equação diofantina $\epi$ não possui soluções inteiras $(n,m,k,x)$ com $x\geq 2$. Em ambos casos, inicialmente são usados resultados como o Teorema de Matveev, para formas lineares em logaritmos e o método de redução de Dujella e Pethö, para limitar as variáveis $n, \; m$ e $x$ em intervalos onde o problema seja computável. Adicionalmente, no caso para $k\geq 3$ é usado que a raiz dominante da sequência de Fibonacci $k$-generalizada e exponencialmente próxima a 2, para limitar $k$, o que é um método desenvolvido por Bravo e Luca.

In dieser Arbeit wird ein Algorithmus zur Lösung von endlichdimensionalen Optimierungsproblemen mit stückweise linearer Zielfunktion und stückweise linearen Nebenbedingungen vorgestellt. Dabei wird angenommen, dass die Funktionen in der sogenannten Abs-Linear Form, einer Matrix-Vektor-Darstellung, vorliegen. Mit Hilfe dieser Form lässt sich der Urbildraum in Polyeder zerlegen, so dass die Nichtglattheiten der stückweise linearen Funktionen mit den Kanten der Polyeder zusammenfallen können. Für die Klasse der abs-linearen Funktionen werden sowohl für den unbeschränkten als auch für den beschränkten Fall notwendige und hinreichende Optimalitätsbedingungen bewiesen, die in polynomialer Zeit verifiziert werden können. Für unbeschränkte stückweise lineare Optimierungsprobleme haben Andrea Walther und Andreas Griewank bereits 2019 mit der Active Signature Method (ASM) einen Lösungsalgorithmus vorgestellt. Aufbauend auf dieser Methode und in Kombination mit der Idee der aktiven Mengen Strategie zur Behandlung von Ungleichungsnebenbedingungen entsteht ein neuer Algorithmus mit dem Namen Constrained Active Signature Method (CASM) für beschränkte Probleme. Beide Algorithmen nutzen die stückweise lineare Struktur der Funktionen explizit aus, indem sie die Abs-Linear Form verwenden. Teil der Analyse der Algorithmen ist der Nachweis der endlichen Konvergenz zu lokalen Minima der jeweiligen Probleme sowie die Betrachtung effizienter Berechnung von Lösungen der in jeder Iteration der Algorithmen auftretenden Sattelpunktsysteme. Die numerische Performanz von CASM wird anhand verschiedener Beispiele demonstriert. Dazu gehören akademische Probleme, einschließlich bi-level und lineare Komplementaritätsprobleme, sowie Anwendungsprobleme aus der Gasnetzwerkoptimierung und dem Einzelhandel. / This thesis presents an algorithm for solving finite-dimensional optimization problems with a piecewise linear objective function and piecewise linear constraints. For this purpose, it is assumed that the functions are in the so-called Abs-Linear Form, a matrix-vector representation. Using this form, the domain space can be decomposed into polyhedra, so that the nonsmoothness of the piecewise linear functions can coincide with the edges of the polyhedra. For the class of abs-linear functions, necessary and sufficient optimality conditions that can be verified in polynomial time are given for both the unconstrained and the constrained case. For unconstrained piecewise linear optimization problems, Andrea Walther and Andreas Griewank already presented a solution algorithm called the Active Signature Method (ASM) in 2019. Building on this method and combining it with the idea of the Active Set Method to handle inequality constraints, a new algorithm called the Constrained Active Signature Method (CASM) for constrained problems emerges. Both algorithms explicitly exploit the piecewise linear structure of the functions by using the Abs-Linear Form. Part of the analysis of the algorithms is to show finite convergence to local minima of the respective problems as well as an efficient solution of the saddle point systems occurring in each iteration of the algorithms. The numerical performance of CASM is illustrated by several examples. The test problems cover academic problems, including bi-level and linear complementarity problems, as well as application problems from gas network optimization and inventory problems.

The territory lies in the City of Brno, in the urban district of Zábrdovice. From the east it is demarcated by the Svitava River, from the south by the Cejl Street, and from the west by the Jan Svoboda Street. Its area is approximately 18,790 m2. From the standpoint of urban planning¸ the intensive house is a separate city block, which is, nevertheless, open in the eastern direction towards the Svitava River, thus communicating with the embankment. In the other direction there is both private and semi-private space inside the block as a relaxation green “garden” in several levels. The mass of the object links up in the prolonged line with the existing neighbouring blocks and thus respects the integrity of the territory. Intensive use of the territory consists in the achievement of high density of built-up area, and at the same time preserving the surroundings of quality for life and habitation. Functions are mixed here and a city of short distances is created here with nearly excluded traffic. In order to achieve such intensity, space urbanism is employed. The principle consists in a simple linear form of masses respecting the shape of the lot, which is vivified with an ellipsoid located in the courtyard, which is an accent towards the linear form. The ellipsoid serves as a small cultural and exhibition centre, at the same time connecting all the arms of the defining block and bringing daylight to the first floor. The block is open in the direction to the river and its half-public space is connected with the embankment with a staircase and verdure. Verdure is also made use of on the roofs. The defining mass is perforated, thus creating covered terraces. The principal architectonic intention was to create an integrated structure, where a significant role is played by the introduction of nature and polyfunctionality to the whole territory, thus increasing its attractivity. From the dispositional perspective the individual functions interpenetrate at al