FIDIC Conditions of Contract and their Application in the Czech Republic Resume in English The goal of this thesis is, mainly by the mutual comparison, to describe the differences between each kinds of contractual forms of FIDIC and also to put the main forms into the context of their application in Czech Republic, especially by enumeration of examples of some modifications which are made through the Particular Conditions by various public employers, including the general assessment of such modifications. The basic structure of the thesis consists of two main sections, which are split into the four chapters. The content of the thesis goes from the general description of the contractual forms of FIDIC, including the brief description of organization's history, thru the international contextualization of its forms and general introduction to their application in Czech Republic, to the comparison and description of the application of each chapters of FIDIC forms in Czech and partially also in the Central Eastern European context. The first main section forms the general introduction into the issue of FIDIC forms by their basic description and by putting them into the context of similar contractual forms around the globe. This first section is split into three chapters, which describe the context of contractual...

One of the fundamental differences between automorphic representations of classical groups like GL(n) and their metaplectic covers is that in the latter case the space of Whittaker functionals usually has a dimension bigger than one. Gelbart and Piatetski-Shapiro called the metaplectic representations, which possess a unique Whittaker model, distinguished and classified them for the double cover of the group GL(2). Later Patterson and Piatetski-Shapiro used a converse theorem to list the distinguished representations for the degree three cover of GL(3). In their milestone paper on general metaplectic covers of GL(n) Kazhdan and Patterson construct examples of non-cuspidal distinguished representations, which come as residues of metaplectic Eisenstein series. These are generalizations of the classical Jacobi theta functions. Despite some impressive local results to date, cuspidal distinguished representations are not classified or even constructed outside rank 1 and 2. In her thesis Wang makes some progress toward the classification in rank 3. In this dissertation we construct the distinguished representations for the degree four metaplectic cover of GL(4), applying a classical converse theorem like Patterson and Piatetski-Shapiro in the case of rank 2. We obtain the necessary local properties of the Rankin-Selberg convolutions at the ramified places and finish the proof of the construction of cuspidal distinguished representations in rank 3.

The large sieve method has been used extensively, beginning with Bombieri in 1965, to provide bounds on the number of zeros of Dirichlet L-functions near the line σ = 1. Using the Kuznetsov trace formula and the work of Deshouillers and Iwaniec on Kloosterman sums, it is possible to derive large sieve inequalities for the Fourier coefficients of Maass cusp forms, which may then similarly be used to study the corresponding Hecke-Maass L-functions. Following an approach developed by Gallagher for Dirichlet L-functions, this thesis shows how the large sieve method may be used to prove a zero density estimate, averaged over the Laplace eigenvalues, for Maass cusp forms of weight zero for the congruence subgroup Γ₀(q) for any positive integer q.

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Let M#k(64) be the space of weakly holomorphic modular forms in level 64 and weight k which can have poles only at infinity, and let S#k(64) be the subspace of M#k(64) consisting of forms which vanish at all cusps other than infinity. We explicitly construct canonical bases for these spaces and show that the coefficients of these basis elements satisfy Zagier duality. We also compute the generating function for the canonical basis.

Let N ∈ {8,9,16,25} and let M#0(N) be the space of level N weakly holomorphic modular functions with poles only at the cusp at infinity. We explicitly construct a canonical basis for M#0(N) indexed by the order of the pole at infinity and show that many of the coefficients of the elements of these bases are divisible by high powers of the prime dividing the level N. Additionally, we show that these basis elements satisfy an interesting duality property. We also give an argument that extends level 1 results on congruences from Griffin to levels 2, 3, 4, 5, 7, 8, 9, 16, and 25.

Let M#k(52) be the space of weight k level 52 weakly holomorphic modular forms with poles only at infinity, and S#k(52) the subspace of forms which vanish at all cusps other than infinity. For these spaces we construct canonical bases, indexed by the order of vanishing at infinity. We prove that the coefficients of the canonical basis elements satisfy a duality property. Further, we give closed forms for the generating functions of these basis elements.

This paper examines the probability that a random polynomial of specific degree over a field has a specific number of distinct roots in that field. Probabilities are found for random quadratic polynomials with respect to various probability measures on the real numbers and p-adic numbers. In the process, some properties of the p-adic integer uniform random variable are explored. The measure Witt ring, a generalization of the canonical Witt ring, is introduced as a way to link quadratic forms and measures, and examples are found for various fields and measures. Special properties of the Haar measure in connection with the measure Witt ring are explored. Higher-degree polynomials are explored with the aid of numerical methods, and some conjectures are made regarding higher-degree p-adic polynomials. Other open questions about measure Witt rings are stated. / Graduation date: 1999

Résumé anglais: This thesis consists of an introductory text followed by two separate parts which may be read independently of each other. In Part I we analyze certain hyperbolic structures arising when studying gravity in the vicinity of spacelike singularities (the BKL-limit). In this limit, spatial points decouple and the dynamics exhibits ultralocal behaviour which may be mapped to an auxiliary problem given in terms of a (possibly chaotic) hyperbolic billiard. In all supergravities arising as low-energy limits of string theory or M-theory, the billiard dynamics takes place within the fundamental Weyl chambers of certain hyperbolic Kac-Moody algebras, suggesting that these algebras generate hidden infinite-dimensional symmetries of gravity. We investigate the modification of the billiard dynamics when the original gravitational theory is formulated on a compact spatial manifold of arbitrary topology, revealing fascinating mathematical structures known as galleries. We further use the conjectured hyperbolic symmetry E10 to generate and classify certain cosmological (S-brane) solutions in eleven-dimensional supergravity. Finally, we show in detail that eleven-dimensional supergravity and massive type IIA supergravity are dynamically unified within the framework of a geodesic sigma model for a particle moving on the infinite-dimensional coset space E10/K(E10). Part II of the thesis is devoted to a study of how (U-)dualities in string theory provide powerful constraints on perturbative and non-perturbative quantum corrections. These dualities are typically given by certain arithmetic groups G(Z) which are conjectured to be preserved in the effective action. The exact couplings are given by moduli-dependent functions which are manifestly invariant under G(Z), known as automorphic forms. We discuss in detail various methods of constructing automorphic forms, with particular emphasis on a special class of functions known as (non-holomorphic) Eisenstein series. We provide detailed examples for the physically relevant cases of SL(2,Z) and SL(3,Z), for which we construct their respective Eisenstein series and compute their (non-abelian) Fourier expansions. We also discuss the possibility that certain generalized Eisenstein series, which are covariant under the maximal compact subgroup K(G), could play a role in determining the exact effective action for toroidally compactified higher derivative corrections. Finally, we propose that in the case of rigid Calabi-Yau compactifications in type IIA string theory, the exact universal hypermultiplet moduli space exhibits a quantum duality group given by the emph{Picard modular group} SU(2,1;Z[i]). To verify this proposal we construct an SU(2,1;Z[i])-invariant Eisenstein series, and we present preliminary results for its Fourier expansion which reveals the expected contributions from D2-brane and NS5-brane instantons. / Résumé francais: Cette thèse est composée d'une introduction suivie de deux parties qui peuvent être lues indépendemment. Dans la première partie, nous analysons des structures hyperboliques apparaissant dans l'étude de la gravité au voisinage d'une singularité de type espace (la limite BKL). Dans cette limite, les points spatiaux se découplent et la dynamique suit un comportement ultralocal qui peut être reformulé en termes d'un billiard hyperbolique (qui peut être chaotique). Dans toutes les supergravités qui sont des limites de basse énergie de théories de cordes ou de la théorie M, la dynamique du billiard prend place à l'intérieur des chambres de Weyl fondamentales de certaines algèbres de Kac-Moody hyperboliques, ce qui suggère que ces algèbres correspondent à des symétries cachées de dimension infinie de la gravité. Nous examinons comment la dynamique du billard est modifiée quand la théorie de gravité originale est formulée sur une variété spatiale compacte de topologie arbitraire, révélant ainsi de fascinantes structures mathématiques appelées galleries. De plus, dans le cadre de la supergravité à onze dimensions, nous utilisons la symétrie hyperbolique conjecturée E10 pour engendrer et classifier certaines solutions cosmologiques (S-branes). Finalement, nous montrons en détail que la supergravité à onze dimensions et la supergravité de type IIA massive sont dynamiquement unifiées dans le contexte d'un modèle sigma géodesique pour une particule se déplaçant sur l'espace quotient de dimension infinie E10/K(E10). La deuxième partie de cette thèse est consacrée à étudier comment les dualités U en théorie des cordes fournissent des contraintes puissantes sur les corrections quantiques perturbatives et non perturbatives. Ces dualités sont typiquement données par des groupes arithmétiques G(Z) dont il est conjecturé qu'ils préservent l'action effective. Les couplages exacts sont donnés par des fonctions des moduli qui sont manifestement invariantes sous G(Z), et qu'on appelle des formes automorphiques. Nous discutons en détail différentes méthodes de construction de ces formes automorphiques, en insistant particulièrement sur une classe spéciale de fonctions appelées séries d'Eisenstein (non holomorphiques). Nous présentons comme exemples les cas de SL(2,Z) et SL(3,Z), qui sont physiquement pertinents. Nous construisons les séries d'Eisenstein correspondantes et leurs expansions de Fourier (non abéliennes). Nous discutons également la possibilité que certaines séries d'Eisenstein généralisées, qui sont covariantes sous le sous-groupe compact maximal, pourraient jouer un rôle dans la détermination des actions effectives exactes pour les théories incluant des corrections de dérivées supérieures compactifiées sur des tores.

The notion of spaces of orderings was introduced by Murray Marshall in the 1970's and provides an abstract framework for studying orderings on fields and the reduced theory of quadratic forms over fields. The structure of a space of orderings (X, G) is completely determined by the group structure of G and the quaternary relation (a_1, a_2) = (a_3, a_4) on G -- the groups with additional structure arising in this way are called reduced special groups. The theory of reduced special groups, in turn, can be conveniently axiomatized in the first order language L_SG. Numerous important notions in this theory, such as isometry, isotropy, or being an element of a value set of a form, make an extensive use of, so called, positive primitive formulae in the language L_SG. Therefore, the following question, which can be viewed as a type of very general and highly abstract local-global principle, is of great importance:<p>Is it true that if a positive primitive formula holds in every finite subspace of a space of orderings, then it also holds in the whole space?<p>This problem is now known as the pp conjecture. The answer to this question is affirmative in many cases, although it has always seemed unlikely that the conjecture has a positive solution in general. In this thesis, we discuss, discovered by us, first counterexamples for which the pp conjecture fails. Namely, we classify spaces of orderings of function fields of rational conics with respect to the pp conjecture, and show for which of such spaces the conjecture fails, and then we disprove the pp conjecture for the space of orderings of the field R(x,y). Some other examples, which can be easily obtained from the developed theory, are also given. In addition, we provide a refinement of the result previously obtained by Vincent Astier and Markus Tressl, which shows that a pp formula fails on a finite subspace of a space of orderings, if and only if a certain family of formulae is verified.