Holography principle and arithmetic of algebraic curves

According to the holography principle (due to G.`t Hooft, L. Susskind, J. Maldacena, et al.), quantum gravity and string theory on certain manifolds with boundary can be studied in terms of a conformal field theory on the boundary. Only a few mathematically exact results corroborating this exciting program are known. In this paper we interpret from this perspective several constructions which arose initially in the arithmetic geometry of algebraic curves. We show that the relation between hyperbolic geometry and Arakelov geometry at arithmetic infinity involves exactly the same geometric data as the Euclidean AdS_3 holography of black holes. Moreover, in the case of Euclidean AdS_2 holography, we present some results on bulk/boundary correspondence where the boundary is a non-commutative space.Comment: AMSTeX 30 pages, 7 figure

The Unification and Decomposition of Processing Structures Using Lattice Theoretic Methods

The purpose of this dissertation is to demonstrate that lattice theoretic methods can be used to decompose and unify computational structures over a variety of processing systems. The unification arguments provide a better understanding of the intricacies of the development of processing system decomposition. Since abstract algebraic techniques are used, the decomposition process is systematized which makes it conducive to the use of computers as tools for decomposition. A general algorithm using the lattice theoretic method is developed to examine the structures and therefore decomposition properties of integer and polynomial rings. Two fundamental representations, the Sino-correspondence and the weighted radix representation, are derived for integer and polynomial structures and are shown to be a natural result of the decomposition process. They are used in developing systematic methods for decomposing discrete Fourier transforms and discrete linear systems. That is, fast Fourier transforms and partial fraction expansions of linear systems are a result of the natural representation derived using the lattice theoretic method. The discrete Fourier transform is derived from a lattice theoretic base demonstrating its independence of the continuous form and of the field over which it is computed. The same properties are demonstrated for error control codes based on polynomials. Partial fraction expansions are shown to be independent of the concept of a derivative for repeated roots and the field used to implement them

Applications of additive combinatorics methods to some multiplicative problems

Wydział Matematyki i InformatykiGłównym celem pracy jest badanie różnych sposobów, w jakie kombinatoryka addytywna może być wykorzystana do radzenia sobie z pewnymi zagadnieniami pojawiającymi się w multiplikatywnej teorii liczb. Najważniejsza część pracy dotyczy następującego problemu: dla pewnej liczby naturalnej n i pewnej liczby pierwszej p jest nam dany zbiór reszt modulo p wszystkich dzielników liczby n i chcielibyśmy stwierdzić, które z nich odpowiadają jej czynnikom pierwszym. Przedstawiony jest algorytm rozwiązujący ten problem dla p i n spełniających pewne naturalne warunki i zostaje pokazane, że jest wiele takich liczb. Interesującą cechą przedstawionego dowodu jest to, że wymaga on użycia kombinatoryki addytywnej. W kolejnej części pracy rozważana jest suma wyrażeń exp(a2r/q ) dla wszystkich r należących do podgrupy multiplikatywnej reszt modulo q generowanej przez element 2. Podajemy górne oszacowanie wartości bezwzględnej z lepszą stałą niż dotychczas znana. W ostatniej części pracy rozważane są oszacowania na wielkość zbioru wszystkich sum postaci c1a1+c2a2+…+ckak, gdzie ci są ustalonymi współczynnikami, zaś ai są elementami zbioru A. Seria oszacowań górnych wielkości tego zbioru jest udowodniona dla A spełniającego |A+A| < K |A|. Najlepsze oszacowania dostajemy w przypadkach, gdy K jest znacznie mniejsze niż h oraz gdy zbiór współczynników ci ma pewną strukturę addytywną.The main aim of this dissertation is the study of different ways in which additive combinatorics may be used to tackle some problems arising in multiplicative number theory. The main part of the thesis deals with the following problem: Suppose that for some natural number n and some prime number p we are given the set of residues mod p of all its divisors and we would like to know which of those residues correspond to prime factors of n. An algorithm which approximately solves this problem for p and n satisfying some natural conditions is presented and it is proved that there are plenty of such numbers. One interesting feature of the proof is that it relies on additive combinatorics. In the next part of the thesis the sum of expressions exp(a2r/q ) over r belonging to multiplicative subgroup of residues modulo q generated by element 2. absolute value of this sum is estimated. The result we obtained in this line of research is the following. We give an upper-bound of absolute value of this sum with a better constant than previously known. In the last part of the thesis bounds for the size of sets of all the sums of the form c1a1+c2a2+…+ckak, where ci are coefficients and ai are elements of the set A. Series of results giving upper-bounds on the size of this set is proved for A satisfying |A+A|<K|A|. The best bounds are obtained in cases when K is much smaller than h and when the set of ci coefficients has some additive structure