Let G = (a,b,c,...) be a groupoid and T a topology for G with U_a denoting an open set in T that contains the element a. The topology T is admissible for G if for every a·b=c and U_c there exist U_a and U_b such that U_a·U_b c U_c. G is said to be topologically trivial if the only admissible topologies for G are the discrete and indiscrete. It is shown that finite groups are topologically trivial if and only if they are simple. It is shown that finite topologically trivial semigroups are necessarily groups. Various classes of topologically trivial groupoids are examine, and it is shown that there exist topologically trivial groupoids of every order. G is said to be right (analogously left) topologically trivial if one can find elements a·b = c in G and U_c in T such that a·U_b ⊈ U_c for all U_b in T whenever T is not trivial. G is said to be totally topologically trivial if one can find a·b = c in G and U_c in T such that a·U_b ⊈ U_c and U_a·b ⊈ U_c for all U_a and U_b in T whenever T is not trivial. Right, left, and total topologically triviality are studies for various algebraic systems. A continuity condition that always holds is exhibited as are new proofs for several old theorems. Consequences of imposing the tower topology on various algebraic systems are examined. If the proper subset I contained in the groupoid G is such that the null set, the set G, and each singleton set of the elements in G-I form the basis for an admissible topology for G, then I is called a generalized ideal in G. Properties of generalized ideals are studied at length. A function t from a groupoid G to another groupoid is called a local homomorphism if for each a and b in G there exist r and s in G such that a·b = r·s and such that t(r·s) = t(r)·t(s). Several properties of local homomorphisms are examined. / Ph. D.

Let <i>K</i> be a compact subset of complex <i>N</i>-dimensional space, where <i>N</i> ≥ 1. Let <i>H</i>(<i>K</i>) denote the functions analytic in a neighborhood of <i>K</i>. Let <i>R</i>(<i>K</i>) denote the closure of <i>H</i>(<i>K</i>) in <i>C</i>(<i>K</i>). We study the problem: What is <i>R</i>(<i>K</i>)? The study of <i>R</i>(<i>K</i>) is contained in the field of rational approximation. In a set of lecture notes, T. Gamelin [6] has shown a certain operator to be essential to the study of rational approximation. We study properties of this operator. Now let <i>K</i> be a compact subset of real <i>N</i>-dimensional space, where <i>N</i> ≥ 2. Let harm<i>K</i> denote those functions harmonic in a neighborhood of <i>K</i>. Let <i>h</i>(<i>K</i>) denote the closure of harm<i>K</i> in <i>C</i>(<i>K</i>). We also study the problem: What is <i>h</i>(<i>K</i>)? The study of <i>h</i>(<i>K</i>) is contained in the field of harmonic approximation. As well as obtaining harmonic analogues to our results in rational approximation, we also produce a harmonic analogue to the operator studied in Gamelin's notes. / Ph. D.

With the proliferation of computers into homes and businesses and the explosive growth rate of the Internet, the ability to conduct secure electronic communications and transactions has become an issue of vital concern. One of the most prominent systems for securing electronic information, known as RSA, relies upon the fact that it is computationally difficult to factor a "large" integer into its component prime integers. If an efficient algorithm is developed that can factor any arbitrarily large integer in a "reasonable" amount of time, the security value of the RSA system would be nullified. The General Number Field Sieve algorithm is the fastest known method for factoring large integers. Research and development of this algorithm within the past five years has facilitated factorizations of integers that were once speculated to require thousands of years of supercomputer time to accomplish. While this method has many unexplored features that merit further research, the complexity of the algorithm prevents almost anyone but an expert from investigating its behavior. We address this concern by first pulling together much of the background information necessary to understand the concepts that are central in the General Number Field Sieve. These concepts are woven together into a cohesive presentation that details each theory while clearly describing how a particular theory fits into the algorithm. Formal proofs from existing literature are recast and illuminated to clarify their inner-workings and the role they play in the whole process. We also present a complete, detailed example of a factorization achieved with the General Number Field Sieve in order to concretize the concepts that are outlined. / Master of Science

京都大学 / 新制・課程博士 / 博士(理学) / 甲第25094号 / 理博第5001号 / 新制||理||1714(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 長谷川 真人, 教授 大木谷 耕司, 准教授 照井 一成 / 学位規則第4条第1項該当 / Doctor of Agricultural Science / Kyoto University / DFAM

This paper discusses the concepts in algebraic and analytic number theory used in the proofs of Dirichlet's and Cheboterev's density theorems. It presents special cases of results due to the latter theorem for which greatly simplified proofs exist. / Master of Science

Let R be a ring with an identity 1. Let A, B, and C be R-modules. The sequence A → [f above arrow] B→[g above arrow] C is exact providing f and g are R-homomorphisms and Im f =Ker g. Let 0 represent the R-module with precisely one element. An R-module J is injective if and only if for every exact sequence 0→A→ [f above arrow] B of R-modules and R-homomorphisms and every R-homomorphism g: A→J there exists an R-homomorphism h: B→J such that hf = g. This is a dual concept to that of a projective R-module. In the second chapter the idea of an injective R-module is studied quite intensively, and several different characterizations of injective · modules are proved. One of the principal results obtained is that every R-module is a submodule of an injective R-module. Further properties of injective R-modules are given in Chapter 3, including the concepts of injective dimension and an injective resolution of an R-module. Using these concepts the Shifting Theorem for injectives is proved. The basic definitions and results necessary for the development of the concept of injective for abstract categories are included in Chapter 4. An injective object is then defined in this general setting. Then the concept of an injective envelope is defined. The problems that arise, in the effort to restrict the category of topological groups to the appropriate subcategory so that the concept of an injective topological group is of interest, are investigated in Chapter 5. The development of the concept for one such restriction concludes this thesis. / Master of Science

As finite element discretizations ever grow in size to address real-world problems, there is an increasing need for fast algorithms. Nowadays there are many GPU/CPU parallel approaches to solve such problems. Multigrid methods can be used to solve large-scale problems, or even better they can be used to precondition the conjugate gradient method, yielding better results in general. Capabilities of multigrid algorithms rely on the effectiveness of the inter-grid transfer operators. In this thesis we focus on the aggregation approach, discussing how different aggregation strategies affect the convergence rate. Based on these discussions, we propose an alternative parallel aggregation algorithm to improve convergence. We also provide numerous experimental results that compare different aggregation approaches, multigrid methods, and conjugate gradient iteration counts, showing that our proposed algorithm performs better in serial and parallel. / Modeling real-world problems incurs a high computational cost because these mathematical models involve large-scale data manipulation. Thus we need fast and efficient algorithms. Nowadays there are many high-performance approaches for these problems. One such method is called the Multigrid algorithm. This approach models a physical domain using a hierarchy of grids, and so the effectiveness of these approaches relies on how well data can be transferred from grid to grid. In this thesis, we focus on the aggregation approach, which clusters a grid’s vertices according to its connections. We also provide an alternative parallel aggregation algorithm to give a faster solution. We show numerous experimental results that compare different aggregation approaches and multigrid methods, showing that our proposed algorithm performs better in serial and parallel than other popular implementations.

The Kostka-Foulkes polynomials describe the change of basis between Schur polynomials and Hall-Littlewood polynomials. In this paper, we extend this idea to the family of BCn Macdonald spherical functions, with multiparameter Kostka-Foulkes polynomials acting as the change of basis from the BC_n spherical functions to the type Cn Schur polynomials. We develop a Kato-Lusztig formula that describes the multiparameter BCn-Kostka-Foulkes polynomials. / Master of Science / The work done in [11] gives a formula to calculate Kostka-Foulkes polynomials that convert between two other forms of polynomials. However, this only applies in specific instances. In this paper, we generalize those ideas to allow for more parameters, and find that a similar formula still holds.

In this thesis we explore the topic of resultants, focusing on the resultants of binomial sys-tems. The study aims to provide an introduction to resultants and explore some of the morerecent research on the relationship between cycles in graphs and the resultant of binomialsystems. This will be done by going over some general theory from algebraic geometry suchas polynomials and ideals, in addition to studying the graphs of the binomial systems and thesurrounding theory. The research includes a comprehensive analysis of these graphs and theircycles, leading to findings that the cycles that appear are not restricted in length. This workprovides exploration of a potential extension of the recent research and encourages furtherresearch on the topic.

In this thesis, we prove a Kudla-Rapoport conjecture for 𝓨-cycles on exotic smooth unitary Rapoport-Zink spaces of odd arithmetic dimension, i.e. the arithmetic intersection numbers for 𝓨-cycles equals the derivatives of local representation density. We also compare 𝓩-cycles and 𝓨-cycles on these RZ spaces. The method is to relate both geometric and analytic sides to the even dimensional case and reduce the conjecture to the results in \cite{LL22}.