Thesis (Ph. D.)--University of Wisconsin, 1964. / Typescript. Vita. Abstracted in Dissertation abstracts, v. 25 (1965) no. 7, p. 4092. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references.

Thesis (Ph. D.)--University of Wisconsin--Madison, 1966. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references.

Thesis (Ph. D.)--State University of New York at Binghamton, Mathematical Sciences Department, 2005. / Includes bibliographical references.

CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior / Este trabalho trata de duas das mais importantes desigualdades da MatemÃtica: a desigualdade entre as mÃdias geomÃtrica e aritmÃtica e a desigualdade de Cauchy-Schwarz. Apresentamos inicialmente diversas demonstraÃÃes para o caso n = 2, apÃs as quais seguem muitas demonstraÃÃes para o caso geral. Nessas demonstraÃÃes utilizamos Ãlgebra elementar, geometria euclidiana, construÃÃes geomÃtricas, geometria analÃtica, induÃÃo matemÃtica, convexidade de funÃÃes, multiplicadores de Lagrange entre outros assuntos. AlÃm disso foram selecionados vinte problemas que visam dar ao leitor uma melhor compreensÃo de como estas desigualdades podem ser aplicadas em diversos assuntos e de diversas formas, estimulando a criatividade dos alunos na resoluÃÃo de problemas. / This paper deals with two of the most important inequalities of Mathematics: the inequality between the geometric and arithmetic and Cauchy-Schwarz. Here several first statements for the case n = 2, after which many statements following for the general case. In these statements we use algebra elementary Euclidean geometry, geometric constructions, analytical geometry, mathematical induction, convexity of functions, Lagrange multipliers among other issues. Also selected were twenty problems that aim to give the reader a better understanding of how these inequalities can be applied in various subjects and in many ways, stimulating students' creativity in problem solving.

O presente trabalho trata de duas importantes desigualdades matemÃticas: a desigualdade triangular e a desigualdade de Jensen. Apresenta inicialmente uma visÃo de como o assunto de desigualdades à tratado de forma inapropriada em livros de matemÃtica de nÃvel mÃdio. Em seguida à explicado o que à uma desigualdade, prossegue mostrando um experimento geomÃtrico empÃrico para que se possa âconcretamenteâ averiguar a veracidade da desigualdade triangular. Dando continuidade ao trabalho, sÃo mostradas sete demonstraÃÃes da desigualdade triangular. Posteriormente sÃo apresentadas trÃs demonstraÃÃes da desigualdade de Jensen. Para realizaÃÃo destas demonstraÃÃes foram necessÃrios conhecimentos de Ãlgebra elementar, geometria euclidiana, construÃÃes geomÃtricas, induÃÃo matemÃtica, convexidade de funÃÃes, desigualdade de Cauchy-Schuwarz, alÃm de vÃrios outros conhecimentos. Foram elencados sete problemas de aplicaÃÃo da desigualdade triangular e quinze da desigualdade de Jensen, com o objetivo de proporcionar ao leitor uma percepÃÃo mais apurada da forma como estas desigualdades podem ser aplicadas para motivar a criatividade dos alunos na resoluÃÃo de problemas. / This paper deals with two important mathematical inequalities: the triangle inequality and Jensen inequality. It first presents an overview of how the issue of inequality is treated improperly in math books for middle level. Then it is explained what is an inequality, continuing an experiment showing empirical geometry so you can "specifically" to ascertain the veracity of the triangle inequality. Continuing the work, seven are shown demonstrations of triangle inequality. Are then presented with three demonstrations of the Jensen inequality. To perform these demonstrations took knowledge of elementary algebra, Euclidean geometry, geometric constructions, mathematical induction, convex of functions, Cauchy-Schuwarz and several other knowledge. Were listed seven issues of application of the triangle inequality and fifteen Jensen inequality, with the aim of providing the reader with a more accurate perception of how these inequalities can be applied to motivate students' creativity in problem solving.

Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico / Os problemas que abordaremos estÃo diretamente associados à existÃncia de elementos no subgrupo derivado que nÃo sÃo comutadores. Nosso objetivo serà apresentar os resultados de Tim Bonner, que sÃo estimativas para a razÃo entre o comprimento do derivado e a ordem do grupo (limitaÃÃo superior e determinaÃÃo do "comportamento assintÃtico"), culminando com uma prova da conjectura de Bardakov. / The problems which we address in this work are directly related to the existence of elements in the derived subgroup that are not commutators. Our purpose is to present the results of Tim Bonner [1]. In his paper, one finds estimates for the ratio between the commutator length and the order of group (more precisely, upper limits and the establishment of its asymptotic behavior), leading to the proof of Bardakov's Conjecture.

When divisorial varieties were first introduced, the question immediately arose whether there are any varieties which are not divisorial. This work answers the question in the affirmative. We prove here that the non-projective variety M defined by Nagata in Memoirs of the College of Science, University of Kyoto, Series A, Vol. XXX, Mathematics No. 3, 1957, pp. 231-235 is, in fact, non-divisorial. The work is organized as follows: We first discuss briefly the concepts relating to the notion of divisorial variety. Next there is a description of Nagata's variety in which we include the proofs of statements which we shall need for the subsequent theorems. The preliminary results are of two types: first we prove several lemmas concerning the dominance of local rings of points on the variety M. Second we prove that divisors whose varieties contain the vertex of the affine cone V used in Nagata's example must intersect the line at infinity of the cone at a point(s) whose local ring dominates the local ring of the vertex of the cone Vσ under the transformation σ defined by Nsgata. This result indicates strongly that the variety M = VU Vσ is not divisorial. For the proof that this is, in fact, the case, we prove in detail a strictly algebraic result to the effect that the (prime) ideals associated with the irreducible components of a divisor which do not contain the vertex P of the cone V are principal. With this result, we finally show by contradiction that M is not divisorial. / Science, Faculty of / Mathematics, Department of / Graduate

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Through composition of isomorphisms from results by Miki and Negut, this paper seeks to simplify calculations of the images of generators of the quantum toroidal algebra. We will be working in the small shuffle algebra, which is isomorphic to the positive part of the quantum toroidal algebra. There, we will be computing commutators, which are equal to images under the Miki automorphism, though are much simpler to compute. / Master of Science / Computing is hard, even for computers. The fewer computations we have to do, the more time we can save to do more math. This paper accomplishes just that. By looking at the quantum toroidal algebra through an automorphism followed by an isomorphism to a small shuffle algebra, we find a way to compute images of generators under the automorphism relatively easily.

Thesis (Ph.D.)--Boston University / PLEASE NOTE: Boston University Libraries did not receive an Authorization To Manage form for this thesis or dissertation. It is therefore not openly accessible, though it may be available by request. If you are the author or principal advisor of this work and would like to request open access for it, please contact us at open-help@bu.edu. Thank you. / Algebraic curves C in the projective space P^n are characterized by their degree d and genus g. We would like to know what g are possible for a curve of degree d in P^n and to study the geometry of such curves. By Castelnuovo's Theorem, the maximum value of g, if deg(C) = d, in P^n is known and is denoted by π(d, n). If g = π(d, n), C lies on a surface S ⊂ P^n such that deg(S) = n - 1. To study other curves C with g < π(d, n), Eisenbud and Harris arranged the possible values of g into intervals πα+1(d, n) < g < πα(d, n) , with α E Z+ and π0(d, n) := π(d, n), where πα (d, n) = maxc⊂s{g(C)} for any normal surface S with deg(S) = n + α - 1. In general, for C ⊂ P^n such that g > πα (d, n), they proved that if n ≥ 8 and d ≥ 2^n+1, then C lies on a surface S ⊂ P^n with deg(S) ≤ n - 2 + α. They also conjectured that the same result should hold for curves of any degree, provided that g > πα(d, n) and proved this conjecture for a = 1. We will focus on the case a = 2. In this case, by the Eisenbud-Harris conjecture, C should lie on a surface of degree n in P^n. We verify this in several special cases for C ⊂ F. To do so, we study the systems cut out by quadric and cubic hypersurfaces on C and prove that C must lie on at least three or four quadrics in P^5. The intersection of such hypersurfaces is a surface S ⊂ P^5 with deg(S) < 7. By analyzing the maximum value of the genus of C for C ⊂ S ⊂ P^n and deg (S) = (n + 1) or (n + 2), we see that the curves C ⊂ P^5 we are analyzing cannot lie on a surface S with deg(S) = 6, 7. / 2999-01-01