A thesis submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Doctor of Philosophy. July 2014. / This is the study of Lie algebraic properties of first integrals of scalar second-, third and higher-order ordinary differential equations (ODEs). The Lie algebraic classification of such differential equations is now well-known from the works of Lie [10] as well as recently Mahomed and Leach [19]. However, the algebraic properties of first integrals are not known except in the maximal cases for the basic first integrals and some of their quotients. Here our intention is to investigate the complete problem for scalar second-order and maximal symmetry classes of higher-order ODEs using Lie algebras and Lie symmetry methods. We invoke the realizations of low-dimensional Lie algebras. Symmetries of the fundamental first integrals for scalar second-order ODEs which are linear or linearizable by point transformations have already been obtained. Firstly we show how one can determine the relationship between the point symmetries and the first integrals of linear or linearizable scalar ODEs of order two. Secondly, a complete classi cation of point symmetries of first integrals of such linear ODEs is studied. As a consequence, we provide a counting theorem for the point symmetries of first integrals of scalar linearizable second-order ODEs. We show that there exists the 0, 1, 2 or 3 point symmetry cases. It is proved that the maximal algebra case is unique. By use of Lie symmetry group methods we further analyze the relationship between the first integrals of the simplest linear third-order ODEs and their point symmetries. It is well-known that there are three classes of linear third-order ODEs for maximal and submaximal cases of point symmetries which are 4, 5 and 7. The simplest scalar linear third-order equation has seven point symmetries. We obtain the classifying relation between the symmetry and the first integral for the simplest equation. It is shown that the maximal Lie algebra of a first integral for the simplest equation y000 = 0 is unique and four-dimensional. Moreover, we show that the Lie algebra of the simplest linear third-order equation is generated by the symmetries of the two basic integrals. We also obtain counting theorems of the symmetry properties of the first integrals for such linear third-order ODEs of maximal type. Furthermore, we provide insights into the manner in which one can generate the full Lie algebra of higher-order ODEs of maximal symmetry from two of their basic integrals. The relationship between rst integrals of sub-maximal linearizable third-order ODEs and their symmetries are investigated as well. All scalar linearizable third-order equations can be reduced to three classes by point transformations. We obtain the classifying relations between the symmetries and the first integral for sub-maximal cases of linear third-order ODEs. It is known, from the above, that the maximum Lie algebra of the first integral is achieved for the simplest equation. We show that for the other two classes they are not unique. We also obtain counting theorems of the symmetry properties of the rst integrals for these classes of linear third-order ODEs. For the 5 symmetry class of linear third-order ODEs, the first integrals can have 0, 1, 2 and 3 symmetries and for the 4 symmetry class of linear third-order ODEs they are 0, 1 and 2 symmetries respectively. In the case of sub-maximal linear higher-order ODEs, we show that their full Lie algebras can be generated by the subalgebras of certain basic integrals. For the n+2 symmetry class, the symmetries of the rst integral I2 and a two-dimensional subalgebra of I1 generate the symmetry algebra and for the n + 1 symmetry class, the full algebra is generated by the symmetries of I1 and a two-dimensional subalgebra of the quotient I3=I2. Finally, we completely classify the first integrals of scalar nonlinear second-order ODEs in terms of their Lie point symmetries. This is performed by first obtaining the classifying relations between point symmetries and first integrals of scalar nonlinear second order equations which admit 1, 2 and 3 point symmetries. We show that the maximum number of symmetries admitted by any first integral of a scalar second-order nonlinear (which is not linearizable by point transformation) ODE is one which in turn provides reduction to quadratures of the underlying dynamical equation. We provide physical examples of the generalized Emden-Fowler, Lane-Emden and modi ed Emden equations.

This thesis consists of two parts. In the first we prove that the Khovanov-Lauda-Rouquier algebras $R_\alpha$ of finite type are (graded) affine cellular in the sense of Koenig and Xi. In fact, we establish a stronger property, namely that the affine cell ideals in $R_\alpha$ are generated by idempotents. This in particular implies the (known) result that the global dimension of $R_\alpha$ is finite. In the second part we use the presentation of the Specht modules given by Kleshchev-Mathas-Ram to derive results about Specht modules. In particular, we determine all homomorphisms from an arbitrary Specht module to a fixed Specht module corresponding to any hook partition. Along the way, we give a complete description of the action of the standard KLR generators on the hook Specht module. This work generalizes a result of James. This dissertation includes previously published coauthored material.

Orientador: Alice Kimie Miwa Libardi / Banca: Renata Zotin Gomes de Oliveira / Banca: Elaine Cristina Catapani Poletti / Resumo: O objetivo dessa dissertação é apresentar algumas observações sobre os tensores de segunda ordem, da forma como são vistos nas engenharias, usando conceitos de Álgebra Linear. Trata-se de um estudo introdutório, que pretende motivar os professores de matemática que atuam em cursos de engenharia a tornarem suas aulas com enfoque maior na área de atuação dos alunos / Abstract: The main goal of this dissertation is to present remarks on tensor, from the engineer's point of view by using Linear Algebra. Actually, the study is quite introductory but aims to motivate mathematics teachers in engineering courses to make their classes more focused on the area of their activity / Mestre

O objetivo principal dessa dissertação é o estudo do comportamento de ideais diferenciais com respeito à importantes temas de álgebra comutativa como decomposição primária e localização. Veremos que dado um ideal diferencial em um anel noetheriano de característica zero, seus primos associados também serão diferenciais e que ele admite uma decomposição primária cujas componentes são diferenciais. Em relação a localização, teremos uma equivalência dos conceitos de ideais diferenciais no anel dado e no localizado, ou seja, um ideal é diferencial se, e somente se, sua localização também o é. / The main goal of this dissertation is to study the behavior of differential ideals regarding important themes of commutative algebra such as primary decomposition and localization. We will see that, given a differential ideal in a noetherian ring of caracteristic zero, its associated primes ideals will also be differentials and we will exhibit a primary decomposition whose components will be differentials too. In relation to localization, we will have an equivalence of the concepts of differential ideals in the given ring and in the localized ring, that is, an ideal is differential if, and only if, its localization is differential too.

by Leung Chi Wai. / Thesis (M.Ph.)--Chinese University of Hong Kong, 1987. / Bibliography: leaf [51]

我们研究了P[subscript g]=0 的复曲面x 上的ADE 向量丛和仿射ADE 向量丛。 / 首先，我们假设x 上有一个ADE 奇异点。这个奇异点在极小分解Y 中的例外轨迹是一条相应形式的ADE 曲线。利用这条ADE 曲线和向量丛的扩张，我们构造了Y 上的一个ADE 向量丛，而且这个向量丛可以下降到x上。此外，我们利用Y 上( -1)- 曲线的组合，描述了他们的极小表示向量丛。 / 其次，我们假设x 是一个椭圆曲面，而且x 上有一个仿射ADE 形式的奇异纤维。类似于以前，我们构造了X 上的一个仿射ADE 向量丛，而且这个向量丛在这条仿射ADE 曲线上的每一个不可约成分上都是平凡的。 / 然后，当X 是P²上突起n ≤9 个点时, x 上有一个典型的En 向量丛。我们详细的研究了x 的几何和这个E[subscript n] 向量丛的可变形性之间的关系。 / We study ADE and affine ADE bundles over complex surfaces X with P[subscript g] = 0. / First, we suppose X admits an ADE singularity. The exceptional locus of this singularity in the minimal resolution Y is an ADE curve of corresponding type. Using this ADE curve and bundle extensions, we construct an ADE bundle over Y which can descend to X. Furthermore, we describe their minuscule representation bundles in terms of configuration of (reducible) (-1)-curves. / Second, we assume X is an elliptic surface with a singular fiber of affine ADE type. Similar to above studies, we construct the affine ADE bundle over X which is trivial on each irreducible component of the affine ADE curve. / Third, when X is the blowup of P² at n ≤9 points, there is a canonical E[subscript n] bundle over it. We give a detailed study of the relationship between the geometry of X and the deformability of this bundle. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Chen, Yunxia. / On t.p. "g" is subscript. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2013. / Includes bibliographical references (leaves 84-87). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts also in Chinese. / Chapter I --- ADE bundles --- p.9 / Chapter 1 --- ADE Lie algebra bundles --- p.10 / Chapter 1.1 --- ADE singularities --- p.10 / Chapter 1.2 --- ADE bundles --- p.12 / Chapter 2 --- Minuscule representations and ( -1)-curves --- p.16 / Chapter 2.1 --- Standard representations --- p.16 / Chapter 2.2 --- Minuscule representations --- p.17 / Chapter 2.3 --- Configurations of ( -1)-curves --- p.17 / Chapter 2.4 --- Minuscule representations from ( -1)-curves --- p.19 / Chapter 2.5 --- Bundles from ( -1)-curves --- p.21 / Chapter 2.6 --- Outline of Proofs for g ≠E₈ --- p.22 / Chapter 3 --- A[subscript n] case --- p.24 / Chapter 3.1 --- A[subscript n] standard representation bundle Lη^(An,Cn+1) --- p.24 / Chapter 3.2 --- An Lie algebra bundle Sη^(An) --- p.28 / Chapter 3.3 --- An minuscule representation bundle Lη^(An,^kCn+1) --- p.28 / Chapter 4 --- Dn case --- p.30 / Chapter 4.1 --- Dn standard representation bundle Lη^(Dn;C2n) --- p.30 / Chapter 4.2 --- Dn Lie algebra bundle Sη^(Dn) --- p.34 / Chapter 4.3 --- Dn spinor representation bundles Lη^(Dn;S±06) --- p.34 / Chapter 5 --- En case --- p.39 / Chapter 5.1 --- E₆ case --- p.39 / Chapter 5.2 --- E₇ case --- p.42 / Chapter 5.3 --- E₈ case --- p.44 / Chapter 6 --- Proof of Theorem 1.2.1 --- p.45 / Chapter II --- Affine ADE bundles --- p.50 / Chapter 7 --- Affine ADE Lie algebra bundles --- p.51 / Chapter 7.1 --- Affine ADE curves --- p.51 / Chapter 7.2 --- Affine ADE bundles --- p.53 / Chapter 8 --- Trivialization of E₀ gover Ci's after deformations --- p.57 / Chapter 8.1 --- Trivializations in loop ADE cases --- p.58 / Chapter 8.2 --- Trivializations in affine ADE cases --- p.60 / Chapter 8.3 --- Proof (except the loop E₈ case) --- p.60 / Chapter 8.4 --- Proof for the loop E₈ case --- p.62 / Chapter III --- Deformability --- p.65 / Chapter 9 --- En-bundle over Xn with n≤9 --- p.66 / Chapter 9.1 --- En-bundle over Xn with n ≤ 9 --- p.66 / Chapter 9.2 --- Deformability of such E₀E₈ --- p.68 / Chapter 9.3 --- Negative curves in X9 --- p.70 / Chapter 9.4 --- Proof of Theorems 9.2.1 and 9.2.2 --- p.75 / Chapter A --- Minuscule configurations --- p.78 / Chapter B --- A ffine Lie algebras --- p.80

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1980. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE. / Vita. / Bibliography: leaves 179-181. / by Stephen Mark Paneitz. / Ph.D.

Thesis. 1976. Ph.D.--Massachusetts Institute of Technology. Dept. of Mathematics. / Microfiche copy available in Archives and Science. / Vita. / Bibliography: leaves 184-186. / by David Vogan. / Ph.D.

No description available.

We calculate the second and third integral homology of arbitrary finite rank Coxeter groups. The first of these calculations refines a theorem of Howlett, the second is entirely new. We then prove that families of Artin monoids, which have the braid monoid as a submonoid, satisfy homological stability. When the K(π,1) conjecture holds this gives a homological stability result for the associated families of Artin groups. In particular, we recover a classic result of Arnol'd.