Browsing by Author "Kailash Misra, Committee Member"

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Classification of K_F-orbits of Unipotent Elements in Symmetric F-varieties of SL(n, F)
(2010-04-07) Wang, Qiang; Robert Bruck, Committee Member; Ernie Stitzinger, Committee Member; Kailash Misra, Committee Member; Aloysius G. Helminck, Committee Chair; Amassa Fauntleroy, Committee Co-Chair
Richardson proved in 1982 that, given an algebraic group G and some involution, we could have only a finite number of K-orbits of unipotent elements in the symmetric variety P = G/K over an algebraically closed field, where K is the fixed point group of the involution. A question arises naturally: what if the field is not algebraically closed? In this thesis we try to answer this question and go a little further by listing all K_F-orbits of unipotent elements in P explicitly. We work on the symmetric F-variety P = G_F/K_F for the special linear group over an arbitrary field F of characteristic not 2. We classify all K_F-orbits of unipotent elements in P for all inner involutions for the special linear group. For Cartan (outer) involution, we classify K-orbits for small n only and illustrate how to get the canonical form for general n. Further proofs are still needed. We also classify G_F-orbits of unipotent elements in G_F.
The Conjugacy Poset of a Reductive Monoid.
(2010-06-08) Therkelsen, Ryan; Mohan Putcha, Committee Chair; Kailash Misra, Committee Member; Ernest Stitzinger, Committee Member; Nathan Reading, Committee Member; Gregory Dawes, Committee Member
Developing a New L-infinity Algebra Using Symmetric Brace Algebras.
(2010-05-04) Capaldi, Mindy; Thomas Lada, Committee Chair; Kailash Misra, Committee Member; Ernest Stitzinger, Committee Member; Amassa Fauntleroy, Committee Member
L(Infinity) Structures on Spaces of Low Dimension
(2004-04-14) Daily, Marilyn Elizabeth; Kailash Misra, Committee Member; Ron Fulp, Committee Member; Jim Stasheff, Committee Member; Tom Lada, Committee Chair
L(Infinity) structures are natural generalizations of Lie algebras, which need satisfy the standard graded Jacobi identity only up to homotopy. They have also been a subject of recent interest in physics, where they occur in closed string theory and in gauge theory. This dissertation classifies all possible L(Infinity) structures which can be constructed on a Z-graded (characteristic 0) vector space of dimension three or less. It also includes necessary and sufficient conditions under which a space with an L(3) structure is a differential graded Lie algebra. Additionally, it is shown that some of these differential graded Lie algebras possess a nontrivial L(n) structure for higher n.
Object-Image Correspondence of Under Projections.
(2010-04-29) Burdis, Joseph; Irina Kogan, Committee Chair; John Seater, Committee Member; Kailash Misra, Committee Member; Ernest Stitzinger, Committee Member
On the Solvability of Nonlinear Boundary Value Problems on Time Scales
(2009-04-27) Kalhorn, Rebecca Isabel Burton; James Selgrade, Committee Member; Ernest Stitzinger, Committee Member; Jesus Rodriguez, Committee Chair; Kailash Misra, Committee Member
In this manuscript we study boundary value problems on time scales. First we will examine weakly nonlinear boundary value problems and analyze problems at resonance; that is, problems where the homogeneous linear boundary value problem has a nontrivial solution space. We establish conditions for the existence of solutions and discuss the dependence of solutions on parameters. Next we consider the existence and properties of solutions to nonlinear dynamic equations of the subject to global boundary conditions. Our most significant result in this section concerns the existence of solutions of problems where the nonlinearity exhibits sublinear growth. Finally we establish sufficient conditions for the solvability of nonlinear scalar two-point boundary value problems at resonance. As a consequence of our results we are able to provide easily verifiable conditions for the existence of periodic behavior for dynamic equations on time scales.
Quantum Symmetric Spaces and Quantum Symplectic Invariants
(2005-07-25) Ray, Robert; Naihuan Jing, Committee Chair; Ernest Stitzinger, Committee Member; Tom Lada, Committee Member; Kailash Misra, Committee Member
Although subgroups of the general linear group are well understood, properties of quantum analogs of these subgroups have been a little more elusive. This is due in part to the fact that these sets are not groups and there does not appear to be a natural way to embed them in a quantum GL(n). With respect to the matrix multiplication, these sets generally fail to be closed, and the elements do not all have inverses. However, the associated quantized regular functions and quantized universal enveloping Lie algebras still retain Hopf algebra structures. It is this structure that was used by Jing and Yamada in 1994 to construct q-analogs of the orthogonal group and the associated q-orthogonal invariants (quantum symmetric algebra). The first two chapters of this text provide the basic background information for the main thesis topic. In chapter 1, we review some basic definitions and properties of Hopf algebras, first discussing algebras and coalgebras (drawing mostly from material by Montgomery and Kassel). Here, the definitions of the regular functions of GLn are recalled and other examples of Hopf algebras are given to illustrate some of the properties. In chapter 2, quantum versions of these Hopf algebras are presented. In their paper, Jing and Yamada use a differential method of defining q-orthogonal invariants of the action of the quantum orthogonal group on Aq(X). In other words, the q-orthogonal invariant subspace is defined as the subspace of Aq(X) that is annihilatad by a q-analog of U(so(n)). In the third chapter, a q-analog of U(sp(n,C)) is constructed and the q-symplectic invariants in Aq(X) are defined relative to the left and right action of this quantum universal symplectic Lie algebra, in a differential fashion similar to Jing and Yamada, where we require n be even. The space of these q-symplectic invariants is then decomposed into right and left irreducible modules and several properties are discussed and we show how these q-symplectic invariants define quantum antisymmetric matrices.
Solving homogeneous linear differential equations of order 4 in terms of equations of smaller order.
(2002-08-20) Person, Axelle Claude; Gabriel Caloz, Committee Member; Michael Singer, Committee Chair; Felix Ulmer, Committee Co-Chair; Kailash Misra, Committee Member; Hoon Hong, Committee Member; Ernie Stitzinger, Committee Member
In this thesis we consider the problem of deciding if a fourth order linear differential equation can be solved in terms of solutions of lower order equations. There is a group theoretic criteria which can be turned into a decision procedure for solving this problem. Once the decision has been made that a certain type of equation can be solved in terms of lower order equations we also give methods for producing the lower order equations used for solving it.
Tannakian Categories and Linear Differential Algebraic Groups
(2007-02-28) Ovchinnikov, Alexey; Irina Kogan, Committee Member; Bojko Bakalov, Committee Member; Kailash Misra, Committee Member; Michael Singer, Committee Chair
Tannaka's Theorem states that a linear algebraic group G is determined by the category of finite dimensional G-modules and the forgetful functor. We extend this result to linear differential algebraic groups by introducing a category corresponding to their representations and show how this category determines such a group. We also provide conditions for a category with a fiber functor to be equivalent to the category of representations of a linear differential algebraic group. This generalizes the notion of a neutral Tannakian category used to characterize the category of representations of a linear algebraic group.
Transformation Semigroups Over Groups
(2008-03-25) Petersen, Richard Francis; Kailash Misra, Committee Member; Ronald Fulp, Committee Member; Tom Lada, Committee Member; Mohan S. Putcha, Committee Chair
The semigroup analogue of the symmetric group, S_{n}, is the full transformation semigroup, T_{n}. T_{n} is the set of all mappings from the set {1,2,..n} to itself. This semigroup has been studied in great detail, especially in connection with automata theory. The wreath product of a group G by S_{n} has been studied for almost one hundred years. In this thesis, we study the wreath product of a group G by T_{n}. These wreath products are expressed as GwrS_{n} and GwrT_{n}, respectively. Many interesting theorems and properties for wreath products will be discussed. For example, the result of John Howie that every element in T_{n} − S_{n} can be expressed as a product of idempotents, is generalized to show that any element of GwrT_{n}- GwrS_{n} can be expressed as a product of idempotents. It will also be shown that GwrT_{n} is unit regular. Chapter five begins with a review of Green's relations for a moniod, M. Green's relations for T_{n} are also reviewed and R and L-classes for the wreath product GwrT_{n} are determined. Finally, in the last two chapters, the conjugacy class structures of GwrT_{n} are determined. Just as the conjugacy classes of GwrS_{n} are indexed by colored partitions, we show that the conjugacy classes of GwrT_{n} are indexed by certain colored directed graphs.