Open Access. Powered by Scholars. Published by Universities.®
Physical Sciences and Mathematics Commons™
Open Access. Powered by Scholars. Published by Universities.®
- Keyword
-
- Value function (2)
- A^1-Homotopy Theory (1)
- Algebraic Geometry (1)
- Bloch Waves (1)
- Cavitation (1)
-
- Conformal mapping (1)
- Correctors (1)
- Curve complex (1)
- Discontinuous Galerkin methods (1)
- Dispersed media (1)
- Graded meshes (1)
- Hamilton-Jacobi (1)
- Hermitian K-Theory (1)
- Homogenization (1)
- Invariance (1)
- J-invariant (1)
- Knot theory (1)
- Koszul duality (1)
- Layered media (1)
- Mapping class group (1)
- Maxwell's equations (1)
- Mayer problem (1)
- Multigrid algorithms (1)
- Nilpotent cone (1)
- Nonconforming finite element methods (1)
- Nonsmooth analysis (1)
- Normalizing field (1)
- Orthogonal Grassmannians (1)
- P-Laplacian (1)
- Periodic Media (1)
Articles 1 - 12 of 12
Full-Text Articles in Physical Sciences and Mathematics
Method Of Riemann Surfaces In Modelling Of Cavitating Flow, Anna Zemlyanova
Method Of Riemann Surfaces In Modelling Of Cavitating Flow, Anna Zemlyanova
LSU Doctoral Dissertations
This dissertation is concerned with the applications of the Riemann-Hilbert problem on a hyperelliptic Riemann surface to problems on supercavitating flows of a liquid around objects. For a two-dimensional steady irrotational flow of liquid it is possible to introduce a complex potential w(z) which allows to apply the powerful methods of complex analysis to the solution of fluid mechanics problems. In this work problems on supercavitating flows of a liquid around one or two wedges have been stated. The Tulin single-spiral-vortex model is employed as a cavity closure condition. The flow domain is transformed into an auxiliary domain with known …
Multigrid Methods For Maxwell's Equations, Jintao Cui
Multigrid Methods For Maxwell's Equations, Jintao Cui
LSU Doctoral Dissertations
In this work we study finite element methods for two-dimensional Maxwell's equations and their solutions by multigrid algorithms. We begin with a brief survey of finite element methods for Maxwell's equations. Then we review the related fundamentals, such as Sobolev spaces, elliptic regularity results, graded meshes, finite element methods for second order problems, and multigrid algorithms. In Chapter 3, we study two types of nonconforming finite element methods on graded meshes for a two-dimensional curl-curl and grad-div problem that appears in electromagnetics. The first method is based on a discretization using weakly continuous P1 vector fields. The second method uses …
Dimer Models For Knot Polynomials, Moshe Cohen
Dimer Models For Knot Polynomials, Moshe Cohen
LSU Doctoral Dissertations
A dimer model consists of all perfect matchings on a (bipartite) weighted signed graph, where the product of the signed weights of each perfect matching is summed to obtain an invariant. In this paper, the construction of such a graph from a knot diagram is given to obtain the Alexander polynomial. This is further extended to a more complicated graph to obtain the twisted Alexander polynomial, which involved "twisting" by a representation. The space of all representations of a given knot complement into the general linear group of a fixed size can be described by the same graph. This work …
Orthogonal Grassmannians And Hermitian K-Theory In A¹-Homotopy Theory Of Schemes, Girja Shanker Tripathi
Orthogonal Grassmannians And Hermitian K-Theory In A¹-Homotopy Theory Of Schemes, Girja Shanker Tripathi
LSU Doctoral Dissertations
In this work we prove that the hermitian K-theory is geometrically representable in the A^1 -homotopy category of smooth schemes over a field. We also study in detail a realization functor from the A^1 -homotopy category of smooth schemes over the field R of real numbers to the category of topological spaces. This functor is determined by taking the real points of a smooth R-scheme. There is another realization functor induced by taking the complex points with a similar description although we have not discussed this other functor in this dissertation. Using these realization functors we have concluded in brief …
Primes Of The Form X² + Ny² In Function Fields, Piotr Maciak
Primes Of The Form X² + Ny² In Function Fields, Piotr Maciak
LSU Doctoral Dissertations
Let n be a square-free polynomial over F_q, where q is an odd prime power. In this work, we determine which irreducible polynomials p in F_q[x] can be represented in the form X^2+nY^2 with X, Y in F_q[x]. We restrict ourselves to the case where X^2+nY^2 is anisotropic at infinity. As in the classical case over Z, the representability of p by the quadratic form X^2+nY^2 is governed by conditions coming from class field theory. A necessary and almost sufficient condition is that the ideal generated by p splits completely in the Hilbert class field H of K=F_q(x,sqrt(-n)) for the …
Hamilton-Jacobi Theory For Optimal Control Problems On Stratified Domains, Richard Charles Barnard
Hamilton-Jacobi Theory For Optimal Control Problems On Stratified Domains, Richard Charles Barnard
LSU Doctoral Dissertations
This thesis studies optimal control problems on stratified domains. We first establish a known proximal Hamilton-Jacobi characterization of the value function for problems with Lipschitz dynamics. This background gives the motivation for our results for systems over stratified domains, which is a system with non-Lipschitz dynamics that were introduced by Bressan and Hong. We provide an example that shows their attempt to derive a Hamilton-Jacobi characterization of the value function is incorrect, and discuss the nature of their error. A new construction of a multifunction is introduced that possesses properties similar to those of a Lipschitz multifunction, and is used …
Optimal Control And Nonlinear Programming, Qingxia Li
Optimal Control And Nonlinear Programming, Qingxia Li
LSU Doctoral Dissertations
In this thesis, we have two distinct but related subjects: optimal control and nonlinear programming. In the first part of this thesis, we prove that the value function, propagated from initial or terminal costs, and constraints, in the form of a differential equation, satisfy a subgradient form of the Hamilton-Jacobi equation in which the Hamiltonian is measurable with respect to time. In the second part of this thesis, we first construct a concrete example to demonstrate conjugate duality theory in vector optimization as developed by Tanino. We also define the normal cones corresponding to Tanino's concept of the subgradient of …
Homogenization Of Nonlinear Partial Differential Equations, Silvia Jiménez
Homogenization Of Nonlinear Partial Differential Equations, Silvia Jiménez
LSU Doctoral Dissertations
This dissertation is concerned with properties of local fields inside composites made from two materials with different power law behavior. This simple constitutive model is frequently used to describe several phenomena ranging from plasticity to optical nonlinearities in dielectric media. We provide the corrector theory for the strong approximation of fields inside composites made from two power law materials with different exponents. The correctors are used to develop bounds on the local singularity strength for gradient fields inside microstructured media. The bounds are multiscale in nature and can be used to measure the amplification of applied macroscopic fields by the …
Power Series Expansions For Waves In High-Contrast Plasmonic Crystals, Santiago Prado Parentes Fortes
Power Series Expansions For Waves In High-Contrast Plasmonic Crystals, Santiago Prado Parentes Fortes
LSU Doctoral Dissertations
In this thesis, a method is developed for obtaining convergent power series expansions for dispersion relations in two-dimensional periodic media with frequency dependent constitutive relations. The method is based on high-contrast expansions in the parameter _x0011_ = 2_x0019_d=_x0015_, where d is the period of the crystal cell and _x0015_ is the wavelength. The radii of convergence obtained are not too small, on the order of _x0011_ _x0019_ 102. That the method applies to frequency dependent media is an important fact, since the majority of the methods available in the literature are restricted to frequency independent constitutive relations. The convergent series …
Koszul Duality For Multigraded Algebras, Fareed Hawwa
Koszul Duality For Multigraded Algebras, Fareed Hawwa
LSU Doctoral Dissertations
Classical Koszul duality sets up an adjoint pair of functors establishing an equivalence of categories. The equivalence is between the bounded derived category of complexes of graded modules over a graded algebra and the bounded derived category of complexes of graded modules over the quadratic dual graded algebra. This duality can be extended in many ways. We consider here two extensions: first we wish to allow a multigraded algebra, meaning that the algebra can be graded by any abelian group (not just the integers). Second, we will allow filtered algebras. In fact we are considering filtered quadratic algebras with an …
Perverse Poisson Sheaves On The Nilpotent Cone, Jared Lee Culbertson
Perverse Poisson Sheaves On The Nilpotent Cone, Jared Lee Culbertson
LSU Doctoral Dissertations
For a reductive complex algebraic group, the associated nilpotent cone is the variety of nilpotent elements in the corresponding Lie algebra. Understanding the nilpotent cone is of central importance in representation theory. For example, the nilpotent cone plays a prominent role in classifying the representations of finite groups of Lie type. More recently, the nilpotent cone has been shown to have a close connection with the affine flag variety and this has been exploited in the Geometric Langlands Program. We make use of the following important fact. The nilpotent cone is invariant under the coadjoint action of G on the …
Subgroups Of The Torelli Group, Leah R. Childers
Subgroups Of The Torelli Group, Leah R. Childers
LSU Doctoral Dissertations
Let Mod(Sg) be the mapping class group of an orientable surface of genus g, Sg. The action of Mod(Sg) on the homology of Sg induces the well-known symplectic representation:
Mod(Sg) ---> Sp(2g, Z).
The kernel of this representation is called the Torelli group, I(Sg).
We will study two subgroups of I(Sg). First we will look at the subgroup generated by all SIP-maps, SIP(Sg). We will show SIP(Sg) is not I(Sg) and is in fact an infinite index subgroup of I(Sg). We will also classify which SIP-maps are in the kernel of the Johnson homomorphism and Birman-Craggs-Johnson homomorphism.
Then we will …