Skein Theory And Topological Quantum Field Theory,
2013
Louisiana State University and Agricultural and Mechanical College
Skein Theory And Topological Quantum Field Theory, Xuanting Cai
LSU Doctoral Dissertations
Skein modules arise naturally when mathematicians try to generalize the Jones polynomial of knots. In the first part of this work, we study properties of skein modules. The Temperley-Lieb algebra and some of its generalizations are skein modules. We construct a bases for these skein modules. With this basis, we are able to compute some gram determinants of bilinear forms on these skein modules. Also we use this basis to prove that the Mahler measures of colored Jones polynomial of a sequence of knots converges to the Mahler measure of some two variable polynomial. The topological quantum field theory constructed …
Multiplicity Formulas For Perverse Coherent Sheaves On The Nilpotent Cone,
2013
Louisiana State University and Agricultural and Mechanical College
Multiplicity Formulas For Perverse Coherent Sheaves On The Nilpotent Cone, Myron Minn-Thu-Aye
LSU Doctoral Dissertations
Arinkin and Bezrukavnikov have given the construction of the category of equivariant perverse coherent sheaves on the nilpotent cone of a complex reductive algebraic group. Bezrukavnikov has shown that this category is in fact weakly quasi-hereditary with Andersen--Jantzen sheaves playing a role analogous to that of Verma modules in category O for a semi-simple Lie algebra. Our goal is to show that the category of perverse coherent sheaves possesses the added structure of a properly stratified category, and to use this structure to give an effective algorithm to compute multiplicities of simple objects in perverse coherent sheaves. The algorithm is …
Mixed Categories, Formality For The Nilpotent Cone, And A Derived Springer Correspondence,
2013
Louisiana State University and Agricultural and Mechanical College
Mixed Categories, Formality For The Nilpotent Cone, And A Derived Springer Correspondence, Laura Joy Rider
LSU Doctoral Dissertations
Recall that the Springer correspondence relates representations of the Weyl group to perverse sheaves on the nilpotent cone. We explain how to extend this to an equivalence between the triangulated category generated by the Springer perverse sheaf and the derived category of di_x000B_erential graded modules over a dg-ring related to the Weyl group
Higher Algebraic K-Theory And Tangent Spaces To Chow Groups,
2013
Louisiana State University and Agricultural and Mechanical College
Higher Algebraic K-Theory And Tangent Spaces To Chow Groups, Sen Yang
LSU Doctoral Dissertations
In this work, using higher algebraic K-theory, we provide an answer to the following question asked by Green-Griffiths in [13]: Can one define the Bloch-Gersten-Quillen sequence Gj on infinitesimal neighborhoods Xj so that Ker(G1 &rarr G0)= TG0, Here TG0 should be the Cousin resolution of TKm(OX) and X is any n-dimensional smooth projective variety over a field k, chark=0. Our main results are as follows. The existence of Gj is discussed in chapter 3, following [8] and [18]. The main theorems are theorem5.2.5, theorem 5.2.6 and theorem …
The Ring Theory And The Representation Theory Of Quantum Schubert Cells,
2013
Louisiana State University and Agricultural and Mechanical College
The Ring Theory And The Representation Theory Of Quantum Schubert Cells, Joel Benjamin Geiger
LSU Doctoral Dissertations
In recent years the quantum Schubert cell algebras, introduced by Lusztig and De Concini--Kac, and Procesi, have garnered much interest as this versatile class of objects are furtive testing grounds for noncommutative algebraic geometry. We unify the two main approaches to analyzing the structure of the torus-invariant prime spectra of quantum Schubert cell algebras, a ring theoretic one via Cauchon's deleting derivations and a representation theoretic characterization of Yakimov via Demazure modules. As a result one can combine the strengths of the two approaches. In unifying the theories, we resolve two questions of Cauchon and Mériaux, one of which involves …
A Characterization Of Almost All Minimal Not Nearly Planar Graphs,
2013
Louisiana State University and Agricultural and Mechanical College
A Characterization Of Almost All Minimal Not Nearly Planar Graphs, Kwang Ju Choi
LSU Doctoral Dissertations
In this dissertation, we study nearly planar graphs, that is, graphs that are edgeless or have an edge whose deletion results in a planar graph. We show that all but finitely many graphs that are not nearly planar and do not contain one particular graph have a well-understood structure based on large Möbius ladders.
Finite Element Methods For Fourth Order Variational Inequalities,
2013
Louisiana State University and Agricultural and Mechanical College
Finite Element Methods For Fourth Order Variational Inequalities, Yi Zhang
LSU Doctoral Dissertations
In this work we study finite element methods for fourth order variational inequalities. We begin with two model problems that lead to fourth order obstacle problems and a brief survey of finite element methods for these problems. Then we review the fundamental results including Sobolev spaces, existence and uniqueness results of variational inequalities, regularity results for biharmonic problems and fourth order obstacle problems, and finite element methods for the biharmonic problem. In Chapter 2 we also include three types of enriching operators which are useful in the convergence analysis. In Chapter 3 we study finite element methods for the displacement …
Large Deviations For Stochastic Navier-Stokes Equations With Nonlinear Viscosities,
2013
Louisiana State University and Agricultural and Mechanical College
Large Deviations For Stochastic Navier-Stokes Equations With Nonlinear Viscosities, Ming Tao
LSU Doctoral Dissertations
In this work, a Wentzell-Freidlin type large deviation principle is established for the two-dimensional stochastic Navier-Stokes equations (SNSE's) with nonlinear viscosities. We fi_x000C_rst prove the existence and uniqueness of solutions to the two-dimensionalstochastic Navier-Stokes equations with nonlinear viscosities using the martingale problem argument and the method of monotonicity. By the results of Varadhan and Bryc, the large deviation principle (LDP) is equivalent to the Laplace-Varadhan principle (LVP) if the underlying space is Polish. Then using the stochastic control and weak convergence approach developed by Budhiraja and Dupuis, the Laplace-Varadhan principle for solutions of stochastic Navier-Stokesequations is obtained in appropriate function …
A Semigroup/Laplace Transform Approach To Approximating Flows,
2013
Louisiana State University and Agricultural and Mechanical College
A Semigroup/Laplace Transform Approach To Approximating Flows, Ladorian Nichele Latin
LSU Doctoral Dissertations
It is well known that all flows in a state space O induce a semigroup of linear operators on an appropriately chosen vector space of functions (observables) from O into a vector space Z (observations). After choosing appropriate continuity assumptions on the flow, the associated semigroup will be strongly continuous and will have a linear, infinitesimal generator A. The purpose of this dissertation is to explore approximation methods for linear semigroups and/or Laplace transform inversion methods in order to reconstruct the flow starting with the linear generator A . In preparing for these investigations, we collect some of the essential …
Refining The Characterization Of Projective Graphs,
2013
Louisiana State University and Agricultural and Mechanical College
Refining The Characterization Of Projective Graphs, Perry K. Iverson
LSU Doctoral Dissertations
Archdeacon showed that the class of graphs embeddable in the projective plane is characterized by a set of 35 excluded minors. Robertson, Seymour and Thomas in an unpublished result found the excluded minors for the class of k-connected graphs embeddable on the projective plane for k = 1,2,3. We give a short proof of that result and then determine the excluded minors for the class of internally 4-connected projective graphs. Hall showed that a 3-connected graph diff_x000B_erent from K5 is planar if and only if it has K3,3 as a minor. We provide two analogous results for projective graphs. For …
Extra Structures On Three-Dimensional Cobordisms,
2013
Louisiana State University and Agricultural and Mechanical College
Extra Structures On Three-Dimensional Cobordisms, Xuanye Wang
LSU Doctoral Dissertations
A Topological Quantum Field Theory (TQFT) is a functor from a cobordism category to the category of vector spaces, satisfying certain properties. An important property is that the vector spaces should be finite dimensional. For the WRT TQFT, the relevant 2 + 1-cobordism category is built from manifolds which are equipped with an extra structure such as a p1-structure, or an extended manifold structure. In chapter 1, we perform the universal construction of [3] on a cobordism category without this extra structure and show that the resulting quantization functor assigns an infinite dimensional vector space to the torus. In chapter …
Project Haiti 2012: Providing An Experiential Learning Experience Through The Design And Delivery Of A Water Purifier In Haiti,
2013
Embry-Riddle Aeronautical University
Project Haiti 2012: Providing An Experiential Learning Experience Through The Design And Delivery Of A Water Purifier In Haiti, Yung Wong, Johnathon Camp, Shavin Pinto, Kyle Fennesy, Marc Compere, Yan Tang
Publications
In this paper, we share our experiences and lessons learned from Project Haiti 2012, a project to design and install a water purification system serving 20,000 people per day in the largest tent city in Haiti. Project Haiti 2012 was the third and largest system we have built for Haitians and represents a huge success for all participants and stakeholders. This paper discusses the unique experiential learning opportunity involved in the design and delivery of the water purifier in a foreign developing country. Multiple positive educational, social, and economic outcomes were achieved including students applying knowledge gained from coursework towards …
Nonlinear Techniques For Stochastic Systems Of Differential Equations,
2013
University of South Florida
Nonlinear Techniques For Stochastic Systems Of Differential Equations, Tadesse G. Zerihun
USF Tampa Graduate Theses and Dissertations
Two of the most well-known nonlinear methods for investigating nonlinear dynamic processes in sciences and engineering are nonlinear variation of constants parameters and comparison method. Knowing the existence of solution process, these methods provide a very powerful tools for investigating variety of problems, for example, qualitative and quantitative properties of solutions, finding error estimates between solution processes of stochastic system and the corresponding nominal system, and inputs for the designing engineering and industrial problems. The aim of this work is to systematically develop mathematical tools to undertake the mathematical frame-work to investigate a complex nonlinear nonstationary stochastic systems of differential …
G-Strands And Peakon Collisions On Diff(R),
2013
Imperial College London
G-Strands And Peakon Collisions On Diff(R), Darryl Holm, Rossen Ivanov
Articles
A G-strand is a map g : R x R --> G for a Lie group G that follows from Hamilton's principle for a certain class of G-invariant Lagrangians. Some G-strands on finite-dimensional groups satisfy 1+1 space-time evolutionary equations that admit soliton solutions as completely integrable Hamiltonian systems. For example, the SO(3)-strand equations may be regarded physically as integrable dynamics for solitons on a continuous spin chain. Previous work has shown that G-strands for diffeomorphisms on the real line possess solutions with singular support (e.g. peakons). This paper studies collisions of such singular solutions of G-strands when G = Diff( …
The Lismullin Enclosure:A Designed Ritual Space,
2013
Technological University Dublin
The Lismullin Enclosure:A Designed Ritual Space, Frank Prendergast
Book/Book Chapter
The discovery in 2007 of a prehistoric post-built enclosure at Lismullin, Co. Meath, during archaeological investigations in advance of the construction of the M3 motorway is, arguably, the most significant Irish archaeological discovery of recent times. This appendix summarises a commissioned specialist report on the spatial and archaeoastronomical features of the enclosure.
Application Of Helmholtz/Hodge Decomposition To Finite Element Methods For Two-Dimensional Maxwell's Equations,
2013
Louisiana State University and Agricultural and Mechanical College
Application Of Helmholtz/Hodge Decomposition To Finite Element Methods For Two-Dimensional Maxwell's Equations, Zhe Nan
LSU Doctoral Dissertations
In this work we apply the two-dimensional Helmholtz/Hodge decomposition to develop new finite element schemes for two-dimensional Maxwell's equations. We begin with the introduction of Maxwell's equations and a brief survey of finite element methods for Maxwell's equations. Then we review the related fundamentals in Chapter 2. In Chapter 3, we discuss the related vector function spaces and the Helmholtz/Hodge decomposition which are used in Chapter 4 and 5. The new results in this dissertation are presented in Chapter 4 and Chapter 5. In Chapter 4, we propose a new numerical approach for two-dimensional Maxwell's equations that is based on …
Full Newton Step Interior Point Method For Linear Complementarity Problem Over Symmetric Cones,
2013
Georgia Southern University
Full Newton Step Interior Point Method For Linear Complementarity Problem Over Symmetric Cones, Andrii Berdnikov
College of Graduate Studies: Theses & Dissertations
In this thesis, we present a new Feasible Interior-Point Method (IPM) for Linear Complementarity Problem (LPC) over Symmetric Cones. The advantage of this method lies in that it uses full Newton-steps, thus, avoiding the calculation of the step size at each iteration. By suitable choice of parameters we prove the global convergence of iterates which always stay in the the central path neighborhood. A global convergence of the method is proved and an upper bound for the number of iterations necessary to find ε-approximate solution of the problem is presented.
Generalized Analytic Fourier-Feynman Transform Of Functionals In A Banach Algebra F_(A1,A2)^(A,B),
2013
Dankook University
Generalized Analytic Fourier-Feynman Transform Of Functionals In A Banach Algebra F_(A1,A2)^(A,B), Jae Gil Choi, David Skough, Seung Jun Chang
Department of Mathematics: Faculty Publications
We introduce the Fresnel type class F_(A1,A2)^(a,b).We also establish the existence of the generalized analytic Fourier-Feynman transform for functionals in the Banach algebra F_(A1,A2)^(a,b).
A Study Of Nonlinear Dynamics In Mathematical Biology,
2013
University of North Florida
A Study Of Nonlinear Dynamics In Mathematical Biology, Joseph Ferrara
UNF Graduate Theses and Dissertations
We first discuss some fundamental results such as equilibria, linearization, and stability of nonlinear dynamical systems arising in mathematical modeling. Next we study the dynamics in planar systems such as limit cycles, the Poincaré-Bendixson theorem, and some of its useful consequences. We then study the interaction between two and three different cell populations, and perform stability and bifurcation analysis on the systems. We also analyze the impact of immunotherapy on the tumor cell population numerically.
Cancer Quasispecies And Stem-Like Adaptive Aneuploidy,
2013
Chapman University
Cancer Quasispecies And Stem-Like Adaptive Aneuploidy, Domenico Napoletani, M. Signore, Daniele C. Struppa
Mathematics, Physics, and Computer Science Faculty Articles and Research
In this paper we develop a theoretical frame to understand self-regulation of aneuploidy rate in cancer and stem cells. This is accomplished building upon quasispecies theory, by leaving its formal mathematical structure intact, but by drastically changing the meaning of its objects. In particular, we propose a novel definition of chromosomal master sequence, as a sequence of physically distinct whole or fragmented chromosomes, whose length is taken to be the sum of the copy numbers of each whole or fragmented chromosome. This fundamental change in the functional objects of quasispecies theory allows us to show that previously measured aneuploidy rates …
