Applied Mathematics Commons^™

Unconstrained L1 Optimization With Applications To Signal And Image Processing, Carlos Andres Ramirez

Open Access Theses & Dissertations

In recent years, the applied mathematical community has witnessed a revolution that is changing the paradigm of classical signal and image processing. Novel and e efficient numerical algorithms have emerged for solving new challenges in large scale signal retrieval, where both constrained and unconstrained L1 minimization methods play a fundamental role.

In this work, we present a new methodology for solving unconstrained L1 minimization problems in the context of image and signal processing. Our approach consists in solving a sequence of relaxed unconstrained minimization problems depending on a positive regularization parameter that converges to zero. The optimality conditions of each …

An Experimental Testbed For Multi-Robot Tracking Of Manifolds And Coherent Structures In Flows, Matthew Michini, Kenneth Mallory, Dennis Larkin, M. Ani Hsieh, Eric Forgoston, Philip A. Yecko

Department of Applied Mathematics and Statistics Faculty Scholarship and Creative Works

In this paper, we describe the development of an experimental testbed capable of producing controllable ocean-like flows in a laboratory setting. The objective is to develop a testbed to evaluate multi-robot strategies for tracking manifolds and Lagrangian coherent structures (LCS) in the ocean. Recent theoretical results have shown that LCS coincide with minimum energy and minimum time optimal paths for autonomous vehicles in the ocean. Furthermore, knowledge of these structures enables the prediction and estimation of the underlying fluid dynamics. The testbed is a scaled flow tank capable of generating complex and controlled quasi-2D flow fields that exhibit wind-driven doublegyre …

Measure-Dependent Stochastic Nonlinear Beam Equations Driven By Fractional Brownian Motion, Mark A. Mckibben

Mathematics Faculty Publications

We study a class of nonlinear stochastic partial differential equations arising in themathematicalmodeling of the transverse motion of an extensible beam in the plane. Nonlinear forcing terms of functional-type and those dependent upon a family of probability measures are incorporated into the initial-boundary value problem (IBVP), and noise is incorporated into the mathematical description of the phenomenon via a fractional Brownian motion process. The IBVP is subsequently reformulated as an abstract second-order stochastic evolution equation driven by a fractional Brownian motion (fBm) dependent upon a family of probability measures in a real separableHilbert space and is studied using the tools …

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 …

Two-Dimensional Hydrodynamic Modeling Of Two-Phase Flow For Understanding Geyser Phenomena In Urban Stormwater System, Zhiyu S. Shao

Theses and Dissertations--Civil Engineering

During intense rain events a stormwater system can fill rapidly and undergo a transition from open channel flow to pressurized flow. This transition can create large discrete pockets of trapped air in the system. These pockets are pressurized in the horizontal reaches of the system and then are released through vertical vents. In extreme cases, the transition and release of air pockets can create a geyser feature.

The current models are inadequate for simulating mixed flows with complicated air-water interactions, such as geysers. Additionally, the simulation of air escaping in the vertical dropshaft is greatly simplified, or completely ignored, in …

When Cp(X) Is Domain Representable, William Fleissner, Lynne Yengulalp

Mathematics Faculty Publications

Let M be a metrizable group. Let G be a dense subgroup of M^X . If G is domain representable, then G = MX . The following corollaries answer open questions. If X is completely regular and C_p(X) is domain representable, then X is discrete. If X is zero-dimensional, T₂ , and C_p(X;D) is subcompact, then X is discrete.

An Implicit Interface Boundary Integral Method For Poisson’S Equation On Arbitrary Domains, Catherine Kublik, Nicolay M. Tanushev, Richard Tsai

Mathematics Faculty Publications

We propose a simple formulation for constructing boundary integral methods to solve Poisson’s equation on domains with smooth boundaries defined through their signed distance function. Our formulation is based on averaging a family of parameterizations of an integral equation defined on the boundary of the domain, where the integrations are carried out in the level set framework using an appropriate Jacobian. By the coarea formula, the algorithm operates in the Euclidean space and does not require any explicit parameterization of the boundaries. We present numerical results in two and three dimensions.

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 …

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

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.

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 G_j on infinitesimal neighborhoods X_j so that Ker(G₁ &rarr G₀)= TG₀, Here TG₀ should be the Cousin resolution of TK_m(O_X) and X is any n-dimensional smooth projective variety over a field k, chark=0. Our main results are as follows. The existence of G_j is discussed in chapter 3, following [8] and [18]. The main theorems are theorem5.2.5, theorem 5.2.6 and theorem …

Clustering Methods And Their Applications To Adolescent Healthcare Data, Morgan Mayer-Jochimsen

Scripps Senior Theses

Clustering is a mathematical method of data analysis which identifies trends in data by efficiently separating data into a specified number of clusters so is incredibly useful and widely applicable for questions of interrelatedness of data. Two methods of clustering are considered here. K-means clustering defines clusters in relation to the centroid, or center, of a cluster. Spectral clustering establishes connections between all of the data points to be clustered, then eliminates those connections that link dissimilar points. This is represented as an eigenvector problem where the solution is given by the eigenvectors of the Normalized Graph Laplacian. Spectral clustering …

Stability Of Large Flocks: An Example, J. J. P. Veerman, F. M. Tangerman

Mathematics and Statistics Faculty Publications and Presentations

The movement of a flock with a single leader (and a directed path from it to every agent) can be stabilized. Nonetheless for large flocks perturbations in the movement of the leader may grow to a considerable size as they propagate throughout the flock and before they die out over time. As an example we consider a string of N+1 oscillators moving in the line. Each one `observes' the relative velocity and position of only its nearest neighbors. This information is then used to determine its own acceleration. Now we fix all parameters except the number of oscillators. We then …

Applications Of Fourier Analysis To Audio Signal Processing: An Investigation Of Chord Detection Algorithms, Nathan Lenssen

CMC Senior Theses

The discrete Fourier transform has become an essential tool in the analysis of digital signals. Applications have become widespread since the discovery of the Fast Fourier Transform and the rise of personal computers. The field of digital signal processing is an exciting intersection of mathematics, statistics, and electrical engineering. In this study we aim to gain understanding of the mathematics behind algorithms that can extract chord information from recorded music. We investigate basic music theory, introduce and derive the discrete Fourier transform, and apply Fourier analysis to audio files to extract spectral data.

Uncertainty Quantification Of Film Cooling Effectiveness In Gas Turbines, Hessam Babaee

LSU Master's Theses

In this study the effect of uncertainty of velocity ratio on jet in crossflow and particual- rly film cooling performance is studied. Direct numerical simulations have been combined with a stochastic collocation approach where the parametric space is discretized using Multi-Element general Polynomial Chaos (ME-gPC) method. Velocity ratio serves as a bifurcation parameter in a jet in a crossflow and the dynamical system is shown to have several bifurcations. As a result of the bifurcations, the target functional is observed to have low-regularity with respect to the paramteric space. In that sense, ME-gPC is particularly effective in discretizing the parametric …

Statistical Classification Problems In Assessment Of Teachers, Xuan Wang

LSU Master's Theses

Classification and regression trees form an important and indispensable tool in data analysis and classification problems. Class trees are described in detail with examples. The method is applied to a data set pertaining to evaluation of teachers. In addition, two other classification methods, bagging and AdaBoost are explained. These methods improve existing classifiers to nearly optimal classifiers.

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 …

Mathematical Modelling Of Internal Heat Recovery In Flash Tank Heat Exchanger Cascades, Andrei Korobeinikov, John E. Mccarthy, Emma Mooney, Krum Semkov, James Varghese

Mathematics Faculty Research

Flash tank evaporation combined with a condensing heat exchanger can be used when heat exchange is required between two streams and where at least one of these streams is difficult to handle (tends severely to scale, foul, causing blockages). To increase the efficiency of heat exchange, a cascade of these units in series can be used. Heat transfer relationships in such a cascade are very complex due to their interconnectivity, thus the impact of any changes proposed is difficult to predict. Moreover, the distribution of loads and driving forces in different stages and the number of designed stages faces tradeoffs …

Group Developed Weighing Matrices, K. T. Arasu, Jeffrey R. Hollon

Mathematics and Statistics Faculty Publications

A weighing matrix is a square matrix whose entries are 1, 0 or −1, such that the matrix times its transpose is some integer multiple of the identity matrix. We examine the case where these matrices are said to be devel- oped by an abelian group. Through a combination of extending previous results and by giving explicit constructions we will answer the question of existence for 318 such matrices of order and weight both below 100. At the end, we are left with 98 open cases out of a possible 1,022. Further, some of the new results provide insight into …

Immersed Finite Element Method For Interface Problems With Algebraic Multigrid Solver, Wenqiang Feng

Masters Theses

"This thesis is to discuss the bilinear and 2D linear immersed finite element (IFE) solutions generated from the algebraic multigrid solver for both stationary and moving interface problems. In contrast to the body-fitting mesh restriction of the traditional finite element methods or finite difference methods for interface problems, a number of numerical methods based on structured meshes independent of the interface have been developed. When these methods are applied to the real world applications, we often need to solve the corresponding large scale linear systems many times, which demands efficient solvers. The algebraic multigrid (AMG) method is a natural choice …