Physical Sciences and Mathematics Commons^™

On The Empirical Balanced Truncation For Nonlinear Systems, Marissa Condon, Rossen Ivanov

Articles

Novel constructions of empirical controllability and observability gramians for nonlinear systems for subsequent use in a balanced truncation style of model reduction are proposed. The new gramians are based on a generalisation of the fundamental solution for a Linear Time-Varying system. Relationships between the given gramians for nonlinear systems and the standard gramians for both Linear Time-Invariant and Linear Time-Varying systems are established as well as relationships to prior constructions proposed for empirical gramians. Application of the new gramians is illustrated through a sample test-system.

On Multivariate Abel-Gontscharoff Interpolation, Tian-Xiao He

Scholarship

By using Gould's annihilation coefficients, we obtain an explicit fundamental polynomials of Multivariate Abel-Gontscharoff Interpolation and its remainder expression.

A Meta-Analysis Of Randomness In Human Behavioral Research, Summer Ann Armstrong

LSU Master's Theses

This work analyzes the concept of randomness in binary sequences from three different perspectives: mathematically, statistically, and psychologically and examines the research on human perception of randomness and the question of whether or not humans can simulate random behavior. Generally, research shows that human subjects have great difficulty producing random sequences, even when they are instructed and motivated. We survey some of the literature and present some leading theoretical proposals. Finally, we present some basic statistical tests that can be used to evaluate randomness in a given binary sequence.

Analysis Of A Multigrid Algorithm For Time Harmonic Maxwell Equations, Jay Gopalakrishnan, Joseph E. Pasciak, Leszek Demkowicz

Mathematics and Statistics Faculty Publications and Presentations

This paper considers a multigrid algorithm suitable for efficient solution of indefinite linear systems arising from finite element discretization of time harmonic Maxwell equations. In particular, a "backslash" multigrid cycle is proven to converge at rates independent of refinement level if certain indefinite block smoothers are used. The method of analysis involves comparing the multigrid error reduction operator with that of a related positive definite multigrid operator. This idea has previously been used in multigrid analysis of indefinite second order elliptic problems. However, the Maxwell application involves a nonelliptic indefinite operator. With the help of a few new estimates, the …

A Characterization Of Hybridized Mixed Methods For Second Order Elliptic Problems, Bernardo Cockburn, Jay Gopalakrishnan

Mathematics and Statistics Faculty Publications and Presentations

In this paper, we give a new characterization of the approximate solution given by hybridized mixed methods for second order self-adjoint elliptic problems. We apply this characterization to obtain an explicit formula for the entries of the matrix equation for the Lagrange multiplier unknowns resulting from hybridization. We also obtain necessary and sufficient conditions under which the multipliers of the Raviart–Thomas and the Brezzi–Douglas–Marini methods of similar order are identical.

Decentralized Control Of Vehicle Formations, Gerardo Lafferriere, Anca Williams, John S. Caughman Iv, J. J. P. Veerman

Mathematics and Statistics Faculty Publications and Presentations

This paper investigates a method for decentralized stabilization of vehicle formations using techniques from algebraic graph theory. The vehicles exchange information according to a pre-specified communication digraph, G. A feedback control is designed using relative information between a vehicle and its in-neighbors in G. We prove that a necessary and sufficient condition for an appropriate decentralized linear stabilizing feedback to exist is that G has a rooted directed spanning tree. We show the direct relationship between the rate of convergence to formation and the eigenvalues of the (directed) Laplacian of G. Various special situations are discussed, including …

On Qualitative Properties And Convergence Of Time-Discretization Methods For Semigroups, Mihaly Kovacs

LSU Doctoral Dissertations

In this dissertation we use functional calculus methods to investigate convergence and qualitative properties of time-discretization methods for strongly continuous semigroups. Stability, convergence, and preservation of contractivity (or norm-bound) of the semigroup under time-discretization is investigated in a Banach space setting. Preservation of positivity, concavity and other qualitative shape properties which can be described via positivity are treated in a Banach lattice framework. The use of the Hille-Phillips (H-P) functional calculus instead of the Dunford-Taylor functional calculus allows us to extend fundamental qualitative results concerning time-discretization methods and simplify their proofs, including results on multi-step schemes and variable step-sizes. We …

A Fast And Simple Algorithm For Computing M-Shortest Paths In State Graph, M. Sherwood, Laxmi P. Gewali, Henry Selvaraj, Venkatesan Muthukumar

Electrical & Computer Engineering Faculty Research

We consider the problem of computing m shortest paths between a source node s and a target node t in a stage graph. Polynomial time algorithms known to solve this problem use complicated data structures. This paper proposes a very simple algorithm for computing all m shortest paths in a stage graph efficiently. The proposed algorithm does not use any complicated data structure and can be implemented in a straightforward way by using only array data structure. This problem appears as a sub-problem for planning risk reduced multiple k-legged trajectories for aerial vehicles.

Class Groups And Norms Of Units, Costel Ionita

LSU Doctoral Dissertations

Our object of study is relative quadratic extensions of algebraic number fields. In 'Class Number Parity', the authors P.E. Conner and J. Hurrelbrink study in detail the cases of real and CM-extensions. In this paper we generalize some of the results without any assumption on the type of the relative quadratic extension.

Orbit Structure On The Silov Boundary Of A Tube Domain And The Plancherel Decomposition Of A Causally Compact Symmetric Space, With Emphasis On The Rank One Case, Troels Roussau Johansen

LSU Doctoral Dissertations

We construct a G-equivariant causal embedding of a compactly causal symmetric space G/H as an open dense subset of the Silov boundary S of the unbounded realization of a certain Hermitian symmetric space G₁/K₁ of tube type. Then S is an Euclidean space that is open and dense in the flag manifold G₁/P', where P' denotes a certain parabolic subgroup of G₁. The regular representation of G on L²(G/H) is thus realized on L²(S), and we use abelian harmonic analysis in the study thereof. In particular, …

Analysis Of Social Aspects Of Migrant Labourers Living With Hiv/Aids Using Fuzzy Theory And Neutrosophic Cognitive Maps, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

Neutrosophic logic grew as an alternative to the existing topics and it represents a mathematical model of uncertainty, vagueness, ambiguity, imprecision, undefined-ness, unknown, incompleteness, inconsistency, redundancy and contradiction. Despite various attempts to reorient logic, there has remained an essential need for an alternative system that could infuse into itself a representation of the real world. Out of this need arose the system of neutrosophy and its connected logic, neutrosophic logic. This new logic, which allows also the concept of indeterminacy to play a role in any real-world problem, was introduced first by one of the authors Florentin Smarandache. In this …

Computational Protein Biomarker Prediction: A Case Study For Prostate Cancer, Michael Wagner, Dayanand N. Naik, Alex Pothen, Srinivas Kasukurti, Raghu Ram Devineni, Bao-Ling Adam, O. John Semmes, George L. Wright Jr.

Mathematics & Statistics Faculty Publications

Background: Recent technological advances in mass spectrometry pose challenges in computational mathematics and statistics to process the mass spectral data into predictive models with clinical and biological significance. We discuss several classification-based approaches to finding protein biomarker candidates using protein profiles obtained via mass spectrometry, and we assess their statistical significance. Our overall goal is to implicate peaks that have a high likelihood of being biologically linked to a given disease state, and thus to narrow the search for biomarker candidates.

Results: Thorough cross-validation studies and randomization tests are performed on a prostate cancer dataset with over 300 patients, obtained …

Biorthogonal Spline Type Wavelets, Tian-Xiao He

Tian-Xiao He

Let ¢ be an orthonormal scaling function with approximation degree p - 1, and let Bn be the B-spline of order n. Then, spline type scaling functions defined by fn = f * Bn (n = 1, 2, ... ) possess higher approximation order, p+n-1, and compact support. The corresponding biorthogonal wavelet functions are also constructed. This technique is extended to the case of biorthogonal scaling function system. As an application of the method supplied in this paper, one can easily construct a sequence of pairs of biorthogonal spline type scaling functions from one pair of biorthogonal scaling functions or …

Microwave Thawing Of Cylinders., Tim Marchant

Tim Marchant

Microwave thawing of a cylinder is examined. The electromagnetic field is governed by Maxwell's equations, where the electrical conductivity and the thermal absorptivity are both assumed to depend on temperature. The forced heat equation governs the absorption and diffusion of heat where convective heating occurs at the surface of the cylinder, while the Stefan condition governs the position of the moving phase boundary. A semi-analytical model, which consists of ordinary differential equations, is developed using the Galerkin method. Semi-analytical solutions are found for the temperature, the electric-field amplitude in the cylinder and the position of the moving boundary. Two examples, …

Asymptotic Solitons For A Third-Order Kortweg-De Vries Equation, Tim Marchant

Tim Marchant

Solitary wave interaction for a higher-order version of the Korteweg–de Vries (KdV) equation is considered. The equation is obtained by retaining third-order terms in the perturbation expansion, where for the KdV equation only first-order terms are retained. The third-order KdV equation can be asymptotically transformed to the KdV equation, if the third-order coefficients satisfy a certain algebraic relationship. The third-order two-soliton solution is derived using the transformation. The third-order phase shift corrections are found and it is shown that the collision is asymptotically elastic. The interaction of two third-order solitary waves is also considered numerically. Examples of an elastic and …

Semi-Analytical Solutions For One - And Two-Dimensional Pellet Problems., Tim Marchant

Tim Marchant

The problem of heat and mass transfer within a porous catalytic pellet in which an irreversible first–order exothermic reaction occurs is a much–studied problem in chemical–reactor engineering. The system is described by two coupled reaction–diffusion equations for the temperature and the degree of reactant conversion. The Galerkin method is used to obtain a semi–analytical model for the pellet problem with both one– and two–dimensional slab geometries. This involves approximating the spatial structure of the temperature and reactant–conversion profiles in the pellet using trial functions. The semi–analytical model is obtained by averaging the governing partial differential equations. As the Arrhenius law …

Cubic Autocatalysis With Michaelis - Menten Kinetics: Semi-Analytical Solutions For The Reaction - Diffusion Cell, Tim Marchant

Tim Marchant

Cubic-autocatalysis with Michaelis–Menten decay is considered in a one-dimensional reaction–diffusion cell. The boundaries of the reactor allow diffusion into the cell from external reservoirs, which contain fixed concentrations of the reactant and catalyst. The Galerkin method is used to obtain a semi-analytical model consisting of ordinary differential equations. This involves using trial functions to approximate the spatial structure of the reactant and autocatalyst concentrations in the reactor. The semi-analytical model is then obtained from the governing partial differential equations by averaging. The semi-analytical model allows steady-state concentration profiles and bifurcation diagrams to be obtained as the solution to sets of …

Characterization And Properties Of Matrices With Generalized Symmetry Or Skew Symmetry, William F. Trench

William F. Trench

No abstract provided.

On Multivariate Abel-Gontscharoff Interpolation, Tian-Xiao He

Tian-Xiao He

By using Gould's annihilation coefficients, we obtain an explicit fundamental polynomials of Multivariate Abel-Gontscharoff Interpolation and its remainder expression.

Existence Of Solutions To A Hamiltonian System Without Convexity Condition On The Nonlinearity, Gregory S. Spradlin

Gregory S. Spradlin

Equilibrium Problems With Equilibrium Constraints Via Multiobjective Optimization, Boris S. Mordukhovich

Mathematics Research Reports

The paper concerns a new class of optimization-related problems called Equilibrium Problems with Equilibrium Constraints (EPECs). One may treat them as two level hierarchical problems, which involve equilibria at both lower and upper levels. Such problems naturally appear in various applications providing an equilibrium counterpart (at the upper level) of Mathematical Programs with Equilibrium Constraints (MPECs). We develop a unified approach to both EPECs and MPECs from the viewpoint of multiobjective optimization subject to equilibrium constraints. The problems of this type are intrinsically nonsmooth and require the use of generalized differentiation for their analysis and applications. This paper presents necessary …

Necessary Conditions In Nonsmooth Minimization Via Lower And Upper Subgradients, Boris S. Mordukhovich

Mathematics Research Reports

The paper concerns first-order necessary optimality conditions for problems of minimizing nonsmooth functions under various constraints in infinite-dimensional spaces. Based on advanced tools of variational analysis and generalized differential calculus, we derive general results of two independent types called lower subdifferential and upper subdifferential optimality conditions. The former ones involve basic/limiting subgradients of cost functions, while the latter conditions are expressed via Frechetjregular upper subgradients in fairly general settings. All the upper subdifferential and major lower subdifferential optimality conditions obtained in the paper are new even in finite dimensions. We give applications of general optimality conditions to mathematical programs with …

Optimal Control Of Delayed Differential-Algebraic Inclusions, Boris S. Mordukhovich, Lianwen Wang

Mathematics Research Reports

This paper concerns constrained dynamic optimization problems governed by delayed differential-algebraic systems. Dynamic constraints in such systems, which are particularly important for engineering applications, are described by interconnected delay-differential inclusions and algebraic equations. We pursue a two-hold goal: to study variational stability of such control systems with respect to discrete approximations and to derive necessary optimality conditions for both delayed differential-algebraic systems and their finite-difference counterparts using modern tools of variational analysis and generalized differentiation. We are not familiar with any results in these directions for differential-algebraic inclusions even in the delay-free case. In the first part of the paper …

The Approximate Maxium Principle In Constrained Optimal Control, Boris S. Mordukhovich, Ilya Shvartsman

Mathematics Research Reports

The paper concerns optimal control problems for dynamic systems governed by a parametric family of discrete approximations of control systems with continuous time. Discrete approximations play an important role in both qualitative and numerical aspects of optimal control and occupy an intermediate position between discrete-time and continuous-time control systems. The central result in optimal control of discrete approximations is the Approximate Maximum Principle (AMP), which is justified for smooth control problems with endpoint constraints under certain assumptions without imposing any convexity, in contrast to discrete systems with a fixed step. We show that these assumptions are essential for the validity …

Kernel Estimation Of Rate Function For Recurrent Event Data, Chin-Tsang Chiang, Mei-Cheng Wang, Chiung-Yu Huang

Johns Hopkins University, Dept. of Biostatistics Working Papers

Recurrent event data are largely characterized by the rate function but smoothing techniques for estimating the rate function have never been rigorously developed or studied in statistical literature. This paper considers the moment and least squares methods for estimating the rate function from recurrent event data. With an independent censoring assumption on the recurrent event process, we study statistical properties of the proposed estimators and propose bootstrap procedures for the bandwidth selection and for the approximation of confidence intervals in the estimation of the occurrence rate function. It is identified that the moment method without resmoothing via a smaller bandwidth …

Optimization And Feedback Control Of Constrained Parabolic Systems Under Uncertain Perturbations, Boris S. Mordukhovich, Ilya Shvartsman

Mathematics Research Reports

This paper concerns a minimax control design problem for a class of parabolic systems with nonregular boundary conditions and uncertain distributed perturbations under pointwise control and state constraints. We deal with boundary controllers acting through Dirichlet boundary conditions that are the most challenging for the parabolic dynamics.

Normal Forms For Nonlinear Discrete Time Control Systems, Boumediene Hamzi, Issa Amadou Tall

Miscellaneous (presentations, translations, interviews, etc)

We study the feedback classification of discrete-time control systems whose linear approximation around an equilibrium is controllable. We provide a normal form for systems under investigation.

Global Solutions To The Lake Equations With Isolated Vortex Regions, Chaocheng Huang

Mathematics and Statistics Faculty Publications

The vorticity formulation for the lake equations in R² is studied.

Unified Cross-Validation Methodology For Selection Among Estimators And A General Cross-Validated Adaptive Epsilon-Net Estimator: Finite Sample Oracle Inequalities And Examples, Mark J. Van Der Laan, Sandrine Dudoit

U.C. Berkeley Division of Biostatistics Working Paper Series

In Part I of this article we propose a general cross-validation criterian for selecting among a collection of estimators of a particular parameter of interest based on n i.i.d. observations. It is assumed that the parameter of interest minimizes the expectation (w.r.t. to the distribution of the observed data structure) of a particular loss function of a candidate parameter value and the observed data structure, possibly indexed by a nuisance parameter. The proposed cross-validation criterian is defined as the empirical mean over the validation sample of the loss function at the parameter estimate based on the training sample, averaged over …

Neumann Boundary Control Of Hyperbolic Equations With Pointwise State Constraints, Boris S. Mordukhovich, Jean-Pierre Raymond

Mathematics Research Reports

We consider optimal control problems for hyperbolic systems with controls in Neumann boundary conditions with pointwise (hard) constraints on control and state functions. Focusing on hyperbolic dynamics governed by the multidimensional wave equation with a nonlinear term, we derive new necessary optimality conditions in the pointwise form of the Pontryagin Maximum Principle for the state-constrained problem under consideration. Our approach is based on modern methods of variational analysis that allows us to obtain refined necessary optimality conditions with no convexity assumptions on integrands in the minimizing cost functional.