Mathematics Commons^™

Multiobjective Optimal Control Problems With Endpoint And State Constraints, Kirsty J. Eisenhart

Dissertations

In this thesis we consider nonsmooth multiobjective optimal control problems in terms of a general preference on [Special characters omitted.]. The optimal control problems considered involve differential inclusion, endpoint constraints and state constraints. No convexity assumption is needed on the differential inclusion. Examples of common preferences are given, and the idea of approximating a preference is introduced. Euler-Lagrange necessary conditions and a form of the maximum principle are developed for closed preferences (and those that can be approximated by closed preferences) in terms of the limiting subdifferential. As a consequence, this is the first result in the literature for lexicographical …

Multi-Symplectic Integrators For Nonlinear Wave Equations, Alvaro Lucas Islas

Mathematics & Statistics Theses & Dissertations

Symplectic (area-preserving) integrators for Hamiltonian ordinary differential equations have shown to be robust, efficient and accurate in long-term calculations. In this thesis, we show how symplectic integrators have a natural generalization to Hamiltonian PDEs by introducing the concept of multi-symplectic partial differential equations (PDEs). In particular, we show that multi-symplectic PDEs have an underlying spatio-temporal multi-symplectic structure characterized by a multi-symplectic conservation law MSCL). Then multi-symplectic integrators (MSIs) are numerical schemes that preserve exactly the MSCL. Remarkably, we demonstrate that, although not designed to do so, MSIs preserve very well other associated local conservation laws and global invariants, such as …

Developing Into Series And Returning From Series: A Note On The Foundations Of Eighteenth-Century Analysis, Giovanni Ferraro, Marco Panza

MPP Published Research

In this paper we investigate two problems concerning the theory of power series in 18th-century mathematics: the development of a given function into a power series and the inverse problem, the return from a given power series to the function of which this power series is the development. The way of conceiving and solving these problems closely depended on the notion of function and in particular on the conception of a series as the result of a formal transformation of a function. After describing the procedures considered acceptable by 18th-century mathematicians, we examine in detail the different strategies—both direct and …

Sandwich Theorem And Calculation Of The Theta Function For Several Graphs, Marcia Ling Riddle

Theses and Dissertations

This paper includes some basic ideas about the computation of a function theta(G), the theta number of a graph G, which is known as the Lovasz number of G. theta(G^c) lies between two hard-to-compute graph numbers omega(G), the size of the largest lique in a graph G, and chi(G), the minimum number of colors need to properly color the vertices of G. Lovasz and Grotschel called this the "Sandwich Theorem". Donald E. Knuth gives four additional definitions of theta, theta_1, theta_2, theta_3, theta_4 and proves that they are all equal.

First I am going to describe the proof of the …

Support Planes And A Wonderful Theorem Of Bishop And Phelps, Joe Diestel

Colloquium

Since the earliest days of duality in linear spaces (i.e., vector spaces), support functionals for convex sets have played a critical role in applications. The Hahn-Banach theorem is an example: any closed bounded convex subset of a normed linear space has support functionals. In the early sixties, Bishop and Phelps gave a much sharper theorem, showing that there are lots of support functionals. This talk will discuss results related to the Bishop-Phelps theorem—results that sometimes fairly amaze even people who work in the area.

On The Local Spectral Properties Of Weighted Shift Operators, Abdellatif Bourhim

Mathematics - All Scholarship

In this paper, we study the local spectral properties for both unilateral and bilateral weighted shift operators.

Pareto Optimal Allocations In Nonconvex Models Of Welfare Economics, Boris S. Mordukhovich

Mathematics Research Reports

The paper is devoted to applications of modern variational analysis to the study of Pareto (as well as weak and strong Pareto) optimal allocations in nonconvex models of welfare economics with infinite-dimensional commodity spaces. Our basic tool is the extremal principle of variational analysis that provides necessary conditions for set extremality and may be viewed as a variational extension of the classical convex separation principle to the case of nonconvex sets. In this way we obtain new versions of the generalized second welfare theorem for nonconvex economies in terms of appropriate concepts of normal cones.

Stochastic Differential Systems With Memory (Spring School On Stochastic Delay Differential Equations), Salah-Eldin A. Mohammed

Miscellaneous (presentations, translations, interviews, etc)

No abstract provided.

Radius Of Convergence Of A Power Series, Todor D. Todorov

Mathematics

We derive two simple and memorizable formulas for the radius of convergence of a power series which seem to be appropriate for teaching in an introductory calculus course.

Evaluating The Performance Of Multiple Classifier Systems: A Matrix Algebra Representation Of Boolean Fusion Rules, Justin M. Hill

Theses and Dissertations

Given a finite collection of classifiers one might wish to combine, or fuse, the classifiers in hopes that the multiple classifier system (MCS) will perform better than the individuals. One method of fusing classifiers is to combine their final decision using Boolean rules (e.g., a logical OR, AND, or a majority vote of the classifiers in the system). An established method for evaluating a classifier is measuring some aspect of its Receiver Operating Characteristic (ROC) curve, which graphs the trade-off between the conditional probabilities of detection and false alarm. This work presents a unique method of estimating the performance of …

Essays In Financial Intermediation., Bappaditya Mukhopadhyay Dr.

Doctoral Theses

No abstract provided.

On The Approximability Of Linear Ordering And Related Np-Optimization Problems., Sounaka Mishra Dr.

Doctoral Theses

We investigate approximability of both maximum and minimum linear ordering problems (MAX-LOP and MIN-LOP) and several related problems such as the well known feedback set problems, acyclie subdigraph problem and several others and their variants.We show that both MAX-LOP and MIN-LOP are strongly NP-complete, and MIN- LOP, MIN-QAP(S) (a special case of minimum quadratic assignment problem) and MIN-W-FAS are equivalent with respect to strict-reduction. The strict-equivalence is also established among these problems as well as MIN-W-FVS, with weights on arcs/vertices bounded by a polynomial, and the unweighted versions of the feedback set. problems. We also show that MAX-LOP is strict-equivalent …

Deformation Theory Of Dialgebras., Anita Majumdar Dr.

Doctoral Theses

The main objective of this thesis is to develop an algebraic deformation theury for associative dialgebras, which are binary quadratic algebras discovered by J.-L. Loday in (16). (17), and subisequently, to derive a G-algebra siructure ou the dialgebra colhomology with cocfticients in itself.Deformation theory dates back at Ieast to Riemann's 1837 memoir on alelian fianetions in which he studied IHanifolds of complex dimension one and calculated the mumber of parameters (called moduli) upon which a deformation depends. The modern theory of deformations of structures on manifolds was developed extensively ly Frolicher-Kodaira-Nijenhnis-Nirenberg-Spencer (13], [14], [15). [25|, [26).The study of deformations of …

Certain Pattern Recognition Tasks For Data Mining Problems., Pabitra Mitra Dr.

Doctoral Theses

Pattern recognition (PR) is an activity that we humans normally excel in. We do it almost all the time, and without conscious effort. We receive information via our various sensory organs, which is processed instantaneously by our brain so that, almost immediately, we are able to identify the source of the information, without having made any perceptible effort. What is even more impressive is the accuracy with which we can perform recognition tasks even under non-ideal conditions, for instance, when the information that needs to be processed is vague, imprecise or even incomplete. In fact, most of our day-to-day activities …

Spectral Triples And Metric Aspects Of Geometry On Some Noncommutative Spaces., Partha Sarathi Chakraborty Dr.

Doctoral Theses

Quantization of mathematical theories is now more than half a century old idea in mathe- matics. It goes back to Gelfand-Naimarks seminal paper [37] in 1943. As the name suggests noncommutative geometry is the quantization" of differential geometry. It is the study of noncommutative algebras as if they were algebras of functions on spaces like the commuta- tive algebras associated to affine algebraic varieties, smooth manifolds, topological spaces. One can trace its roots in the Gelfand-Naimark theorems (1943, 37]). In modern terminol- ogy their theorem says there is an antiequivalence between the category of (locally) compact Hausdorff spaces and (proper, …

Bounding The Number Of Graphs Containing Very Long Induced Paths, Steven Kay Butler

Theses and Dissertations

Induced graphs are used to describe the structure of a graph, one such type of induced graph that has been studied are long paths.

In this thesis we show a way to represent such graphs in terms of an array with two colors and a labeled graph. Using this representation and the techniques of Polya counting we will then be able to get upper and lower bounds for graphs containing a long path as an induced subgraph.

In particular, if we let P(n,k) be the number of graphs on n+k vertices which contains P_n, a path on n vertices, as …

The Cohomology Of The Steendrod Algebra And Representations Of The General Linear Groups, Nguyen H. V. Hu'ng

Mathematics Research Reports

Let Tr_k be the algebraic transfer that maps from the coinvariants of certain GL_k-representation to the cohomology of the Steenrod algebra. This transfer was defined by W. Singer as an algebraic version of the geometrical transfer tr_k : pi_*^S((B[doublestrike V]_k)_+) --> pi_*^S(S^0). It has been shown that the algebraic transfer is highly nontrivial, more precisely, that Tr_k is an isomorphism for k = 1, 2, 3 and that T_r = ⊕_k(Tr_k) is a homomorphism of algebras.

In this paper, we first recognize the phenomenon that if we start from any degree d, and apply Sq^0 repeatedly at most (k- 2) …

Designing Radiotherapy Plans With Elastic Constraints And Interior Point Methods, Allen G. Holder

Mathematics Faculty Research

A new linear programming model used to aid in the design of radiotherapy plans is introduced. This model incorporates elastic constraints, and when solved with a path following interior point method, produces favorable plans. A sound mathematical analysis shows how to interpret the solution, and hence, the treatment planner receives meaningful knowledge about the radiotherapy plan being developed. Preliminary experiments are conducted.

Consensus-Halving Via Theorems Of Borsuk-Ulam And Tucker, Forrest W. Simmons, Francis E. Su

All HMC Faculty Publications and Research

In this paper we show how theorems of Borsuk-Ulam and Tucker can be used to construct a consensus-halving: a division of an object into two portions so that each of n people believes the portions are equal. Moreover, the division takes at most n cuts, which is best possible. This extends prior work using methods from combinatorial topology to solve fair division problems. Several applications of consensus-halving are discussed.

A Sign-Changing Solution For A Superlinear Dirichlet Problem, Ii, Alfonso Castro, Pavel Drabek, John M. Neuberger

All HMC Faculty Publications and Research

In previous work by Castro, Cossio, and Neuberger [2], it was shown that a superlinear Dirichlet problem has at least three nontrivial solutions when the derivative of the nonlinearity at zero is less than the first eigenvalue of -Δ with zero Dirichlet boundry condition. One of these solutions changes sign exactly-once and the other two are of one sign. In this paper we show that when this derivative is between the k-th and k+1-st eigenvalues there still exists a solution which changes sign at most k times. In particular, when k=1 the sign-changing exactly-once solution persists although one-sign solutions no …

Noise-Induced Unstable Dimension Variability And Transition To Chaos In Random Dynamical Systems, Ying-Cheng Lai, Zonghua Liu, Lora Billings, Ira B. Schwartz

Department of Mathematics Faculty Scholarship and Creative Works

Results are reported concerning the transition to chaos in random dynamical systems. In particular, situations are considered where a periodic attractor coexists with a nonattracting chaotic saddle, which can be expected in any periodic window of a nonlinear dynamical system. Under noise, the asymptotic attractor of the system can become chaotic, as characterized by the appearance of a positive Lyapunov exponent. Generic features of the transition include the following: (1) the noisy chaotic attractor is necessarily nonhyperbolic as there are periodic orbits embedded in it with distinct numbers of unstable directions (unstable dimension variability), and this nonhyperbolicity develops as soon …

Stage Based Interventions For Low Fat Diet With Middle School Students, Marilyn Frenn, Shelly Malin, Naveen K. Bansal

College of Nursing Faculty Research and Publications

Preventing obesity and cardiovascular disease at early ages is important; however, few effective interventions for early adolescents have been reported. In this study, low-income, culturally diverse students from an urban middle school (n = 60) received four classroom interventions with the use of a combined Health Promotion/Transtheoretical Model to control fat in diet and increase physical activity. A control group (n = 57) received the usual classroom education. Pretest percentage fat in diet was regressed on demographics, access to low-fat foods, perceived self-efficacy, benefits/barriers, and stage of change with results as proposed by the model [F(9,64) …

Closed-Loop Control Of Vortex Shedding By Means Of Lorentz Force, Xiaoyun Sun

Dissertations

When an incompressible fluid flows past a circular cylinder, vortex shedding occurs as soon as the Reynolds number exceeds about 40. Vortex shedding is usually undesirable, as it generates a significant increase in drag, as well an oscillating lift force on the cylinder leading to cross-stream structural vibrations. Flow control to either delay the appearance of vortex shedding or fully suppress it has attracted much attention during the last decade. The focus of this dissertation is to control vortex shedding from a circular cylinder by applying an external electromagnetic field. As in previous works, the latter is generated by electrodes …

Every Three-Point Set Is Zero Dimensional, David L. Fearnley, J. W. Lamoreaux, David L. Fearnley

Faculty Publications

This paper answers a question of Jan J. Dijkstra by giving a proof that all three-point sets are zero dimensional. It is known that all two-point sets are zero dimensional, and it is known that for all n > 3, there are n-point sets which are not zero dimensional, so this paper answers the question for the last remaining case.

Khovanov Homology And Conway Mutation, Stephan Wehrli

Mathematics - All Scholarship

We present an easy example of mutant links with different Khovanov homology. The existence of such an example is important because it shows that Khovanov homology cannot be defined with a skein rule similar to the skein relation for the Jones polynomial.

Odd Perfect Numbers Have A Prime Factor Exceeding 10^7, Paul M. Jenkins

Faculty Publications

It is proved that every odd perfect number is divisible by a prime greater than 10^7.

Two-Point Boundary Value Problems For Higher-Order Linear Differential Equations With Strong Singularities, Ravi P. Agarwal, Ivan T. Kiguradze

Mathematics and System Engineering Faculty Publications

For strongly singular higher-order linear differential equations together with two-point conjugate and right-focal boundary conditions, we provide easily verifiable best possible conditions which guarantee the existence of a unique solution. Copyright © 2006 Hindawi Publishing Corporation. All rights reserved.

Approximation By Piecewise Constant Functions In A Bv Metric, Pavel Bělík, Mitchell Luskin

Faculty Authored Articles

Westudytheapproximationpropertiesofpiecewiseconstantfunc- tions with respect to triangular and rectangular finite elements in a metric defined on functions of bounded variation. We apply our results to a thin film model for martensitic crystals and to the approximation of deformations with microstructure.

On The Nonembeddability And Crossing Numbers Of Some Kleinical Polyhedral Maps On The Torus, Adrian Riskin, Judy L. Klein

Mathematics

We designed and constructed a sundial for the purpose of observing the declination of the sun and thus marking solar seasonal variation. The 122 × 122 cm vertical sundial on the south-facing wall of our library has two unusual features: a nodus on the gnomon that casts a shadow of a point for marking the height of the sun and a large blank working space for students to mark the shadow of the nodus at different hours of the day and to connect the marks of 1 day in a line of declination. We discuss the design of a dial …

An Oscillation Theorem For Discrete Eigenvalue Problems, Martin Bohner, Ondřej Došlý, Werner Kratz

Mathematics and Statistics Faculty Research & Creative Works

In this paper we consider problems that consist of symplectic difference systems depending on an eigenvalue parameter, together with self-adjoint boundary conditions. Such symplectic difference systems contain as important cases linear Hamiltonian difference systems and also Sturm-Liouville difference equations of second and of higher order. The main result of this paper is an oscillation theorem that relates the number of eigenvalues to the number of generalized zeros of solutions.