Physical Sciences and Mathematics Commons^™

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 …

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.

The Second Hull Of A Knotted Curve, Jason Cantarella, Greg Kuperberg, Robert B. Kusner, John M. Sullivan

Robert Kusner

The convex hull of a set K in space consists of points which are, in a certain sense, "surrounded" by K. When K is a closed curve, we define its higher hulls, consisting of points which are "multiply surrounded" by the curve. Our main theorem shows that if a curve is knotted then it has a nonempty second hull. This provides a new proof of the Fary/Milnor theorem that every knotted curve has total curvature at least 4pi.

Drill 3.1, Vadim Ponomarenko

Mathematics Faculty Research

Distance learning can offer a solution to a long-standing challenge of undergraduate education: how to assign an appropriate amount of work to each student, and how to assess this work efficiently. In this paper I describe the Depository of Repetitive Internet-based probLems and Lessons (DRILL), an online system for providing education and assessment for precalculus, which can substitute for routine problems given in homework and as exam questions. Its special features include:

• adaptive testing,
• on-the-fly question generation,
• instant assessment,
• context-sensitive help,
• question balancing, and
• two-dimensional test design.

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.

Deconstructing Bases: Fair, Fitting, And Fast Bases, Thomas Q. Sibley

Mathematics Faculty Publications

No abstract provided.

How Secondary School Mathematics Teachers Construct An Understanding Of "Appropriate Use" Of Graphing Calculators In The Context Of Collegial Inquiry, Marcia L. Weller Weinhold

Dissertations

No abstract provided.

Uniform Approximation Of Continuous Functions On A Compact Riemann Surface By Elliptic Modular Forms, Michael Berg

Mathematics, Statistics and Data Science Faculty Works

We show that the graded algebra of elliptic modular forms and their conjugates comprises a uniformly dense subspace of the space of all continuous functions on the compactification of the fundamental domain for the action of SL₂(Z) on the complex upper half-plane by fractional linear transformations.

A Solution To Einstein’S Field Equations For A Tachyonic Gas: Possible Astrophysical Applications, Kris H. Green, W. John Cocke

Mathematical and Computing Sciences Faculty/Staff Publications

In this paper we show that a change in the signs of some of the metric components of the solution of the field equations for the classical cosmic string results in a solution which we interpret as a time-dependent wall composed of tachyons. We show that the walls have the property of focusing the paths of particles which pass through them. As an illustration of this focusing, we demonstrate the results of a simple simulation of the interaction between one such tachyon wall and a rotating disk of point masses. This interaction leads to the temporary formation of spiral structures. …

Objectivity, Information, And Maxwell's Demon, Steven Weinstein

Dartmouth Scholarship

This paper examines some common measures of complexity, structure, and information, with an eye toward understanding the extent to which complexity or information‐content may be regarded as objective properties of individual objects. A form of contextual objectivity is proposed which renders the measures objective, and which largely resolves the puzzle of Maxwell's Demon.

A Class Of Functions That Are Quasiconvex But Not Polyconvex, Catherine S. Remus

Masters Theses

In 1991 V. Sverak [11] gave an example of a function that was invariant and quasiconvex but not polyconvex. We have generalized this example to a wide class of functions that meet certain ellipticity and growth conditions. Quasiconvexity is one necessary and sufficient condition for the existence of solutions to the minimization problem in elliptic P.D.E. theory. Invariance is frequently a requirement of the stored energy function in Calculus of Variation approaches to elasticity problems.

Results Of Soy-Based Meal Replacement Formula On Weight, Anthropometry, Serum Lipids & Blood Pressure During A 40-Week Clinical Weight Loss Trial, Kevin R. Fontaine, Dongyan Yang, Gary L. Gadbury, Stanley Heshka, Linda G. Schwartz, Radha Murugesan, Jennifer L. Kraker, Moonseong Heo, Steven B. Heymsfield, David B. Allison

Mathematics and Statistics Faculty Research & Creative Works

Background: to evaluate the intermediate-term health outcomes associated with a soy-Based meal replacement, and to compare the weight loss efficacy of two distinct patterns of caloric restriction. Methods: Ninety overweight/obese (28 < BMI ≤ 41 kg/m2) adults received a single session of dietary counseling and were randomized to either 12 weeks at 1200 kcal/day, 16 weeks at 1500 kcal/d and 12 weeks at 1800 kcal/d (i.e., the 12/15/18 diet group), or 28 weeks at 1500 kcal/d and 12 weeks at 1800 kcal/d (i.e., the 15/18 diet group). Weight, body fat, waist circumference, blood pressure and serum lipid concentrations were measured at 4-week intervals throughout the 40-week trial. Results: Subjects in both treatments showed statistically significant improvements in outcomes. a regression model for weight change suggests that subjects with larger baseline weights tended to lose more weight and subjects in the 12/15/18 group tended to experience, on average, an additional 0.9 kg of weight loss compared with subjects in the 15/18 group. Conclusion: Both treatments using the soy-Based meal replacement program were associated with significant and comparable weight loss and improvements on selected health variables.

Statistical Pronouncements Ii, Shlomo S. Sawilowsky

Theoretical and Behavioral Foundations of Education Faculty Publications

No abstract provided.

Deconstructing Arguments From The Case Against Hypothesis Testing, Shlomo S. Sawilowsky

Theoretical and Behavioral Foundations of Education Faculty Publications

The main purpose of this article is to contest the propositions that (1) hypothesis tests should be abandoned in favor of confidence intervals, and (2) science has not benefited from hypothesis testing. The minor purpose is to propose (1) descriptive statistics, graphics, and effect sizes do not obviate the need for hypothesis testing, (2) significance testing (reporting p values and leaving it to the reader to determine significance) is subjective and outside the realm of the scientific method, and (3) Bayesian and qualitative methods should be used for Bayesian and qualitative research studies, respectively.

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.

Determining Error Bars In Measurements Of Ultrashort Laser Pulses, Ziyang Wang, Erik Zeek, Rick Trebino, Paul Kvam

Department of Math & Statistics Faculty Publications

We present a simple and automatic method for determining the uncertainty in the retrieved intensity and phase versus time (and frequency) due to noise in a frequency-resolved optical-gating trace, independent of noise source. It uses the ‘‘bootstrap’’ statistical method and also yields an automated method for phase blanking (omitting the phase when the intensity is too low to determine it).

Some Combinatorial Design In Vlsi Architectures And Statistics., Soumen Maity Dr.

Doctoral Theses

In this dissertation, we consider the following combinatorial problems: some character- ization, enumeration, construction and optimization problems in both VLSI linear and VLSI two-dimensional arrays; and construction of two combinatorial designs as used by statisticians: nearly strongly balanced uniform repeated measurements designs (NSBUR- MDs) and balanced near uniform repeated measurements designs (BNURMDS). We give below, chapter-wise, the problems considered and a brief outline of the solutions.1.1 Enumerating Catastrophic Fault Patterns in VLSI Linear Arrays with Bidirectional or Unidirectional LinksSystolic systems consist, of a large mimber of identical and elementary processing element locally conuccted in a regular fashion. Each element receives …

Sylvester: Ushering In The Modern Era Of Research On Odd Perfect Numbers, Steven Gimbel, John Jaroma

Philosophy Faculty Publications

In 1888, James Joseph Sylvester (1814-1897) published a series of papers that he hoped would pave the way for a general proof of the nonexistence of an odd perfect number (OPN). Seemingly unaware that more than fifty years earlier Benjamin Peirce had proved that an odd perfect number must have at least four distinct prime divisors, Sylvester began his fundamental assault on the problem by establishing the same result. Later that same year, he strengthened his conclusion to five. These findings would help to mark the beginning of the modern era of research on odd perfect numbers. Sylvester's bound stood …

On Textured Image Analysis Using Wavelets., Mausumi Acharyya Dr.

Doctoral Theses

In image processing and computer vision research, we aim to derive better tools that give us different perspectives on the same image, allowing us to understand not only its content, but also its meaning and significance. Image processing can not compete with the human eye in terms of accuracy but it can outperform the latter easily on observational consistency, and ability to carry out detailed mathematical estimations. With time, image processing research has broadened from the basic pixel-based low- level operations to high-level analysis, that now includes the use of artificially intelligent techniques for image interpretation and understanding. These new …

Auroral Source Region: Plasma Properties Of The High-Latitude Plasma Sheet, C. A. Kletzing, J. D. Scudder, E. E. Dors, Carina Curto

Department of Mathematics: Faculty Publications

Statistical results from a survey of 93 passes through the high-latitude extension of the plasma sheet of electron data from the Hydra instrument on the Polar spacecraft show that the values for electron density can range from 0.01 to 0.5 cm^–3 with an average value around 0.1 cm^–3 on the poleward side and 0.3 cm^–3 on the equatorward side. Electron mean energy is found to have an average value near 900 eV on the equatorward side and 400 eV on the poleward side but varies from 100 eV to 4 keV. These values for density and mean …

Controllability And Local Accessibility—A Normal Form Approach, Wei Kang, Mingqing Xiao, Issa Amadou Tall

Articles and Preprints

Given a system with an uncontrollable linearization at the origin, we study the controllability of the system at equilibria around the origin. If the uncontrollable mode is nonzero, we prove that the system always has other equilibria around the origin. We also prove that these equilibria are linearly controllable provided a coefficient in the normal form is nonzero. Thus, the system is qualitatively changed from being linearly uncontrollable to linearly controllable when the equilibrium point is moved from the origin to a different one. This is called a bifurcation of controllability. As an application of the bifurcation, systems with a …

Dirichlet Boundary Control Of Hyperbolic Equations In The Presence Of State Constraints, Boris S. Mordukhovich, Jean-Pierre Raymond

Mathematics Research Reports

We study optimal control problems for hyperbolic equations (focusing on the multidimensional wave equation) with control functions in the Dirichlet boundary conditions under hard/pointwise control and state constraints. Imposing appropriate convexity assumptions on the cost integral functional, we establish the existence of optimal control and derive new necessary optimality conditions in the integral form of the Pontryagin Maximum Principle for hyperbolic state-constrained systems.

Exact Multiplicity For Periodic Solutions Of Duffing Type, Hongbin Chen, Yi Li, Xiaojie Hou

Yi Li

In this paper, we study the following Duffing-type equation: x″+cx′+g(t,x)=h(t),

where g(t,x) is a 2π-periodic continuous function in t and concave–convex in x, and h(t) is a small continuous 2π-periodic function. The exact multiplicity and stability of periodic solutions are obtained.

Exact Multiplicity For Periodic Solutions Of Duffing Type, Hongbin Chen, Yi Li, Xiaojie Hou

Mathematics and Statistics Faculty Publications

In this paper, we study the following Duffing-type equation:

x″+cx′+g(t,x)=h(t),

where g(t,x) is a 2π-periodic continuous function in t and concave–convex in x, and h(t) is a small continuous 2π-periodic function. The exact multiplicity and stability of periodic solutions are obtained.

Modeling Of The Inverse Heat -Conduction Problem With Application To Laser Chemical Vapor Deposition And Bioheat Transfer, Peng Zhen

Doctoral Dissertations

This dissertation consists of two parts. Part one deals with three-dimensional laser induced chemical vapor deposition (3D-LCVD), whereas part two deals with a Pennes model of a 3D skin structure. LCVD is an important technique in manufacturing complex micro-structures with high aspect ratio. In part one, a numerical model was developed for simulating kinetically-limited growth of an axisymmetric cylindrical rod by pre-specifying the surface temperature distribution required for growing the rod and then by obtaining optimized laser power that gives rise to the pre-specified temperature distribution. The temperature distribution at the surface of the rod was assumed to be at …

A Combinatorial Approach To Hyperharmonic Numbers, Arthur T. Benjamin, David Gaebler '04, Robert Gaebler '04

All HMC Faculty Publications and Research

Hyperharmonic numbers arise by taking repeated partial sums of harmonic numbers. These numbers can be expressed in terms of r-Stirling numbers, leading to combinatorial interpretations of many interesting identities.

Brownian Dynamics Simulation Studies Of Ion Throughput During Cellular Electroporation, Juan A. Gonzalez-Cuevas

Electrical & Computer Engineering Theses & Dissertations

The purpose of the present research work is to analyze the flow of ions through pores formed in the plasma membrane of biological cells during electroporation. In this regard, simulations have been conducted to track the movement of ions in 3-D space based on a Brownian dynamics model, which is mostly deterministic but has some stochastic properties to account for ionic scattering. From the Brownian dynamics simulations, it has been possible to estimate the pore's conductance as well as transport parameters such as the diffusion coefficient and mobility within the pore region, of which there is no experimental data currently …

Invertibility Of Current Density From Near-Field Electromagnetic Data, Evangelos A. Coutsias, Daniel Sheltraw

Branch Mathematics and Statistics Faculty and Staff Publications

The problem of determining a current density confined to a volume from measurements of the magnetic and electric fields it produces exterior to that volume is known to have nonunique solutions. Despite the nonuniqueness of the inversion we show that one may nevertheless uniquely determine certain moments of the vector spherical harmonic expansion of the current. It is demonstrated that the determination of these moments allows for the unique inversion of a current density confined to a spherical shell. Although unique the inversion may be ill conditioned and require a regularization of the inversion as demonstrated in an example numerical …

Rank And K-Nullity Of Contact Manifolds, Philippe Rukimbira

Department of Mathematics and Statistics

We prove that the dimension of the 1-nullity distribution N(1) on a closed Sasakian manifold M of rankl is at least equal to 2l−1 provided that M has an isolated closed characteristic. The result is then used to provide some examples of k-contact manifolds which are not Sasakian. On a closed, 2n+1-dimensional Sasakian manifold of positive bisectional curvature, we show that either the dimension of N(1) is less than or equal to n+1 or N(1) is the entire tangent bundle TM. In the latter case, the Sasakian manifold Mis isometric to a quotient of the Euclidean sphere under a finite …