Applied Mathematics Commons^™

Discrete Maximum Principle For Nonsmooth Optimal Control Problems With Delays, Boris S. Mordukhovich, Ilya Shvartsman

Mathematics Research Reports

We consider optimal control problems for discrete-time systems with delays. The main goal is to derive necessary optimality conditions of the discrete maximum principle type in the case of nonsmooth minimizing functions. We obtain two independent forms of the discrete maximum principle with transversality conditions described in terms of subdifferentials and superdifferentials, respectively. The superdifferential form is new even for non-delayed systems and may be essentially stronger than a more conventional subdifferential form in some situations.

How Many Symmetries Does Admit A Nonlinear Single-Input Control System Around An Equilibrium?, Witold Respondek, Issa Amadou Tall

Miscellaneous (presentations, translations, interviews, etc)

We describe all symmetries of a single-input nonlinear control system, that is not feedback linearizable and whose first order approximation is controllable, around an equilibrium point. For a system such that a feedback transformation, bringing it to the canonical form, is analytic we prove that the set of all local symmetries of the system is exhausted by exactly two 1-parameter families of symmetries, if the system is odd, and by exactly one 1-parameter family otherwise. We also prove that the form of the set of symmetries is completely described by the canonical form of the system: possessing a nonstationary symmetry, …

Modeling Control Of Hiv Infection Through Structured Treatment Interruptions With Recommendations For Experimental Protocol, Shannon Kubiak, Heather Lehr, Rachel Levy, Todd Moeller, Albert Parker, Edward Swim

All HMC Faculty Publications and Research

Highly Active Anti-Retroviral Therapy (HAART) of HIV infection has significantly reduced morbidity and mortality in developed countries. However, since these treatments can cause side effects and require strict adherence to treatment protocol, questions about whether or not treatment can be interrupted or discontinued with control of infection maintained by the host immune system remain to be answered. We present sensitivity analysis of a compartmental model for HIV infection that allows for treatment interruptions, including the sensitivity of the compartments themselves to our parameters as well as the sensitivity of the cost function used in parameter estimation. Recommendations are made about …

Ultraconvergence Of Zz Patch Recovery At Mesh Symmetry Points, Zhimin Zhang, Runchang Lin

Mathematics Research Reports

Ultraconvergence property of the Zienkiewicz-Zhu gradient patch recovery technique based on local discrete least squares fitting is established for a large class of even-order finite elements. The result is valid at all rectangular mesh symmetry points. Different smoothing strategies are discussed. Superconvergence recovery for the Q8 element is proved and ultraconvergence numerical examples are demonstrated.

Monotone Solutions Of A Nonautonomous Differential Equation For A Sedimenting Sphere, Andrew Belmonte, Jon T. Jacobsen, Anandhan Jayaraman

All HMC Faculty Publications and Research

We study a class of integrodifferential equations and related ordinary differential equations for the initial value problem of a rigid sphere falling through an infinite fluid medium. We prove that for creeping Newtonian flow, the motion of the sphere is monotone in its approach to the steady state solution given by the Stokes drag. We discuss this property in terms of a general nonautonomous second order differential equation, focusing on a decaying nonautonomous term motivated by the sedimenting sphere problem

Control Of Error Rates In Adaptive Analysis Of Orthogonal Saturated Designs, Weizhen Wang, Daniel T. Voss

Mathematics and Statistics Faculty Publications

Individual and simultaneous confidence intervals using the data adaptively are constructed for the effects in orthogonal saturated designs under the assumption of effect sparsity. The minimum coverage probabilities of the intervals are equal to the nominal level 1 - α.

Automatic Closure Of Invariant Linear Manifolds For Operator Algebras, Allan P. Donsig, Alan Hopenwasser, David R. Pitts

Department of Mathematics: Faculty Publications

Kadison's transitivity theorem implies that, for irreducible representations of C*-algebras, every invariant linear manifold is closed. It is known that CSL algebras have this property if, and only if, the lattice is hyperatomic (every projection is generated by a nite number of atoms). We show several other conditions are equivalent, including the condition that every invariant linear manifold is singly generated.

We show that two families of norm closed operator algebras have this property. First, let L be a CSL and suppose A is a norm closed algebra which is weakly dense in Alg L and is a bimodule over …

A Construction Of Compactly-Supported Biorthogonal Scaling Vectors And Multiwavelets On $R^2$, Bruce Kessler

Mathematics Faculty Publications

In \cite{K}, a construction was given for a class of orthogonal compactly-supported scaling vectors on $\R^{2}$, called short scaling vectors, and their associated multiwavelets. The span of the translates of the scaling functions along a triangular lattice includes continuous piecewise linear functions on the lattice, although the scaling functions are fractal interpolation functions and possibly nondifferentiable. In this paper, a similar construction will be used to create biorthogonal scaling vectors and their associated multiwavelets. The additional freedom will allow for one of the dual spaces to consist entirely of the continuous piecewise linear functions on a uniform subdivision of the …

Constructing Critical Indecomposable Codes, Judy L. Walker

Department of Mathematics: Faculty Publications

Critical indecomposable codes were introduced by Assmus, who also gave a recursive construction for these objects. One of the key ingredients in the construction is an auxiliary code, which is an indecomposable code of minimum distance at least 3. In terms of actually being able to construct all critical indecomposable codes, however, Assmus leaves many unanswered questions about these auxiliary codes. In this paper, we provide answers to these questions, including a description of when two equivalent auxiliary codes can yield inequivalent critical indecomposable codes, and results on both the minimum length and the maximum number of critical columns of …

Mathematics Memory Verses: Weekly Devotionals For Math Class, Mark Colgan

ACMS Conference Proceedings 2001

Each Monday during the semester I start class with a short devotional on a verse that relates in some way to mathematics. After three weeks I choose one of the three at random for students to write out on their quiz for a possible bonus point. This encourages students to practice memorizing Scripture and it gives us the opportunity to discuss biblical principles that relate to some of the topics we are studying in the course.

I would like to share some of the Bible verses and weekly devotionals I have used in my mathematics classes. These can be organized …

Parables For Mathematicians: With Good News For Curved Beings, Ashley Reiter Ahlin

ACMS Conference Proceedings 2001

Because we often lack the language for talking about such deep matters, the things of God can be hard to understand or talk about. The things that we do see and know were made by the same God of whom we speak. Thus, they are reflections of His nature, purposes, and ways and can help us to think and take about Him. This presentation expresses a parable using the language of math.

Three Problems From Number Theory, Robert Brabenec

ACMS Conference Proceedings 2001

This paper discusses the experiences of Wheaton College mathematics and computer science department colloquium as they explored open-ended problems.

Theism & Mathematical Realism, John Byl

ACMS Conference Proceedings 2001

This paper examines connections between theism and mathematical realism. Mathematical realism, which offers the best account of mathematics, strongly supports theism. Theism, in turn, supports mathematical realism. Theism readily explains the intricate relations between mathematics, matter, and mind. The attributes of the biblical God provide justification for classical mathematics.

What Mathematical Paradoxes Teach Us About Paradoxes In Christianity, Paul Bialek

ACMS Conference Proceedings 2001

In Christian academic circles, we talk about the integration of our faith and learning. That is, we seek to discover and develop connections between our Christian faith and our particular discipline. This is notoriously difficult when the discipline is mathematics. I have found that asking myself these three questions has helped me to integrate my Christian faith with mathematics, although they could be applied to any discipline: (1) How does the fact that I am a Christian affect the way I view mathematics? (2) How does the fact that I am a mathematician affect the way I view Christianity? (3) …

Why Natural Selection Can't Design Anything, William A. Dembski

ACMS Conference Proceedings 2001

In The Fifth Miracle Paul Davies suggests that any laws capable of explaining the origin of life must be radically different from scientific laws known to date? The problem, as he sees it, with currently known scientific laws, like the laws of chemistry and physics, is that they cannot explain the key feature of life that needs to be explained. That feature is specified complexity. Life is both complex and specified. The basic institution here is straightforward. Davies rightly notes, laws (that is, necessities of nature) can explain specification but not complexity. Once life (or more generally some self-replicator) …

The Soviet Concept Of The Correlation Of Forces, James Bradley

ACMS Conference Proceedings 2001

This paper takes a look at the Soviet Union’s accumulation of nuclear weapons during the Cold War and what mathematical strategy they employed to make their choices.

Mathematics As Worship, David J. Stucki

ACMS Conference Proceedings 2001

In keeping with the mission of this organization to explore the relationship of faith to our discipline, I would like to take this opportunity to investigate the relationship, if any, between mathematics and worship. There have been throughout history, at least since Pythagoras, connections made between the mathematical and the theological. Many of these such efforts have followed the Pythagorean cult in deifying number, thus making mathematics the object of worship. Othes have effectively situated theology in subservience to mathematical reason. However, these are not the only alternatives.

Once we admit the possibility of a connection between mathematics and theology, …

On Periodic Points On Maps Of Trees And The Expansive Property, Fred Worth

ACMS Conference Proceedings 2001

In this paper, we consider the expansive property (A homeomorphism, f, of a metric space, X, onto itself is called expansive if there is a positive number, ε, such that if x and y are distinct points of X, then there exists an integer, n = n(x,y), such that d(f n(x), f n(y)) > ε. It should be noted that n may be negative.) and how it relates to shift homeomorphisms of a tree with a single, surjective bonding map. We also consider some results regarding the periodicity of points in self-maps of trees.

Thml: Theological Markup Language For The Christian Classics Ethereal Library, Harry Plantinga

ACMS Conference Proceedings 2001

This document describes the Theological Markup Language (ThML), an XML markup language for theological texts. ThML was developed for use in the Christian Classics Ethereal Library (CCEL), but it is hoped that the language will serve as a royalty-free format for theological texts in other applications. Key design goals are that the language should be (1) rich enough to represent information needed for digital libraries and for theological study involving multiple, related texts, including cross-reference, synchronization, indexing, and scripture references, (2) based on XML and usable with World Wide Web tools, (3) automatically convertible to other common formats, and (4) …

Gravitational Acceleration In Hades, Andrew Simoson

ACMS Conference Proceedings 2001

Does acceleration due to gravity increase or decrease upon descending from Earth’s surface? The answer—as we show—depends on one’s model for Earth’s density. For our Earth, gravity increases before it collapses to zero at Earth center.

Cost Domination In Graphs, David John Erwin

Dissertations

Let G be a connected graph having order at least 2. A function f : V (G) —> {0 , 1 , . . . , diam G} for which f ( v ) < e(v) for every vertex v of G is a cost function on G. A vertex v with f ( v ) > 0 is an f-dominating vertex, and the set Vj~ = {v 6 V(G) : f(v) > 0} of f-dominating vertices is the f-dominating set. An /-dominating vertex v is said to f-dominate every vertex u with d(n, v) < f(u ), while …

Introduction (2001), Association Of Christians In The Mathematical Sciences

ACMS Conference Proceedings 2001

Thirteenth ACMS Conference on Mathematics from a Christian Perspective

Schedule (2001), Association Of Christians In The Mathematical Sciences

ACMS Conference Proceedings 2001

Thirteenth ACMS Conference on Mathematics from a Christian Perspective

Table Of Contents (2001), Association Of Christians In The Mathematical Sciences

ACMS Conference Proceedings 2001

Thirteenth ACMS Conference on Mathematics from a Christian Perspective

Separability Of Tilings, Nicholas Baeth, Jason Deblois, Lisa Powell

Mathematical Sciences Technical Reports (MSTR)

A tiling by triangles of an orientable surfaces is called kaleidoscopic if the local reflection in any edge of a triangle extends to a global isometry of the surface. Given such a global reflection the fixed point subset of the reflection consists of embedded circles (ovals) whose union is called the mirror of the reflection. The reflection is called separating if removal of the mirror disconnects the surface into two components. We consider surfaces such that the orientation preserving subgroup of the tiling group generated by the reflection is cyclic or abelian. A complete classification of those surfaces with separating …

Topologically Pure Extensions, Peter Loth

Mathematics Faculty Publications

A proper short exact sequence 0→H →G→K→0 (*) in the category of locally compact abelian groups is said to be topologically pure if the induced sequence 0→nH→nG→nK→0 is proper short exact for all positive integers n. Some characterizations of topologically pure sequences in terms of direct decompositions, pure extensions and tensor products are established. A simple proof is given for a theorem on pure subgroups by Hartman and Hulanickl. Using topologically pure extensions, we characterize those splitting locally compact abelian groups whose torsion part is a direct sum of a compact …

Efficient Traitor Tracing Algorithms Using List Decoding, Alice Silverberg, Jessica Staddon, Judy L. Walker

Department of Mathematics: Faculty Publications

We use powerful new techniques for list decoding error-correcting codes to efficiently trace traitors. Although much work has focused on constructing traceability schemes, the complexity of the tracing algorithm has received little attention. Because the TA tracing algorithm has a runtime of O(N) in general, where N is the number of users, it is inefficient for large populations.We produce schemes for which the TA algorithm is very fast. The IPP tracing algorithm, though less efficient, can list all coalitions capable of constructing a given pirate. We give evidence that when using an algebraic structure, the ability to …

Decay Estimates Of Heat Transfer To Melton Polymer Flow In Pipes With Viscous Dissipation, Dongming Wei, Zhenbu Zhang

Mathematics Faculty Publications

In this work, we compare a parabolic equation with an elliptic equation both of which are used in modeling temperature profile of a power-law polymer flow in a semi-infinite straight pipe with circular cross section. We show that both models are well-posed and we derive exponential rates of convergence of the two solutions to the same steady state solution away from the entrance. We also show estimates for difference between the two solutions in terms of physical data.

Biharmonic Maps On V-Manifolds, Yuan-Jen Chiang, Hongan Sun

Mathematics

We generalize biharmonic maps between Riemannian manifolds into the case of the domain being V-manifolds. We obtain the first and second variations of biharmonic maps on V-manifolds. Since a biharmonic map from a compact V-manifold into a Riemannian manifold of nonpositive curvature is harmonic, we construct a biharmonic non-harmonic map into a sphere. We also show that under certain condition the biharmonic property of f implies the harmonic property of f. We finally discuss the composition of biharmonic maps on V-manifolds.

A Critical Look At Self-Dual Codes, Judy L. Walker

Department of Mathematics: Faculty Publications

We investigate self-dual codes from a structural point of view. In particular, we study properties of critical indecomposable codes which appear in the spectrum of a self-dual code. As an application of the results we obtain, we revisit the study of self-dual codes of dimension at most 10.

In the late 1950’s, Slepian [4] became the first to take an abstract approach to the study of error-correcting codes. He introduced a structure theory for binary linear codes, developing in particular the idea of an indecomposable code; that is, a code which is not isomorphic to a nontrivial direct sum of …