Applied Mathematics Commons^™

Radiotherapy Optimal Design: An Academic Radiotherapy Treatment Design System, R Acosta, W Brick, A Hanna, Allen Holder, D Lara, G Mcquillen, D Nevin, P Uhlig, B Salter

Mathematical Sciences Technical Reports (MSTR)

Optimally designing radiotherapy and radiosurgery treatments to increase the likelihood of a successful recovery from cancer is an important application of operations research. Researchers have been hindered by the lack of academic software that supports head-to-head comparisons of different techniques, and this article addresses the inherent difficulties of designing and implementing an academic treatment planning system. In particular, this article details the algorithms and the software design of Radiotherapy optimAl Design (RAD).

The Barycenter Of The Numerical Range Of A Matrix, Sean A. Broughton, Roger G. Lautzenheiser, Thomas Werne

Mathematical Sciences Technical Reports (MSTR)

The numerical range W(A) of an nxn matrix A is the totality of the scalar products <Ax,x> as x varies over all unit vectors in Cⁿ The barycenter (center of mass) of the numerical range is defined geometrically as the center of mass of W(A) considered as a planar lamina with variable density and also as a limit of sample averages (<Ax₁,x₁>+...+<Ax_N,x_N>)/N. Under a wide range the sampling schemes it is shown that the barycenter is the average of the spectrum …

Utilizing Thermal Testing For Recovering, James Preciado, Thomas Werne

Mathematical Sciences Technical Reports (MSTR)

Given a two-dimensional region that contains one or more circular voids, we develop mathematical methods to locate the center and radius of the voids based on thermal boundary data. These methods can be readily applied in the field of non-destructive evaluation.

Thermal Imaging To Recover A Defect In Three Dimensional Objects, Breanne Baker

Mathematical Sciences Technical Reports (MSTR)

This paper focuses on the inverse problem of identifying an internal void in a bounded two- or three-dimensional region. Information, in form of a heat flux and temperature, is assumed to be obtainable only on the external boundary of the region. The reciprocity gap approach with a suitable test functions is used in both the two- and three-dimensional cases.

Non-Destructive Recovery Of Voids Within A Three Dimensional Domain Using Thermal Imaging, Victor B. Oyeyemi

Mathematical Sciences Technical Reports (MSTR)

We develop an algorithm capable of detecting the presence of spherical voids in a thermally conducting object. In addition, the process recovers both the radii and locations of each void. Our method involves the application of a known steady state heat flux to the object's boundary and measurement of the induced steady state temperature on the boundary.

Foundations Of Generalized Cwatsets, Jesse Beder

Mathematical Sciences Technical Reports (MSTR)

We present a new, abstract definition for a generalized cwatset that produces notions of subcwatset and quotient cwatset that behave naturally. We use small cancellation theory to prove a result analogous to the statement that every group is isomorphic to some permutation group.

Thermal Imaging Of Circular Inclusions Within A Two-Dimensional Region, Shannon Talbott, Hilary Spring

Mathematical Sciences Technical Reports (MSTR)

The ability to study the interior of an object without destroying it is an important industrial tool. One method of recent interest is steady-state thermal or impedance imaging. In this paper we will use the steady-state heat equation to locate one or more circular inclusions within a two-dimensional region, where the boundaries of the inclusions have partially disbonded from the surrounding material; this disbond is modelled as between the heat flux and jump discontinuity at the disbonded interface.

Time-Dependent Thermal Imaging Of Circular Inclusions, Donald L. Brouwn, Mark Hubenthal

Mathematical Sciences Technical Reports (MSTR)

This paper considers the inverse problem of locating one or more circular inclusions in a two-dimensional domain using thermal boundary data, specifically, the input heat flux and measured boundary temperature. The forward problem is governed by the heat equation. We show how the position and size of such defects can be recovered using the boundary data and various approximations of the solution to the forward problem. We also consider the stability of the algorithm involved to recover the defects.

Reconstruction Of Partially Conductive Cracks Using Boundary Data, David Mccune, Janine Haugh

Mathematical Sciences Technical Reports (MSTR)

This paper develops an algorithm for finding one or more non-insulated, pair-wise disjoint, linear cracks in a two dimensional region using boundary measurements.

Non-Destructive Testing Of Thermal Resistances For A Single Inclusion In A 2-Dimensional Domain, Nicholas Christian, Mathew A. Johnson

Mathematical Sciences Technical Reports (MSTR)

In this paper we examine the inverse problem of determining the amount of corrosion/disbonding which has occurred on the boundary of a single circular (or nearly circular) inclusion D in a two dimensional domain W using Cauchy data for the steady-state heat equation. We develop an algorithm for reconsructing a function which qunatifies the level of corrosion/disbonding at each point in ¶W. We also address the issue of well-posedness and develop a simple regularization scheme. Then we provide several numerical examples. We shall show a simple procedure for recovering the center of D assuming that the boundary of W and …

Reconstruction Of An Unknown Boundary Portion From Cauchy Data In N-Dimensions, Kurt M. Bryan, Lester Caudill

Mathematical Sciences Technical Reports (MSTR)

We consider the inverse problem of determining the shape of some inacces­ sible portion of the boundary of a region in n dimensions from Cauchy data for the heat equation on an accessible portion of the boundary. The inverse problem is quite ill-posed, and nonlinear. We develop a Newton-like algorithm for solving the problem, with a simple and efficient means for computing the required derivatives, develop methods for regularizing the process, and provide computational examples

Determining The Length Of A One-Dimensional Bar, Natalya Yarlikina, Holly Walrath

Mathematical Sciences Technical Reports (MSTR)

In this paper we examine the inverse problem of determining the length of a one-dimensional bar from thermal measurements (temperature and heat flux) at one end of the bar (the "accessible" end); the other inaccessible end of the bar is assumed to be moving. We develop two different approaches to estimating the length of the bar, and show how one approach can also be adapted to find unknown boundary conditions at the inaccessible end of the bar.

When Abelian Groups Split, Rachel M. Thomas, Robert C. Rhoades

Mathematical Sciences Technical Reports (MSTR)

Let S be a hyperbolic surface tiled by kaleidoscopic triangles. Let R_e denote the set of fixed points by the reflection in an edge, e, of a triangle. We say that R_e is separating if S-R_e has two components. Once we have a tiling, we can define a group of orientation preserving transformations, G. We develop a method for determining when a reflection is separating using the group algebra of G. Using this method we give necessary and sufficient conditions for a mirror to be separating when G is abelian. We also conjecture, that …

Fixed Point And Two-Cycles Of The Discrete Logarithm, Joshua Holden

Mathematical Sciences Technical Reports (MSTR)

We explore some questions related to one of Brizolis: does every prime p have a pair (g, h) such that h is a fixed point for the discrete logarithm with base g? We extend this question to ask about not only fixed points but also two-cycles. Campbell and Pomerance have not only answered the fixed point question for sufficiently large p but have also rigorously estimated the number of such pairs given certain conditions on g and h. We attempt to give heuristics for similar estimates given other conditions on g and h and also in the case …

A Review Of Selected Works On Crack Indentification, Kurt M. Bryan

Mathematical Sciences Technical Reports (MSTR)

We give a short survey of some of the results obtained within the last 10 years or so concerning crack identification using impedance imaging techniques. We touch upon uniqueness results, continuous dependence results, and computational algorithms.

Characterizing A Defect In A One-Dimensional Bar, Cynthia Gangi, Sameer Shah

Mathematical Sciences Technical Reports (MSTR)

We examine the inverse problem of locating and describing an internal point defect in a one­ dimensional rod W by controlling the heat inputs and measuring the subsequent temperatures at the boundary of W. We use a variation of the forward heat equation to model heat flow through W, then propose algorithms for locating an internal defect and quantifying the effect the defect has on the heat flow. We implement these algorithms, analyze the stability of the procedures, and provide several computational examples.

Reconstruction Of Cracks With Unknown Transmission Condition From Boundary Data, F Ronald Ogborne Iii, Melissa E. Vellela

Mathematical Sciences Technical Reports (MSTR)

We examine the problem of Identifying both the location and constitutive law governing electrical current flow across a one-dimensional linear crack in a two dimensional region when the crack only partially blocks the flow of current. We develop a a constructive numerical procedure for solving the inverse problem and provide computational examples.

Fast Reconstruction Of Cracks Using Boundary Measurements, Nicholas A. Trainor, Rachel M. Krieger

Mathematical Sciences Technical Reports (MSTR)

This paper develops a fast algorithm for locating one or more perfectly insulating, pair-wise disjoint, linear cracks in a homogeneous two-dimensional electrical conductor, using boundary measurements.

The Order At 2 Of The Odd-Partition Function, Chris Mehelich

Mathematical Sciences Technical Reports (MSTR)

Abstract. We evaluate the odd-partition function p₂(n) modulo 4 by elementary methods and analyze the asymptotic distribution of p₂(n) modulo 4. We use the theory of modular forms to obtain necessary and sufficient conditions for the order at 2 of p₂(n) to equal any given value between 0 and 4 inclusive.

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 …

Solvability Of A Parabolic Boundary Value Problem With Internal Jump Condition, Kurt M. Bryan, Lester Caudill

Mathematical Sciences Technical Reports (MSTR)

We examine a model for the propagation of heat through a one-dimensional object with an interior ''flaw". The flaw is modeled as a nonlinear relationship between the flux and temperature jump at an interior point of the object. Under realistic hypotheses, the resulting nonlinear initial boundary value problem is shown to have a unique and suitably smooth solution.

Singular Solutions To A Nonlinear Elliptic Boundary Value Problem Originating From Corrosion Modeling, Kurt M. Bryan, Michael Vogelius

Mathematical Sciences Technical Reports (MSTR)

We consider a nonlinear elliptic boundary value problem on a planar domain. The exponential type nonlinearity in the boundary condition is one that frequently appears in the modeling of electrochemical systems. For the case of a disk we construct a family of exact solutions that exhibit limiting logarithmic singularities at certain points on the boundary. Based on these solutions we develop two criteria that we believe predict the possible locations of the boundary singularities on quite general domains.

Maximally Disjoint Solutions Of The Set Covering Problem, David J. Rader, Peter L. Hammer

Mathematical Sciences Technical Reports (MSTR)

This paper is concerned with finding two solutions of a set covering problem that have a minimum number of variables in common. We show that this problem is NP­ complete, even in the case where we are only interested in completely disjoint solutions. We describe three heuristic methods based on the standard greedy algorithm for set covering problems. Two of these algorithms find the solutions sequentially, while the third finds them simultaneously. A local search method for reducing the overlap of the two given solutions is then described. This method involves the solution of a reduced set covering problem. Finally, …

Stability And Reconstruction For An Inverse Problem For The Heat Equations, Kurt M. Bryan, Lester Caudill

Mathematical Sciences Technical Reports (MSTR)

We examine the inverse problem of determining the shape of some unknown portion of the boundary of a region W from measurements of the Cauchy data for solutions to the heat equation on W. By suitably linearizing the inverse problem we obtain uniqueness and continuous dependence results. We propose an algorithm for recovering estimates of the unknown portion of the surface and use the insight gained from a detailed analysis of the inverse problem to regularize the inversion. Several computational examples are presented.

Uniqueness For A Boundary Identification Problem In Thermal Imaging, Kurt M. Bryan, Lester F. Caudill

Mathematical Sciences Technical Reports (MSTR)

An inverse problem for a parabolic initial-boundary value problem is considered. The goal is to determine an unknown portion of the boundary of a region in Rⁿ from measurements of Dirichlet data on a known portion of the boundary. It is shown that under reasonable hypotheses uniqueness results hold.

Effective Behavior Of Clusters Of Microscopic Cracks Inside A Homogeneous Conductor, Kurt M. Bryan, Michael Vogelius

Mathematical Sciences Technical Reports (MSTR)

We study the effective behaviour of a periodic array of microscopic cracks inside a homoge­neous conductor. Special emphasis is placed on a rigorous study of the case in which the corresponding effective conductivity becomes nearly singular, due to the fact that adjacent cracks nearly touch. It is heuristically shown how thin clusters of such extremely close cracks may macroscopically appear as a single crack. The results have implications for our earlier work on impedance imaging.

Population Genetics: Estimation Of Distributions Through Systems Of Non-Linear Differential Equations, Nacer E. Abrouk, Robert J. Lopez

Mathematical Sciences Technical Reports (MSTR)

In stochastic population genetics, the fundamental quantity used for describing the genetic composition of a Mendelian population is the gene frequency. The process of change in the gene frequency is generally modeled as a stochastic process satisfying a stochastic differential equation. The drift and diffusion coefficients in this equation reflect such mechanisms as mutation, selection, and migration that affect the population. Except in very simple cases, it is difficult to determine the probability law of the stochastic process of change in gene frequency. We present a method for obtaining approximations of this process, enabling us to study models more realistic …

A Variable Time-Step Midpoint Scheme For Hamiltonian Systems, Yosi Shibberu

Mathematical Sciences Technical Reports (MSTR)

A smooth time-step selection formula for the midpoint method is derived which minimize deviations in the Hamiltonian function along piecewise-linear phase space trajectories of autonomous Hamiltonian systems. The time-step formula is implemented in a second order pre­dictor/corrector scheme and applied to Kepler's problem. The formula significantly improves energy conservation as well as the accuracy of the configuration space trajectory. Peak errors in position and momentum coordinates are not significantly reduced, but the time behavior of the errors is markedly more regular.

An Elementary Proof That Finite Groups Lack Unique Product Structures, Matthew Cushman

Mathematical Sciences Technical Reports (MSTR)

We provide an elementary proof that no nontrivial finite group has a unique m-element product structure for any m greater than or equal to 2.

Distinct Products Of Triples In Finite Groups, Curtis Z. Mitchell

Mathematical Sciences Technical Reports (MSTR)

Let G be a finite group and let D_i(G) be the proportion of triples ( x , y , z ) of elements in G such that the cardinality of { xyz , xzy , yxz, yzx , zxy , zyx } is i. In this paper we show that:

i) The average value of D_i is either 1 or at least 53/32.

ii) D₂= 0 ==> D₃ = D₄ = D₅ = D₆ = 0;

iii) D₃= 0 ==> D₄ = D₅ = 0