Physical Sciences and Mathematics Commons^™

Multiscale Registration Of Planning Ct And Daily Cone Beam Ct Images For Adaptive Radiation Therapy, Dana C. Paquin, Doron Levy, Lei Xing

Mathematics

Adaptive radiation therapy (ART) is the incorporation of daily images in the radiotherapy treatment process so that the treatment plan can be evaluated and modified to maximize the amount of radiation dose to the tumor while minimizing the amount of radiation delivered to healthy tissue. Registration of planning images with daily images is thus an important component of ART. In this article, the authors report their research on multiscale registration of planning computed tomography (CT) images with daily cone beam CT (CBCT) images. The multiscale algorithm is based on the hierarchical multiscale image decomposition of E. Tadmor, S. Nezzar, and …

An Optimal Method To Combine Results From Different Experiments, Theodore P. Hill, Jack Miller

Research Scholars in Residence

This article describes an optimal method (conflation) to consolidate data from different experiments, and illustrates the advantages of conflation by graphical examples involving gaussian input distributions, and by a concrete numerical example involving the values of lattice spacing of silicon crystals used in determination of the current values of Planck's constant and the Avogadro constant.

Eigenvalue Inequalities For A Family Of Spherically Symmetric Riemannian Manifolds, Julie Miker

University of Kentucky Doctoral Dissertations

This thesis considers two isoperimetric inequalities for the eigenvalues of the Laplacian on a family of spherically symmetric Riemannian manifolds. The Payne-Pólya-Weinberger Conjecture (PPW) states that for a bounded domain Ω in Euclidean space Rⁿ, the ratio λ1(Ω)/λ0(Ω) of the first two eigenvalues of the Dirichlet Laplacian is bounded by the corresponding eigenvalue ratio for the Dirichlet Laplacian on the ball B_Ωof equal volume. The Szegö-Weinberger inequality states that for a bounded domain Ω in Euclidean space Rⁿ, the first nonzero eigenvalue of the Neumann Laplacian μ1(Ω) is maximized on the ball B_{Ω …}

Collapsible Graphs And Reductions Of Line Graphs, Zhi-Hong Chen, Peter C.B. Lam, Wai-Chee Shiu

Scholarship and Professional Work - LAS

A graph G is collapsible if for every even subset X ⊆ V ( G ) , G has a subgraph such that G − E ( Γ ) is connected and the set of odd-degree vertices of Γ is X . A graph obtained by contracting all the non-trivial collapsible subgraphs of G is called the reduction of G . In this paper, we characterize graphs of diameter two in terms of collapsible subgraphs and investigate the relationship between the line graph of the reduction and the reduction of the line graph. Our results extend former results in [H.-J. …

Analysis Of Connections Between Pseudocodewords, Nathan Axvig, Deanna Dreher, Katherine Morrison, Eric T. Psota, Lance C. Pérez, Judy L. Walker

Department of Mathematics: Faculty Publications

The role of pseudocodewords in causing noncodeword outputs in linear programming (LP) decoding, graph cover decoding, and iterative message-passing decoding is investigated. The three main types of pseudocodewords in the literature — linear programming pseudocodewords, graph cover pseudocodewords, and computation tree pseudocodewords — are reviewed and connections between them are explored. Some discrepancies in the literature on minimal and irreducible pseudocodewords are highlighted and clarified, and a value for the minimal degree cover necessary to realize an LP pseudocodeword is found. Additionally, some conditions for the existence of connected realizations of graph cover pseudocodewords are given. This allows for further …

The Erdős-Lovász Tihany Conjecture For Quasi-Line Graphs, J. Balogh, A. V. Kostochka, N. Prince, M. Stiebitz

Faculty Publications & Research

Erdős and Lovász conjectured in 1968 that for every graph G with χ(G) > ω(G) and any two integers s, t ≥ 2 with s + t = χ(G) + 1, there is a partition (S,T) of the vertex set V(G) such that χ(G[S]) ≥ s and χ(G[T]) ≥ t . Except for a few cases, this conjecture is still unsolved. In this note we prove the conjecture for quasi-line graphs and for graphs with independence number 2.

The Hamiltonian Index Of Graphs, Yi Hong, Jian-Liang Lin, Zhi-Sui Tao, Zhi-Hong Chen

Scholarship and Professional Work - LAS

The Hamiltonian index of a graph G is defined as h ( G ) = min { m : L m ( G ) is Hamiltonian } . In this paper, using the reduction method of Catlin [P.A. Catlin, A reduction method to find spanning Eulerian subgraphs, J. Graph Theory 12 (1988) 29–44], we constructed a graph H ̃ ( m ) ( G ) from G and prove that if h ( G ) ≥ 2 , then h ( G ) = min{ m : H ̃ ( m ) ( G ) has a spanning Eulerian subgraph …

Discrete Mathematics With Proof Second Edition, Eric Gossett

Faculty Books

Discrete mathematics has become increasingly popular in recent years due to its growing applications in the field of computer science. Discrete Mathematics with Proof, Second Edition continues to facilitate an up-to-date understanding of this important topic, exposing readers to a wide range of modern and technological applications. The book begins with an introductory chapter that provides an accessible explanation of discrete mathematics. Subsequent chapters explore additional related topics including counting, finite probability theory, recursion, formal models in computer science, graph theory, trees, the concepts of functions, and relations.

Introduction To The "Prisoners And Guards" Game, Eugen J. Ionascu, Tim Howard

Faculty Bibliography

We study the half-dependent problem for the king graph Kn. We give proofs to establish the values h(Kn) for n ∈ {1, 2, 3, 4, 5, 6} and an upper bound for h(Kn) in general. These proofs are independent of computer assisted results. Also, we introduce a two-player game whose winning strategy is tightly related with the values h(Kn). This strategy is analyzed here for n = 3 and some facts are given for the case n = 4. Although the rules of the game are very simple, the winning strategy is highly complex even for n = 4.

Solution Of Problem 878, Eugen J. Ionascu

Faculty Bibliography

No abstract provided.

Galois Structure And De Rhan Invariants Of Elliptic Curves, Darren B. Glass, Sonin Kwon

Math Faculty Publications

Let K be a number field with ring of integers OK. Suppose a finite group G acts numerically tamely on a regular scheme X over OK. One can then define a de Rham invariant class in the class group Cl(OK[G]), which is a refined Euler characteristic of the de Rham complex of X. Our results concern the classification of numerically tame actions and the de Rham invariant classes. We first describe how all Galois etale G-covers of a K-variety may be built up from finite Galois extensions of K and from geometric covers. When X is a curve of positive …

Reliability Confidence Intervals For Oil Spills In The Gulf Of Mexico, William V. Harper, Thomas R. James, Ted G. Eschenbach

Mathematics Faculty Scholarship

An extensive study [Eschenbach and Harper (2006)] of offshore oil spills in the Gulf of Mexico with extensions to the northern seas of Alaska involved the estimation of the likelihood of oil spill volumes in the Gulf of Mexico for both pipeline and platform spills. This paper develops both maximum likelihood based reliability and percentile confidence intervals for the 3- parameter Weibull distribution. The statistical aspects are discussed along with applications of developed Excel VBA functions. The Excel routines are available free on the web at http://faculty.otterbein.edu/WHarper/. The functions are illustrated with Gulf of Mexico oil spill data.

Algebras Having Bases Consisting Entirely Of Units, Jeremy Moore

Mathematics Faculty Scholarship

We introduce a hierarchy of notions about algebras having a basis B consisting entirely of units. Such a basis is called an invertible basis and algebras that have invertible bases are said to be invertible algebras...

Degree Spectra Of The Successor Relation Of Computable Linear Orderings, Jennifer Chubb, Andrey Frolov, Valentina Harizanov

Mathematics

We establish that for every computably enumerable (c.e.) Turing degree b, the upper cone of c.e. Turing degrees determined by b is the degree spectrum of the successor relation of some computable linear ordering. This follows from our main result, that for a large class of linear orderings, the degree spectrum of the successor relation is closed upward in the c.e. Turing degrees.

Generalized Helmholtz-Kirchhoff Model For Two-Dimensional Distributed Vortex Motion, Raymond J. Nagem, Guido Sandri, David Uminsky, C. Eugene Wayne

Mathematics

The two-dimensional Navier-Stokes equations are rewritten as a system of coupled nonlinear ordinary differential equations. These equations describe the evolution of the moments of an expansion of the vorticity with respect to Hermite functions and of the centers of vorticity concentrations. We prove the convergence of this expansion and show that in the zero viscosity and zero core size limit we formally recover the Helmholtz-Kirchhoff model for the evolution of point vortices. The present expansion systematically incorporates the effects of both viscosity and finite vortex core size. We also show that a low-order truncation of our expansion leads to the …

Cooperation In An Evolutionary Prisoner's Dilemma On Networks With Degree-Degree Correlations, Stephen Devlin, T Treloar

Mathematics

We study the effects of degree-degree correlations on the success of cooperation in an evolutionary prisoner's dilemma played on a random network. When degree-degree correlations are not present, the standardized variance of the network's degree distribution has been shown to be an accurate analytical measure of network heterogeneity that can be used to predict the success of cooperation. In this paper, we use a local-mechanism interpretation of standardized variance to give a generalization to graphs with degree-degree correlations. Two distinct mechanisms are shown to influence cooperation levels on these types of networks. The first is an intrinsic measurement of base-line …

Evolution Of Cooperation Through The Heterogeneity Of Random Networks, Stephen Devlin, T Treloar

Mathematics

We use the standardized variance (nu_{st}) of the degree distribution of a random network as an analytic measure of its heterogeneity. We show that nu_{st} accurately predicts, quantitatively, the success of cooperators in an evolutionary prisoner's dilemma. Moreover, we show how the generating functional expression for nu_{st} suggests an intrinsic interpretation for the heterogeneity of the network that helps explain local mechanisms through which cooperators thrive in heterogeneous populations. Finally, we give a simple relationship between nu_{st} , the cooperation level, and the epidemic threshold of a random network that reveals an appealing connection between epidemic disease models and the …

Biwave Maps Into Manifolds, Yuan-Jen Chiang

Mathematics

We generalize wave maps to biwave maps. We prove that the composition of a biwave map and a totally geodesic map is a biwave map. We give examples of biwave nonwave maps. We show that if f is a biwave map into a Riemannian manifold under certain circumstance, then f is a wave map. We verify that if f is a stable biwave map into a Riemannian manifold with positive constant sectional curvature satisfying the conservation law, then f is a wave map. We finally obtain a theorem involving an unstable biwave map.

Effect Of Dlk1 And Rtl1 But Not Meg3 Or Meg8 On Muscle Gene Expression In Callipyge Lambs, Jolena N. Fleming-Waddell, Gayla R. Olbricht, Tasia M. Taxis, Jason D. White, Tony Vuocolo, Bruce A. Craig, Ross L. Tellam, Mike K. Neary, Noelle E. Cockett, Christopher A. Bidwell

Mathematics and Statistics Faculty Research & Creative Works

Callipyge sheep exhibit extreme postnatal muscle hypertrophy in the loin and hindquarters as a result of a single nucleotide polymorphism (SNP) in the imprinted DLK1-DIO3 domain on ovine chromosome 18. The callipyge SNP up-regulates the expression of surrounding transcripts when inherited in cis without altering their allele-specific imprinting status. The callipyge phenotype exhibits polar overdominant inheritance since only paternal heterozygous animals have muscle hypertrophy. Two studies were conducted profiling gene expression in lamb muscles to determine the down-stream effects of over-expression of paternal allele-specific DLK1 and RTL1 as well as maternal allele-specific MEG3, RTL1AS and MEG8, using Affymetrix bovine expression …

A Finite Element Splitting Extrapolation For Second Order Hyperbolic Equations, Xiaoming He, Tao Lü

Mathematics and Statistics Faculty Research & Creative Works

Splitting extrapolation is an efficient technique for solving large scale scientific and engineering problems in parallel. This article discusses a finite element splitting extrapolation for second order hyperbolic equations with time-dependent coefficients. This method possesses a higher degree of parallelism, less computational complexity, and more flexibility than Richardson extrapolation while achieving the same accuracy. By means of domain decomposition and isoparametric mapping, some grid parameters are chosen according to the problem. The multiparameter asymptotic expansion of the d-quadratic finite element error is also established. The splitting extrapolation formulas are developed from this expansion. An approximation with higher accuracy on a …

Characterization Of Partial Derivatives With Respect To Boundary Conditions For Solutions Of Nonlocal Boundary Value Problems For Nth Order Differential Equations, Jeffrey W. Lyons, Johnny Henderson

Mathematics Faculty Articles

Under certain conditions, solutions of the nonlocal boundary value problem, y⁽ⁿ⁾ = f(x, y, y', ... , y^{(n- 1)}), y(x_i) = Y_i for 1 £ i £ n- 1, and y(x_n) - Σ^m_k=1 Υ_iy (n_i) = y _n, are differentiated with respect to boundary conditions, where a < X₁ < X₂ < · · · < X_n-1 < n₁ < · · · < n_m < X_n < b, r₁, ... , r_m, Y1, ... , Y_n ∈ R .

The Dixmier-Douady Invariant For Dummies, Claude Schochet

Mathematics Faculty Research Publications

The Dixmier-Douady invariant is the primary tool in the classification of continuous trace C*-algebras. These algebras have come to the fore in recent years because of their relationship to twisted K-theory and via twisted K-theory to branes, gerbes, and string theory.

This note sets forth the basic properties of the Dixmier-Douady invariant using only classical homotopy and bundle theory. Algebraic topology enters the scene at once since the algebras in question are algebras of sections of certain fibre bundles.

Banach Algebras And Rational Homotopy Theory, Gregory Lupton, N. Christopher Phillips, Claude Schochet, Samuel B. Smith

Mathematics Faculty Research Publications

Let A be a unital commutative Banach algebra with maximal ideal space Max(A). We determine the rational H-type of GL_n(A), the group of invertible n x n matrices with coefficients in A, in terms of the rational cohomology of Max(A). We also address an old problem of J. L. Taylor. Let Lc_n(A) denote the space of "last columns" of GL_n(A). We construct a natural isomorphism

Ȟ^s(Max(A);ℚ) ≅ π_2n-1-s(Lc_n(A)) ⊗ ℚ …

Hybrid Approximate Proximal Method With Auxiliary Variational Inequality For Vector Optimization, L C. Ceng, Boris S. Mordukhovich, Jen-Chih Yao

Mathematics Research Reports

This paper studies the general vector optimization problem of finding weakly efficient points for mappings in a Banach space Y, with respect to the partial order induced by a closed, convex, and pointed cone C C Y with nonempty interior. In order to find a solution of this problem, we introduce an auxiliary variational inequality problem for monotone, Lipschitz-continuous mapping. The approximate proximal method in vector optimization is extended to develop a hybrid approximate proximal method for the general vector optimization problem by the combination of extragradient method for finding a solution to the variational inequality problem and approximate proximal …

A Newton Root-Finding Algorithm For Estimating The Regularization Parameter For Solving Ill-Conditioned Least Squares Problems, Jodi Mead, Rosemary Renaut

Mathematics Faculty Publications and Presentations

We discuss the solution of numerically ill-posed overdetermined systems of equations using Tikhonov a-priori-based regularization. When the noise distribution on the measured data is available to appropriately weight the fidelity term, and the regularization is assumed to be weighted by inverse covariance information on the model parameters, the underlying cost functional becomes a random variable that follows a X² distribution. The regularization parameter can then be found so that the optimal cost functional has this property. Under this premise a scalar Newton root-finding algorithm for obtaining the regularization parameter is presented. The algorithm, which uses the singular value …

The Logic Of Bailout Strategies, Rudolf Kaehr

Rudolf Kaehr

Some thoughts about/of the logic, blend, chiasm and diamond of bailout strategies. Eliciting aspects of the maxim: “Without insurrection, no resurrection".

Diamond Semiotic Short Studies, Rudolf Kaehr

Rudolf Kaehr

A collection of papers on semiotics, polycontexturality and diamond theory

The Fundamentals Of Local Fractional Derivative Of The One-Variable Non-Differentiable Functions, Yang Xiaojun

Xiao-Jun Yang

Based on the theory of Jumarie’s fractional calculus, local fractional derivative is modified in detail and its fundamentals of local fractional derivative are proposed in this paper. The uniqueness of local fractional derivative is obtained and the Rolle’s theorem, the mean value theorem, the Cauchy’s generalized mean value theorem and the L’Hospital’s rules are proved.

Local Fractional Newton’S Method Derived From Modified Local Fractional Calculus, Yang Xiao-Jun

Xiao-Jun Yang

A local fractional Newton’s method, which is derived from the modified local fractional calculus , is proposed in the present paper. Its iterative function is obtained and the convergence of the iterative function is discussed. The comparison between the classical Newton iteration and the local fractional Newton iteration has been carried out. It is shown that the iterative value of the local fractional Newton method better approximates the real-value than that of the classical one.

A Tikz Tutorial: Generating Graphics In The Spirit Of Tex, Andrew Mertz, William Slough

Andrew Mertz

TikZ is a system which can be used to specify graphics of very high quality. For example, accurate place- ment of picture elements, use of TEX fonts, ability to incorporate mathematical typesetting, and the possi- bility of introducing macros can be viewed as positive factors of this system. The syntax uses an amal- gamation of ideas from METAFONT, METAPOST, PSTricks, and SVG, allowing its users to \program" their desired graphics. The latest revision to TikZ introduces many new features to an already feature- packed system, as evidenced by its 560-page user manual. Here, we present a tutorial overview of this …