Mathematics Commons^™

Branching Exponent Heterogeneity And Wall Shear Stress Distribution In Vascular Trees, Kelly Lynn Karau, Gary S. Krenz, Christopher A. Dawson

Mathematics, Statistics and Computer Science Faculty Research and Publications

A bifurcating arterial system with Poiseuille flow can function at minimum cost and with uniform wall shear stress if the branching exponent (z) = 3 [where z is defined by (D ₁)^z = (D ₂)^z + (D ₃)^z; D ₁ is the parent vessel diameter and D ₂ and D ₃ are the two daughter vessel diameters at a bifurcation]. Because wall shear stress is a physiologically transducible force, shear stress-dependent control over vessel diameter would appear to provide a means for preserving this optimal structure through maintenance …

Mapping Quantitative Trait Loci In Humans: Some Statistical Contributions., Saurabh Ghosh Dr.

Doctoral Theses

Maty qualitative tralts - such an, milk yield la cows, blood pressure in lumans --are known to be determined primarly, though zot exclusively, by inherited genetic luctora. It ls the of coasklerable impartance to identify chromosontal locations of tho genes that control a quantitative character. Linkage analysis (Ou 1990), which deals with the deduction of linkagn and estimation of recombination fractions among the loci controlling a qualitative/quantitative character and major loci wkoo poertions are knows aprfori, is widely used for localisation of gens. Although statistical methodologies for magplag gemen determining dichotomos qualitative charactes in humans aro well-developed, the demicrant of …

Boolean Functions With Important Cryptographic Properties., Subhamoy Maitra Dr.

Doctoral Theses

In this thesis we concentrate on properties of cryptographically significant Boolean functions.The techniques are mainly combinstorial and provide new resulta on enumeration and construction of such functions. Initially we concentrate on a partieular subset of Boolean functions called the symmetric Boolean functions. A closed form expression for the Walsh transform of an arbitrary symmetric Boolean function is presented. We completely characterize the symmetric functions with maximum nonlinearity and show that the maximum nonlinearity of n-variable symmetrie function can be 2n-1-2[n-1l2], Moreover, new classes of symmetric balanced and symmetric correlation immune functions are considered.We provide a randomised heuristic to construct balanced …

Contributions To Random Interactions And Dynamical Systems., Santanu Chakraborty Dr.

Doctoral Theses

In recent years random iterations of maps on Polish spaces has gained prominence. They are. nice examples of Markov processes whose invariant measures can be used in Computer imaging (see Berger ( 1). They also arise as random perturabations of deterministic dynamical systems.Let S be a Polish space with its Borel a-field. Let r be a collection of Borel maps from S to S. Let P be a probability on r. Then starting with a point z in S, we choose a map yn er according to the law P and move to ya(x). Then we choose 2 € r …

Triangular Surface Tiling Groups For Low Genus, Sean A. Broughton, Robert M. Dirks, Maria Sloughter, C. Ryan Vinroot

Mathematical Sciences Technical Reports (MSTR)

Consider a surface, S, with a kaleidoscopic tiling by non-obtuse triangles (tiles), i.e., each local reflection in a side of a triangle extends to an isometry of the surface, preserving the tiling. The tiling is geodesic if the side of each triangle extends to a closed geodesic on the surface consisting of edges of tiles. The reflection group G*, generated by these reflections, is called the tiling group of the surface. This paper classifies, up to isometry, all geodesic, kaleidoscopic tilings by triangles, of hyperbolic surfaces of genus up to 13. As a part of this classification the tiling groups …

Lengths Of Systoles On Tileable Hyperbolic Surfaces, Kevin Woods

Mathematical Sciences Technical Reports (MSTR)

The same triangle may tile geometrically distinct surfaces of the same genus, and these tilings may determine isomorphic tiling groups. We determine if there are geometric differences in the surfaces that can be found using group theoretic methods. Specifically, we determine if the systole, the shortest closed geodesic on a surface, can distinguish a certain families of tilings. For example, there are three tilings of surfaces of genus 14 by the hyperbolic triangle with angles π/2 , π/3 , and π/7 whose tiling groups are all PSL₂(13). These tilings can be distinguished by the lengths of their systoles.

Discrepancy Convergence For The Drunkard's Walk On The Sphere, Francis E. Su

All HMC Faculty Publications and Research

We analyze the drunkard's walk on the unit sphere with step size θ and show that the walk converges in order C/sin²(θ) steps in the discrepancy metric (C a constant). This is an application of techniques we develop for bounding the discrepancy of random walks on Gelfand pairs generated by bi-invariant measures. In such cases, Fourier analysis on the acting group admits tractable computations involving spherical functions. We advocate the use of discrepancy as a metric on probabilities for state spaces with isometric group actions.

Multi-Mode Cavity Effects In The Microwave Heating Of A Ceramic Slab, Stuart J. Walker

Dissertations

In order to gain insight into hot spot development in microwave heated ceramics, a partially insulated, two dimensional ceramic slab situated in a TE_M01 cavity is modeled in the small Biot number limit. If the electrical conductivity is an exponential function of temperture and E₀ is the strength of the incident mode, then the relationship between the spatially uniform, steady state leading order temperature, v₈, and E₀¹ is characterized by the well known bi-stable, or S shaped, response curve. The steady state second order temperature, v₁, is described by a boundary …

Analysis Of Discrete Dynamical System Models For Competing Species, Jerry J. Chen

Dissertations

A discrete version of the Lotka-Volterra (LV) differential equations for competing population species is analyzed in detail, much the same way as the discrete form of the logistic equation has been investigated as a source of bifurcation phenomena and chaotic dynamics. Another related system, namely, the Exponentially Self Regulating (ESR) population model, is also thoroughly analyzed. It is found that in addition to logistic dynamics - ranging from the very simple to manifestly chaotic regimes in terms of the governing parameters - the discrete LV model and the ESR model exhibit their own brands of bifurcation and chaos that are …

Topological Conjugacy And Rigidity Of Affine Actions., Siddhartha Bhattacharya Dr.

Doctoral Theses

The main theme of this thesis is topological classiication of aline Hows on homogeneous spaces and rigidity of equivariant continuous maps between such dows. Both these aspects have been extensively studied in the literature for subgroup actions (cf. [Be), [B-Dj and [Wi) and for automorphism flows of the cyclic group (cf. [Ar]. [K-R] and [C-S). We will consider similar questions in more general situations. A detailed outline is given below. For a topological group T, by a r-flow we mean pair (X,p), where X is a opological space and p is a continuous action of I on X. For any …

Routing Algorithms For Channels, Switchboxes And Mcm's In Vlsi Layout Design., Sandip Das Dr.

Doctoral Theses

The term Very Large Scale Integration (VLSI) reflects the capability of semi- conductor industry to fabricate a complex electronic circuit consisting of millions of components on a single silicon substrate. The growth of semiconductor technol- ogy in recent years has been described by "Moore's law", enunciated in the late 1960s, which projected quadrupling of components in a chip in every three to four years. Several factors contributed to this tremendous growth : (i) reduction of line width of the basic device and interconnection wires due to the development of high- resolution lithographic techniques and improved processing capabilities, (ii) increase in …

An Ellam Scheme For Multidimensional Advection-Reaction Equations And Its Optimal-Order Error Estimate, Hong Wang, Xiquan Shi, Richard E. Ewing

Faculty Publications

We present an Eulerian-Lagrangian localized adjoint method (ELLAM) scheme for initial-boundary value problems for advection-reaction partial differential equations in multiple space dimensions. The derived numerical scheme is not subject to the Courant-Friedrichs-Lewy condition and generates accurate numerical solutions even if large time steps are used. Moreover, the scheme naturally incorporates boundary conditions into its formulation without any artificial outflow boundary conditions needed, and it conserves mass. An optimal-order error estimate is proved for the scheme. Numerical experiments are performed to verify the theoretical estimate.

Group Closures Of One-To-One Transformations, Inessa Levi

Faculty Bibliography

For a semigroup 5 of transformations of an infinite set X let Gs be the group of all the permutations of X that preserve 5 under conjugation. Fix a permutation group H on X and a transformation / of X, and let (/ : H) = ({hfh~l : h 6 H}) be the //-closure of /. We find necessary and sufficient conditions on a one-to-one transformation / and a normal subgroup H of the symmetric group on X to satisfy G(f-.H) = H. We also show that if S is a semigroup of one-to-one transformations of X and Gs contains …

Rank-One Perturbations Of Diagonal Operators., Eugen J. Ionascu

Faculty Bibliography

We study rank-one perturbations of diagonal Hilbert space operators mainly from the standpoint of invariant subspace problem. In addition to proving some general properties of these operators, we identify the normal operators and contractions in this class. We show that two well known results about the eigenvalues of rank-one perturbations and one-codimension compressions of self-adjoint compact operators are equivalent. Sufficient conditions are given for existence of nontrivial invariant subspaces for this class of operators.

The Breakdown Of Synchronization In Systems Of Non-Identical Chaotic Oscillators: Theory And Experiment, Jennifer Chubb, Ernest Barreto, Paul So, Bruce J. Gluckman

Mathematics

The synchronization of chaotic systems has received a great deal of attention. However, most of the literature has focused on systems that possess invariant manifolds that persist as the coupling is varied. In this paper, we describe the process whereby synchronization is lost in systems of nonidentical coupled chaotic oscillators without special symmetries. We qualitatively and quantitatively analyze such systems in terms of the evolution of the unstable periodic orbit structure. Our results are illustrated with data from physical experiments.

Modular Miracles, John Stillwell

Mathematics

Over the last 20 years, the modular function has become widely known through its miraculous intervention in two great mathematical achievements: the proof of Fermat's last theorem and the "moonshine" of the monster simple group. In both cases, the modular function appears where no one expected it, and it bridges a chasm between seemingly unrelated fields. It is probably fair to say that, in these two cases, we do not yet fully understand how the modular magic works.

However, it can at least be said that these are not the first modular miracles. Ever since its discovery, in the early …

On The Nonembeddability And Crossing Numbers Of Some Toroidal Graphs On The Klein Bottle, Adrian Riskin

Mathematics

We show that toroidal polyhedral maps with four or more disjoint homotopic noncontractiblecircuits are not embeddable on the projective plane and that toroidal polyhedral maps with -veor more disjoint homotopic noncontractible circuits are not embeddable on the Klein bottle. Wealso show that the Klein bottle crossing numbers ofCm×Cn(m6n) form=3;4;5;6 are 1,2,4,and 6, respectively, and give upper bounds for all other values ofn. These crossing numbersdispla yat ypical behavior in that the value depends onl yonminstead of on bothmandnas isthe case for the plane and projective plane.

A Degree Of Nonlocal Connectedness, J. J. Charatonik, W. J. Charatonik

Mathematics and Statistics Faculty Research & Creative Works

To any continuum X weassign an ordinal number (or the symbol ∞) s(X), called the degree of nonlocal connectedness of X. We show that (1) the degree cannot be increased under continuous surjections; (2) for hereditarily unicoherent continua X, the degree of a subcontinuum of X is less than or equal to s(X); (3) s(C(X)) ≤ s(X), where C(X) denotes the hyperspace of subcontinua of a continuum X. We also investigate the degrees of Cartesian products and inverse limits. As an application weconstruct an uncountable family of metric continua X homeomorphic to C(X).

Open To Reason, David L. Neuhouser

Books by Taylor faculty, staff, and administrators

Open to Reason is a comparison of the roles of reason, experience, imagination, intuition, faith, love, humility, and obedience in mathematics and religion.

Mathematics As Worship, David J. Stucki

Mathematics Faculty Scholarship

This paper treats worship in a broad, inclusive sense as the primary and necessary response of human beings to their creator. It provides a brief overview of the relationship between mathematics and theology from the Pythagoreans to the present. It argues that all knowledge is contingent upon faith and thus that contemplation of mathematical insights can lead to worship of God.

Jkarelrobot: A Case Study In Supporting Levels Of Cognitive Development In The Computer Science Curriculum, Duane Buck, David J. Stucki

Mathematics Faculty Scholarship

We introduce a new software tool, JKarelRobot, for supporting an Inside/Out pedagogy in introductory programming courses. Extending the original conception of "Karel the Robot", with Bloom's Taxonomy of Educational Objectives as a guiding principle, we have provided a mechanism for designing exercises that are cognitively appropriate to the developmental levels of our students. JKarelRobot is platform independent (written in Java) and language/paradigm independent, supporting Pascal, Java, and Lisp style environments.

Boundary Layers In Channel Flow With Injection And Suction, R. Temam, X. Wang

Mathematics and Statistics Faculty Research & Creative Works

We present a rigorous result regarding the boundary layer associated with the incompressible Newtonian channel flow with injection and suction. © 2000 Elsevier Science Ltd. All rights reserved.

Transient Anomalous Diffusion In Poiseuille Flow, Marco Latini '01, Andrew J. Bernoff

All HMC Faculty Publications and Research

We revisit the classical problem of dispersion of a point discharge of tracer in laminar pipe Poiseuille flow. For a discharge at the centre of the pipe we show that in the limit of small non-dimensional diffusion, D, tracer dispersion can be divided into three regimes. For small times (t [double less-than sign] D^−1/3), diffusion dominates advection yielding a spherically symmetric Gaussian dispersion cloud. At large times (t [dbl greater-than sign] D⁻¹), the flow is in the classical Taylor regime, for which the tracer is homogenized transversely across the pipe and diffuses with …

No-Nonsense Guide To Csab/Csac Accreditation, Pete Sanderson

Mathematics Faculty Scholarship

CSAB/CSAC provides professional accreditation of computer science bachelor's degree programs in the United States. As of October 2000, 159 institutions held this accreditation. By our count, over 80% of the accredited programs were offered by departments which also offer graduate programs in computer science. This means that few small colleges are represented. Our intent in this work is to give the small college audience an up-to-date guide to the recently-revised CSAB/CSAC accreditation standards. The guide is not comprehensive; we emphasize those issues we believe to be of greatest interest to small colleges and address them from the perspective we have …

Discrete-Time Approximations Of Stochastic Differential Systems With Memory, Yaozhong Hu, Salah-Eldin A. Mohammed, Feng Yan

Articles and Preprints

In this paper, we develop two discrete-time strong approximation schemes for solving stochastic differential systems with memory: strong Euler-Maruyama schemes for stochastic delay differential equations (SDDE's) and stochastic functional differential equations (SFDE's) with continuous memory, and a strong Milstein scheme for SDDE's. The convergence orders of the Euler-Maruyama and Milstein schemes are 0.5 and 1 respectively. In order to establish the Milstein scheme, we prove an infinite-dimensional Itô formula for "tame" functions acting on the segment process of the solution of an SDDE. It is interesting to note that the presence of the memory in the SDDE requires the use …

A Sequential Search Distribution: Proofreading, Russian Roulette, And The Incomplete Q-Eulerian Polynomials, Travis Herbranson, Don Rawlings

Mathematics

The distribution for the number of searches needed to find k of n lost objects is expressed in terms of a refinement of the q-Eulerian polynomials, for which formulae are developed involving homogeneous symmetric polynomials. In the case when k = n and the find probability remains constant, relatively simple and efficient formulas are obtained. From our main theorem, we further (1) deduce the inverse absorption distribution and (2) determine the expected number of times the survivor pulls the trigger in an n-player game of Russian roulette.

Back To Classics: Teaching Limits Through Infinitesimals, Todor D. Todorov

Mathematics

The usual ϵ, δ-definition of the limit of a function (whether presented at a rigorous or an intuitive level) requires a “candidate L” for the limit value. Thus, we have to start our first calculus course with “guessing” instead of “calculating”. In this paper we criticize the method of using calculators for the purpose of selecting candidates for L. We suggest an alternative: a working formula for calculating the limit value L of a real function in terms of infinitesimals. Our formula, if considered as a definition of limit, is equivalent to the usual ϵ, δ-definition but does not involve …

A Coin Flipping Game With Non Transitive Odds, Stacy Jurgens

Honors Capstones

Capstone submitted as a graduation requirement for the BSU Honors Program.

Optimal Token Allocations In Solitaire Knock 'M Down, Arthur Benjamin, Matthew T. Fluet, Mark L. Huber

All HMC Faculty Publications and Research

In the game Knock ’m Down, tokens are placed in N bins. At each step of the game, a bin is chosen at random according to a fixed probability distribution. If a token remains in that bin, it is removed. When all the tokens have been removed, the player is done. In the solitaire version of this game, the goal is to minimize the expected number of moves needed to remove all the tokens. Here we present necessary conditions on the number of tokens needed for each bin in an optimal solution, leading to an asymptotic solution. MR Subject Classifications: …

The Renner Monoids And Cell Decompositions Of The Classical Algebraic Monoids, Zhenheng Li

Digitized Theses

The Renner monoids, cross section lattices and cell decompositions of the classical algebraic monoids are studied.

The Renner monoid is extremely important in the theory of reductive algebraic monoids. It is well know that the Renner monoid [Special characters omitted.] of M_n (K ) is the monoid of all zero-one matrices which have at most one entry equal to one in each row and column, i.e., [Special characters omitted.] consists of injective partial maps on a set of n elements. We obtain that the Renner monoids of the symplectic algebraic monoids and special orthogonal algebraic monoids turn out …