Mathematics Commons^™

Dendrites And Light Open Mappings, J. J. Charatonik, W. J. Charatonik, Pawel Krupski

Mathematics and Statistics Faculty Research & Creative Works

It is shown that a metric continuum X is a dendrite if and only if for every compact space Y and for every light open mapping f : Y ->f(Y ) such that X c f(Y ) there is a copy X1 of X in Y for which the restriction fjX1 : X1 ->X is a homeomorphism. Another characterization of dendrites in terms of continuous selections of multivalued functions is also obtained.

Computational Geometry Column 37, Erik D. Demaine, Joseph O'Rourke

Computer Science: Faculty Publications

Open problems from the 15th Annual ACM Symposium on Computational Geometry.

Numerical Aspects Of Discrete And Continuum Hybrid Models In Cell Biology, J. C. Dallon

Faculty Publications

In this paper we introduce a method of modeling which mixes continuum and discrete variables, and explain two models in cell biology that use this method. The first application deals with wound healing, more specifically the collagen alignment in scar tissue formation and the second models early aggregation in the cellular slime mold Dictyostelium discoideum. We solve these models using numerical techniques similar to the particle-in-cell method which requires that the discrete and continuum variables are interpolated one to the other. The implementational and numerical details are discussed in an informal and practical manner with particular attention given to the …

A Mathematical Model Of Wheelchair Racing, Susan J. Schenk

Dissertations

Wheelchair racing strokes are very complicated movements, which involve a coupling between the athlete and his or her racing chair. Each body segment, as well as the wheel, follows a distinct trajectory as the motion is performed. Understanding the kinematics and kinetics of various stroke techniques would provide the athletes and their coaches with information, which could help guide the racers toward improved performances.

In this thesis, a mathematical model is developed, which is capable of providing such valuable kinematic and kinetic information. This two-dimensional model represents the body segments as a coupled pendulum system of point masses and the …

Perturbed Laplacian Matrix And The Structure Of A Graph., Sukanta Pati Dr.

Doctoral Theses

Laplacian matrices Let G be a connected simple graph with vertex set V = {1,2,.,n), edge set E and let each edge be associated with a positive number, the weight of the edge. The above graph is called a weighted graph. An unweighted graph is just a weighted graph with each of the edges bearing weight 1. All the graphs considered are weighted and simple, unless specified otherwise; all the matrices considered are real. The adjacency matrix A(G) related to this graph is defined as A(G) = (aij), whereaij, if (i, j] € E and the weight of the edge …

Pushpush Is Np-Hard In 2d, Erik D. Demaine, Martin L. Demaine, Joseph O'Rourke

Computer Science: Faculty Publications

We prove that a particular pushing-blocks puzzle is intractable in 2D, improving an earlier result that established intractability in 3D [OS99]. The puzzle, inspired by the game *PushPush*, consists of unit square blocks on an integer lattice. An agent may push blocks (but never pull them) in attempting to move between given start and goal positions. In the PushPush version, the agent can only push one block at a time, and moreover, each block, when pushed, slides the maximal extent of its free range. We prove this version is NP-hard in 2D by reduction from SAT.

Dynamics Of Stochastic Systems With Memory (Mathematics And Statistics Colloquium, Wright State University), Salah-Eldin A. Mohammed

Miscellaneous (presentations, translations, interviews, etc)

We describe an approach to the dynamics of stochastic systems with finite memory using multiplicative cocycles in Hilbert space. We introduce the notion of hyperbolicity for stationary solutions of the stochastic differential system. We then establish the existence of smooth stable and unstable manifolds in a neighborhood of a hyperbolic stationary solution. The stable and unstable manifolds are stationary and asymptotically invariant under the stochastic semiflow. The proof uses ideas from infinite-dimensional multiplicative ergodic theory and interpolation arguments.

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.

Inverse Limits On [0,1] Using Piecewise Linear Unimodal Bonding Maps, William Thomas Ingram

Mathematics and Statistics Faculty Research & Creative Works

In this paper we investigate inverse limits on [0,1] using a single bonding map chosen from a two-parameter family of piecewise linear unimodal bonding maps. This investigation focuses on the parameter values at the boundary between a hereditarily decomposable inverse limit and an inverse limit containing an indecomposable continuum. © 1999 American Mathematical Society.

Design Early Considered Harmful: Graduated Exposure To Complexity And Structure Based On Levels Of Cognitive Development, Duane Buck, David J. Stucki

Mathematics Faculty Scholarship

We have recognized that the natural tendency to teach according to the structure of one’s own understanding runs contrary to established models of cognitive development. Bloom’s Taxonomy has provided a basis for establishing a more efficacious pedagogy. Emphasizing a hierarchical progression of skill sets and gradual learning through example, our approach advocates teaching software development from the inside/out rather than beginning with either console apps or monolithic designs.

Cwatset Isomorphism And Its Consequences, Carolyn M. Girod, Matthew Lipinski, Joseph R. Mileti, Jennifer R. Paulhus

Mathematical Sciences Technical Reports (MSTR)

We explore the consequences of cwatset isomorphism (there are a finite number of non-isomorphic cwatsets of each order) and consider parallels between the theory of groups and the theory of cwatsets (cwatsets of prime order are cyclic but direct sums of isomorphic cwatsets aren't necessarily isomorphic).

Remarks On The Prandtl Equation For A Permeable Wall, R. Temam, X. Wang

Mathematics and Statistics Faculty Research & Creative Works

The goal of this article is to study the boundary layer for a flow in a channel with permeable walls. Observing that the Prandtl equation can be solved almost exactly in this case, we are able to derive rigorously a number of results concerning the boundary layer and the convergence of the Navier-Stokes equations to the Euler equations. We indicate also how to derive higher order terms in the inner and outer expansions with respect to the kinematic viscosity v.

Nonlinear Eigenvalue Problems For Higher Order Lidstone Boundary Value Problems, Paul W. Eloe

Mathematics Faculty Publications

In this paper, we consider the Lidstone boundary value problem y(t) = λa(t)f(y(t), . . . , y(t), . . . y(t)), 0 < t < 1, y(0) = 0 = y(1), i = 0, . . . , m − 1, where (−1)f > 0 and a is nonnegative. Growth conditions are imposed on f and inequalities involving an associated Green’s function are employed which enable us to apply a well-known cone theoretic fixed point theorem. This in turn yields a λ interval on which there exists a nontrivial solution in a cone for each λ in that interval. The methods of the paper are known. The emphasis here is that f depends upon higher order derivatives. Applications are made to …

A Look At Biseparating Maps From An Algebraic Point Of View, Melvin Henriksen, Frank A. Smith

All HMC Faculty Publications and Research

In [ABN], Araujo, Beckenstein, and Narici add the capstone to a series of papers by several groups of authors by showing that if ρ is a biseparating map between two algebras of all real or complex-valued functions on realcompact spaces, then it is a continuous multiple of an isomorphism between these rings. Their proof uses relatively powerful analytic and topological techniques. In what follows, the extent to which such a result can be generalized to a wider class of algebras using algebraic techniques is investigated. We are unable, however to obtain the main result of [ABN] using these techniques.

When Is |C(X X Y)| = |C(X)||C(Y)|?, O. T. Alas, W. W. Comfort, S. Garcia-Ferreira, Melvin Henriksen, R. G. Wilson, R. G. Woods

All HMC Faculty Publications and Research

Sufficient conditions on the Tychonoff spaces X and Y are found that imply that the equation in the title holds. Sufficient conditions on the Tychonoff space X are found that ensure that the equation holds for every Tychonoff space Y . A series of examples (some using rather sophisticated cardinal arithmetic) are given that witness that these results cannot be generalized much.

A Primer For Applying Service Learning To Computer Science, Pete Sanderson, Kenneth Vollmar

Mathematics Faculty Scholarship

Service learning is an educational philosophy that promotes active learning through community service. We have recently applied this approach in our computer science curriculum, specifically to our software engineering course. In order that other computer science departments can benefit from our experience, we have developed a primer one can follow to establish a program for service learning in the computer sciences. We also describe and assess our experience after one year of applying service learning to software engineering.

Affine Varieties, Groebner Basis, And Applications, Eui Won James Byun

Theses Digitization Project

No abstract provided.

Solutions To The Chinese Postman Problem, Kenneth Peter Cramm

Theses Digitization Project

Considering the Chinese Postman Problem, in which a mailman must deliver mail to houses in a neighborhood. The mailman must cover each side of the street that has houses, at least once. The focus of this paper is our attempt to discover the optimal path, or the least number of times each street is walked. The integration of algorithms from graph theory and operations research form the method used to explain solutions to the Chinese Postman Problem.

Investigation Of The Effectiveness Of Interface Constraints In The Solution Of Hyperbolic Second-Order Differential Equations, Paul Jerome Silva

Theses Digitization Project

Solutions to differential equations describing the behavior of physical quantities (e.g., displacement, temperature, electric field strength) often only have finite range of validity over a subdomain. Interest beyond the subdomain often arises. As a result, the problem of making the solution compatible across the connecting subdomain interfaces must be dealt with. Four different compatibility methods are examined here for hyperbolic (time varying) second-order differential equations. These methods are used to match two different solutions, one in each subdomain along the connecting interface. The entire domain that is examined here is a unit square in the Cartesian plane. The four compatibility …

On The Baroclinic Instability Of Axisymmetric Rotating Gravity Currents With Bottom Slope, Paul F. Choboter, Gordone E. Swaters

Mathematics

The baroclinic stability characteristics of axisymmetric gravity currents in a rotating system with a sloping bottom are determined. Laboratory studies have shown that a relatively dense fluid released under an ambient fluid in a rotating system will quickly respond to Coriolis effects and settle to a state of geostrophic balance. Here we employ a subinertial two-layer model derived from the shallow-water equations to study the stability characteristics of such a current after the stage at which geostrophy is attained. In the model, the dynamics of the lower layer are geostrophic to leading order, but not quasi-geostrophic, since the height deflections …

The Edge-Isoperimetric Problem For The Square Tessellation Of Plane, Sunmi Lee

Theses Digitization Project

The solution for the edge-isoperimetric problem (EIP) of the square tessellation of plane is investigated and solved. Summaries of the stabilization theory and previous research dealing with the EIP are stated. These techniques enable us to solve the EIP of the cubical tessellation.

Visually Building Smale Flows In S3, Michael C. Sullivan

Articles and Preprints

A Smale flow is a structurally stable flow with one dimensional invariant sets. We use information from homology and template theory to construct, visualize and in some cases, classify, Smale flows in the 3-sphere.

Nonstandard Analysis In Topology: Nonstandard And Standard Compactifications, S. Salbany, Todor D. Todorov

Mathematics

Let (X, T) be a topological space, and ^∗X a non–standard extension of X. There is a natural “standard” topology ^ST on ^∗X generated by ^∗G,where G ∈ T . The topological space (^∗X,^ST) will be used to study, in a systematic way, compactifications of (X, T).

Constructible Circles On The Unit Sphere, Blaga Slavcheva Pauley

Theses Digitization Project

In this paper we show how to give an intrinsic definition of a constructible circle on the sphere. The classical definition of constructible circle in the plane, using straight edge and compass is there by translated in ters of so called Lenart tools. The process by which we achieve our goal involves concepts from the algebra of Hermitian matrices, complex variables, and Sterographic projection. However, the discussion is entirely elementary throughout and hopefully can serve as a guide for teachers in advanced geometry.

Steiner Systems Of The Mathieu Group M₁₂, Kristin Marie Dillard

Theses Digitization Project

A Steiner system T with parameters (5,6,12) is a collection of 6-element sets, called hexads, of a 12-element set [omega], such that any 5 of the 12 elements belong to exactly one hexad. In this project we construct a graph whose vertices are the orbits of S₁₂ on T x T, where T is the set of all Steiner systems S(5,6,12). Two vertices are joined if an orbit is taken into another under the action of a transposition. The number of hexads common to two Steiner systems are also given. We also prove that any two Steiner systems with parameters …

The Proof Of Fermat's Last Theorem, Mohamad Trad

Theses Digitization Project

Fermat, Pierre de, is perhaps the most famous number theorist who ever lived. Fermat's Last Theorem states that the equation xn + yn = zn has no non-zero integer solutions for x, y and z when n>2.

Super-Brownian Limits Of Voter Model Clusters, Maury Bramzon, J. Theodore Cox, Jean-Francois Le Gall

Mathematics - All Scholarship

The voter model is one of the standard interacting particle systems. Two related problems for this process are to analyze its behavior, after large times t, for the sets of sites (a) sharing the same opinion as the site 0, and (b) having the opinion that was originally at 0. Results on the sizes of these sets were given in [Sa79] and [BG80]. Here, we investigate the spatial structure of these sets in d ≥ 2, which we show converge to quantities associated with super-Brownian motion, after suitable normalization. The main theorem from [CDP98] serves as an important tool for …

On Simplicial Commutative Algebras With Vanishing André-Quillen Homology, James M. Turner

University Faculty Publications and Creative Works

In this paper, we study the André-Quillen homology of simplicial commutative ℓ-algebras, ℓ a field, having certain vanishing properties. When ℓ has non-zero characteristic, we obtain an algebraic version of a theorem of J.-P. Serre and Y. Umeda that characterizes such simplicial algebras having bounded homotopy groups. We further discuss how this theorem fails in the rational case and, as an application, indicate how the algebraic Serre theorem can be used to resolve a conjecture of D. Quillen for algebras of finite type over Noetherian rings, having non-zero characteristic.

Review: Cake-Cutting Algorithms: Be Fair If You Can, Francis E. Su

All HMC Faculty Publications and Research

No abstract provided in this article.

A Leveque-Type Lower Bound For Discrepancy, Francis E. Su

All HMC Faculty Publications and Research

A sharp lower bound for discrepancy on R / Z is derived that resembles the upper bound due to LeVeque. An analogous bound is proved for discrepancy on R^k / Z^k. These are discussed in the more general context of the discrepancy of probablity measures. As applications, the bounds are applied to Kronecker sequences and to a random walk on the torus.