Mathematics Commons^™

Almost Periodic Sets And Subactions In Topological Dynamics, Ethan Akin, Joseph Auslander

Mathematics and Statistics Faculty Research & Creative Works

Let Γ be a group with subgroup λ. We show, under certain conditions, that an almost periodic point for the action of λ is also almost periodic for Γ. This is applied to obtain a theorem of Glasner.

Randomization Tests For Small Samples: An Application For Genetic Expression Data, Gary L. Gadbury, Grier P. Page, Moonseong Heo, John D. Mountz, David B. Allison

Mathematics and Statistics Faculty Research & Creative Works

An advantage of randomization tests for small samples is that an exact P-value can be computed under an additive model. a disadvantage with very small sample sizes is that the resulting discrete distribution for P-values can make it mathematically impossible for a P-value to attain a particular degree of significance. We investigate a distribution of P-values that arises when several thousand randomization tests are conducted simultaneously using small samples, a situation that arises with microarray gene expression data. We show that the distribution yields valuable information regarding groups of genes that are differentially expressed between two groups: A treatment group …

Modern Statistical Methods For Handling Missing Repeated Measurements In Obesity Trial Data: Beyond Locf, Gary L. Gadbury, C. S. Coffey, D. B. Allison

Mathematics and Statistics Faculty Research & Creative Works

This paper brings together some modern statistical methods to address the problem of missing data in obesity trials with repeated measurements. Such missing data occur when subjects miss one or more follow-up visits or drop out early from an obesity trial. a common approach to dealing with missing data because of dropout is 'last observation carried forward' (LOCF). This method, although intuitively appealing, requires restrictive assumptions to produce valid statistical conclusions. We review the need for obesity trials, the assumptions that must be made regarding missing data in such trials, and some modern statistical methods for analyzing data containing missing …

Cause-Effect Relationships In Analytical Surveys: An Illustration Of Statistical Issues, Gary L. Gadbury, Hans T. Schreuder

Mathematics and Statistics Faculty Research & Creative Works

Establishing cause-effect is critical in the field of natural resources where one may want to know the impact of management practices, wildfires, drought, etc. on water quality and quantity, wildlife, growth and survival of desirable trees for timber production, etc. Yet, key obstacles exist when trying to establish cause-effect in such contexts. Issues involved with identifying a causal hypothesis, and conditions needed to estimate a causal effect or to establish cause-effect are considered. Ideally one conducts an experiment and follows with a survey, or vice versa. in an experiment, the population of inference may be quite limited and in surveys, …

An Oscillation Theorem For Discrete Eigenvalue Problems, Martin Bohner, Ondřej Došlý, Werner Kratz

Mathematics and Statistics Faculty Research & Creative Works

In this paper we consider problems that consist of symplectic difference systems depending on an eigenvalue parameter, together with self-adjoint boundary conditions. Such symplectic difference systems contain as important cases linear Hamiltonian difference systems and also Sturm-Liouville difference equations of second and of higher order. The main result of this paper is an oscillation theorem that relates the number of eigenvalues to the number of generalized zeros of solutions.

Prediction Intervals For The Binomial Distribution With Dependent Trials, Florian Sebastian Rueck

Masters Theses

"A generalization of a prediction interval procedure for the binomial distribution to the case of the binomial distribution with dependent trials is considered. Several different methods have been developed for obtaining prediction intervals for the binomial distribution. An unpublished study by Vlieger and Samaranayake has shown that two of these methods achieve coverage probabilities close to nominal levels. The proposed method is an extension of one of these methods and is based on the maximum likelihood predictive density proposed by Lejeune and Faulkenberry. A simulation study was carried out to investigate the coverage probabilities of the proposed prediction bounds.

This …

Invariant Sets And Inverse Limits, William Thomas Ingram

Mathematics and Statistics Faculty Research & Creative Works

In this paper we investigate the nature of inverse limits from the point of view of invariant sets. We then introduce a special class of examples of inverse limits on [0,1] using Markov bonding maps determined by members of the group of permutations on n elements. © 2002 Elsevier Science B.V. All rights reserved.

Boundary Layers Associated With Incompressible Navier-Stokes Equations: The Noncharacteristic Boundary Case, R. Temam, X. Wang

Mathematics and Statistics Faculty Research & Creative Works

The goal of this article is to study the boundary layer of wall bounded flows in a channel at small viscosity when the boundaries are uniformly non-characteristic, i.e., there is injection and/or suction everywhere at the boundary. Following earlier work on the boundary layer for linearized Navier-Stokes equations in the case where the boundaries are characteristic (non-slip at the boundary and non-permeable), we consider here the case where the boundary is permeable and thus non-characteristic. the form of the boundary layer and convergence results are derived in two cases: linearized equation and full nonlinear equations. We prove that there exists …

An Adaptive Analysis Of Covariance Using Tree-Structured Regression, Gary L. Gadbury, H. K. Iyer, H. T. Schreuder

Mathematics and Statistics Faculty Research & Creative Works

In this article, we propose an adaptive procedure for testing for the effect of a factor of interest in the presence of one or more confounding variables in observational studies. It is especially relevant for applications where the factor of interest has a suspected causal relationship with a response. This procedure is not tied to linear modeling or normal distribution theory, and it offers a valuable alternative to traditional methods. It is suitable for applications where a factor of interest is categorical, and the response is continuous. Confounding variables may be continuous or categorical. The method is comprised of two …

On Size Mappings, W. J. Charatonik, Alicja Samulewicz

Mathematics and Statistics Faculty Research & Creative Works

A real-valued mapping r from the hyperspace of all compact subsets of a givenmetric space X is called a size mapping if r({x}) = 0 for x ∈ X and r(A) ≤ r(B) if a ⊂ B. We investigate what continua admit an open or a monotone size mapping. Special attention is paid to the diameter mappings.

On The Existence Of Nontrivial Solutions To Some Elliptic Variational Inequalities, Vy Khoi Le, Klaus Schmitt

Mathematics and Statistics Faculty Research & Creative Works

The paper is concerned with the existence of nontrivial solutions of the obstacle problem: u ε K: ∫Ω ▽u▽ (v - u) dx - λ ∫ Ω u (v - u) dx ≥ ∫ Ω p (x, u) (v - u) dx ∀x ε K, where K = {v ε Ho1(Ω): v ≤ Ψ a.e. on Ω}. By using a generalized mountain pass theorem for inequalities, we prove, subject to some restrictions on the obstacle Ψ, the existence of nontrivial solutions of the above inequality.

Canonical Thurston Obstructions, Kevin M. Pilgrim

Mathematics and Statistics Faculty Research & Creative Works

We refine Douady and Hubbard's proof of Thurston's topological characterization of rational functions by proving the following theorem. Let f: S2→S2 be a branched covering with finite postcritical set Pf and hyperbolic orbifold. Let Γc denote the set of all homotopy classes γ of nonperipheral, simple closed curves in S2-Pf such that the length of the unique geodesic homotopic to γ tends to zero under iteration of the Thurston map induced by f on Teichmüller space. Then either Γc is empty, and f is equivalent to a rational function, or else Γc is a Thurston obstruction. © 2001 Academic Press.

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).

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.

Residual Properties And Almost Equicontinuity, Ethan Akin, Eli Glasner

Mathematics and Statistics Faculty Research & Creative Works

A property P of a compact dynamical system (X, f) is called a residual property if it is inherited by factors, almost one-to-one lifts and surjective inverse limits. Many transitivity properties are residual. Weak disjointness from all property P systems is a residual property provided P is a residual property stronger than transitivity. Here two systems are weakly disjoint when their product is transitive. Our main result says that for an almost equicontinuous system (X, f) with associated monothetic group Λ, (X, f) is weakly disjoint from all P systems if the only P systems upon which Λ acts are …

Power Connector, Stephen L. Clark, Joseph B. Shuey, Jose L. Ortega, John B. Brown Iii

Mathematics and Statistics Faculty Research & Creative Works

A pair of mating connectors includes a receptacle having an insulative housing and at least one conductive receptacle contact with a pair of spaced walls forming a plug contact receiving space. The plug connector has an insulative housing and at least one conductive contact having a pair of spaced walls which converge to form a projection engageable in the plug receiving space of the receptacle contact. In each case, the spaced walls are joined by a bridging structure that unites the walls. The plug and receptacle contacts are retained in the respective housings by engagement of opposed lateral edge portions …

Openness Of Induced Projections, J. J. Charatonik, W. J. Charatonik, Alejandro Illanes

Mathematics and Statistics Faculty Research & Creative Works

For continua X and Y it is shown that if the projection f : X x Y ->X has its induced mapping C(f) open, then X is C*-smooth. As a corollary, a characterization of dendrites in these terms is obtained.

A Census Of Rational Maps, Eva Brezin, Rosemary Byrne, Joshua Levy, Kevin M. Pilgrim, Kelly Plummer

Mathematics and Statistics Faculty Research & Creative Works

We discuss the general combinatorial, topological, algebraic, and dynamical issues underlying the enumeration of postcritically finite rational functions, regarded as holomorphic dynamical systems on the Riemann sphere. We present findings from our creation of a census of all degree two and three hyperbolic nonpolynomial maps with four or fewer postcritical points. Our data is tabulated in detail at. © 1999 American Mathematical Society.

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.

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.

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.

Openness And Monotoneity Of Induced Mappings, W. J. Charatonik

Mathematics and Statistics Faculty Research & Creative Works

It is shown that for locally connected continuum X if the induced mapping C(f) : C(X) ->C(Y) is open, then f is monotone. As a corollary it follows that if the continuum X is hereditarily locally connected and C(f) is open, then f is a homeomorphism. An example is given to show that local connectedness is essential in the result.

Invariant Measures For Set-Valued Dynamical Systems, Walter Miller, Ethan Akin

Mathematics and Statistics Faculty Research & Creative Works

A continuous map on a compact metric space, regarded as a dynamical system by iteration, admits invariant measures. For a closed relation on such a space, or, equivalently, an upper semicontinuous set-valued map, there are several concepts which extend this idea of invariance for a measure. We prove that four such are equivalent. In particular, such relation invariant measures arise as projections from shift invariant measures on the space of sample paths. There is a similarly close relationship between the ideas of chain recurrence for the set-valued system and for the shift on the sample path space. ©1999 American Mathematical …

Attractors For Non-Compact Semigroups Via Energy Equations, Ioana Moise, Ricardo Rosa, Xiaoming Wang

Mathematics and Statistics Faculty Research & Creative Works

The energy-equation approach used to prove the existence of the global attractor by establishing the so-called asymptotic compactness property of the semigroup is considered, and a general formulation that can handle a number of weakly damped hyperbolic equations and parabolic equations on either bounded or unbounded spatial domains is presented. as examples, three specific and physically relevant problems are considered, namely the flows of a second-grade fluid, the flows of a Newtonian fluid in an infinite channel past an obstacle, and a weakly damped, forced Korteweg-de Vries equation on the whole line.

Attractor Dimension Estimates For Two-Dimensional Shear Flows, Charles R. Doering, Xiaoming Wang

Mathematics and Statistics Faculty Research & Creative Works

We study the large time behavior of boundary and pressure-gradient driven incompressible fluid flows in elongated two-dimensional channels with emphasis on estimates for their degrees of freedom, i.e., the dimension of the attractor for the solutions of the Navier-Stokes equations. for boundary driven shear flows and flux driven channel flows we present upper bounds for the degrees of freedom of the form ca Re^3/2 where c is a universal constant, a denotes the aspect ratio of the channel (length/width), and Re is the Reynolds number based on the channel width and the imposed "outer" velocity scale. for fixed pressure …

Atomoicity Of Mappings, J. J. Charatonik, W. J. Charatonik

Mathematics and Statistics Faculty Research & Creative Works

A mapping f:X→Y between continua X and Y is said to be atomic at a subcontinuumK of the domain X provided that f(K) is nondegenerate and K=f^-1(f(K)). The set of subcontinua at which a given mapping is atomic, considered as a subspace of the hyperspace of all subcontinua of X, is studied. The introduced concept is applied to get new characterizations of atomic and monotone mappings. Some related questions are asked.

Arc Approximation Property And Confluence Of Induced Mappings, W. J. Charatonik

Mathematics and Statistics Faculty Research & Creative Works

We say that a continuum X has the arc approximation property if every subcontinuum K of X is the limit of a sequence of arcwise connected subcontinua of X all containing a fixed point of K. This property is applied to exhibit a class of continua Y such that confluence of a mapping f : X - Y implies confluence of the induced mappings 2^f : 2^x - @^y and C(f) : C(x) - C(y). The converse implications are studied and similar interrelations are considered for some other classes of mappings, related to confluent ones.

Exponential Dichotomy And Mild Solutions Of Nonautonomous Equations In Banach Spaces, Y. Latushkin, Timothy W. Randolph, R. Schnaubelt

Mathematics and Statistics Faculty Research & Creative Works

We prove that the exponential dichotomy of a strongly continuous evolution family on a Banach space is equivalent to the existence and uniqueness of continuous bounded mild solutions of the corresponding inhomogeneous equation. This result addresses nonautonomous abstract Cauchy problems with unbounded coefficients. The technique used involves evolution semigroups. Some applications are given to evolution families on scales of Banach spaces arising in center manifolds theory. © 1998 Plenum Publishing Corporation.

Some Harmonic N-Slit Mappings, Michael Dorff

Mathematics and Statistics Faculty Research & Creative Works

The class SH consists of univalent, harmonic, and sense-preserving functions f in the unit disk, Δ, such that f = h+ḡ where h(z) = z + ∑2∞ akzk g(z) = ∑1∞ bkzk. SHO will denote the subclass with b1 = 0. We present a collection of n-slit mappings (n ≥ 2) and prove that the 2-slit mappings are in SH while for n ≥ 3 the mappings are in SHO. Finally, we show that these mappings establish the sharpness of a previous theorem by Clunie and Sheil-Small while disproving a conjecture about the inner mapping radius. ©1998 American Mathematical Society.

Further Properties Of An Extremal Set Of Uniqueness, David E. Grow, Matt Insall

Mathematics and Statistics Faculty Research & Creative Works

Consider the circle group T = R mod 2_ as the interval [0, 1). Then each x 2 T has a binary expansion: x =P1 k=1 xk2−k where each xk is 0 or 1. Let S be the set of x with a binary expansionsuch that the number of 1's does not exceed the number of the leading zeros by more than one. The authors prove that the countable compact set S cannot be expressed as the union of a finite number of Dirichlet sets.