Mathematics Commons^™

Algebraic Characterizations Of Graph Imbeddability In Surfaces And Pseudosurfaces, Lowell Abrams, Dan Slilaty

Mathematics and Statistics Faculty Publications

Given a finite connected graph G and specifications for a closed, connected pseudosurface, we characterize when G can be imbedded in a closed, connected pseudosurface with the given specifications. The specifications for the pseudosurface are: the number of face-connected components, the number of pinches, the number of crosscaps and handles, and the dimension of the first Z2-homology group. The characterizations are formulated in terms of the existence of a dual graph G ∗ on the same set of edges as G which satisfies algebraic conditions inspired by homology groups and their intersection products.

Restricted Signed Permutations Counted By The Schröder Numbers, Eric S. Egge

Mathematics and Statistics Faculty Work

Gire, West, and Kremer have found ten classes of restricted permutations counted by the large Schröder numbers, no two of which are trivially Wilf-equivalent. In this paper we enumerate eleven classes of restricted signed permutations counted by the large Schröder numbers, no two of which are trivially Wilf-equivalent. We obtain five of these enumerations by elementary methods, five by displaying isomorphisms with the classical Schröder generating tree, and one by giving an isomorphism with a new Schröder generating tree. When combined with a result of Egge and a computer search, this completes the classification of restricted signed permutations counted by …

Differential Inclusions On Proximate Retracts Of Separable Hilbert Spaces, Ravi P. Agarwal, Donal O'Regan

Mathematics and System Engineering Faculty Publications

New existence results are presented which guarantee the existence of viable solutions to differential inclusions in separable Hilbert spaces. Our results rely on the existence of maximal solutions for an appropriate differential equation in the real case. Copyright ©2006 Rocky Mountain Mathematics Consortium.

A Positive Solution For Singular Discrete Boundary Value Problems With Sign-Changing Nonlinearities, Haishen Lü, Donal O'Regan, Ravi P. Agarwal

Mathematics and System Engineering Faculty Publications

This paper presents new existence results for the singular discrete boundary value problem - Δ²u(k - 1) = g(k, u(k)) + λh(k,u(k)), k ∈ [1,T], u(0) = 0 = u(T + 1). In particular, our nonlinearity may be singular in its dependent variable and is allowed to change sign.

On Adaptive Testing In Orthogonal Saturated Designs, Daniel T. Voss, Weizhen Wang

Mathematics and Statistics Faculty Publications

Adaptive, size-a step-down tests are provided for the analysis of orthogonal saturated designs. The tests work effectively under effect sparsity, and include as special cases the individual nonadaptive tests of Berk and Picard (1991) and the simultaneous nonadaptive tests of Voss (1988). The approach is similar to that used by Wang and Voss (2003) to construct adaptive confidence intervals, but testing is simpler because one can use the same denominator for all statistics. Step-down tests also have a clear power advantage over simultaneous confidence intervals and analogous single-step tests, as is demonstrated theoretically and assessed via simulation.

Temporal Processing In The Exponential Integrate-And-Fire Model Is Nonlinear, Joanna R. Wares, Todd W. Troyer

Department of Math & Statistics Faculty Publications

The exponential integrate-and-fire (EIF) model was introduced by Fourcaud-Trocme et al. (2003) as an extension of the standard leaky integrate-and-fire model (LIF). Here, the nonlinearity in the EIF model’s temporal response to square-wave inputs is investigated. Comparing the time course of onset and offset responses revealed that offset responses have a steeper initial slope, but a slower approach to equilibrium. A linear systems analysis performed for these square-wave inputs indicates that at frequencies above ~40 Hz, gain was slightly smaller for square-wave inputs, but phase did not change significantly relative to simulations in which the corresponding sinusoids were presented in …

Adaptive Finite Element Methods For Elliptic Pdes Based On Conforming Centroidal Voronoi-Delaunay Triangulations, Lili Ju, Max Gunzburger, Weidong Zhao

Faculty Publications

A new triangular mesh adaptivity algorithm for elliptic PDEs that combines a posteriori error estimation with centroidal Voronoi–Delaunay tessellations of domains in two dimensions is proposed and tested. The ability of the first ingredient to detect local regions of large error and the ability of the second ingredient to generate superior unstructured grids result in a mesh adaptivity algorithm that has several very desirable features, including the following. Errors are very well equidistributed over the triangles; at all levels of refinement, the triangles remain very well shaped, even if the grid size at any particular refinement level, when viewed globally, …

Writing Projective Representations Over Subfields, Stephen P. Glasby, C. R. Leedham-Green, E. A. O'Brien

All Faculty Scholarship for the College of the Sciences

Let G=〈X〉be an absolutely irreducible subgroup of GL(d, K), and let F be a proper subfield of the finite field K. We present a practical algorithm to decide constructively whether or not G is conjugate to a subgroup of GL(d, F).K×, where K× denotes the centre of GL(d, K). If the derived group of G also acts absolutely irreducibly, then the algorithm is Las Vegas and costs O(|X|d³+d²log|F|) arithmetic operations in K. This work forms part of a recognition project based on Aschbacher’s classification of maximal subgroups of GL(d, K).

Epidemiological Models For Mutating Pathogens With Temporary Immunity, Neeta Singh

Electronic Theses and Dissertations

Significant progress has been made in understanding different scenarios for disease transmissions and behavior of epidemics in recent years. A considerable amount of work has been done in modeling the dynamics of diseases by systems of ordinary differential equations. But there are very few mathematical models that deal with the genetic mutations of a pathogen. In-fact, not much has been done to model the dynamics of mutations of pathogen explaining its effort to escape the host's immune defense system after it has infected the host. In this dissertation we develop an SIR model with variable infection age for the transmission …

The Integral Cohomology Of The Group Of Loops, Craig Jensen, Jon Mccammond, John Meier

Mathematics Faculty Publications

No abstract provided.

Application Of The Empirical Likelihood Method In Proportional Hazards Model, Bin He

Electronic Theses and Dissertations

In survival analysis, proportional hazards model is the most commonly used and the Cox model is the most popular. These models are developed to facilitate statistical analysis frequently encountered in medical research or reliability studies. In analyzing real data sets, checking the validity of the model assumptions is a key component. However, the presence of complicated types of censoring such as double censoring and partly interval-censoring in survival data makes model assessment difficult, and the existing tests for goodness-of-fit do not have direct extension to these complicated types of censored data. In this work, we use empirical likelihood (Owen, 1988) …

Effects Of Atmospheric Turbulence On The Propagation Of Flattened Gaussian Optical Beams, Doris Cowan

Electronic Theses and Dissertations

In an attempt to mitigate the effects of the atmosphere on the coherence of an optical (laser) beam, interest has recently been shown in changing the beam shape to determine if a different power distribution at the transmitter will reduce the effects of the random fluctuations in the refractive index. Here, a model is developed for the field of a flattened Gaussian beam as it propagates through atmospheric turbulence, and the resulting effects upon the scintillation of the beam and upon beam wander are determined. A comparison of these results is made with the like effects on a standard TEM00 …

Intrinsic Linking And Knotting Of Graphs In Arbitrary 3–Manifolds, Erica Flapan, Hugh Howards, Don Lawrence, Blake Mellor

Mathematics, Statistics and Data Science Faculty Works

We prove that a graph is intrinsically linked in an arbitrary 3–manifold MM if and only if it is intrinsically linked in S³. Also, assuming the Poincaré Conjecture, we prove that a graph is intrinsically knotted in M if and only if it is intrinsically knotted in S³.

Intersection Graphs For String Links, Blake Mellor

Mathematics, Statistics and Data Science Faculty Works

We extend the notion of intersection graphs for knots in the theory of finite type invariants to string links. We use our definition to develop weight systems for string links via the adjacency matrix of the intersection graph, and show that these weight systems are related to the weight systems induced by the Conway and Homfly polynomials.

Chord Diagrams And Gauss Codes For Graphs, Thomas Fleming, Blake Mellor

Mathematics, Statistics and Data Science Faculty Works

Chord diagrams on circles and their intersection graphs (also known as circle graphs) have been intensively studied, and have many applications to the study of knots and knot invariants, among others. However, chord diagrams on more general graphs have not been studied, and are potentially equally valuable in the study of spatial graphs. We will define chord diagrams for planar embeddings of planar graphs and their intersection graphs, and prove some basic results. Then, as an application, we will introduce Gauss codes for immersions of graphs in the plane and give algorithms to determine whether a particular crossing sequence is …

Tree Diagrams For String Links, Blake Mellor

Mathematics, Statistics and Data Science Faculty Works

In previous work, the author defined the intersection graph of a chord diagram associated with string links (as in the theory of finite type invariants). In this paper, we classify the trees which can be obtained as intersection graphs of string link diagrams.

Linked Exact Triples Of Triangulated Categories And A Calculus Of T-Structures, Michael Berg

Mathematics, Statistics and Data Science Faculty Works

We introduce a new formalism of exact triples of triangulated categories arranged in certain types of diagrams. We prove that these arrangements are well-behaved relative to the process of gluing and ungluing t-structures defined on the indicated categories and we connect our con. structs to· a problem (from number theory) involving derived categories. We also briefly address a possible connection with a result of R. Thomason.

On The Use Of Gaussian Filter Functions For Adaptive Optics, Merfit Assad

Electronic Theses and Dissertations

For adaptive optic systems, the use of aperture filter functions calculated using various Zernike modes can be useful in removing lower-order aberrations caused by atmospheric turbulence. Traditionally, these filter functions are calculated using the step function depicting a hard aperture that introduces integrals that are sometimes difficult to integrate and must be done numerically. The Gaussian method can be used in place of the conventional method for calculating the aperture filter functions. Evaluation of the Gaussian approximation for modeling a finite receiver aperture can be made by comparison of reduction in phase variance with results achieved using the conventional method. …

Quasi Separation Axioms, Mohammad S. Sarsak

Turkish Journal of Mathematics

In [5], Maheshwari et al. introduced and studied some new separation axioms, namely, quasi semi T_i axioms where i \in {0, 1, 2}, the quasi semi T_{1/2} axiom was then introduced and investigated by Gyu-Ihn et al. in [2]. In the present paper we introduce and study quasi T_i axioms, i \in {0, 1 / 2, 1, 2} as a special variety of quasi semi T_i axioms, the class of quasi T_{1/2} (respectively, quasi T_1) bitopological spaces is placed between quasi T_0 (respectively, quasi T_{1/2}) bitopological spaces and quasi T_1 (respectively, quasi T_2) bitopological spaces. Among several counter examples we …

The Restriction And The Continuity Properties Of Potentials Depending On \Lambda-Distance, M. Zeki̇ Sarikaya, Hüseyi̇n Yildirim

Turkish Journal of Mathematics

In this study we establish theorems on the restriction and continuity of the generalized Riesz potentials with the non-isotropic kernels depending on \lambda-distance.

The Radius Of Starlikeness P-Valently Analytic Functions In The Unit Disc, Yaşar Polatoğlu, Meti̇n Bolcal, Arzu Şen, H. Esra Özkan

Turkish Journal of Mathematics

In the present paper we shall give the radius of starlikeness for the classes of p-valent analytic functions in the unit disc D = { z z < 1 }.

Some Random Fixed Point Theorems For Non-Self Nonexpansive Random Operators, Poom Kumam, Somyot Plubtieng

Turkish Journal of Mathematics

Let (\Omega, \Sigma) be a measurable space, with \sum a sigma-algebra of subsets of \Omega, and let E be a nonempty bounded closed convex and separable subset of a Banach space X, whose characteristic of noncompact convexity is less than 1. We prove that a multivalued nonexpansive, non-self operator T: \Omega \times E \rightarrow KC(X) satisfying an inwardness condition and itself being a 1-\chi-contractive nonexpansive mapping has a random fixed point. We also prove that a multivalued nonexpansive, non-self operator T:\Omega\times E\rightarrow KC(X) with a uniformly convex X satisfying an inwardness condition has a random fixed point.

Remarks About Some Weierstrass Type Results, Mihai Turinici

Turkish Journal of Mathematics

The Weierstrass type results of Gajek and Zagrodny [7] are not in general retainable in the precise context. Our first aim in this exposition is to show that a completion of the imposed conditions may be offered so that these results be true. As a second aim, alternate proofs of the statements in question are performed, via ordering principles comparable with the one in Brezis and Browder [3].

Multi-Dimensional Seismic Imaging Using The Inverse Scattering Series, Fang Liu, Arthur B. Weglein, Kristopher A. Innanen, Bogdan Nita

Department of Mathematics Faculty Scholarship and Creative Works

The inverse scattering series (ISS) is a comprehensive multidimensional theory for processing and inverting seismic reflection data, that may be task-separated such that meaningful sub-problems of the seismic inverse problem may be accomplished individually, each without an accurate velocity model. We describe a task-separated subseries of the ISS geared towards accurate location in depth of reflectors, in particular the mechanisms of the series that act in multiple dimensions. We show that some 2D ISS imaging terms have analogs in previously developed 1D ISS imaging theory (e.g., Weglein et al., 2002; Shaw, 2005) and others do not; the former are used …

$H$-Vectors Of Generalized Associahedra And Noncrossing Partitions, Colum Watt, Thomas Brady, Christos A. Athanasiadis, Jon Mccammond

Articles

A uniform proof is given that the entries of the $h$-vector of the cluster complex $\Delta (\Phi)$, associated by S. Fomin and A. Zelevinsky to a finite root system $\Phi$, count elements of the lattice $\mathbf{L}$ of noncrossing partitions of corresponding type by rank. Similar interpretations for the $h$-vector of the positive part of $\Delta (\Phi)$ are provided. The proof utilizes the appearance of the complex $\Delta (\Phi)$ in the context of the lattice $\mathbf{L}$ in recent work of two of the authors, as well as an explicit shelling of $\Delta (\Phi)$.

Countable Short Recursively Saturated Models Of Arithmetic, Erez Shochat

Dissertations, Theses, and Capstone Projects

Short recursively saturated models of arithmetic are exactly the elementary initial segments of recursively saturated models of arithmetic. Since any countable recursively saturated model of arithmetic has continuum many elementary initial segments which are already recursively saturated, we turn our attention to the (countably many) initial segments which are not recursively saturated. We first look at properties of countable short recursively saturated models of arithmetic and show that although these models cannot be cofinally resplendent (an expandability property slightly weaker than resplendency), these models have non-definable expansions which are still short recursively saturated.

Infinitely Often Dense Bases And Geometric Structure Of Sumsets, Jaewoo Lee

Dissertations, Theses, and Capstone Projects

We'll discuss two problems related to sumsets.

Nathanson constructed bases of integers with prescribed representation functions, then asked how dense bases for integers can be in such cases. Let A(-x, x) be the number of elements of A whose absolute value is less than or equal to x, then it's easy to see that A(-x, x) << x1/2 if its representation function is bounded, giving us a general upper bound. Chen constructed unique representation bases for integers with A(-x, x) ≥ x1/2-epsilon infinitely often. In the first chapter, we'll construct bases for integers with a prescribed representation function with A(-x, x) > x1/2/&phis;(x) infinitely often where &phis;(x) is any nonnegative real-valued function which tends to infinity.

In the second chapter, we'll see how sumsets appear geometrically. Assume A is a finite set of lattice points and h*D=h˙x:x∈conv A is a full dimensional polytope. Then we'll see …

An Euler-Type Formula For Ζ(2k +1), Tian-Xiao He, Michael Dancs

Scholarship

In this short paper, we give several new formulas for ζ(n) when n is an odd positive integer. The method is based on a recent proof, due to H. Tsumura, of Euler’s classical result for even n. Our results illuminate the similarities between the even and odd cases, and may give some insight into why the odd case is much more difficult.

Numerical Approximation To Ζ(2n+1), Tian-Xiao He, Michael Dancs

Scholarship

In this short paper, we establish a family of rapidly converging series expansions ζ(2n +1) by discretizing an integral representation given by Cvijovic and Klinowski [3] in Integral representations of the Riemann zeta function for odd-integer arguments, J. Comput. Appl. Math. 142 (2002) 435–439. The proofs are elementary, using basic properties of the Bernoulli polynomials.

Multivariate Expansion Associated With Sheffer-Type Polynomials And Operators, Tian-Xiao He, Leetsch Hsu, Peter Shiue

Scholarship

With the aid of multivariate Sheffer-type polynomials and differential operators, this paper provides two kinds of general expansion formulas, called respectively the first expansion formula and the second expansion formula, that yield a constructive solution to the problem of the expansion of A(ˆt)f([g(t)) (a composition of any given formal power series) and the expansion of the multivariate entire functions in terms of multivariate Sheffer-type polynomials, which may be considered an application of the first expansion formula and the Sheffer-type operators. The results are applicable to combinatorics and special function theory.