Physical Sciences and Mathematics Commons^™

Splitting Of Vector Bundles On Punctured Spectrum Of Regular Local Rings, Mahdi Majidi-Zolbanin

Dissertations, Theses, and Capstone Projects

In this dissertation we study splitting of vector bundles of small rank on punctured spectrum of regular local rings. We give a splitting criterion for vector bundles of small rank in terms of vanishing of their intermediate cohomology modules Hⁱ(U, E)_{2_i_n−3}, where n is the dimension of the regular local ring. This is the local analog of a result by N. Mohan Kumar, C. Peterson, and A. Prabhakar Rao for splitting of vector bundles of small rank on projective spaces.

As an application we give a positive answer (in a special case) to a conjecture …

Mapping Properties Of Co-Existentially Closed Continua, Paul Bankston

Mathematics, Statistics and Computer Science Faculty Research and Publications

A continuous surjection between compacta is called co-existential if it is the second of two maps whose composition is a standard ultracopower projection. A continuum is called co-existentially closed if it is only a co-existential image of other continua. This notion is not only an exact dual of Abraham Robinson's existentially closed structures in model theory, it also parallels the definition of other classes of continua defined by what kinds of continuous images they can be. In this paper we continue our study of co-existentially closed continua, especially how they (and related continua) behave in certain mapping situations.

Weak Factorizations, Fractions And Homotopies, Alexander Kurz, Jiří Rosický

Engineering Faculty Articles and Research

We show that the homotopy category can be assigned to any category equipped with a weak factorization system. A classical example of this construction is the stable category of modules. We discuss a connection with the open map approach to bisimulations proposed by Joyal, Nielsen and Winskel.

On The Regulation Of Networks As Complex Systems: A Graph Theory Approach, Daniel F. Spulber, Christopher S. Yoo

All Faculty Scholarship

The dominant approach to regulating communications networks treats each network component as if it existed in isolation. In so doing, the current approach fails to capture one of the essential characteristics of networks, which is the complex manner in which components interact with one another when combined into an integrated system. In this Essay, Professors Daniel Spulber and Christopher Yoo propose a new regulatory framework based on the discipline of mathematics known as graph theory, which better captures the extent to which networks represent complex systems. They then apply the insights provided by this framework to a number of current …

Bootstrap Prediction Intervals For Multivariate Time Series, Florian Sebastian Rueck

Doctoral Dissertations

"The theory and methodology of obtaining bootstrap prediction intervals for univariate time series using the forward representation of the series is extended to vector autoregressive (VAR) models. Kim has shown that simultaneous prediction intervals based on the Bonferroni method and the backward representation of the time series achieve coverage close to nominal when the parameter estimates are corrected for small sample bias. To utilize his method, it is necessary to assume that the innovations are normally distributed to maintain independence of the innovations associated with the backward representation of the time series. This assumption is not necessary if the forward …

Separately Nowhere Constant Functions; N-Cube And \Alpha-Prism Densities, Krzysztof Ciesielski

Faculty & Staff Scholarship

A function f from a countable product X of of Polish spaces X_i into a Polish space is separately nowhere constant provided it is nowhere constant on every section of X. We show that every continuous separately nowhere constant function is one-to-one on a product of perfect subsets of X_i's. This result is used to distinguish between n-cube density notions for different n\leq\omega, where \omega-cube density is a basic notion behind the Covering Property Axiom CPA formulated by Ciesielski and Pawlikowski. We will also distinguish, for different values of \alpha<\omega₁, between the notions of \alpha-prism …

The Convergence Of Difference Boxes, Christopher Kribs, Antonio Behn, Vadim Ponomarenko

Mathematics Faculty Publications

We consider an elementary mathematical puzzle known as a "difference box" in terms of a discrete map from R⁴ to R⁴ or , canonically, from a subset of the first R² into itself. We identify the map's unique canonical fixed point and answer more generally the question of how many interactions a given "difference box" takes to reach zero. (The number is finite except for boxes corresponding to the fixed point.)

Positive Solutions For A Fourth Order Boundary Value Problem, Bo Yang

Faculty Articles

We consider a boundary value problem for the beam equation, in which the boundary conditions mean that the beam is embedded at one end and free at the other end. Some new estimates to the positive solutions to the boundary value problem are obtained. Some sufficient conditions for the existence of at least one positive solution for the boundary value problem are established. An example is given at the end of the paper to illustrate the main results.

Curvature Of The Weinhold Metric For Thermodynamical Systems With 2 Degrees Of Freedom, Manuel Santoro, Serge Preston

Mathematics and Statistics Faculty Publications and Presentations

In this work the curvature of Weinhold (thermodynamical) metric is studied in the case of systems with two thermodynamical degrees of freedom. Conditions for the Gauss curvature R to be zero, positive or negative are worked out. Signature change of the Weinhold metric and the corresponding singular behavior of the curvature at the phase boundaries are studied. Cases of systems with the constant C_v, including Ideal and Van der Waals gases, and that of Berthelot gas are discussed in detail.

Actively Engaging Students In Constructing Knowledge About Two-Dimensional Shapes Using Shape Makers, Shannon Driskell

Shannon O.S. Driskell

Driskell, S. (PI), University of Dayton Research Council Seed Grant, $3,900

Topological Symmetry Groups Of Embedded Graphs In The 3-Sphere, Ramin Naimi, Erica Flapan, James Pommersheim, Harry Tomvakis

Ramin Naimi

No abstract provided.

The Cover Pebbling Number Of Graphs, Lara Pudwell, Betsy Crull, Tammy Cundiff, Paul Feltman, Glenn Hurlbert, Zsuzsanna Szaniszlo, Zsolt Tuza

Lara K. Pudwell

No abstract provided.

Constraint Programming, John Hooker

John Hooker

No abstract provided.

Mra Frame Wavelets With Certain Regularities Associated With The Refinable Generators Of Shift Invariant Spaces, Tian-Xiao He

Tian-Xiao He

In this paper, we will start the discussion with the refinable generators of the shift invariant (SI) spaces in L² (R) that possess the largest possible regularities and required vanishing moments. For the pseudo-scaling generators, the corresponding MRA frame wavelets with certain regularities are constructed. In addition, the stability of the refinable SI spaces and the corresponding complementary spaces, biorthogonality of the SI spaces, and the approximation property of the spaces are also discussed.

A Symbolic Operator Approach To Several Summation Formulas For Power Series, Tian-Xiao He, Leetsch Hsu, Peter Shiue, D. Torney

Tian-Xiao He

This paper deals with the summation problem of power series of the form Sba (f; x) = ∑a ≤ k ≤ b f(k) xk, where 0≤ a < b ≤ ∞, and {f(k)} is a given sequence of numbers with k Є [a, b) or f(t) is a differentiable function defined on [a, b). We present a symbolic summation operator with its various expansions, and construct several summation formulas with estimable remainders for Sba (f; x), by the aid of some classical interpolation series due to Newton, Gauss and Everett, respectively.

Visualizing Patterns In The Integers Relating To The Abundancy Index, Judy Holdener

Judy Holdener

Systematic Studies In Pattern Avoidance, Lara Pudwell, Shalosh Ekhad, Vince Vatter

Lara K. Pudwell

No abstract provided.

On An Extension Of Abel-Gontscharoff's Expansion Formula, Tian-Xiao He, Leetsch C. Hsu, Peter J.-S. Shiue

Tian-Xiao He

We present a constructive generalization of Abel-Gontscharoff’s series expansion to higher dimensions. A constructive application to a problem of multivariate interpolation is also investigated. In addition, two algorithms for the constructing the basis functions of the interpolants are given.

Characterization And Properties Of $(R,S)$-Symmetric, $(R,S)$-Skew Symmetric, And $(R,S)$-Conjugate Matrices, William F. Trench

William F. Trench

No abstract provided.

On Matrices With Rotative Symmetries, William F. Trench

William F. Trench

No abstract provided.

Asymptotic Relationships Between Singular Values Of Structured Matrices Similarly Generated By Different Formal Expansions Of A Rational Function, William F. Trench

William F. Trench

No abstract provided.

Multilevel Matrices With Involutory Symmetries And Skew Symmetries, William F. Trench

William F. Trench

No abstract provided.

An Explicit Fusion Algebra Isomorphism For Twisted Quantum Doubles Of Finite Groups, Christopher Goff

Christopher Goff

Vortices In Bose-Einstein Condensates: Some Recent Developments, Panos Kevrekidis, R. Carretero-Gonzalez, D. J. Frantzeskakis, I. G. Kevrekidis

Panos Kevrekidis

In this brief review we summarize a number of recent developments in the study of vortices in Bose-Einstein condensates, a topic of considerable theoretical and experimental interest in the past few years. We examine the generation of vortices by means of phase imprinting, as well as via dynamical instabilities. Their stability is subsequently examined in the presence of purely magnetic trapping, and in the combined presence of magnetic and optical trapping. We then study pairs of vortices and their interactions, illustrating a reduced description in terms of ordinary differential equations for the vortex centers. In the realm of two vortices …

Partitioning Regular Polygons Into Circular Pieces Ii: Nonconvex Partitions, Mirela Damian, Joseph O'Rourke

Computer Science: Faculty Publications

We explore optimal circular nonconvex partitions of regular k-gons. The circularity of a polygon is measured by its aspect ratio: the ratio of the radii of the smallest circumscribing circle to the largest inscribed disk. An optimal circular partition minimizes the maximum ratio over all pieces in the partition. We show that the equilateral triangle has an optimal 4-piece nonconvex partition, the square an optimal 13-piece nonconvex partition, and the pentagon has an optimal nonconvex partition with more than 20 thousand pieces. For hexagons and beyond, we provide a general algorithm that approaches optimality, but does not achieve it.

The Topology, Geometry And Conformal Structure Of Properly Embedded Minimal Surfaces, Pascal Collin, Robert Kusner, William H. Meeks, Harold Rosenberg

Robert Kusner

This paper develops new tools for understanding surfaces with more than one end and infinite topology which are properly minimally embedded in Euclidean three-space. On such a surface, the set of ends forms a totally disconnected compact Hausdorff space, naturally ordered by the relative heights of the ends in space. One of our main results is that the middle ends of the surface have quadratic area growth, and are thus not limit ends. This implies that the surface can have at most two limit ends, which occur at the top and bottom of the ordering, and thus only a countable …

On A Theorem Of Peters On Automorphisms Of Kahler Surfaces, Weimin Chen Chen

Weimin Chen

For any K¨ahler surface which admits no nonzero holomorphic vectorfields, we consider the group of holomorphic automorphisms which induce identity on the second rational cohomology. Assuming the canonical linear system is without base points and fixed components, C.A.M. Peters [12] showed that this group is trivial except when the K¨ahler surface is of general type and either c21 = 2c2 or c21 = 3c2 holds. Moreover, this group is a 2-group in the former case, and is a 3-group in the latter. The purpose of this note is to give further information about this group. In particular, we show that …

Hyperbolic Sets That Are Not Locally Maximal, Todd L. Fisher

Faculty Publications

This paper addresses the following topics relating to the structure of hyperbolic sets: First, hyperbolic sets that are not contained in locally maximal hyperbolic sets. Second, the existence of a Markov partition for a hyperbolic set. We construct new examples of hyperbolic sets which are not contained in locally maximal hyperbolic sets. The first example is robust under perturbations and can be constructed on any compact manifold of dimension greater than one. The second example is robust, topologically transitive, and constructed on a 4-dimensional manifold. The third example is volume preserving and constructed on R4. We show that every hyperbolic …

Wave Scattering From Infinite Cylindrical Obstacles Of Arbitrary Cross-Section, Matthew B. Weber

Theses and Dissertations

In this work the scattering of an incident plane wave propagating along a plane perpendicular to the xy-plane is studied. The wave is scattered from an infinitely long cylindrical object of arbitrary cross-section. Due to the arbitrary geometry of the obstacle, a finite differences numerical method is employed to approximate the solution of the scattering problems. The wave equation is expressed in terms of generalized curvilinear coordinates. Boundary conforming grids are generated using elliptic grid generators. Then, a explicit marching in time scheme is implemented over these grids. It is found that as time grows the numerical solution converges to …

Strict Feedforward Form And Symmetries Of Nonlinear Control Systems, Witold Respondek, Issa Amadou Tall

Miscellaneous (presentations, translations, interviews, etc)

We establish a relation between strict feedforward form and symmetries of nonlinear control systems. We prove that a system is feedback equivalent to the strict feedforward form if and only if it gives rise to a sequence of systems, such that each element of the sequence, firstly, possesses an infinitesimal symmetry and, secondly, it is the factor system of the preceding one, i.e., is reduced from the preceding one by its symmetry. We also propose a strict feedforward normal form and prove that a smooth strict feedforward system can be smoothly brought to that form.