Mathematics Commons^™

Homeomorphisms Between Pairs Of Genus Two Handlebodies And Separating Circles In Their Boundaries, Daniel Orlando Vazquez

Open Access Theses & Dissertations

A known result in low-dimensional topology is that every incompressible surface properly embedded in a genus two handlebody is boundary compressible. This result produces a pair consisting of a genus two handlebody and a separating curve in its boundary.

We study particular examples of genus two handlebodies and essential simple closed curves which separate their boundaries. The aim of this paper is to give our own construction of separating curves on the boundary of genus two handlebodies and compare them with previously studied pairs. These constructions involve taking 2n arcs on either side of a waist disk of a genus …

Comparison Of Second Order Conformal Symplectic Schemes With Linear Stability Analysis, Dwayne Floyd

Electronic Theses and Dissertations

Numerical methods for solving linearly damped Hamiltonian ordinary differential equations are analyzed and compared. The methods are constructed from the well-known Störmer-Verlet and implicit midpoint methods. The structure preservation properties of each method are shown analytically and numerically. Each method is shown to preserve a symplectic form up to a constant and are therefore conformal symplectic integrators, with each method shown to accurately preserve the rate of momentum dissipation. An analytical linear stability analysis is completed for each method, establishing thresholds between the value of the damping coefficient and the step-size that ensure stability. The methods are all second order …

On Clifford Analysis For Holomorphic Mappings, M. E. Luna-Elizarrarás, M. Shapiro, Daniele C. Struppa

Mathematics, Physics, and Computer Science Faculty Articles and Research

In the classical theory of several complex variables, holomorphic mappings are just n-tuples of holomorphic functions in m variables, with arbitrary n and m, and no relations between these functions are assumed. Some 30 years ago John Ryan introduced complex, or complexified, Clifford analysis which is, in a sense, the study of certain classes of holomorphic mappings where the components are not independent, and instead obey the relations generated by the Cauchy– Riemann and Dirac-type operators. In this paper, we take a closer look at this theory emphasizing some additional properties that holomorphic mappings satisfy in this context. Our attention …

Topological Duality And Lattice Expansions, I: A Topological Construction Of Canonical Extensions, M. Andrew Moshier, Peter Jipsen

Mathematics, Physics, and Computer Science Faculty Articles and Research

The two main objectives of this paper are (a) to prove purely topological duality theorems for semilattices and bounded lattices, and (b) to show that the topological duality from (a) provides a construction of canonical extensions of bounded lattices. In previously known dualities for semilattices and bounded lattices, the dual spaces are compact 0-dimensional spaces with additional algebraic structure. For example, semilattices are dual to 0-dimensional compact semilattices. Here we establish dual categories in which the spaces are characterized purely in topological terms, with no additional algebraic structure. Thus the results can be seen as generalizing Stone's duality for distributive …

An Incremental Reseeding Strategy For Clustering, Xavier Bresson, Huiyi Hu, Thomas Laurent, Arthur Szlam, James Von Brecht

Mathematics, Statistics and Data Science Faculty Works

In this work we propose a simple and easily parallelizable algorithm for multiway graph partitioning. The algorithm alternates between three basic components: diffusing seed vertices over the graph, thresholding the diffused seeds, and then randomly reseeding the thresholded clusters. We demonstrate experimentally that the proper combination of these ingredients leads to an algorithm that achieves state-of-the-art performance in terms of cluster purity on standard benchmarks datasets. Moreover, the algorithm runs an order of magnitude faster than the other algorithms that achieve comparable results in terms of accuracy. We also describe a coarsen, cluster and refine approach similar to GRACLUS and …

Crossed Modules Of Racks, Alissa S. Crans, Friedrich Wagemann

Mathematics, Statistics and Data Science Faculty Works

We generalize the notion of a crossed module of groups to that of a crossed module of racks. We investigate the relation to categorified racks, namely strict 2-racks, and trunk-like objects in the category of racks, generalizing the relation between crossed modules of groups and strict 2-groups. Then we explore topological applications. We show that by applying the rack-space functor, a crossed module of racks gives rise to a covering. Our main result shows how the fundamental racks associated to links upstairs and downstairs in a covering fit together to form a crossed module of racks.

Complete Bipartite Graphs Whose Topological Symmetry Groups Are Polyhedral, Blake Mellor

Mathematics, Statistics and Data Science Faculty Works

We determine for which n, the complete bipartite graph K_n,n has an embedding in S³ whose topological symmetry group is isomorphic to one of the polyhedral groups: A₄, A₅, or S₄.

Robust Noise Attenuation Under Stochastic Noises And Worst-Case Unmodelled Dynamics, Araz Hashemi, Ben G. Fitzpatrick, Le Yi Wang, George Yin

Mathematics, Statistics and Data Science Faculty Works

This paper investigates noise attenuation problems for systems with unmodelled dynamics and unknown noise characteristics. A unique methodology is introduced that employs signal estimation in one phase, followed by control design for noise rejection. The methodology enjoys certain advantages in its simple control design process, accommodation of unmodelled dynamics, and non-conservative noise rejection performance. Under mild information on unmodelled dynamics, we first derive robust performance bounds on noise attenuation with respect to unmodelled dynamics without noise estimation errors. Then more general results are presented for systems that are subject to both stochastic signal estimation errors and unmodelled dynamics. Examples are …

Hom Quandles, Alissa S. Crans, Sam Nelson

Mathematics, Statistics and Data Science Faculty Works

If A is an abelian quandle and Q is a quandle, the hom set Hom(Q,A) of quandle homomorphisms from Q to A has a natural quandle structure. We exploit this fact to enhance the quandle counting invariant, providing an example of links with the same counting invariant values but distinguished by the hom quandle structure. We generalize the result to the case of biquandles, collect observations and results about abelian quandles and the hom quandle, and show that the category of abelian quandles is symmetric monoidal closed.

Simulation Of The Sampling Distribution Of The Mean Can Mislead, Ann E. Watkins, Anna E. Bargagliotti, Christine Franklin

Mathematics, Statistics and Data Science Faculty Works

Although the use of simulation to teach the sampling distribution of the mean is meant to provide students with sound conceptual understanding, it may lead them astray. We discuss a misunderstanding that can be introduced or reinforced when students who intuitively understand that “bigger samples are better” conduct a simulation to explore the effect of sample size on the properties of the sampling distribution of the mean. From observing the patterns in a typical series of simulated sampling distributions constructed with increasing sample sizes, students reasonably—but incorrectly—conclude that, as the sample size, n, increases, the mean of the (exact) sampling …

Measures Of Tactical Efficiency In Water Polo, James Graham, John Mayberry

College of the Pacific Faculty Articles

We present a notational analysis of offensive tactics commonly employed in elite men's water polo and address three questions related to this objective: which tactics are most effective?, which tactical performance indicators best classify the winning team?, and how accurate are predictive models based on these performance indicators? We define a new statistic, Efficiency Rating, which quantifies the importance of a tactic via a weighted average of direct and indirect goals generated by its use. By this measure, direct shot is the most efficient even strategy despite being employed far less frequently than centre or perimeter tactics. We address our …

Generalized Derivations Centralizing On Jordan Ideals Of Rings With Involution, Lahcen Oukhtite, Abdellah Mamouni

Turkish Journal of Mathematics

A classical result of Posner states that the existence of a nonzero centralizing derivation on a prime ring forces the ring to be commutative. In this paper we extend Posner's result to generalized derivations centralizing on Jordan ideals of rings with involution and discuss the related results. Moreover, we provide examples to show that the assumed restriction cannot be relaxed.

Central Configurations In The Collinear 5-Body Problem, Muhammad Shoaib, Anoop Sivasankaran, Abdulrehman Kashif

Turkish Journal of Mathematics

We study the inverse problem of central configuration of collinear general 4- and 5-body problems. A central configuration for n-body problems is formed if the position vector of each particle with respect to the center of mass is a common scalar multiple of its acceleration. In the 3-body problem, it is always possible to find 3 positive masses for any given 3 collinear positions given that they are central. This is not possible for more than 4-body problems in general. We consider a collinear 5-body problem and identify regions in the phase space where it is possible to choose positive …

Monomial Ideals Of Linear Type, Monica La Barbiera, Paola Lea Stagliano'

Turkish Journal of Mathematics

Let S=K[x_1,…,x_n;y_1,…,y_m] be the polynomial ring in 2 sets of variables over a field K. We investigate some classes of monomial ideals of S in order to classify ideals of the linear type.

Existence And Multiplicity Of Positive Solutions For Discrete Anisotropic Equations, Marek Galewski, Renata Wieteska

Turkish Journal of Mathematics

In this paper we consider the Dirichlet problem for a discrete anisotropic equation with some function \alpha , a nonlinear term f, and a numerical parameter \lambda :\Delta (\alpha (k) \Delta u(k-1) ^{p(k-1)-2}\Delta u(k-1)) + \lambda f(k,u(k))=0, k\in [1,T] . We derive the intervals of a numerical parameter \lambda for which the considered BVP has at least 1, exactly 1, or at least 2 positive solutions. Some useful discrete inequalities are also derived.

Asymptotic Analysis Of The 2-Dimensional Soliton Solutions For The Nizhnik--Veselov--Novikov Equations, Meti̇n Ünal

Turkish Journal of Mathematics

In this paper we present a direct approach to determining a class of solutions, the asymptotic analysis of the dromion solutions, and their asymptotic properties of the Nizhnik--Veselov--Novikov equations by means of Pfaffians. The form of the solution obtained allows a detailed asymptotic analysis of the dromion solutions and compact expression for the phase shifts and changes of amplitude as a result of interaction of the dromions to be determined.

Pullback Diagram Of H^*-Algebras, Mahnaz Khanehgir, Maryam Amyari, Marzieh Moradian Khibary

Turkish Journal of Mathematics

In this paper we obtain some properties for the pullback diagram of H^*-algebras. More precisely, we prove that the commutative diagram of H^*-algebras and morphisms A_1 @>\varphi_1>> B_1 @VV\psi_1V @VV\psi_2V A_2 @>\varphi_2>> B_2 is pullback and \psi_1 is an injection if and only if \psi_1 is a surjection, \psi_2 is an injection, and \ker \varphi_1 \cap \ker \psi_1 = \{0\}.

Semi-Cotangent Bundle And Problems Of Lifts, Furkan Yildirim, Arif Salimov

Turkish Journal of Mathematics

Using the fiber bundle M over a manifold B, we define a semi-cotangent (pull-back) bundle t^{\ast}B, which has a degenerate symplectic structure. We consider lifting problem of projectable geometric objects on M to the semi-cotangent bundle. Relations between lifted objects and a degenerate symplectic structure are also presented.

On Kapranov's Description Of \Overline{M}_{0,N} As A Chow Quotient, Noah Giansiracusa, William Danny Gillam

Turkish Journal of Mathematics

We provide a direct proof, valid in arbitrary characteristic, of the result originally proven by Kapranov over C that the Hilbert quotient (P^1)^n//_HPGL_2 and Chow quotient (P^1)^n//_{Ch}PGL_2 are isomorphic to \overline{M}_{0,n}. In both cases this is done by explicitly constructing the universal family of orbit closures and then showing that the induced morphism is an isomorphism onto its image. The proofs of these results in many ways reduce to the case n = 4; in an appendix we outline a formalism of this phenomenon relating to certain operads.

Some Rings For Which The Cosingular Submodule Of Every Module Is A Direct Summand, Derya Keski̇n Tütüncü, Ni̇l Orhan Ertaş, Patrick F. Smith, Rachid Tribak

Turkish Journal of Mathematics

The submodule \overline{Z}(M) = \cap {N M/N is small in its injective hull} was introduced by Talebi and Vanaja in 2002. A ring R is said to have property (P) if \overline{Z}(M) is a direct summand of M for every R-module M. It is shown that a commutative perfect ring R has (P) if and only if R is semisimple. An example is given to show that this characterization is not true for noncommutative rings. We prove that if R is a commutative ring such that the class {M \in Mod-R \overline{Z}_{R}(M) = 0} is closed under factor modules, then …