Mathematics Commons^™

Relation Algebras As Expanded Fl-Algebras, Nikolaos Galatos, Peter Jipsen

Mathematics, Physics, and Computer Science Faculty Articles and Research

This paper studies generalizations of relation algebras to residuated lattices with a unary De Morgan operation. Several new examples of such algebras are presented, and it is shown that many basic results on relation algebras hold in this wider setting. The variety qRA of quasi relation algebras is defined and shown to be a conservative expansion of involutive FL-algebras. Our main result is that equations in qRA and several of its subvarieties can be decided by a Gentzen system, and that these varieties are generated by their finite members.

Counting Links And Knots In Complete Graphs, Loren Abrams, Blake Mellor, Lowell Trott

Mathematics, Statistics and Data Science Faculty Works

We investigate the minimal number of links and knots in embeddings of complete partite graphs in S³. We provide exact values or bounds on the minimal number of links for all complete partite graphs with all but 4 vertices in one partition, or with 9 vertices in total. In particular, we find that the minimal number of links in an embedding of K_4,4,1 is 74. We also provide exact values or bounds on the minimal number of knots for all complete partite graphs with 8 vertices.

Congruence Classes Of 2-Adic Valuations Of Stirling Numbers Of The Second Kind, Curtis Bennett, Edward Mosteig

Mathematics, Statistics and Data Science Faculty Works

We analyze congruence classes of S(n,k), the Stirling numbers of the second kind, modulo powers of 2. This analysis provides insight into a conjecture posed by Amdeberhan, Manna and Moll, which those authors established for k at most 5. We provide a framework that can be used to justify the conjecture by computational means, which we then complete for values of k between 5 and 20.

Twisted Alexander Polynomials Of 2-Bridge Knots, Jim Hoste, Patrick D. Shanahan

Mathematics, Statistics and Data Science Faculty Works

We investigate the twisted Alexander polynomial of a 2-bridge knot associated to a Fox coloring. For several families of 2-bridge knots, including but not limited to, torus knots and genus-one knots, we derive formulae for these twisted Alexander polynomials. We use these formulae to confirm a conjecture of Hirasawa and Murasugi for these knots.

Torsion In One-Term Distributive Homology, Alissa S. Crans, Józef H. Przytycki, Krzysztof K. Putyra

Mathematics, Statistics and Data Science Faculty Works

The one-term distributive homology was introduced by J.H.Przytycki as an atomic replacement of rack and quandle homology, which was first introduced and developed by R.Fenn, C.Rourke and B.Sanderson, and J.S.Carter, S.Kamada and M.Saito. This homology was initially suspected to be torsion-free, but we show in this paper that the one-term homology of a finite spindle can have torsion. We carefully analyze spindles of block decomposition of type (n,1) and introduce various techniques to compute their homology precisely. In addition, we show that any finite group can appear as the torsion subgroup of the first homology of some finite spindle. Finally, …

Cohomology Of Frobenius Algebras And The Yang-Baxter Equation, J. Scott Carter, Alissa S. Crans, Mohamed Elhamdadi, Enver Karadayi, Masahico Saito

Mathematics, Statistics and Data Science Faculty Works

A cohomology theory for multiplications and comultiplications of Frobenius algebras is developed in low dimensions in analogy with Hochschild cohomology of bialgebras based on deformation theory. Concrete computations are provided for key examples. Skein theoretic constructions give rise to solutions to the Yang-Baxter equation using multiplications and comultiplications of Frobenius algebras, and 2-cocycles are used to obtain deformations of R-matrices thus obtained.

A Method Based On Total Variation For Network Modularity Optimization Using The Mbo Scheme, Huiyi Hu, Thomas Laurent, Mason A. Porter, Andrea L. Bertozzi

Mathematics, Statistics and Data Science Faculty Works

The study of network structure is pervasive in sociology, biology, computer science, and many other disciplines. One of the most important areas of network science is the algorithmic detection of cohesive groups of nodes called “communities.” One popular approach to finding communities is to maximize a quality function known as modularity to achieve some sort of optimal clustering of nodes. In this paper, we interpret the modularity function from a novel perspective: we reformulate modularity optimization as a minimization problem of an energy functional that consists of a total variation term and an $\ell_2$ balance term. By employing numerical techniques …

On The Krull Dimension Of Endo-Bounded Modules, Ahmad Haghany, Majid Mazrooei, Mohamad Reza Vedadi

Turkish Journal of Mathematics

Modules in which every essential submodule contains an essential fully invariant submodule are called endo-bounded. Let M be a nonzero module over an arbitrary ring R and X = Spec_2(M_R), the set of all fully invariant L_2-prime submodules of M_R. If M_R is a quasi-projective L_2-Noetherian such that (M/P)_R is endo-bounded for any P \in X, then it is shown that the Krull dimension of M_R is at most the classical Krull dimension of the poset X. The equality of these dimensions and some applications are obtained for certain modules. This gives a generalization of a well-known result on right …

On The Existence Of Nonzero Injective Covers And Projective Envelopes Of Modules, Xiaoxiang Zhang, Xianmei Song

Turkish Journal of Mathematics

In general, the injective cover (projective envelope) of a simple module can be zero. A ring R is called a weakly left V-ring (strongly left Kasch ring) if every simple left R-module has a nonzero injective cover (projective envelope). It is proven that every nonzero left R-module has a nonzero injective cover if and only if R is a left Artinian weakly left V-ring. Dually, every nonzero left R-module has a nonzero projective envelope if and only if R is a left perfect right coherent strongly left Kasch ring. Some related rings and examples are considered.

An Exhaustive Computer Search For Finding New Curves With Many Points Among Fibre Products Of Two Kummer Covers Over F_5 And F_7, Ferruh Özbudak, Burcu Gülmez Temür, Oğuz Yayla

Turkish Journal of Mathematics

In this paper we make an exhaustive computer search for finding new curves with many points among fibre products of 2 Kummer covers of the projective line over F_5 and F_7. At the end of the search, we have 12 records and 6 new entries for the current Table of Curves with Many Points. In particular, we observe that the fibre product y_1^3 = \frac{5(x + 2)(x + 5)}{x}, y_2^3 = \frac{3x^2(x + 5)}{x + 3} over F_7 has genus 7 with 36 rational points. As this coincides with the Ihara bound, we conclude that the maximum number N_7(7) of …

On The Pollard Decomposition Method Applied To Some Jacobi--Sobolev Expansions, Francisco Marcellán, Yamilet Quintana, Alejandro Urieles

Turkish Journal of Mathematics

Let {q_n^{(\alpha,\beta)}}_{n \geq 0} be the sequence of polynomials orthonormal with respect to the Sobolev inner product \langle f,g\rangle_S:=\int_{-1}^1f(x)g(x)w^{(\alpha,\beta)}(x)dx+\int_{-1}^1f'(x)g'(x)w^{(\alpha+1,\beta+1)}(x)dx, where w^{(\alpha,\beta)}(x)=(1-x)^{\alpha}(1+x)^{\beta}, x\in [-1,1] and \alpha,\beta>-1. This paper explores the convergence in the W^{1,p}\left((-1,1), (w^{(\alpha,\beta)},w^{(\alpha+1,\beta+1)})\right) norm of the Fourier expansion in terms of {q_n^{(\alpha,\beta)}}_{n\geq 0} with 1< p

Some Results On Cyclic Codes Over The Ring R_{2,M}, Reza Sobhani, Maryam Molakarimi

Turkish Journal of Mathematics

Let R_{k,m} be the ring F_{2^m}[u_1,u_2,...,u_k]/\gen u_i^2, u_iu_j-u_ju_i. In this paper, cyclic codes of arbitrary length n over the ring R_{2,m} are completely characterized in terms of unique generators and a way for determination of these generators is investigated. A F_{2^m}-basis for these codes is also derived from this representation. Moreover, it is proven that there exists a one-to-one correspondence between cyclic codes of length 2n, n odd, over the ring R_{k-1,m} and cyclic codes of length n over the ring R_{k,m}. By determining the complete structure of cyclic codes of length 2 over R_{2,m}, a mass formula for the …

How Ideas Grow: Critical Mass In The Linear Threshold Model, Hossein Alidaee

Mathematics, Statistics, and Computer Science Honors Projects

We study how ideas spread through a social network using the Linear Threshold Model. Each node i on the complete graph K_n is given a threshold Ɵ_i chosen uniformly at random from (0, 1]. This threshold indicates the fraction of the social network that must be active (or believe the idea) prior to node i becoming active. We start with an activated group of early adopters, called the seed set. Considering various scenarios, we use the probabilistic method to find lower bounds on size of a seed set which guarantees that all nodes become active with high …

Proceedings Of The Seventh Annual Meeting Of The Georgia Association Of Mathematics Teacher Educators Introductory Texts

Proceedings of the Annual Meeting of the Georgia Association of Mathematics Teacher Educators

Contents of 7th Annual GAMTE Proceedings Front Matter:

• Proceedings Committee
• Officers of GAMTE
• Purposes and Goals of GAMTE
• Table of Contents
• Letter from President

How Differing Interpretations Of Making Mathematics Fun Influence Teaching Practice, Amanda Sawyer

Proceedings of the Annual Meeting of the Georgia Association of Mathematics Teacher Educators

I investigated an experienced teacher and a beginning teacher who held similar beliefs about making mathematics learning fun yet held different interpretations of implementation. I found when a teacher equated fun with problem solving, her classroom practice included activities with higher-level thinking skills. In contrast, a teacher who defined fun as students’ enjoyment layered manipulatives and group work on top of procedures. Therefore, teachers need opportunities to reflect on the nature of student understanding as a precursor to shaping their views of fun.

Negative phrases and attitudes toward mathematics have become commonplace in popular culture. Perhaps in response to this …

The Weierstrass Approximation Theorem, Larita Barnwell Hipp

Theses and Dissertations

In this thesis we will consider the work began by Weierstrass in 1855 and several generalization of his approximation theorem since. Weierstrass began by proving the density of algebraic polynomials in the space of continuous real-valued functions on a finite interval in the uniform norm. His theorem has been generalized to an arbitrary compact Hausdorff space and the approximation with elements from more general algebras of continuous real-valued functions. We will consider proofs that use brute force and proofs based on convolutions and approximate identities, trudge through probability and the use of the Bernstein polynomials, and become intimately close to …

The Mc-Dagum Distribution And Its Statistical Properties With Applications, Broderick O. Oluyede, Sasith Rajasooriya

Mathematical Sciences: Faculty Publications

In this paper, a new class of distributions called Mc-Dagum distribution is proposed. This class of distributions contains several distributions such as beta-Dagum, beta-Burr III, beta-Fisk, Dagum, Burr III and Fisk distributions as special cases. The hazard function, reverse hazard function, moments, mean residual life function, Renyi entropy and Fisher information are obtained. Lorenz, Bonferroni and Zenga curves are derived. Maximum likelihood estimates of the model parameters and numerical examples are given to illustrate the usefulness of the proposed class of distributions.

Generalized Analytic Fourier-Feynman Transform Of Functionals In A Banach Algebra F_(A1,A2)^(A,B), Jae Gil Choi, David Skough, Seung Jun Chang

Department of Mathematics: Faculty Publications

We introduce the Fresnel type class F_(A1,A2)^(a,b).We also establish the existence of the generalized analytic Fourier-Feynman transform for functionals in the Banach algebra F_(A1,A2)^(a,b).

Creating An Interdisciplinary Research Course In Mathematical Ecology, Glenn Ledder, Brigitte Tenhumberg

School of Biological Sciences: Faculty Publications

An integrated interdisciplinary research course in biology and mathematics is useful for recruiting students to interdisciplinary research careers, but there are difficulties involved in creating and implementing it. We describe the genesis, objectives, design policies, and structure of the Research Skills in Theoretical Ecology course at the University of Nebraska–Lincoln and discuss the difficulties that can arise in designing and implementing interdisciplinary courses.

An Interdisciplinary Research Course In Theoretical Ecology For Young Undergraduates, Glenn Ledder, Brigitte Tenhumberg, G. Travis Adams

School of Biological Sciences: Faculty Publications

As part of an interdepartmental effort to attract promising young students to research at the interface between mathematics and biology, we created a course in which groups of recent high school graduates and first-year college students conducted a research project in insect population dynamics. The students set up experiments, collected data, used the data to develop mathematical models, tested their models against further experiments, and prepared their results for dissemination. The course was self-contained in that the lecture portion developed the mathematical, statistical, and biological background needed for the research. A special writing component helped students learn the principles of …