Mathematics Commons^™

Quantum Cluster Algebras At Roots Of Unity, Poisson-Lie Groups, And Discriminants, Kurt Malcolm Trampel Iii

LSU Doctoral Dissertations

This dissertation studies quantum algebras at roots of unity in regards to cluster structure and Poisson structure. Moreover, quantum cluster algebras at roots of unity are rigorously defined. The discriminants of these algebras are described, in terms of frozen cluster variables for quantum cluster algebras and Poisson primes for specializations of quantum algebras. The discriminant is a useful invariant for representation theoretic and algebraic study, whose laborious computation deters direct evaluation. The discriminants of quantum Schubert cells at roots of unity will be computed from the two distinct approaches. These methods can be applied to many other quantum algebras.

The Last Digits Of Infinity (On Tetrations Under Modular Rings), William Stowe

Celebration of Learning

A tetration is defined as repeated exponentiation. As an example, 2 tetrated 4 times is 2^(2^(2^2)) = 2^16. Tetrated numbers grow rapidly; however, we will see that when tetrating where computations are performed mod n for some positive integer n, there is convergent behavior. We will show that, in general, this convergent behavior will always show up.

A “Rule-Of-Five” Framework For Models And Modeling To Unify Mathematicians And Biologists And Improve Student Learning, C. Diaz Eaton, H. C. Highlander, K. D. Dahlquist, G. Ledder, M.D. Lamar, R.C. Schugart

Department of Mathematics: Faculty Publications

Despite widespread calls for the incorporation of mathematical modeling into the undergraduate biology curriculum, there is lack of a common understanding around the definition of modeling, which inhibits progress. In this paper, we extend the “Rule-of-Four,” initially used in calculus reform efforts, to a “Rule-of-Five” framework for models and modeling that is inclusive of varying disciplinary definitions of each. This unifying framework allows us to both build on strengths that each discipline and its students bring, but also identify gaps in modeling activities practiced by each discipline. We also discuss benefits to student learning and interdisciplinary collaboration.

An Optimal Medium-Strength Regularity Algorithm For 3-Uniform Hypergraphs, John Theado

USF Tampa Graduate Theses and Dissertations

Szemere´di’s Regularity Lemma [32, 33] is an important tool in combinatorics, with numerous appli- cations in combinatorial number theory, discrete geometry, extremal graph theory, and theoretical computer science.

The Regularity Lemma hinges on the following concepts. Let G = (V, E) be a graph and let ∅ /= X, Y ⊂ V be a pair of disjoint vertex subsets. We define the density of the pair (X, Y ) by d_G(X, Y ) = |E[X, Y ]|/(|X||Y |) where E[X, Y ] denotes the …

The Mystery Of Frobenius Symmetry, Maciej Piwowarczyk

DePaul Discoveries

In this project we studied the mathematical concept of the Frobenius number and some curious patterns that come with it. One common example of the Frobenius number is the Coin Problem: If handed two denominations of coins, say 4¢ and 5¢, and asked to create all possible values, we will eventually find ourselves in a position where we can make any value. With 4¢ and 5¢ coins, we can create any value above 11¢, but not 11¢ itself. So, that makes 11 the Frobenius number of 4 and 5. What we explore in this paper is a pattern we call …

Computational Number Theory: Modular Forms, Paul Jenkins

Journal of Undergraduate Research

In 2017 and 2018, the following students participated in the BYU Computational Number Theory research group under my direction and produced the following deliverables.

Design Of Metamaterials For Optics, Abiti Adili

LSU Doctoral Dissertations

First part of this dissertation studies the problem of designing metamaterial crystals with double negative effective properties for applications in optics by investigating the conditions necessary for generating novel dispersion properties in a metamaterial crystal with subwavelength microstructure. This provides novel optical properties created through local resonances tied to the geometry of the media in subwavelength regime.

In the second part, this dissertation studies the representation formula used to describe band structures in photonic crystals with plasmonic inclusions. By using layer potential techniques, a magnetic dipole operator describing the tangential component of the electrical field generated by magnetic distribution is …

Metric Geometry And The Determination Of The Bohmian Quantum Potential, Paul Bracken

School of Mathematical & Statistical Sciences Faculty Publications

A geometric approach to quantum mechanics which is formulated in terms of Finsler geometry is developed. It is shown that quantum mechanics can be formulated in terms of Finsler configuration space trajectories which obey Newton-like evolution but in the presence of an additional kind of potential. This additional quantum potential which was obtained first by Bohm has the consequence of contributing to the forces driving the system. This geometric picture accounts for many aspects of quantum dynamics and leads to a more natural interpretation. It is found for example that dynamics can be accounted for by incorporating quantum effects into …

Dehn Functions Of Bestvina-Brady Groups, Yu-Chan Chang

LSU Doctoral Dissertations

In this dissertation, we prove that if the flag complex on a finite simplicial graph is a 2-dimensional triangulated disk, then the Dehn function of the associated Bestvina--Brady group depends on the maximal dimension of the simplices in the interior of the flag complex. We also give some examples where the flag complex on a finite simplicial graph is not 2-dimensional, and we establish a lower bound for the Dehn function of the associated Bestvina--Brady group.

Navigating Around Convex Sets, J. J. P. Veerman

Mathematics and Statistics Faculty Publications and Presentations

We review some basic results of convex analysis and geometry in Rn in the context of formulating a differential equation to track the distance between an observer flying outside a convex set K and K itself.

Cocyclic Hadamard Matrices: An Efficient Search Based Algorithm, Jonathan S. Turner

Theses and Dissertations

This dissertation serves as the culmination of three papers. “Counting the decimation classes of binary vectors with relatively prime fixed-density" presents the first non-exhaustive decimation class counting algorithm. “A Novel Approach to Relatively Prime Fixed Density Bracelet Generation in Constant Amortized Time" presents a novel lexicon for binary vectors based upon the Discrete Fourier Transform, and develops a bracelet generation method based upon the same. “A Novel Legendre Pair Generation Algorithm" expands upon the bracelet generation algorithm and includes additional constraints imposed by Legendre Pairs. It further presents an efficient sorting and comparison algorithm based upon symmetric functions, as well …

Graph Pebbling Algorithms And Lemke Graphs, Charles A. Cusack, Aaron Green, Airat Bekmetjev, Mark Powers

Faculty Publications

Given a simple, connected graph, a pebbling configuration (or just configuration) is a function from its vertex set to the nonnegative integers. A pebbling move between adjacent vertices removes two pebbles from one vertex and adds one pebble to the other. A vertex r is said to be reachable from a configuration if there exists a sequence of pebbling moves that places at least one pebble on r. A configuration is solvable if every vertex is reachable. The pebbling number π(G) of a graph G is the minimum integer such that every configuration of size π(G) on G …

Constructing General Solution Of An Hyperbolic Equation With Two Singular Coefficients, A O’Rinov, А. Rafiqov

Scientific journal of the Fergana State University

equation of hyperbolic type, singular coefficient, general solution.

A Mixed Problem For A Fourth Order Equation With Singular Coefficients In A Rectangular, A.S V. Berdishev, M. Azizov

Scientific journal of the Fergana State University

In the present work with spectral method it was proved the uniqueness and existence of the solution of the mixed problem for a fourth order equation with singular coefficients in a rectangular domain.

The Use Of Interactive Methods To Increase The Efficiency Of Studying Physics In Secondary School, O Dehqonova, F Yusupova

Scientific journal of the Fergana State University

The article emphasizes interactive methods of teaching physics and their connections in the study of the subject’s of technology in secondary schools.

Constructing General Solution Of An Hyperbolic Equation With Two Singular Coefficients, A O’Rinov, А. Rafiqov

Scientific journal of the Fergana State University

equation of hyperbolic type, singular coefficient, general solution.

The Use Of Interactive Methods To Increase The Efficiency Of Studying Physics In Secondary School, O Dehqonova, F Yusupova

Scientific journal of the Fergana State University

The article emphasizes interactive methods of teaching physics and their connections in the study of the subject’s of technology in secondary schools.

A Mixed Problem For A Fourth Order Equation With Singular Coefficients In A Rectangular, A.S V. Berdishev, M. Azizov

Scientific journal of the Fergana State University

In the present work with spectral method it was proved the uniqueness and existence of the solution of the mixed problem for a fourth order equation with singular coefficients in a rectangular domain.

Inquiry In Inquiry: A Classification Of The Learning Theories Underlying Inquiry-Based Undergraduate Number Theory Texts, Rebecca L. Butler

Honors Projects

While undergraduate inquiry-based texts in number theory share similar approaches with respect to learning as the embodiment of professional practice, this does not entail that these texts all operate from the same fundamental understanding of what it means to learn mathematics. In this paper, the instructional design of several texts of the aforementioned types are analyzed to assess the theory of learning under which they operate. From this understanding of the different theories of learning employed in an inquiry-based mathematical setting, one can come to understand the popular model of what it is to learn number theory in a meaningful …

String Oscillations With Impulse Effects, K Yelgondiyev

Karakalpak Scientific Journal

The problem of existence of periodic solutions to the equation of oscillations of a pulse stuny impact soft moments. Necessary and sufficient conditions of existence of periodic solutions in such oscillatory systems.