Physical Sciences and Mathematics Commons^™

Diving Deeper Into Supercuspidal Representations, Prerna Agarwal

LSU Doctoral Dissertations

In 2013, Reeder and Yu introduced certain low positive depth supercuspidal representations of $p$-adic groups called \textit{epipelagic} representations. These representations generalize the simple supercuspidal representations of Gross and Reeder, which have the lowest possible depth. Epipelagic representations also arise in recent work on the Langlands correspondence; for example, simple supercuspidals appear in the automorphic data corresponding to the Kloosterman $l$-adic sheaf. In this thesis, we take a first step towards the construction of ``\textit{mesopelagic} representation (of Iwahori type)'' which are the higher depth analogues of simple supercuspidal representations. We see that these constructions can be done in a similar way …

Multi-Symplectic Method For The Two-Component Camassa–Holm (2ch) System, Xiaojian Xi, Weipeng Hu, Bo Tang, Pingwei Deng, Zhijun Qiao

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

In this paper, the multi-symplectic formulations of the two-component Camassa–Holm system are presented. Both the multi-symplectic structure and two local conservation laws of the generalized two-component Camassa–Holm model are proposed for its first-order canonical form. Then, combining the Fourier pseudo-spectral method in the spatial domain with the midpoint method in the time dimension, the multi-symplectic Fourier pseudo-spectral scheme is constructed for the first-order canonical form. Meanwhile, the discrete scheme of the residuals of the multi-symplectic structure and two local conservation laws are also provided. By using the multi-symplectic Fourier pseudo-spectral scheme, the evolution of one- and two-soliton solutions for the …

Limit Theorems For Increments Of Branching Particle Systems With Linear Rates And Poisson Initial Condition, Alexander Kreinin, Vladimir V. Vinogradov

Journal of Stochastic Analysis

No abstract provided.

Asymptotic Formula For Scattering Problems Related To Thin Metasurfaces, Zachary Jermain

LSU Doctoral Dissertations

The goal of this work is to develop an asymptotic formula for the behavior of a scattered electromagnetic field in the presence of a thin metamaterial known as a metasurface. By using a carefully chosen Green’s function and the single and double layer potentials we analyze the perturbed scattering problem in the presence of the metamaterial and a background scattering problem. By using Lippman-Schwinger type representation formulas for the two fields we develop the asymptotic formula for the perturbed field. From here we prove the asymptotic formula holds up to a specific error term based on the size of the …

P-Adic Quantum Mechanics, The Dirac Equation, And The Violation Of Einstein Causality, Wilson A. Zuniga-Galindo

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

This article studies the breaking of the Lorentz symmetry at the Planck length in quantum mechanics. We use three-dimensional ℘-adic vectors as position variables, while the time remains a real number. In this setting, the Planck length is 1/℘, where ℘ is a prime number, and the Lorentz symmetry is naturally broken. In the framework of the Dirac-von Neumann formalism for quantum mechanics, we introduce a new ℘-adic Dirac equation that predicts the existence of particles and antiparticles and charge conjugation like the standard one. The discreteness of the ℘-adic space imposes substantial restrictions on the solutions of the new …

The B-Chromatic Number Of Super Cycles, Blair E. Johnson

DePaul Discoveries

The b-chromatic number, a characteristic studied in graph theory, was introduced less than 25 years ago, and while some statistics have been gathered of its properties, the b-chromatic number has yet to be explored in many graphs and circumstances. Before the Undergraduate Summer Research Program, I had no exposure to graph theory, let alone b-colorings. Throughout the program, I developed a fundamental understanding of the subject and began to pursue my own research inquiries. While this objective was difficult to attain at the beginning, I learned strategies and methods to hone my ability to delve into different graphs, and soon …

S-Preclones And The Galois Connection SPol–SInv, Part I, Peter Jipsen, Erkko Lehtonen, Reinhard Pöschel

Mathematics, Physics, and Computer Science Faculty Articles and Research

We consider S-operations f : Aⁿ → A in which each argument is assigned a signum s ∈ S representing a “property” such as being order- preserving or order-reversing with respect to a fixed partial order on A. The set S of such properties is assumed to have a monoid structure reflecting the behaviour of these properties under the composition of S-operations (e.g., order-reversing composed with order-reversing is order- preserving). The collection of all S-operations with prescribed properties for their signed arguments is not a clone (since it is not closed under arbitrary identification of arguments), …

Properties Of Gyrogroups Induced By Groups Whose Central Quotients Being 2-Engel, Jaturon Wattanapan, Teerapong Suksumran

Turkish Journal of Mathematics

A group Γ is said to be CCII if the quotient Γ/Z(Γ) is 2-Engel or, equivalently, commutator-inversion invariant, where Z(Γ) is the center of Γ. In this article, we prove algebraic and topological properties of gyrogroups that are induced by CCII groups. Then, using a classification of non-abelian groups of order n with n < 32, we determine all finite CCII groups of order less than 32.

Positive And Decreasing Solutions For Higher Order Caputo Boundary Valueproblems With Sign-Changing Green’S Function, Rian Yan, Yige Zhao, Xuan Leng, Yabing Li

Turkish Journal of Mathematics

In this paper, Caputo boundary value problems of order 3 < ζ ≤ 4 are investigated on the interval [0, 1] .By Guo-Krasnoselskii fixed point theorem, some criteria of existence and multiplicity of positive and decreasing solutionsare established. The main novelty of the paper lies in its capability to achieve positive solutions while the correspondingGreen’s function changes sign. Finally, two examples are provided to illustrate the application of these results.

On Vietoris’ Hybrid Number Sequence, Nurten Gürses, Gülsüm Yeli̇z Saçli, Sali̇m Yüce

Turkish Journal of Mathematics

This work is intended to establish a relation between Vietoris’ sequence, which is a rational sequence, andhybrid numbers. Then it provides some characteristic properties of the hybrid numbers with Vietoris’ number coefficients.Some relations between this hybrid number and its norm, the recurrence relations, the generating function, Binet-likeformula and Catalan-like identities are also indicated. Furthermore, a determinantal approach is presented to obtainelements of Vietoris’ hybrid number sequence.

Stacks In Einstein Gravity, Kadri̇ İlker Berktav

Turkish Journal of Mathematics

In this paper, we examine stacky structures in certain Einstein gravity theories. In brief, using the classicalformulation of (vacuum) gravity, with vanishing cosmological constant, we first construct the stack of solutions to Einsteinfield equations on any given fixed manifold. Using a similar approach and setup, we also study Einstein’s gravity onfamilies of manifolds and define another stack encoding this situation. Later on, we focus on the gauge theoreticalinterpretation of 3D gravity and provide a natural stack associated with that interpretation. Finally, in a particularsetup, we give a natural morphism between the two stacks arising from different descriptions of 3D gravity.

The Flow-Geodesic Curvature And The Flow-Evolute Of Spherical Curves, Mircea Crasmareanu

Turkish Journal of Mathematics

We introduce and study a deformation of the geodesic curvature for a given spherical curve γ . Also, wedefine a new type of evolute and two Fermi-Walker type derivatives for γ . Some concrete examples are detailed with aspecial attention towards space curves with a constant torsion.

Some Qualitative Results For Nonlocal Dynamic Boundary Value Problem Of Thermistor Type, Svetlin G. Georgiev, Mahammad Khuddush, Sanket Tikare

Turkish Journal of Mathematics

This paper is concerned with second-order nonlocal dynamic thermistor problem with two-point boundary conditions on time scales. By utilizing the fixed point theorems due to Schaefer and Rus, we establish some sufficient conditions for the existence and uniqueness of solutions. Further, we discuss the continuous dependence of solutions and four types of Ulam stability. We provide examples to support the applicability of our results.

Pt-Symmetry-Enabled Stable Modes In A Multicore Fiber, Tamara Gratcheva, Yogesh N. Joglekar, Jay Gopalakrishnan

Mathematics and Statistics Faculty Publications and Presentations

Open systems with balanced gain and loss, described by parity-time (PT -symmetric) Hamiltonians have been deeply explored over the past decade. Most explorations are limited to finite discrete models (in real or reciprocal spaces) or continuum problems in one dimension. As a result, these models do not leverage the complexity and variability of two-dimensional continuum problems on a compact support. Here, we investigate eigenvalues of the Schrödinger equation on a disk with zero boundary condition, in the presence of constant, PT -symmetric, gain-loss potential that is confined to two mirror-symmetric disks. We find a rich variety of exceptional points, re-entrant …

Curves As Slant Submanifolds Of An Almost Product Riemannian Manifold, Pablo Alegre, Alfonso Carriazo

Turkish Journal of Mathematics

In this paper, we show that in an almost product manifold there exist curves that are slant submanifolds. Wecharacterize these curves and study them in two and three-dimensional locally product manifolds. Finally, we constructcurves in a hypersurface of a Kaehler manifold.

Bernstein-Nikol’Skii-Markov-Type Inequalities For Algebraic Polynomials In Aweighted Lebesgue Space In Regions With Cusps, Uğur Değer, Fahreddi̇n Abdullayev

Turkish Journal of Mathematics

In this paper, we study Bernstein-Nikol’skii-Markov type inequalities for arbitrary algebraic polynomials withrespect to a weighted Lebesgue space, where the weight functions have some singularities on a given contour. We considercurves which can contain a finite number of exterior and interior corners with power law tangency of the boundary arcs atthose points where the weight functions have both zeros and poles of finite order. The estimates are given for the growthof the module of derivatives for algebraic polynomials on the closure of a region bounded by a given curve, dependingon the behavior of weight functions, on the property of curve, …

New Oscillation Criteria For First-Order Differential Equations With General Delay Argument, Emad R. Attia, Irena Jadlovska

Turkish Journal of Mathematics

This paper is concerned with the oscillation of solutions to a class of first-order differential equations withvariable coefficients and a general delay argument. New oscillation criteria are established, which improve and extendmany known results reported in the literature. A couple of illustrative examples are given to show the efficiency of thenewly obtained results. In particular, it is shown that our criteria partially fulfill a remaining gap in a recent sharp resultby Pituk et al. [31].

On A Class Of Permutation Trinomials Over Finite Fields, Burcu Gülmez Temür, Buket Özkaya

Turkish Journal of Mathematics

In this paper, we study the permutation properties of the class of trinomials of the form f(x) = x4q+1 +λ1xq+4 +λ2x2q+3 ∈ Fq2 [x] , where λ1, λ2 ∈ Fq and they are not simultaneously zero. We find all necessary and sufficientconditions on λ1 and λ2 such that f(x) permutes Fq2 , where q is odd and q = 22k+1, k ∈ N.

Interior Schauder-Type Estimates For M − Th Order Elliptic Operators Inrearrangement-Invariant Sobolev Spaces, Emi̇nağa M. Mamedov, Şeyma Çeti̇n

Turkish Journal of Mathematics

In this study, we investigate the m-th order elliptic operators on n-dimensional bounded domain Ω ⊂ Rnwith discontinuous coefficients in the rearrangement-invariant Sobolev space WmX (Ω). In general, the consideredrearrangement-invariant spaces are not separable, so the use of classical methods in these spaces requires substantialmodification of classical methods and a lot of preparation, concerning correctness of substitution operator, problemsrelated to the extension operator in such spaces, etc. For this purpose, the corresponding separable subspaces of thesespaces, in which the set of compact supported infinitely differentiable functions is dense, are introduced based on theshift operator. We establish interior Schauder-type estimates in …

Holomorphic Functional Calculus Approach To The Characteristic Function Of Quantum Observables, Andreas Boukas

Journal of Stochastic Analysis

No abstract provided.

Torus Surgery, Fibrations, Multisections, And Spun 4-Manifolds, Nicholas Paul Meyer

Dissertations and Doctoral Documents from University of Nebraska-Lincoln, 2023–

A compact n-manifold X is fibered if it is a fiber bundle where the fiber F and base space B are manifolds. Fibered manifolds are particularly nice, as they are essentially classified by their monodromy maps. Two common examples of 4-dimensional fibered manifolds are surface bundles over surfaces and 3-manifold bundles over the circle.

The main focus of this dissertation is to investigate fibered 4-manifolds whose boundaries are the 3-torus and how these manifolds glue together to give new closed, fibered 4-manifolds. In particular, suppose W is diffeomorphic to S¹ × E_Y (K) where Y …

Nonlocal Frameworks For Nonlinear Conservation Laws And Advection-Diffusion Processes, Anh Thuong Vo

Dissertations and Doctoral Documents from University of Nebraska-Lincoln, 2023–

Conservation laws are fundamental principles that play an important role in modeling various phenomena in physics, chemistry, and biology. However, their limitations, such as the development of shocks despite smooth initial conditions, are well known. The nonlocal model framework can be used to overcome these challenges. Nonlocal frameworks utilize integral operators that mimic differential operators but also incorporate long-range interactions within a finite horizon. This approach not only allows for non-smooth solutions, but also provides flexibility in modeling different phenomena. This study investigates the convergence of nonlocal divergence operators, defined with a general flux density function, to their classical counterparts. …

Stochastic Dominance: Cases Of Interval And P-Box Uncertainty, Kittawit Autchariyapanikul, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Traditional decision theory recommendation about making a decision assume that we know both the probabilities of different outcomes of each possible decision, and we know the utility function -- that describes the decision maker's preferences. Sometimes, we can make a recommendation even when we only have partial information about utility. Such cases are known as cases of stochastic dominance. In other cases, in addition to not knowing the utility function, we also only have partial information about the probabilities of different outcomes. For example, we may only known bounds on the outcomes (case of interval uncertainty) or bounds on the …

If Subsequent Results Are Too Easy To Obtain, The Proof Most Probably Has Errors: Explanation Of The Empirical Observation, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Many modern mathematical proofs are very complex, checking them is difficult; as a result, errors sneak into published proofs, even into proofs published in highly reputable journals. Sometimes, the errors are repairable, but sometimes, it turns out that the supposedly proven result is actually wrong. When the error is not noticed for some time, the published result is used to prove many other results -- and when the error is eventually found, all these new results are invalidated. This happened several times. Since it is not realistic to more thoroughly check all the proofs, and we want to minimize the …

For 2 X N Cases, Proportional Fitting Problem Reduces To A Single Equation, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical situations, for each of two classifications, we know the probabilities that a randomly selected object belong to different categories. For example, we know what proportion of people are below 20 years old, what proportion is between 20 and 30, etc., and we also know what proportion of people earns less than 10K, between 10K and 20K, etc. In such situations, we are often interested in proportion of people who are classified by two classifications into two given categories. For example, we are interested in the proportion of people whose age is between 20 and 30 and whose …

Is Alaska Negative-Tax Arrangement Fair? Almost: Mathematical Analysis, Chon Van Le, Vladik Kreinovich

Departmental Technical Reports (CS)

In the State of Alaska there is no state income tax. Instead, there is a negative tex: every year every resident gets some money from the state. At present, every resident -- from the poorest to the richest -- gets the exact same amount of money: in 2024, it is expected to be around $1500. A natural question is: Is this fair? Maybe poor people should get more since their needs are greater? Maybe the rich people should get proportionally more, since fairness means equal added happiness for all, and for rich people, extra $1500 is barely noticeable? There have …

Why Angles Between Galactic Center Filaments And Galactic Plane Follow A Bimodal Distribution: A Symmetry-Based Explanation, Julio C. Urenda, Vladik Kreinovich

Departmental Technical Reports (CS)

Recent observations have shown that the angles between the Galaxy Center filaments and the Galactic plane follow a bimodal distribution: a large number of filaments are approximately orthogonal to the Galactic plane, a large number of filaments are approximately parallel to the Galactic plane, and much fewer filaments have other orientations. In this paper, we show this bimodal distribution can be explained by natural geometric symmetries.

Why Seismicity In Ireland Is Low: A Possible Geometric Explanation, Julio C. Urenda, Aaron Velasco, Vladik Kreinovich

Departmental Technical Reports (CS)

For each geographic location, its seismicity level is usually determined by how close this location is to the boundaries of tectonic plates. However, there is one notable exception: while Ireland and Britain are at approximately the same distance from such boundaries, the seismicity level in Ireland is much lower than in Britain. A recent paper provided a partial explanation for this phenomenon: namely, it turns out that the lithosphere under Ireland is unusually thick, and this can potentially lead to lower seismicity. However, the current explanation of the relation between the lithosphere thickness and seismicity level strongly depends on the …

Mega-Influencers And Brand Dynamics: Shaping Attitudes Toward Leading And Challenger Brandsthrough Electronic Word Of Mouth, Riccardo Rialto, Lamberto Zollo, Kacy Kim, Sukki Yoon

Mathematics and Economics Faculty Journal Articles

The aim of this paper is to explore how mega‐influencers' electronic word of mouth (eWOM) messages on social media influence consumers' brand attitudes in duopolistic markets. Through three experimental studies, we observe that when mega‐ influencers send positive (vs. negative) eWOM messages about a leading brand, followers form positive (vs. negative) brand attitudes, but these effects fail to occur when influencers back challenger brands. The findings are consistent across three duopolistic market rivals (Apple vs. Samsung; UPS vs. FedEx; Nike vs. Adidas), three social media platforms (Facebook, Instagram, and X), and four mega‐influencers (Marques Brownlee, Gary Vaynerchuk, Kanye West, and …

Math 75: Introduction To Linear Algebra, Sarah K. Merz

Pacific Open Texts

This text is intended to use in a first course of Linear Algebra with a prerequisite of Calculus 1. Topics covered include systems of linear equations, matrix operations and inverses, linear transformations, Markov chains, determinants, eigenvalues and eigenvectors, diagonalization, vector geometry, projections and planes, homogeneous coordinates, subspaces, spanning sets, linear independence, orthogonality, fundamental subspaces, and least squares.