Physical Sciences and Mathematics Commons^™

Local Existence Of Solutions To A Nonlinear Autonomous Pde Model For Population Dynamics With Nonlocal Transport And Competition, Michael R. Lindstrom

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

Highlights

• Partial differential equation models are ubiquitous in applied sciences.

• A partial differential equation based in ecology is studied for solution existence.

• Energy methods and convergence analysis lead to local classical solutions.

Abstract

In this paper, we prove that a particular nondegenerate, nonlinear, autonomous parabolic partial differential equation with nonlocal mass transfer admits the local existence of classical solutions. The equation was developed to qualitatively describe temporal changes in population densities over space through accounting for location desirability and fast, long-range travel. Beginning with sufficiently regular initial conditions, through smoothing the PDE and employing energy arguments, we obtain a sequence …

Weighted Ehrhart Theory: Extending Stanley's Nonnegativity Theorem, Esme Bajo, Robert Davis, Jesús A. De Loera, Alexey Garber, Sofía Garzón Mora, Katharina Jochemko, Josephine Yu

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

We generalize R. P. Stanley's celebrated theorem that the h∗-polynomial of the Ehrhart series of a rational polytope has nonnegative coefficients and is monotone under containment of polytopes. We show that these results continue to hold for weighted Ehrhart series where lattice points are counted with polynomial weights, as long as the weights are homogeneous polynomials decomposable as sums of products of linear forms that are nonnegative on the polytope. We also show nonnegativity of the h∗-polynomial as a real-valued function for a larger family of weights.

We generalize R. P. Stanley's celebrated theorem that the h ⁎ -polynomial of …

Ramanujan Type Congruences For Quotients Of Klein Forms, Timothy Huber, Nathaniel Mayes, Jeffery Opoku, Dongxi Ye

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

In this work, Ramanujan type congruences modulo powers of primes p≥5 are derived for a general class of products that are modular forms of level p. These products are constructed in terms of Klein forms and subsume generating functions for t-core partitions known to satisfy Ramanujan type congruences for p=5,7,11. The vectors of exponents corresponding to products that are modular forms for Γ1(p) are subsets of bounded polytopes with explicit parameterizations. This allows for the derivation of a complete list of products that are modular forms for Γ1(p) of weights 1≤k≤5 for primes 5≤p≤19 and whose Fourier coefficients …

Study Of Hybrid Nanofluid Flow In A Stationary Cone-Disk System With Temperature-Dependent Fluid Properties, A. S. John, Mahanthesh Basavarajappa, G. Lorenzini

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

Cone-disk systems find frequent use such as conical diffusers, medical devices, various rheometric, and viscosimetry applications. In this study, we investigate the three-dimensional flow of a water-based Ag-MgO hybrid nanofluid in a static cone-disk system while considering temperature-dependent fluid properties. How the variable fluid properties affect the dynamics and heat transfer features is studied by Reynolds’s linearized model for variable viscosity and Chiam’s model for variable thermal conductivity. The single-phase nanofluid model is utilized to describe convective heat transfer in hybrid nanofluids, incorporating the experimental data. This model is developed as a coupled system of convective-diffusion equations, encompassing the conservation …

The Invariantring Package For Macaulay2, Luigi Ferraro, Federico Galetto, Francesca Gandini, Hang Huang, Matthew Mastroeni, Xianglong Ni

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

We describe a significant update to the existing InvariantRing package for Macaulay2. In addition to expanding and improving the methods of the existing package for actions of finite groups, the updated package adds functionality for computing invariants of diagonal actions of tori and finite abelian groups, as well as invariants of arbitrary linearly reductive group actions. The implementation of the package has been completely overhauled with the aim of serving as a unified resource for invariant theory computations in Macaulay2.

Algebraic Structures On Parallelizable Manifolds, Sergey Grigorian

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

In this paper we explore algebraic and geometric structures that arise on parallelizable manifolds. Given a parallelizable manifold L, there exists a global trivialization of the tangent bundle, which defines a map ρp : l −→ TpL for each point p ∈ L, where l is some vector space. This allows us to define a particular class of vector fields, known as fundamental vector fields, that correspond to each element of l. Furthermore, flows of these vector fields give rise to a product between elements of l and L, which in turn induces a local loop structure (i.e. a non-associative …

Numerical Simulations For Fractional Differential Equations Of Higher Order And A Wright-Type Transformation, Mariana Nacianceno, Tamer Oraby, Hansapani Rodrigo, Y. Sepulveda, Josef A. Sifuentes, Erwin Suazo, T. Stuck, J. Williams

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

In this work, a new relationship is established between the solutions of higher fractional differential equations and a Wright-type transformation. Solutions could be interpreted as expected values of functions in a random time process. As applications, we solve the fractional beam equation, fractional electric circuits with special functions as external sources, and derive d’Alembert’s formula for the fractional wave equation. Due to this relationship, we present two methods for simulating solutions of fractional differential equations. The two approaches use the interpretation of the Caputo derivative of a function as a Wright-type transformation of the higher derivative of the function. In …

Finite Element Solution Of Crack-Tip Fields For An Elastic Porous Solid With Density-Dependent Material Moduli And Preferential Stiffness, Hyun C. Yoon, S. M. Mallikarjunaiah, Dambaru Bhatta

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

In this paper, the finite element solutions of crack-tip fields for an elastic porous solid with density-dependent material moduli are presented. Unlike the classical linearized case in which material parameters are globally constant under a small strain regime, the stiffness of the model presented in this paper can depend upon the density with a modeling parameter. The proposed constitutive relationship appears linear in the Cauchy stress and linearized strain independently. From a subclass of the implicit constitutive relation, the governing equation is bestowed via the balance of linear momentum, resulting in a quasi-linear partial differential equation (PDE) system. Using the …

Using A Two-Way Engagement Community- And Family-Centered Pedagogy To Prepare Pre-Service Mathematics Teachers In A Hispanic-Serving Institution, Olga Ramirez, Mayra Ortiz Galarza, Luis M. Fernandez

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

Research on effective methods to prepare pre-service teachers (PSTs) in teaching mathematics to K-12 Latin* students has been gaining significant momentum. These efforts have focused, in part, on promoting pedagogical practices that recognize and incorporate the culture and language that K-12 Latin* students and their communities share. As teacher educators, we argue that if we are to further prepare PSTs to serve the needs of such increasingly diversifying K-12 student population, the same pedagogical focus on the learner’s cultural wealth should also be applied to the preparation of PSTs themselves, especially among Latin* PSTs in Hispanic Serving Institutions (HSI) like …

Functional Data Learning Using Convolutional Neural Networks, Jose Galarza, Tamer Oraby

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

In this paper, we show how convolutional neural networks (CNNs) can be used in regression and classification learning problems for noisy and non-noisy functional data (FD). The main idea is to transform the FD into a 28 by 28 image. We use a specific but typical architecture of a CNN to perform all the regression exercises of parameter estimation and functional form classification. First, we use some functional case studies of FD with and without random noise to showcase the strength of the new method. In particular, we use it to estimate exponential growth and decay rates, the bandwidths of …

Gauss Circle Problem Over Smooth Integers, Ankush Goswami

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

For a positive integer 𝑛, let 𝑟2(𝑛) be the number of representations of 𝑛 as sums of two squares (of integers), where the convention is that different signs and different orders of the summands yield distinct representations. A famous result of Gauss shows that 𝑅(𝑥) ∶= ∑ 𝑛≤𝑥 𝑟2(𝑛) ∼ 𝜋𝑥. Let 𝑃(𝑛) denote the largest prime factor of 𝑛 and let 𝑆(𝑥, 𝑦) ∶= {𝑛 ≤ 𝑥 ∶ 𝑃(𝑛) ≤ 𝑦}. In this paper, we study the asymptotic behavior of 𝑅(𝑥, 𝑦) ∶= ∑ 𝑛∈𝑆(𝑥,𝑦) 𝑟2(𝑛) for various ranges of 2 ≤ 𝑦 ≤ 𝑥. For 𝑦 in a …

Conditional Optimal Sets And The Quantization Coefficients For Some Uniform Distributions, Evans Nyanney, Megha Pandey, Mrinal Kanti Roychowdhury

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

Bucklew and Wise (1982) showed that the quantization dimension of an absolutely continuous probability measure on a given Euclidean space is constant and equals the Euclidean dimension of the space, and the quantization coefficient exists as a finite positive number. By giving different examples, in this paper, we have shown that the quantization coefficients for absolutely continuous probability measures defined on the same Euclidean space can be different. We have taken uniform distribution as a prototype of an absolutely continuous probability measure. In addition, we have also calculated the conditional optimal sets of n-points and the nth conditional quantization errors …

A Finite Element Model For Hydro-Thermal Convective Flow In A Porous Medium: Effects Of Hydraulic Resistivity And Thermal Diffusivity, S. M. Mallikarjunaiah, Dambaru Bhatta

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

In this article, a finite element model is implemented to analyze hydro-thermal convective flow in a porous medium. The mathematical model encompasses Darcy’s law for incompressible fluid behavior, which is coupled with a convection-diffusion-type energy equation to characterize the temperature in the porous medium. The current investigation presents an efficient, stable, and accurate finite element discretization for the hydro-thermal convective flow model. The well-posedness of the proposed discrete Galerkin finite element formulation is guaranteed due to the decoupling property and the linearity of the numerical method. Computational experiments confirm the optimal convergence rates for a manufactured solution. Several numerical results …

On Modeling Arterial Blood Flow With Or Without Solute Transport And In Presence Of Atherosclerosis, Daniel N. Riahi, Saulo Orizaga

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

In this article, we review previous studies of modeling problems for blood flow with or without transport of a solute in a section of arterial blood flow and in the presence of atherosclerosis. Moreover, we review problems of bio-fluid dynamics within the field of biophysics. In most modeling cases, the presence of red blood cells in the plasma is taken into account either by using a two-phase flow approach, where blood plasma is considered as one phase and red blood cells are counted as another phase, or by using a variable viscosity formula that accounts for the amount of hematocrit …

Constrained Quantization For The Cantor Distribution With A Family Of Constraints, Megha Pandey, Mrinal Kanti Roychowdhury

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

In this paper, for a given family of constraints and the classical Cantor distribution we determine the constrained optimal sets of n-points, nth constrained quantization errors for all positive integers n. We also calculate the constrained quantization dimension and the constrained quantization coefficient, and see that the constrained quantization dimension D(P) exists as a finite positive number, but the D(P)-dimensional constrained quantization coefficient does not exist.

Order-2 Delaunay Triangulations Optimize Angles, Herbert Edelsbrunner, Alexey Garber, Morteza Saghafian

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

The local angle property of the (order-1) Delaunay triangulations of a generic set in R2 asserts that the sum of two angles opposite a common edge is less than π. This paper extends this property to higher order and uses it to generalize two classic properties from order-1 to order-2: (1) among the complete level-2 hypertriangulations of a generic point set in R2, the order-2 Delaunay triangulation lexicographically maximizes the sorted angle vector; (2) among the maximal level-2 hypertriangulations of a generic point set in R2, the order-2 Delaunay triangulation is the only one that has the local angle property. …

Modeling The Effect Of Observational Social Learning On Parental Decision-Making For Childhood Vaccination And Diseases Spread Over Household Networks, Tamer Oraby, Andras Balogh

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

In this paper, we introduce a novel model for parental decision-making about vaccinations against a childhood disease that spreads through a contact network. This model considers a bilayer network comprising two overlapping networks, which are either Erdős–Rényi (random) networks or Barabási–Albert networks. The model also employs a Bayesian aggregation rule for observational social learning on a social network. This new model encompasses other decision models, such as voting and DeGroot models, as special cases. Using our model, we demonstrate how certain levels of social learning about vaccination preferences can converge opinions, influencing vaccine uptake and ultimately disease spread. In addition, …

Conditional Constrained And Unconstrained Quantization For Probability Distributions, Megha Pandey, Mrinal Kanti Roychowdhury

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

In this paper, we present the idea of conditional quantization for a Borel probability measure P on a normed space Rk. We introduce the concept of conditional quantization in both constrained and unconstrained scenarios, along with defining the conditional quantization errors, dimensions, and coefficients in each case. We then calculate these values for specific probability distributions. Additionally, we demonstrate that for a Borel probability measure, the lower and upper quantization dimensions and coefficients do not depend on the conditional set of the conditional quantization in both constrained and unconstrained quantization.

Structure Of Fine Selmer Groups In Abelian P-Adic Lie Extensions, Debanjana Kundu, Filippo Alberto Edoardo Nuccio Mortarino Majno Di Capriglio, Sujatha Ramdorai

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

This paper studies fine Selmer groups of elliptic curves in abelian p -adic Lie extensions. A class of elliptic curves are provided where both the Selmer group and the fine Selmer group are trivial in the cyclotomic Z p -extension. The fine Selmer groups of elliptic curves with complex multiplication are shown to be pseudonull over the trivializing extension in some new cases. Finally, a relationship between the structure of the fine Selmer group for some CM elliptic curves and the Generalized Greenberg's Conjecture is clarified.

Deep Neural Networks: A Formulation Via Non-Archimedean Analysis, Wilson A. Zuniga-Galindo

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

We introduce a new class of deep neural networks (DNNs) with multilayered tree-like architectures. The architectures are codified using numbers from the ring of integers of non-Archimdean local fields. These rings have a natural hierarchical organization as infinite rooted trees. Natural morphisms on these rings allow us to construct finite multilayered architectures. The new DNNs are robust universal approximators of real-valued functions defined on the mentioned rings. We also show that the DNNs are robust universal approximators of real-valued square-integrable functions defined in the unit interval.

Conditional Quantization For Uniform Distributions On Line Segments And Regular Polygons, Pigar Biteng, Mathieu Caguiat, Tsianna Dominguez, Mrinal Kanti Roychowdhury

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

Quantization for a Borel probability measure refers to the idea of estimating a given probability by a discrete probability with support containing a finite number of elements. If in the quantization some of the elements in the support are preselected, then the quantization is called a conditional quantization. In this paper, we have investigated the conditional quantization for the uniform distributions defined on the unit line segments and m-sided regular polygons, where m≥3, inscribed in a unit circle.

Extensions Of Polynomial Plank Covering Theorems, Alexey Glazyrin, Roman Karasev, Alexandr Polyanskii

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

We prove the complex polynomial plank covering theorem for not necessarily homogeneous polynomials. As the consequence of this result, we extend the complex plank theorem of Ball to the case of planks that are not necessarily centrally symmetric and not necessarily round. We also prove a weaker version of the spherical polynomial plank covering conjecture for planks of different widths.

Symmetries And Integrable Systems, Sen-Yue Lou, Bao-Feng Feng

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

Symmetry plays key roles in modern physics especially in the study of integrable systems because of the existence of infinitely many local and nonlocal generalized symmetries. In addition to the fundamental role to find exact group invariant solutions via Lie point symmetries, some important new developments on symmetries and conservation laws are reviewed. The recursion operator method is important to find infinitely many local and nonlocal symmetries of (1+1)-dimensional integrable systems. In this paper, it is pointed out that a recursion operator may be obtained from one key symmetry, say, a residual symmetry. For (2+1)-dimensional integrable systems, the master-symmetry approach …

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

We introduce a new p-adic Dirac equation that predicts the existence of particles and antiparticles and charge conjugation like the standard one. The new equation shares many properties with the old one. However, the space's discrete (p-adic) nature imposes substantial restrictions on the solutions of the new equation. This equation admits localized solutions, which is impossible in the standard case. Finally, we show that a quantum system whose evolution is controlled by the p-adic Dirac equation does not satisfy the Einstein causality.

Exact Solutions Of Stochastic Burgers-Kdv Equation With Variable Coefficients, Kolade Adjibi, Allan Martinez, Miguel Mascorro, Carlos Montes, Tamer Oraby, Rita Sandoval, Erwin Suazo

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

We will present exact solutions for three variations of stochastic Korteweg de Vries-Burgers (KdV-Burgers) equation featuring variable coefficients. In each variant, white noise exhibits spatial uniformity, and the three categories include additive, multiplicative, and advection noise. Across all cases, the coefficients are time-dependent functions. Our discovery indicates that solving certain deterministic counterparts of KdV-Burgers equations and composing the solution with a solution of stochastic differential equations leads to the exact solution of the stochastic Korteweg de Vries-Burgers (KdV-Burgers) equations.

Turing Patterns In A P-Adic Fitzhugh-Nagumo System On The Unit Ball, L. F. Chacón-Cortés, C. A. Garcia-Bibiano, Wilson A. Zuniga-Galindo

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

We introduce discrete and p-adic continuous versions of the FitzHugh-Nagumo system on the one-dimensional p-adic unit ball. We provide criteria for the existence of Turing patterns. We present extensive simulations of some of these systems. The simulations show that the Turing patterns are traveling waves in the p-adic unit ball.

Limit Distributions Of Products Of Independent And Identically Distributed Random 2 × 2 Stochastic Matrices: A Treatment With The Reciprocal Of The Golden Ratio, Santanu Chakraborty

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

Consider a sequence (Xn)n≥1 of i.i.d. 2×2 stochastic matrices with each Xn distributed as μ. This μ is described as follows. Let (Cn,Dn)T denote the first column of Xn and for a given real r with 012 is very challenging. Considering the extreme nontriviality of this case, we stick to a very special such r, namely, r=√5−12 (the reciprocal of the golden ratio), briefly mention the challenges in this nontrivial case, and completely identify λ for a very special situation.

The Tor Algebra Of Trimmings Of Gorenstein Ideals, Luigi Ferraro, Alexis Hardesty

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

Let (R,\mathfrak m,\Bbbk ) be a regular local ring of dimension 3. Let I be a Gorenstein ideal of R of grade 3. Buchsbaum and Eisenbud proved that there is a skew-symmetric matrix of odd size such that I is generated by the sub-maximal pfaffians of this matrix. Let J be the ideal obtained by multiplying some of the pfaffian generators of I by \mathfrak m; we say that J is a trimming of I. Building on a recent paper of Vandebogert, we construct an explicit free resolution of R/J and compute a partial DG algebra structure on this …

Time Series Based Road Traffic Accidents Forecasting Via Sarima And Facebook Prophet Model With Potential Changepoints, Edmund F. Agyemang, Joseph A. Mensah, Eric Ocran, Enock Opoku, Ezekiel N.N. Nortey

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

Road traffic accident (RTA) is a critical global public health concern, particularly in developing countries. Analyzing past fatalities and predicting future trends is vital for the development of road safety policies and regulations. The main objective of this study is to assess the effectiveness of univariate Seasonal Autoregressive Integrated Moving Average (SARIMA) and Facebook (FB) Prophet models, with potential change points, in handling time-series road accident data involving seasonal patterns in contrast to other statistical methods employed by key governmental agencies such as Ghana's Motor Transport and Traffic Unit (MTTU). The aforementioned models underwent training with monthly RTA data spanning …

Rigidity Of Ext And Tor Via Flat–Cotorsion Theory, Lars Winther Christensen, Luigi Ferraro, Peder Thompson

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

Let p be a prime ideal in a commutative noetherian ring R and denote by k(p) the residue field of the local ring R_p. We prove that if an R-module M satisfies Ext_R^n(k(p),M) = 0 for some n >= dim R, then Ext_R^i(k(p),M) = 0 holds for all i >= n. This improves a result of Christensen, Iyengar, and Marley by lowering the bound on n. We also improve existing results on Tor-rigidity. This progress is driven by the existence of minimal semi-flat-cotorsion replacements in the derived category as recently proved by Nakamura and Thompson.