Mathematics | Open Access Articles | Digital Commons Network™

Graphs Without A 2c3-Minor And Bicircular Matroids Without A U3,6-Minor, Daniel Slilaty

Mathematics and Statistics Faculty Publications

In this note we characterize all graphs without a 2C3-minor. A consequence of this result is a characterization of the bicircular matroids with no U3,6-minor.

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The Multiset Partition Algebra: Diagram-Like Bases And Representations, Alexander N. Wilson

Dartmouth College Ph.D Dissertations

There is a classical connection between the representation theory of the symmetric group and the general linear group called Schur--Weyl Duality. Variations on this principle yield analogous connections between the symmetric group and other objects such as the partition algebra and more recently the multiset partition algebra. The partition algebra has a well-known basis indexed by graph-theoretic diagrams which allows the multiplication in the algebra to be understood visually as combinations of these diagrams. My thesis begins with a construction of an analogous basis for the multiset partition algebra. It continues with applications of this basis to constructing the irreducible …

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Adapted Metrics For A Modified Coulomb/Newton’S Potential, Lucian M. Ionescu, Cristina-Liliana Pripoae, Gabriel-Teodor Pripoae

Faculty Publications – Mathematics

Modified Theories of Gravity include spin dependence in General Relativity, to account for additional sources of gravity instead of dark matter/energy approach. The spin-spin interaction is already included in the effective nuclear force potential, and theoretical considerations and experimental evidence hint to the hypothesis that Gravity originates from such an interaction, under an averaging process over spin directions. This invites to continue the line of theory initiated by Einstein and Cartan, based on tetrads and spin effects modeled by connections with torsion. As a first step in this direction, the article considers a new modified Coulomb/Newton Law accounting for the …

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Predation And Harvesting In Spatial Population Models, Connor R. Shrader

Honors Undergraduate Theses

Predation and harvesting play critical roles in maintaining biodiversity in ecological communities. Too much harvesting may drive a species to extinction, while too little harvesting may allow a population to drive out competing species. The spatial features of a habitat can also significantly affect population dynamics within these communities. Here, we formulate and analyze three ordinary differential equation models for the population density of a single species. Each model differs in its assumptions about how the species is harvested. We then extend each of these models to analogous partial differential equation models that more explicitly describe the spatial habitat and …

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Clustering Of Diverse Multiplex Networks, Yaxuan Wang

Electronic Theses and Dissertations, 2020-2023

This dissertation introduces the DIverse MultiPLEx Generalized Dot Product Graph (DIMPLE-GDPG) network model where all layers of the network have the same collection of nodes and follow the Generalized Dot Product Graph (GDPG) model. In addition, all layers can be partitioned into groups such that the layers in the same group are embedded in the same ambient subspace but otherwise all matrices of connection probabilities can be different. In common particular cases, where layers of the network follow the Stochastic Block Model (SBM) and Degree Corrected Block Model (DCBM), this setting implies that the groups of layers have common community …

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Finding The Common Denominator: Understanding The Shared Experiences Of Female Math Majors, Abigail R. Rosenbaum

Honors Theses

Despite efforts to increase gender diversity in STEM fields, women remain underrepresented in mathematics, especially in advanced academic and research positions. This study aimed to explore the experiences of female math majors as they attempt to navigate this male-dominated space. Through qualitative interviews with seven female math majors, two female math professors, and a focus group with education majors at Woodbridge College, small liberal arts college in the United States, several common themes were identified that define the experiences of female math majors. The findings suggest that math is held at an elevated status in society and that there is …

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Elliptic Curves Over Finite Fields, Christopher S. Calger

Honors Theses

The goal of this thesis is to give an expository report on elliptic curves over finite fields. We begin by giving an overview of the necessary background in algebraic geometry to understand the definition of an elliptic curve. We then explore the general theory of elliptic curves over arbitrary fields, such as the group structure, isogenies, and the endomorphism ring. We then study elliptic curves over finite fields. We focus on the number of F_q-rational solutions, Tate modules, supersingular curves, and applications to elliptic curves over Q. In particular, we approach the topic largely through the use …

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Advances In Differentially Methylated Region Detection And Cure Survival Models, Daniel Ahmed Alhassan

Doctoral Dissertations

"This dissertation focuses on two areas of statistics: DNA methylation and survival analysis. The first part of the dissertation pertains to the detection of differentially methylated regions in the human genome. The varying distribution of gaps between succeeding genomic locations, which are represented on the microarray used to quantify methylation, makes it challenging to identify regions that have differential methylation. This emphasizes the need to properly account for the correlation in methylation shared by nearby locations within a specific genomic distance. In this work, a normalized kernel-weighted statistic is proposed to obtain an optimal amount of "information" from neighboring locations …

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Minimal Sets, Union-Closed Families, And Frankl's Conjecture, Christopher S. Flippen

Theses and Dissertations

The most common statement of Frankl's conjecture is that for every finite family of sets closed under the union operation, there is some element which belongs to at least half of the sets in the family. Despite its apparent simplicity, Frankl's conjecture has remained open and highly researched since its first mention in 1979. In this paper, we begin by examining the history and previous attempts at solving the conjecture. Using these previous ideas, we introduce the concepts of minimal sets and minimally-generated families, some ideas related to viewing union-closed families as posets, and some constructions of families involving poset-defined …

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The Mceliece Cryptosystem As A Solution To The Post-Quantum Cryptographic Problem, Isaac Hanna

Senior Honors Theses

The ability to communicate securely across the internet is owing to the security of the RSA cryptosystem, among others. This cryptosystem relies on the difficulty of integer factorization to provide secure communication. Peter Shor’s quantum integer factorization algorithm threatens to upend this. A special case of the hidden subgroup problem, the algorithm provides an exponential speedup in the integer factorization problem, destroying RSA’s security. Robert McEliece’s cryptosystem has been proposed as an alternative. Based upon binary Goppa codes instead of integer factorization, his cryptosystem uses code scrambling and error introduction to hinder decrypting a message without the private key. This …

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Analytical And Numerical Negative Boundedness Of Fractional Differences With Mittag–Leffler Kernel, Pshtiwan Othman Mohammed, Rajendra Dahal, Christopher S. Goodrich, Y. S. Hamed, Dumitru Baleanu

Mathematics and Statistics

We show that a class of fractional differences with Mittag–Leffler kernel can be negative and yet monotonicity information can still be deduced. Our results are complemented by numerical approximations. This adds to a growing body of literature illustrating that the sign of a fractional difference has a very complicated and subtle relationship to the underlying behavior of the function on which the fractional difference acts, regardless of the particular kernel used.

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Duality For Spaces Of Polynomials On Tree Spaces, Christopher Boyd, Costas Poulios, Milena Venkova

Articles

In this paper we study the duality of certain spaces of polynomials on a class of tree spaces. In particular, we are interested in establishing isometric isomorphisms between the spaces of integral and nuclear polynomials and in determining when all polynomials are weakly continuous on bounded sets. This will provide us with new examples of what are known as Q-reflexive spaces.

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On The Cubic Equation With Its Siebeck–Marden–Northshield Triangle And The Quartic Equation With Its Tetrahedron, Emil Prodanov

Articles

The real roots of the cubic and quartic polynomials are studied geometrically with the help of their respective Siebeck–Marden–Northshield equilateral triangle and regular tetrahedron. The Viète trigonometric formulæ for the roots of the cubic are established through the rotation of the triangle by variation of the free term of the cubic. A very detailed complete root classification for the quartic 𝑥4 + 𝑎𝑥3 + 𝑏𝑥2 + 𝑐𝑥 + 𝑑 is proposed for which the conditions are imposed on the individual coefficients 𝑎, 𝑏, 𝑐, and 𝑑. The maximum and minimum lengths of the interval containing the four real roots of …

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A Study Of The Local Deep Galerkin Method For The Modified Cahn Hilliard Equation, Shi Wen Wong

Electronic Theses and Dissertations

Solving higher order partial differential equations (PDEs) can often prove to be a challenging task due to the involvement of higher-order derivatives of the unknown function, particularly for complex problems. The higher the order of the PDE, the more challenging it becomes to obtain an analytical solution. In such cases, alternative numerical methods are often used, such as finite element method or finite difference method. However, these methods can be computationally expensive and require a significant amount of mathematical expertise to implement. In recent times, there has been significant progress in applying neural networks to various fields, including the solution …

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Characterizations And Representations Of Weak Core Inverses And $M$-Weak Group Inverses, Wende Li, Jianlong Chen, Yukun Zhou

Turkish Journal of Mathematics

In a ring with an involution, we first present some necessary and sufficient conditions for the existence of the $m$-weak group inverse and expression. As an application, we prove that a regular element $a$ is $(m+1)$-weak group invertible if and only if $a^2a^{-}$ is $m$-weak group invertible, where $a^{-}$ is an inner inverse of $a$. The relevant results for weak core inverses and for pseudocore inverses are given. In addition, we present some new characterizations of weak core inverses, and also investigate maximal classes

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The Adjoint Reidemeister Torsion For Compact 3-Manifolds Admitting A Unique Decomposition, Esma Di̇ri̇can Erdal

Turkish Journal of Mathematics

Let $M$ be a triangulated, oriented, connected compact $3$-manifold with a connected nonempty boundary. Such a manifold admits a unique decomposition into $\triangle$-prime $3$-manifolds. In this paper, we show that the adjoint Reidemeister torsion has a multiplicative property on the disk sum decomposition of compact $3$-manifolds without a corrective term.

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On The Modelling Of Short And Intermediate Water Waves, Rossen I. Ivanov

Articles

The propagation of water waves of finite depth and flat bottom is studied in the case when the depth is not small in comparison to the wavelength. This propagation regime is complementary to the long-wave regime described by the famous KdV equation. The Hamiltonian approach is employed in the derivation of a model equation in evolutionary form, which is both nonlinear and nonlocal, and most likely not integrable. Possible implications for the numerical solutions are discussed.

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Math Department 2022-2023 Newsletter, Mathematics Department

Mathematics Newsletter

Welcome to our annual Mathematics Department Newsletter. We had an exciting year with many activities and accomplishments. Our students have received awards, attended conferences, conducted research, and have really contributed to the life of the department. Our faculty have had many accomplishments in teaching and scholarship. We conducted several events such as Alumni Night, hosted a guest speaker for the university community, had our Pi Mu Epsilon induction and awards ceremony, and much more. We have heard from several alumni and are happy to report what they are doing. Our Math Club has been becoming more active. On the curriculum …

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Generalization Of Statistical Limit-Cluster Points And The Concepts Of Statistical Limit Inferior-Superior On Time Scales By Using Regular Integral Transformations, Ceylan Yalçin

Turkish Journal of Mathematics

With the aid of regular integral operators, we will be able to generalize statistical limit-cluster points and statistical limit inferior-superior ideas on time scales in this work. These two topics, which have previously been researched separately from one another sometimes only in the discrete case and other times in the continuous case, will be studied at in a single study. We will investigate the relations of these concepts with each other and come to a number of new conclusions. On some well-known time scales, we shall analyze these ideas using examples.

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Novel Results On Trapezoid-Type Inequalities For Conformable Fractional Integrals, Fati̇h Hezenci̇, Hüseyi̇n Budak

Turkish Journal of Mathematics

This paper establishes an identity for the case of differentiable $s-$convex functions with respect to the conformable fractional integrals. By using this identity, sundry trapezoid-type inequalities are proven by $s-$convex functions with the help of the conformable fractional integrals. Several important inequalities are acquired with taking advantage of the convexity, the Hölder inequality, and the power mean inequality. Moreover, an example using graph is given in order to show that our main results are correct. By using the special choices of the obtained results, we present several new results connected with trapezoid-type inequalities.

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Twisted Dirac Operators And The Kastler-Kalau-Walze Type Theorem For Five Dimensional Manifolds With Boundary, Tong Wu, Sining Wei, Yong Wang

Turkish Journal of Mathematics

In this paper, we prove the Kastler-Kalau-Walze type theorems for twisted Dirac operators on 5-dimensional manifolds with boundary.

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A Note On $Ss$-Supplement Submodules, Emi̇ne Önal Kir

Turkish Journal of Mathematics

In this paper, we describe $ss$-supplement submodules in terms of a special class of endomorphisms. Let $R$ be a ring with semisimple radical and $P$ be a projective $R-$module. We show that there is a bijection between ss-supplement submodules of $P$ and ss-supplement submodules of $End_{R}(P)$. Moreover, we define radical-s-projective modules as a generalization of projective modules. We prove that every $ss$-supplement submodule of a projective $R-$module is radical-s-projective over the ring $R$ with semisimple radical. We show that over $SSI$-ring $R$, every radical-s-projective $R-$module is projective. We provide that over a ring $R$ with semisimple radical, every $ss$-supplement submodule …

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Qualitative Study Of A Second Order Difference Equation, Messaoud Berkal, Juan Francisco Navarro

Turkish Journal of Mathematics

In this paper, we study a second order rational difference equation. We analyze the stability of the unique positive equilibrium of the equation and prove the existence of a Neimark-Sacker bifurcation, validating our theoretical analysis via a numerical exploration of the system.

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Numerical Solution For Benjamin-Bona-Mahony-Burgers Equation With Strang Time-Splitting Technique, Meli̇ke Karta

Turkish Journal of Mathematics

In the present manuscript, the Benjamin-Bona-Mahony-Burgers (BBMB) equation will be handled numerically by Strang time-splitting technique. While applying this technique, collocation method based on quintic B-spline basis functions is applied. In line with our purpose, after splitting the BBM-Burgers equation given with appropriate initial boundary conditions into two subequations containing the derivative in terms of time, the quintic B-spline based collocation finite element method (FEM) for spatial discretization and the suitable finite difference approaches for time discretization is applied to each subequation and hereby two different systems of algebraic equations are obtained. Four test problems are utilized to test the …

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Bi-Periodic Incomplete Horadam Numbers, Eli̇f Tan, Mehmet Dağli, Amine Belkhir

Turkish Journal of Mathematics

In this paper, we introduce bi-periodic~incomplete~Horadam numbers as a natural generalization of incomplete Horadam numbers. We study their basic properties and provide recurrence relations. In particular, we derive the generating function of these numbers.

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On Solvability Of Homogeneous Riemann Boundary Value Problems In Hardy-Orlicz Classes, Yusuf Zeren, Fi̇dan A. Ali̇zadeh, Feyza Eli̇f Dal

Turkish Journal of Mathematics

This work deals with the Orlicz space and the Hardy-Orlicz classes generated by this space, which consist of analytic functions inside and outside the unit disk. The homogeneous Riemann boundary value problems with piecewise continuous coefficients are considered in these classes. New characteristic of Orlicz space is defined which depends on whether the power function belongs to this space or not. Relationship between this characteristic and Boyd indices of Orlicz space is established. The concept of canonical solution of homogeneous problem is defined, which depends on the argument of the coefficient. In terms of the above characteristic, a condition on …

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On Max-Min Solutions Of Fuzzy Games With Nonlinear Memberships Functions, Adem Cengi̇z Çevi̇kel

Turkish Journal of Mathematics

In this paper, we deal with two-person zero-sum games with fuzzy goals. We investigated the cases where the membership functions of the players are nonlinear. We examined how the solutions should be if the membership functions of players were exponential functions. In case players' membership functions are exponential, we developed a new method for the maximin solution according to a degree of attainment of the fuzzy goals. An application was made to show the effectiveness of the method.

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On Conditions Of Regular Solvability For Two Classes Of Third-Order Operator-Differential Equations In A Fourth-Order Sobolev-Type Space, Araz R. Aliev, Nazila L. Muradova

Turkish Journal of Mathematics

In this paper, we study two classes of operator-differential equations of the third order with a multiple characteristic, considered on the whole axis. We introduce the concept of a smooth regular solution of order 1 and obtain sufficient conditions for the "smoothly" regular solvability of these equations.

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Geodesics And Natural Complex Magnetic Trajectories On Tangent Bundles, Mohamed Tahar Kadaoui Abbassi, Noura Amri

Turkish Journal of Mathematics

In this paper, we investigate geodesics of the tangent bundle $TM$ of a Riemannian manifold $(M,g)$ endowed with an arbitrary pseudo-Riemannian $g$-natural metric of Kaluza-Klein type. Then considering a class of naturally defined almost complex structures on $TM$, constructed by V. Oproiu, we construct a class of magnetic fields and we characterize the corresponding magnetic curves on $TM$, when $(M,g)$ is a space form.

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Curves And Stick Figures Not Contained In A Hypersurface Of A Given Degree, Edoardo Ballico

Turkish Journal of Mathematics

A stick figure $X\subset \mathbb{P}^r$ is a nodal curve whose irreducible components are lines. For fixed integers $r\ge 3$, $s\ge 2$ and $d$ we study the maximal arithmetic genus of a connected stick figure (or any reduced and connected curve) $X\subset \mathbb{P}^r$ such that $\deg (X)=d$ and $h^0(\mathcal{I}_X(s-1))=0$. We consider Halphen's problem of obtaining all arithmetic genera below the maximal one.

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Full-Text Articles in Mathematics

Graphs Without A 2c3-Minor And Bicircular Matroids Without A U3,6-Minor, Daniel Slilaty

Mathematics and Statistics Faculty Publications

The Multiset Partition Algebra: Diagram-Like Bases And Representations, Alexander N. Wilson

Dartmouth College Ph.D Dissertations

Adapted Metrics For A Modified Coulomb/Newton’S Potential, Lucian M. Ionescu, Cristina-Liliana Pripoae, Gabriel-Teodor Pripoae

Faculty Publications – Mathematics

Predation And Harvesting In Spatial Population Models, Connor R. Shrader

Honors Undergraduate Theses

Clustering Of Diverse Multiplex Networks, Yaxuan Wang

Electronic Theses and Dissertations, 2020-2023

Finding The Common Denominator: Understanding The Shared Experiences Of Female Math Majors, Abigail R. Rosenbaum

Honors Theses

Elliptic Curves Over Finite Fields, Christopher S. Calger

Honors Theses

Advances In Differentially Methylated Region Detection And Cure Survival Models, Daniel Ahmed Alhassan

Doctoral Dissertations

Minimal Sets, Union-Closed Families, And Frankl's Conjecture, Christopher S. Flippen

Theses and Dissertations

The Mceliece Cryptosystem As A Solution To The Post-Quantum Cryptographic Problem, Isaac Hanna

Senior Honors Theses

Analytical And Numerical Negative Boundedness Of Fractional Differences With Mittag–Leffler Kernel, Pshtiwan Othman Mohammed, Rajendra Dahal, Christopher S. Goodrich, Y. S. Hamed, Dumitru Baleanu

Mathematics and Statistics

Duality For Spaces Of Polynomials On Tree Spaces, Christopher Boyd, Costas Poulios, Milena Venkova

Articles

On The Cubic Equation With Its Siebeck–Marden–Northshield Triangle And The Quartic Equation With Its Tetrahedron, Emil Prodanov

Articles

A Study Of The Local Deep Galerkin Method For The Modified Cahn Hilliard Equation, Shi Wen Wong

Electronic Theses and Dissertations

Characterizations And Representations Of Weak Core Inverses And $M$-Weak Group Inverses, Wende Li, Jianlong Chen, Yukun Zhou

Turkish Journal of Mathematics

The Adjoint Reidemeister Torsion For Compact 3-Manifolds Admitting A Unique Decomposition, Esma Di̇ri̇can Erdal

Turkish Journal of Mathematics

On The Modelling Of Short And Intermediate Water Waves, Rossen I. Ivanov

Articles

Math Department 2022-2023 Newsletter, Mathematics Department

Mathematics Newsletter

Generalization Of Statistical Limit-Cluster Points And The Concepts Of Statistical Limit Inferior-Superior On Time Scales By Using Regular Integral Transformations, Ceylan Yalçin

Turkish Journal of Mathematics

Novel Results On Trapezoid-Type Inequalities For Conformable Fractional Integrals, Fati̇h Hezenci̇, Hüseyi̇n Budak

Turkish Journal of Mathematics

Twisted Dirac Operators And The Kastler-Kalau-Walze Type Theorem For Five Dimensional Manifolds With Boundary, Tong Wu, Sining Wei, Yong Wang

Turkish Journal of Mathematics

A Note On $Ss$-Supplement Submodules, Emi̇ne Önal Kir

Turkish Journal of Mathematics

Qualitative Study Of A Second Order Difference Equation, Messaoud Berkal, Juan Francisco Navarro

Turkish Journal of Mathematics

Numerical Solution For Benjamin-Bona-Mahony-Burgers Equation With Strang Time-Splitting Technique, Meli̇ke Karta

Turkish Journal of Mathematics

Bi-Periodic Incomplete Horadam Numbers, Eli̇f Tan, Mehmet Dağli, Amine Belkhir

Turkish Journal of Mathematics

On Solvability Of Homogeneous Riemann Boundary Value Problems In Hardy-Orlicz Classes, Yusuf Zeren, Fi̇dan A. Ali̇zadeh, Feyza Eli̇f Dal

Turkish Journal of Mathematics

On Max-Min Solutions Of Fuzzy Games With Nonlinear Memberships Functions, Adem Cengi̇z Çevi̇kel

Turkish Journal of Mathematics

On Conditions Of Regular Solvability For Two Classes Of Third-Order Operator-Differential Equations In A Fourth-Order Sobolev-Type Space, Araz R. Aliev, Nazila L. Muradova

Turkish Journal of Mathematics

Geodesics And Natural Complex Magnetic Trajectories On Tangent Bundles, Mohamed Tahar Kadaoui Abbassi, Noura Amri

Turkish Journal of Mathematics

Curves And Stick Figures Not Contained In A Hypersurface Of A Given Degree, Edoardo Ballico

Turkish Journal of Mathematics