Mathematics

Finding Pythagoras In The Pythagoreans, Brandon Barnes

Classics Ancient Science Fair

This presentation for the Ancient Science Fair deals with the Pythagorean Theorem. Since Pythagoras himself did not write down his work, early scholars had to work to find the knowledge and unify it among differing sources.

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Codes, Cryptography, And The Mceliece Cryptosystem, Bethany Matsick

Senior Honors Theses

Over the past several decades, technology has continued to develop at an incredible rate, and the importance of properly securing information has increased significantly. While a variety of encryption schemes currently exist for this purpose, a number of them rely on problems, such as integer factorization, that are not resistant to quantum algorithms. With the reality of quantum computers approaching, it is critical that a quantum-resistant method of protecting information is found. After developing the proper background, we evaluate the potential of the McEliece cryptosystem for use in the post-quantum era by examining families of algebraic geometry codes that allow …

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Discrete Morse Theory By Vector Fields: A Survey And New Directions, Matthew Nemitz

All Graduate Theses, Dissertations, and Other Capstone Projects

We synthesize some of the main tools in discrete Morse theory from various sources. We do this in regards to abstract simplicial complexes with an emphasis on vector fields and use this as a building block to achieve our main result which is to investigate the relationship between simplicial maps and homotopy. We use the discrete vector field as a catalyst to build a chain homotopy between chain maps induced by simplicial maps.

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Classification Of The Real Roots Of The Quartic Equation And Their Pythagorean Tunes, Emil Prodanov

Articles

Presented is a two-tier analysis of the location of the real roots of the general quartic equation x4+ax3+bx2+cx+d=0 with real coefficients and the classification of the roots in terms of a, b, c, and d, without using any numerical approximations. Associated with the general quartic, there is a number of subsidiary quadratic equations (resolvent quadratic equations) whose roots allow this systematization as well as the determination of the bounds of the individual roots of the quartic. In many cases the root isolation intervals are found. The second tier of the analysis uses two subsidiary cubic equations (auxiliary cubic equations) and …

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Model-Based Cluster Analysis Of Indiana Social Security Beneficiary Data, Gwendolyn Spencer

Undergraduate Honors Thesis Collection

Annual reports of the U.S. Old-Age, Survivors, and Disability Insurance (OASDI) program, published by the Social Security Administration, detail the aggregate information about the program for each U.S. Postal ZIP code. This information includes the types of beneficiaries and monthly benefits received. These reports present the opportunity for contemporary analysis of the aggregate information about the OASDI program. To better capture the significance of the most-recent report for 2018, this project will use model-based cluster analysis, the unsupervised machine-learning process of grouping similar data points, to compare the 2017 and 2018 data. Due to the large amount of data, the …

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Stochastic Technique For Solutions Of Non-Linear Fin Equation Arising In Thermal Equilibrium Model, Iftikhar Ahmad, Hina Qureshi, Muhammad Bilal, Muhammad Usman

Mathematics Faculty Publications

In this study, a stochastic numerical technique is used to investigate the numerical solution of heat transfer temperature distribution system using feed forward artificial neural networks. Mathematical model of fin equation is formulated with the help of artificial neural networks. The effect of the heat on a rectangular fin with thermal conductivity and temperature de-pendent internal heat generation is calculated through neural networks optimization with optimizers like active set technique, interior point technique, pattern search, genetic algorithm and a hybrid approach of pattern search - interior point technique, genetic algorithm - active set technique, genetic algorithm - interior point technique, …

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The Relative Gromov Width Of Lagrangian Cobordisms Between Legendrians, Sabloff M. Joshua, Lisa Traynor

Mathematics Faculty Research and Scholarship

We obtain upper and lower bounds for the relative Gromov width of Lagrangian cobordisms between Legendrian submanifolds. Upper bounds arise from the existence of J-holomorphic disks with boundary on the Lagrangian cobordism that pass through the center of a given symplectically embedded ball. The areas of these disks - and hence the sizes of these balls - are controlled by a real-valued fundamental capacity, a quantity derived from the algebraic structure of filtered linearized Legendrian Contact Homology of the Legendrian at the top of the cobordism. Lower bounds come from explicit constructions that use neighborhoods of Reeb chords in the …

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Saperi: Approaching Gender Gap Using Spatial Ability Training Week In High-School Context, Maria Giulia Ballatore, Gavin Duffy, Sheryl Sorby, Anita Tabacco

Articles

Maria Giulia Ballatore, Gavin Duffy, Sheryl Sorby, and Anita Tabacco. 2020. SAperI: approaching gender gap using Spatial Ability training week in high-school context. In Eighth International Conference on Technological Ecosystems for Enhancing Multiculturality (TEEM’20), October 21–23, 2020, Salamanca, Spain. ACM, New York, NY, USA, 7 pages. https://doi.org/10.1145/3434780.3436577

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Swirling Fluid Flow In Flexible, Expandable Elastic Tubes: Variational Approach, Reductions And Integrability, Rossen Ivanov, Vakhtang Putkaradze

Articles

Many engineering and physiological applications deal with situations when a fluid is moving in flexible tubes with elastic walls. In real-life applications like blood flow, a swirl in the fluid often plays an important role, presenting an additional complexity not described by previous theoretical models. We present a theory for the dynamics of the interaction between elastic tubes and swirling fluid flow. The equations are derived using a variational principle, with the incompressibility constraint of the fluid giving rise to a pressure-like term. In order to connect this work with the previous literature, we consider the case of inextensible and …

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Designing A National Blended Learning Program For “Out‑Of‑Field” Mathematics Teacher Professional Development, Merrilyn Goos, John O’Donoghue, Máire Ní Ríordáin, Fiona Faulkner, Tony Hall, Niamh O'Meara

Articles

“Out-of-field” teaching refers to the practice of assigning secondary school teachers to teach subjects that do not match their training or education. This practice is an issue of concern in many countries around the world, and seems particularly prevalent in the teaching of mathematics. The aim of this paper is to analyse the design principles underpinning the development and delivery of a blended learning program of professional development for out-of-field teachers of secondary school mathematics in Ireland. Three theoretical frameworks inform our analysis of the blended learning design. The first identifies critical dimensions of blended learning environments as a boundary …

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A Mosquito Population Model With Four Positive Equilibrium Points And Sterile Insect Release, Maxwell Fox

Summer Community of Scholars Posters (RCEU and HCR Combined Programs)

No abstract provided.

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Kähler Metrics Via Lorentzian Geometry In Dimension Four, Amir Babak Aazami, Gideon Maschler

Given a semi-Riemannian 4-manifold (M, g) with two distinguished vector fields satisfying properties determined by their shear, twist and various Lie bracket relations, a family of Kähler metrics gK is constructed, defined on an open set in M, which coincides with M in many typical examples. Under certain conditions g and gK share various properties, such as a Killing vector field or a vector field with a geodesic flow. In some cases the Kähler metrics are complete. The Ricci and scalar curvatures of gK are computed under certain assumptions in terms of data associated to g. Many examples are described, …

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The Adjunction Inequality For Weyl-Harmonic Maps, Robert Ream

Mathematics

In this paper we study an analog of minimal surfaces called Weyl-minimal surfaces in conformal manifolds with a Weyl connection (M4, c, D). We show that there is an Eells-Salamon type correspondence between nonvertical '1-holomorphic curves in the weightless twistor space and branched Weyl-minimal surfaces. When (M, c, J) is conformally almost-Hermitian, there is a canonical Weyl connection. We show that for the canonical Weyl connection, branched Weyl-minimal surfaces satisfy the adjunction inequality. The ±J-holomorphic curves are automatically Weyl-minimal and satisfy the corresponding equality. These results generalize results of Eells-Salamon and Webster for minimal surfaces in Kähler 4-manifolds as well …

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Maxwell's Equations And Yang-Mills Equations In Complex Variables : New Perspectives, Sachin Munshi

Legacy Theses & Dissertations (2009 - 2024)

Maxwell's equations, named after James C. Maxwell, are a U(1) gauge theory describing the interactions between electric and magnetic fields. They lie at the heart of classical electromagnetism and electrodynamics. Yang-Mills equations, named after C. N. Yang and Robert Mills, generalize Maxwell's equations and are associated with a non-abelian gauge theory called Yang-Mills theory. Yang-Mills theory unified the electroweak interaction with the strong interaction (QCD), and it is the foundation of the Standard Model in particle physics.

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The Unrolled Quantum Group Of Sl(2) With Connections To Topological Quantum Field Theory And Vertex Operator Algebras, James F. Clark

Legacy Theses & Dissertations (2009 - 2024)

This dissertation expands on the work of Constantino, Geer, and Patureau-Mirand \cite{CGP}, as well as Creutzig, Milas, and Rupert \cite{CMR} on exploring the unrolled quantum group of $\mathfrak{sl}(2)$ (denoted by $\overline{U_q}^H(\mathfrak{sl}_2)$) and the category of finite dimensional weight $\overline{U_q}^H(\mathfrak{sl}_2)$-modules (denoted by $\mathscr{C}$) for any $p$\textsuperscript{th} root of unity where $p>2$. This unrolled quantum group and category has connections to topological quantum field theory (TQFT) and to vertex operator algebras (VOAs). Significant results of this dissertation include proving that the $\mathscr{C}$ is a ribbon category, finding all simple and projective modules of $\mathscr{C}$ and proving that they form a full …

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Asymptotic Properties Of Solutions To Second-Order Difference Equations, Janusz Migda

Turkish Journal of Mathematics

In this paper the second-order difference equations of the form \[ \Delta^2 x_n=a_nf(n,x_{\sigma(n)})+b_n \] are considered. We establish sufficient conditions for the existence of solutions with prescribed asymptotic behavior. In particular, we present conditions under which there exists an asymptotically linear solution. Moreover, we study the asymptotic behavior of solutions.

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Submanifolds Of Almost Poly-Norden Riemannian Manifolds, Selcen Yüksel Perktaş

Turkish Journal of Mathematics

Our aim in the present paper is to initiate the study of submanifolds in an almost poly-Norden Riemannian manifold, which is a new type of manifold first introduced by Şahin [17]. We give fundamental properties of submanifolds equipped with induced structures provided by almost poly-Norden Riemannian structures and find some conditions for such submanifolds to be totally geodesics. We introduce some subclasses of submanifolds in almost poly-Norden Riemannian manifolds such as invariant and antiinvariant submanifolds. We investigate conditions for a hypersurface of almost poly-Norden Riemannian manifolds to be invariant and totally geodesic, respectively, by using the components of the structure …

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Nonabelian Cocycles And The Spectrum Of A Symmetric Monoidal Category, Antonio M. Cegarra

Turkish Journal of Mathematics

We present an Eilenberg-MacLane-type description for the first, second and third spaces of the spectrum defined by a symmetric monoidal category.

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Galerkin-Like Method For Solving Linear Functional Differential Equations Underinitial Conditions, Şuayi̇p Yüzbaşi, Murat Karaçayir

Turkish Journal of Mathematics

In this paper, we present a weighted residual Galerkin method to solve linear functional differential equations. We consider the problem with variable coefficients under initial conditions. Assuming the exact solution of the problem has a Taylor series expansion convergent in the relevant domain, we seek a solution of the given problem in the form of a polynomial having degree $N$ of our choice. Substituting this polynomial with unknown coefficients in the given equation yields an expression linear in these coefficients. We then proceed as in the weighted residual method and take inner product of this expression with monomials up to …

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Coefficient Estimation Of A Certain Subclass Of Bi-Close-To-Convex Functions Analytic In The Exterior Of The Unit Disc, Sarbeswar Barik

Turkish Journal of Mathematics

In this paper, we introduce two new subclasses of biunivalent functions analytic in the exterior of the unit disc. The bounds obtained for the $zero^{th}$, first and second coefficient improves upon earlier known results. The results are obtained by refining the well-known estimates for the initial coefficients of the Carth$\acute{e}$odory functions.

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Weighted Composition Operators From The Bloch Space To Nth Weighted-Typespaces, Ebrahim Abbasi, Songxiao Li, Hamid Vaezi

Turkish Journal of Mathematics

In this work, we characterize the boundedness of weighted composition operators from the Bloch space and the little Bloch space to $n$th weighted-type spaces. Some estimates for the essential norm of these operators are also given. As a corollary, we obtain some characterizations for the compactness of weighted composition operators from the Bloch space and the little Bloch space to $n$th weighted-type spaces.

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The $2$-Adic And $3$-Adic Valuation Of The Tripell Sequence And An Application, Jhon Jairo Bravo, Maribel Díaz, José Luis Ramírez

Turkish Journal of Mathematics

Let $(T_n)_{n\geq 0}$ denote the Tripell sequence, defined by the linear recurrence $T_n=2T_{n-1} + T_{n-2}+T_{n-3}$ for $n\geq 3$ with $T_0=0$, $T_{1}=1$ and $T_2=2$ as initial conditions. In this paper, we study the $2$-adic and $3$-adic valuation of the Tripell sequence and, as an application, we determine all Tripell numbers which are factorials.

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Locally $D_\Delta$-Connected And Locally $D_\Delta$-Compact Spaces, Harshit Mathur, Davinder Singh

Turkish Journal of Mathematics

The local analogues of the notions of $d_\delta$-connectedness and $D_\delta$-compactness of a topological space are introduced, and are named respectively, locally $d_\delta$-connectedness and locally $D_\delta$-compactness. Several properties including characterization of the concepts are discussed.

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Geometric Properties Of Partial Sums Of Generalized Koebe Function, Erhan Deni̇z, Erkan Yalçin

Turkish Journal of Mathematics

The aim of the present paper is to investigate the starlikeness, convexity, and close-to-convexity of some partial sums of the generalized Koebe function. Furthermore, we give some special results related with special cases of $c$ constant. The results, which are presented in this paper, would generalize those in related works of several earlier authors.

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General Coefficient Estimates For Bi-Univalent Functions: A New Approach, Oqlah Alrefai, Mohammed Ali

Turkish Journal of Mathematics

We prove for univalent functions $f(z)=z+\sum_{k=n}^{\infty}a_k z^k;(n\geq 2)$ in the unit disk $\mathbb{U}=\{z:\; z

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A Reduced Computational Matrix Approach With Convergence Estimation For Solving Model Differential Equations Involving Specific Nonlinearities Of Quartic Type, Ömür Kivanç Kürkçü

Turkish Journal of Mathematics

This study aims to efficiently solve model differential equations involving specific nonlinearities of quartic type by proposing a reduced computational matrix approach based on the generalized Mott polynomial. This method presents a reduced matrix expansion of the generalized Mott polynomial with the parameter-$\alpha$, matrix equations, and Chebyshev--Lobatto collocation points. The simplicity of the method provides fast computation while eliminating an algebraic system of nonlinear equations, which arises from the matrix equation. The method also scrutinizes the consistency of the solutions due to the parameter-$\alpha$. The oscillatory behavior of the obtained solutions on long time intervals is simulated via a coupled …

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Conformally Flat Willmore Spacelike Hypersurfaces In $\Mathbb{R}^{N+1}_1$, Zonggang Deng, Tongzhu Li

Turkish Journal of Mathematics

In this paper, we give the equation satisfied by umbilics-free Willmore spacelike hypersurfaces using the conformal invariants in Lorentzian space forms. At the same time, we give the equation satisfied by hyperelastic spacelike curves in $2$-dimensional Lorentzian space forms and classify the closed hyperelastic spacelike curves. Finally conformally flat Willmore spacelike hypersurfaces are classified in terms of the hyperelastic spacelike curves in $2$-dimensional Lorentzian space forms.

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Vector Fields And Planes In $\Mathbb{E}^4$ Which Play The Role Of Darboux Vector, Mustafa Düldül

Turkish Journal of Mathematics

In this paper, we define some new vector fields along a space curve with nonvanishing curvatures in Euclidean 4-space. By using these vector fields we determine some new planes, curves, and ruled hypersurfaces. We show that the determined new planes play the role of the Darboux vector. We also show that, contrary to their definitions, osculating curves of the first kind and rectifying curves in Euclidean 4-space can be considered as space curves whose position vectors always lie in a two-dimensional subspace. Furthermore, we construct developable and nondevelopable ruled hypersurfaces associated with the new vector fields in which the base …

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Direct And Inverse Approximation Theorems In The Weighted Orlicz-Type Spaces With A Variable Exponent, Fahreddi̇n Abdullayev, Stanislav Chaichenko, Meerim Imash Kyzy, Andrii Shidlich

Turkish Journal of Mathematics

In weighted Orlicz-type spaces ${\mathcal S}_{_{\scriptstyle \mathbf p,\,\mu}}$ with a variable summation exponent, the direct and inverse approximation theorems are proved in terms of best approximations of functions and moduli of smoothness of fractional order. It is shown that the constant obtained in the inverse approximation theorem is the best in a certain sense. Some applications of the results are also proposed. In particular, the constructive characteristics of functional classes defined by such moduli of smoothness are given. Equivalence between moduli of smoothness and certain Peetre $K$-functionals is shown in the spaces ${\mathcal S}_{_{\scriptstyle \mathbf p,\,\mu}}$.

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Evaluation Of Sums Of Products Of Gaussian $Q$-Binomial Coefficients With Rational Weight Functions, Talha Arikan, Emrah Kiliç, Helmut Prodinger

Turkish Journal of Mathematics

Generalizing earlier results, sums over the products of the Gaussian $q$-binomial coefficients are computed. Some applications of the results for special choices of $q$ are emphasized. The results are obtained by the elementary technique of partial fraction decomposition.

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Full-Text Articles in Mathematics

Finding Pythagoras In The Pythagoreans, Brandon Barnes

Classics Ancient Science Fair

Codes, Cryptography, And The Mceliece Cryptosystem, Bethany Matsick

Senior Honors Theses

Discrete Morse Theory By Vector Fields: A Survey And New Directions, Matthew Nemitz

All Graduate Theses, Dissertations, and Other Capstone Projects

Classification Of The Real Roots Of The Quartic Equation And Their Pythagorean Tunes, Emil Prodanov

Articles

Model-Based Cluster Analysis Of Indiana Social Security Beneficiary Data, Gwendolyn Spencer

Undergraduate Honors Thesis Collection

Stochastic Technique For Solutions Of Non-Linear Fin Equation Arising In Thermal Equilibrium Model, Iftikhar Ahmad, Hina Qureshi, Muhammad Bilal, Muhammad Usman

Mathematics Faculty Publications

The Relative Gromov Width Of Lagrangian Cobordisms Between Legendrians, Sabloff M. Joshua, Lisa Traynor

Mathematics Faculty Research and Scholarship

Saperi: Approaching Gender Gap Using Spatial Ability Training Week In High-School Context, Maria Giulia Ballatore, Gavin Duffy, Sheryl Sorby, Anita Tabacco

Articles

Swirling Fluid Flow In Flexible, Expandable Elastic Tubes: Variational Approach, Reductions And Integrability, Rossen Ivanov, Vakhtang Putkaradze

Articles

Designing A National Blended Learning Program For “Out‑Of‑Field” Mathematics Teacher Professional Development, Merrilyn Goos, John O’Donoghue, Máire Ní Ríordáin, Fiona Faulkner, Tony Hall, Niamh O'Meara

Articles

A Mosquito Population Model With Four Positive Equilibrium Points And Sterile Insect Release, Maxwell Fox

Summer Community of Scholars Posters (RCEU and HCR Combined Programs)

Kähler Metrics Via Lorentzian Geometry In Dimension Four, Amir Babak Aazami, Gideon Maschler

Mathematics

The Adjunction Inequality For Weyl-Harmonic Maps, Robert Ream

Mathematics

Maxwell's Equations And Yang-Mills Equations In Complex Variables : New Perspectives, Sachin Munshi

Legacy Theses & Dissertations (2009 - 2024)

The Unrolled Quantum Group Of Sl(2) With Connections To Topological Quantum Field Theory And Vertex Operator Algebras, James F. Clark

Legacy Theses & Dissertations (2009 - 2024)

Asymptotic Properties Of Solutions To Second-Order Difference Equations, Janusz Migda

Turkish Journal of Mathematics

Submanifolds Of Almost Poly-Norden Riemannian Manifolds, Selcen Yüksel Perktaş

Turkish Journal of Mathematics

Nonabelian Cocycles And The Spectrum Of A Symmetric Monoidal Category, Antonio M. Cegarra

Turkish Journal of Mathematics

Galerkin-Like Method For Solving Linear Functional Differential Equations Underinitial Conditions, Şuayi̇p Yüzbaşi, Murat Karaçayir

Turkish Journal of Mathematics

Coefficient Estimation Of A Certain Subclass Of Bi-Close-To-Convex Functions Analytic In The Exterior Of The Unit Disc, Sarbeswar Barik

Turkish Journal of Mathematics

Weighted Composition Operators From The Bloch Space To Nth Weighted-Typespaces, Ebrahim Abbasi, Songxiao Li, Hamid Vaezi

Turkish Journal of Mathematics

The $2$-Adic And $3$-Adic Valuation Of The Tripell Sequence And An Application, Jhon Jairo Bravo, Maribel Díaz, José Luis Ramírez

Turkish Journal of Mathematics

Locally $D_\Delta$-Connected And Locally $D_\Delta$-Compact Spaces, Harshit Mathur, Davinder Singh

Turkish Journal of Mathematics

Geometric Properties Of Partial Sums Of Generalized Koebe Function, Erhan Deni̇z, Erkan Yalçin

Turkish Journal of Mathematics

General Coefficient Estimates For Bi-Univalent Functions: A New Approach, Oqlah Alrefai, Mohammed Ali

Turkish Journal of Mathematics

A Reduced Computational Matrix Approach With Convergence Estimation For Solving Model Differential Equations Involving Specific Nonlinearities Of Quartic Type, Ömür Kivanç Kürkçü

Turkish Journal of Mathematics

Conformally Flat Willmore Spacelike Hypersurfaces In $\Mathbb{R}^{N+1}_1$, Zonggang Deng, Tongzhu Li

Turkish Journal of Mathematics

Vector Fields And Planes In $\Mathbb{E}^4$ Which Play The Role Of Darboux Vector, Mustafa Düldül

Turkish Journal of Mathematics

Direct And Inverse Approximation Theorems In The Weighted Orlicz-Type Spaces With A Variable Exponent, Fahreddi̇n Abdullayev, Stanislav Chaichenko, Meerim Imash Kyzy, Andrii Shidlich

Turkish Journal of Mathematics

Evaluation Of Sums Of Products Of Gaussian $Q$-Binomial Coefficients With Rational Weight Functions, Talha Arikan, Emrah Kiliç, Helmut Prodinger

Turkish Journal of Mathematics