Zeta Function Regularization And Its Relationship To Number Theory,
2021
East Tennessee State University
Zeta Function Regularization And Its Relationship To Number Theory, Stephen Wang
Electronic Theses and Dissertations
While the "path integral" formulation of quantum mechanics is both highly intuitive and far reaching, the path integrals themselves often fail to converge in the usual sense. Richard Feynman developed regularization as a solution, such that regularized path integrals could be calculated and analyzed within a strictly physics context. Over the past 50 years, mathematicians and physicists have retroactively introduced schemes for achieving mathematical rigor in the study and application of regularized path integrals. One such scheme was introduced in 2007 by the mathematicians Klaus Kirsten and Paul Loya. In this thesis, we reproduce the Kirsten and Loya approach to …
On Elliptic Curves,
2021
Missouri State University
On Elliptic Curves, Montana S. Miller
Graduate Theses/Dissertations
An elliptic curve over the rational numbers is given by the equation y2 = x3+Ax+B. In our thesis, we study elliptic curves. It is known that the set of rational points on the elliptic curve form a finitely generated abelian group induced by the secant-tangent addition law. We present an elementary proof of associativity using Maple. We also present a relatively concise proof of the Mordell-Weil Theorem.
A History And Translation Of Lagrange's "Sur Quelques Problèmes De L'Analyse De Diophante'',
2021
University of the Pacific
A History And Translation Of Lagrange's "Sur Quelques Problèmes De L'Analyse De Diophante'', Christopher Goff, Michael Saclolo
Euleriana
Among Lagrange's many achievements in number theory is a solution to the problem posed and solved by Fermat of finding a right triangle whose legs sum to a perfect square and whose hypotenuse is also a square. This article chronicles various appearances of the problem, including multiple solutions by Euler, all of which inadequately address completeness and minimality of solutions. Finally, we summarize and translate Lagrange's paper in which he solves the problem completely, thus successfully proving the minimality of Fermat's original solution.
Theory And Application Of Hypersoft Set,
2021
University of New Mexico
Theory And Application Of Hypersoft Set, Florentin Smarandache, Muhammad Saeed, Muhammad Saqlain, Mohamed Abdel-Baset
Branch Mathematics and Statistics Faculty and Staff Publications
Aims and Scope Florentin Smarandache generalize the soft set to the hypersoft set by transforming the function �� into a multi-argument function. This extension reveals that the hypersoft set with neutrosophic, intuitionistic, and fuzzy set theory will be very helpful to construct a connection between alternatives and attributes. Also, the hypersoft set will reduce the complexity of the case study. The Book “Theory and Application of Hypersoft Set” focuses on theories, methods, algorithms for decision making and also applications involving neutrosophic, intuitionistic, and fuzzy information. Our goal is to develop a strong relationship with the MCDM solving techniques and to …
Computational Thinking In Mathematics And Computer Science: What Programming Does To Your Head,
2021
Education Development Center
Computational Thinking In Mathematics And Computer Science: What Programming Does To Your Head, Al Cuoco, E. Paul Goldenberg
Journal of Humanistic Mathematics
How you think about a phenomenon certainly influences how you create a program to model it. The main point of this essay is that the influence goes both ways: creating programs influences how you think. The programs we are talking about are not just the ones we write for a computer. Programs can be implemented on a computer or with physical devices or in your mind. The implementation can bring your ideas to life. Often, though, the implementation and the ideas develop in tandem, each acting as a mirror on the other. We describe an example of how programming and …
A Case Study On Hooley's Conditional Proof Of Artin's Primitive Root Conjecture,
2021
University of Warwick, Coventry
A Case Study On Hooley's Conditional Proof Of Artin's Primitive Root Conjecture, Shalome Kurian
Rose-Hulman Undergraduate Mathematics Journal
Artin’s Primitive Root Conjecture represents one of many famous problems in elementary number theory that has resisted complete solution thus far. Significant progress was made in 1967, when Christopher Hooley published a conditional proof of the conjecture under the assumption of a certain case of the Generalised Riemann Hypothesis. In this survey we present a description of the conjecture and the underlying algebraic theory, and provide a detailed account of Hooley’s proof which is intended to be accessible to those with only undergraduate level knowledge. We also discuss a result concerning the qx+1 problem, whose proof requires similar techniques to …
Mathematical Magic: A Study Of Number Puzzles,
2021
Maryville College
Mathematical Magic: A Study Of Number Puzzles, Nicasio M. Velez
Rose-Hulman Undergraduate Mathematics Journal
Within this paper, we will briefly review the history of a collection of number puzzles which take the shape of squares, polygons, and polyhedra in both modular and nonmodular arithmetic. Among other results, we develop construction techniques for solutions of both Modulo and regular Magic Squares. For other polygons in nonmodular arithmetic, specifically of order 3, we present a proof of why there are only four Magic Triangles using linear algebra, disprove the existence of the Magic Tetrahedron in two ways, and utilizing the infamous 3-SUM combinatorics problem we disprove the existence of the Magic Octahedron.
Three Imprimitive Character Sums,
2021
CUNY New York City College of Technology
Three Imprimitive Character Sums, Brad Isaacson
Publications and Research
We express three imprimitive character sums in terms of generalized Bernoulli numbers. These sums are generalizations of sums introduced and studied by Arakawa, Berndt, Ibukiyama, Kaneko and Ramanujan in the context of modular forms and theta function identities. As a corollary, we obtain a formula for cotangent power sums considered by Apostol.
A Twisted Generalization Of The Classical Dedekind Sum,
2021
CUNY New York City College of Technology
A Twisted Generalization Of The Classical Dedekind Sum, Brad Isaacson
Publications and Research
In this paper, we express three different, yet related, character sums in terms of generalized Bernoulli numbers. Two of these sums are generalizations of sums introduced and studied by Berndt and Arakawa–Ibukiyama–Kaneko in the context of the theory of modular forms. A third sum generalizes a sum already studied by Ramanujan in the context of theta function identities. Our methods are elementary, relying only on basic facts from algebra and number theory.
Solving Neutrosophic Linear Equations Systems Using Symbolic Computation (Resolucion De Sistemas De Ecuaciones Lineales Neutrosóficas Mediante Computación Simbólica),
2021
University of New Mexico
Solving Neutrosophic Linear Equations Systems Using Symbolic Computation (Resolucion De Sistemas De Ecuaciones Lineales Neutrosóficas Mediante Computación Simbólica), Maykel Leyva-Vazquez, Florentin Smarandache
Branch Mathematics and Statistics Faculty and Staff Publications
In this paper, we apply the concept of neutrosophic numbers to solve a systems of neutrophic linear equations using symbolic computation. Also, we utilize Jupyter, which is supported in Google Colaboratory for performing symbolic computation. The sympy library of Python is used to perform the process of neutrosophic computation. Systems of neutrosophic linear equations are solved through symbolic computation in Python. A case study was developed for the determination of vehicular traffic with indeterminacy. This king of computation opens new ways to deal with indeterminacy in real-world problems.
The Plus-Minus Davenport Constant Of Finite Abelian Groups,
2021
University of Kentucky
The Plus-Minus Davenport Constant Of Finite Abelian Groups, Darleen S. Perez-Lavin
Theses and Dissertations--Mathematics
Let G be a finite abelian group, written additively. The Davenport constant, D(G), is the smallest positive number s such that any subset of the group G, with cardinality at least s, contains a non-trivial zero-subsum. We focus on a variation of the Davenport constant where we allow addition and subtraction in the non-trivial zero-subsum. This constant is called the plus-minus Davenport constant, D±(G). In the early 2000’s, Marchan, Ordaz, and Schmid proved that if the cardinality of G is less than or equal to 100, then the D±(G) …
The Smallest Solution Of An Isotropic Quadratic Form,
2021
University of Kentucky
The Smallest Solution Of An Isotropic Quadratic Form, Deborah H. Blevins
Theses and Dissertations--Mathematics
An isotropic quadratic form f(x1,...,xn) = ∑ ni=1 ∑ nj=1 fijxixj defined on a Z- lattice has a smallest solution, where the size of the solution is measured using the infinity norm (∥ ∥∞), the l1 norm (∥ ∥1), or the Euclidean norm (∥ ∥2). Much work has been done to find the least upper bound and greatest lower bound on the smallest solution, beginning with Cassels in the mid-1950’s. Defining F := (f11,...,f …
Solubility Of Additive Forms Over Local Fields,
2021
University of Kentucky
Solubility Of Additive Forms Over Local Fields, Drew Duncan
Theses and Dissertations--Mathematics
Michael Knapp, in a previous work, conjectured that every additive sextic form over $\mathbb{Q}_2(\sqrt{-1})$ and $\mathbb{Q}_2(\sqrt{-5})$ in seven variables has a nontrivial zero. In this dissertation, I show that this conjecture is true, establishing that $$\Gamma^*(6, \mathbb{Q}_2(\sqrt{-1})) = \Gamma^*(6, \mathbb{Q}_2(\sqrt{-5})) = 7.$$ I then determine the minimal number of variables $\Gamma^*(d, K)$ which guarantees a nontrivial solution for every additive form of degree $d=2m$, $m$ odd, $m \ge 3$ over the six ramified quadratic extensions of $\mathbb{Q}_2$. We prove that if $$K \in \{\mathbb{Q}_2(\sqrt{2}), \mathbb{Q}_2(\sqrt{10}), \mathbb{Q}_2(\sqrt{-2}), \mathbb{Q}_2(\sqrt{-10})\},$$ then $$\Gamma^*(d,K) = \frac{3}{2}d,$$ and if $$K \in \{\mathbb{Q}_2(\sqrt{-1}), \mathbb{Q}_2(\sqrt{-5})\},$$ then $$\Gamma^*(d,K) = …
Tiling Representations Of Zeckendorf Decompositions,
2021
Claremont Colleges
Tiling Representations Of Zeckendorf Decompositions, John Lentfer
HMC Senior Theses
Zeckendorf’s theorem states that every positive integer can be decomposed uniquely into a sum of non-consecutive Fibonacci numbers (where f1 = 1 and f2 = 2). Previous work by Grabner and Tichy (1990) and Miller and Wang (2012) has found a generalization of Zeckendorf’s theorem to a larger class of recurrent sequences, called Positive Linear Recurrence Sequences (PLRS’s). We apply well-known tiling interpretations of recurrence sequences from Benjamin and Quinn (2003) to PLRS’s. We exploit that tiling interpretation to create a new tiling interpretation specific to PLRS’s that captures the behavior of the generalized Zeckendorf’s theorem.
On Properties Of Positive Semigroups In Lattices And Totally Real Number Fields,
2021
Claremont Colleges
On Properties Of Positive Semigroups In Lattices And Totally Real Number Fields, Siki Wang
CMC Senior Theses
In this thesis, we give estimates on the successive minima of positive semigroups in lattices and ideals in totally real number fields. In Chapter 1 we give a brief overview of the thesis, while Chapters 2 – 4 provide expository material on some fundamental theorems about lattices, number fields and height functions, hence setting the necessary background for the original results presented in Chapter 5. The results in Chapter 5 can be summarized as follows. For a full-rank lattice L ⊂ Rd, we are concerned with the semigroup L+ ⊆ L, which denotes the set of all vectors with nonnegative …
Amm Problem #12219,
2021
CUNY New York City College of Technology
On Generating Functions In Additive Number Theory, Ii: Lower-Order Terms And Applications To Pdes,
2020
University of Gothenburg
On Generating Functions In Additive Number Theory, Ii: Lower-Order Terms And Applications To Pdes, J. Brandes, Scott T. Parsell, C. Poulias, G. Shakan, R. C. Vaughn
Mathematics Faculty Publications
We obtain asymptotics for sums of the form
Sigma(p)(n=1) e(alpha(k) n(k) + alpha(1)n),
involving lower order main terms. As an application, we show that for almost all alpha(2) is an element of [0, 1) one has
sup(alpha 1 is an element of[0,1)) | Sigma(1 <= n <= P) e(alpha(1)(n(3) + n) + alpha(2)n(3))| << P3/4+epsilon,
and that in a suitable sense this is best possible. This allows us to improve bounds for the fractal dimension of solutions to the Schrodinger and Airy equations.
Sum Of Cubes Of The First N Integers,
2020
California State University, San Bernardino
Sum Of Cubes Of The First N Integers, Obiamaka L. Agu
Electronic Theses, Projects, and Dissertations
In Calculus we learned that Sum^{n}_{k=1} k = [n(n+1)]/2 , that Sum^{n}_{k=1} k^2 = [n(n+1)(2n+1)]/6 , and that Sum^{n}_{k=1} k^{3} = (n(n+1)/2)^{2}. These formulas are useful when solving for the area below quadratic or cubic function over an interval [a, b]. This tedious process, solving for areas under a quadratic or a cubic, served as motivation for the introduction of Riemman integrals. For the overzealous math student, these steps were replaced by a simpler method of evaluating antiderivatives at the endpoints a and b. From my recollection, a former instructor informed us to do the value of memorizing these formulas. …
On The Local Theory Of Profinite Groups,
2020
Western Michigan University
On The Local Theory Of Profinite Groups, Mohammad Shatnawi
Dissertations
Let G be a finite group, and H be a subgroup of G. The transfer homomorphism emerges from the natural action of G on the cosets of H. The transfer was first introduced by Schur in 1902 [22] as a construction in group theory, which produce a homomorphism from a finite group G into H/H' an abelian group where H is a subgroup of G and H' is the derived group of H. One important first application is Burnside’s normal p-complement theorem [5] in 1911, although he did not use the transfer homomorphism explicitly to prove it. …
The Name Tag Problem,
2020
Boise State University
The Name Tag Problem, Christian Carley
Rose-Hulman Undergraduate Mathematics Journal
The Name Tag Problem is a thought experiment that, when formalized, serves as an introduction to the concept of an orthomorphism of $\Zn$. Orthomorphisms are a type of group permutation and their graphs are used to construct mutually orthogonal Latin squares, affine planes and other objects. This paper walks through the formalization of the Name Tag Problem and its linear solutions, which center around modular arithmetic. The characterization of which linear mappings give rise to these solutions developed in this paper can be used to calculate the exact number of linear orthomorphisms for any additive group Z/nZ, which is demonstrated …
