First Days In Number, 2020 Livre de Lyon

#### First Days In Number, James H. Porter

*Science and Mathematical Science*

This book is designed to furnish systematic written work for the pupil, in which he finds expressed in written words, figures, and signs the facts he must have discovered and expressed understandingly in the oral lessons. Number work, as well as every phase of primary education, should be made a delight and the children should be happy in its accomplishment.

Introduction To Neutroalgebraic Structures And Antialgebraic Structures (Revisited), 2020 University of New Mexico

#### Introduction To Neutroalgebraic Structures And Antialgebraic Structures (Revisited), Florentin Smarandache

*Mathematics and Statistics Faculty and Staff Publications*

In all classical algebraic structures, the Laws of Compositions on a given set are well-defined. But this is a restrictive case, because there are many more situations in science and in any domain of knowledge when a law of composition defined on a set may be only partially-defined (or partially true) and partially-undefined (or partially false), that we call NeutroDefined, or totally undefined (totally false) that we call AntiDefined. Again, in all classical algebraic structures, the Axioms (Associativity, Commutativity, etc.) defined on a set are totally true, but it is again a restrictive case, because similarly there are numerous situations ...

The Mobius Function And Mobius Inversion, 2020 Ursinus College

#### The Mobius Function And Mobius Inversion, Carl Lienert

*Number Theory*

No abstract provided.

Quadruple Neutrosophic Theory And Applications Volume I, 2020 University of New Mexico

#### Quadruple Neutrosophic Theory And Applications Volume I, Florentin Smarandache, Memet Şahin, Vakkas Uluçay, Abdullah Kargin

*Mathematics and Statistics Faculty and Staff Publications*

Neutrosophic set has been derived from a new branch of philosophy, namely Neutrosophy. Neutrosophic set is capable of dealing with uncertainty, indeterminacy and inconsistent information. Neutrosophic set approaches are suitable to modeling problems with uncertainty, indeterminacy and inconsistent information in which human knowledge is necessary, and human evaluation is needed. Neutrosophic set theory firstly proposed in 1998 by Florentin Smarandache, who also developed the concept of single valued neutrosophic set, oriented towards real world scientific and engineering applications. Since then, the single valued neutrosophic set theory has been extensively studied in books and monographs introducing neutrosophic sets and its applications ...

The Inﬁnite Is The Chasm In Which Our Thoughts Are Lost: Reﬂections On Sophie Germain's Essays, 2020 California State University, Fullerton

#### The Inﬁnite Is The Chasm In Which Our Thoughts Are Lost: Reﬂections On Sophie Germain's Essays, Adam Glesser, Bogdan D. Suceavă, Mihaela Vajiac

*Mathematics, Physics, and Computer Science Faculty Articles and Research*

"Sophie Germain (1776–1831) is quite well-known to the mathematical community for her contributions to number theory [17] and elasticity theory (e.g., see [2, 5]). On the other hand, there have been few attempts to understand Sophie Germain as an intellectual of her time, as an independent thinker outside of academia, and as a female mathematician in France, facing the prejudice of the time of the First Empire and of the Bourbon Restoration, while pursuing her thoughts and interests and writing on them. Sophie Germain had to face a double challenge: the mathematical difficulty of the problems she approached ...

N-Cycle Splines Over Sexy Rings, 2020 Bard College

#### N-Cycle Splines Over Sexy Rings, Jacob Tilden Cummings

*Senior Projects Spring 2020*

In this project we abstract the work of previous bard students by introducing the concept of splines over non-integers, non-euclidean domains, and even non-PIDs. We focus on n-cycles for some natural number n. We show that the concept of flow up class bases exist in PID splines the same way they do in integer splines, remarking the complications and intricacies that arise when abstracting from the integers to PIDs. We also start from scratch by finding a flow up class basis for n-cycle splines over the real numbers adjoin two indeterminates, denoted R[x,y] which necessitate more original techniques.

Simultaneous Zeros Of A System Of Two Quadratic Forms, 2020 University of Kentucky

#### Simultaneous Zeros Of A System Of Two Quadratic Forms, Nandita Sahajpal

*Theses and Dissertations--Mathematics*

In this dissertation we investigate the existence of a nontrivial solution to a system of two quadratic forms over local fields and global fields. We specifically study a system of two quadratic forms over an arbitrary number field. The questions that are of particular interest are:

- How many variables are necessary to guarantee a nontrivial zero to a system of two quadratic forms over a global field or a local field? In other words, what is the
*u*-invariant of a pair of quadratic forms over any global or local field? - What is the relation between
*u*-invariants of a ...

Solutions To Systems Of Equations Over Finite Fields, 2020 University of Kentucky

#### Solutions To Systems Of Equations Over Finite Fields, Rachel Petrik

*Theses and Dissertations--Mathematics*

This dissertation investigates the existence of solutions to equations over finite fields with an emphasis on diagonal equations. In particular:

- Given a system of equations, how many solutions are there?
- In the case of a system of diagonal forms, when does a nontrivial solution exist?

Many results are known that address (1) and (2), such as the classical Chevalley--Warning theorems. With respect to (1), we have improved a recent result of D.R. Heath--Brown, which provides a lower bound on the total number of solutions to a system of polynomials equations. Furthermore, we have demonstrated that several of our lower ...

An Exploration Of The Use Of The Fibonacci Sequence In Unrelated Mathematics Disciplines, 2020 University of New Hampshire, Durham

#### An Exploration Of The Use Of The Fibonacci Sequence In Unrelated Mathematics Disciplines, Molly E. Boodey

*Honors Theses and Capstones*

No abstract provided.

Codes, Cryptography, And The Mceliece Cryptosystem, 2020 Liberty University

#### 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 ...

Complex Powers Of I Satisfying The Continued Fraction Functional Equation Over The Gaussian Integers, 2019 Illinois Mathematics and Science Academy

#### Complex Powers Of I Satisfying The Continued Fraction Functional Equation Over The Gaussian Integers, Matthew Niemiro '20

*Exemplary Student Work*

We investigate and then state the conditions under which* i ^{z} *satisfies the simple continued fraction functional equation for real and then complex

*z*over the Gaussian integers.

Elliptic Curves And Power Residues, 2019 University of Massachusetts Amherst

#### Elliptic Curves And Power Residues, Vy Thi Khanh Nguyen

*Doctoral Dissertations*

Let *E*_{1} x *E*_{2} over **Q** be a fixed product of two elliptic curves over **Q** with complex multiplication. I compute the probability that the *p*th Fourier coefficient of *E*_{1} x *E _{2}*, denoted as

*a*

_{p}(

*E*

_{1}) +

*a*(

_{p}*E*), is a square modulo

_{2}*p*. The results are 1/4, 7/16, and 1/2 for different imaginary quadratic fields, given a technical independence of the twists. The similar prime densities for cubes and 4th power are 19/54, and 1/4, respectively. I also compute the probabilities without the technical assumption on ...

On The Equality Case Of The Ramanujan Conjecture For Hilbert Modular Forms, 2019 Portland State University

#### On The Equality Case Of The Ramanujan Conjecture For Hilbert Modular Forms, Liubomir Chiriac

*Mathematics and Statistics Faculty Publications and Presentations*

The generalized Ramanujan Conjecture for cuspidal unitary automorphic representations π on GL(2) asserts that |av(π)| ≤ 2. We prove that this inequality is strict if π is generated by a CM Hilbert modular form of parallel weight two and v is a finite place of degree one. Equivalently, the Satake parameters of πv are necessarily distinct. We also give examples where the equality case does occur for primes of degree two.

Comparison Of Three Dimensional Selfdual Representations By Faltings-Serre Method, 2019 University of Massachusetts Amherst

#### Comparison Of Three Dimensional Selfdual Representations By Faltings-Serre Method, Lian Duan

*Doctoral Dissertations*

In this thesis, we prove that, a selfdual 3-dimensional Galois representation constructed by van Geemen and Top is isomorphic to a quadratic twist of the symmetric square of the Tate module of an elliptic curve. This is an application of our refinement of the Faltings-Serre method to 3-dimensional Galois representations with ground field not equal to Q. The proof makes use of the Faltings-Serre method, $\ell$-adic Lie algebra, and Burnside groups.

Bruhat-Tits Buildings And A Characteristic P Unimodular Symbol Algorithm, 2019 University of Massachusetts Amherst

#### Bruhat-Tits Buildings And A Characteristic P Unimodular Symbol Algorithm, Matthew Bates

*Doctoral Dissertations*

Let k be the finite field with q elements, let F be the field of Laurent series in the variable 1/t with coefficients in k, and let A be the polynomial ring in the variable t with coefficients in k. Let SLn(F) be the ring of nxn-matrices with entries in F, and determinant 1. Given a polynomial g in A, let Gamma(g) subset SLn(F) be the full congruence subgroup of level g. In this thesis we examine the action of Gamma(g) on the Bruhat-Tits building Xn associated to SLn(F) for n equals 2 and ...

Winnability Of The Group Labeling Lights Out Game On Complete Bipartite Graphs, 2019 Grand Valley State University

#### Winnability Of The Group Labeling Lights Out Game On Complete Bipartite Graphs, Christian J. Miller

*McNair Scholars Manuscripts*

For an arbitrary graph, we can play Lights Out on it if we assign a number label to each of the vertices of a graph G, representing states of on/off in the original Lights Out game, with the edges connecting those vertices representing the buttons that are adjacent to each other. This project is focused on a slightly modifed version of the game's original rules, with the labels for the vertices coming from the group Zn. It is not always possible to win the game. We will be investigating the values of n for which this group labeling ...

Zeta Functions Of Classical Groups And Class Two Nilpotent Groups, 2019 The Graduate Center, City University of New York

#### Zeta Functions Of Classical Groups And Class Two Nilpotent Groups, Fikreab Solomon Admasu

*Dissertations, Theses, and Capstone Projects*

This thesis is concerned with zeta functions and generating series associated with two families of groups that are intimately connected with each other: classical groups and class two nilpotent groups. Indeed, the zeta functions of classical groups count some special subgroups in class two nilpotent groups.

In the first chapter, we provide new expressions for the zeta functions of symplectic groups and even orthogonal groups in terms of the cotype zeta function of the integer lattice. In his paper on universal $p$-adic zeta functions, J. Igusa computed explicit formulae for the zeta functions of classical algebraic groups. These zeta ...

A Few Firsts In The Epsilon Years Of My Career, 2019 Brooklyn College (CUNY)

#### A Few Firsts In The Epsilon Years Of My Career, Heidi Goodson

*Journal of Humanistic Mathematics*

In this essay, I describe the unexpected ways I achieved some milestones in the early years of my career.

The Last Digits Of Infinity (On Tetrations Under Modular Rings), 2019 Augustana College, Rock Island Illinois

#### The Last Digits Of Infinity (On Tetrations Under Modular Rings), William Stowe

*Celebration of Learning*

A tetration is defined as repeated exponentiation. As an example, 2 tetrated 4 times is 2^(2^(2^2)) = 2^16. Tetrated numbers grow rapidly; however, we will see that when tetrating where computations are performed mod n for some positive integer n, there is convergent behavior. We will show that, in general, this convergent behavior will always show up.

Inquiry In Inquiry: A Classification Of The Learning Theories Underlying Inquiry-Based Undergraduate Number Theory Texts, 2019 Seattle Pacific University

#### Inquiry In Inquiry: A Classification Of The Learning Theories Underlying Inquiry-Based Undergraduate Number Theory Texts, Rebecca L. Butler

*Honors Projects*

While undergraduate inquiry-based texts in number theory share similar approaches with respect to learning as the embodiment of professional practice, this does not entail that these texts all operate from the same fundamental understanding of what it means to learn mathematics. In this paper, the instructional design of several texts of the aforementioned types are analyzed to assess the theory of learning under which they operate. From this understanding of the different theories of learning employed in an inquiry-based mathematical setting, one can come to understand the popular model of what it is to learn number theory in a meaningful ...