Integer Generalized Splines On The Diamond Graph, 2016 Bard College

#### Integer Generalized Splines On The Diamond Graph, Emmet Reza Mahdavi

*Senior Projects Spring 2016*

In this project we extend previous research on integer splines on graphs, and we use the methods developed on n-cycles to characterize integer splines on the diamond graph. First, we find an explicit module basis consisting of flow-up classes. Then we develop a determinantal criterion for when a given set of splines forms a basis.

Visual Properties Of Generalized Kloosterman Sums, 2016 Pomona College

#### Visual Properties Of Generalized Kloosterman Sums, Paula Burkhardt '16, Alice Zhuo-Yu Chan '14, Gabriel Currier '16, Stephan Ramon Garcia, Florian Luca, Hong Suh '16

*Pomona Faculty Publications and Research*

For a positive integer *m* and a subgroup A of the unit group (*Z*/*mZ*)^{x}, the corresponding *generalized Kloosterman sum* is the function *K*(*a, b, m, A*) = **Σ**_{uEA} *e(**au+bu*^{-1}*/m)*. Unlike classical Kloosterman sums, which are real valued, generalized Kloosterman sums display a surprising array of visual features when their values are plotted in the complex plane. In a variety of instances, we identify the precise number-theoretic conditions that give rise to particular phenomena.

Automated Conjecturing Approach To The Discrete Riemann Hypothesis, 2016 Virginia Commonwealth University

#### Automated Conjecturing Approach To The Discrete Riemann Hypothesis, Alexander Bradford

*Theses and Dissertations*

This paper is a study on some upper bounds of the Mertens function, which is often considered somewhat of a ``mysterious" function in mathematics and is closely related to the Riemann Hypothesis. We discuss some known bounds of the Mertens function, and also seek new bounds with the help of an automated conjecture-making program named CONJECTURING, which was created by C. Larson and N. Van Cleemput, and inspired by Fajtowicz's Dalmatian Heuristic. By utilizing this powerful program, we were able to form, validate, and disprove hypotheses regarding the Mertens function and how it is bounded.

Adinkras And Arithmetical Graphs, 2016 Harvey Mudd College

#### Adinkras And Arithmetical Graphs, Madeleine Weinstein

*HMC Senior Theses*

Adinkras and arithmetical graphs have divergent origins. In the spirit of Feynman diagrams, adinkras encode representations of supersymmetry algebras as graphs with additional structures. Arithmetical graphs, on the other hand, arise in algebraic geometry, and give an arithmetical structure to a graph. In this thesis, we will interpret adinkras as arithmetical graphs and see what can be learned.

Our work consists of three main strands. First, we investigate arithmetical structures on the underlying graph of an adinkra in the specific case where the underlying graph is a hypercube. We classify all such arithmetical structures and compute some of the corresponding ...

The Monochromatic Column Problem: The Prime Case, 2016 Eastern Kentucky University

#### The Monochromatic Column Problem: The Prime Case, Loran Elizabeth Crowell

*Online Theses and Dissertations*

Let p1, p2, . . . , pn be pairwise coprime positive integers and let P = p1p2 · · · pn. Let 0,1,...,m−1 be a sequence of m different colors. Let A be an n×mP matrix of colors in which row i consists of blocks of pi consecutive entries of the same color, with colors 0 through m − 1 repeated cyclically. The Monochromatic Column problem is to determine the number of columns of A in which every entry is the same color. A partial solution for the case when m is prime is given.

A Computational And Theoretical Exploration Of The St. Petersburg Paradox, 2016 Butler University

#### A Computational And Theoretical Exploration Of The St. Petersburg Paradox, Alexander Olivero

*Undergraduate Honors Thesis Collection*

This thesis displays a sample distribution, generated from both a simulation (for large n) by computer program and explicitly calculated (for smaller n), that is not governed by the Central Limit Theorem and, in fact seems to display chaotic behavior. To our knowledge, the explicit calculation of the sample distribution function is new. This project outlines the results that have found a relation to number theory in a probabilistic game that has perplexed mathematicians for hundreds of years.

An Exposition Of The Eisenstein Integers, 2016 Eastern Illinois University

#### An Exposition Of The Eisenstein Integers, Sarada Bandara

*Masters Theses*

In this thesis, we will give a brief introduction to number theory and prime numbers. We also provide the necessary background to understand how the imaginary ring of quadratic integers behaves.

An example of said ring are complex numbers of the form ℤ[*ω*] = {*a*+*bω* ∣ *a*, *b* ∈ ℤ} where *ω*^{2} + *ω* + 1 = 0. These are known as the Eisenstein integers, which form a triangular lattice in the complex plane, in contrast with the Gaussian integers, ℤ[*i*] = {*a* + *bi* ∣ *a*, *b* ∈ ℤ} which form a square lattice in the complex plane. The Gaussian moat problem, first posed by ...

Mathematics. Possible Subjects For The High School Entrance Examination And The Capacity Examination In Romania, 2016 University of New Mexico

#### Mathematics. Possible Subjects For The High School Entrance Examination And The Capacity Examination In Romania, Florentin Smarandache, Constantin Coanda, Ionuț Ivanescu

*Mathematics and Statistics Faculty and Staff Publications*

The present book tries to offer students and teachers knowledge evaluation tools for all the chapters from the current Romanian mathematics syllabus. In the evolution of teenagers, the phase of admission in high schools mobilizes particular efforts and emotions. The present workbook aims to be a permanent advisor in the agitated period starting with the capacity examination and leading to the admittance to high school. The tests included in this workbook have a complementary character as opposed to the many materials written with the purpose to support all those who prepare for such examinations and they refer to the entire ...

Nidus Idearum. Scilogs, I: De Neutrosophia, 2016 University of New Mexico

#### Nidus Idearum. Scilogs, I: De Neutrosophia, Florentin Smarandache

*Mathematics and Statistics Faculty and Staff Publications*

Welcome into my scientific lab! My lab[oratory] is a virtual facility with noncontrolled conditions in which I mostly perform scientific meditation and chats: a nest of ideas (nidus idearum, in Latin). I called the jottings herein scilogs (truncations of the words scientific, and gr. Λόγος – appealing rather to its original meanings "ground", "opinion", "expectation"), combining the welly of both science and informal (via internet) talks (in English, French, and Romanian). In this first books of scilogs collected from my nest of ideas, one may find new and old questions and solutions, some of them already put at work, others ...

Nidus Idearum. Scilogs, Ii: De Rerum Consectatione, 2016 University of New Mexico

#### Nidus Idearum. Scilogs, Ii: De Rerum Consectatione, Florentin Smarandache

*Mathematics and Statistics Faculty and Staff Publications*

Welcome into my scientific lab! My lab[oratory] is a virtual facility with noncontrolled conditions in which I mostly perform scientific meditation and chats: a nest of ideas (nidus idearum, in Latin). I called the jottings herein scilogs (truncations of the words scientific, and gr. Λόγος – appealing rather to its original meanings "ground", "opinion", "expectation"), combining the welly of both science and informal (via internet) talks (in English, French, and Romanian). In this second book of scilogs collected from my nest of ideas, one may find new and old questions and solutions, some of them already put at work, others ...

Luhn Primes Of Order Ω, 2016 University of New Mexico

#### Luhn Primes Of Order Ω, Florentin Smarandache, Octavian Cira

*Mathematics and Statistics Faculty and Staff Publications*

A prime p, that added to its reverse results in a new prime is called a Luhn prime . The number 229 is a Luhn prime because 229 + 922 = 1151 and 1151 is also a prime . We extend the definition of Luhn primes, present some of the of their properties and ways to determine them .

Various Arithmetic Functions And Their Applications, 2016 University of New Mexico

#### Various Arithmetic Functions And Their Applications, Florentin Smarandache, Octavian Cira

*Mathematics and Statistics Faculty and Staff Publications*

Over 300 sequences and many unsolved problems and conjectures related to them are presented herein. These notions, deﬁnitions, unsolved problems, questions, theorems corollaries, formulae, conjectures, examples, mathematical criteria, etc. on integer sequences, numbers, quotients, residues, exponents, sieves, pseudo-primes squares cubes factorials, almost primes, mobile periodicals, functions, tables, prime square factorial bases, generalized factorials, generalized palindromes, so on, have been extracted from the Archives of American Mathematics (University of Texas at Austin) and Arizona State University (Tempe): "The Florentin Smarandache papers" special collections, University of Craiova Library, and Arhivele Statului (Filiala Vâlcea & Filiala Dolj, România). The book is based on various ...

Applying Dijkstra Algorithm For Solving Neutrosophic Shortest Path Problem, 2016 University of New Mexico

#### Applying Dijkstra Algorithm For Solving Neutrosophic Shortest Path Problem, Florentin Smarandache, Luige Vladareanu, Said Broumi, Assia Bakali, Muhammad Akram

*Mathematics and Statistics Faculty and Staff Publications*

The selection of shortest path problem is one the classic problems in graph theory. In literature, many algorithms have been developed to provide a solution for shortest path problem in a network. One of common algorithms in solving shortest path problem is Dijkstra’s algorithm. In this paper, Dijkstra’s algorithm has been redesigned to handle the case in which most of parameters of a network are uncertain and given in terms of neutrosophic numbers. Finally, a numerical example is given to explain the proposed algorithm.

Special Type Of Fixed Point Pairs Using Mod Rectangular Matrix Operators, 2016 University of New Mexico

#### Special Type Of Fixed Point Pairs Using Mod Rectangular Matrix Operators, Florentin Smarandache, W.B. Vasantha Kandasamy, K. Ilanthenral

*Mathematics and Statistics Faculty and Staff Publications*

In this book authors for the first time define a special type of fixed points using MOD rectangular matrices as operators. In this case the special fixed points or limit cycles are pairs which is arrived after a finite number of iterations. Such study is both new and innovative for it can find lots of applications in mathematical modeling. Since all these Zn or I nZ or 〈Zn ∪ g〉 or 〈Zn ∪ g〉I or C(Zn) or CI(Zn) are all of finite order we are sure to arrive at a MOD fixed point pair or a MOD limit cycle ...

Single Valued Neutrosophic Graphs: Degree, Order And Size, 2016 University of New Mexico

#### Single Valued Neutrosophic Graphs: Degree, Order And Size, Florentin Smarandache, Said Broumi, Mohamed Talea, Assia Bakali

*Mathematics and Statistics Faculty and Staff Publications*

The single valued neutrosophic graph is a new version of graph theory presented recently as a generalization of fuzzy graph and intuitionistic fuzzy graph. The single valued neutrosophic graph (SVN-graph) is used when the relation between nodes (or vertices) in problems are indeterminate. In this paper, we examine the properties of various types of degrees, order and size of single valued neutrosophic graphs and a new definition for regular single valued neutrosophic graph is given.

Problems On Mod Structures, 2016 University of New Mexico

#### Problems On Mod Structures, Florentin Smarandache, W.B. Vasantha Kandasamy, K. Ilanthenral

*Mathematics and Statistics Faculty and Staff Publications*

In this book authors for the first time give several types of problems on MOD structures happens to be an interesting field of study as it makes the whole 4 quadrant plane into a single quadrant plane and the infinite line into a half closed open interval. So study in this direction will certainly yield several interesting results. The law of distributivity is not true. Further the MOD function in general do not obey all the laws of integration or differentiation. Likewise MOD polynomials in general do not satisfy the basic properties of polynomials like its roots etc. Thus over ...

Kronecker's Theory Of Binary Bilinear Forms With Applications To Representations Of Integers As Sums Of Three Squares, 2016 University of Kentucky

#### Kronecker's Theory Of Binary Bilinear Forms With Applications To Representations Of Integers As Sums Of Three Squares, Jonathan A. Constable

*Theses and Dissertations--Mathematics*

In 1883 Leopold Kronecker published a paper containing “a few explanatory remarks” to an earlier paper of his from 1866. His work loosely connected the theory of integral binary bilinear forms to the theory of integral binary quadratic forms. In this dissertation we discover the statements within Kronecker's paper and offer detailed arithmetic proofs. We begin by developing the theory of binary bilinear forms and their automorphs, providing a classification of integral binary bilinear forms up to equivalence, proper equivalence and complete equivalence.

In the second chapter we introduce the class number, proper class number and complete class number ...

A Measure Theoretic Approach To Problems Of Number Theory With Applications To The Proof Of The Prime Number Theorem, 2016 Minnesota State University Mankato

#### A Measure Theoretic Approach To Problems Of Number Theory With Applications To The Proof Of The Prime Number Theorem, Russell Lee Jahn

*All Graduate Theses, Dissertations, and Other Capstone Projects*

In this paper we demonstrate how the principles of measure theory can be applied effectively to problems of number theory. Initially, necessary concepts from number theory will be presented. Next, we state standard concepts and results from measure theory to which we will need to refer. We then develop our repertoire of measure theoretic machinery by constructing the needed measures and defining a generalized version of the multiplicative convolution of measures. A suitable integration by parts formula, one that is general enough to handle various combinations of measures, will then be derived. At this juncture we will be ready to ...

Rainbow Arithmetic Progressions, 2016 Iowa State University

#### Rainbow Arithmetic Progressions, Steve Butler, Craig Erickson, Leslie Hogben, Kirsten Hogenson, Lucas Kramer, Richard Kramer, Jephian C. H. Lin, Ryan R. Martin, Derrick Stolee, Nathan Warnberg, Michael Young

*Mathematics Publications*

In this paper, we investigate the anti-Ramsey (more precisely, anti-van der Waerden) properties of arithmetic progressions. For positive integers n and k, the expression aw([n]; k) denotes the smallest number of colors with which the integers f1; : : : ; ng can be colored and still guarantee there is a rainbow arithmetic progression of length k. We establish that aw([n]; 3) = (log n) and aw([n]; k) = n1o(1) for k 4. For positive integers n and k, the expression aw(Zn; k) denotes the smallest number of colors with which elements of the cyclic group of order n can be ...

A Partition Function Connected With The Göllnitz-Gordon Identities, 2016 Georgia Southern University

#### A Partition Function Connected With The Göllnitz-Gordon Identities, Nicolas A. Smoot

*Electronic Theses and Dissertations*

Nearly a century ago, the mathematicians Hardy and Ramanujan established their celebrated circle method to give a remarkable asymptotic expression for the unrestricted partition function. Following later improvements by Rademacher, the method was utilized by Niven, Lehner, Iseki, and others to develop rapidly convergent series representations of various restricted partition functions. Following in this tradition, we use the circle method to develop formulas for counting the restricted classes of partitions that arise in the Gollnitz-Gordon identities. We then show that our results are strongly supported by numerical tests. As a side note, we also derive and compare the asymptotic behavior ...