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A Tale Of Two Workshops: Two Workshops, Three Papers, New Ideas, Gizem Karaali 2012 Pomona College

A Tale Of Two Workshops: Two Workshops, Three Papers, New Ideas, Gizem Karaali

Pomona Faculty Publications and Research

No abstract provided.


On A Quantum Form Of The Binomial Coefficient, Eric J. Jacob 2012 University of Tennessee Space Institute

On A Quantum Form Of The Binomial Coefficient, Eric J. Jacob

Masters Theses

A unique form of the quantum binomial coefficient (n choose k) for k = 2 and 3 is presented in this thesis. An interesting double summation formula with floor function bounds is used for k = 3. The equations both show the discrete nature of the quantum form as the binomial coefficient is partitioned into specific quantum integers. The proof of these equations has been shown as well. The equations show that a general form of the quantum binomial coefficient with k summations appears to be feasible. This will be investigated in future work.


Elliptic Curves Of High Rank, Cecylia Bocovich 2012 Macalester College

Elliptic Curves Of High Rank, Cecylia Bocovich

Mathematics, Statistics, and Computer Science Honors Projects

The study of elliptic curves grows out of the study of elliptic functions which dates back to work done by mathematicians such as Weierstrass, Abel, and Jacobi. Elliptic curves continue to play a prominent role in mathematics today. An elliptic curve E is defined by the equation, y2 = x3 + ax + b, where a and b are coefficients that satisfy the property 4a3 + 27b2 = 0. The rational solutions of this curve form a group. This group, denoted E(Q), is known as the Mordell-Weil group and was proved by Mordell to be isomorphic to Zr ⊕ E ...


Circular Units Of Function Fields, Frederick Harrop 2012 Rhode Island College

Circular Units Of Function Fields, Frederick Harrop

Frederick F Harrop

A unit index-class number formula is proved for subfields of cyclotomic function fields in analogy with similar results for subfields of cyclotomic number fields.


Computing Local Constants For Cm Elliptic Curves, Sunil Chetty, Lung Li 2012 College of Saint Benedict/Saint John's University

Computing Local Constants For Cm Elliptic Curves, Sunil Chetty, Lung Li

Sunil Chetty

Let E/k be an elliptic curve with CM by O. We determine a formula for (a generalization of) the arithmetic local constant of Mazur-Rubin at almost all primes of good reduction. We apply this formula to the CM curves defined over Q and are able to describe extensions F/Q over which the O-rank of E grows.


Splitting Fields And Periods Of Fibonacci Sequences Modulo Primes, Sanjai Gupta, Parousia Rockstroh '08, Francis E. Su 2012 Irvine Valley College

Splitting Fields And Periods Of Fibonacci Sequences Modulo Primes, Sanjai Gupta, Parousia Rockstroh '08, Francis E. Su

All HMC Faculty Publications and Research

We consider the period of a Fibonacci sequence modulo a prime and provide an accessible, motivated treatment of this classical topic using only ideas from linear and abstract algebra. Our methods extend to general recurrences with prime moduli and provide some new insights. And our treatment highlights a nice application of the use of splitting fields that might be suitable to present in an undergraduate course in abstract algebra or Galois theory.


An Interesting Opportunity: The Gilbreath Conjecture, Kyle Sturgill-Simon 2012 Carroll College, Helena, MT

An Interesting Opportunity: The Gilbreath Conjecture, Kyle Sturgill-Simon

Mathematics, Engineering and Computer Science Undergraduate Theses

Unsolved mathematical conjectures are a rich source of more than just frustration. By looking at the Gilbreath conjecture (unproven since 1878), we explore the learning opportunities that these unsolved problems present. Through explanation of the workings of the conjecture and the previous attempts by mathematicians the reader can develop an understanding of the number theory involved and even get started on a project of their own. This paper provides context for the Gilbreath conjecture so that an entry level mathematician (or person of any study) can get acquainted with an important but relatively unknown portion of mathematics. It also summarizes ...


Prove It!, Kenny W. Moran 2012 Pomona College

Prove It!, Kenny W. Moran

Journal of Humanistic Mathematics

A dialogue between a mathematics professor, Frank, and his daughter, Sarah, a mathematical savant with a powerful mathematical intuition. Sarah's intuition allows her to stumble into some famous theorems from number theory, but her lack of academic mathematical background makes it difficult for her to understand Frank's insistence on the value of proof and formality.


Ramanujan Sums As Supercharacters, Christopher F. Fowler '12, Stephan Ramon Garcia, Gizem Karaali 2012 Pomona College

Ramanujan Sums As Supercharacters, Christopher F. Fowler '12, Stephan Ramon Garcia, Gizem Karaali

Pomona Faculty Publications and Research

The theory of supercharacters, recently developed by Diaconis-Isaacs and Andre, can be used to derive the fundamental algebraic properties of Ramanujan sums. This machinery frequently yields one-line proofs of difficult identities and provides many novel formulas. In addition to exhibiting a new application of supercharacter theory, this article also serves as a blueprint for future work since some of the abstract results we develop are applicable in much greater generality.


Introduction Aux Méthodes D’Intégrale De Chemin Et Applications, Nour-Eddiine Fahssi 2012 SelectedWorks

Introduction Aux Méthodes D’Intégrale De Chemin Et Applications, Nour-Eddiine Fahssi

Nour-Eddine Fahssi

These lecture notes are based on a master course given at University Hassan II - Agdal in spring 2012.


Computing Local Constants Of Cm Elliptic Curves, Sunil Chetty, Lung Li 2011 College of Saint Benedict/Saint John's University

Computing Local Constants Of Cm Elliptic Curves, Sunil Chetty, Lung Li

Sunil Chetty

Let E/k be an elliptic curve with CM by O. We determine a formula for (a generalization of) the arithmetic local constant of Mazur-Rubin at almost all primes of good reduction. We apply this formula to the CM curves defined over Q and are able to describe extensions F/Q over which the O-rank of E grows.


Resolutions Of The Steinberg Module For Gl(N), Avner Ash, Paul E. Gunnells, Mark McConnell 2011 University of Massachusetts - Amherst

Resolutions Of The Steinberg Module For Gl(N), Avner Ash, Paul E. Gunnells, Mark Mcconnell

Paul Gunnells

We give several resolutions of the Steinberg representation St_n for the general linear group over a principal ideal domain, in particular over Z. We compare them, and use these results to prove that the computations in [AGM4] are definitive. In particular, in [AGM4] we use two complexes to compute certain cohomology groups of congruence subgroups of SL(4,Z). One complex is based on Voronoi's polyhedral decomposition of the symmetric space for SL(n,R), whereas the other is a larger complex that has an action of the Hecke operators. We prove that both complexes allow us to compute ...


Constructing Large Numbers With Cheap Computers, Andrew Shallue, Steven Hayman 2011 Illinois Wesleyan University

Constructing Large Numbers With Cheap Computers, Andrew Shallue, Steven Hayman

Scholarship

No abstract provided.


Constructing Large Numbers With Cheap Computers, Andrew Shallue, Steven Hayman 2011 Illinois Wesleyan University

Constructing Large Numbers With Cheap Computers, Andrew Shallue, Steven Hayman

Andrew Shallue

No abstract provided.


Infinitary Equivalence Of Zp- Modules With Nice Decomposition Bases, Rüdiger Göbel, Katrin Leistner, Peter Loth, Lutz Strüngmann 2011 Universitat Duisburg-Essen, Germany

Infinitary Equivalence Of Zp- Modules With Nice Decomposition Bases, Rüdiger Göbel, Katrin Leistner, Peter Loth, Lutz Strüngmann

Mathematics Faculty Publications

Warfield modules are direct summands of simply presented Zp - modules, or, alternatively, are Zp - modules possessing a nice decomposition basis with simply presented cokernel. They have been classified up to isomorphism by theor Ilm-Kaplansky and Warfield invariants. Taking a model theoretic point of view and using infinitary languages we give here a complete theoretic characterization of a large class of Zp - modules having a nice decomposition basis. As a corollary, we obtain the classical classification of countable Warfield modules. This generalizes results by Barwise and Eklof.


On Cyclotomic Primality Tests, Thomas Francis Boucher 2011 University of Tennessee, Knoxville

On Cyclotomic Primality Tests, Thomas Francis Boucher

Masters Theses

In 1980, L. Adleman, C. Pomerance, and R. Rumely invented the first cyclotomicprimality test, and shortly after, in 1981, a simplified and more efficient versionwas presented by H.W. Lenstra for the Bourbaki Seminar. Later, in 2008, ReneSchoof presented an updated version of Lenstra's primality test. This thesis presents adetailed description of the cyclotomic primality test as described by Schoof, along withsuggestions for implementation. The cornerstone of the test is a prime congruencerelation similar to Fermat's \little theorem" that involves Gauss or Jacobi sumscalculated over cyclotomic fields. The algorithm runs in very nearly polynomial time.This primality test ...


Higher Derivatives Of The Hurwitz Zeta Function, Jason Musser 2011 Western Kentucky University

Higher Derivatives Of The Hurwitz Zeta Function, Jason Musser

Masters Theses & Specialist Projects

The Riemann zeta function ζ(s) is one of the most fundamental functions in number theory. Euler demonstrated that ζ(s) is closely connected to the prime numbers and Riemann gave proofs of the basic analytic properties of the zeta function. Values of the zeta function and its derivatives have been studied by several mathematicians. Apostol in particular gave a computable formula for the values of the derivatives of ζ(s) at s = 0. The Hurwitz zeta function ζ(s,q) is a generalization of ζ(s). We modify Apostolʼs methods to find values of the derivatives of ζ(s ...


The Elliptic Curve Discrete Logarithm And Functional Graphs, Christopher J. Evans 2011 University of Arkansas

The Elliptic Curve Discrete Logarithm And Functional Graphs, Christopher J. Evans

Mathematical Sciences Technical Reports (MSTR)

The discrete logarithm problem, and its adaptation to elliptic curves, called the elliptic curve discrete logarithm problem (ECDLP) is an open problem in the field of number theory, and its applications to modern cryptographic algorithms are numerous. This paper focuses on a statistical analysis of a modification to the ECDLP, called the x-ECDLP, where one is only given the xcoordinate of a point, instead of the entire point. Focusing only on elliptic curves whose field of definition is smaller than the number of points, this paper attempts to find a statistical indication of underlying structure (or lack thereof) in the ...


The Combinatorialization Of Linear Recurrences, Arthur T. Benjamin, Halcyon Derks, Jennifer J. Quinn 2011 Harvey Mudd College

The Combinatorialization Of Linear Recurrences, Arthur T. Benjamin, Halcyon Derks, Jennifer J. Quinn

All HMC Faculty Publications and Research

We provide two combinatorial proofs that linear recurrences with constant coefficients have a closed form based on the roots of its characteristic equation. The proofs employ sign-reversing involutions on weighted tilings.


Cohomology Of Congruence Subgroups Of Sl4(Z). Iii, A Ash, PE Gunnells, M McConnell 2011 University of Massachusetts - Amherst

Cohomology Of Congruence Subgroups Of Sl4(Z). Iii, A Ash, Pe Gunnells, M Mcconnell

Paul Gunnells

In two previous papers we computed cohomology groups for a range of levels , where is the congruence subgroup of consisting of all matrices with bottom row congruent to mod . In this note we update this earlier work by carrying it out for prime levels up to . This requires new methods in sparse matrix reduction, which are the main focus of the paper. Our computations involve matrices with up to 20 million nonzero entries. We also make two conjectures concerning the contributions to for prime coming from Eisenstein series and Siegel modular forms.


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