Embedding Fractional Quantum Hall Solitons In M-Theory Compactifications, 2011 Selected Works

#### Embedding Fractional Quantum Hall Solitons In M-Theory Compactifications, Nour-Eddiine Fahssi

*Nour-Eddine Fahssi*

No abstract provided.

Resolutions Of The Steinberg Module For Gl(N), 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, 2011 Illinois Wesleyan University

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

*Scholarship*

No abstract provided.

Constructing Large Numbers With Cheap Computers, 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, 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 **Z**_{p }- modules, or, alternatively, are **Z**_{p }- 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** Z**_{p }- 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.

Higher Derivatives Of The Hurwitz Zeta Function, 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 ...

On Cyclotomic Primality Tests, 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 ...

The Elliptic Curve Discrete Logarithm And Functional Graphs, 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, 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, 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.

Cohomology Of Congruence Subgroups Of Sl(4, Z) Ii, 2011 University of Massachusetts - Amherst

#### Cohomology Of Congruence Subgroups Of Sl(4, Z) Ii, A Ash, Pe Gunnells, M Mcconnell

*Paul Gunnells*

In a previous paper [3] we computed cohomology groups H5(..0(N),C), where ..0(N) is a certain congruence subgroup of SL(4,Z), for a range of levels N. In this note we update this earlier work by extending the range of levels and describe cuspidal cohomology classes and additional boundary phenomena found since the publication of [3]. The cuspidal cohomology classes in this paper are the first cuspforms for GL(4) concretely constructed in terms of Betti cohomology.

Algebraic Properties Of A Family Of Generalized Laguerre Polynomials, 2011 University of Massachusetts - Amherst

#### Algebraic Properties Of A Family Of Generalized Laguerre Polynomials, F Hajir

*Farshid Hajir*

We study the algebraic properties of Generalized Laguerre Polynomials for negative integral values of the parameter. For integers r,n≥0 , we conjecture that L(−1−n−r)n(x)=∑nj=0(n−j+rn−j)xj/j! is a \Q -irreducible polynomial whose Galois group contains the alternating group on n letters. That this is so for r=n was conjectured in the 1950's by Grosswald and proven recently by Filaseta and Trifonov. It follows from recent work of Hajir and Wong that the conjecture is true when r is large with respect to n≥5 . Here we ...

Specializations Of One-Parameter Families Of Polynomials, 2011 University of Massachusetts - Amherst

#### Specializations Of One-Parameter Families Of Polynomials, F Hajir, S Wong

*Farshid Hajir*

Let K be a number field, and suppose λ(x,t)∈K[x,t] is irreducible over K(t). Using algebraic geometry and group theory, we describe conditions under which the K-exceptional set of λ, i.e. the set of α∈K for which the specialized polynomial λ(x,α) is K-reducible, is finite. We give three applications of the methods we develop. First, we show that for any fixed n≥10, all but finitely many K-specializations of the degree n generalized Laguerre polynomial L n (t) (x) are K-irreducible and have Galois group S n . Second, we study specializations ...

On The Galois Group Of Generalized Laguerre Polynomials, 2011 University of Massachusetts - Amherst

#### On The Galois Group Of Generalized Laguerre Polynomials, F Hajir

*Farshid Hajir*

Using the theory of Newton Polygons, we formulate a simple criterion for the Galois group of a polynomial to be “large.” For a fixed α∈ℚ-ℤ <0 , Filaseta and Lam have shown that the nth degree Generalized Laguerre Polynomial L n (α) (x)=∑ j=0 n n+α n-j(-x) j /j! is irreducible for all large enough n. We use our criterion to show that, under these conditions, the Galois group of L n (α) (x) is either the alternating or symmetric group on n letters, generalizing results of Schur for α=0,1,±1 2,-1-n.

Finitely Ramified Iterated Extensions, 2011 University of Massachusetts - Amherst

#### Finitely Ramified Iterated Extensions, W Aitken, F Hajir, C Maire

*Farshid Hajir*

Let K be a number field, t a parameter, F = K(t), and φ(x)∈ K [x] a polynomial of degree d ≥ 2. The polynomial Φn(x,t) = φ^n (x) − t ∈ F[x], where φ^ n = φ ^ φ ^ … ^ φ is the n-fold iterate of φ, is irreducible over F; we give a formula for its discriminant. Let F be the field obtained by adjoining to F all roots (in a fixed ) of Φn(x,t) for all n ≥ 1; its Galois group Gal(Fφ/F) is the iterated monodromy group of φ. The iterated extension Fφ is finitely ramified ...

Modular Forms And Elliptic Curves Over The Field Of Fifth Roots Of Unity, 2011 University of Massachusetts - Amherst

#### Modular Forms And Elliptic Curves Over The Field Of Fifth Roots Of Unity, Pe Gunnells, F Hajir, Dan Yasaki

*Farshid Hajir*

Let F be the cyclotomic field of fifth roots of unity. We computationally investigate modularity of elliptic curves over F

Distribution Of Prime Numbers,Twin Primes And Goldbach Conjecture, 2011 SelectedWorks

#### Distribution Of Prime Numbers,Twin Primes And Goldbach Conjecture, Subhajit Kumar Ganguly

*Subhajit Kumar Ganguly*

The following paper deals with the distribution of prime numbers, the twin prime numbers and the Goldbach conjecture. Starting from the simple assertion that prime numbers are never even, a rule for the distribution of primes is arrived at. Following the same approach, the twin prime conjecture and the Goldbach conjecture are found to be true.

Statistical Properties Of A Convoluted Beta-Weibull Distribution, 2011 Marshall University

#### Statistical Properties Of A Convoluted Beta-Weibull Distribution, Jianan Sun

*Theses, Dissertations and Capstones*

A new class of distributions recently developed involves the logit of the beta distribution. Among this class of distributions are the beta-normal (Eugene et.al. (2002)); beta-Gumbel (Nadarajah and Kotz (2004)); beta-exponential (Nadarajah and Kotz (2006)); beta-Weibull (Famoye et al. (2005)); beta-Rayleigh (Akinsete and Lowe (2008)); beta-Laplace (Kozubowski and Nadarajah (2008)); and beta-Pareto (Akinsete et al. (2008)), among a few others. Many useful statistical properties arising from these distributions and their applications to real life data have been discussed in the literature. One approach by which a new statistical distribution is generated is by the transformation of random variables having ...

Infinitesimal Time Scale Calculus, 2011 Marshall University

#### Infinitesimal Time Scale Calculus, Tom Cuchta

*Theses, Dissertations and Capstones*

Calculus has historically been fragmented into multiple distinct theories such as differential calculus, difference calculus, quantum calculus, and many others. These theories are all about the concept of what it means to "change", but in various contexts. Time scales calculus (introduced by Stefan Hilger in 1988) is a synthesis and extension of all the various calculi into a single theory. Calculus was originally approached with "infinitely small numbers" which fell out of use because the use of these numbers could not be justified. In 1960, Abraham Robinson introduced hyperreal numbers, a justification for their use, and therefore the original approach ...

Elliptic Curves: Minimally Spanning Prime Fields And Supersingularity, 2011 Bard College

#### Elliptic Curves: Minimally Spanning Prime Fields And Supersingularity, Travis Mcgrath

*Senior Projects Spring 2011*

Elliptic curves are cubic curves that have been studied throughout history. From Diophantus of Alexandria to modern-day cryptography, Elliptic Curves have been a central focus of mathematics. This project explores certain geometric properties of elliptic curves defined over finite fields.

Fix a finite field. This project starts by demonstrating that given enough elliptic curves, their union will contain every point in the affine plane. We then find the fewest curves possible such that their union still contains all these points. Using some of the tools discussed in solving this problem, we then explore what can be said about the number ...