Showing 51 - 60 of 94 Items
On the Dirichlet L-functions and the L-functions of Cusp Forms
Date: 2021-01-01
Creator: Nawapan Wattanawanichkul
Access: Open access
- The main objects of our study are L-functions, which are meromorphic functions on the complex plane that analytically continue from the series of the form \sum_{n=1}^{\infty} \frac{a_n}{n^s}, where {a_n} is a sequence of complex numbers. In particular, we are interested in two families of L-functions: ''The Dirichlet L-functions" and ''the L-functions of cusp forms." The former refers to the L-functions whose a_n's are determined by Dirichlet characters, whereas cusp forms determine the latter. We begin our study with the celebrated Riemann zeta function, the simplest Dirichlet L-function, and discuss some of its well-known properties: the Euler product, analytic continuation, functional equation, Riemann hypothesis, and Euler's formula for its critical values. Then, we generalize our exploration to the Dirichlet L-functions and point out some analogous properties to those of the Riemann zeta function. Moreover, we present our original work on computing the critical values of the Dirichlet L-function associated with the primitive character mod 4, or what is known as the Dirichlet beta function. Lastly, we establish some knowledge of the theory of modular forms and cusp forms, which are nicely-behaved modular forms, and discuss some properties of the L-functions of cusp forms.
Combinatorial and metric properties of Thompson's group t
Date: 2009-02-01
Creator: José Burillo, Sean Cleary, Melanie Stein, Jennifer Taback
Access: Open access
- We discuss metric and combinatorial properties of Thompson's group T, including normal forms for elements and unique tree pair diagram representatives. We relate these properties to those of Thompson's group F when possible, and highlight combinatorial differences between the two groups. We define a set of unique normal forms for elements of T arising from minimal factorizations of elements into natural pieces. We show that the number of carets in a reduced representative of an element of T estimates the word length, that F is undistorted in T, and we describe how to recognize torsion elements in T. © 2008 American Mathematical Society Reverts to public domain 28 years from publication.
Extinction in competitive lotka-volterra systems
Date: 1995-01-01
Creator: Mary Lou Zeeman
Access: Open access
- It is well known that for the two species autonomous competitive Lotka-Volterra model with no fixed point in the open positive quadrant, one of the species is driven to extinction, whilst the other population stabilises at its own carrying capacity. In this paper we prove a generalisation of this result to arbitrary finite dimension. That is, for the n-species autonomous competitive Lotka-Volterra model, we exhibit simple algebraic criteria on the parameters which guarantee that all but one of the species is driven to extinction, whilst the one remaining population stabilises at its own carrying capacity. © 1995 American Mathematical Society.
An alternative almost sure construction of Gaussian stochastic processes in the L2([0,1]) space
Date: 2019-05-01
Creator: Kevin Chen
Access: Open access
Higher rank lamplighter groups are graph automatic
Date: 2018-02-15
Creator: Sophie Bérubé, Tara Palnitkar, Jennifer Taback
Access: Open access
- We show that the higher rank lamplighter groups, or Diestel–Leader groups Γd(q) for d≥3, are graph automatic. This introduces a new family of graph automatic groups which are not automatic.
Sign changes of Fourier coefficients of Hilbert modular forms
Date: 2014-01-01
Creator: Jaban Meher, Naomi Tanabe
Access: Open access
- Sign changes of Fourier coefficients of various modular forms have been studied. In this paper, we analyze some sign change properties of Fourier coefficients of Hilbert modular forms, under the assumption that all the coefficients are real. The quantitative results on the number of sign changes in short intervals are also discussed. © 2014 Elsevier Inc.
Nonlinear excitations in magnetic lattices with long-range interactions
Date: 2019-06-24
Creator: Miguel Molerón, C. Chong, Alejandro J. Martínez, Mason A. Porter, P. G., Kevrekidis, Chiara Daraio
Access: Open access
- We study - experimentally, theoretically, and numerically - nonlinear excitations in lattices of magnets with long-range interactions. We examine breather solutions, which are spatially localized and periodic in time, in a chain with algebraically-decaying interactions. It was established two decades ago (Flach 1998 Phys. Rev. E 58 R4116) that lattices with long-range interactions can have breather solutions in which the spatial decay of the tails has a crossover from exponential to algebraic decay. In this article, we revisit this problem in the setting of a chain of repelling magnets with a mass defect and verify, both numerically and experimentally, the existence of breathers with such a crossover.
Thompson's group F is not almost convex
Date: 2003-12-01
Creator: Sean Cleary, Jennifer Taback
Access: Open access
- We show that Thompson's group F does not satisfy Cannon's almost convexity condition AC(n) for any positive integer n with respect to the standard generating set with two elements. To accomplish this, we construct a family of pairs of elements at distance n from the identity and distance 2 from each other, which are not connected by a path lying inside the n-ball of length less than k for increasingly large k. Our techniques rely upon Fordham's method for calculating the length of a word in F and upon an analysis of the generators' geometric actions on the tree pair diagrams representing elements of F. © 2003 Elsevier Inc. All rights reserved.
Tempered, invariant, positive-definite distributions on SU(1, 1)/{± 1}
Date: 1984-01-01
Creator: William H. Barker
Access: Open access
Knuth relations for the hyperoctahedral groups
Date: 2009-06-01
Creator: Thomas Pietraho
Access: Open access
- C. Bonnafé, M. Geck, L. Iancu, and T. Lam have conjectured a description of Kazhdan-Lusztig cells in unequal parameter Hecke algebras of type B which is based on domino tableaux of arbitrary rank. In the integer case, this generalizes the work of D. Garfinkle. We adapt her methods and construct a family of operators which generate the equivalence classes on pairs of arbitrary rank domino tableaux described in the above conjecture. © 2008 Springer Science+Business Media, LLC.