Number Theory Seminar Schedule

Math and Stat Home Colloquia and Seminars Home

The Number Theory Seminar meets on Wednesdays at 3:30 PM, in the Chase Building. If you would like to give a talk, send an email to Rob Noble. We also maintain a mailing list for seminar announcements. To subscribe, send an email here.

Past Talks

Wednesday, April 9, 2014, 3:30 pm
Location:Chase 319
Speaker: Karl Dilcher
Dalhousie University
Title: Zeros and irreducibility of polynomials with gcd powers as coefficients
Abstract: We study the family of self-inversive polynomials of degree n, whose jth coefficient is gcd(n,j)^k, for each fixed integer k>0. We prove that these polynomials have all of their roots on the unit circle, with uniform angular distribution. In the process we prove some new results on Jordan's totient function. We also prove that these polynomials are irreducible, apart from an obvious linear factor, whenever n is a power of a prime, and conjecture that this holds for all n. (Joint work with Sinai Robins, Nanyang Technological University, Singapore).


Thursday, March 19, 2014, 3:30 pm
Location:Chase 319
Speaker: John Cosgrave
Dublin, Ireland
Title: Gauss factorials, Jacobi primes, and prime factors of generalized Fermat numbers
Abstract: In a just published paper at Acta Mathematica Hungarica, Karl Dilcher and I introduced the notion of a Gauss prime; these primes play an absolutely fundamental role in connection with our study of 'quarter Gauss factorials'. In a future paper we will introduce the notion of a Jacobi prime, and analogously these primes play the same (but much more complex) role with respect to one-third and one-sixth Gauss factorials. In this talk I shall attempt to convey some of the basic ideas. No previous knowledge of Gauss factorials will be presumed.


Thursday, January 29, 2014, 3:30 pm
Location:Chase 319
Speaker: Abdullah Al-Shaghay
Dalhousie University
Title: The Local-Global Principle: A Brief Introduction to the p-adic Numbers
Abstract: We will discuss the Local-Global Principle (also known as the Hasse Principle) in Number Theory to motivate an introduction to the p-adic numbers. Our main focus will be the Hasse-Minkowski theorem and its proof. We will then briefly discuss two well-known counter-examples to the Hasse Principle in Number Theory. If time permits, we will also discuss other examples of the Hasse-Principle outside of Number Theory.


Thursday, April 29, 2013, 10:00 am
Location:Chase 319
Speaker: Matilde Lalin
Université de Montreal
Title: Mahler measure and special values of L-functions
Abstract: The Mahler measure of a Laurent polynomial P is defined as the integral of log|P| over the unit torus with respect to the Haar measure. For multivariate polynomials, it often yields special values of L-functions. In this talk I will discuss some of these relationships and the meaning behind them and present some results involving L(E,3) for E an elliptic curve.


Thursday, February 21, 2013, 2:30 pm
Location:Chase 319
Speaker: Karl Dilcher
Dalhousie University
Title: Some number theoretic results due to Euler
Abstract: In the first part of this talk I will speak about Euler's importance in the development of number theory as a mathematical subject. I will then deal in greater detail with two very well-known results of Euler, related to his work on Fermat numbers and on Fermat's little theorem. This talk is a rerun, with some small changes, of the one I gave during the 2007 Euler symposium, and is different from my recent historical/ biographical presentation about Euler.


Thursday, January 24, 2013, 2:30 pm
Location:Chase 319
Speaker: Abdullah Al-Shaghay
Dalhousie University
Title: The Prime Number Theorem and Dirichlet's Theorem
Abstract: Last semester I completed a reading course which was an introduction to analytic number theory. In this presentation, we will have a look at what I felt were the two main theorems: The Prime Number Theorem and Dirichlet's Theorem on Primes in Arithmetic Progression. This presentation is meant to be an overview and no previous knowledge is assumed/required.


Monday, January 7, 2013, 12:30 pm
Location:Chase 319
Speaker: Timothy Caley
University of Waterloo
Title: A new algorithm for the Prouhet-Tarry-Escott problem
Abstract: Given natural numbers n and k, the Prouhet-Tarry-Escott (PTE) asks for integers x_1, ... , x_n and y_1, ... , y_n such that the sums of the first k powers are equal. This problem has connections to combinatorics and theoretical computer science, as well as to other areas of number theory, such as Waring's problem. The most interesting case is when k=n-1, which is called ideal. A major open problem is determining whether ideal PTE solutions exist for a given n, as well as characterizing those that do exist. Computational techniques have been used to search for PTE solutions. In this talk, we present a new algorithm to find PTE solutions, and explain how the results yield more information than other computational searches in the literature.


Thursday, November 29, 2012, 10:00 am
Location:Chase 227
Speaker: Kira Scheibelhut
Dalhousie University
Title: A Classification of Integer-valued Polynomials using a Generalized Factorial Function
Abstract: An integer-valued polynomial is a polynomial with rational coefficients that takes an integer value when evaluated at an integer. It is clear that every polynomial with integer coefficients is integer-valued. However, it is also possible for an integer-valued polynomial to have rational coefficients. For example, the polynomial x(x-1)/2 still maps the integers to the integers, since one of x and x-1 must be even. During my talk, I will introduce a classification of integer-valued polynomials on the integers and then show how to generalize this classification to subsets of the integers. In order to do this, I will first define a generalized factorial function using the associated p-sequence of the subset.


Thursday, October 25, 2012, 10:30 am
Location:Chase 227
Speaker: John Cosgrave
Dublin.
Title: The multiplicative orders of certain Gauss factorials (again)
Abstract: Departing from the well-known Wilson's Theorem of elementary number theory and a generalization due to Gauss, I will discuss more general ``Gauss factorials''. These are products of integers from 1 to (n-1)/M (mod n) and relatively prime to n, where n is congruent to 1 modulo M. In particular, I will present improved results on the multiplicative orders (mod n) of these products, where n is a power of a prime and M = 3, 4 and 6. These improvements - in particular a more elegant and much faster test for 'exceptionality' - relate to our extensions of the classic binomial coefficient congruences of Gauss and Jacobi, and a more recent one due to Hudson and Kenneth Williams. (Joint work with Karl Dilcher.)


Thursday, September 27, 2012, 10:00 am
Location:Chase 227
Speaker: Ram Murty
Queen's University
Title: Automorphy and the Sato-Tate conjecture
Abstract: The Sato-Tate conjecture for modular forms was proved recently using the formidable tools of Galois representations and automorphic L-functions. However, Langlands conjecture that the symmetric power L-functions attached to the conjecture should be automorphic has not yet been proved. We will show how a super special case of the Langlands functoriality conjecture along with some ideas of Ramanujan will imply this automorphy. Once this is in place, one can derive error terms in the Sato-Tate conjecture/theorem. This is joint work with Kumar Murty.


Thursday, March 22, 2012, 11:30 am
Location:Chase 227
Speaker: Tatiana Hessami Pilehrood
Shahrekord University and Dalhousie University
Title: Congruences concerning Jacobi polynomials, II
Abstract: In this talk we will prove some polynomial congruences modulo a power of a prime in terms of the finite polylogarithms. This refines several recent results of Z. W. Sun on supercongruences. As a consequence, we also obtain some numerical congruences involving central binomial coefficients 'and Fibonacci or Lucas numbers. (Joint work with Kh. Hessami Pilehrood and R. Tauraso)


Thursday, March 1, 2012, 11:30 am
Location:Chase 227
Speaker: Tony Vargas
Department of Mathematics and Statistics, Dalhousie University
Title: Zeros of Partial Sums of Power Series
Abstract: Given a power series which converges everywhere, what can we say about the zeros of the partial sums of the series? In this talk we will investigate this question from the viewpoint of asymptotic analysis. A brief overview will be given of a few (of the few) examples which have been previously studied, including the exponential function and the confluent hypergeometric functions. We'll then present a new result which partially answers the question for the Bessel functions.


Thursday, February 2, 2012, 11:30 am
Location:Chase 227
Speaker: Tatiana Hessami Pilehrood
Shahrekord University and Dalhousie University.
Title: Congruences concerning Jacobi polynomials and Apery-like formulae
Abstract: In this talk we will discuss how to obtain congruences for finite central binomial sums arising from the truncation of Apery-type series for zeta values. Applying functional properties of Jacobi polynomials we prove some polynomial congruences. We also establish a generalization of Morley's congruence for the central binomial coefficient binom{p-1}{(p-1)/2}, where p>5 is a prime, to modulo p^6. (Joint work with Kh. Hessami Pilehrood and R. Tauraso.)


Thursday, January 26, 2012, 11:30 am
Location:Chase 227
Speaker: Karl Dilcher
Department of Mathematics and Statistics, Dalhousie University.
Title: The Gauss-Wilson Theorem for Partial Products
Abstract: For positive integers M > 1 and n = 1 (mod M) we define the *Gauss factorial* ((n-1)/M)_n! to be the product of all integers up to (n-1)/M and relatively prime to n, a terminology suggested by Gauss's generalization of Wilson's theorem. While the multiplicative orders (mod n) of Gauss factorials are completely determined when M = 2, the general case presents numerous interesting challenges. After some general results, this talk will concentrate on the special cases M = 3 and M = 4. The binomial coefficient theorems of Gauss and Jacobi are important tools, as are certain Pell equations and their solutions. Some large-scale computations are also involved. (Joint work with John B. Cosgrave.)


Wednesday, January 11, 2012, 3:30 pm
Location:Chase 319
Speaker: Abdullah Al-Shaghay
Department of Mathematics and Statistics, Dalhousie University.
Title: An Irreducibility Criterion of Arthur Cohn
Abstract: We begin by introducing an interesting irreducibility criterion, which is attributed to Arthur Cohn, for polynomials belonging to the ring Z[x]. We will then take a look at generalizations of this criterion and discuss their consequences and applications in determining whether a polynomial is irreducible over a ring.


Wednesday, December 21, 2011, 11:00 am
Location:Chase 319
Speaker: Doug Staple
TU Dresden, Germany
Title: Introduction to Perfect Numbers
Abstract: Perfect numbers are a classic topic in mathematics, predating Euclid. Here we will review fundamental results due to Euclid and Euler, as well as modern efforts focused on the existence or nonexistence of odd perfect numbers. Classic results give necessary and sufficient conditions for an even number to be perfect, and 47 such numbers have been found to date. Much modern work centers around the question: Do odd perfect numbers exist? J. J. Sylvester revitalized and popularized this question with a series of papers in 1888, which showed that an odd perfect number would have to be divisible by at least five distinct primes. Since Sylvester, countless assaults have yielded numerous partial results, but the basic question remains unanswered.


Wednesday, November 16, 2011, 3:30 pm
Location:Chase 319
Speaker: Dante Manna
Department of Mathematics and Statistics, Dalhousie University.
Title: A new q-analogue for Bernoulli numbers
Abstract: We define and give properties of a q-analogue for the Bernoulli numbers B_n. Our definition is inspired by the development of Strodt numbers, a class of sequences whose generating function is the multiplicative inverse of the moment generating function of a probability distribution. The talk will begin with an outline of Strodt numbers, followed by a discussion of our q-Bernoulli numbers, including their properties and closed forms. We compare our results to a different q-analogue of B_n considered by Carlitz in the 1950s. This is joint work with O-Yeat Chan.


Wednesday, October 26, 2011, 3:30 pm
Location:Chase 319
Speaker: John Cosgrave
Dublin.
Title: Quarter Gauss Factorials assuming simplest value
Abstract: Karl Dilcher and I study Gauss factorials (they are beautiful). We are completing our work on the special case of quarter Gauss factorials, and of particular interest are the integers for which such factorials assume their simplest value: 1. These integers divide into two classes: the standard and non-standard. The *least* non-standard is 205479813, and there are *exactly* five other small ones (the largest of which is n1 = 133303423608267). We have proved a complete characterisation of such integers, as a result of which we can construct others, the least *known* of which (N1) has 155 decimal digits, and we can construct others, the largest *known* of which has 14306 decimal digits. There could be others between n1 and N1; but we do not know. How can we have a 'complete characterisation' and *not* know this? All will be made clear. No previous knowledge of number theory will be needed to follow this talk.


Wednesday, October 5, 2011, 3:30 pm
Location:Chase 319
Speaker: Keith Johnson
Department of Mathematics and Statistics, Dalhousie University.
Title: Integer valued polynomials on matrices
Abstract: Let M_n(Z) denote the ring of n x n matrices with integer coefficients. If f(x) is a polynomial with rational coefficients and M is a member of M_n(Z) then f(M) is again an n x n matrix. When is it in M_n(Z), i.e. when are its entries all integers? We will present some recent results, both computational results when f(x) is of low degree and general results for some subrings of M_n(Z).


Wednesday, September 28, 2011, 3:30 pm
Location:Chase 319
Speaker: Tony Vargas
Department of Mathematics and Statistics, Dalhousie University.
Title: Zeros and limit functions of Stern polynomials
Abstract: In this talk I'll present the results of my research on the Stern polynomials. In the first half of the talk I'll describe the asymptotic character of their zeros using a combination modern and classical tools. In the second half I will switch to a more structural point of view, examining closely the sequence's recursive definition to deduce a "useful" form for what I consider the important subsequences. Using this I will show that these subsequences (there are uncountably many of them!) have unique limit functions which are analytic on the open unit disk. This expands the class of limit functions of the sequence beyond the two Fibonacci-inspired selections presented by Dr. Dilcher last week--in fact we will find his functions among those considered in this talk. I'll conclude with a few new open problems relating to these results.


Wednesday, September 21, 2011, 3:30 pm
Location:Chase 319
Speaker: Karl Dilcher
Department of Mathematics and Statistics, Dalhousie University.
Title: Stern polynomials and continued fractions
Abstract: The Stern (diatomic) sequence is a little-known but fascinating and important sequence of positive integers. In this talk I define a polynomial analogue of the Stern sequence and derive various identities. I then define two subsequences of these polynomials and obtain various properties for these two interrelated subsequences which can be seen as extensions or analogues of the Fibonacci numbers. I also define two analytic functions as limits of these sequences. As an application we obtain evaluations of certain finite and infinite continued fractions whose partial quotients are doubly exponential. In a case of particular interest, the set of convergents has exactly two limit points.


Wednesday, March 30, 2011, 3:30 pm
Location:Chase 319
Speaker: Hester Graves
Department of Mathematics and Statistics, Queen's University.
Title: Euclidean Ideals
Abstract: Lenstra introduced Euclidean ideals, a generalization of Euclidean rings. I will give some background on what has been discovered in the last couple decades about Euclidean rings and how it relates to my research with M. Ram Murty on Euclidean ideals. This work uses both algebraic and analytic methods, so there should be something for everyone.


Wednesday, March 9, 2011, 3:30 pm
Location:Chase 319
Speaker: Karen Chandler
Department of Mathematics and Statistics, Dalhousie University.
Title: The Regularity of Singularity, III
Abstract: See here


Wednesday, March 2, 2011, 3:30 pm
Location:Chase 319
Speaker: Karen Chandler
Department of Mathematics and Statistics, Dalhousie University.
Title: The Regularity of Singularity, II
Abstract: See here


Wednesday, February 16, 2011, 3:30 pm
Location:Chase 319
Speaker: Karen Chandler
Department of Mathematics and Statistics, Dalhousie University.
Title: The Regularity of Singularity
Abstract: See here


Monday, December 6, 2010, 9:30 am
Location:Chase 227
Speaker: Karl Dilcher
Department of Mathematics and Statistics, Dalhousie University.
Title: Fermat's Last Theorem
Abstract: Fermat's Last Theorem states that for a fixed integer n > 2 the equation x^n + y^n = z^n has no solution in integers x, y, z. It is one of the most famous (no longer) unsolved problems in mathematics; in fact, attempts at solving it accelerated the development of whole areas of mathematics. I will begin this talk with a very brief historical account, and then give an overview of the ideas and methods that went into the eventual proof by Wiles and Taylor of 1993. This seminar talk is joint with the final class of MATH 4070/5070: Topics in Number Theory. However, no specific background in number theory is required; the talk is suitable for honours students and graduate students.


Wednesday, November 24, 2010, 3:30 pm
Location:Chase 319
Speaker: Rob Noble
Department of Mathematics and Statistics, Dalhousie University.
Title: Conjugate algebraic numbers on plane curves
Abstract: If an algebraic number lies with all of its conjugates on a single line in the complex plane, then the number must either be totally real (so that the line is the real axis) or have rational real part and totally real imaginary part (so that the line consists of those complex numbers with a fixed rational real part). Moving to the next level of complexity, we arrive at the conic sections. Roots of unity lie with all of their conjugates on the unit circle, and by a classical result of Kronecker, are the only examples of algebraic integers that satisfy this property. Work of Robinson, Ennola and Smyth completes the picture for circles, and Smyth and Berry have taken care of the other conic sections. It would be natural to next tackle the case of elliptic curves, but it may be easier to first treat curves of higher degree that arise as conic sections with respect to non-euclidean L^p norms. In this talk, an overview of the literature on conjugate algebraic numbers on lines and conics with respect to the euclidean norm, as well as some preliminary results for the unit circle with respect to non-euclidean L^p norms will be given.


Wednesday, November 3, 2010, 2:30 pm
Location:Chase 319
Speaker: Ping Zhou
Department of Mathematics, Statistics and Computer Science, St. Francis Xavier University.
Title: Some arithmetical results on certain multivariate power series
Abstract: We discuss and show that the irrationality and/or transcendence properties of some multivariate power series, such as multivariate exponential series, logarithmic series, partial theta series, and many others, can be obtained by using the linear independent properties of the values of the one variable projection of the multivariate series. Some of the results are published and some are to appear.


Wednesday, October 27, 2010, 3:30 pm
Location:Chase 319
Speaker: Karen A. Chandler
Title: Counting Polynomials with Higher-Order Singularities
Abstract: Please click here


Wednesday, June 30, 2010, 1:30 pm
Location:Chase 319
Speaker: Rob Noble
Department of Mathematics and Statistics, Dalhousie University.
Title: Asymptotics of a family of binomial sums
Abstract: Using a recent method of Pemantle and Wilson, the asymptotics of a family of combinatorial sums that involve products of two binomial coefficients and includes both alternating and non-alternating sums will be determined. With the exception of finitely many cases the main terms will be obtained explicitly, while the existence of a complete asymptotic expansion is established. A recent method by Flajolet and Sedgewick will be used to establish the existence of a full asymptotic expansion for the remaining cases, and the main terms will again be obtained explicitly. Among several specific examples, generalizations of the central Delannoy numbers and their alternating analogues will be considered.


Wednesday, June 30, 2010, 2:00 pm
Location:Chase 319
Speaker: Karl Dilcher
Department of Mathematics and Statistics, Dalhousie University.
Title: A mod p^3 analogues of a theorem of Gauss on binomial coefficients
Abstract: The theorem of Gauss that gives a modulo p evaluation of a certain central binomial coefficient was extended modulo p^2 by Chowla, Dwork, and Evans. In this talk I present a further extension to a congruence modulo p^3, with a similar extension of a theorem of Jacobi. This is done by first obtaining congruences to arbitrarly high powers of p for certain quotients resembling binomial coefficients and related to the p-adic gamma function. These congruences are of a very simple form and involve Catalan numbers as coefficients. As another consequence we obtain complete p-adic expansions for certain Jacobi sums. (Joint work with J.B. Cosgrave).


Wednesday, June 30, 2010, 2:30 pm
Location:Chase 319
Speaker: Karyn McLellan
Department of Mathematics and Statistics, Dalhousie University.
Title: Growth Rates of Recurrence Sequences with Periodic Coefficients
Abstract: This talk will extend some ideas from Viswanath's work on random Fibonacci sequences by looking at non-random cases. Specifically, I will look at second order linear recurrence sequences whose coefficients belong to the set {1, -1} and form periodic cycles. I will analyze the growth of such sequences and develop criteria for determining whether a given sequence is bounded, grows linearly or grows exponentially. Also, I will introduce an equivalence relation on the sequences such that each equivalence class has a common growth rate and consider the number of such classes for a given cycle length.


Wednesday, June 30, 2010, 3:00 pm
Location:Chase 319
Speaker: Chester Weatherby
Department of Mathematical Sciences, University of Delaware.
Title: Transcendence of Infinite Series with Applications of Baker's Theorem
Abstract: Determining whether or not a given series is algebraic or transcendental appears as early as Euler's theorem that the zeta function evaluated at even values is related to powers of pi times a rational number (and therefore transcendental). We will examine two families of (simple) infinite series and attempt to characterize whether nor not a given series is transcendental. Baker's Theorem on linear forms in logarithms is used heavily in the characterization and will be reviewed.


Wednesday, March 31, 2010, 3:30 pm
Location:Chase 319
Speaker: Karl Dilcher
Department of Mathematics and Statistics, Dalhousie University.
Title: The multiplicative orders of certain Gauss factorials
Abstract: Departing from the well-known Wilson's Theorem of elementary number theory and a generalization due to Gauss, I will discuss more general "Gauss factorials". These are products of integers from 1 to (n-1)/M (mod n) and relatively prime to n, where n = 1 (mod M). In particular, I will present results on the multiplicative orders (mod n) of these products, where n is a power of a prime and M = 2, 3, 4 and 6. Connections with certain Diophantine equations will also be discussed.


Wednesday, March 24, 2010, 3:30 pm
Location:Chase 319
Speaker: Eva Curry
Department of Mathematics and Statistics, Acadia University.
Title: Similar Matrices and Topological Properties of Self-Affine Tiles
Abstract: Multidimensional radix representations generate self-affine sets that tile $R^n$ under translation by integer vectors. These sets are of interest to people studying iterated function systems, as well as to wavelet theorists. Avra Laarakker and I have recently made progress in investigating topological properties of these tile sets. In this talk, I will present our results and methods, and discuss the possibility of solving some related open problems with our approach.


Wednesday, March 17, 2010, 3:30 pm
Location:Chase 319
Speaker: Dante Manna
Department of Mathematics and Computer Science, Virginia Wesleyan College
Title: Integrals of powers of loggamma
Abstract: We present properties of the integrals of the powers of $\log \Gamma x$ from 0 to 1.


Wednesday, March 10, 2010, 3:30 pm
Location:Chase 319
Speaker: Rob Noble
Department of Mathematics and Statistics, Dalhousie University
Title: Zero terms in linear recurrence sequences
Abstract: The theorem of Skolem-Mahler-Lech describes the possible sets of zero terms in sequences that satisfy linear recurrence relations with constant coefficients. In characteristic zero, they are comprised, up to a finite set, of finitely many infinite arithmetic progressions. Work of Bézivin and Methfessel shows that this result still holds true under suitable conditions for sequences that satisfy linear recurrence relations with coefficients that are generally nonconstant, assuming we replace the exceptional finite set with a set of density zero. In the constant coefficient case, we can eliminate the possibility of arithmetic progressions in case no two distinct eigenvalues of our sequence share a common power. In the general case, we can eliminate the possibility of arithmetic progressions in case the sequence cannot be sectioned to obtain sequences of lower order. In this talk, it will be shown that this latter condition reduces to the former in case of constant coefficients and a unified result will be obtained.


Wednesday, March 3, 2010, 3:30 pm
Location:Chase 319
Speaker: Karyn McLellan
Department of Mathematics and Statistics, Dalhousie University
Title: Growth Rates of Recurrence Sequences with Periodic Coefficients
Abstract: This talk will extend some ideas from Viswanath's work on random Fibonacci sequences by looking at non-random cases. Specifically, I will look at second order linear recurrence sequences whose coefficients belong to the set {1, -1} and form periodic cycles. I will analyze the growth of such sequences and develop criteria for determining whether a given sequence is bounded, grows linearly or grows exponentially. Also, I will introduce an equivalence relation on the sequences such that each equivalence class has a common growth rate and consider the number of such classes for a given cycle length.


Wednesday, February 10, 2010, 3:30 pm
Location:Chase 319
Speaker: Mark Pavlovski
Department of Mathematics and Statistics, Dalhousie University
Title: Prime Factorization in Hurwitz Quaternions.
Abstract: In this talk I will explore prime factorization of elements in this interesting non-commutative ring.


Wednesday, February 3, 2010, 3:30 pm
Location:Chase 319
Speaker: Keith Johnson
Department of Mathematics and Statistics, Dalhousie University
Title: Limits and Integer Valued Polynomials
Abstract: The study of algebras of rational polynomials taking integer values on a specified subset of the integers has a long history and a many applications. This talk will present an elementary limit formula for integer sequences which gives a useful numerical invariant of such algebras.


Wednesday, January 27, 2010, 3:30 pm
Location:Chase 319
Speaker: Karl Dilcher
Department of Mathematics and Statistics, Dalhousie University
Title: Dedekind Sums and Uniform Distributions
Abstract: Dedekind sums have important applications in various parts of number theory, and they have been studied quite extensively. By far the most important property is the reciprocity law, which has many consequences and also allows for an easy computation of Dedekind sums. In this talk I will give a new elementary proof of the reciprocity law, based on uniform distributions of integers in subintervals of the real line.


Wednesday, January 20, 2010, 3:30 pm
Location:Chase 319
Speaker: Dante Manna
Department of Mathematics and Computer Science, Virginia Wesleyan College
Title: Extensions of Strodt Polynomials and Numbers
Abstract: We report recent attempts to extend properties of Strodt Polynomials and Numbers. A sequence of Strodt Polynomials is an Appell sequence whose generating function comes from the moment generating function of a probability distribution. These generalize well-known sequences of polynomials such as Bernoulli, Euler, Hermite. Strodt Numbers are the constant terms of the polynomials.


Wednesday, July 15, 2009, 3:30 pm
Location:Chase 319
Speaker: Theodore Kolokolnikov
Department of Mathematics and Statistics, Dalhousie University
Title: Combinatorics and PDEs
Abstract: Let a(n) denote the number of sign choices + and - such that +/-1 +/- 2 +/- 3 +- ・ ・ ・+/- n = 0. For example when n=3 we have 1+2-3=0 and -1-2+3=0 so a(3)=2. We are interested to know how a(n) grows as a function of n. In the limit of large n, we will derive an asymptotic formula for a(n) by using the fundamental solution of the heat equation. We will also investigate a more general question: given integers n,m, let b(m,n) be the number of partitions of the set {0, 1, 2, ..., n} that add up to m. We derive an asymptotic formula for b(m,n) when n>>1 and m=O(n^2).


Wednesday, Jun 17, 2009, 3:30 pm
Location:Chase 319
Speaker: Richard McIntosh
Department of Mathematics and Statistics, University of Regina
Title: Mock Theta Functions
Abstract: In his last letter to G.H. Hardy, Ramanujan explained his discovery of the mock theta functions and listed 17 examples to which he assigned orders 3, 5 and 7. Some functions with even order were found in his "lost notebook". In the early 90's, Gordon and McIntosh discovered some 8th order mock theta functions and obtained their modular transformation laws. They went on to show that all of the classical mock theta functions can be expressed in terms of ordinary theta functions and specializations of the function g2(x, q). In the 21st century the subject experienced a renaissance. Among other things this led to connections with L-series, Donaldson invariants and such physical concepts as Maass wave forms and string theory. A brief description of the development of the theory of mock theta functions will be given in this presentation.


Wednesday, Jun 3, 2009, 3:30 pm
Location:Chase 319
Speaker: Rob Noble
Department of Mathematics and Statistics, Dalhousie University
Title: Asymptotics of holonomic sequences
Abstract: Let F(z) be the ordinary generating function of a sequence a(n). One expects the asymptotic behaviour of a(n) as n tends to infinity to be dictated by the asymptotic behaviour of F(z) near its singularities. a(n) is called holonomic if it satisfies a linear recurrence relation with polynomial coefficients. In this case F(z) satisfies a linear differential equation with polynomial coefficients. The behaviour of F(z) near its singularities can then be determined by applying singularity analysis to this differential equation. In addition, when F(z) is algebraic, we can also look at the singularities of an associated algebraic curve to get more info. Finally, if a(n) belongs to a third class, which will be described in the talk, then the singularities of F(z) can be analyzed directly. In this talk, I will illustrate what information each of these viewpoints can provide by proving a recent conjecture of Marc Chamberland and Karl Dilcher regarding the asymptotics of a particular sequence of binomial sums. I will then discuss the extent to which the asymptotics of this sequence are understood, and describe what remains to be determined.


Wednesday, Apr 8, 2009, 3:30 pm
Location:Chase 319
Speaker: Franklin Mendivil
Department of Mathematics and Statistics, Acadia University
Title: Random iteration of functions
Abstract: The dynamics of the iteration of some function is a classical area of study within dynamical systems, iteration of complex polynomials being a particularly striking example. It is no surprise then that adding randomness is a natural thing to do, particularly since any Markov chain can be viewed a random iteration of functions.

In this talk we give a brief background on general results in the area of convergence theorems for random iteration of functions, with particular attention to the case of families of random contractions. After this we concentrate on some results on a simple model of time inhomogeneous random iteration. As is the case for Markov chains, allowing the dynamics to vary with time presents new complications.


Wednesday, Jan 14, 2009, 3:30 pm
Location:Chase 319
Speaker: Tony Thompson
Department of Mathematics and Statistics, Dalhousie University
Title: Some of the other Busemann-Petty problems
Abstract: What became known as the Busemann-Petty problem was the first of 10 in their paper in Math. Scand 4 (1956). I would like to draw attention to three others: problems 5,7 and 10. In the talk I will outline what is known about these and offer some possible approaches to Problem 5.


Wednesday, Nov 26, 2008, 3:30 pm
Location:Chase 319
Speaker: Karl Dilcher
Department of Mathematics and Statistics, Dalhousie University
Title: The theorems of Gauss and Jacobi on binomial coefficients
Abstract: The theorem of Gauss that gives a modulo $p$ evaluation of a certain central binomial coefficient has been extended modulo $p^2$ by Chowla, Dwork, and Evans. In this talk we extend it further to a congruence modulo $p^3$. We derive a similar extension of a theorem of Jacobi. In the process we prove congruences to arbitrarily high powers of $p$ for certain quotients resembling binomial coefficients and related to the $p$-adic gamma function. These congruences are of a very simple form and involve Catalan numbers as coefficients. As another consequence we obtain complete $p$- adic expansions for certain Jacobi sums.

(Joint work with John B. Cosgrave.)


Wednesday, Oct 29, 2008, 3:30 pm
Location:Chase 319
Speaker: Tony Thompson
Department of Mathematics and Statistics, Dalhousie University
Title: How large is the unit ball?
Abstract: It is fairly well known that the perimeter of a unit disc in a normed plane lies in the interval [6,8]. A natural question is: what are the bounds for the surface area of the unit ball in a three dimensional normed space? in $R^n$? The talk will describe what is known about these questions.


Friday, Oct 24, 2008, 1:30 pm
Location:Chase 227
Speaker: Daniel Klain
Department of Mathematical Sciences, University of Massachusetts Lowell
Title: Containment and Inscribed Simplices
Abstract: A theorem of Lutwak asserts that a convex body K can be translated inside another convex body L if and only if K can be translated inside every circumscribing simplex of L. The proof of this result involves an application of Helly's theorem, and can be generalized in a number of ways. We will present a variation of this result that uses inscribed (instead of circumscribed) simplices, and then extend this result to questions about coverings of lower dimensional shadows. In particular, we will see that the m-dimensional shadows of a convex body K can be translated inside corresponding shadows of a convex body L if and only if every m-dimensional simplex inside K can be translated inside L. Additional consequences are then explored.

This talk will expand on some themes mentioned in my Thursday (colloquium) talk, however the two talks are independent, and listeners need not attend one talk in order to follow the other.


Wednesday, Oct 1, 2008, 3:30 pm
Location:Chase 319
Speaker: Karl Dilcher
Department of Mathematics and Statistics, Dalhousie University
Title: On Multiple Zeros of Bernoulli Polynomials
Abstract: Building on results of Brillhart (1969), it is shown that Bernoulli polynomials have no multiple zeros. The proof uses the theorem of von Staudt and Clausen as well as congruences for certain sums of binomial coefficients.


Wednesday, Sep 3, 2008, 3:30 pm
Location:Chase 319
Speaker: Lutz G. Lucht
Institut für Mathematik, Technische Universität Clausthal, Germany
Title: Inequalities in Abstract Prime Number Theory
Abstract: Let $\mathcal{N}\subset[1,\infty)$ be a discrete multiplicative semigroup of real numbers with $1\in\mathcal{N}$ and a countable set $\mathcal{P}\subset\mathcal{N}$ of free generators (prime elements). Basic results concerning the distribution of primes in $\mathcal{N}$ remain valid in arithmetic semigroups $\mathcal{N}$ satisfying
$$ \#\{n\in\mathcal{N}:n\leq x\}=ax+\textrm{O}\big(x\log^{-\eta}(ex)\big) \qquad (x\geq 1) $$
with constants $a>0$ and $\eta>1$. The talk presents new proofs for abstract versions of Chebyshev's inequality and Mertens' estimate.


Thursday, Aug 28, 2008, 2:00 pm
Location:Chase 319
Speaker: Lutz G. Lucht
Institut für Mathematik, Technische Universität Clausthal, Germany
Title: Banach Algebra Techniques in the Theory of Arithmetic Functions
Abstract: For infinite discrete additive semigroups $X\subset[0,\infty)$ we study normed algebras of arithmetic functions $g : X \rightarrow \mathbb{C}$ endowed with the linear operations and the convolution. In particular, we investigate the problem of refining the normal classification based on absolute convergence of associated general Dirichlet series in some right half plane. This involves an extension of Banach algebras of arithmetic functions by introducing weight functions and proving a weighted inversion theorem of Wiener type in the frame of Gelfond's theory of commutative Banach algebras.


Friday, Aug 8, 2008, 2:00 pm
Location:Chase 319
Speaker: Hans J. H. Tuenter
Mathematical Finance Program, University of Toronto
Title: A Characterization of the Frobenius Problem and its Applications
Abstract: In the Frobenius problem we are given a set of coprime, positive integers A={a1, a2,...,ak}, and are interested in the set of positive numbers NR that have no representation as a linear, nonnegative integer combination of the elements of A. We give a functional relationship that completely characterizes the set NR, and apply it to several problem instances that have been studied in the literature. The talk is in two parts. In the first part, we will discuss the two-variable Frobenius problem, that traces its history back to a problem that Sylvester posed in 1883, and show how its solution allows one to derive new recurrences for the Bernoulli numbers. In the second part, we show how the characterization relates to the Hilbert series of an additive numerical semigroup, apply the concepts to the Frobenius problem for arithmetic progressions and derive a simple, compact representation for the corresponding Hilbert series.


Thursday, May 8, 2008, 1:00 pm
Location:Chase 319
Speaker: Karl Dilcher
Department of Mathematics and Statistics, Dalhousie University
Title: Stern Polynomials and Continued Fractions
Abstract: We derive new identities for a polynomial analogue of the Stern sequence and define two subsequences of these polynomials. We obtain various properties for these two interrelated subsequences which have 0-1 coefficients and can be seen as extensions or analogues of the Fibonacci numbers. We also define two analytic functions as limits of these sequences. As an application we obtain evaluations of certain finite and infinite continued fractions whose partial quotients are doubly exponential. Finally we prove transcendence results for some of the infinite continued fractions. (Joint work with K.B. Stolarsky).


Thursday, Apr 17, 2008, 2:30 pm
Location:Chase 319
Speaker: Neil McKay
Department of Mathematics and Statistics, Dalhousie University
Title: Enumerating Magic Squares using Ehrhart Theory
Abstract: A 'traditional magic square' is an n x n array whose entries are all the integers from 1 to n^2, such that each row, column and both diagonals sum to the same number, called the 'magic sum'. However, 'magic squares' have been defined in many ways, the constant among all definitions is that all rows and columns have a common sum. Enumerating traditional magic squares has largely been a computational exercise. On the other hand, some magic square variants are enumerable with the help of Ehrhart Theory, which has application in counting integer points in polyhedra. By considering the Birkhoff-von Neumann Polytopes (which consists of nonnegative real matrices, in which all rows and columns sum to one) and its t-dilates, we can derive polynomials or 'quasi-polynomials' in t that enumerate magic squares with magic sum t.


Wednesday, Mar 26, 2008
Location:Chase 319
Speaker: O-Yeat Chan
Department of Mathematics and Statistics, Dalhousie University
Title: Duality in Tails of Multiple Zeta Values
Abstract: Multiple zeta-values (MZVs), or Euler-Zagier sums, are higher dimensional analogues of the Riemann zeta function. They are defined by a sum over a $k$-dimensional simplex:
$$\zeta(a_1,\dots,a_k) := \sum_{n_1>n_2>\cdots>n_k>0} \prod_{i=1}^k \frac{1}{n_i^{a_i}}.$$
Connection formulas between an MZV and other MZVs of lower dimension, also known as reduction formulas, are analytically, combinatorially, and computationally interesting. One fundamental reduction formula is the MZV duality formula. In this talk, we will give an elementary proof of a generalization to MZV duality. We will also discuss the discovery process for this formula and how to extract the structure from a special case to reveal the general identity.


Wednesday, Mar 12, 2008
Location:Chase 319
Speaker: Rob Noble
Department of Mathematics and Statistics, Dalhousie University
Title: Period Collapse in Ehrhart Quasi-Polynomials
Abstract: A convex polytope is the convex hull of finitely many points in Euclidean space. Given a convex polytope, P, we are interested in the lattice point enumerator function, LP , which associates to each nonnegative integer, t, the number of integer points in the t-th dilate of P. When the points that define P have integer or rational coordinates, LP takes on a special form. In the integral case LP is a polynomial in t (the Ehrhart polynomial of P), and in the rational case LP is a "quasi-polynomial" in t (the Ehrhart quasi-polynomial of P). Here a quasi-polynomial is a linear combination of nonnegative powers of t where we allow the coefficients to be periodic functions instead of forcing them to be constant. Taking a common multiple of the periods of the coefficients gives us a period for LP. The least integer that clears all of the denominators of the coordinates of the points that define P is a period for LP. It is not, however, always the minimum period. We will outline a recent paper of T. B. McAllister and K. M. Woods that investigates the conditions under which such a "period collapse" is possible.


Wednesday, Jan 16, 2008
Location:Chase 319
Speaker: Gary Walsh
Department of Mathematics and Statistics, University of Ottawa
Title: Fibonacci numbers, the ABC conjecture, and Fermat's Last Theorem
Abstract: The well known Fibonacci numbers are an example of a linear recurrence sequence. Such sequences appear to have fairly predictable arithmetical behaviour, although the results in the literature fall well short of the observed truth. We will discuss this behaviour, connections to what is now referred to as the ABC conjecture, and show how related questions are connected to extensions of Fermat's Last Theorem. This talk is intended for a general audience.


Wednesday, Nov 21, 2007
Location:Chase 319
Speaker: John Cosgrave
Title: Why I believe - contrary to orthodoxy - that there could exist a sixth Fermat prime.
Abstract: Fermat believed the Fermat numbers {F[n]} (F[n] = 2^(2^n) + 1) to be prime for all n = 0, 1, 2, 3, 4, 5, ..., but, as is well known, they are prime for n = 0, 1, 2, 3 and 4, and for no other known value of n. F[n] is known to be composite for all n between 5 and 32. The 4933 decimal digit F[14] (Selfridge and Hurwitz, 1963) is the smallest composite Fermat number with no known factor. F[2478782] (a number so large that, if written in decimal notation at 4 digits per inch in the horizontal and vertical directions, would require a sheet of paper having its side length greater than 10^373075) - found by the speaker in 2003 (using remarkable software of Yves Gallot, George Woltman and Paul Jobling) - is the largest known composite Fermat number, having smallest prime factor 3*2^2478785 + 1 (746190 decimal digits).

An orthodoxy has developed that F[n] is composite for all n > 4, but with not the remotest sign of any proof. In March 1999 I rediscovered a generally unknown unification of Fermat and Mersenne numbers, an outcome of which was to make an observation which - I believe (though my point of view has opponents) - should cause open-minded people to question the above orthodoxy. This talk will be of an entirely elementary nature (and indeed would be suitable for undergraduate students).


Wednesday, Nov 7, 2007
Location:Chase 319
Speaker: Karl Dilcher
Department of Mathematics and Statistics, Dalhousie University
Title: A polynomial analogue to the Stern sequence
Abstract: We extend the Stern sequence, sometimes also called Stern's diatomic sequence, to polynomials with coefficients 0 and 1 and derive various properties, including a generating function. A simple iteration for quotients of consecutive terms of the Stern sequence, recently obtained by Moshe Newman, is extended to this polynomial sequence. Finally we establish connections with Stirling numbers and Chebyshev polynomials, extending some results of Carlitz. In the process we also obtain some new results and new proofs for the classical Stern sequence.


Wednesday, Oct 31, 2007
Location:Chase 319
Speaker: Dante Manna
Department of Mathematics and Statistics, Dalhousie University
Title: Euler-Boole Summation Revisited
Abstract: Among formulas that connect an integral of a function with the sum of its values, the Euler-MacLaurin summation formula is very popular, or more so than its 'alternating cousin' (called the Boole summation formula). As per a suggestion by W. Strodt in his 1960 article, we explain both of these summation formulas as special cases of a single phenomenon. The key is a definition of a polynomial sequence, a kind of Appell sequence for which the denominator is the moment generating function of some cumulative distribution function. From here, we are able to re-derive or generalize known properties of Euler and Bernoulli Polynomials.

(Joint work with Jonathan Borwein and Neil Calkin)


Friday, May 11, 2007
Location:Chase 319
Speaker: O-Yeat Chan
Department of Mathematics and Statistics, Dalhousie University
Title: Calculating Bessel Functions via the Exp-arc Method
Abstract: This is joint work with David Borwein and Jonathan Borwein.

The standard method for computing values of Bessel functions has been to use the well-known ascending series for small argument $|z|$, and to use an asymptotic series for large $|z|$. In a recent paper, D. Borwein, J. Borwein, and R. Crandall derived a series for an ``exp-arc'' integral which gave rise to an absolutely convergent series for the $J$ and $I$ Bessel functions with integral order. Such series can be rapidly evaluated via recursion and elementary operations, and provides a viable alternative to the conventional ascending-asymptotic switching. In the present work, we extend the method to deal with Bessel functions of general (non-integral) order, as well as to deal with the $Y$ and $K$ Bessel functions.

Familiarity with Bessel Functions not required. Background and motivation will be provided at the talk (but bring your own drinks).


Wednesday, Apr 4, 2007
Location:Chase 319
Speaker: Fok-Shuen Leung
Mathematical Institute, University of Oxford
Title: The density of rational points on del Pezzo surfaces
Abstract: The study of rational points on algebraic varieties is one of math's oldest and most fundamental themes. In 1989 Yuri Manin made a precise conjecture about the density of rational points on the large family of varieties known as Fano varieties. We will look at the state of the conjecture for two-dimensional Fano varieties, called del Pezzo surfaces, and give a brief exposition of a new result on a family of smooth quartic surfaces.


Wednesday, Mar 28, 2007
Location:Chase 319
Speaker: O-Yeat Chan
Department of Mathematics and Statistics, Dalhousie University
Title: The Hardy-Ramanujan-Rademacher and Hardy-Littlewood Circle Methods
Abstract: The Circle Method, first used by Hardy and Ramanujan in their 1918 paper "Asymptotic Formulae in Combinatory Analysis", is a powerful tool in modern additive number theory. In this talk, I will describe how the method is used to obtain strong results for Waring's Problem as well as its application to asymptotics of the partition function. I will also give a general overview of the steps of the method and indicate when its use might be appropriate.


Wednesday, Mar 21, 2007
Location:Chase 319
Speaker: Karl Dilcher
Department of Mathematics and Statistics, Dalhousie University
Title: Bernoulli Numbers and Confluent Hypergeometric Functions
Abstract: If the reciprocal of the confluent hypergeometric function $M(s+1;r+s+2;z)$ is taken as an exponential generating function, the resulting sequences of numbers have many properties resembling those of the Bernoulli numbers, which arise when $r=s=0$. Other special cases include van der Pol numbers and generalized van der Pol numbers (for positive integers $r=s$), as well as various sequences of combinatorial numbers and linear recurrence sequences of arbitrary orders. Explicit expressions resembling Euler's formula involve sums of powers of the zeros of the corresponding confluent hypergeometric functions.


Wednesday, Mar 14, 2007
Location:D-Drive Lab, 2nd Floor Computer Science
Speaker: Jonathan Borwein
Department of Mathematics and Statistics, Dalhousie University
Title: The Life of Pi --- a talk for Pi Day
Abstract: The desire, and originally need, to calculate ever more accurate values of Pi, the ratio of the circumference of a circle to its diameter, has challenged mathematicians for centuries and, especially recently, Pi has provided fascinating examples of computational math. Pi, is also uniquely part of the popular imagination. This talk will be accessible to all undergraduates.

References: This link contains the talk and this link an associated book chapter.


Wednesday, Mar 7, 2007
Location:Chase 319
Speaker: Dante Manna
Department of Mathematics and Statistics, Dalhousie University
Title: Divisibility Properties of Stirling Numbers of the Second Kind
Abstract: The definition of Stirling Numbers of the Second Kind as the number of ways to partition a set of n elements into k nonempty subsets is over 250 years old. Given their longevity and widespread influence in various mathematical disciplines, it is unusual that very basic questions of the divisibility of these integers by primes remains unsolved. Even in the case p = 2, a full characterization is lacking. Our attempts to fill this hole in the literature, while not completely realized, do lead to full descriptions of special cases. Mining for ideas among previously successful approaches brings us to a new characterization of residues of Stirling Numbers modulo 4 and a corresponding matrix rule.

This is joint work with T. Amdeberhan and V. Moll, Tulane University and O-Y. Chan, Dalhousie University.


Wednesday, Feb 28, 2007
Location:Chase 319
Speaker: O-Yeat Chan
Department of Mathematics and Statistics, Dalhousie University
Title: Uniform Bounds for the Complementary Incomplete Gamma Function
Abstract: (Paper at http://www.oyeat.com/papers/Gamma_Submit.pdf)

Euler's Gamma Function is a classical object in mathematics with a wide range of applications. Related to Euler's Gamma are the Incomplete Gamma Function $\gamma(a,z)$ and its complement $\Gamma(a,z)$, both of which show up in many different contexts. In this talk, we will describe some history and basic facts about the Complementary Incomplete Gamma Function as well as known inequalities. We then prove new upper and lower bounds for $\Gamma(a,z)$ with complex parameters $a$ and $z$. In particular, the bounds within the circular hyperboloid of one sheet $\{(a,z):|z|>c|a-1|\}$ with $a$ real and $z$ complex are very elegant. Our results show that within this region, $|\Gamma(a,z)|$ is of order $|z|^{a-1}e^{-\Re(z)}$, and extends an upper estimate of Natalini and Palumbo to complex values of $z$.

Graduate students and advanced undergraduates are encouraged to attend.

This is joint work with Jon Borwein.


Wednesday, Feb 14, 2007
Location:D-Drive Lab, 2nd Floor Computer Science
Speaker: Jonathan Borwein
Department of Mathematics and Statistics, Dalhousie University
Title: Effective Laguerre Asymptotics, Part II
Abstract: (Paper at http://locutus.cs.dal.ca:8088/archive/00000334/)

This is a continuation of the Talk on January 31.

Special functions form a large and rambling, beautiful yet horrifying stew of classical mathematics and mathematical physics. In these two lectures, I will describe some of the ideas in a frecently completed work with Richard Crandall (Apple and Reed College) and D. Borwein (Western Ontario). Our overarching goal is to provide implementable asymptotic algorithms for a variety of special functions.

It is known that the generalized Laguerre polynomials can enjoy sub-exponential growth for large primary index. Specifically, for certain fixed parameter pairs $(a,z)$ one has the large-$n$ asymptotic $$L_n^{(-a)}(-z) \sim C(a,z) n^{-a/2-1/4} e^{2\sqrt {nz}}.$$ We introduce a computationally-motivated contour integral that allows highly efficient numerical evaluation of $L_n$, yet also leads to general asymptotic series over the full domain for sub-exponential behavior.

We eventually lay out a fast algorithm for generation of the rather formidable expansion coefficients. Along the way we address the difficult problem of establishing effective (i.e. rigorous and explicit) error bounds on the general expansion. To this end, we avoid classical stationary-phase and steepest-descent techniques in favor of an ``exp-arc" method that amounts to a natural bridge between converging series and effective asymptotics. Finally, we exhibit an absolutely convergent exp-arc series for Bessel-function evaluation as an alternative to conventional ascending-asymptotic switching.

Assisted by Maple, I hope to give the flavour of our research rather than to bury the audience in details.


Wednesday, Feb 7, 2007
Note: This is joint with the Graph Theory Seminar and will begin at 4 pm
Location:Chase 227
Speaker:Matt Hurshman
Department of Mathematics and Statistics, Acadia University
Title: Zeta Functions and Directed Graphs
Abstract: In this talk the exciting worlds of graph theory and zeta function theory will collide!

As inspiration a general discussion of the Riemann zeta function will be provided. We will then present the Ihara Zeta Function for graphs. The reciprocal of this function turns out to be a polynomial which we will call the Ihara Polynomial. Next, a new directed graph polynomial will be introduced. The meaning of the coefficients of this polynomial and a recursive formula to build the polynomial using smaller graphs will be discussed.

We will then see that when a directed graph has a specific construction our new digraph polynomial coincides with the Ihara Polynomial. What we know about our digraph polynomial will then be used to generate general Ihara Polynomials for families of graphs.


Wednesday, Jan 31, 2007
Location:D-Drive Lab, 2nd Floor Computer Science
Speaker: Jonathan Borwein
Department of Mathematics and Statistics, Dalhousie University
Title: Effective Laguerre Asymptotics, Part I
Abstract: (Paper at http://locutus.cs.dal.ca:8088/archive/00000334/)

Special functions form a large and rambling, beautiful yet horrifying stew of classical mathematics and mathematical physics. In these two lectures, I will describe some of the ideas in a frecently completed work with Richard Crandall (Apple and Reed College) and D. Borwein (Western Ontario). Our overarching goal is to provide implementable asymptotic algorithms for a variety of special functions.

It is known that the generalized Laguerre polynomials can enjoy sub-exponential growth for large primary index. Specifically, for certain fixed parameter pairs $(a,z)$ one has the large-$n$ asymptotic $$L_n^{(-a)}(-z) \sim C(a,z) n^{-a/2-1/4} e^{2\sqrt {nz}}.$$ We introduce a computationally-motivated contour integral that allows highly efficient numerical evaluation of $L_n$, yet also leads to general asymptotic series over the full domain for sub-exponential behavior.

We eventually lay out a fast algorithm for generation of the rather formidable expansion coefficients. Along the way we address the difficult problem of establishing effective (i.e. rigorous and explicit) error bounds on the general expansion. To this end, we avoid classical stationary-phase and steepest-descent techniques in favor of an ``exp-arc" method that amounts to a natural bridge between converging series and effective asymptotics. Finally, we exhibit an absolutely convergent exp-arc series for Bessel-function evaluation as an alternative to conventional ascending-asymptotic switching.

Assisted by Maple, I hope to give the flavour of our research rather than to bury the audience in details.


Wednesday, Jan 24, 2007
Location:Chase 319
Speaker: Neil Calkin
Department of Mathematical Sciences, Clemson University
Title: Tossing coins, and a random shooting game of Lampert and Slater
Abstract: In the American Mathematical Monthly in 1998, Lampert and Slater presented the following game: $n$ players each have a gun: each round of the game, each remaining player randomly shoots one of the other players, knocking them out of the game. The game continues until there is either one player (the ``winner'') remaining, or there are no players left. They asked for the behaviour of the probability that a game starting with $n$ players terminates in a win, and in particular, whether it tends to a limit.

Using analytic, combinatorial and computational methods, we analyze a simpler game, involving tossing coins, and use the insight we obtain to answer Lampert and Slater's original question.

This is joint work with E. Rodney Canfield and Herbert S. Wilf.


Wednesday, Jan 17, 2007
Location:Chase 319
Speaker: Karl Dilcher
Department of Mathematics and Statistics, Dalhousie University
Title: Multiple zeros of Bernoulli polynomials
Abstract: Bernoulli polynomials occur in various different areas of mathematics, including number theory, numerical analysis, and the analysis of finite differences.

While a lot is known about the geometry and analysis of the zeros of Bernoulli polynomials, some important algebraic questions remain unsolved. One such question is whether Bernoulli polynomials can have multiple zeros. A negative answer has been given in most cases. In this talk I will reduce the remaining cases to a very strong number theoretic conjecture which is almost certainly true and can be verified up to very large indices of the Bernoulli polynomials.


Wednesday, Dec 13, 2006
Location:Chase 319
Speaker: Gary Walsh
Department of Mathematics and Statistics, University of Ottawa
Title: Effective measures of approximation, Thue equations, and some classical Diophantine problems
Abstract: We first show how open problems arising from the classical work of Wilhelm Ljunggren on Diophantine equations can be reformulated in terms of integer solutions to families of Thue equations. Once these classical problems have been casted in this way, it is often the case that one can apply the method of Thue and Siegel to obtain an effective measure of approximation to the roots of the corresponding dehomogenized polynomial. We describe the process of doing so, and how this approach lends itself to improvements of those classical theorems.

This is joint work with Michael Bennett and Alain Togbe.


Wednesday, Nov 29, 2006
Location:Chase 319
Speaker: Dante Manna
Department of Mathematics and Statistics, Dalhousie University
Title: Divisibility Properties of Certain Integer Sequences
(with V. Moll and T. Amdeberhan, Tulane University)
Abstract: Our work in this subject began when a 2002 SIMU undergraduate research group noticed a surprising pattern in plotting the 2-adic valuations of an integer sequence that comes from a hypergeometric-type sum. A few more surprises and fortunate accidents later, we arrived at a characterization of these sequences via a remarkable combinatorial interpretation of the 2-adic valuations of Pochhammers. We wish to find a similar structure for Stirling numbers of the second kind. A characterization of their 2-adic valuations is lacking in the literature, and their plots lack the symmetry of the former sequence. Describing their behavior modulo powers of 2 may be the key here; they exhibit a connection to binomial coefficients.


Wednesday, Nov 22, 2006
Location:Chase 319
Speaker: O-Yeat Chan
Department of Mathematics and Statistics, Dalhousie University
Title: Two Bessel Function Identities in Ramanujan's Lost Notebook
Abstract:On page 335 of Ramanujan's Lost Notebook, he states a pair of identities relating double series of Bessel functions to finite trigonometric series. The first of the two identities had been proved by Berndt and Zaharescu, while the second is still open. In this talk, I will discuss the proof of the first identity as well as the work that has been done on the second identity. It is the hope of the speaker that this talk will create more interest in this problem.


Wednesday, Nov 15, 2006
Location:Chase 319
Speaker: Karl Dilcher
Department of Mathematics and Statistics, Dalhousie University
Title: Divisibility Properties of Some Classes of Binomial Sums.
Abstract:In this talk I will present congruence and divisibility properties of two different classes of combinatorial sums. The first class involves products of powers of two binomial coefficients; we will see that even though in general there is no evaluation in closed form, the sums behave in certain respects like single binomial coefficients. This is evident through a result similar to Wolstenholme's theorem, and through the fact that under certain conditions the sums are divisible by all primes in specific intervals.

The second class of combinatorial sums is the alternating version of a well-known sum that was used in the theory of Bernoulli numbers. This new sum is evaluated modulo an odd prime, and as an application it is shown that the $n$-th Bernoulli polynomial cannot have multiple zeros unless $n$ is extremely large. Related to this, I will also give strong evidence that Bernoulli polynomials can never have multiple zeros.

(Talk is partly based on joint work with Marc Chamberland.)


Wednesday, Nov 8, 2006
Location:Chase 319
Speaker: Keith Taylor
Department of Mathematics and Statistics, Dalhousie University
Title: Smooth Points in Operator Algebras
Abstract:We provide characterizations of points in C*-algebras (and von Neumann algebras) where the norm is differentiable.


Wednesday, Nov 1, 2006
Location:D-DRIVE Lab, Computer Science
Speaker: Jonathan Borwein
Faculty of Computer Science, Dalhousie University
Title: Effective Error Bounds for Euler-Maclaurin Based Numerical Quadrature, Part II
Abstract:We analyze the behaviour of Euler-Maclaurin-based integration schemes with the intention of deriving accurate and economic estimations of the error. These schemes typically provide very high-precision results (hundreds or thousands of digits), in reasonable run time, even when the integrand function has a blow-up singularlity or infinite derivative at an endpoint. Heretofore, researchers using these schemes have relied mostly on ad hoc error estimation schemes to project the estimated error of the present iteration. In this talk, we seek to develop some more rigorous, yet highly usable schemes to estimate these errors.

This is the continuation of Part I of this talk given on October 25.


Wednesday, Oct 25, 2006
Location:D-DRIVE Lab, Computer Science
Speaker: Jonathan Borwein
Faculty of Computer Science, Dalhousie University
Title: Effective Error Bounds for Euler-Maclaurin Based Numerical Quadrature, Part I
Abstract:We analyze the behaviour of Euler-Maclaurin-based integration schemes with the intention of deriving accurate and economic estimations of the error. These schemes typically provide very high-precision results (hundreds or thousands of digits), in reasonable run time, even when the integrand function has a blow-up singularlity or infinite derivative at an endpoint. Heretofore, researchers using these schemes have relied mostly on ad hoc error estimation schemes to project the estimated error of the present iteration. In this talk, we seek to develop some more rigorous, yet highly usable schemes to estimate these errors.


Wednesday, Oct 18, 2006
Location:Chase 319
Speaker: O-Yeat Chan
Department of Mathematics and Statistics, Dalhousie University
Title: Weighted Trigonometric Sums over a Half-Period
Abstract:Let $f(z)$ be a quotient of trigonometric functions. Suppose $f(z)$ has period $k$, an odd integer. We use a contour integral to evaluate the sum $$\sum a_i f(i)$$ for any sequence of complex numbers $a_i$ in terms of the Fourier coefficients of $f$. Several examples and applications will be shown for illustration.


Wednesday, Oct 11, 2006
Location:D-DRIVE Lab, Computer Science
Speaker: Mason Macklem
Faculty of Computer Science, Dalhousie University
Title: Direct Search Methods and Updating Search Directions
Abstract: Classical line-search optimization algorithms typically involve three stages: at the current iterate, evaluate higher-order information, use this information to locate a descent direction, and determine an amount to move in that direction. Another approach, which does not require the use of higher-order information, is to use only objective function values on a grid of points around each iterate, and to expand or contract the grid based on the success or failure in improving the best objective function value found. This approach has most famously been formulated in the GPS algorithms of Torczon, which proved convergence results for a general class of direct search algorithms, designed to contain a variety of classical algorithm including Hooke-Jeeves and the standard coordinate search algorithm; however, this formulation left open the approach for selecting the grid directions, and restricted the ability to update the grid directions as the algorithm learns the local behaviour of the objective function. This talk will discuss the background and history behind direct search methods, and will discuss generally some approaches to updating the grid directions within a specific class of these methods.


Wednesday, Oct 4, 2006
Location:Chase 319
Speaker: Dante Manna
Department of Mathematics and Statistics, Dalhousie University
Title: Rational Landen Transformations
Abstract: The Landen Transformation $(a,b) \mapsto ((a+b)/2,\sqrt{ab})$ preserves the value of an elliptic integral and (with proper restrictions on initial values) its iterates limit to the classical arithmetic-geometric mean AGM$(a,b)$~. We present an analogous transformation for rational integrals on the whole real line.


Wednesday, Sept 27, 2006
Location:Chase 319
Speaker: Rob Noble
Department of Mathematics and Statistics, Dalhousie University
Title: Local and Global Complex Multiplication
Abstract: Kronecker's Jugendtraum ("youthful dream") was to construct the abelian extensions of an arbitrary number field. This problem has been immortalized as one of Hilbert's famous problems, (#12) and the goal of the talk is to shed some light on some partial solutions provided by the theory of elliptic curves (complex multiplication) and the theory of commutative formal groups (Lubin-Tate theory). The theory of complex multiplication will provide us with a full solution for the case where we replace $\mathbb{Q}$ by an imaginary quadratic field whereas the Lubin-Tate theory will provide us with a complete solution to the analogous ``local'' version of this problem obtained by replacing $\mathbb{Q}$ by a local field. Here, a local field is a finite extension of the field of $p$-adic numbers $\mathbb{Q}_{p}$ for some prime $p$ in characteristic 0, or a finite extension of the field of formal power series $\mathbb{F}_{p}[[X]]$ in characteristic $p>0$.


Wednesday, Sept 20, 2006
Location: Chase 319
Speaker: Songping Zhou
Title: Convergence Problems for Trigonometric Series: The Ultimate Condition to Generalize Monotonicity for the Uniform Convergence
Abstract: It is well known that Fourier analysis plays an important role in pure mathematics and has many important applications in science and technology. Convergence problems of Fourier (trigonometric) series are very fundamental to establish a solid basis for Fourier analysis.

This talk gives a brief review of the history of convergence problems of Fourier (trigonometric) series. Finally we present a very recent development: the ultimate condition to generalize monotonicity for the uniform convergence.