CARMA SEMINAR Speaker: Prof Richard Brent, CARMA, The University of Newcastle Title: Some Identities involving Products of Gamma Functions: a Case Study in Experimental Mathematics Location: Room V205, Mathematics Building (Callaghan Campus) The University of Newcastle Time and Date: 2:00 pm, Tue, 27th Oct 2015 Abstract: We consider identities satisfied by discrete analogues of Mehta-like integrals. The integrals are related to Selberg’s integral and the Macdonald conjectures. Our discrete analogues have the form $$S_{\alpha,\beta,\delta} (r,n) := \sum_{k_1,...,k_r\in\mathbb{Z}} \prod_{1\leq i < j\leq r} |k_i^\alpha - k_j^\alpha|^\beta \prod_{j=1}^r |k_j|^\delta \binom{2n}{n+k_j},$$ where $\alpha,\beta,\delta,r,n$ are non-negative integers subject to certain restrictions. In the cases that we consider, it is possible to express $S_{\alpha,\beta,\delta} (r,n)$ as a product of Gamma functions and simple functions such as powers of two. For example, if $1 \leq r \leq n$, then $$S_{2,2,3} (r,n) = \prod_{j=1}^r \frac{(2n)!j!^2}{(n-j)!^2}.$$ The emphasis of the talk will be on how such identities can be obtained, with a high degree of certainty, using numerical computation. In other cases the existence of such identities can be ruled out, again with a high degree of certainty. We shall not give any proofs in detail, but will outline the ideas behind some of our proofs. These involve $q$-series identities and arguments based on non-intersecting lattice paths. This is joint work with Christian Krattenthaler and Ole Warnaar. [Permanent link] CARMA DISCRETE MATHEMATICS INSTRUCTIONAL SEMINAR Speaker: Prof Richard Brent, CARMA, The University of Newcastle Title: Bounds on the Hadamard maximal determinant problem using the Lovasz local lemma Location: Room V101, Mathematics Building (Callaghan Campus) The University of Newcastle Time and Date: 3:00 pm, Thu, 18th Jul 2013 Abstract: I will explain how the probabalistic method can be used to obtain lower bounds for the Hadamard maximal determinant problem, and outline how the Lovasz local lemma (Alon and Spencer, Corollary 5.1.2) can be used to improve the lower bounds. This is a continuation of last semester's lectures on the probabilistic method, but is intended to be self-contained. [Permanent link] CARMA DISCRETE MATHEMATICS INSTRUCTIONAL SEMINAR Speaker: Prof Richard Brent, CARMA, The University of Newcastle Title: The Probabilistic Method Location: Room V101, Mathematics Building (Callaghan Campus) The University of Newcastle Time and Date: 3:00 pm, Thu, 6th Jun 2013 Abstract: This will be a short course of lectures. See http://maths-people.anu.edu.au/~brent/probabilistic.html. [Permanent link] CARMA DISCRETE MATHEMATICS INSTRUCTIONAL SEMINAR Speaker: Prof Richard Brent, CARMA, The University of Newcastle Title: The Probabilistic Method continues Location: Room V101, Mathematics Building (Callaghan Campus) The University of Newcastle Time and Date: 3:00 pm, Thu, 23rd May 2013 Abstract: We continue on the Probabilistic Method, looking at Chapter 4 of Alon and Spencer. We will consider the second moment, the Chebyshev's inequality, Markov's inequality and Chernoff's inequality. [Permanent link] CARMA DISCRETE MATHEMATICS INSTRUCTIONAL SEMINAR Speaker: Prof Richard Brent, CARMA, The University of Newcastle Title: Probabilistic Method applied to the Hadamard Maximal Determinant Problem Location: Room V101, Mathematics Building (Callaghan Campus) The University of Newcastle Time and Date: 3:00 pm, Thu, 9th May 2013 Re-scheduled from last week. Abstract: This will be a short course of lectures. See http://maths-people.anu.edu.au/~brent/probabilistic.html. [Permanent link] CARMA DISCRETE MATHEMATICS INSTRUCTIONAL SEMINAR Speaker: Prof Richard Brent, CARMA, The University of Newcastle Title: The Probabilistic Method continues... Location: Room V101, Mathematics Building (Callaghan Campus) The University of Newcastle Time and Date: 3:00 pm, Thu, 18th Apr 2013 (Updated Thursday 11 am: reverting to usual time and place.) Abstract: This will be a short course of lectures. See http://maths-people.anu.edu.au/~brent/probabilistic.html [Permanent link] CARMA DISCRETE MATHEMATICS INSTRUCTIONAL SEMINAR Speaker: Prof Richard Brent, CARMA, The University of Newcastle Title: The Probabilistic Method Location: Room V101, Mathematics Building (Callaghan Campus) The University of Newcastle Time and Date: 3:00 pm, Thu, 11th Apr 2013 Abstract: This will be a short course of lectures. See http://maths-people.anu.edu.au/~brent/probabilistic.html. [Permanent link] CARMA DISCRETE MATHEMATICS INSTRUCTIONAL SEMINAR Speaker: Prof Richard Brent, CARMA, The University of Newcastle Title: The Probabilistic Method III Location: Room V101, Mathematics Building (Callaghan Campus) The University of Newcastle Time and Date: 3:00 pm, Thu, 28th Mar 2013 Abstract: This will be a short course of lectures. See http://maths-people.anu.edu.au/~brent/probabilistic.html [Permanent link] CARMA DISCRETE MATHEMATICS INSTRUCTIONAL SEMINAR Speaker: Prof Richard Brent, CARMA, The University of Newcastle Title: The Probabilistic Method I Location: Room V27, Mathematics Building (Callaghan Campus) The University of Newcastle Time and Date: 2:00 pm, Thu, 14th Mar 2013 Abstract: This will be a short course of lectures. See http://maths-people.anu.edu.au/~brent/probabilistic.html. [Permanent link] CARMA DISCRETE MATHEMATICS INSTRUCTIONAL SEMINAR Speaker: Prof Richard Brent, CARMA, The University of Newcastle Title: The Probabilistic Method I Location: Room V206, Mathematics Building (Callaghan Campus) The University of Newcastle Time and Date: 3:00 pm, Thu, 7th Mar 2013 Abstract: This will be a short course of lectures. See http://maths-people.anu.edu.au/~brent/probabilistic.html [Permanent link]