 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, 27^{th} Oct 2015
 Abstract:
We consider identities satisfied by discrete analogues of Mehtalike integrals.
The integrals are related to Selbergâ€™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 nonnegative 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
$$S_{2,2,3} (r,n) =
\prod_{j=1}^r
\frac{(2n)!j!^2}{(nj)!^2}.$$
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 nonintersecting lattice paths.
This is joint work with Christian Krattenthaler and Ole Warnaar.
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 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, 18^{th} 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 selfcontained.
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 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, 23^{rd} 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.
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