 CARMA SEMINAR
 Speaker: Prof Richard Brent, CARMA, The University of Newcastle
 Title: Jonathan Borwein and Pi
 Location: Room V205, Mathematics Building (Callaghan Campus) The University of Newcastle
 Time and Date: 1:59 pm, Wed, 14^{th} Mar 2018
 Abstract:
The late Professor Jonathan Borwein was fascinated by the constant
$\pi$. Some of his talks on this topic can be found on the CARMA website.
This homage to Jon is based on my talk at the Jonathan Borwein Commemorative
Conference. I will describe some algorithms for the highprecision
computation of $\pi$ and the elementary functions, with particular reference
to the book Pi and the AGM by Jon and his brother Peter Borwein.
Here "AGM" is the arithmeticgeometric mean
of Gauss and Legendre. Because the AGM has secondorder convergence, it
can be combined with FFTbased fast multiplication algorithms to give fast
algorithms for the \hbox{$n$bit} computation of $\pi$.
I will survey a few of the results and algorithms that were of interest to
Jon. In several cases they were either discovered or improved by him. If
time permits, I will also mention some new results that would have been of
interest to Jon.
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 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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