• CARMA SEMINAR
  • CARMA Special Semester in Computation and Visualisation
  • Speaker: Prof Richard Brent, CARMA, The University of Newcastle
  • Title: Algorithms for the Multiplication Table Problem
  • Location: Room V205, Mathematics Building (Callaghan Campus) The University of Newcastle
  • Time and Date: 3:00 pm, Tue, 29th May 2018
  • To participate remotely, connect to the ViewMe meeting called "carmaspecial" (you can enter that name, or the meeting number 1689883675). This will be persistant for future talks in this series. The ViewMe client is free and you do not need an account. You can install ViewMe on a computer or phone to take part, or use the web interface (Firefox or Chrome) at https://viewme.ezuce.com/webrtc/?meetingID=1689883675. It's quite easy to use, but for assistance please contact Andrew.Danson@newcastle.edu.au. Some guides are available at https://viewme.ezuce.com/support/guides-tutorials/.
  • Abstract:

    Let $M(n)$ be the number of distinct entries in the multiplication table for integers smaller than $n$. More precisely, $M(n) := |\{ij \mid\ 0<= i,j <n\}|$. The order of magnitude of $M(n)$ was established in a series of papers by various authors, starting with Erdös (1950) and ending with Ford (2008), but an asymptotic formula for $M(n)$ is still unknown. After describing some of the history of $M(n)$ I will consider two algorithms for computing $M(n)$ exactly for moderate values of $n$, and several Monte Carlo algorithms for estimating $M(n)$ accurately for large $n$. This leads to consideration of algorithms, due to Bach (1985-88) and Kalai (2003), for generating random factored integers - integers $r$ that are uniformly distributed in a given interval, together with the complete prime factorisation of $r$. The talk will describe ongoing work with Carl Pomerance (Dartmouth, New Hampshire) and Jonathan Webster (Butler, Indiana).

    Bio: Richard Brent is a graduate of Monash and Stanford Universities. His research interests include analysis of algorithms, computational complexity, parallel algorithms, structured linear systems, and computational number theory. He has worked at IBM Research (Yorktown Heights), Stanford, Harvard, Oxford, ANU and the University of Newcastle (NSW). In 1978 he was appointed Foundation Professor of Computer Science at ANU, and in 1983 he joined the Centre for Mathematical Analysis (also at ANU). In 1998 he moved to Oxford, returning to ANU in 2005 as an ARC Federation Fellow. He was awarded the Australian Mathematical Society Medal (1984), the Hannan Medal of the Australian Academy of Science (2005), and the Moyal Medal (2014). Brent is a Fellow of the Australian Academy of Science, the Australian Mathematical Society, the IEEE, ACM, IMA, SIAM, etc. He has supervised twenty PhD students and is the author of two books and about 270 papers. In 2011 he retired from ANU and moved to Newcastle to join CARMA, at the invitation of the late Jon Borwein.

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  • 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, 14th 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 high-precision 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 arithmetic-geometric mean of Gauss and Legendre. Because the AGM has second-order convergence, it can be combined with FFT-based 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, 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.

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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, 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.

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  • 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.

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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, 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.

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  • 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.

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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, 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

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  • 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.

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  • 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

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  • 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.

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  • 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

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