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Paul Leopardi

Research Fellow

Real representation matrix for neutral Clifford algebra R_{2,2} Recursive zonal equal area partition of S^2 into 33 regions Paul Leopardi

Email address

paul.leopardi {AT} anu.edu.au

Postal address

Centre for Bioinformation Science
Mathematical Sciences Institute Building 27
Australian National University
Canberra ACT Australia 0200

Phone

+61 2 6125 0023

Brief history

Academic: BSc (Hons in Computer Science), University of New South Wales, 1983. MCom (Information Systems), University of New South Wales, 1990. Master of Science and Technology by coursework in Mathematics, University of New South Wales, 2002. Doctor of Philosophy in Applied Mathematics, University of New South Wales, 2007, supervised by Professor Ian Sloan and Associate Professor Rob Womersley. Grad. Cert. in Higher Education, Australian National University, 2012.
Working: Telecom Australia 1983-1986 - Computer Systems Officer. Memorex-Telex 1986-1990 - Systems Engineer. Travel Industries Automated Systems 1990-1995 - Systems Analyst. Accenture 1995-2001 - Consultant. UNSW 2001-2002 - Research assistant programmer, Mathematics. University of Sydney 2005-2007 - Scientific Computing Officer, Physics. ANU 2007-2012 - Postdoctoral Research Fellow, Mathematics.


Research interests

Clifford algebras, Clifford analysis, object-oriented numerical analysis, parallel linear algebra using ScaLAPACK, approximation and quadrature on the sphere, sparse grids, random number generation and testing, statistics and combinatorics on strings, new constructions for Hadamard matrices.


Teaching

  • 2014 Jan-Feb: AMSI Summer School - Bioinformatics (convened by Conrad Burden).
  • MATH2307/MATH6100 Bioinformatics and Biological Modelling 2013 Semester 2 (convened by Conrad Burden).
  • 2013, 2012: Reading course and seminar on Clifford algebras.
  • MATH3512/MATH6112 Matrix Computations
    • 2011 Semester 2.
  • MATH3512/MATH6112 Matrix Computations and Optimization
    • 2010 Semester 2 (with Richard Brent).
    • 2009 Semester 2 (with Richard Brent and Vikram Sunkara).
  • Approximation Theory

PhD thesis

Distributing points on the sphere: Partitions, separation, quadrature and energy , UNSW, 2007. (Citations).
Accompanying Thesis/Dissertation Sheet.


Publications and preprints

  1. Paul Leopardi, "Conversion of a Sphere Optimization Program from LAPACK to ScaLAPACK", (unpublished draft, 2002).

    Describes methods used to parallelize code used in optimization on the sphere.

  2. Paul Leopardi, "A generalized FFT for Clifford algebras", Bulletin of the Belgian Mathematical Society - Simon Stevin, Volume 11, Number 5, 2005, pp. 663-688.
    MR 2130632. (Citations).
    Preprint: UNSW Applied Mathematics Report AMR04/17, March 2004.

    Describes algorithms used in the GluCat C++ software library, for the real representations of real Clifford algebras, having the same order of complexity as the generalized FFTs on finite groups.

  3. Paul Leopardi, "A partition of the unit sphere into regions of equal area and small diameter", Electronic Transactions on Numerical Analysis, Volume 25, 2006, pp. 309-327.
    MR 2280380, (Citations).
    Preprint: UNSW Applied Mathematics Report AMR05/18, May 2005, revised June 2006.

    Describes the algorithm used in the EQSP software package, which partitions a finite dimensional unit sphere into regions of equal area and small diameter.

  4. Paul Leopardi, "Positive weight quadrature on the sphere and monotonicities of Jacobi polynomials", DWCAA06 proceedings, Numerical Algorithms, Volume 45, Numbers 1-4 / August, 2007, pp. 75-87.
    DOI 10.1007/s11075-007-9073-7 , MR 2355973, (Citations).
    Preprint: UNSW Applied Mathematics Report AMR06/41, December 2006, revised Febrary 2007.

    Examines the relationship, for a positive weight quadrature rule on the unit sphere, between the the total quadrature weight on any spherical cap and the area of that cap. Uses conjectures from [5] to give improved estimates.

  5. Walter Gautschi and Paul Leopardi, "Conjectured inequalities for Jacobi polynomials and their largest zeros", DWCAA06 proceedings, Numerical Algorithms, Volume 45, Numbers 1-4 / August, 2007, pp. 217-230.
    DOI 10.1007/s11075-007-9067-5 , MR 2355984, (Citations).
    Preprint: UNSW Applied Mathematics Report AMR07/2, February 2007.

    Describes new conjectures on monotonicities of the values and the zeros of functions related to Jacobi polynomials with fixed \alpha and \beta and increasing degree.

  6. Kerstin Hesse and Paul Leopardi, "The Coulomb energy of spherical designs on S^2", Advances in Computational Mathematics, Volume 28, Number 4 / May, 2008 , pp. 331-354.
    DOI 10.1007/s10444-007-9026-7 , MR 2390282, (Citations).
    Preprint: UNSW Applied Mathematics Report AMR04/34, December 2004, revised January 2006.

    Gives bounds for the Coulomb energy of a sequence of well separated spherical designs on the unit sphere, including a conjectured bound comparable to the minimum Coulomb energy.

  7. Paul Leopardi and Rob Womersley, "Porting a sphere optimization program from LAPACK to ScaLAPACK", ANZIAM Journal, 50 (CTAC 2008), November 2008, pp. C204-C219.
    Preprint: Revised October 2008. Figure 1: TG, TF. Figure 2: TI, TT.

    Describes methods used to parallelize code used in optimization on the sphere, and analyzes performance of the code in relation to the topology of the computer cluster used for testing.

  8. Paul Leopardi, "Diameter bounds for equal area partitions of the unit sphere", Electronic Transactions on Numerical Analysis, Volume 35, 2009, pp. 1-16.
    Preprint: December 2007, revised January 2009

    Proves diameter bounds for the sphere partition described in [3], and a modified version of the construction of Feige and Schechtman.

  9. Paul Leopardi, "Testing the tests: using random number generators to improve empirical tests", Monte Carlo and Quasi-Monte Carlo Methods 2008, Pierre L' Ecuyer, Art B. Owen (Eds.) Springer, 2009 pp. 501--512. ISBN: 978-3-642-04106-8, MR 2743916, (Citations).
    Preprint: Revised July 2009.

    Examines implementations of the overlapping serial tests of Marsaglia and Zaman, and improves them, using accurate calculation of the mean and variance of the number of missing words in a random string.

  10. Paul Leopardi, "Approximating the square root and logarithm functions in Clifford algebras: what to do in the case of negative eigenvalues?", (extended abstract) AGACSE 2010, June 2010.

    Describes how the Clifford algebras over the real numbers can be treated as real matrices, except in the case of negative real eigenvalues, when the square root and logarithm functions may take values in a larger Clifford algebra.

  11. Markus Hegland and Paul Leopardi, "The rate of convergence of sparse grid quadrature on the torus", ANZIAM Journal, 52 (CTAC 2010), June 2011, pp. C500--C517.
    Preprint: January 2011, revised June 2011.

    Describes a dimension adaptive algorithm for sparse grid quadrature on reproducing kernel Hilbert spaces on the unit torus, and compares this algorithm to the WTP algorithm of Wasilkowski and Wozniakowski.

  12. Paul Leopardi, "Discrepancy, separation and Riesz energy of finite point sets on the unit sphere", Advances in Computational Mathematics, Volume 39, Issue 1, July 2013, pp. 27-43, (published online December 2011).
    DOI 10.1007/s10444-011-9266-4 .
    Preprint: June 2010, revised December 2011, revised November 2012 (Proof of Lemma 4.6 was incorrect, "Stolarsky" was spelled incorrectly, minor reformatting).

    Shows that a sequence of spherical codes with a well behaved upper bound on discrepancy and a well behaved lower bound on separation, satisfies an upper bound on Riesz s-energy.

  13. Paul Leopardi, "Can compatible discretization, finite element methods, and discrete Clifford analysis be fruitfully combined?", Clifford Analysis, Clifford Algebras and their applications (CACAA), Volume 7, Issue 1, 2012, pp. 57-64. ISSN 2050- 0300 (print), 2050-0319 (online).
    Preprint: January 2012.
    Conference paper: 9th International Conference on Clifford Algebras and their Applications (ICCA 9), July 2011.
    Preprint: May 2011, revised July 2011.

    Describes work in progress, towards the formulation, implementation and testing of compatible discretization of differential equations, using a combination of Finite Element Exterior Calculus and discrete Geometric Calculus / Clifford analysis.

  14. Paul Leopardi, "Constructions for Hadamard matrices, Clifford algebras, and their relation to amicability - anti-amicability graphs", Australasian Journal of Combinatorics, Volume 58(2) (2014), pp. 214–248.
    Preprints: December 2011, revised August 2012, second revision November 2012, third revision October 2013, corrected proof, January 2014.
    Supplementary material can be found in the Hadamard directory, and the Hadamard-fractious Github repository.

    Describes how the pattern of commuting and anticommuting pairs of basis elements of a real Clifford algebra, and their representation theory, can be used in the construction of Hadamard matrices.

  15. Markus Hegland and Paul Leopardi, "Sparse grid quadrature on products of spheres", in progress.
    Preprints: January 2012, revised June 2012, second revision April 2013, arXiv:1202.5710v3 [math.NA].

    Describes sparse grid quadrature on products of spheres, giving the initial and asymptotic rates of convergence.

  16. Conrad Burden, Paul Leopardi and Sylvain Fôret, "Word match counts between Markovian sequences", to appear in "Biomedical Engineering Systems and Technologies 6th International Joint Conference, BIOSTEC 2013", Springer series on Communications in Computer and Information Science.
    Preprint of conference paper: ("The distribution of short word match counts between Markovian sequences"), presented at International Conference on Bioinformatics Models, Methods and Algorithms (Bioinformatics 2013). November 2012.
    Preprint of revised and extended paper: June 2013.

    Examines the D2 statistic, which counts the number of word matches between two given sequences, under the assumptions of periodic boundary conditions and Markovian dependence. Includes the calculation of the mean of D2 for all Markov orders and the variance for Markov order 1.

  17. Conrad Burden, Paul Leopardi and Sylvain Fôret, "The distribution of word matches between Markovian sequences with periodic boundary conditions", accepted (14 August 2013) by Journal of Computational Biology.
    Preprints: November 2012, revised May 2013.

    Further examines the D2 statistic, which counts the number of word matches between two given sequences, under the assumptions of periodic boundary conditions and Markovian dependence. Includes the calculation of the mean of D2 for all Markov orders and the variance for all Markov orders up to and including the word length. Also includes a comparison of synthetic data with DNA data from human chromosome 1.

  18. Paul Leopardi, "Discrepancy, separation and Riesz energy of finite point sets on compact connected Riemannian manifolds", Submitted to Dolomites Research Notes on Approximation.
    Preprints: December 2012, revised April 2013, second revision January 2014.

    Proves that, for a smooth compact connected d-dimensional Riemannian manifold M, if 0 <= s <= d then an asymptotically equidistributed sequence of finite subsets of M that is also well-separated yields a sequence of Riesz s-energies that converges to the energy double integral.

  19. Paul Leopardi and Ari Stern, "The abstract Hodge-Dirac operator and its stable discretization", to be submitted to Foundations of Computational Mathematics, January 2014.
    Preprints: arXiv:1401.1576 [math.NA].

    Adapts the techniques of finite element exterior calculus to study and discretize the abstract Hodge-Dirac operator, a square root of the abstract Hodge-Laplace operator considered by Arnold, Falk, and Winther.


Presentations

Clifford algebras and Clifford analysis

Approximation and quadrature on the sphere

Random number generation and testing; combinatorics on strings

Other topics


Software


Integer sequences

  • A129337: Maximal possible degree of a Chebyshev-type quadrature formula with n nodes, in the case of the constant weight function on [ -1,1], May 2007.
  • A152139: Correlation classes of pairs of different words, November 2008.
  • A152959: Number of correlation classes for pairs of different words in an alphabet of size 4, December 2008.

Research project proposals and grants


Conference, session and user group organization


Citations

Mentions in acknowledgements and elsewhere


Links

Updated: 31 January 2014/ Responsible Officer:  Director, MSI / Page Contact:  Paul Leopardi