Publications

PhD thesis

Distributing points on the sphere: Partitions, separation, quadrature and energy , UNSW, 2007. (Citations).

Clean PDF file. Accompanying Thesis/Dissertation Sheet.

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.

Publications and preprints

  1. 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.
  2. 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.
  3. 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 [4] to give improved estimates.
  4. 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.
  5. 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.
  6. Paul Leopardi and Rob Womersley, "Porting a sphere optimization program from LAPACK to ScaLAPACK", ANZIAM Journal, 50 (CTAC 2008), November 2008, pp. C204-C219. Draft: Paul Leopardi, "Conversion of a Sphere Optimization Program from LAPACK to ScaLAPACK", (unpublished draft, 2002). Preprint: Revised October 2008.
    • 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.
  7. Paul Leopardi, "Diameter bounds for equal area partitions of the unit sphere", Electronic Transactions on Numerical Analysis, Volume 35, 2009, pp. 1-16, (Citations). 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.
  8. 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.
  9. 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.
  10. 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.
  11. 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: 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.
  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. DOI 10.1007/s10444-011-9266-4, (Citations). 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. Conrad Burden, Paul Leopardi and Sylvain Fôret, "The distribution of word matches between Markovian sequences with periodic boundary conditions", Journal of Computational Biology, 21(1), January 2014, pp. 41--63. DOI 10.1089/cmb.2012.0277, (Citations). Preprint: 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.
  14. Conrad Burden, Paul Leopardi and Sylvain Fôret, "Word match counts between Markovian sequences", "Biomedical Engineering Systems and Technologies 6th International Joint Conference, BIOSTEC 2013", Springer series on Communications in Computer and Information Science, 2014, pp. 147-161. 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.
  15. 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. Preprint: Revised January 2014. Supplementary material can be found in 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.
  16. Paul Leopardi, "Discrepancy, separation and Riesz energy of finite point sets on compact connected Riemannian manifolds", Dolomites Research Notes on Approximation, Volume 6, July 2014, pp. 120-129. Preprint: arXiv:1403.6550 [math.NA].
    • 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.
  17. Markus Hegland and Paul Leopardi, "Sparse grid quadrature on products of spheres", Numerical Algorithms, Volume 70, Issue 3, 2015, pp. 485-517. DOI 10.1007/s11075-015-9958-9. Preprint: arXiv:1202.5710 [math.NA].
    • Describes sparse grid quadrature on products of spheres, giving the initial and asymptotic rates of convergence.
  18. Paul Leopardi, "Twin bent functions and Clifford algebras", in C. Colbourn (ed.) Algebraic Design Theory and Hadamard Matrices (ADTHM 2014), Springer, 2015, pp. 189-199. Preprint: arXiv:1501.05477 [math.CO].
    • Examines a pair of bent functions on and their relationship to a necessary condition for the existence of an automorphism of an edge-coloured graph, whose colours are defined by the properties of a canonical basis for the real representation of a real Clifford algebra.
  19. Paul Leopardi and Ari Stern, "The abstract Hodge-Dirac operator and its stable discretization", SIAM Journal on Numerical Analysis, 54(6), 2016, pp. 3258–3279, (Citations). Preprint: 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.
  20. Giacomo Gigante and Paul Leopardi, "Diameter bounded equal measure partitions of Ahlfors regular metric measure spaces", Discrete and Computational Geometry, 57 (2), 2017, pp. 419–430. Preprint: arXiv:1510.05236 [math.NA].
    • Combines the Feige and Schechtman construction with David's and Christ's dyadic cubes to yield a partition algorithm for connected Ahlfors regular metric measure spaces of finite measure.
  21. Paul Leopardi, "Twin bent functions, strongly regular Cayley graphs, and Hurwitz-Radon theory", Journal of Algebra Combinatorics Discrete Structures and Applications, 4 (3) , 2017, pp. 271-280. Preprint: arXiv:1504.02827 [math.CO].
    • Uses a theorem of Radon to prove that the corresponding graphs in the two sequences of strongly regular graphs considered in [15] and [18] are not isomorphic, except in the first 3 cases.
  22. Paul Leopardi, Alvise Sommariva, Marco Vianello, "Optimal polynomial meshes and Caratheodory-Tchakaloff submeshes on the sphere", Dolomites Research Notes on Approximation, Volume 10 Special Issue, 2017, pp. 18-24. Preprint: arXiv:1612.04952 [math.NA].
    • Using the notion of Dubiner distance, gives an elementary proof that good covering point configurations on the 2-sphere are optimal polynomial meshes, and extracts Caratheodory-Tchakaloff submeshes for compressed least squares fitting.
  23. Paul Leopardi, "Classifying bent functions by their Cayley graphs", rejected by INTEGERS: The Electronic Journal of Combinatorial Number Theory, November 2018. Preprint: arXiv:1705.04507 [math.CO]. Revised, December, 2018.
    • Explores the connections between bent functions and their Cayley graphs, as well as projective two-weight linear codes, and symmetric designs with the symmetric difference property, through various equivalence classes. Includes exhaustive classifications of bent functions up to 8 dimensions and degree 3, with selected examples of degree 4.
  24. Paul Leopardi, "Gastineau-Hills' quasi-Clifford algebras and plug-in constructions for Hadamard matrices". Advances in Applied Clifford Algebras (2019) 29: 48. Preprint: arXiv:1804.09454 [math.CO].
    • Applies the representation theory of quasi-Clifford algebras, as described by Gastineau-Hills, to the plug in constructions for Hadamard matrices described in [15].