Points on Spheres and Manifolds

Research Articles on the Sphere and other Manifolds

(242) Mesh Ratios for Best-Packing and Limits of Minimal Energy Configurations (with A. Bondarenko and D. Hardin),accepted for publication in Acta Mathematica Hungarica.. [PDF]

(240) QMC Designs: Optimal Order Quasi Monte Carlo Integration Schemes on the Sphere (with J.S. Brauchart, I.H. Sloan, and R.S. Womersley), accepted for publication in Math. Comp. [PDF]

(239) Polarization Optimality of Equally Spaced Points on the Circle for Discrete Potentials (with D. Hardin and A. Kendall), accepted for publication in Discrete & Computational Geometry. arXiv number: 1208.5261v1 [PDF]

(238) Riesz polarization in higher dimensions (with T. Erdelyi), J. Approx. Theory, vol. 171 (2013), 128-147. [PDF]

(235) The Next-Order Term for Minimal Riesz and Logarithmic Energy Asymptotics on the Sphere (with J.S. Brauchart and D.P. Hardin), Contemp. Math., Vol. 578 (2012), 31-61. [PDF]

(233) Quasi-uniformity of Minimal Weighted Energy Points (with D. Hardin and T. Whitehouse), Journal of Complexity. Vol. 28, Issue 2, (2012), 177-191. [PDF]

(232) A Fascinating Polynomial Sequence Arising From An Electrostatics Problem on the Sphere (with J.S. Brauchart, P.D. Dragnev, and C.E. Van de Woestijne), Acta Mathematica Hungarica: Volume 137, Issue 1 (2012), Page 10-26. [PDF]

(229) Minimal N-Point Diameters and f-Best-Packing Constants in R^d (with A.V. Bondarenko and D.P. Hardin), accepted for publication, Proceedings American Mathematics Society. [PDF]

(228) Discrete Energy Asymptotics on a Riemannian Circle (with J.S. Brauchart and D.P. Hardin), Uniform Distribution Theory, vol. 7, no. 2, (2012), 77-108. [PDF]

(226) Riesz Extremal Measures on the Sphere for Axis-Supported External Fields (with J.S. Brauchart and P.D. Dragnev), J. Math. Anal. Appl., 356 (2009), 769-792. [PDF]

(225) Asymptotics of Greedy Energy Points (with Abey Lopez), Math. Comp. vol. 79 (2010), 2287-2316. [PDF]

(223) The Riesz Energy of the N-th Roots of Unity: An Asymptotic Expansion for Large N (with J.S. Brauchart and D.P. Hardin), Bull. London Math. Soc., 41 (2009), 621-633. [PDF]

(221) Riesz Energy and Sets of Revolution in R^3 (with Johann S. Brauchart and Douglas P. Hardin), In Functional Analysis and Complex Analysis, 47-57, Contemp. Math., 481, Amer. Math. Soc., Providence, RI, 2009.[PDF]

(217) Asymptotics of Weighted Best-Packing on Rectifiable Sets (with S. V. Borodachov and D. P. Hardin), (Russian) Mat.Sb. 199 (2008), no. 11, 3-20; translation in Sb. Math. 199 (2008), no. 11-12, 1579-1595. [PDF]

(216) The Support of the Limit Distribution of Optimal Riesz Energy Points on Sets of Revolution in R3­ (with J. Brauchart and D. Hardin), J. Math. Phys, 48 (2007), no. 12, 122901, 24 pp. [PDF]

(215) Menke Points on the Real Line and Their Connection to Classical Orthogonal Polynomials (with P. Mathur and J.S. Brauchart), J. Comput. Appl. Math., 233 (2010), 1416-1431. [PDF]

(212) Asymptotics of Best-Packing on Rectifiable Sets (with S.V. Borodachov and D.P. Hardin), Proc. Amer. Math. Soc., Vol. 135 (2007), pp. 2369-2380. [PDF]

(211) Riesz Spherical Potentials with External Fields and Minimal Energy Points Separation (with P. Dragnev), Potential Anal., Vol. 26, No. 2 (2007), pp. 139-162. [PDF]

(210) The Support of the Logarithmic Equilibrium Measure on Sets of Revolution in R3 (with D. Hardin and H. Stahl), J. Math. Phys., Vol. 48, No. 2 (2007), 022901, 14 pp. [PDF]

(208) Asymptotics for Discrete Weighted Minimal Riesz Energy Problems on Rectifiable Sets (with S.V. Borodachov and D.P. Hardin), Trans. Amer. Math. Soc., Vol. 360 (2008), pp. 1559-1580. [PDF]

(205) On Separation of Minimal Riesz Energy Points on Spheres in Euclidean Spaces (with A. B. J. Kuijlaars and X. Sun), Journal Comp. & Applied Math., Vol 199, No. 1 (2007), pp. 172-180. [PDF]

(201) Discretizing Manifolds via Minimum Energy Points
(with Doug Hardin), Notices of the American Mathematical Society, Vol. 51, No. 10 (2004), pp. 1186-1194. [PDF]

(199) Minimal Riesz Energy Point Configurations for Rectifiable d-Dimensional Manifolds
(with Doug Hardin), Advances in Mathematics, Vol. 193, No. 1 (2005), pp. 174-204. [PDF]

(196) Asymptotics for Minimal Discrete Riesz Energy on Curves in Rd
(with A. Martínez-Finkelshtein, V. Maymeskul, and E.A. Rakhmanov), Canadian Journal of Mathematics, Vol. 56 (2004), pp. 529-552. [PDF]

(192) Note on d-extremal configurations for the Sphere in R(d+1)
(with M. Goetz), Recent progress in multivariate approximation, Internat. Ser. Numer. Math., Birkhauser, Basel, Vol. 137  (2001), pp. 159-162. [PDF]

(161) Distributing Many Points on a Sphere
(with A.B.J. Kuijlaars), The Mathematical Intelligencer, Vol. 19, No. 1 (1997), pp. 5-11. [PDF][/p>

(159) Asymptotics for Minimal Discrete Energy on the Sphere
(with A.B.J. Kuijlaars), Transactions of the American Mathematical Society, Vol. 350, No. 2 (1998), pp. 523-538. [PDF]

(156) Electrons on the Sphere
(with E.A. Rakhmanov and Y.M. Zhou), Computational Methods and Function Theory, (R. M. Ali, S. Ruscheweyh, and E. B. Saff, eds.), World Scientific, (1995), pp. 111-127. [PDF]

(155) Minimal Discrete Energy on the Sphere
(with E.A. Rakhmanov and Y.M. Zhou), Mathematical Research Letters, Vol. 1 (1994), pp. 647-662. [PDF]


Spiral Points on the Sphere

The spiral point algorithm developed by Rahkmanov, Saff, and Zhou has been improved by Knud Thomsen as follows:

Initialize:
  p = 1/2
  a = 1 - 2*p/(n-3)
  b = p*(n+1)/(n-3)
  r(1)     = 0
  theta(1) = pi
  phi(1))  = 0
Then for k stepping by 1 from 2 to n-1:
  k'       = a*k + b
  h(k)     = -1 + 2*(k'-1)/(n-1)
  r(k)     = sqrt(1-h(k)^2)
  theta(k) = arccos(h(k))
  phi(k)   = [phi(k-1) + 3.6/sqrt(n)*2/(r(k-1)+r(k))] (mod 2*pi)
Finally:
  theta(n) = 0
  phi(n)   = 0

See
http://groups.google.com/group/sci.math/browse_thread/thread/983105fb1ced42c/e803d9e3e9ba3d23#e803d9e3e9ba3d23

Code for Equal-Area Points on Sphere

EQ Sphere Partititions and Recursive Zonal Equal Area (EQ) Sphere Partitioning Toolbox: http://eqsp.sourceforge.net


Poppy-Seed Bagel Theorem

Science & Vie Profile
Science & Vie, July 2005.

Poppy Seed Bagel Math
National Public Radio, Weekend Edition, December 11, 2004.

The Poppy-Seed Bagel Theorem
Exploration, The Online Research Journal of Vanderbilt University, November 30, 2004.

Discretizing Manifolds via Minimum Energy Points
Notices of the American Mathematical Society, November, 2004.

Rob Womersley's Visualization of Minimum Energy Points on the Torus