# Points on Spheres and Manifolds

(277) Next levels universal bounds for spherical codes: the Levenshtein framework lifted (with P. Boyvalenkov, P. Dragnev, D.P. Hardin and M. Stoyanova), submitted.

(276) Unconstrained Polarization (Chebyshev) Problems: Basic Properties and Riesz Kernel Asymptotics (with D. Hardin, and M. Petrache) submitted.

**(273)** Energy Bounds for Codes in Polynomial Metric Spaces (with P. Boyvalenkov, P. Dragnev, D.P. Hardin and M. Stoyanova), submitted.

**(272)** Asymptotic Linear Programming Lower Bounds for the Energy of Minimizing Riesz and Gauss Configurations (with D.P. Hardin and T.J. Michaels), Mathematika, 65, 2019, no. 1, 157-180

**(271)** On spherical codes with inner products in a prescribed interval (with P. Boyvalenkov, P. Dragnev, D. Hardin, and M. Stoyanova), Des. Codes Cryptogr. (2018). https://doi.org/10.1007/s10623-018-0524-z. [PDF]

**(268)** Local properties of Riesz minimal energy configurations and equilibrium measures (with D. Hardin, A. Reznikov and A. Volberg), IMRN, rnx262, https://doi.org/10.1093/imrn/rnx262 [PDF]

**(267)** Logarithmic and Riesz equilibrium for multiple sources on the sphere-the exceptional case (with J. Brauchart, P. Dragnev, and R. Womersley), accepted In: Contemporary Computational Mathematics – a celebration of the 80th birthday of Ian Sloan (J. Dick, F. Y. Kuo, H. Wozniakowski, eds.),

Springer-Verlag, 2018.

[PDF]

**(266)** Large deviation principles for hypersingular Riesz gases (with D.P. Hardin, T. Leblé, and S. Serfaty), Constr. Approx., 48(1), 2018, 61–100. [PDF]

**(265)** Covering and separation of Chebyshev points for non-integrable Riesz potentials

(with A. Reznikov and A. Volberg), J. Complexity, 46 (2018), 19-44 [PDF]

**(264) ** A Minimum Principle for Potentials with Application to Chebyshev Constants (with A. Reznikov), Potential Analysis, 47 (2017), no. 2, 235–244 [PDF]

**(262)** A Comparison of Popular Point Configurations on S^2 (with D. Hardin and T. Michaels), Dolomites Research Notes on Approximation, vol.9, 2016, 16-49.

[PDF]

**(261)** Optimal discrete measures for Riesz potentials, (with Sergiy V Borodachov, Douglas P. Hardin, Alexander Reznikov) Trans. Amer. Math. Soc., 370(10), 2018, 6973-6993. [PDF]

**(260)** Generating Point Configurations via Hypersingular Riesz Energy with an External Field (with D. Hardin and O. Vlasiuk) SIAM J. Math. Analysis, 49 (2017), no. 1, 646–673. [PDF]

**(259)** Random Point Sets on the Sphere-Hole Radii, Covering, and Separation (with J. Brauchart, A. Reznikov, I. Sloan, Y.G. Wang and R. Womersley) Experimental Math, 27 (2018), no. 1, 62-81. [PDF]

**(258)** Universal Upper and Lower Bounds on Energy of Spherical Designs (with P.G. Boyvalenkov, P. Dragnev, D. Hardin, and M. Stoyanova, Dolomites Research Notes on Approximation, vol. 8, 2015, 51-65. [PDF]

**(257)** Energy Bounds for Codes and Designs in Hamming Spaces (with P.G. Boyvalenkov, P. Dragnev, D. Hardin, and M. Stoyanova) Designs, Codes and Cryptography, Des. Codes Cryptogr. 82 (2017), no. 1-2, 411–433. 94B65 . [PDF]

**(256)** Next Order Energy Asymptotics for Riesz Potentials on Flat Tori”

(with D. Hardin, E. Saff, B. Simanek and Y. Su) IMRN (to appear) .[PDF]

**(255)** Minimum Riesz Energy Problems for a Condenser with “Touching Plates” (with P. Dragnev, B.Fuglede, D. Hardin, and N. Zorii), Potential Analysis (to appear) [PDF]

**(254)** The Covering Radius of Randomly Distributed Points on a Manifold (with A. Reznikov), IMRN (to appear)[PDF]

**(253)** Universal Lower Bounds for Potential Energy of Spherical Codes (with P. Boyvalenkov, P. Dragnev, D. Hardin, and M. Stoyanova) Constructive Approximation (to appear) [PDF]

**(252)** Covering of Spheres by Spherical Caps and Worst-Case Error for Equal

Weight Cubature in Sobolev Spaces (with J. S. Brauchart, J. Dick, I. H. Sloan, YG. Wang and R. S. Womersley),* J. Math. Anal. Appl.* 431 (2015), no. 2, 782–811. [PDF]

**(249) **Periodic Discrete Energy for Long-Range Potentials (with D. Hardin and B. Simanek), J. Math Physics, 55, 123509 (2014) [PDF]

**(248) **An Electrostatics Problem on the Sphere Arising from a Nearby Point Charge (with J. Brauchart and P. Dragnev), In: Constructive Theory of Functions (ed. K. Ivanov, G. Nikolov, R. Uluchev), Sozopol 2013, pp. 11-55, Drinov Academic Publishing House, Sofia, (2014) [PDF]

**(247) **Riesz External Field Problems on the Hypersphere and Optimal Point Separation (with J. Brauchart and P. Dragnev), Potential Analysis, vol. 41 (2014), 647–678.

[PDF]

**(246) **Inverse Bernstein Inequalities and Min-Max-Min Problems on the Unit Circle (with T. Erdelyi and D. Hardin), Mathematika, vol. 61 (2015), no. 3, 581–590. [PDF]

**(245) **Low Complexity Methods for Discretizing Manifolds via Riesz Energy Minimization (with S. Borodachov and D. Hardin), *Found. Comput. Math. * 14 (2014), no. 6, 1173–1208.[PDF]

**(244) **Reverse Triangle Inequalities for Riesz Potentials and Connections with Polarization (with I. Pritsker and W. Wise), *J. Math. Anal. Appl.* 410 (2014), no. 2, 868–881.[PDF]

**(242) **Mesh Ratios for Best-Packing and Limits of Minimal Energy Configurations (with A. Bondarenko and D. Hardin), *Acta Math. Hungar.* 142 (2014), no. 1, 118–131. [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), Math. Comp. 83 (2014), no. 290, 2821–2851. *Math. Comp.* 83 (2014), no. 290, 2821–2851.[PDF]

**(239) **Polarization Optimality of Equally Spaced Points on the Circle for Discrete Potentials (with D. Hardin and A. Kendall), Discrete Comput. Geom. 50 (2013), no. 1, 236–243. [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), Proceedings American Mathematics Society. vol. 142, Issue 3 (2014), 981-988.[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 R^{3} (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 R^{3} (with D. Hardin and H. Stahl), J. Math. Phys., Vol. 48, No. 2 (2007), 122901, 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 R^{d}

(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. 293-309. [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

100 Equal – Area Points on a Sphere

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1000 Equal – Area Points on a Sphere

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10000 Equal – Area Points on a Sphere

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## Poppy-Seed Bagel Theorem

*Wikipedia
*

Science & Vie, July 2005.

Poppy Seed Bagel Math

National Public Radio, Weekend Edition, December 11, 2004.

*Science & Vie*ProfileScience & 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*

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