Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Hostname: page-component-7479d7b7d-qs9v7 Total loading time: 0 Render date: 2024-07-13T13:32:07.013Z Has data issue: false hasContentIssue false

1 - The State of the Art in Smale's 7th Problem

Published online by Cambridge University Press:  05 December 2012

By
C. Beltrán
Edited by
Felipe Cucker ,
Teresa Krick ,
Allan Pinkus  and
Agnes Szanto
C. Beltrán
Affiliation:
Universidad de Cantabria
Felipe Cucker
Affiliation:
City University of Hong Kong
Teresa Krick
Affiliation:
Universidad de Buenos Aires, Argentina
Allan Pinkus
Affiliation:
Technion - Israel Institute of Technology, Haifa
Agnes Szanto
Affiliation:
North Carolina State University
Get access

Summary

A very brief historical note

Smale's 7th problem is the computational version of an old problem dating back to Thomson [30] and Tammes [29], see Whyte's early review [32] for its history, namely, the sensible distribution of points in the two- dimensional sphere. In Whyte's paper different possible definitions of “well-distributed points in the sphere” are suggested:

  1. 1. Points which maximise the product of their mutual distances (called elliptic Fekete points after [14]).

  2. 2. Points which minimise the sum of the inverse of their mutual distances (Thomson's problem), and more generally which minimise some sum of potentials which depend on the mutual distances (like Riesz potentials).

  3. 3. Points which maximise the least distance between any pair.

  4. 4. Points which are the center of the optimal packing problem, that is, the problem of finding the smallest radius of a sphere such that one can place on its surface k non-overlapping circles of a given radius.

This beautiful problem is terribly challenging! A first shocking result by Leech [19] showed that even though the set of N particles on the sphere which are critical points for the problem in item (2) for every possible potential can be completely described, this description is not enough to solve the problem for any particular potential. Namely, solving problem (2) for some particular potential may be completely meaningless for solving problem (2) for another, different potential.

Type
Chapter
Information
Foundations of Computational Mathematics, Budapest 2011 , pp. 1 - 15
Publisher: Cambridge University Press
Print publication year: 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] N. N., Andreev, An extremal property of the icosahedron. East J. Approx., 2, 459–462, 1996.Google Scholar
[2] D., Armentano, C., Beltrán, and M., Shub, Minimizing the discrete logarithmic energy on the sphere: The role of random polynomials. Trans. Amer. Math. Soc., 363, 2955–2965, 2011.
[3] B., Beauzamy, E., Bombieri, P., Enflo, and H. L., Montgomery, Products of polynomials in many variables. J. Number Theory, 36, 219–245, 1990.Google Scholar
[4] B., Beauzamy, V., Trevisan, and P. S., Wang, Polynomial factorization: sharp bounds, effcient algorithms. J. Symbolic Comput., 15, 393–413, 1993.Google Scholar
[5] C., Beltrán, Harmonic properties of the logarithmic potential and the computability of elliptic Fekete points. To appear in Constr. Approx. (DOI: 10.1007/s00365-012-9158-y).
[6] E., Bendito, A., Carmona, A. M., Encinas, J. M., Gesto, A., Gómez, C., Mouriño, and M. T., Sánchez, Computational cost of the Fekete problem. I. The forces method on the 2-sphere. J. Comput. Phys., 228, 3288n–3306, 2009.Google Scholar
[7] B., Bergersen, D., Boal, and P., Palffy-Muhoray, Equilibrium configurations of particles on a sphere: the case of logarithmic interactions. J. Phys. A: Math. Gen., 27, 2579–2586, 1994.Google Scholar
[8] L., Blum, F., Cucker, M., Shub, and S., Smale, Complexity and Real Computation. Springer-Verlag, New York, 1998.
[9] L., Blum, M., Shub, and S., Smale, On a theory of computation and complexity over the real numbers: NP-completeness, recursive functions and universal machines. Bull. Amer. Math. Soc. (N.S.), 21, 1–46, 1989.Google Scholar
[10] J. S., Brauchart, Optimal logarithmic energy points on the unit sphere. Math. Comp., 77, 1599–1613, 2008.Google Scholar
[11] P. D., Dragnev, On the separation of logarithmic points on the sphere. In Approximation Theory X, (St. Louis, MO, 2001), Innov. Appl. Math., p. 137–144, Vanderbilt Univ. Press, Nashville, TN, 2002.
[12] P. D., Dragnev, D. A., Legg, and D. W., Townsend, Discrete logarithmic energy on the sphere. Pacific J. Math., 207, 345–358, 2002.Google Scholar
[13] A., Dubickas, On the maximal product of distances between points on a sphere. Liet. Mat. Rink., 36, 303–312, 1996.Google Scholar
[14] M., Fekete, Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten. Math. Z., 17, 228–249, 1923.Google Scholar
[15] D., Hardin and E. B., Saff, Discretizing manifolds via minimum energy points. Notices Amer. Math. Soc., 51, 1186–1194, 2004.Google Scholar
[16] J.-B., Hiriart-Urruty, A new series of conjectures and open questions in optimization and matrix analysis. In ESAIM: Control, Optimisation and Calculus of Variations, p. 454–470, 2009.
[17] A. V., Kolushov and V. A., Yudin, Extremal dispositions of points on the sphere. Anal. Math., 23, 25–34, 1997.Google Scholar
[18] A. B. J., Kuijlaars and E. B., Saff, Distributing many points on a Sphere. Math. Int., 19, 5–11, 1997.Google Scholar
[19] J., Leech, Equilibrium of sets of particles on a sphere. Math. Gaz., 41, 81–90, 1957.Google Scholar
[20] J., Pintér, Globally optimized spherical point arrangements: Model variants and illustrative results. Ann. Op. Res., 104, 213–230, 2001.Google Scholar
[21] E. A., Rakhmanov, E. B., Saff, and Y. M., Zhou, Minimal discrete energy on the sphere. Math. Res. Letters, 1, 647–662, 1994.Google Scholar
[22] E. A., Rakhmanov, E. B., Saff, and Y. M., Zhou, Electrons on the sphere. In Computational Methods and Function Theory 1994 (Penang), volume 5 of Ser. Approx. Decompos., p. 293–309, World Sci. Publ., River Edge, NJ, 1995.
[23] E., Schmutz, Rational points on the unit sphere. Cent. Eur. J. Math., 6, 482–487, 2008.Google Scholar
[24] M., Shub and S., Smale, Complexity of Bézout's theorem. I. Geometric aspects. J. Amer. Math. Soc., 6, 459–501, 1993.Google Scholar
[25] M., Shub and S., Smale, Complexity of Bezout's theorem. II. Volumes and probabilities. In Computational Algebraic Geometry (Nice, 1992), volume 109 of Progr. Math., p. 267–285, Birkhäuser Boston, Boston, MA, 1993.
[26] M., Shub and S., Smale, Complexity of Bezout's theorem. III. Condition number and packing. J. Complexity, 9, 4–14, 1993. Festschrift for Joseph F. Traub, Part I.Google Scholar
[27] S., Smale, Mathematical problems for the next century. In Mathematics: Frontiers and Perspectives, p. 271–294. Amer. Math. Soc., Providence, RI, 2000.
[28] W., Stortelder, J., Swart, and J., Pintér, Finding elliptic Fekete points sets: two numerical solution approaches. J. Comp. App. Math., 130, 205–216, 2001.Google Scholar
[29] P. M. L., Tammes, On the origin of number and arrangement of the places of exit on the surface of pollen-grains. Recueil des travaux botaniques neerlandais 27, Diss. Groningen., 1930.
[30] J. J., Thomson, On the structure of the atom. Phil. Mag., 7, 237–265, 1904.Google Scholar
[31] G., Wagner, On the product of distances to a point set on a sphere. J. Austral. Math. Soc. Ser.A, 47, 466–482, 1989.Google Scholar
[32] L. L., Whyte, Unique arrangements of points on a sphere. Amer. Math. Monthly, 59, 606–611, 1952.Google Scholar
[33] S., Zelditch and Q., Zhong, Addendum to “Energies of zeros of random sections on Riemann surfaces”. Indiana Univ. Math. J., 59, 2001–2006, 2010.Google Scholar
[34] Q., Zhong, Energies of zeros of random sections on Riemann surfaces. Indiana Univ. Math. J., 57, 1753–1780, 2008.Google Scholar
[35] Y., Zhou, Arrangements of Points on the Sphere. Ph. D. Thesis, Math. Department, University of South Florida, 1995.

Save book to Kindle

To save this book to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×