nach oben

Foundations of Computational Mathematics

Erschienen in:

01.02.2015

A Facility Location Formulation for Stable Polynomials and Elliptic Fekete Points

verfasst von: Carlos Beltrán

Erschienen in: Foundations of Computational Mathematics | Ausgabe 1/2015

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Aus

Abstract

A breakthrough paper written in 1993 by Shub and Smale unveiled the relationship between stable polynomials and points which minimize the discrete logarithmic energy on the Riemann sphere (a.k.a. elliptic Fekete points). This relationship has inspired advances in the study of both concepts, many of whose main properties are not well known yet. In this paper I prove an equivalent formulation for the problem of elliptic Fekete points and some consequences, including a (nonsharp) reciprocal of Shub and Smale’s result and some novel nontrivial claims about these classical problems.

Vorheriger Artikel Effect of Islands in Diffusive Properties of the Standard Map for Large Parameter Values

Nächster Artikel Degeneracy Loci and Polynomial Equation Solving

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

C. Aistleitner, J. S. Brauchart, and J. Dick, Point sets on the sphere \({\mathbb{S}}^2\) with small spherical cap discrepancy, Discrete Comput. Geom. 48 (2012), no. 4, 990–1024.MATHMathSciNet

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 (2011), no. 6, 2955–2965.CrossRefMATHMathSciNet

B. Beauzamy, E. Bombieri, P. Enflo, and H. L. Montgomery, Products of polynomials in many variables, J. Number Theory 36 (1990), no. 2, 219–245.CrossRefMATHMathSciNet

J. Beck, Sums of distances between points on a sphere-an application of the theory of irregularities of distribution to discrete geometry, Mathematika 31 (1984), no. 1, 33–41.CrossRefMATHMathSciNet

C. Beltrán, Harmonic properties of the logarithmic potential and the computability of elliptic Fekete points, Constr. Approx. 37 (2013), no. 1, 135–165.CrossRefMATHMathSciNet

L. Blum, F. Cucker, M. Shub, and S. Smale, Complexity and Real Computation, Springer-Verlag, New York, 1998.CrossRef

J. S. Brauchart, Optimal logarithmic energy points on the unit sphere, Math. Comp. 77 (2008), no. 263, 1599–1613.CrossRefMATHMathSciNet

J. S. Brauchart, D. P. Hardin, and E. B. Saff, The next-order term for optimal Riesz and logarithmic energy asymptotics on the sphere, Recent advances in orthogonal polynomials, special functions, and their applications, Contemp. Math., vol. 578, Amer. Math. Soc., Providence, RI, 2012, pp. 31–61.

Jean-Pierre Dedieu, Estimations for the separation number of a polynomial system, J. Symbolic Comput. 24 (1997), no. 6, 683–693.CrossRefMATHMathSciNet

10.

P. D. Dragnev, On the separation of logarithmic points on the sphere, Approximation Theory, X (St. Louis, MO, 2001), Innov. Appl. Math., Vanderbilt Univ. Press, Nashville, TN, 2002, pp. 137–144.

11.

Z. Drezner and H. W. Hamacher (eds.), Facility location, Springer-Verlag, Berlin, 2002, Applications and theory.

12.

A. Dubickas, On the maximal product of distances between points on a sphere, Liet. Mat. Rink. 36 (1996), no. 3, 303–312.MathSciNet

13.

M. Götz, On the distribution of weighted extremal points on a surface in \({ R}^d, d\ge 3\), Potential Anal. 13 (2000), no. 4, 345–359.CrossRefMATHMathSciNet

14.

W. Habicht and B. L. van der Waerden, Lagerung von Punkten auf der Kugel, Math. Ann. 123 (1951), 223–234.CrossRefMATHMathSciNet

15.

A. B. J. Kuijlaars and E. B. Saff, Distributing many points on a Sphere, Math. Int. 19 (1997), no. 1, 5–11.CrossRefMATHMathSciNet

16.

J. Leech, Equilibrium of sets of particles on a sphere, Math. Gaz. 41 (1957), 81–90.CrossRefMATHMathSciNet

17.

P. Leopardi, Discrepancy, separation and Riesz energy of finite point sets on the unit sphere, Adv. Comput. Math. 39 (2013), no. 1, 27–43.CrossRefMATHMathSciNet

18.

J. Marzo and J. Ortega-Cerdà, Equidistribution of Fekete points on the sphere, Constr. Approx. 32 (2010), no. 3, 513–521.CrossRefMATHMathSciNet

19.

E. A. Rakhmanov, E. B. Saff, and Y. M. Zhou, Minimal discrete energy on the sphere, Math. Res. Letters 1 (1994), 647–662.CrossRefMATHMathSciNet

20.

M. Reymer, Constructive theory of multivariate functions, Bibliographisches Institut, Mannheim, 1990.

21.

M. Shub, Complexity of Bézout’s theorem. VI: Geodesics in the condition (number) metric, Found. Comput. Math. 9 (2009), no. 2, 171–178.CrossRefMATHMathSciNet

22.

M. Shub and S. Smale, Complexity of Bézout’s theorem. I. Geometric aspects, J. Amer. Math. Soc. 6 (1993), no. 2, 459–501.MATHMathSciNet

23.

I. H. Sloan and R. S. Womersley, Complexity of Bezout’s theorem. II. Volumes and probabilities, Computational Algebraic Geometry (Nice, 1992), Progr. Math., vol. 109, Birkhäuser Boston, Boston, MA, 1993, pp. 267–285.

24.

I. H. Sloan and R. S. Womersley, Complexity of Bezout’s theorem. III. Condition number and packing, J. Complexity 9 (1993), no. 1, 4–14, Festschrift for Joseph F. Traub, Part I.

25.

I. H. Sloan and R. S. Womersley, Extremal systems of points and numerical integration on the sphere, Adv. Comput. Math. 21 (2004), no. 1–2, 107–125.CrossRefMATHMathSciNet

26.

S. Smale, Mathematical problems for the next century, Mathematics: Frontiers and Perspectives, Amer. Math. Soc., Providence, RI, 2000, pp. 271–294.

27.

J. J. Thomson, On the structure of the atom, Phil. Mag. 7 (1904), no. 6, 237–265.CrossRefMATHMathSciNet

28.

G. Wagner, On the product of distances to a point set on a sphere, J. Austral. Math. Soc. Ser. A 47 (1989), no. 3, 466–482.CrossRefMATHMathSciNet

29.

G. Wagner, Erdős-Turán inequalities for distance functions on spheres, Michigan Math. J. 39 (1992), no. 1, 17–34.CrossRefMATHMathSciNet

30.

L. L. Whyte, Unique arrangements of points on a sphere, Amer. Math. Monthly 59 (1952), 606–611.CrossRefMATHMathSciNet

Titel: A Facility Location Formulation for Stable Polynomials and Elliptic Fekete Points
verfasst von: Carlos Beltrán
Publikationsdatum: 01.02.2015
Verlag: Springer US
Erschienen in: Foundations of Computational Mathematics / Ausgabe 1/2015
Print ISSN: 1615-3375
Elektronische ISSN: 1615-3383
DOI: https://doi.org/10.1007/s10208-014-9213-0

Springer Professional

Abstract

Bitte loggen Sie sich ein, um Zugang zu Ihrer Lizenz zu erhalten.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Technik"

Springer Professional "Wirtschaft"

Weitere Artikel der Ausgabe 1/2015

A Complexity Theory of Constructible Functions and Sheaves

A Randomized Homotopy for the Hermitian Eigenpair Problem

Overdetermined Systems of Sparse Polynomial Equations

A -Conjecture for Newton Polygons

Foreword

Effect of Islands in Diffusive Properties of the Standard Map for Large Parameter Values