Closing the Case t = 3 for 3-D Spherical t-Designs Using a Result-Verifying Nonlinear Solver

The question if there exists an N-point spherical t-design is not yet settled for all combinations of t and N. Using our framework SONIC for the solution of nonlinear systems, we were able to close the two remaining open cases for t = 3. More precisely, a computational proof revealed that there are no spherical 3-designs with N = 7 or N = 9 points. We describe how these results were obtained and comment on the open cases for larger values of t.

[1]  Aaas News,et al.  Book Reviews , 1893, Buffalo Medical and Surgical Journal.

[2]  A. Frommer,et al.  Symbolic–Numeric Techniques for Solving Nonlinear Systems , 2005 .

[3]  G. Wildenhain.  Bronstein, I. N., und K. A. Semendjajew: Taschenbuch der Mathematik. 5. Aufl., B. G. Teubner Verlagsgesellschaft, Leipzig 1962; 584 S., 430 Abb., Plastikeinband, DM 22,50 , 1964 .

[4]  R. B. Kearfott.  Rigorous Global Search: Continuous Problems , 1996 .

[5]  M.A. Stadtherr,et al.  Parallel Interval-Newton Using Message Passing: Dynamic Load Balancing Strategies , 2001, ACM/IEEE SC 2001 Conference (SC'01).

[6]  Luc Jaulin,et al.  Applied Interval Analysis , 2001, Springer London.

[7]  F. Bertrand.  Plans sphériques de force t et applications en statistique , 2007 .

[8]  Thomas Beelitz,et al.  Effiziente Methoden zum Verifizierten Lösen von Optimierungsaufgaben und Nichtlinearen Gleichungssystemen , 2006 .

[9]  I N Bronstein,et al.  Taschenbuch der Mathematik , 1966 .

[10]  E. M. Rains,et al.  Codes and invariant theory , 2003, math/0311046.

[11]  Richard J Morris,et al.  An evaluation of spherical designs for molecular-like surfaces. , 2006, Journal of molecular graphics & modelling.

[12]  Gabriele Nebe,et al.  On tight spherical designs , 2012, 1201.1830.

[13]  Christian H. Bischof,et al.  Efficient Task Scheduling in the Parallel Result-Verifying Solution of Nonlinear Systems , 2006, Reliab. Comput..

[14]  Christian H. Bischof,et al.  A Hybrid Approach for Efficient Robust Design of Dynamic Systems , 2007, SIAM Rev..

[15]  N. J. A. Sloane,et al.  McLaren’s improved snub cube and other new spherical designs in three dimensions , 1996, Discret. Comput. Geom..

[16]  J. Seidel,et al.  SPHERICAL CODES AND DESIGNS , 1991 .

[17]  Pham Huu Tiep,et al.  Symmetric Squares, Spherical Designs, and Lattice Minima , 2001 .

[19]  Eero Hyvönen,et al.  Constraint Reasoning Based on Interval Arithmetic: The Tolerance Propagation Approach , 1992, Artif. Intell..