|
|
COMMENTS
|
a(d) is number of parts d-dimensional space (x_1,...,x_d) is split into by set of (2^d - 1) hyperplanes c_1 x_1 + c_2 x_2 + ...+ c_d x_d =0 where c_j are 0 or +1 and we exclude case with all c=0. It is also number of independent real-time Green functions of Quantum Field Theory produced when analytically continuing from euclidean time/energy (d+1 = number of energy/time variables). These are also known as Generalized retarted functions. The numbers upto d=6 were first produced by T. S. Evans using a PASCAL programme, strictly as upper bounds only. M. van Eijck wrote a C programme using a direct enumeration of hyperplanes which confirmed these and produced the value for d=7. Kamiya et al. showed how to find these numbers and some associated polynomials using more sophisticated methods, giving results upto d=7. T. S. Evans added the last number 1st August 2011 using an updated version of van Eijck's programme which took 7 days on a standard desktop computer.
|
|
|
REFERENCES
|
L. J. Billera, J. T. Moore, C. D. Moraites, Y. Wang and K. Williams, Maximal unbalanced families, arXiv preprint arXiv:1209.2309, 2012. - From N. J. A. Sloane, Dec 26 2012
T. S. Evans, N-point finite temperature expectation values at real times, Nuclear Physics B 374 (1992) 340-370.
T. S. Evans, What is being calculated with thermal field theory?, in `Particle Physics and Cosmology: Proceedings of the Ninth Lake Louise Winter School', World Scientific, 1995 (ISBN 9810221002), preprint arXiv:hep-ph/9404262.
H. Kamiya, A. Takemura and H. Terao, Ranking patterns of unfolding models of codimension one, Advances in Applied Mathematics 47 (2011) 379 - 400.
M. van Eijck, Thermal Field Theory and Finite-Temperature Renormalisation Group, PhD thesis, Univ. Amsterdam, 4th Dec. 1995.
|
