The EM Algorithm applied to determining new limit points of Mahler measures

• Autorzy: El Otmani S. , Rhin G. , Sac-Epee J. M.
• Języki publikacji: EN

Abstrakty

EN

In this work, we propose new candidates expected to be limit points of Mahler measures of polynomials. The tool we use for determining these candidates is the Expectation-Maximization algorithm, whose goal is to optimize the likelihood for the given data points, i.e. the known Mahler measures up to degree 44, to be generated by a specific mixture of Gaussians. We will give the mean (which is a candidate to be a new limit point) and the relative amplitude of each component of the more likely gaussian mixture.

Słowa kluczowe

EN

Mahler measure; EM algorithm

• Wydawca: Systems Research Institute, Polish Academy of Sciences
• Czasopismo: Control and Cybernetics
• Rocznik: 2010
• Tom: Vol. 39, no 4
• Strony: 1185--1192
• Opis fizyczny: Bibliogr. 17 poz., wykr.

Twórcy

autor

El Otmani S.

autor

Rhin G.

autor

Sac-Epee J. M.

• Universite Paul Verlaine-Metz, LMAM, CNRS UMR 7122, He du Saulcy, METZ, F-57045, France,

Bibliografia

• BOYD, D.W. (1981) Speculations concerning the range of Mahler’s measure. Canad. Math. Bull. 24, 453-469.
• BOYD, D.W. (1980) Reciprocal polynomials having small measure. Math. Comp. 35, 1361-1377.
• BOYD, D.W. (1989) Reciprocal polynomials having small measure II. Math. Comp. 53, 355-357, S1-S5.
• BOYD, D.W. (2005) M. J. Mossinghoff, Small Limit Points of Mahler’s Measure. Experimental Mathematics 14, Part 4, 403-414.
• DASGUPTA, S. and SCHULMAN, L.J. (2000) A Two-Round Variant of EM for Gaussian Mixtures. 16th Conference on Uncertainty in Artificial Intelligence, 152-159.
• DEMPSTER, A.P., LAIRD, N.M. and RUBIN, D.B. (1977) Maximum Likelihood from Incomplete Data via the EM Algorithm. J. Royal Statist. Soc. Ser. B., 39, 1-38.
• FLAMMANG, V., GRANDCOLAS, M. and RHIN, G. (1999) Small Salem numbers. Number Theory in Progress, 1 (Zakopane-Kościelisko, 1997), de Gruyter, Berlin, 165-168.
• FLAMMANG, V., RHIN, G. and SAC-ÉPÉE, J.M. (2006) Integer Transfmite Diameter and polynomials with small Mahler measure. Math. Comp. 75, 1527-1540.
• LEHMER, D.H. (1933) Factorization of certain cyclotomic functions. Ann. of Math. 2 (34), 461-479.
• MCLACHLAN, G.J. and KRISHNAN, T. (2008) The EM Algorithm and Extensions. Wiley Series in Probability and Statistics (Second, ed.). Wiley-Interscience, Hoboken, NJ.
• MOSSINGHOFF, M.J. (1998) Polynomials with small Mahler measure. Math. Comp. 67, 1697-1705.
• MOSSINGHOFF, M.J., RHIN, G. and WU, Q. (2008) Minimal Mahler Measures. Experiment. Math., 17 (4), 451-458.
• RHIN, G. and SAC-ÉPÉE J.M. (2003) New methods providing high degree polynomials with small Mahler measure. Experiment. Math., 12 (4), 457-461.
• TAGARE, H.D. (1998) A Gentle Introduction to the EM Algorithm, Part I: Theory. Available on the web.
• XU, L. and JORDAN, M.I. (1996) On Convergence Properties of the EM Algorithm for Gaussian Mixtures. Neural Computation 8, 129-151.
• WU, C.F.J. (1983) On the Convergence Properties of the EM Algorithm. The Annals of Statistics 11 (1), 95-103.
• SMYTH, C. (2008) The Mahler measure of algebraic numbers: A survey. Number Theory and Polynomials, London Math. Soc. Lecture Note Ser. 352, Cambridge Univ. Press, 322-349.