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Persistent homology and random models of the Gaussian primes. (English) Zbl 1439.11239
Summary: I. Vardi introduced a probabilistic model for the distribution of Gaussian primes in [Exp. Math. 7, No. 3, 275–289 (1998; Zbl 0926.11065)]. We use persistent homology to test how well Vardi’s model and a symmetrized variant of Vardi’s model capture the geometry of the Gaussian primes. An analysis of the persistent first homology of point clouds produced by these models provides statistical evidence that the models miss some fundamental geometric features of the Gaussian primes.
MSC:
11N05 Distribution of primes
55N35 Other homology theories in algebraic topology
55N31 Persistent homology and applications, topological data analysis
Software:
Dionysus; PHAT
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References:
[1] Michaª Adamaszek and Henry Adams,The VietorisRips complexes of a circle, Pacic J. Math. 290 (2017), no. 1, 140. · Zbl 1366.05124
[2] Michaª Adamaszek, Henry Adams, and Samadwara Reddy, On VietorisRips complexes of ellipses. J. Topol. Anal. in press. https://doi.org/10.1142/S1793525319500274 · Zbl 1426.05182
[3] Peter Bubenik, Statistical topological data analysis using persistence landscapes, J. Mach. Learn. Res. 16 (2015), 77102. · Zbl 1337.68221
[4] Erin W. Chambers, Vin de Silva, Je Erickson, and Robert Ghrist, VietorisRips complexes of planar point sets, Discrete Comput. Geom. 44 (2010), no. 1, 7590. · Zbl 1231.05306
[5] Herbert Edelsbrunner, A Short Course in Computational Geometry and Topology. SpringerBriefs in Applied Sciences and Technology. Cham: Springer, 2014.
[6] Brittany T. Fasy, Jisu Kim, Fabrizio Lecci, Clement Maria, David L. Millman, and Vincent Rouvreau. The included GUDHI is authored by Clement Maria, Dionysus by Dmitriy Morozov, and PHAT by Ulrich Bauer, Michael Kerber, and Jan Reininghaus. TDA: Statistical Tools for Topological Data Analysis. Version 1.6.5. 2017. Available at https://rdrr.io/cran/TDA/
[7] Ellen Gethner, Stan Wagon, and Brian Wick, A stroll through the Gaussian primes, Amer. Math. Monthly 105 (1998), no. 4, 327337. · Zbl 0946.11002
[8] Robert Ghrist, Barcodes: The persistent topology of data, Bull. Amer. Math. Soc. (N.S.) 45 (2008), no. 1, 6175. · Zbl 1391.55005
[9] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 5th ed. New York: The Clarendon Press, Oxford University Press, 1979. · Zbl 0423.10001
[10] Allen Hatcher, Algebraic Topology. Cambridge: Cambridge University Press, 2002.
[11] Konstantin Mischaikow and Vidit Nanda, Morse theory for ltrations and ecient computation of persistent homology, Discrete Comput. Geom. 50 (2013), no. 2, 330353. · Zbl 1278.57030
[12] Vanessa Robins and Katharine Turner, Principal component analysis of persistent homology rank functions with case studies of spatial point patterns, sphere packing and colloids, Phys. D 334 (2016), 99117. · Zbl 1415.60052
[13] Andrew Robinson and Katharine Turner, Hypothesis testing for topological data analysis, J. Appl. Comput. Topol. 1 (2017), no. 2, 241261. · Zbl 1396.62085
[14] John G. Saw, Mark C. K. Yang, and Tse Chin Mo, Chebyshev inequality with estimated mean and variance, Amer. Statist. 38 (1984), no. 2, 130132.
[15] Ilan Vardi, Prime percolation, Experiment. Math. 7 (1998), no. 3, 275289. · Zbl 0926.11065
[16] Afra Zomorodian and Gunnar Carlsson, Computing persistent homology, Discrete Comput. · Zbl 1375.68174
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