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16 - Topics on Mathematical Crystallography

Published online by Cambridge University Press:  20 July 2017

Toshikazu Sunada
Affiliation:
School of Interdisciplinary Mathematical Sciences, Meiji University, Nakano 4-21-1, Nakano-ku, Tokyo, 164-8525 Japan
Tullio Ceccherini-Silberstein
Affiliation:
Università degli Studi del Sannio, Italy
Maura Salvatori
Affiliation:
Università degli Studi di Milano
Ecaterina Sava-Huss
Affiliation:
Cornell University, New York
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Print publication year: 2017

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References

[1] M., Baake, Solution of the coincidence problem in dimensions d ≤ 4, in R. V., Moody, ed., The Mathematics of Long-Range Aperiodic Order, Kluwer, Dordrecht, 1997, pp. 199–237.
[2] R., Bacher, P. De La, Harpe, and T., Nagnibeda, The lattice of integral flows and the lattice of integral cuts on a finite graph, Bull. Soc. Math. France, 125 (1997), 167–98.Google Scholar
[3] M., Baker and S., Norine, Riemann-Roch and Abel-Jacobi theory on a finite graph, Adv. in Math., 215 (2007), 766–88.Google Scholar
[4] N. L., Biggs, Algebraic Graph Theory, Cambridge University Press, 1993.
[5] N. L., Biggs, Algebraic potential theory on graphs, Bull. London Math. Soc., 29 (1997), 641–82.Google Scholar
[6] A., Borel, Introduction aux groupes arithmetiques, Hermann 1969.
[7] K. S., Brown, Building, Springer, 1989.
[8] L.S., Charlap, Bieberbach Groups and Flat Manifolds, Springer-Verlag, 1986.
[9] S.J., Chung, T., Hahn, and W.E., Klee, Nomenclature and generation of three-periodic nets: the vector method, Acta. Cryst., A40 (1984), 42–50.Google Scholar
[10] J.H., Conway, A characterisation of Leech's lattice, Invent. math., 7 (1969), 137–42.Google Scholar
[11] J.H., Conway, H., Burgiel, C., Goodman-Strauss, The Symmetries of Things, A K Peters Ltd, 2008.
[12] H. S. M., Coxeter, Regular Polytopes, Dover 1973.
[13] P.R., Cromwell, Polyhedra, Cambridge University Press, 1997.
[14] O., Delgado-Friedrichs and M., O'Keeffe, Identification of and symmetry computation for crystal nets, Acta Cryst., A59 (2003), 351–60.Google Scholar
[15] R.J., Duffin and A.C., Schaeffer. A class of nonharmonic Fourier series, Trans. Amer. Math. Soc., 72 (1952) 341–66.Google Scholar
[16]O., Delgado-Friedrichs, Barycentric drawings of periodic graphs, LNCS 2912 (2004), 178–189.Google Scholar
[17] W., Ebeling, Lattices and Codes, Vieweg, 1994.
[18] J., Eells and J.H., Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math., 86 (1964) 109–160.Google Scholar
[19] Y.C., Eldar, Optimal tight frames and quantum measurement, Information Theory, IEEE Transactions on 48 (2002), 599–610.Google Scholar
[20] J-G., Eon, Archetypes and other embeddings of periodic nets generated by orthogonal projection, J. Solid State Chem. 147 (1999), 429–37.Google Scholar
[21] J-G., Eon, Euclidean embeddings of periodic nets: definition of a topologically induced complete set of geometric descriptors for crystal structures, Acta Cryst. A67 (2011), 68–86.Google Scholar
[22] V.K., Goyal, J., Kovacevic, and J.A., Kelner, Quantized frame expansions with erasures, Applied and Computational Harmonic Analysis 10 (2001), 203–33.Google Scholar
[23] H., Grimmer, Coincidence rotations for cubic lattices, Scripta Metallurgica 7 (1973), 1295–1300.Google Scholar
[24] J.E., Humphreys, Introduction to Lie Algebra and Representation theory, Springer, 1970.
[25] M., Kotani and T., Sunada, Standard realizations of crystal lattices via harmonic maps, Trans. Amer. Math. Soc., 353 (2000), 1–20.Google Scholar
[26] M., Kotani and T., Sunada, Jacobian tori associated with a finite graph and its abelian covering graphs, Advances in Apply. Math., 24 (2000), 89–110.Google Scholar
[27] M., Kotani and T., Sunada, Albanese maps and off diagonal long time asymptotics for the heat kernel, Comm. Math. Phys., 209 (2000), 633–70.Google Scholar
[28] M., Kotani and T., Sunada, Spectral geometry of crystal lattices, Contemporary Math., 338 (2003), 271–305.Google Scholar
[29] G., McColm, Generating geometric graphs using automorphisms, J. of Graph Algorithms and Appl., 16 (2012), 507–41.Google Scholar
[30] J., Milnor and D., Husemoller, Symmetric Bilinear Forms, Springer 1973.
[31] T., Nagnibeda, The Jacobian of a finite graph, Contemporary Math., 206 (1997), 149–51.Google Scholar
[32] M., Nespolo, Does mathematical crystallography still have a role in XXI century?, Acta Cryst., A64 (2008), 96–111.Google Scholar
[33] P.M., Neumann, G.A., Stoy, and E.C., Thompson, Groups and Geometry, Oxford University Press, 1994.
[34] T., Sunada, Why do Diamonds Look so Beautiful?, (in Japanese) Springer Japan, 2006.
[35] T., Sunada, Crystals that nature might miss creating, Notices Amer. Math. Soc., 55 (2008), 208–15.Google Scholar
[36] T., Sunada, Discrete geometric analysis, Proceedings of Symposia in Pure Mathematics, (ed. by P., Exner, J.P., Keating, P., Kuchment, T., Sunada, A., Teplyaev), 77 (2008), 51–86.Google Scholar
[37] T., Sunada, Lecture on topological crystallography, Japan. J. Math. 7 (2012), 1–39.Google Scholar
[38] T., Sunada, Topological crystallography –With a View Towards Discrete Geometric Analysis–, Surveys and Tutorials in the Applied Mathematical Sciences, Vol. 6, Springer, 2012.
[39] T., Sunada, Standard 2D crystalline patterns and rational points in complex quadrics, arXiv: submit/0620196 [math.CO] 23 Dec 2012.
[40] R., Vale and S., Waldron, Tight frames and their symmetries, Constructive Approximation, 21 (2004), 83–112.Google Scholar
[41] S., Waldron, Generalised Welch bound equality sequences are tight frames, Information Theory, IEEE Transactions on, 49, 9 (2003), 2307–9.Google Scholar
[42] A.F., Wells, The geometrical basis of crystal chemistry, Acta Cryst. 7 (1954), 535.Google Scholar
[43] A.F., Wells, Three Dimensional Nets and Polyhedra, Wiley (1977).
[44] Y.M., Zou, Structures of coincidence symmetry groups, Acta Cryst., A62 (2006), 109–14.Google Scholar
[45] Y.M., Zou, Indices of coincidence isometries of the hypercubic lattice Zn, Acta Cryst., A62 (2006), 454–8.Google Scholar

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