On symmetries of crystals with defects related to a class of solvable groups \((S_{2})\). (English) Zbl 1458.74030
Summary: We find the geometrical symmetries of discrete structures which generalize the perfect lattices of crystallography so as to account for the existence of continuous distributions of defects.
MSC:
| 74E15 | Crystalline structure |
| 22E70 | Applications of Lie groups to the sciences; explicit representations |
References:
| [1] | Auslander, Flows on Homogeneous Spaces (1963) · Zbl 0106.36802 · doi:10.1515/9781400882021 |
| [2] | Thurston, Three Dimensional Geometry and Topology, Vol. 1 (1997) · Zbl 0873.57001 |
| [3] | Parry, Reconciliation of local and global symmetries for a class of crystals with defects, Journal of Elasticity 107 pp 81– (2012) |
| [4] | Nicks, On symmetries of crystals with defects related to a class of solvable groups, (S1), Mathematics and Mechanics of Solids · Zbl 1308.74020 |
| [5] | Bachmuth, Automorphisms of free metabelian groups, Transactions of the American Mathematical Society 118 pp 93– (1965) · Zbl 0131.02101 |
| [6] | Olver, Equivalence, Invariants, and Symmetry (1996) |
| [7] | Davini, A Proposal for a continuum theory of defective crystals, Archive for Rational Mechanics and Analysis 96 pp 295– (1986) · Zbl 0623.73002 |
| [8] | Davini, On defect preserving deformations in crystals, International Journal of Plasticity 5 pp 337– (1989) · Zbl 0697.73036 |
| [9] | Parry, Elastic scalar invariants in the theory of defective crystals, Proc. R. Soc. Lond. A 455 pp 4333– (1999) · Zbl 0954.74013 |
| [10] | Pontryagin, Topological Groups, 2. ed. (1955) |
| [11] | Elzanowski, Material symmetry in a theory of continuously defective crystals, Journal of Elasticity 74 pp 215– (2004) |
| [12] | Jacobson, Lie Algebras (1979) |
| [13] | Cermelli, The structure of uniform discrete defective crystals, Continuum Mechanics and Thermodynamics 18 pp 47– (2006) · Zbl 1101.74019 |
| [14] | Rossmann, Lie Groups. An Introduction Through Linear Groups (2002) |
| [15] | Conrad K Decomposing S L 2 ( R ) S L 2 ( R ) |
| [16] | Magnus, Combinatorial Group Theory (1976) |
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