On the generators of the symplectic modular group

Hua, Loo-Keng; Reiner, Irving

doi:10.1090/s0002-9947-1949-0029942-0

Cited by 42 publications

(26 citation statements)

References 2 publications

(3 reference statements)

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“…We finally establish that for any pair of Bravais lattices with maximal point symmetry there are reconstructive transformations which generate the entire group G. Indeed, it is readily verified that for suitable pairs of subgroups in G belonging to the four arithmetic classes with maximal point symmetry one can produce all the generators of G, i. e. a suitable reflection, permutation, and simple shear 30 . …”

Section: Methodsmentioning

confidence: 72%

Crystal symmetry and the reversibility of martensitic transformations

Bhattacharya

Conti

Zanzotto

et al. 2004

Nature

311

201

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§ All authors contributed equally to this work.Martensitic transformations are diffusionless solid-to-solid phase transitions characterized by a rapid change of crystal structure, observed in metals, alloys, ceramics, and proteins 1,2 . Phenomenologically, they come in two widely different classes. In steels, quenching generates a microstructure which remains essentially unchanged upon subsequent loading or heating; the transformation is not reversible 1 . In shape-memory alloys on the other hand the microstructures formed on cooling are easily manipulated by loads and disappear upon reheating, and the transformation is reversible 3,4 . Here we explain these sharp differences on the basis of the change in crystal symmetry during the transition. In particular, we show that martensitic transformations fall into two categories. In one case the energy barrier to plastic deformation (via lattice-invariant shears, as in twinning or slip) is no higher than the barrier to the phase change itself. These transformations are therefore irreversible, as observed in steels. In the other case, the energy barrier to lattice-invariant shears can be much higher than that pertaining to the phase change. Consequently, transformations of this type can occur with virtually no plasticity and can be reversible, as for shape-memory alloys.Martensitic transformations are at the basis of numerous technological applications. Most notable amongst these is in steel, where the transformation induced by quenching (fast cooling) is exploited for enhancing the alloy's strength 1 . Another is the fascinating shape-memory effect in alloys like Nitinol, used in medical and engineering devices 3 . Martensitic phase changes are also exploited to toughen structural ceramics 5 such as zirconia, and observed in biological systems such as the tail sheath of the T4 bacteriophage virus 6 . Ideas originating from the study of these transformations have led to improved materials for actuation (ferromagnetic shape-memory alloys 7,8 and ferroelectrics 9 ) and to candidates for artificial muscles 10 . Finally, the rich microstructure (distinctive patterns developed at scales ranging from a few nanometers to a few microns) that accompanies these transformations, has made this a valuable theoretical sand-box for the development of multi-scale modeling tools 11 .

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Section: Methodsmentioning

confidence: 72%

Crystal symmetry and the reversibility of martensitic transformations

Bhattacharya

Conti

Zanzotto

et al. 2004

Nature

311

201

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“…The generators of these groups are known (on Sp(2g, Z) see for example [12], on Sp(2g, Z 2 ) see for example [7,Chap.3]), and these generators are induced by the action of…”

Section: Figurementioning

confidence: 99%

Surfaces in the complex projective plane and their mapping class groups

Hirose

2005

Algebr. Geom. Topol.

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“…Auf die Koeffizienten an den Stellen (3,1), (3,2) kann man durch Transformation mit A1, B 1 einen Euklidischen Aigorithmus ausfiben, ohne dab sonst aul3er an den Stellen (1,4), (2,4) etwas ge£ndert wird. Wegen (all, asl ) -1 folgt also…”

Section: N)unclassified

“…Falls m = 2, und %3 = --1, so multipliziere man A mit I-2eaa--2e44. Die letztere Matrix liegt offenbar in Q~n+2,,, mid in der Produktmatrix steht an der S$elle (3,3) der Koeffizient + 1. Also dfirfen wir annehmen a 42 # 0.…”

Section: N)unclassified

Zur Theorie der Siegelschen Modulgruppe

Mennicke¹

1965

Math. Ann.

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On the generators of the symplectic modular group

Cited by 42 publications

References 2 publications

Crystal symmetry and the reversibility of martensitic transformations

Crystal symmetry and the reversibility of martensitic transformations

Surfaces in the complex projective plane and their mapping class groups

Zur Theorie der Siegelschen Modulgruppe

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