The electrogyration effect linear in the electric field occurs in crystals of all point groups of symmetry except for the three cubic m3m, 432 and \ overline { 4 } 3m \,.
22.
In Newtonian mechanics, which can be viewed as a limiting case of special relativity in which the speed of light is infinite, inertial frames of reference are related by the Galilean group of symmetries.
23.
In the terms of Felix Klein's Erlangen programme, we read off from this that Euclidean geometry, the geometry of the Euclidean group of symmetries, is therefore a specialisation of affine geometry.
24.
According to this definition, supplemented with the constancy of the speed of light, inertial frames of reference transform among themselves according to the Poincar?group of symmetry transformations, of which the Lorentz transformations are a subgroup.
25.
Because General Relativity is a diffeomorphism invariant theory, it has an infinite continuous group of symmetries rather than a finite-parameter group of symmetries, and hence has the wrong group structure to guarantee a conserved energy.
26.
Because General Relativity is a diffeomorphism invariant theory, it has an infinite continuous group of symmetries rather than a finite-parameter group of symmetries, and hence has the wrong group structure to guarantee a conserved energy.
27.
In number theory, a "'Poincar?series "'is a mathematical series generalizing the classical theta series that is associated to any discrete group of symmetries of a complex domain, possibly of several complex variables.
28.
But the most geometrically symmetrical version of all is the original Sudanese M�bius band in the three-sphere " S " 3, where its full group of symmetries is isomorphic to the Lie group O ( 2 ).
29.
The group of symmetries of the projective space \ mathbb P ^ n _ { \ mathbf k } is the group of projectivized linear automorphisms \ mathrm { PGL } _ { n + 1 } ( \ mathbf k ).
30.
When & beta; is monotone, its packing density is clearly 1, and packing densities are invariant under the group of symmetries generated by inverse and reverse, so for permutations of length three, there is only one nontrivial packing density.
