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1978
The idea of a group is one of the great unifying ideas of mathematics. It arises in the study of symmetries, both of mathematical and of scientific objects. Very surprisingly, the examination of these symmetries leads to deep insights which are not available by direct inspection: while the notion of a group is very easy to explain, the applications of ...
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The idea of a group is one of the great unifying ideas of mathematics. It arises in the study of symmetries, both of mathematical and of scientific objects. Very surprisingly, the examination of these symmetries leads to deep insights which are not available by direct inspection: while the notion of a group is very easy to explain, the applications of ...
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1978
Let us now consider more formally the question of groups and symmetries. Mathematically a group is defined by a set of axioms, and for the sake of completeness these axioms are stated here.
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Let us now consider more formally the question of groups and symmetries. Mathematically a group is defined by a set of axioms, and for the sake of completeness these axioms are stated here.
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1988
Those discrete groups which play the central role in solid-state physics are the point groups and their extensions (double, colour groups), the translation groups, and the combination of both (the space groups). These groups and the meaning of their elements are discussed in the following sections.
Wolfgang Ludwig, Claus Falter
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Those discrete groups which play the central role in solid-state physics are the point groups and their extensions (double, colour groups), the translation groups, and the combination of both (the space groups). These groups and the meaning of their elements are discussed in the following sections.
Wolfgang Ludwig, Claus Falter
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2009
Up to this point, structures of mostly finite objects have been discussed. Thus, point groups were applicable to their symmetries. A simplified classification of various symmetries was presented in Chapter 2 (cf., Figure 2-31 and Table 2-2). Point-group symmetries are characterized by the lack of periodicity in any direction.
Magdolna Hargittai, István Hargittai
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Up to this point, structures of mostly finite objects have been discussed. Thus, point groups were applicable to their symmetries. A simplified classification of various symmetries was presented in Chapter 2 (cf., Figure 2-31 and Table 2-2). Point-group symmetries are characterized by the lack of periodicity in any direction.
Magdolna Hargittai, István Hargittai
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1974
A mathematical fact of very great significance for relativity theory is the existence of the familiar homomorphism* between the group SL(2,C) of complex unimodular (2 × 2) matrices and the Lorentz group 0(1,3). This homomorphism $$SL\left( {2,C} \right) \to 0\left( {1,3} \right)$$ (1.1) is a local isomorphism and maps SL(2,C) onto the ...
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A mathematical fact of very great significance for relativity theory is the existence of the familiar homomorphism* between the group SL(2,C) of complex unimodular (2 × 2) matrices and the Lorentz group 0(1,3). This homomorphism $$SL\left( {2,C} \right) \to 0\left( {1,3} \right)$$ (1.1) is a local isomorphism and maps SL(2,C) onto the ...
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1982
Thus far the treatment of symmetry has been restricted to the proper rotational and reflection symmetries of space lattices. The discussion of the symmetry of crystalline solids does not end with the presentation of the 14 Bravais lattice types because the symmetry of a solid is the symmetry of its three-dimensionally periodic particle density, and ...
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Thus far the treatment of symmetry has been restricted to the proper rotational and reflection symmetries of space lattices. The discussion of the symmetry of crystalline solids does not end with the presentation of the 14 Bravais lattice types because the symmetry of a solid is the symmetry of its three-dimensionally periodic particle density, and ...
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2007
Abstract The previous two chapters dealt with the geometrical properties of a three-dimensional point lattice, its admissible rotational symmetries and their classification in terms of point groups, crystal systems and Bravais lattices. The symmetry of an ideal, three-dimensional crystal, that is, a triply periodic (infinite) assembly of
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Abstract The previous two chapters dealt with the geometrical properties of a three-dimensional point lattice, its admissible rotational symmetries and their classification in terms of point groups, crystal systems and Bravais lattices. The symmetry of an ideal, three-dimensional crystal, that is, a triply periodic (infinite) assembly of
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2014
Abstract The study of symmetry elements in two and three dimensions is followed by point groups, their derivation and recognition, including an interactive program for point group recognition. Euler’s theorem on the combination of rotations is discussed, and the physical properties of crystals and molecules in relation to their point ...
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Abstract The study of symmetry elements in two and three dimensions is followed by point groups, their derivation and recognition, including an interactive program for point group recognition. Euler’s theorem on the combination of rotations is discussed, and the physical properties of crystals and molecules in relation to their point ...
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