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Showing papers on "Symmetry (geometry) published in 1974"


Journal ArticleDOI
TL;DR: In this paper, it was shown that MS3 point groups are reducible in the form Q⊕∊. 1, where 1 is the unit 2 × 2 matrix, and ∊ = ± 1.
Abstract: A modulated structure can be depicted as a section through a four-dimensional periodic structure. In the latter, each atom is represented by a string continuing endlessly in the overall direction (e4) of the normal to R3, R3 being the hyperplane of the section. The strings have periodic bends or densifications for displacive and substitutional modulation respectively. Formulae for structure factors can be derived from this picture with little effort. The pseudo-symmetry of modulated structures can be described conveniently in this picture. Each four-dimensional space group to which the four-dimensional structure can belong is a possible MS3 (modulated three-dimensional structure) group of pseudo-symmetry, and is called an MS3 space group. It is shown that MS3 point groups are reducible in the form Q⊕∊. 1, where 1 is the unit 2 × 2 matrix, and ∊ = ± 1. A list is presented of these 31 groups written as black-and-white or colourless groups of three-dimensional symmetry. The MS3 space groups are discussed briefly. As an example of the peculiar differentiations caused by e4 being a unique direction, the 23 MS2 space groups are listed explicitly. Finally, it is shown that MS groups are essential for the description of MS symmetry, because very often the latter cannot be represented completely and unambiguously by the normal space group of an approximate superstructure.

376 citations


Journal Article
TL;DR: In this article, it was shown that MS3 point groups are reducible in the form Q⊕∊. 1, where 1 is the unit 2 × 2 matrix, and ∊ = ± 1.
Abstract: A modulated structure can be depicted as a section through a four-dimensional periodic structure. In the latter, each atom is represented by a string continuing endlessly in the overall direction (e4) of the normal to R3, R3 being the hyperplane of the section. The strings have periodic bends or densifications for displacive and substitutional modulation respectively. Formulae for structure factors can be derived from this picture with little effort. The pseudo-symmetry of modulated structures can be described conveniently in this picture. Each four-dimensional space group to which the four-dimensional structure can belong is a possible MS3 (modulated three-dimensional structure) group of pseudo-symmetry, and is called an MS3 space group. It is shown that MS3 point groups are reducible in the form Q⊕∊. 1, where 1 is the unit 2 × 2 matrix, and ∊ = ± 1. A list is presented of these 31 groups written as black-and-white or colourless groups of three-dimensional symmetry. The MS3 space groups are discussed briefly. As an example of the peculiar differentiations caused by e4 being a unique direction, the 23 MS2 space groups are listed explicitly. Finally, it is shown that MS groups are essential for the description of MS symmetry, because very often the latter cannot be represented completely and unambiguously by the normal space group of an approximate superstructure.

360 citations






Journal ArticleDOI
TL;DR: The concept of population weighted-symmetry (a generalization of population symmetry) is discussed in this article, and a characterization of weighted symmetry is provided for the special case of population symmetry.
Abstract: The concept of population weighted-symmetry (a generalization of population symmetry) is discussed and a characterization of weighted-symmetry provided. For the special case of population symmetry, this characterization yields a strong converse to the well-known result: if the distribution of a continuous random variable X is symmetric about θ, then Z= |X – θ| and the indicator function for the event {X ≥ θ} are stochastically independent. Implications and possible applications of the general weighted-symmetry results are considered.

19 citations


Journal ArticleDOI
TL;DR: In recent years there was a growing interest in the problem of symmetry of Banach algebras as mentioned in this paper, and very substantial progress has been made towards a solution of the characterizations of such locally compact groups G for which the group algebra Lι(G) is symmetric.
Abstract: In recent years there was a growing interest in the problem of symmetry of involutive Banach algebras. In particular very substantial progress has been made towards a solution of the problem of characterizations of such locally compact groups G for which the group algebra Lι(G) is symmetric. The most striking results in this direction are due to J. Jenkins, who first proved that the discrete "a% + 6 "-group has a nonsymmetric algebra [3] and that the same

17 citations



Journal ArticleDOI
01 Dec 1974

12 citations


Journal ArticleDOI
TL;DR: In this paper, two methods for the deduction of asymmetric units are proposed and have been applied to cubic space groups based on the knowledge of the Dirichlet domains (Wirkungsbereiche) for special sets of equivalent points.
Abstract: Two different methods for the deduction of asymmetric units are proposed and have been applied to cubic space groups. Both these methods are based on the knowledge of the Dirichlet domains (Wirkungsbereiche) for special sets of equivalent points: (1) The Dirichlet domains for points in general positions directly give rise to asymmetric units. For the limiting cases, where higher symmetry is simulated by relations between the coordinates, these Dirichlet domains are known as those of special positions in supergroups. (2) According to point symmetry the Dirichlet domains for special positions may be split into asymmetric units for the space group under consideration. Selection of the simplest asymmetric unit for each space group leads to 15 different polyhedra for all cubic space groups. The part of the border of the asymmetric unit that belongs to the asymmetric unit is specified for each space group.




Journal ArticleDOI
TL;DR: In this article, the symmetry and identification problems for time-optimal linear autonomous control systems in Euclidean space were studied, where a symmetry is a nonsingular linear transformation which reproduces each of the reachable sets.
Abstract: This paper is concerned with symmetry and identification problems for time-optimal linear autonomous control systems in Euclidean space. A symmetry is a nonsingular linear transformation which reproduces each of the reachable sets. Under rather general conditions all the symmetries of a control system are described constructively (Theorem 4). In an analogous situation, it is determined when two control systems have the same reachable sets; the identification problem (Theorem 6). Both of these are special cases of a wider result (Theorem 3) which may well be amenable to further generalization.


Journal ArticleDOI
TL;DR: A re-examination of the conditions for the diffraction enhancement of symmetry for the structures of types 1 and 2 [Iwasaki H. as discussed by the authors ] has brought out the existence of some additional solutions.
Abstract: A re-examination of the conditions for the diffraction enhancement of symmetry for the structures of types 1 and 2 [Iwasaki H. (1972). Acta Cryst. A28, 253-260] has brought out the existence of some additional solutions. The conditions for these types of structures have been systematically tabulated.

Journal ArticleDOI
TL;DR: In this paper, the 3.39 μm methane line has been denoted F 1 or F 2, depending on the author, and it is shown that either notation may be used, but that it is necessary in all theories to employ some trick to avoid the use of translational wavefunctions.

Journal ArticleDOI
TL;DR: Using the theory of representation analysis, Bertaut et al. as discussed by the authors presented the stereographic projections for all the magnetic symmetry groups, which are useful in studying the properties of magnetically ordered crystals.
Abstract: Using the theory of representation analysis [Bertaut, E. F. (1968). Acta Cryst. A24, 217-231] and with the aid of some newly introduced symmetry symbols we present the stereographic projections for all the magnetic symmetry groups. These groups are useful in studying the properties of magnetically ordered crystals.


Journal ArticleDOI
TL;DR: In this paper, a system with C s symmetry differing in detail from the "quasi D 2d " form proposed by the discoverers is found to be the stable form.

Journal ArticleDOI
B. B. Phadke1
TL;DR: In this paper, it was shown that in two-dimensional straight G-spaces with a not necessarily symmetric distance, the symmetry of distance is in fact implied by the flatness of bisectors.
Abstract: H. Busemann proved that the elementary spaces are characterized by the flatness of bisectors among all G-spaces with symmetric distance. In this paper we prove that in two dimensional straight G-spaces with a not necessarily symmetric distance the symmetry of distance is in fact implied by the flatness of bisectors. Thus Busemann's characterizations of the euclidean and hyperbolic planes are shown to hold among all straight two dimensional G-spaces with a not necessarily symmetric distance.

Book ChapterDOI
01 Jan 1974


Journal ArticleDOI
TL;DR: In this paper, the defining equations (5) and (6), respectively, for coordinates of the first integrals of Eq. (1) and the third-order symmetry operators of Eqs. (2) were obtained in covariant form.
Abstract: In the study we have obtained, in covariant form, the defining equations (5) and (6), respectively, for coordinates of the first integrals of Eq. (1) and the third-order symmetry operators of Eq. (2).

Journal ArticleDOI
TL;DR: In this paper, the relationship between positional variances and covariances that refer to different choices of origin is presented, and the situation is considered in which the uncertainties in the positioning of atoms are direction-independent and the only covariance terms are those that arise from polar aspects and nonorthogonality of the coordinate axes and from symmetry.
Abstract: Relationships between positional variances and covariances that refer to different choices of origin are presented, and the situation is considered in which the uncertainties in the positioning of atoms are direction-independent and the only covariance terms are those that arise from polar aspects and nonorthogonality of the coordinate axes and from symmetry. Covariance terms can often be made small by placing the origin at a suitable centroid.

Journal ArticleDOI
TL;DR: In this paper, the exterior formulation of the Helmholtz equation for a scattered wave is reduced to a line integral via the Maggi transformation, which is performed over the line dividing a three-dimensional scatterer into an illuminated region and a shadow region.
Abstract: The exterior formulation of the Helmholtz equation for a scattered wave is reduced to a line integral via the Maggi transformation. This integration is performed over the line dividing a three‐dimensional scatterer into an illuminated region and a “shadow region.” The scattered wave was found for “shadow lines” with shapes of regular polygons and a circle. As the symmetry is broken, the amplitude of the scattered wave directly in back of the scatterer (on the symmetry axis) decreases.

Book ChapterDOI
01 Jan 1974
TL;DR: The logic of the plausible is the transmission of plausibility by symmetry or transitivity, wherever it is correct for it to be done.
Abstract: The logic of the plausible is the transmission of plausibility by symmetry or transitivity, wherever it is correct for it to be done.


Book ChapterDOI
01 Jan 1974
TL;DR: The Bifurcation Sets of the Reduced Double Cusp (BDS) as mentioned in this paper were drawn to investigate the possibility of the existence of three-, four-and five-fold symmetry in the projected envelope.
Abstract: The Bifurcation Sets of the Reduced Double Cusp V=A(x4+y4−6x2y2)+B(x3−3xy2)+C (x2+y2) -ux-vy were drawn to investigate the possibility of the existence of three-, four- and five-fold symmetry in the projected envelope While three-and four-fold symmetry could be demonstrated, it was impossible to demonstrate five-fold symmetry However, modifying one of the parametic equations in a heuristic way did permit the production of a figure with five-fold symmetry