Showing papers on "Symmetry (geometry) published in 1973"
TL;DR: The symmetrical arrangements of monomers into such cylindrical structures as microfilaments of actin, flagella of bacteria, microtubules of many organisms, and the protein coats of viruses can be specified by citing the index numbers of two or three sets of contact parastichies, or helical ranks ofmonomers, as has been done in classical studies of phyllotaxis.
Abstract: The symmetrical arrangements of monomers into such cylindrical structures as microfilaments of actin, flagella of bacteria, microtubules of many organisms, and the protein coats of viruses can be specified by citing the index numbers of two or three sets of contact parastichies, or helical ranks of monomers, as has been done in classical studies of phyllotaxis. This specification has the form k(m, n) or k(m, n, m+n), where m, n, and (m+n) are parastichy numbers specifying screw displacements, and k is the jugacy, or frequency of rotational symmetry. For simple structures, k = 1. This notation has the advantage of terseness and of indicating the basic isometries of these helically symmetrical structures. Theoretical models of the packing of spheres whose centers lie on the surface of a cylinder have been investigated geometrically. Their symmetry properties are discussed. Parameters of these models, such as the angular divergence, α, the longitudinal displacement between successive spheres, h, the radius of the cylinder, and the angles of inclination of the parastichies, have been computed for representative patterns. The ultrastructural symmetry of several biological structures of this sort has been inferred by comparison with these models. Actin, for example, has the symmetry (1, 2), Salmonella flagella, 2(2, 3, 5), the tobacco mosaic virus, (1, 16, 17) and the microtubules of many higher organisms, (6, 7, 13).
119 citations
TL;DR: In this paper, the symmetry properties of the rotational and torsional levels of two identical XY2 groups connected by a symmetrical linear chain of atoms, with an arbitrary barrier to internal rotation, are investigated.
54 citations
TL;DR: In this article, the structure of all possible symmetry groups of the geometry is described in case that the space-like sections ℊ and ℳ are compact orientable surfaces.
Abstract: Inner geometry and embedding formulas for a totally geodesic null hypersurface ℳ in an electrovacuum space-time are given. The structure of all possible symmetry groups of the geometry is described in case that the space-like sections ℊ and ℳ are compact orientable surfaces and ℳ is, topologically, ℊ ×R1. The result is
, where
are the well-known isometry groups of ℊ, and ℋ is an at most two-dimensional group acting along rays, which is fully specified in the paper. It is not shown that all these symmetry types exist, but this will be done in the next papers where all horizons of a given symmetry type will be explicitly written down.
42 citations
TL;DR: A simple rule that defines which of the symbols of Fig. 1 fall into group A and which into group B can be found in this paper, but the reader is unlikely to guess it even if we tell him that the only relevant differences between the patterns are in symmetry, dot colour and dot shape.
Abstract: THERE is a simple rule that defines which of the symbols of Fig. 1 fall into group A and which into group B. It can be expressed in thirteen words, but the reader is unlikely to guess it—even if we tell him that the only relevant differences between the patterns are in symmetry, dot colour and dot shape.
37 citations
TL;DR: In this paper, an image plane technique of crystallographic analysis which achieves spatial resolution of structural defects and diffraction information, reveals a lack of a center of symmetry, requires only minute crystals and is rapidly executed.
Abstract: We have discovered an image plane technique of crystallographic analysis which achieves spatial resolution of structural defects and diffraction information, reveals a lack of a centre of symmetry, requires only minute crystals and is rapidly executed.
36 citations
TL;DR: The V symmetry coupling coefficients for the icosahedral double group are generated from the behaviour of a minimum number of |JM> ket vectors where the symmetry coupling coefficient are defined as analogues of the Racah V coefficients as mentioned in this paper.
Abstract: The V symmetry coupling coefficients for the icosahedral double group are generated from the behaviour of a minimum number of |JM> ket vectors where the symmetry coupling coefficients are defined as analogues of the Racah V coefficients. The phases are determined from the way the irreducible representations for the specific J values are defined. An investigation of the symmetry properties of the system by a translation of the |JM> ket vectors for integral J values is examined. The handling of the irreducible-tensor method in group notation is discussed briefly.
29 citations
25 citations
TL;DR: The finite, canonical symmetry transformations of the negative energy motions of the classical Kepler problem are constructed by solving the fundamental differential equations of the dynamical invariance group as mentioned in this paper, and the geometric interpretation of the transformations is discussed.
Abstract: The finite, canonical symmetry transformations of the negative energy motions of the classical Kepler problem are constructed by solving the fundamental differential equations of the dynamical invariance group. The geometric interpretation of the transformations is discussed.
24 citations
TL;DR: In this paper, it was shown that the application of a projection operator from a given group to a function is equivalent to the successive application of projection operators from factor groups of the starting group to that function.
Abstract: It is shown that the application of a projection operator from a given group to a function is equivalent to the successive application of projection operators from factor groups of the starting group to that function. When used with the factor groups representing the site symmetry of a position and the simplest group of interchanges of positions, this concept provides a very simple method for obtaining symmetry adapted linear combinations of basis functions.
22 citations
TL;DR: In this article, a unique standardization of phase is proposed by the requirement, in Racah's lemma, (j1α1a1,j2α2a2|j Г ab) ≧ 0 and real.
Abstract: Useful approaches to the calculation of symmetry coupling coefficients 〈Г1 γ1Г2/2|Гγb〉 are reviewed. Since a common phase factor always remains undetermined for each trio ofГ1,Г2, andГ, a unique standardization of phase is proposed by the requirement, in Racah's lemma, (j1Г1a1,j2Г2a2|j Г ab) ≧ 0and real. In conjunction with the basis relations and the phase convention for Wigner coefficients, a novel method is suggested for the calculation of symmetry coupling coefficients in the group G from those in the subgroupG ⊂SU(2) orR3. The results apply in full generality to any point groupG, single or double group.
TL;DR: In this paper, the authors studied the geodesic flow on homogeneous spaces with a Riemannian metric invariant under the group action, as one more application of Smale's theory.
Abstract: The geodesic flow on a homogeneous space with an invariant metric can be naturally considered within the framework of Smale's mechanical systems with symmetry. In this way we have at our disposal the whole method of Smale for studying such systems. We prove that certain sets E', E, Im, a and Re which play an important role in the global behavior of those systems, have a particularly simple structure in our case, and we also find some geometrical implications about the geodesics. The results obtained are especially powerful for the case of Lie groups, as in the rigid body problem. I. Introduction. Since Smale developed his theory of mechanical systems with symmetry to study the plane n-body problem from a global viewpoint (121, (131, a few applications have been considered to other mechanical systems admitting some symmetry. We study here from the global analysis viewpoint geodesics on homogeneous spaces with a Riemannian metric invariant under the group action, as one more application of Smale's theory. Roughly speaking, the paper is carried out as follows. The sets 2', 2, Im, a and Re associated with any mechanical system with symmetry furnish an important part of its global structure (?2). We show that those sets have particularly simple structure for our problem under consideration, and then we find some geometri- cal implications about the corresponding geodesics. The rigid body problem is the most classical example where our general results apply, and lacob (61 has already studied some aspects of it by using Smale's theory (see end of ?3). ?2 is devoted to notation and a quick review of Smale's theory. In ?3 we study properties of structure and invariance under the group action for the five above- mentioned sets. By exploiting the transitivity we find that in some precise sense they can be generated from much simpler sets. In particular, that simpler set Re'
TL;DR: In this article, a general method for generating sequence-adapted molecular tensors using finite group algebra is formulated, and a catalog of irreducible representations of G(24) adapted to a member of each of the eight sequence classes is given together with the transformations which generate representations adapted to all other sequences.
Abstract: Tensorial sets adapted to sequences of finite subgroups are applied to the crystal field problem, and a general method for generating sequence-adapted molecular tensors using finite group algebra is formulated. All subgroup sequences of the abstract finite group G(24), isomorphic to the octahedral, O, tetrahedral, Td, and symmetric, S(4), groups are tabulated with explicit isomorphisms provided. The sequences fall into eight equivalence classes. A catalog of irreducible representations of G(24) adapted to a member of each of the eight sequence classes is given together with the transformations which generate representations adapted to all other sequences. With this data it is possible to systematically generate tensorial sets adapted to any sequence of a realization of G(24). Unitary transformations which adapt conventional forms of first- and second-rank irreducible tensorial sets of the rotation group to the eight sequences of the octahedral group are provided. Forms suitable for use with magnetic fields are included. The problem of a d1 ion in a trigonal crystal field is treated with sequence-adapted molecular tensors, and the utility of different sequences for descent in symmetry is discussed.
TL;DR: In this article, the authors collected some of their results and extended them where possible, demonstrating the broad applicability of the extended results and presenting several examples demonstrating their applicability for various classes of statistics.
Abstract: Hogg [2, 3, 4], Hollander [5], and Randies and Hogg [10] have obtained interesting results about zero correlation and certain symmetry properties for classes of statistics. One of the purposes of this article is to collect some of their results and extend them where possible. Several examples are presented demonstrating the broad applicability of the extended results.
TL;DR: In this article, it was shown that the analysis of nonunitary groups can be made mainly in terms of representations of its unitary part and additional conditions due to the anti-unitary symmetry.
Abstract: It is shown that calculating Clebsch‐Gordan coefficients of a nonunitary group can be reduced to formulas containing only representations of the unitary subgroup and additional conditions due to the antiunitary symmetry. This is another example demonstrating that, in applications involving corepresentations of nonunitary groups, the analysis can be made mainly in terms of representations of its unitary part.
TL;DR: In this article, the authors follow Fock's analysis of the hydrogen atom who projected it on a 4-dimensional hypersphere and obtain the top's symmetry group G providing representations corresponding exactly to the periods of Mendeleev.
Abstract: Many questions concerning the theoretical understanding of the table of the elements still remain open. In particular the existence of atomic magic numbers associated with noble gas behaviour has led to a search for approximate hidden symmetries. Such a symmetry is obtained here without any postulates. Instead, the authors follow Fock's analysis of the hydrogen atom who projected it on a 4-dimensional hypersphere. In this case the corresponding mapped system is the quantal top in 4-dimensions. The top's symmetry group G provides representations corresponding exactly to the periods of Mendeleev. The chain G contains/implies O(4) contains/implies SO(3) yields the quantum numbers n, l necessary for predicting the correct subshell ordering through a modified Aufbau scheme. Finally the non-invariance groups of G are obtained. Every step in this procedure is justified purely through group theoretical considerations.
TL;DR: The structure of the title compound has been determined from X-ray diffraction data collected on an automatic single-crystal diffractometer with monochromated Cu Ke radiation as discussed by the authors.
Abstract: The structure of the title compound has been determined from X-ray diffraction data collected on an automatic single-crystal diffractometer with monochromated Cu Ke radiation. The crystals are quadratic with a = b = 10.33 + 0"01 and c = 20-38 + 0.02 A; space group 141/acd with Z= 8. The structure has been refined by least-squares methods to R=0.058. The [Mg(OHz)6] z+ cations have 222 symmetry and the POzH2 anion has symmetry 2, with P-O bond lengths of 1.507 (3)/~, and an O-P-O angle of 116.2 (3) °. The two independent Mg-O bond lengths are 2.044 (3) and 2.066 (5) A. All the water molecules are hydrogen-bonded with hypophosphite oxygen and their oxygens are sp 2 hybridized. The three O-H. • • O independent bond lengths are 2"752, 2.749 and 2"762 A and the O...O...O angles are 119.5 and 117 °.
TL;DR: The theory of the irreducible representations of space groups and the symmetry projection of crystal wave functions can be written in a more concise form, when the IR-representations of arbitrary finite groups can be calculated as mentioned in this paper.
Journal Article•
TL;DR: In this paper, the symmetry elements of the four-dimensional hypercube are represented in a hyper-stereogram, which is used to classify the special and general directions in holosymmetric hyper-cubic symmetry.
Abstract: The symmetry elements of the four-dimensional hyper-cube are represented in a hyper-stereogram. This is used to classify the special and general directions in holosymmetric hyper-cubic symmetry. Projections of the hyper-cube in these various directions are constructed, with the incorporation of colour perspective, and the two-colour Shubnikov symmetry of these projections is tabulated and related to the four-dimensional symmetry elements on which the projection directions lie. The general representation of other four-dimensional symmetry elements in the hyper-stereogram is discussed, and the convenience of the hyper-stereogram for facilitating the evaluation of the matrices of symmetry operations in non-standard orientations is demonstrated.
TL;DR: In this paper, a symmetry adapted analysis for the C∞v and D∞h infinite groups of linear molecules is presented. But it is not applicable for the finite groups only, since one has to analyze the different subgroups of the linear molecules indirectly and correlate them with the irreducible representation of D∆h and C∆v.
Abstract: Symmetry groups of the linear molecules belong to the C∞v and D∞h infinite groups. The symmetry adapted analysis of such types of molecule, is usually not systematically performed in the text book or paper. Since the standard formulas of symmetry adapted analysis are usually applicable for the finite groups only, one has to analyze the different subgroups of the linear molecules indirectly and correlates them with the irreducible representation of D∞h and C∞v. In this work, a systematic symmetry adapted analysis are introduced for the C∞v and D∞h molecules. It is a uniquely convenient way for molecular orbital calculations and vibrational normal mode analysis of the linear molecules.
TL;DR: It is known from a paper by Das that multiform total symmetry is equivalent to linearity and a simple proof is given to show that even for multiform partial symmetries the corresponding subfunctions are linear sub functions.
Abstract: It is known from a paper by Das that multiform total symmetry is equivalent to linearity. In this correspondence the results of Das are generalized and a simple proof is given to show that even for multiform partial symmetries the corresponding subfunctions are linear subfunctions.
Patent•
08 Nov 1973TL;DR: A beamforming network having zero boresight error comprising a network hng symmetry about the network centerline is defined in this article, where the beamforming is defined as a symmetric beamforming.
Abstract: A beam-forming network having zero boresight error comprising a network hng symmetry about the network centerline.
TL;DR: In this paper, the third-order aberrations of thin lenses with a pair of mutually orthogonal planes of symmetry were derived, provided that the power of a lens in each of its two planes of symmetrized symmetry is fixed.
Abstract: Explicit expressions for the third-order aberrations of thin lenses which have a pair of mutually orthogonal planes of symmetry are derived. Provided that the power of a lens in each of its two planes of symmetry is fixed, the aberration coefficients are polynomials of at most the third degree in the principal curvatures cy and cz of one of the surfaces of the lens.