Showing papers on "Symmetry (geometry) published in 1984"

Codes and information

Abstract: CONTENTS Introduction Chapter I. Algebraic curves (computational aspects) Chapter II. Error-correcting codes Chapter III. Information and symmetry References

Force of symmetry in form perception.

Abstract: Many objects, natural and manufactured, have at least one axis of symmetry; thus, the detection of symmetry could facilitate the detection and representation of objects. Literature is reviewed that supports the notion that humans have effective and efficient symmetry-detection ability. The question addressed in the present research is whether symmetry detection leads to biases in representations of visual forms. Two types of experimental tasks were used: a similarity-judgment task and a matching-figures task in which reaction time to find identical figures in a display was measured. Stimuli varied in degree of measured symmetry. The results of the experiments reported here indicate that nearly symmetric standard forms are judged to be more similar to, and are more confusable with, even more symmetric forms than they are with less symmetric forms. The pull toward a more symmetric form does not occur for standard forms of lower symmetry. These findings can be accounted for by a two-stage process. First, the perceiver quickly determines the presence of overall symmetry. Then, if the form is perceived as having overall symmetry, the form is assumed, sometimes incorrectly, to have symmetry at the local level as well.

Corner piece for frames

Abstract: Corner piece for frames for joining together frame sides situated adjacent the corner piece, which sides are tubular or profile-shaped and adapted in their inner cavity to receive legs which are angularly arranged on the corner piece (1) and interconnected by means of an intermediate part (4). Each leg consists of two parallel shanks and each shank is shaped so as to be complementary to the cavity of the frame side profile (5, 8) engaging the respective shank so that the shank can be inserted guidingly and with a certain friction in the cavity of the profile. The corner piece is composed of two identical corner parts (2,3) interconnected by fastening means, the legs and the intermediate part of the corner piece (1) having, on one hand, a first symmetry plane coinciding with the frame plane and, on the other hand, a second simmetry plane or bisector plane through the bisector to the angle between the legs of the corner piece (1), said second plane being perpendicular to the first symmetry plane.

Peccei-Quinn symmetry as flavor symmetry and grand unification

Abstract: We consider the possibility of identifying the Peccei-Quinn (PQ) symmetry as also the flavor symmetry in multigenerational grand unification schemes. The essential ingredient, a global, axial U(1) symmetry in the PQ mechanism to avoid the strong $\mathrm{CP}$-violation problem provides useful constraints on the fermion\char22{}Higgs-boson couplings in the theory, thereby leading to identical "canonical" forms for fermion mass matrices in both the charged sectors. These forms are the conjectured Fritzsch-type matrices exhibiting the "nearest-neighbor" interactions in generation space. From among the popular schemes for grand unification, SO(10) emerges as one which has several advantages over the others for constructing multigenerational grand unification models. Reasonable assumptions regarding the quark masses lead to unique PQ quantum-number assignments for the fermionic generations. These quantum numbers combined with the hierarchy in quark masses lead to a picture in which the lighter generations are composite in nature. Once can then show qualitatively that the hierarchy is such that $logm$ varies linearly with respect to the generation index.

Generalised Fulton-Gouterman transformation for systems of Abelian symmetry

Abstract: A two-level system which is coupled to a multitude of oscillators, such that the total system formally or geometrically is governed by a mirror symmetry, can be exactly diagonalised with respect to the two-level subsystem (FG transformation). This has been thoroughly exploited for the two-site exciton localisation problem as well as for two-site quantum transport. A generalisation to the N-level system is given, provided the N levels establish a regular representation of an abelian group, and if the latter symmetry also governs the multi-oscillatory subsystem. Implications for the quantum transport problem are discussed.

Polytopes and symmetry

Abstract: Preface Synopsis 1. The space of polytopes 2. Combinatorial structure 3. Symmetry equivalence 4. Products and sums 5. Polygons 6. Polyhedra Concluding remarks Bibliography Index of symbols Index of names General index.

Superspace groups and representations of ordinary space groups: Alternative approaches to the symmetry of incommensurate crystal phases

Abstract: Incommensurate crystal phases may be described either with irreducible representations of space groups or with so-called superspace groups which are space groups in more than three dimensions. It is shown that from those representations the superspace groups can be determined and vice versa. However, there are some differences between the two approaches.

An aid to the structure analysis of incommensurate phases

Abstract: Group theory is used to establish three results likely to be useful in solving the crystal structures of complicated incommensurate phases. In the first of these it is demonstrated that an incommensurate structure with paired scattering vectors ± q must contain two different component structures, one modulated with cos q.r and the other with sin q.r. The second theorem states that the two components have different but related symmetries if the average structure has at least one element in its space group which turns q into -q. In that case, each aspect of the modulation is assigned uniquely by symmetry to either the cosine or sine factor. The third result concerns the Patterson function that may be constructed from the intensity scattered by the incommensurate modulation. This is also necessarily two-dimensional, the plus difference Patterson function being the sum of the Patterson functions obtained separately for the two component structures, while the minus difference Patterson function contains cross terms between the two components. Other symmetry arguments are mentioned, including symmetry signatures in Patterson functions, and systematic equalities in satellite intensities which arise from systematic extinctions in the scattering from one component or the other.

Nonlinear analogs of linear group actions on spheres

Abstract: Introduction. This article is based upon a principle which is so standard that it is almost a cliche: The first step to understanding a nonlinear phenomenon is to define and study a suitable linear approximation. To be more specific, we shall describe some applications of this to the symmetry questions of topological transformation groups. Given a topological space X, let Homeo(^) denote the set of self-homeomorphisms of X. This is a group under composition of mappings. If G is an arbitrary group, then a group action of G on I is a homomorphism Homeo( X). Frequently we wish to impose some weak assumptions on

3 Hypothesis of symmetry

Abstract: Publisher Summary This chapter focuses on the hypothesis of symmetry. The hypothesis of symmetry is the hypothesis corresponding to a nonparametric variant of the classical testing problem on paired samples that can be formulated as follows: (Zj, Yj), j = 1, . . . , n, are pairs of independent, identically distributed, p-dimensional random vectors; the cumulative distribution function H(z, y) of (Zj, Yj) fulfills some symmetry property. Here Zj = (Zj1 . . . . . Zjp)′ = (Yj1 . . . . . Yjp)′; z' denotes the transpose of z.

Bounding Finite Groups Acting on 3-Manifolds

Abstract: In Problem 3·39 (B) and (C) of Kirby's collection [10], Giffen and Thurston asked whether, for a closed 3-manifold M, the order of finite subgroups of Diff M is bounded, so that it contains no infinite torsion subgroups unless M admits a circle action. In this paper, we answer this question affirmatively for homotopy geometric manifolds, and then discuss some hyperbolic 3-manifolds with only a few actions as examples showing poor symmetry of 3-manifolds in general.

Domain independent object description and decomposition

Abstract: A significant problem in image understanding (IU) is to represent objects as models stored in a machine environment for IU systems to use in model driven pattern matching for object recognition. This paper presents a technique for autonomous machine description of objects presented as spatial data, i.e., data presented as point sets in Euclidean n-space. This general definition of objects as spatial data encompasses the cases of explicit listings of points, lines or other spatial features, objects defined by light pen in a CAD system, generalized cone representations, polygonal boundary representations, quad-trees, etc. The description technique decomposes an object into component sub-parts which are meaningful to a human being. It is based upon a measure of symmetry of point sets. Most spatial data has no global symmetry. In order to arrive at a reasonable description of a point set, we attempt to decompose the data into the fewest subsets each of which is as symmetric as possible. The technique is based upon statistics which capture the opposing goals of fewest pieces and most symmetry. An algorithm is proposed which operates sequentially in polynomial time to reach an optimal (but not necessarily unique) decomposition. The semantic content of the descriptions which the technique produces agrees with results of experiments on qualitative human perception of spatial data. In particular, the technique provides a step toward a quantitative measure of the old perceptual Gestalt school of psychology's concept of "goodness of figure".

On the reconstruction of a star-shaped body from its “half-volumes”

Abstract: The problem is the reconstruction of the shape of an object, whose shell is a surface star-shaped with respect to a point 0, from the knowledge of the volume of every “half-object” obtained by taking any plane through 0 Conditions for the existence and uniqueness of the solution are given The main result consists in showing that any uniform a-priori bound on the mean curvature of the shell reestablishes continuous dependence on the data for bodies satisfying a certain symmetry condition

Symmetry-restricted phase transitions in two-dimensional solids

Abstract: Group-theoretical criteria which determine the possibility of group-subgroup phase transitions have been implemented on a computer. Lower-symmetry groups are determined by the subduction and chain criteria. We list all possible such symmetry-restricted transitions in two dimensions corresponding to k points of symmetry. We indicate relative origins and orientations of the prototype group and subgroup sufficient to obtain a wide class of useful experimental information.

A retabulation of the 80 layer groups for electron diffraction usage

Abstract: The 80 space groups of infinitely extended layers are identified as the defining groups for transmission electron diffraction symmetries obtained from lamellar crystals. These layer groups are retabulated using a notation which characterizes the symmetries of convergent-beam diffraction patterns. The new tabulation provides a means for determining the three-dimensional space group of a particular structure from one or more convergent-beam zone-axis patterns. This is a two-stage process, involving determination of a layer-group corresponding to a principal zone-axis pattern, followed by identification, where necessary, of the appropriate class-equivalent subgroup. For this final identification, the three-dimensional extinction conditions, determined from an alternative set of convergent-beam zone-axis patterns, are required. In terms of standard group symbols [e.g. Vainshtein (1981). Modern Crystallography I. Berlin: Springer], the one-way interrelation between patterns and space groups G32 → G33 (space group analysis) is given. The inverse problem, equally accessible from group theory, of providing the interrelation G33 → G32 (pattern symmetry prediction) is left for separate tabulation.

Symmetry transformations in indefinite metric spaces: A generalization of Wigner's theorem

Abstract: We formulate and prove a generalization to indefinite metric spaces of Uhlhorn's version of Wigner's theorem.

Symmetries of the Dirac equation

Abstract: The symmetry of the free Dirac equation recently discussed by Radford is regarded as a subgroup of a larger SL(2,C) group of transformations. The Radford Lagrangian is shown to be unsatisfactory for a field theory quantized according to Fermi-Dirac statistics, and is replaced by a suitable alternative. The symmetry of the Dirac equation, of the equal-time field anticommutation relations, and of the Lagrangian, is studied under SL(2,C) and various subgroups. In particular, it is found that the Radford subgroup is not a symmetry of the quantized field theory.

The symmetry of convergent-beam electron diffraction patterns from bicrystals: erratum. By F. W.

Abstract: Table 1 of Schapink, Forghany & Buxton [Acta Cryst. (1983), A39, 805-813] contains several printing errors. The correct table is given. Table 1. The relation between the diffraction groups and the dichromatic point groups for bicrystals with ~, > 1

Local black holes are type D on the Horizon

Abstract: For the most general static and axisymmetric local black hole with (topologically) spherical or toroidal horizon we calculate the Riemann tensor, we construct a Newman-Penrose null tetrad, we evaluate Weyl scalars and we investigate the algebraic type of the solutions. We find that the spherical black holes are of Petrov typeD on the horizon and on the axis of azimuthal symmetry, while the toroidal ones are of typeD on the horizon and on the plane of the reflectional symmetry.

Erratum: Group-subgroup structural phase transitions: A comparison with existing tables

Abstract: We have implemented group-theoretical methods on the computer for the description of possible structural phase transitions. All isotropy groups (corresponding to $\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}$ points of symmetry) for each of the 230 three-dimensional space groups as well as the Landau and Lifshitz conditions for each representation have been obtained. Here we compare our results with previous tables and list the errors and necessary corrections to those tables.

Space-group algorithms. I. The space group and its symmetry elements

Abstract: A well-defined and efficient algorithm, which enables the derivation of any crystallographic space group and full characterization of its symmetry operations and elements, is described and illustrated. The algorithm is based on a representation of crystallographic point groups in terms of cyclic groups, and on isomorphism relations between the point groups and the corresponding factor-group representations of the space groups. The characterization of the symmetry operations and the corresponding symmetry elements is also presented in an algorithmic manner, with particular emphasis on the orientation of the axes of rotation and their location vectors. The above algorithms have been implemented in a computer program, an application of which to the space group Pa\bar 3 is shown and some relevant programming considerations are given. The input to this general program can be fully adapted to the space-group tables in Vol. A of International Tables for Crystallography [(1983). Dordrecht: Reidel].

Symmetric Distributions for Dependent Unit Vectors

Abstract: This paper introduces several notions of symmetry for the joint distribution of two dependent unit vectors Bivariate generalizations of $\mathscr{L}$-symmetry (Rivest, 1984) and rotational symmetry are introduced If the joint distribution of two unit vectors is at least $\mathscr{L}$-symmetric the information matrix for the parameters indexing it is shown to have a simple shape