Showing papers on "Symmetry (geometry) published in 1987"

Symmetries and Integrability

Abstract: On considere les symetries des equations d'evolution non lineaires integrables. On donne une definition de l'integrabilite basee sur des considerations de symetrie

Generalized Symmetry Conditions at a Core Point

Abstract: Previous analyses have shown that if a point x is to be a core of a majority rule voting game in Euclidean space, when preferences are smooth, then the utility gradients must satisfy certain restrictive symmetry conditions. In this paper these results are generalized to the case of an arbitrary voting rule, and necessary and sufficient conditions, expressed in terms of "pivotal" coalitions, are obtained.

Symmetry-curvature duality

Abstract: Several studies have shown the importance of two very different descriptors for shape: symmetry structure and curvature extrema. The main theorem proved by this paper, i.e., the Symmetry-Curvature Duality Theorem , states that there is an important relationship between symmetry and curvature extrema: If we say that curvature extrema are of two opposite types, either maxima or minima, then the theorem states: Any segment of a smooth planar curve, bounded by two consecutive curvature extrema of the same type, has a unique symmetry axis, and the axis terminates at the curvature extremum of the opposite type. The theorem is initially proved using Brady's SLS as the symmetry analysis. However, the theorem is then generalized for any differential symmetry analysis. In order to prove the theorem, a number of results are established concerning the symmetry structure of Hoffman's and Richards' codons. All results are obtained first by observing that any codon is a string of two, three, or four spirals, and then by reducing the theory of codons to that of spirals. We show that the SLS of a codon is either (1) an SAT, which is a more restricted symmetry analysis that was introduced by Blum, or (2) an ESAT, which is a symmetry analysis that is introduced in the present paper and is dual to Blum's SAT.

Fully automated microcomputer calculation of vibrational spectra

Abstract: A fully automated method for computing frequencies and atomic displacements of normal modes, giving synthetic infrared and Raman spectra, has been developed for use on small computers. No expertise in group theory or the mathematics of normal-mode calculation are required to use the computer program. The method takes full account of symmetry and is applicable to any crystal or molecule. Force constants can be specified in terms of any two-atom “bonds” or three-atom angles. The essential steps in the computer program are: (1) Locate all atoms in the unit cell or molecule and compute displacement vectors for each internal coordinate; (2) Convert the basis of the force constants from bonds and angles to cartesian displacements; (3) Construct the full-matrix irreducible representations of the point or factor group in question, using appropriate symmetry matrices and polynomial basis functions; (4) Derive the symmetry coordinates in terms of cartesian displacements using the projection/transfer-operator technique; (5) construct secular equations for each species with Wilson's f–g method; (6) solve for frequencies and atomic motions; and (7) use simple models of infrared and Raman intensities to calculate spectra.

A real‐space computer‐based symmetry algebra

Abstract: A computer-based symmetry algebra is described which permits the reconstruction of an infinite bond network from the asymmetric connectivity without an a priori knowledge of atomic coordinates. The algebra requires not only an algorithmic ordering of the Wyckoff groups but the designation of one site in each Wyckoff group as a special-position representative (SPR) site. The algebra is designed to be used for analysing the bonding network of compounds appearing in the Inorganic Crystal Structure Database.

The semigroup of values of a curve singularity with several branches

Abstract: We give an explicit description of the semigroup of values of a plane curve singularity with several branches in terms of the usual invariants of the equisingularity type in the sense of Zariski. The main tool is the set of elements called maximals, specially the absolute and the relative ones. First, we describe the semigroup in terms of the relative maximals and these ones in terms of the absolute maximals by means of a symmetry property which generalizes the well known property of symmetry for the singularities with only one branch. Then the absolute maximals are described in terms of the theory of maximal contact of higher genus developed by Lejeune.

Symmetry classification of states in high temperature superconductors

Abstract: The symmetry classification of possible singlet and also triplet states in the case of the 2-dim square lattice, and 3-dim tetragonal and orthorhombic lattices is examined. If particle-hole symmetry is present then an additional symmetry classification is possible. However in the lower symmetry crystal structures that actually occur, no distinction on symmetry grounds can be drawn between usuals-wave, extendeds-wave and some of thed-wave states.

Symmetry theorems on the forward and backward scattering Mueller matrices for light scattering from a nonspherical dielectric scatterer.

Abstract: The symmetry theorems on the complete forward and backward scattering Mueller matrices for light scattering from a single dielectric scatterer (as opposed to an ensemble of scatterers) are systematically and thoroughly analyzed. Symmetry operations considered include discrete rotations about the incident direction and mirror planes not coinciding with the scattering plane. For forward scattering we find sixteen different symmetry shapes (not including the totally asymmetric one), which may be classified into five symmetry classes, with identical reductions in the forward scattering matrices for all symmetry shapes that fall into the same symmetry class. For backward scattering we find only four different symmetry shapes, which may be classified into only two symmetry classes. The forward scattering symmetry theorems also lead to a symmetry theorem on the total extinction cross section. Based on the conclusions of this work it should be possible to design quick and nondestructive methods for the identification of certain small objects, when suitable partial information about the objects to be identified is already available. A promising practical example is given.

The representation of orientation distributions

Abstract: Textures are properly described as densities in a three-dimensional orientation space. The appropriate coordinates of this space are the Euler angles describing relative rotations or, equivalently, the location and azimuth of a vector on a sphere (a boat on the earth). The space spanned up by these angular coordinates has conventionally been chosen Cartesian, and the orientation distribution function (ODF) is then represented in two-dimensional (square) sections through this space, with contour lines. It is proposed that an alternative representation be used that maintains a closer connection to pole figures and inverse pole figures, which areprojections of orientation space. There are advantages if ODFsections are also plotted in polar coordinates. The equivalents to pole figures are ’partial pole figures’ or COD’s (crystal orientation distributions); the equivalents to inverse pole figures are partial inverse pole figures or SOD’s (sample orientation distributions). These representations are less distorted than conventional ODF’s and display the symmetry properties in a more obvious way. In addition, both projections and sections should be plotted inequal-area projection; then, equal volume elements in orientation space are displayed by equal area elements in the representations. Common operations with these diagrams are outlined and illustrated using a rolled copper sample.

N=2, d=2 supergeometry

Abstract: N=2, d=2 supergeometry is formulated in superspaces with U(1)*U(1) and U(1) internal symmetry factors in the tangent space group. The N=2 geometry is compared with the N=1 and (p, q)=(1, 0) geometries, and the superconformal transformations are derived from the superconformal Killing equation.

Using the bootstrap in testing symmetry versus asymmetry

Abstract: One approach to testing the hypothesis of symmetry versus asymmetry is to use estimates of the center of symmetry in nonparametric tests about a known center. The problem with this approach is that the resulting tests are not distribution-free. Hence, one cannot compute critical values, p-values, or probabilities of errors for these tests. We propose to sidestep this problem with a symmetric bootstrap procedure which uses bootstrap samples from (a smoothed version of) the closest symmetric distribution to the empirical distribution of the data to estimate these values. In this paper, we report simulation experience with this approach in testing the hypothesis of symmetry using the Schuster and Narvarte estimator of the center of symmetry in the so-called Butler sup norm test for symmetry about a known center.

Maximally symmetric polyhedral realizations of Dyck's regular map

Abstract: We construct realizations of Dyck's regular map of genus three as polyhedra in ℝ3. One of these has one axis of symmetry of order three and three axes of symmetry of order two. The other polyhedra have three axes of symmetry. We show that a polyhedron realizing Dyck's regular map cannot have a symmetry group of order larger than six. Thus the symmetry groups of our realizations are maximal.

Correction to "Meromorphic Functions that Share Four Values"

Abstract: It has been brought to my attention by Norbert Steinmetz that the symmetry reasoning that is used in the proof of Theorem 2 in [1] to conclude that ''d1 is symmetric in a and b" in the third line from the bottom on page 564, and also to conclude that (123) holds on page 565, is not valid. We note that the symmetry reasoning that is used in the sentence in lines 8 and 9 from the bottom on page 564 is valid, because this sentence is meant in the sense of the following sentence: By considering a b-point that is simple for f and double for 9, a similar argument to the above argument (with (61) instead of (60)) will produce

ODF Calculation by Series Expansion from Incompletely Measured Pole Figures Using the Positivity Condition Part I—Cubic Crystal Symmetry

Abstract: The calculation of orientation distribution functions (ODF) from incomplete pole figures can be carried out by an iterative procedure taking into account the positivity condition for all pole figures. This method strongly reduces instabilities which may occasionally occur in other methods.

3D Tomographic Reconstruction from 2D Data Using Spherical Harmonics

Abstract: Tomographic reconstruction of a 3D object in terms of spherical harmonics from a small number of 2D data sets is shown to be possible in cases of objects of high symmetry and/or low shape anisotropy. A test case using an object of cubic symmetry shows that the reconstruction can work well in such high symmetry cases when only two or three data directions are available. Numerical tests suggest that reconstructions are best done from data taken in low symmetry directions.

Asymptotic symmetry and the global structure of future null infinity

Abstract: Space-times for whichI+ (future null infinity) is not necessarily homeomorphic toR×S2 are considered. It is shown that, depending on the global conformal structure ofI+, a given space-time either (1) possesses an asymptotic symmetry group with a normal subgroup of supertranslations, similar in structure to the BMS group, or (2) possesses a simpler kind of asymptotic symmetry group, not involving supertranslations, or (3) has no asymptotic symmetry. The setting is Newman and Unti's approach to asymptotically flat space-times and use is made of the characterization of the asymptotic symmetry transformation as a conformal motion ofI+ that preserves null angles.

Global theorems for symmetry sets of smooth curves and polygons in the plane

Abstract: Etablissement des formules globales de structure des ensembles de symetrie a l'aide d'un theoreme d'Ozawa. Etude des transitions sur de tels ensembles. Cas de polygones plans

Symmetry of the EL2 defect in GaAs

Abstract: From experiments on the stress-induced splitting of the zero-phonon line associated with the optical-absorption band originating from photoexcitation of the EL2 center in GaAs, Kaminska et al. concluded that this center has tetrahedral symmetry, which would account for its simple point-defect structure. An alternative explanation of the observed splitting pattern, which involves an orthorhombic center of ${C}_{2v}$ symmetry, is discussed in this paper. This would account for both the recently revealed complex structure of EL2 and its apparent high symmetry.