Showing papers on "Symmetry (geometry) published in 1988"

Backtrack Searching in the Presence of Symmetry

Abstract: Methods from computational group theory are used to improve the speed of backtrack searching on problems with symmetry. The symmetry testing algorithm, which is similar to a color automorphism algorithm, takes the symmetry group as input and uses it to avoid searching equivalent portions of the search space. The algorithm permits dynamic search rearrangement in conjunction with symmetry testing. Experimental results confirm that the algorithm saves a considerable amount of time on some search problems.

Gorenstein curves and symmetry of the semigroup of values

Abstract: LetO be the local ring of a irreducible algebroid curve and S its semigroup of values, Kunz in [7] proves thatO is a Gorenstein ring if and only if S is symmetrical. In this paper we give a generalization of this fact for the case of reduced curves with an arbitrary number of branches, d. For it we introduce a concept of symmetry for the semigroup of values S⊂ℤ+ d which generalizes the well known symmetry for d=1 (i.e. the irreducible case). This concept of symmetry is also closely related to the symmetry introduced by Garcia in [4] (for the d=2 case) and the author in [3] (for arbitrary d) with the main goal of the explicit determination of S (in the case of plane curves).

More space-group changes

Abstract: Revisions are made to the space groups in ten published X-ray diffraction studies. In five cases the revisions entail increases in the Laue symmetry and, except for repositioning a nitrate group in one compound, there are no important changes in the interatomic distances and angles. In four other cases, centers of symmetry have been added; in one of these cases further refinement has led to appreciable changes in the bond lengths and angles, while in the other three cases the original mtens1ty data were not available and further refinement could not be carried out. In the tenth case, both the Laue symmetry has been increased and a center of symmetry has been added, and the structure further refined.

Recognition of local symmetries in gray value images by harmonic functions

Abstract: A method for modeling symmetries of the neighborhoods in gray-value images is derived. It is based on the form of the iso-gray-value curves. For every neighborhood a complex number is obtained through a convolution of a complex-valued image with a complex-valued filter. The magnitude of the complex number is the degree of symmetry with respect to the a priori chosen harmonic function pair. The degree of symmetry has a clear definition which is based on the error in the Fourier domain. The argument of the complex number is the angle representing the relative dominance of one of the pair of harmonic functions compared to the other. >

Z 2 × S 6 symmetry of the two-loop diagram

Abstract: The generic two-loop diagram of massless scalar field theory ind dimensions, with propagators raised to powers {α i |i=1,5} is shown to have a symmetry groupZ 2×S 6 corresponding to reflection and permutation of six linear combinations of the six parameters. The expansion about α i =d/4=1 is given to the level required for six-loop renormalization. All but five of the coefficients are obtained from products of one-loop diagrams. All but one are expressed in terms of the Riemann zeta function.

Symmetry Finding Algorithms

Abstract: Several algorithms for finding symmetries of geometrical objects have recently appeared. In this paper the results are surveyed, and some problems which remain open are described.

SL(3,R) as the group of symmetry transformations for all one‐dimensional linear systems

Abstract: The converse problem of similarity analysis is solved in general for the finite symmetry transformations of any inhomogeneous ordinary linear differential equation of the second order x+f2(t)x+f1(t)x =f0(t). The eight‐parameter realizations of the symmetry group are obtained in the form F−1P2 F, where F stands for transformations of (t,x) that depend exclusively on the fundamental solutions of the equation, and where P2 is an arbitrary projective transformation in the plane. Thus it is shown that the full point symmetry group corresponds to SL(3,R) indeed, without recourse to the Lie algebra. Also, a technique is obtained for calculating the finite point symmetry realization of SL(3,R) for any given one‐dimensional linear system. Some miscellaneous examples are given.

Correlation averaging and 3-D reconstruction of 2-D crystalline membranes and macromolecules

Abstract: Publisher Summary The correlation averaging determines the position of each unit cell via cross-correlation of the image with a representative reference, comprising one or more unit cells. In principle the reference is centered on each pixel of the source image, and the correlation coefficient between the reference and the corresponding subframe of the image is calculated. The two independent averages are extracted and smoothly masked to comprise one morphological complex (unit cell) each. Corresponding Fourier coefficients of the two transforms representing identical spatial frequencies (on rings) are summed according to the function of the resolution criterion applied and displayed as a function of the spatial frequency. The intersection of the experimental curve with a threshold level function is taken as a measure of resolution. According to the symmetry properties of the average, symmetrization is performed by rotation about the symmetry center and/or by reflection at the mirror axes. For symmetrization purposes the axis of symmetry must be correctly centered; the axes can be conveniently determined by correlation techniques.

Symmetry-adapted classification of aberrations

Abstract: Optical systems produce canonical transformations on phase space that are nonlinear. When a power expansion of the coordinates is performed around a chosen optical axis, the linear part is the paraxial approximation, and the nonlinear part is the ideal of aberrations. When the optical system has axial symmetry, its linear part is the symplectic group Sp(2, R) represented by 2 × 2 matrices. It is used to provide a classification of aberrations into multiplets of spin that are irreducible under the group, in complete analogy with the quantum harmonic-oscillator states. The “magnetic” axis of the latter may be chosen to adapt to magnifying systems or to optical fiberlike media. There seems to be a significant computational advantage in using the symplectic classification of aberrations.

Symmetries of planar growth functions

Abstract: Chapter 4, Exercises]. If G is also a Weyl group, then via the Bruhat decomposition, this symmetry reflects Poincar6 duality on a homogeneous space (see [9, P. 123-124]). If G is an infinite Coxeter group, Bourbaki (and also Steinberg [I4, p. 12]) showed that f(z) is the power series of a rational function, which we will also call f [1, p. 45]. In the case that G is compact hyperbolic or irreduc-

Patterson-oriented automatic structure determination: getting a good start

Abstract: On the basis of a generalized symmetry minimum function several computer-oriented methods for interpreting Patterson functions and for locating the position of heavy-atom fragments in crystals belonging to space groups of higher symmetry than P1 have been developed. The methods utilize cross vectors for finding relationships among the peaks of the symmetry minimum function. This approach has the advantage of suppressing false peaks of the symmetry minimum function, in locating more than one atom and in revealing the correct solution with greater probability. The heavy-atom fragment can be extended by superposition or Fourier methods. The methods are valid for all space groups, are simple to apply and form the basis for fully automated structure determination. In contrast to many other Patterson methods no a priori structural information is necessary. A few selected examples demonstrate the power of the new version of the computer program XFPS.

Symmetry and classification of energy bands in crystals

Abstract: An energy band in a solid contains an infinite number of states which transform linearly as a space group representation induced from a finite dimensional representation of the isotropy group of a point in space. A band representation is elementary if it cannot be decomposed as a direct sum of band representations; it describes a single band. We give a complete classification of the inequivalent elementary band representations.

Model-based matching using skewed symmetry information

Abstract: A model-based approach is proposed for recovering the 3-D position of a symmetrical polygon in the models from its single image. First, a Hough-type voting method detects skewed symmetry in the image. Matching of the skewed symmetry axis to symmetry axes of models then makes both selection of models and recovering their orientation easy. The method is shown to possess robustness of matching from occlusion. >

Pseudo-symmetry curvature conditions on hypersurfaces of Euclidean spaces and on Kahlerian manifolds

Abstract: On etudie des varietes de Riemann pseudo-symetriques, qui sont des generalisations des espaces symetriques et semi-symetriques. On classe les hypersurfaces pseudosymetriques d'un espace euclidien. On demontre qu'il n'y a pas de variete kahlerienne pseudo-symetrique et non semi-symetrique

Symmetry Coordinates of Molecular Vibrations of "Footballene" C60

Abstract: A complete set of the 174 symmetry coordinates for the C molecular model referred to as footballene is reported. The model is the truncated icosahedron (symmetry Ih ). Hence in addition to triple degeneracy also quadruple and quintuple degeneracies occur.