Showing papers on "Symmetry (geometry) published in 1997"

[21] Data collection strategy

Abstract: Publisher Summary This chapter emphasizes the importance of determining the outer resolution limit to define the parameters to be used in data collection. The chapter describes the behavior of the reciprocal lattice during rotation photography, the effect of crystal mosaicity, the myth of the blind region, and the ways in which crystal symmetry can help. To design a data-collection protocol so that a data set is complete is not straightforward. Geometric considerations must be taken into account that depend on a crystal's orientation, its cell dimensions, and its symmetry. The geometric principles of the rotation method and individual cases of different symmetries of a crystal are discussed in the chapter. As the crystal is rotated during exposure of X-rays, the diffraction geometry may be explained by the Ewald-sphere construction, which, in turn, illustrates Bragg's law in three dimensions. The Ewald sphere with the radius 1/λ represents the radiation and therefore is stationary, whereas the crystal is represented by the reciprocal lattice with the origin at the point where the primary beam leaves the sphere. The crystal is rotated around one axis, usually perpendicular to the beam.

3D symmetry detection using the extended Gaussian image

Abstract: Symmetry detection is important in the area of computer vision. A 3D symmetry detection algorithm is presented in this paper. The symmetry detection problem is converted to the correlation of the Gaussian image. Once the Gaussian image of the object has been obtained, the algorithm is independent of the input format. The algorithm can handle different kinds of images or objects. Simulated and real images have been tested in a variety of formats, and the results show that the symmetry can be determined using the Gaussian image.

World-Volume Action of the M Theory Five-Brane

Abstract: This paper presents a 6d world-volume action that describes the dynamics of the M theory five-brane in a flat 11d space-time background. The world-volume action has global 11d super-Poincare invariance, as well as 6d general coordinate invariance and kappa symmetry, which are realized as local symmetries. The paper mostly considers a formulation in which general coordinate invariance is not manifest in one direction. However, it also describes briefly an alternative formulation, due to Pasti, Sorokin, and Tonin, in which general coordinate invariance is manifest. The latter approach requires auxiliary fields and new gauge invariances.

[34] Rotation function calculations with GLRF program

Abstract: Publisher Summary This chapter discusses the rotation function calculation with the GLRF program. The calculation of rotation functions represents the first step in structure determination by the molecular-replacement method. Rotation functions can be used to determine the orientation of a noncrystallographic axis in a crystal or the orientation of a search model relative to the molecules in a crystal. In the second step, rotational parameters can be used to determine the position of the noncrystallographic symmetry in the crystal or the position of the search model in the crystal. The information on the orientation and position of a search model leads to an initial solution for the crystal structure. Finally, the orientation and position of the noncrystallographic symmetry axes can be used to determine an initial phasing set, as well as to improve and extend the phase information.

Generating Families in the Restricted Three-Body Problem

Abstract: Definitions and Properties.- Generating Orbits of the First Species.- Generating Orbits of the Second Species.- Generating Orbits of the Third Species.- Bifurcation Orbits.- Junctions: Symmetry.- Junctions: Broucke's Principle.- Fragments.- Generating Families.

Automatic detection of the mid-sagittal plane in 3-D brain images

Abstract: This article presents a detailed description of an algorithm for the automatic detection of the mid-sagittal plane in three-dimensional (3-D) brain images. The algorithm seeks the plane with respect to which the image exhibits maximum symmetry. For a given plane, symmetry is measured by the cross-correlation between the image sections lying on either side. The search for the plane of maximum symmetry is performed by using a multiresolution approach which substantially decreases computational time. The choice of the starting plane was found to be an important issue in optimization. A method for selecting the initial plane is presented. The algorithm has been tested on brain images from various imaging modalities in both humans and animals. Results were evaluated by visual inspection by neuroradiologists and were judged to be consistently correct.

Mirror symmetry via 3-tori for a class of Calabi-Yau threefolds

Abstract: We give an example of the recent proposed mirror construction of Strominger, Yau and Zaslow in ``Mirror Symmetry is T-duality,'' hep-th/9606040. The paper first considers mirror symmetry for K3 surfaces in light of this construction. We then consider the example of mirror symmetry for Calabi-Yau threefolds of the type considered by Voisin and Borcea, of the form SxE/involution where S is a K3 surface with involution, and E is an elliptic curve. We show how dualizing a family of special Lagrangian real 3-tori does actually produce the mirrors in these examples.

The Symmetry, Color, and Morphology of Galaxies

Abstract: The structural symmetry of forty-three face-on galaxy images in the R(650 nm) and J(450 nm) bands are measured to determine the usefulness of symmetry as a morphological parameter Each galaxy image is rotated by 180° and subtracted from the original to obtain a quantitative value for its structural symmetry The symmetry numbers computed for the sample are then compared with RC3 morphological types, color & absolute blue magnitudes A strong correlation between color and symmetry is found, and the RC3 Hubble sequence is found to be one of increasing asymmetry The use of symmetry as a morphological parameter, and the possible causes of the asymmetries are discussed

Behavior of Einstein-Rosen waves at null infinity

Abstract: The asymptotic behavior of Einstein-Rosen waves at null infinity in 4 dimensions is investigated in {\it all} directions by exploiting the relation between the 4-dimensional space-time and the 3-dimensional symmetry reduction thereof. Somewhat surprisingly, the behavior in a generic direction is {\it better} than that in directions orthogonal to the symmetry axis. The geometric origin of this difference can be understood most clearly from the 3-dimensional perspective.

Hecke Symmetries and Characteristic Relations on Reflection Equation Algebras

Abstract: We analyze the relation between the properties of Hecke symmetry (i.e., Hecke type R-matrix) and the algebraic structure of the corresponding reflection equation (RE) algebra. Analogues of the Newton relations and Cayley–Hamilton theorem for the matrix of generators of the RE algebra associated with a finite rank even Hecke symmetry are derived.

Asymmetric Multidimensional Scaling of Two-Mode Three-Way Proximities

Abstract: An asymmetric multidimensional scaling model and an associated nonmetric algorithm to analyze two-mode three-way proximities (object × object × source) are introduced. The model consists of a common object configuration and two kinds of weights, i.e., for both symmetry and asymmetry. In the common object configuration, each object is represented by a point and a circle (sphere, hypersphere) in a Euclidean space. The common object configuration represents pairwise proximity relationships between pairs of objects for the ‘group’ of all sources. Each source has its own symmetry weight and a set of asymmetry weights. Symmetry weights represent individual differences among sources of data in symmetric proximity relationships, and asymmetry weights represent individual differences among sources in asymmetric proximity relationships. The associated nonmetric algorithm, based on Kruskal’s (1964b) nonmetric multidimensional scaling algorithm, is an extension of the algorithm for the asymmetric multidimensional scaling of one mode two-way proximities developed earlier (Okada and Imaizumi 1987). As an illustrative example, we analyze intergenerational occupational mobility from 1955 to 1985 in Japan among eight occupational categories.

Comments on "Symmetry as a Continuous Feature"

Abstract: We point out the existence of a theoretical difficulty that underlies the symmetry detection studied by Zabrodsky et al. (1995) and present a possible solution to it.

Exponents of orientable maps

Abstract: We generalize the idea of reflexibility of a map on a surface by introducing certain integers as its ”exponents“. An exponent is any integer with the property that changing the cyclic permutation of edges around each vertex induced by the map to its -th power gives rise to an isomorphic map. The exponents reduced modulo the least common multiple of the vertex valencies form an Abelian group, the exponent group of the map . Along with the automorphism group, the group in fact provides an additional measure of symmetry of ~.The paper is devoted to developing the fundamentals of the theory of exponent groups of maps. Motivation comes from the problem of classification of regular maps with a given underlying graph. To this end, we prove that the number of non-isomorphic regular maps (if any) with a given underlying graph and the same map automorphism group is . We calculate the exponent groups for some interesting families of regular maps including complete maps. Special attention is paid to the problem of how the antipodality of a map is reflected by its exponent group. In the final section we discuss several open problems.1991 Mathematics Subject Classification: 05C10, 05C25, 20F32.

A Guide to the Symmetry Structure of Quasiperiodic Tiling Classes

Abstract: Based upon the torus parametrization which was introduced recently, we present a recipe allowing for a complete analysis of the symmetry structure of quasiperiodic local isomorphism classes in one and two dimensions. A number of results is provided explicitly, including some widely used planar tiling classes with 8-, 10- and 12-fold rotational symmetry.

The structure of area-minimizing double bubbles

Abstract: We show that the least area required to enclose two volumes in ℝn orS n forn ≥ 3 is a strictly concave function of the two volumes. We deduce that minimal double bubbles in ℝn have no empty chambers, and we show that the enclosed regions are connected in some cases. We give consequences for the structure of minimal double bubbles in ℝn. We also prove a general symmetry theorem for minimal enclosures ofm volumes in ℝn, based on an idea due to Brian White.

The intersection seam between the 11A′ and 21A′ states of ozone

Abstract: The intersection seam between the two lowest 1A′ states of ozone has been determined. The potential energy surfaces and the seam are calculated and discussed in perimetric coordinates which exhibit the full three-dimensional symmetry. The seam is shown to form a closed curve which crosses the C 2 v -restricted coordinate planes at six points. Three of these correspond to the previously determined intersection, the starting point of the present search. The other three correspond to highly repulsive regions on the potential energy surface where two atoms approach each other to within two-thirds of the O2 bond length. At the former three points both states have 1 A 1 symmetry, but at the latter three points one state has 1 A 1 symmetry whereas the other has 1 B 2 symmetry. Consequently, there exist three additional branches of the intersection seam between these two states. Each of these branches lies entirely in one C 2 v -restricted coordinate plane and connects to the previously discussed C s -seam at one point. The existence of a further intersection seam is established. A novel method for determining intersection points is described.

Hypersurface-orthogonal generators of an orthogonally transitive transitive $G_2I$, topological identifications, and axially and cylindrically symmetric spacetimes

Abstract: A criterion given by Castejon-Amenedo and MacCallum (1990) for the existence of (locally) hypersurface-orthogonal generators of an orthogonally-transitive two-parameter Abelian group of motions (a $G_2I$) in spacetime is re-expressed as a test for linear dependence with constant coefficients between the three components of the metric in the orbits in canonical coordinates. In general, it is shown that such a relation implies that the metric is locally diagonalizable in canonical coordinates, or has a null Killing vector, or can locally be written in a generalized form of the `windmill' solutions characterized by McIntosh. If the orbits of the $G_2I$ have cylindrical or toroidal topology and a periodic coordinate is used, these metric forms cannot in general be realized globally as they would conflict with the topological identification. The geometry then has additional essential parameters, which specify the topological identification. The physical significance of these parameters is shown by their appearance in global holonomy and by examples of exterior solutions where they have been related to characteristics of physical sources. These results lead to some remarks about the definition of cylindrical symmetry.

Noncrystallographic symmetry averaging in phase refinement and extension.

Abstract: Publisher Summary A well-known method of noise reduction in image processing is the addition of multiple, independent images of the same object to generate an improved, averaged image. Using this method, one can increase the signal-to-noise ratio by a factor of N 1/2 , when N -independent images are being averaged. The technique has been used with great success in the processing of transmission electron microscopy images, where object recognition and alignment are performed before the averaging operation. A similar situation is encountered frequently in macromolecular crystallography, where the presence of multiple copies of the same object in the asymmetrical unit is known as “noncrystallographic symmetry (n.c.s).” The potential uses of such redundancy in helping to generate improved electron-density maps have long been known. Two separate types of n.c.s exist: (1) proper symmetry, in which the set of n.c.s, operations forms a closed group so that they apply equally to all objects in the assembly and (2) improper symmetry, in which the set of n.c.s, operations applying to object 1 is different from that applying to object 2 and to object 3.

Three-Dimensional Bilateral Symmetry Bias in Judgments of Figural Identity and Orientation

Abstract: The two experiments reported explored a bias toward symmetry in judging identity and orientation of indeterminate two-dimensional shapes Subjects viewed symmetric and asymmetric filled, random polygons and described “what each figure looks like” and its orientation Viewers almost universally interpreted the shapes as silhouettes of bilaterally symmetric three-dimensional (3-D) objects This assumption of 3-D symmetry tended to constrain perceived vantage of the identified objects such that symmetric shapes were interpreted as straight-on views, and asymmetric shapes as profile or oblique views Because most salient objects in the world are bilaterally symmetric, these findings are consistent with the view that assuming 3-D symmetry can be a robust heuristic for constraining orientation when identifying objects from indeterminate patterns

Symmetry-based vehicle location for AHS

Abstract: Video systems can locate, identify and track vehicles. A video-based vehicle detection and location method is presented, which exploits the symmetry of vehicles seen from behind. The method can account for a range of symmetry types. These include (1) simple pixel presence in a binary edge image, (2) gray level of the edge pixel, (3) color value of the edge pixel, and (4) connectedness structure of the (binary) pixels around an edge pixel. A fast algorithm for the generation of a symmetry histogram is presented, whose (sufficiently strong) peaks indicate the likely presence and approximate location of a vehicle. The speed of the algorithm results from its data driven nature. Typical results for this algorithm in dense urban traffic are presented, using symmetry type (1). The generalization of the algorithm to skew symmetric images is shown. A potential application of the algorithm to automated roadways and fatigue detection is sketched out. Robustifying extensions in the spirit of (2), (3) and (4) are proposed.