Showing papers on "Symmetry (geometry) published in 2002"
TL;DR: In this article, it was shown that a diagonalizable Hamiltonian H is pseudo-Hermitian if and only if it has an antilinear symmetry, i.e., a symmetry generated by an invertible antilINear operator, and the eigenvalues of H are real or come in complex conjugate pairs.
Abstract: We show that a diagonalizable (non-Hermitian) Hamiltonian H is pseudo-Hermitian if and only if it has an antilinear symmetry, i.e., a symmetry generated by an invertible antilinear operator. This implies that the eigenvalues of H are real or come in complex conjugate pairs if and only if H possesses such a symmetry. In particular, the reality of the spectrum of H implies the presence of an antilinear symmetry. We further show that the spectrum of H is real if and only if there is a positive-definite inner-product on the Hilbert space with respect to which H is Hermitian or alternatively there is a pseudo-canonical transformation of the Hilbert space that maps H into a Hermitian operator.
617 citations
TL;DR: In this paper, a continuous symmetry study of the structures of transition metal six-vertex polyhedra is presented, considering both molecular models and experimental structural data, including the Bailar twist that interconverts one into another.
Abstract: A continuous symmetry study of the structures of transition metal six-vertex polyhedra is presented, considering both molecular models and experimental structural data. The concept of symmetry map is introduced, consisting of a scatterplot of the symmetry measures relative to two alternative ideal polyhedra. In the case of hexacoordinated complexes, we take as reference shapes the octahedron and the equilateral trigonal prism and study different distortions from these two extremes, including the Bailar twist that interconverts one into another. Such a symmetry map allows us to establish trends in the structural chemistry of the coordination sphere of hexacoordinated transition metal atoms, including the effects of several factors, such as the electron configuration or the presence of bidentate, terdentate or encapsulating ligands. Also introduced is the concept of a symmetry constant, which identifies a distortive route that preserves the minimum distance to two reference
symmetries. A wide variety of model distortions are analyzed, and the models are tested against experimental structural data of a wide variety of six-coordinated complexes.
361 citations
275 citations
Posted Content•
TL;DR: In this article, the symmetry algebra of the Laplacian on Euclidean space was identified as an explicit quotient of the universal enveloping algebra of conformal motions, and analogues of these symmetries on a general conformal manifold were constructed.
Abstract: Using the AdS/CFT correspondence, we identify the symmetry algebra of the Laplacian on Euclidean space as an explicit quotient of the universal enveloping algebra of the Lie algebra of conformal motions. We construct analogues of these symmetries on a general conformal manifold.
178 citations
Book•
01 Jun 2002
TL;DR: The role of pattern outline in Bilateral Symmetry Detection with Briefly Flashed Dot Patterns was discussed in this article, where it was shown that second-order pattern processing throughout the visual field is the dominant process for symmetry detection.
Abstract: Contents: Part I:Introduction. C.W. Tyler, Human Symmetry Perception. Part II:Empirical Evaluation of Symmetry Perception. J. Wagemans, Detection of Visual Symmetries. P. Wenderoth, The Role of Pattern Outline in Bilateral Symmetry Detection With Briefly Flashed Dot Patterns. K.E. Higgins, A. Arditi, K. Knoblauch, Detection and Identification of Mirror-Image Letter Pairs in Central and Peripheral Vision. P.T. Quinlan, Evidence for the Use of Scene-Based Frames of Reference in Two-Dimensional Shape Recognition. G. Leone, M. Lipshits, J. McIntyre, V. Gurfinkel, Independence of Bilateral Symmetry Detection From a Gravitational Reference Frame. J.P. Szlyk, I. Rock, C.B. Fisher, Level of Processing in the Perception of Symmetrical Forms Viewed From Different Angles. S. Hong, M. Pavel, Determinants of Symmetry Perception. C.W. Tyler, L. Hardage, Mirror Symmetry Detection: Predominance of Second-Order Pattern Processing Throughout the Visual Field. P.J. Passmore, A. Johnston, Human Discrimination of Surface Slant in Fractal and Related Textured Images. Part III:Theoretical Issues in Symmetry Analysis. S.C. Dakin, R.J. Watt, Detection of Bilateral Symmetry Using Spatial Filters. C. Latimer, W. Joung, C. Stevens, Modelling Symmetry Detection With Back-Propagation Networks. M.A. Kurbat, A Network Model for Generating Differential Symmetry Axes of Shapes Via Receptive Fields. I. Rentschler, E. Barth, T. Caelli, C. Zetzsche, M. Juttner, On the Generalization of Symmetry Relations in Visual Pattern Classification. F. Labonte, Y. Shapira, P. Cohen, J. Faubert, A Model for Global Symmetry Detection in Dense Images. H. Zabrodsky, D. Algom, Continuous Symmetry: A Model for Human Figural Perception. Y. Bonneh, D. Reisfeld, Y. Yeshurun, Quantification of Local Symmetry: Application to Texture Discrimination. J.S. Joseph, J.D. Victor, A Continuum of Non-Gaussian Self-Similar Image Ensembles With White Power Spectra. L.L. Kontsevich, Symmetry as a Depth Cue. T. Vetter, T. Poggio, Symmetric 3D Objects Are an Easy Case for 2D Object Recognition. L. Matin, W. Li, Mirror Symmetry and Parallelism: Two Opposite Rules for the Identity Transform in Space Perception and Their Unified Treatment by the Great Circle Model. J.R. Pani, The Generalized Cone in Human Spatial Organization.
141 citations
28 May 2002
TL;DR: A new reflective symmetry descriptor is introduced that represents a measure of reflective symmetry for an arbitrary 3D voxel model for all planes through the model's center of mass (even if they are not planes of symmetry).
Abstract: Computing reflective symmetries of 2D and 3D shapes is a classical problem in computer vision and computational geometry. Most prior work has focused on finding the main axes of symmetry, or determining that none exists. In this paper, we introduce a new reflective symmetry descriptor that represents a measure of reflective symmetry for an arbitrary 3D voxel model for all planes through the model's center of mass (even if they are not planes of symmetry). The main benefits of this new shape descriptor are that it is defined over a canonical parameterization (the sphere) and describes global properties of a 3D shape. Using Fourier methods, our algorithm computes the symmetry descriptor in O(N4 logN) time for an N × N × N voxel grid, and computes a multiresolution approximation in O(N3 logN) time. In our initial experiments, we have found the symmetry descriptor to be useful for registration, matching, and classification of shapes.
134 citations
TL;DR: In this paper, it was shown that the pseudo-hermiticity property of diagonalizable pseudo-Hermitian operators is strictly connected with the existence of an antilinear symmetry.
Abstract: We inquire into some properties of diagonalizable pseudo-Hermitian operators, showing that their definition can be relaxed and that the pseudo-Hermiticity property is strictly connected with the existence of an antilinear symmetry. This result is then illustrated by considering the particular case of the complex Morse potential.
108 citations
TL;DR: In this paper, an exponential-type PT-symmetric potential was constructed, which includes the PT symmetric versions of the Rosen-Morse well and Scarf potential, and the complex PT-invariant potential well V(x) = q2 tanh2 αx+i(q 1/2) sech αx tanh αx + q 0, q 2 > 0.
101 citations
TL;DR: In this paper, it is argued that the holographic direction is the light-cone coordinate and the degrees of freedom live on a codimension-one screen at fixed $u$.
Abstract: Homogeneous gravitational wave backgrounds arise as infinite momentum limits of many geometries with a well-understood holographic description. General global aspects of these geometries are discussed. Using exact CFT techniques, strings in pp-wave backgrounds supported by a Neveu-Schwarz flux are quantized. As in Euclidean $AdS_3$, spectral flow and associated long strings are shown to be crucial in obtaining a complete spectrum. Holography is investigated using conformally flat coordinates analogous to those of the Poincar\\e patch in AdS. It is argued that the holographic direction is the light-cone coordinate $u$, and that the holographic degrees of freedom live on a codimension-one screen at fixed $u$. The usual conformal symmetry on the boundary is replaced by a representation of a Heisenberg-type algebra $H_D\\times H_D$, hinting at a new class of field theories realizing this symmetry. A sample holographic computation of 2 and 3-point functions is provided and Ward identities are derived. A complementary screen at fixed $v$ is argued to be necessary in order to encode the vacuum structure.
97 citations
TL;DR: In this article, a birational realization of affine Weyl group of type A(m−1 × A(1)n−1) was given and applied to construct some discrete integrable systems and discrete Painleve equations.
Abstract: We give a birational realization of affine Weyl group of type A(1)m−1 × A(1)n−1. We apply this representation to construct some discrete integrable systems and discrete Painleve equations. Our construction has a combinatorial counterpart through the ultra-discretization procedure.
TL;DR: Visual evoked potentials are used to measure the time course of neural events associated with the extraction of symmetry in random dot fields and results are consistent with the hypothesis that the symmetry property is extracted by processing in extrastriate cortex.
Abstract: Symmetry is a highly salient feature of animals, plants, and the constructed environment. Although the perceptual phenomenology of symmetry processing is well understood, little is known about the underlying neural mechanisms. Here we use visual evoked potentials to measure the time course of neural events associated with the extraction of symmetry in random dot fields. We presented sparse random dot patterns that were symmetric about both the vertical and horizontal axes. Symmetric patterns were alternated with random patterns of the same density every 500 msec, using new exemplars of symmetric and random patterns on each image update. Random/random exchanges were used as a control. The response to updates of random patterns was multiphasic, consisting of P65, N90, P110, N140 and P220 peaks. The response to symmetric/random sequences was indistinguishable from that for random/random sequences up to about 220 msec, after which the response to symmetric patterns became relatively more negative. Symmetry in random dot patterns thus appears to be extracted after an initial response phase that is indifferent to configuration. These results are consistent with the hypothesis (Lee, Mumford, Romero, & Lamme, 1998; Tyler & Baseler, 1998) that the symmetry property is extracted by processing in extrastriate cortex. Keywords: evoked potentials, shape, form
TL;DR: It is shown that symmetry discrimination during brief presentations is best for RF2+RF3 but becomes impossible for RF1+RF7, and that Human V4 is hypothesized to be the site for symmetry discrimination of RF patterns but not of random dot patterns.
TL;DR: In this paper, the authors used translations isometries of pp-waves to construct D4 and D3 brane solutions, using T-duality transformations, in exactly solvable pp-wave background originating from AdS 3 × S 3 geometry.
TL;DR: In this article, it was shown that if a Hamiltonian H has an unbroken PT symmetry, then it also possesses a hidden symmetry represented by the linear operator C. The inner product with respect to CPT is associated with a positive norm and the quantum theory built on the associated Hilbert space is unitary.
Abstract: In a recent paper it was shown that if a Hamiltonian H has an unbroken PT symmetry, then it also possesses a hidden symmetry represented by the linear operator C. The operator C commutes with both H and PT. The inner product with respect to CPT is associated with a positive norm and the quantum theory built on the associated Hilbert space is unitary. In this paper it is shown how to construct the operator C for the non-Hermitian PT-symmetric Hamiltonian $H={1\over2}p^2+{1\over2}x^2 +i\epsilon x^3$ using perturbative techniques. It is also shown how to construct the operator C for $H={1\over2}p^2+{1\over2}x^2-\epsilon x^4$ using nonperturbative methods.
TL;DR: In this paper, the symmetry rank of a Riemannian manifold is the rank of the isometry group and the fundamental groups of all closed positively curved n-manifolds with almost maximal symmetry rank (n−1)/2) (n≠ 6, 7).
Abstract: The symmetry rank of a Riemannian manifold is the rank of the isometry group. We determine precisely which closed simply connected 5-manifolds admit positively curved metrics with (almost maximal) symmetry rank two. We also determine the precise Euler characteristic and the fundamental groups of all closed positively curved n-manifolds with almost maximal symmetry rank [(n−1)/2] (n≠ 6, 7).
TL;DR: In this paper, the relation between the form invariance and Lie symmetry of non-holonomic systems is studied. But the relation is not restricted to holonomic systems, and it is not only restricted to nonholonomic but also restricted to homomorphic systems.
Abstract: In this paper, we study the relation between the form invariance and Lie symmetry of non-holonomic systems. Firstly, we give the definitions and criteria of the form invariance and Lie symmetry in the systems. Next, their relation is deduced. We show that the structure equation and conserved quantity of the form invariance and Lie symmetry of non-holonomic systems have the same form. Finally, we give an example to illustrate the application of the result.
TL;DR: In this article, unique affine extensions Haff2, Haff3 and Haff4 are determined for the non-crystallographic Coxeter groups H2,H3 andH4, and used for the construction of new mathematical models for quasicrystal fragments with tenfold symmetry.
Abstract: Unique affine extensions Haff2, Haff3 and Haff4 are determined for the noncrystallographic Coxeter groups H2, H3 and H4. They are used for the construction of new mathematical models for quasicrystal fragments with tenfold symmetry. The case of Haff2 corresponding to planar point sets is discussed in detail. In contrast to the cut-and-project scheme we obtain by construction finite point sets, which grow with a model specific growth parameter.
Patent•
26 Sep 2002
TL;DR: A plate for fixing the bones of a joint, in particular of a metatarso-phalangeal joint, for the purpose of performing arthodesis is described in this article.
Abstract: A plate for fixing the bones of a joint, in particular of a metatarso-phalangeal joint, for the purpose of performing arthodesis. The plate comprises two sections, respectively a proximal section and a distal section, each section having a respective longitudinal axis of symmetry S 1 , S 2 such that the projection onto a horizontal plane of the axis of symmetry S 2 of the distal section presents an angle of inclination relative to the projection of the axis of symmetry S 1 of the proximal portion, the projections inserting at a point A. The projection onto a vertical plane of the axis of symmetry S 2 presents an angle of inclination relative to the projection of the axis of symmetry S 1 , their intersection taking place at a point A 2 which is distinct from the point A 1 .
TL;DR: In this paper, an analysis of the domain symmetry involved and a full set of materials properties measured in crystals having only two of the four possible domains based on mm2 symmetry is presented.
Abstract: The crystal symmetry of a 0.955Pb(Zn1/3Nb2/3)O3-0.045PbTiO3 (PZN-4.5%PT) single crystal is rhombohedral 3m. It is in multidomain state after being poled along [001] of its original cubic coordinates. Although most studies on this system assumed the tetragonal 4mm effective symmetry, experimental observations show that many of the crystals actually have effective orthorhombic mm2 symmetry. Some crystal domain patterns may even become monoclinic m or even lower in symmetry. In other words, domains form a hierarchical symmetry, which controls the effective materials properties of the multidomain system. We report an analysis of the domain symmetry involved and a full set of materials properties measured in crystals having only two of the four possible domains based on mm2 symmetry.
TL;DR: In this paper, the authors used translations isometries of pp-waves to construct D4 and D3 brane solutions, using T-duality transformations, in exactly solvable pp-wave background originating from $AdS_3\times S^3$ geometry.
Abstract: We use recently proposed translations isometries of pp-waves to construct D4 and D3 brane solutions, using T-duality transformations, in exactly solvable pp-wave background originating from $AdS_3\times S^3$ geometry. A unique property of the new brane solutions is the breaking of SO(10-p-1) symmetry in the transverse direction of the branes due to the presence of constant NS-NS and R-R background fluxes. We verify that the our `localized' solutions satisfy the field equations and explicitly present the corresponding Killing spinors. We also show the connection of our results to certain M5-branes in pp-wave geometry.
Patent•
18 Jan 2002TL;DR: In this article, a method for extracting a midsagittal plane MSP of human brains from radiological images is presented, which is suitable for use in determining symmetry in images, namely 2D images or 3D images, which comprise 2D image slices.
Abstract: The present invention is suitable for use in determining symmetry in images, namely 2D images or 3D images, which comprise 2D images slices. In a preferred form, a method is disclosed for extracting a midsagittal plane MSP of human brains from radiological images. The method includes the steps of: 1) determining axial slices to be processed from said radiological images; 2) analysing said axial slices to determine fissure line segments; and 3) calculating plane equation of MSP from said fissure line segments.
01 Jun 2002
TL;DR: The prominent symmetry points (core-points) in the fingerprint are suggested from the complex orientation field estimated from the global structure of the fingerprint, i.e. the overall pattern of the ridges and valleys.
Abstract: For the alignment of two fingerprints position of certain landmarks are needed. These should be automatically extracted with low misidentification rate. As landmarks we suggest the prominent symmetry points (core-points) in the fingerprint. They are extracted from the complex orientation field estimated from the global structure of the fingerprint, i.e. the overall pattern of the ridges and valleys. Complex filters, applied to the orientation field in multiple resolution scales, are used to detect the symmetry and the type of symmetry. Experimental results are reported.
TL;DR: The results suggest that (1) symmetry detection is spatially imprecise, and (2) attentional gating can operate prior to symmetry detection in the visual pathway.
TL;DR: In this paper, the authors show that if a compact Lie group acts effectively, locally linearly, and homologically trivially on a closed, simply connected four-manifold M with b 2 (M )⩾3, then it must be isomorphic to a subgroup of S 1 × S 1, and the action must have nonempty fixed-point set.
TL;DR: The traditional way of deriving the secular equation for surface waves propagating in the direction of the x1axis in an anisotropic elastic halfspace x20 is to find a general steadystate solution f....
Abstract: The traditional way of deriving the secular equation for surface waves propagating in the direction of the x1axis in an anisotropic elastic halfspace x20 is to find a general steadystate solution f...
01 Jun 2002
TL;DR: In this article, Gromov-Witten invariants of a rational elliptic surface using holomorphic anomaly equation in [HST1]-HST2 have been derived for affine $E_8$ Weyl group symmetry.
Abstract: We study Gromov-Witten invariants of a rational elliptic surface using holomorphic anomaly equation in [HST1](hep-th/9901151). Formulating invariance under the affine $E_8$ Weyl group symmetry, we determine conjectured invariants, the number of BPS states, from Gromov-Witten invariants. We also connect our holomorphic anomaly equation to that found by Bershadsky,Cecotti,Ooguri and Vafa [BCOV1](hep-th/9302103).
TL;DR: In this paper, a set of data supposed to give possible axioms for spacetimes with a sufficient number of isometries in spectral geometry is given, which are shown to be sufficient to obtain 1+1 dimensional de Sitter spacetime.
Abstract: A set of data supposed to give possible axioms for spacetimes with a sufficient number of isometries in spectral geometry is given. These data are shown to be sufficient to obtain 1+1 dimensional de Sitter spacetime. The data rely at the moment somewhat on the guidance given by a required symmetry, in part to allow explicit calculations in a specific model. The framework applies also to the noncommutative case. Finite spectral triples are discussed as an example.
TL;DR: In this article, the authors generalize this analytic continuation map to the case of rotating wormholes and obtain Euclidean manifolds whose boundary is the Schottky double of the geometry of the t = 0 plane.
Abstract: We have previously proposed that asymptotically AdS 3D wormholes and black holes can be analytically continued to the Euclidean signature The analytic continuation procedure was described for non-rotating spacetimes, for which a plane t = 0 of time symmetry exists The resulting Euclidean manifolds turned out to be handlebodies whose boundary is the Schottky double of the geometry of the t = 0 plane In the present paper we generalize this analytic continuation map to the case of rotating wormholes The Euclidean manifolds we obtain are quotients of the hyperbolic space by a certain quasi-Fuchsian group The group is the Fenchel–Nielsen deformation of the group of the non-rotating spacetime The angular velocity of an asymptotic region is shown to be related to the Fenchel–Nielsen twist This solves the problem of classification of rotating black holes and wormholes in (2 + 1) dimensions: the spacetimes are parametrized by the moduli of the boundary of the corresponding Euclidean spaces We also comment on the thermodynamics of the wormhole spacetimes
TL;DR: In this article, the authors characterize a K3 surface of Klein-Mukai type in terms of its symmetry, and show that the symmetry of the surface is a function of the density of the surfaces.
Abstract: The aim of this note is to characterize a K3 surface of Klein-Mukai type in terms of its symmetry.