Showing papers on "Symmetry (geometry) published in 1999"

Classical 6j{symbols and the tetrahedron

Abstract: A classical 6j {symbol is a real number which can be associated to a labelling of the six edges of a tetrahedron by irreducible representations of SU(2). This abstract association is traditionally used simply to express the symmetry of the 6j {symbol, which is a purely algebraic object; however, it has a deeper geometric signicance. Ponzano and Regge, expanding on work of Wigner, gave a striking (but unproved) asymptotic formula relating the value of the 6j { symbol, when the dimensions of the representations are large, to the volume of an honest Euclidean tetrahedron whose edge lengths are these dimensions. The goal of this paper is to prove and explain this formula by using geometric quantization. A surprising spin-o is that a generic Euclidean tetrahedron gives rise to a family of twelve scissors-congruent but non-congruent tetrahedra.

Five-branes, Seven-branes and Five-dimensional E_n field theories

Abstract: We generalize the (p,q) 5-brane web construction of five-dimensional field theories by introducing (p,q) 7-branes, and apply this construction to theories with a one-dimensional Coulomb branch. The 7-branes render the exceptional global symmetry of these theories manifest. Additionally, 7-branes allow the construction of all E_n theories up to n=8, previously not possible in 5-brane configurations. The exceptional global symmetry in the field theory is a subalgebra of an affine symmetry on the 7-branes, which is necessary for the existence of the system. We explicitly determine the quantum numbers of the BPS states of all E_n theories using two simple geometrical constraints.

Orientations and Rotations: Computations in Crystallographic Textures

Abstract: 1 Preliminaries.- 2 Parameterizations.- 3 Geometry of the Rotation Space.- 4 More on Small Orientation Changes.- 5 Some Statistical Issues.- 6 Symmetry.- 7 Misorientation Angle and Axis Distributions.- 8 Crystalline Interfaces and Symmetry.- 9 Crystallographic Textures.- 10 Diffraction Geometry.- 11 Effective Elastic Properties of Polycrystals.- References.

From Asymmetry to Full Symmetry: New Techniques for Symmetry Reduction in Model Checking

Abstract: It is often the case that systems are "nearly symmetric"; they exhibit symmetry in a part of their description but are, nevertheless, globally asymmetric. We formalize several notions of near symmetry and show how to obtain the benefits of symmetry reduction when applied to asymmetric systems which are nearly symmetric. We show that for some nearly symmetric systems it is possible to perform symmetry reduction and obtain a bisimilar (up to permutation) symmetry reduced system. Using a more general notion of "sub-symmetry" we show how to generate a reduced structure that is simulated (up to permutation) by the original asymmetric program. In the symbolic model checking paradigm, representing the symmetry reduced quotient structure entails representing the BDD for the orbit relation. Unfortunately, for many important symmetry groups, including the full symmetry group, this BDD is provably always intractably large, of size exponential in the number of bits in the state space. In contrast, under the assumption of full symmetry, we show that it is possible to reduce a textual program description of a symmetric system to a textual program description of the symmetry reduced system. This obviates the need for building the BDD representation of the orbit relation on the program states under the symmetry group. We establish that the BDD representing the reduced program is provably small, essentially polynomial in the number of bits in the state space of the original program.

Almost commuting elements in compact Lie groups

Abstract: We describe the components of the moduli space of conjugacy classes of commuting pairs and triples of elements in a compact Lie group. This description is in terms of the extended Dynkin diagram of the simply connected cover, together with the coroot integers and the action of the fundamental group. In the case of three commuting elements, we compute Chern-Simons invariants associated to the corresponding flat bundles over the three-torus, and verify a conjecture of Witten which reveals a surprising symmetry involving the Chern-Simons invariants and the dimensions of the components of the moduli space.

Symmetry detection by generalized complex (GC) moments: a close-form solution

Abstract: This paper presents a unified method for detecting both reflection-symmetry and rotation-symmetry of 2D images based on an identical set of features, i.e., the first three nonzero generalized complex (GC) moments. This method is theoretically guaranteed to detect all the axes of symmetries of every 2D image, if more nonzero GC moments are used in the feature set. Furthermore, we establish the relationship between reflectional symmetry and rotational symmetry in an image, which can be used to check the correctness of symmetry detection. This method has been demonstrated experimentally using more than 200 images.

Mean values of L-functions and symmetry

Abstract: Recently Katz and Sarnak introduced the idea of a symmetry group attached to a family of L-functions, and they gave strong evidence that the symmetry group governs many properties of the distribution of zeros of the L-functions. We consider the mean-values of the L-functions and the mollified mean-square of the L-functions and find evidence that these are also governed by the symmetry group. We use recent work of Keating and Snaith to give a complete description of these mean values. We find a connection to the Barnes-Vign\'eras $\Gamma_2$-function and to a family of self-similar functions.

Exploiting Incommensurate Symmetry Numbers: Rational Design and Assembly of M2M3′L6 Supramolecular Clusters with C3h Symmetry

Abstract: Mixed-metal mesocates [M2 Pd3 Br6 L6 ]4- (M=TiIV , SnIV ; L=4-diphenylphosphanyl-catecholate) have been synthesized, in which the two incommensurate symmetry elements generated by the different metal ions are linked by a rigid, bifunctional ligand to generate a C3h -symmetrical cluster (see picture).

Symmetry and reduction in implicit generalized Hamiltonian systems

Abstract: In this paper the notion of symmetry for implicit generalized Hamiltonian systems will be studied and a reduction theorem, generalizing the `classical' reduction theorems of symplectic and Poisson-Hamiltonian systems, will be derived.

The Eightfold way : the beauty of Klein's quartic curve

Abstract: Felix Klein discovered in the 1870's that the simple and elegant equation x3y + y3z + z3x (in complex projective coordinates) describes a surface having many remarkable properties, including 336-fold symmetry -- the maximum possible for any surface of this genus. Since then this object has come up in different guises in several areas of mathematics.

Magnetoresistance Anomalies in Antiferromagnetic YBa 2 Cu 3 O 6+x : Fingerprints of Charged Stripes

Abstract: We report novel features in the in-plane magnetoresistance (MR) of heavily underdoped YBa_2Cu_3O_{6+x}, which unveil a developed ``charged stripe'' structure in this system. One of the striking features is an anisotropy of the MR with a "d-wave" symmetry upon rotating the magnetic field H within the ab plane, which is caused by the rotation of the stripes with the external field. With decreasing temperature, a hysteresis shows up below ~20 K in the MR curve as a function of H and finally below 10 K the magnetic-field application produces a persistent change in the resistivity. This "memory effect" is caused by the freezing of the directionally-ordered stripes.

On the Grushin operator and hyperbolic symmetry

Abstract: Complexity of geometric symmetry for differential operators with mixed homogeniety is examined here. Sharp Sobolev estimates are calculated for the Grushin operator in low dimensions using hyperbolic symmetry and conformal geometry.

Fast Reflectional Symmetry Detection Using Orientation Histograms

Abstract: A simple and fast reflectional symmetry detection algorithm has been developed in this paper. The algorithm employs only the original gray scale image and the gradient information of the image, and it is able to detect multiple reflectional symmetry axes of an object in the image. The directions of the symmetry axes are obtained from the gradient orientation histogram of the input gray scale image by using the Fourier method. Both synthetic and real images have been tested using the proposed algorithm.

No role for colour in symmetry perception

Abstract: Bilateral colour symmetry, such as that evident in a Siberian tiger's face (Fig. 1a), is relevant to many animals1,2, including humans3,4. We examined the role of colour in symmetry perception by asking observers to detect colour symmetry in regular grids of coloured squares (a colour-symmetrical image has regions of the same colour located equidistantly from a vertical axis). Our results suggest, unexpectedly, that the mechanisms of symmetry perception are inherently colour-blind: although observers can verify colour symmetry, they do so only by shifting attention from one colour to the next and assessing the symmetry of regions of that colour.

Proof of the symmetry of the off-diagonal Hadamard/Seeley-deWitt's coefficients in $C^{\infty}$ Lorentzian manifolds by a local Wick rotation

Abstract: Completing the results obtained in a previous paper, we prove the symmetry of Hadamard/Seeley-deWitt off-diagonal coefficients in smooth $D$-dimensional Lorentzian manifolds. To this end, it is shown that, in any Lorentzian manifold, a sort of ``local Wick rotation'' of the metric can be performed provided the metric is a locally analytic function of the coordinates and the coordinates are ``physical''. No time-like Killing field is necessary. Such a local Wick rotation analytically continues the Lorentzian metric in a neighborhood of any point, or, more generally, in a neighborhood of a space-like (Cauchy) hypersurface, into a Riemannian metric. The continuation locally preserves geodesically convex neighborhoods. In order to make rigorous the procedure, the concept of a complex pseudo-Riemannian (not Hermitian or K\"ahlerian) manifold is introduced and some features are analyzed. Using these tools, the symmetry of Hadamard/Seeley-deWitt off-diagonal coefficients is proven in Lorentzian analytical manifolds by analytical continuation of the (symmetric) Riemannian heat-kernel coefficients. This continuation is performed in geodesically convex neighborhoods in common with both the metrics. Then, the symmetry is generalized to $C^\infty$ non analytic Lorentzian manifolds by approximating Lorentzian $C^{\infty}$ metrics by analytic metrics in common geodesically convex neighborhoods. The symmetry requirement plays a central r\^{o}le in the point-splitting renormalization procedure of the one-loop stress-energy tensor in curved spacetimes for Hadamard quantum states.

Symmetry and lattices of single-wall nanotubes

Abstract: The full Euclidean symmetry groups for all the single-wall carbon nanotubes are non-Abelian non-symorphic line groups, enlarging the groups reported in the literature. For the chiral tubes (n1,n2) (n1>n2>0) the groups are Lqp22 = TrqDn, where n is the greatest common divisor of n1 and n2, q = 2(n21+n1n2+n22)/n, while the parameters r and p are expressed in the closed forms as functions of n1 and n2. The number is three if n1-n2 is a multiple of 3n and one otherwise; it divides the tubes into two bijective classes. The line group uniquely determines the tube, unless q = 2n (then r = 1), when both the zig-zag (n,0) ( = 1) and the armchair (n,n) ( = 3) tubes are obtained, with the line group L(2n)n/mcm = Tn2nDnh having additional mirror planes. Some physical consequences are discussed: metallic tubes, quantum numbers and related selection rules, electronic and phonon bands, and their degeneracy, and applications to tensor properties.

Symmetry in Science: An Introduction to the General Theory.

Abstract: This is the most fascinating book on symmetry I have ever read. Everyone who applies the symmetry concept should read it.[...]

Proof of the Symmetry of the Off-Diagonal Heat-Kernel and Hadamard's Expansion Coefficients in General C ∞ Riemannian Manifolds

Abstract: We consider the problem of the symmetry of the off-diagonal heat-kernel coefficients as well as the coefficients corresponding to the short-distance-divergent part of the Hadamard expansion in general smooth (analytic or not) manifolds. The requirement of such a symmetry played a central role in the theory of the point-splitting one-loop renormalization of the stress tensor in either Riemannian or Lorentzian manifolds. Actually, the symmetry of these coefficients has been assumed as a hypothesis in several papers concerning these issues without an explicit proof. The difficulty of a direct proof is related to the fact that the considered off-diagonal heat-kernel expansion, also in the Riemannian case, in principle, may be not a proper asymptotic expansion. On the other hand, direct computations of the off-diagonal heat-kernel coefficients are impossibly difficult in nontrivial cases and thus no case is known in the literature where the symmetry does not hold. By approximating C∞ metrics with analytic metrics in common (totally normal) geodesically convex neighborhoods, it is rigorously proven that in general C∞ Riemannian manifolds, any point admits a geodesically convex neighborhood where the off-diagonal heat-kernel coefficients, as well as the relevant Hadamard expansion coefficients, are symmetric functions of the two arguments.

On the intrinsic reconstruction of shape from its symmetries

Abstract: We address the issue of the use of symmetry-based representations, such as the medial axis and an augmented form of it, the shock structure, to regenerate shapes. First, we address pointwise reconstruction of the boundary from points of the medial axis. As classified into three generic types (A/sup 2//1 mid-branch, A/sub 3/ end point of a branch, and A/sub 1//sup 3/ junction). Second, we examine the intrinsic reconstruction of shape when differential properties of the axis are also available. We show the surprising result that the tangent and curvature of the medial axis, coupled with the speed and acceleration of the shock flowing along the's axis, i.e., first and second order properties, are sufficient to determine the boundary tangents and curvatures at corresponding points of the boundary. This implies that for a rather coarse sampling of the symmetry axis, the location together with its tangent, curvature: speed, and acceleration is sufficient to accurately regenerate a local neighborhood of shape at this point. Together with reconstruction properties at junction (A/sup 3//sub 1/) and end points (A/sub 3/), these results lead to the full intrinsic regeneration of a shape from a representation of it as a directed planar graph (where the links represent curvature and acceleration functions, and where the nodes contain tangent and speed information): a representation ideally suited for the design and manipulation of free-form shape.

Generalized Quadrangles with a Spread of Symmetry

Abstract: We present a common construction for some known infinite classes of generalized quadrangles. Whether this construction yields other (unknown) generalized quadrangles is an open problem. The class of generalized quadrangles obtained this way is characterized in two different ways. On the one hand, they are exactly the generalized quadrangles having a spread of symmetry. On the other hand, they can be characterized in terms of the group of projectivities with respect to a spread. We explore some properties of these generalized quadrangles. All these results can be applied to the theory of the glued near hexagons, a class of near hexagons introduced by the author in De Bruyn (1998) On near hexagons and spreads of generalized quadrangles, preprint.

Symmetry and symmetry breaking : an algebraic approach to the genetic code

Abstract: We give a comprehensive review of the algebraic approach to the genetic code originally proposed by two of the present authors, which aims at explaining the degeneracies encountered in the genetic code as the result of a sequence of symmetry breakings that have occurred during its evolution. We present the relevant background material from molecular biology and from mathematics, including the representation theory of (semi) simple Lie groups/algebras, together with considerations of general nature.

Accurate Robust Symmetry Estimation

Abstract: There are various applications, both in medical and non-medical image analysis, which require the automatic detection of the line (2D images) or plane (3D) of reflective symmetry of objects. There exist relatively simple methods of finding reflective symmetry when object images are complete (i.e., completely symmetric and perfectly segmented from image “background”). A much harder problem is finding the line or plane of symmetry when the object of interest contains asymmetries, and may not have well defined edges.

A geometric theory of form, profile, and orientation tolerances

Abstract: This article develops a geometric theory which unifies the formulation and computation of form (straightness, flatness, cylindricity, and circularity), profile and orientation tolerances stipulated in ANSI Y14.5M standard. The theory rests on an important observation that a toleranced feature exhibits a symmetry subgroup G 0 under the action of the Euclidean group, SE (3). Thus, the configuration space of a toleranced (or a symmetric) feature can be identified with the homogeneous space SE (3)/ G 0 of the Euclidean group. Geometric properties of SE (3)/ G 0 , especially its exponential coordinates carried over from that of SE (3), are analyzed. We show that all cases of form, profile and orientation tolerances can be formulated as a minimization or constrained minimization problem on the space SE (3)/ G 0 , with G 0 being the symmetry subgroup of the underlying feature. We develop a simple geometric algorithm, called the Symmetric Minimum Zone (SMZ) algorithm, to unify the computation of form, profile, and orientation tolerances. Finally, we use numerical simulation results comparing the performances of the SMZ algorithm against the best known algorithms in the literature.

Substituent effects and the noncrossing rule: The importance of reduced symmetry subspaces. I. The quenching of OH(A 2Σ+) by H2

Abstract: The effects of substituent substitution on the locus of a seam of conical intersection and the importance of conical intersections in the associated low symmetry subspaces are considered. For molecules with more than three atoms and with some symmetry the seam of conical intersection may well include an accidental symmetry-allowed portion involving two states of different symmetry. However, in regions of reduced point group symmetry, conical intersections involving two states of the same symmetry may exist. This later class of conical intersections is rarely considered although it could significantly alter the predicted outcome of a nonadiabatic process. The efficient quenching of OH(A 2Σ+)by H2, a consequence of OH–H2 conical intersections, is particularly compelling in this regard. Previous analyses have considered only the C2v2A1–2B2 accidental symmetry-allowed portion of the seam of conical intersection. It is demonstrated that when intersections of states of the same symmetry are considered conical i...