Showing papers on "Symmetry (geometry) published in 2008"

Dual superconformal symmetry from AdS5xS5 superstring integrability

Abstract: We discuss $2d$ duality transformations in the classical ${\mathrm{AdS}}_{5}\ifmmode\times\else\texttimes\fi{}{S}^{5}$ superstring and their effect on the integrable structure. $T$-duality along four directions in the Poincar\'e parametrization of ${\mathrm{AdS}}_{5}$ maps the bosonic part of the superstring action into itself. On the bosonic level, this duality may be understood as a symmetry of the first-order (phase space) system of equations for the coset components of the current. The associated Lax connection is invariant modulo the action of an $\mathfrak{s}\mathfrak{o}(2,4)$-automorphism. We then show that this symmetry extends to the full superstring, provided one supplements the transformation of the bosonic components of the current with a transformation on the fermionic ones. At the level of the action, this symmetry can be seen by combining the bosonic duality transformation with a similar one applied to part of the fermionic superstring coordinates. As a result, the full superstring action is mapped into itself, albeit in a different $\ensuremath{\kappa}$-symmetry gauge. One implication is that the dual model has the same superconformal symmetry group as the original one, and this may be seen as a consequence of the integrability of the superstring. The invariance of the Lax connection under the duality implies a map on the full set of conserved charges that should interchange some of the Noether (local) charges with hidden (nonlocal) ones and vice versa.

Unique horizontal symmetry of leptons

Abstract: There is a group-theoretical connection between fermion mixing matrices and minimal horizontal symmetry groups. Applying this connection to the tribimaximal neutrino mixing matrix, we show that the minimal horizontal symmetry group for leptons is uniquely S{sub 4}, the permutation group of four objects.

Paying Attention to Symmetry

Abstract: Humans are very sensitive to symmetry in visual patterns. Symmetry is detected and recognized very rapidly. While viewing symmetrical patterns eye fixations are concentrated along the axis of symmetry or the symmetrical center of the patterns. This suggests that symmetry is a highly salient feature. Existing computational models of saliency, however, have mainly focused on contrast as a measure of saliency. These models do not take symmetry into account. In this paper, we discuss local symmetry as measure of saliency. We developed a number of symmetry models an performed an eye tracking study with human participants viewing photographic images to test the models. The performance of our symmetry models is compared with the contrast saliency model of Itti et al. [1]. The results show that the symmetry models better match the human data than the contrast model. This indicates that symmetry is a salient structural feature for humans, a finding which can be exploited in computer vision.

Kahler geometry on toric manifolds, and some other manifolds with large symmetry

Abstract: This is an expository article. Among other topics, we discuss the existence of Kahler-Ricci soliton metrics on toric Fano manifolds, and Kahler-Einstein metrics on deformations of the Mukai-Umemura 3-fold

Extremal vacuum black holes in higher dimensions

Abstract: We consider extremal black hole solutions to the vacuum Einstein equations in dimensions greater than five. We prove that the near-horizon geometry of any such black hole must possess an SO(2,1) symmetry in a special case where one has an enhanced rotational symmetry group. We construct examples of vacuum near-horizon geometries using the extremal Myers-Perry black holes and boosted Myers-Perry strings. The latter lead to near-horizon geometries of black ring topology, which in odd spacetime dimensions have the correct number rotational symmetries to describe an asymptotically flat black object. We argue that a subset of these correspond to the near-horizon limit of asymptotically flat extremal black rings. Using this identification we provide a conjecture for the exact “phase diagram” of extremal vacuum black rings with a connected horizon in odd spacetime dimensions greater than five.

The MOND limit from space-time scale invariance

Abstract: The MOND limit is shown to follow from a requirement of space-time scale invariance of the equations of motion for nonrelativistic, purely gravitational systems; i.e., invariance of the equations of motion under (t,r) goes to (qt,qr), in the limit a0 goes to infinity. It is suggested that this should replace the definition of the MOND limit based on the low-acceleration behavior of a Newtonian-MOND interpolating function. In this way, the salient, deep-MOND results--asymptotically flat rotation curves, the mass-rotational-speed relation (baryonic Tully-Fisher relation), the Faber-Jackson relation, etc.--follow from a symmetry principle. For example, asymptotic flatness of rotation curves reflects the fact that radii change under scaling, while velocities do not. I then comment on the interpretation of the deep-MOND limit as one of "zero mass": Rest masses, whose presence obstructs scaling symmetry, become negligible compared to the "phantom", dynamical masses--those that some would attribute to dark matter. Unlike the former masses, the latter transform in a way that is consistent with the symmetry. Finally, I discuss the putative MOND-cosmology connection in light of another, previously known symmetry of the deep-MOND limit. In particular, it is suggested that MOND is related to the asymptotic de Sitter geometry of our universe. It is conjectured, for example, that in an exact de Sitter cosmos, deep-MOND physics would exactly apply to local systems. I also point out, in this connection, the possible relevance of a de Sitter-conformal-field-theory (dS/CFT) duality.

Performance evaluation of state-of-the-art discrete symmetry detection algorithms

Abstract: Symmetry is one of the important cues for human and machine perception of the world. For over three decades, automatic symmetry detection from images/patterns has been a standing topic in computer vision. We present a timely, systematic, and quantitative performance evaluation of three state of the art discrete symmetry detection algorithms. This evaluation scheme includes a set of carefully chosen synthetic and real images presenting justified, unambiguous single or multiple dominant symmetries, and a pair of well-defined success rates for validation. We make our 176 test images with associated hand-labeled ground truth publicly available with this paper. In addition, we explore the potential contribution of symmetry detection for object recognition by testing the symmetry detection algorithm on three publicly available object recognition image sets (PASCAL VOC'07, MSRC and Caltech-256). Our results indicate that even after several decades of effort, symmetry detection in real-world images remains a challenging, unsolved problem in computer vision. Meanwhile, we illustrate its future potential in object recognition.

Symmetry of planar four-body convex central configurations

Abstract: We study the relationship between the masses and the geometric properties of central configurations. We prove that, in the planar four-body problem, a convex central configuration is symmetric with respect to one diagonal if and only if the masses of the two particles on the other diagonal are equal. If these two masses are unequal, then the less massive one is closer to the former diagonal. Finally, we extend these results to the case of non-planar central configurations of five particles.

Reconstruction of an object from its symmetry-averaged diffraction pattern.

Abstract: The problem of reconstructing an object from diffraction data that has been incoherently averaged over a discrete group of symmetries is considered. A necessary condition for such data to uniquely specify the object is derived in terms of the object support and the symmetry group. An algorithm is introduced for reconstructing objects from symmetry-averaged data and its use with simulations is demonstrated. The results demonstrate the feasibility of structure determination using a recent proposal for aligning molecules by means of their anisotropic dielectric interaction with an intense light field

Optimal packings of superdisks and the role of symmetry.

Abstract: Almost all studies of the densest particle packings consider convex particles. Here, we provide exact constructions for the densest known two-dimensional packings of superdisks whose shapes are defined by jx1j 2pj x2j 2p � 1 and thus contain a large family of both convex (p � 0:5) and concave (0

Globally Optimal Grouping for Symmetric Closed Boundaries by Combining Boundary and Region Information

Abstract: Many natural and man-made structures have a boundary that shows a certain level of bilateral symmetry, a property that plays an important role in both human and computer vision. In this paper, we present a new grouping method for detecting closed boundaries with symmetry. We first construct a new type of grouping token in the form of symmetric trapezoids by pairing line segments detected from the image. A closed boundary can then be achieved by connecting some trapezoids with a sequence of gap-filling quadrilaterals. For such a closed boundary, we define a unified grouping cost function in a ratio form: the numerator reflects the boundary information of proximity and symmetry, and the denominator reflects the region information of the enclosed area. The introduction of the region-area information in the denominator is able to avoid a bias toward shorter boundaries. We then develop a new graph model to represent the grouping tokens. In this new graph model, the grouping cost function can be encoded by carefully designed edge weights, and the desired optimal boundary corresponds to a special cycle with a minimum ratio-form cost. We finally show that such a cycle can be found in polynomial time using a previous graph algorithm. We implement this symmetry-grouping method and test it on a set of synthetic data and real images. The performance is compared to two previous grouping methods that do not consider symmetry in their grouping cost functions.

Alternative models for two crystal structures of bovine rhodopsin

Abstract: The space-group symmetry of two crystal forms of rhodopsin (PDB codes 1gzm and 2j4y; space group P31) can be re-interpreted as hexagonal (space group P64). Two molecules of the G protein-coupled receptor are present in the asymmetric unit in the trigonal models. However, the noncrystallographic twofold axes parallel to the c axis can be treated as crystallographic symmetry operations in the hexagonal space group. This halves the asymmetric unit and makes all of the protein molecules equivalent in these structures. Corrections for merohedral twinning were also applied in the refinement in the higher symmetry space group for one of the structures (2j4y).

Emergence of symmetry in complex networks.

Abstract: Many real networks have been found to have a rich degree of symmetry, which is a universal structural property of complex networks, yet has been rarely studied so far. One of the fascinating problems related to symmetry is exploration of the origin of symmetry in real networks. For this purpose, we summarized the statistics of local symmetric motifs that contribute to local symmetry of networks. Analysis of these statistics shows that the symmetry of complex networks is a consequence of similar linkage pattern, which means that vertices with similar degrees tend to share common neighbors. An improved version of the Barabaśi-Albert model integrating similar linkage pattern successfully reproduces the symmetry of real networks, indicating that similar linkage pattern is the underlying ingredient that is responsible for the emergence of symmetry in complex networks.

On Galilean and Lorentz invariance in pilot-wave dynamics

Abstract: It is argued that the natural kinematics of the pilot-wave theory is Aristotelian. Despite appearances, Galilean invariance is not a fundamental symmetry of the low-energy theory. Instead, it is a fictitious symmetry that has been artificially imposed. It is concluded that the search for a Lorentz-invariant extension is physically misguided.

An exceptional algebraic origin of the AdS/CFT Yangian symmetry

Abstract: In the (2|2) spin chain motivated by the AdS/CFT correspondence, a novel symmetry extending the superalgebra (2|2) into (2|2) was found. We pursue the origin of this symmetry in the exceptional superalgebra (2, 1; e), which recovers (2|2) when the parameter e is taken to zero. Especially, we rederive the Yangian symmetries of the AdS/CFT spin chain using the exceptional superalgebra and find that the e-correction corresponds to the novel symmetry. Also, we reproduce the non-canonical classical r-matrix of the AdS/CFT spin chain expressed with this symmetry from the canonical one of the exceptional algebra.

Rotation symmetry group detection via frequency analysis of frieze-expansions

Abstract: We present a novel and effective algorithm for rotation symmetry group detection from real-world images. We propose a frieze-expansion method that transforms rotation symmetry group detection into a simple translation symmetry detection problem. We define and construct a dense symmetry strength map from a given image, and search for potential rotational symmetry centers automatically. Frequency analysis, using discrete Fourier transform (DFT), is applied to the frieze-expansion patterns to uncover the types and the cardinality of multiple rotation symmetry groups in an image, concentric or otherwise. Furthermore, our detection algorithm can discriminate discrete versus continuous and cyclic versus dihedral symmetry groups, and identify the corresponding supporting regions in the image. Experimental results on over 80 synthetic and natural images demonstrate superior performance of our rotation detection algorithm in accuracy and in speed over the state of the art rotation detection algorithms.

Exactly solvable `discrete' quantum mechanics; shape invariance, Heisenberg solutions, annihilation-creation operators and coherent states

Abstract: Various examples of exactly solvable `discrete' quantum mechanics are explored explicitly with emphasis on shape invariance, Heisenberg operator solutions, annihilation-creation operators, the dynamical symmetry algebras and coherent states. The eigenfunctions are the (q-)Askey-scheme of hypergeometric orthogonal polynomials satisfying difference equation versions of the Schr\"odinger equation. Various reductions (restrictions) of the symmetry algebra of the Askey-Wilson system are explored in detail.

Tables of crystallographic properties of magnetic space groups.

Abstract: Tables of crystallographic properties of the reduced magnetic superfamilies of space groups, i.e. the 7 one-dimensional, 80 two-dimensional and 1651 three-dimensional group types, commonly referred to as magnetic space groups, are presented. The content and format are similar to that of non-magnetic space groups and subperiodic groups given in International Tables for Crystallography. Additional content for each representative group of each magnetic space-group type includes a diagram of general positions with corresponding general magnetic moments, Seitz notation used as a second notation for symmetry operations, and general and special positions listed with the components of the corresponding magnetic moments allowed by symmetry.

Breaking value symmetry

Abstract: Symmetry is an important factor in solving many constraint satisfaction problems. One common type of symmetry is when we have symmetric values. In a recent series of papers, we have studied methods to break value symmetries (Walsh 2006a; 2007). Our results identify computational limits on eliminating value symmetry. For instance, we prove that pruning all symmetric values is NP-hard in general. Nevertheless, experiments show that much value symmetry can be broken in practice. These results may be useful to researchers in planning, scheduling and other areas as value symmetry occurs in many different domains.

A novel method for alignment of 3D models

Abstract: In this paper we present a new method for alignment of 3D models. This approach is based on symmetry properties, and uses the fact that the principal components analysis (PCA) have good properties with respect to the planar reflective symmetry. The fast search of the best optimal alignment axes within the PCA-eigenvectors is an essential first step in our alignment process. The plane reflection symmetry is used as a criterion for selection. This pre-processing transforms the alignment problem into an indexing scheme based on the number of the retained PCA-axes. We also introduce a local translational invariance cost (LTIC) that captures a measure of the local translational symmetries of a shape with respect to a given direction. Experimental results show that the proposed method finds the rotation that best aligns a 3D mesh.

Random matrices beyond the Cartan classification

Abstract: It is known that hermitean random matrix ensembles can be identified with symmetric coset spaces of Lie groups, or else with tangent spaces of the same. This results in a classification of random matrix ensembles as well as applications in practical calculations of physical observables. In this paper, we show that a large number of non-hermitean random matrix ensembles defined by physically motivated symmetries—chiral symmetry, time-reversal invariance, space-rotation invariance, particle–hole symmetry or different reality conditions—can likewise be identified with symmetric spaces. We give explicit representations of the random matrix ensembles identified with lateral algebra subspaces, and of the corresponding symmetric subalgebras spanning the group of invariance. Among the ensembles listed we identify as special cases all the hermitean ensembles identified with the Cartan classes of symmetric spaces and the three Ginibre ensembles with complex eigenvalues.

Detection of skewed symmetry.

Abstract: This study examined the ability of human observers to discriminate between symmetric and asymmetric planar figures from perspective and orthographic images. The first experiment showed that the discrimination is reliable in the case of polygons, but not dotted patterns. The second experiment showed that the discrimination is facilitated when the projected symmetry axis or projected symmetry lines are known to the subject. A control experiment showed that the discrimination is more reliable with orthographic, than with perspective images. Based on these results, we formulated a computational model of symmetry detection. The model measures the asymmetry of the presented polygon based on its single orthographic or perspective image. Performance of the model is similar to that of the subjects.

Coupled Painleve VI Systems in Dimension Four with Affine Weyl Group Symmetry of Type D (1) , II

Abstract: We give a reformulation of a six-parameter family of coupled Painleve VI systems with affine Weyl group symmetry of type D (1) from the viewpoint of its symmetry and holomorphy properties.

Detecting approximate symmetries of discrete point subsets

Abstract: Detecting approximate symmetries of parts of a model is important when attempting to determine the geometrical design intent of approximate boundary-representation (B-rep) solid models produced e.g. by reverse engineering systems. For example, such detected symmetries may be enforced exactly on the model to improve its shape, to simplify its analysis, or to constrain it during editing. We give an algorithm to detect local approximate symmetries in a discrete point set derived from a B-rep model: the output comprises the model's potential local symmetries at various automatically detected tolerance levels. Non-trivial symmetries of subsets of the point set are found as unambiguous permutation cycles, i.e. vertices of an approximately regular polygon or an anti-prism, which are sufficiently separate from other points in the point set. The symmetries are detected using a rigorous, tolerance-controlled, incremental approach, which expands symmetry seed sets by one point at a time. Our symmetry cycle detection approach only depends on inter-point distances. The algorithm takes time O(n^4) where n is the number of input points. Results produced by our algorithm are demonstrated using a variety of examples.

Islamic geometrical patterns indexing and classification using discrete symmetry groups

Abstract: In this article, we propose a general computational model for the extraction of symmetry features of Islamic geometrical patterns' (IGP) images. We describe IGP images using the discrete symmetry groups theory. Our model contains the three following steps. (1) By noting that these patterns fall into three major categories, we begin our indexation process by classifying every pattern into one of these categories. The first pattern category describes all the patterns generated by translation along one direction. Every pattern of this category can be classified into one of the seven Frieze groups. The second type of pattern contains translational symmetries in two independent directions. Patterns of this category can be classified into one of the seventeen Wallpaper groups. The last type, called rosettes, describes patterns which begin at a central point and grow radially outward. We use rosette symmetry groups to classify patterns of this latter category. (2) For every pattern, we extract the symmetry features, namely, the symmetry group and the fundamental region, which is a representative region in the image from which the whole image can be regenerated. But for rosette groups, we can also compute the number of folds. (3) Finally, we describe the fundamental region by a simple color histogram and build the feature vector which is a combination of the symmetry feature (defined in the second step) and histogram information. Experiments show promising results for either IGP images' classification or indexing. Efforts for the subsequent task of classifying Islamic geometrical patterns' images can be significantly reduced.

Higher symmetries of the square of the Laplacian

Abstract: The symmetry operators for the Laplacian in flat space were recently described and here we consider the same question for the square of the Laplacian. Again, there is a close connection with conformal geometry. There are three main steps in our construction. The first is to show that the symbol of a symmetry is constrained by an overdetermined partial differential equation. The second is to show existence of symmetries with specified symbol (using a simple version of the AdS/CFT correspondence). The third is to compute the composition of two first order symmetry operators and hence determine the structure of the symmetry algebra. There are some interesting differences as compared to the corresponding results for the Laplacian.

Symbolic representation and classification of integrable systems

Abstract: This is a review paper of recent results in the perturbative symmetry approach in the symbolic representation.

The Neumann problem for the Hénon equation, trace inequalities and Steklov eigenvalues

Abstract: We consider the Neumann problem for the Henon equation. We obtain existence results and we analyze the symmetry properties of the ground state solutions. We prove that some symmetry and variational properties can be expressed in terms of eigenvalues of a Steklov problem. Applications are also given to extremals of certain trace inequalities.