Showing papers on "Symmetry (geometry) published in 2009"

Non-Linear Symmetry-Preserving Observers on Lie Groups

Abstract: In this technical note, we give a geometrical framework for the design of observers on finite-dimensional Lie groups for systems which possess some specific symmetries. The design and the error (between true and estimated state) equation are explicit and intrinsic. We consider also a particular case: left-invariant systems on Lie groups with right equivariant output. The theory yields a class of observers such that the error equation is autonomous. The observers converge locally around any trajectory, and the global behavior is independent from the trajectory, which is reminiscent of the linear stationary case.

Bernstein and De Giorgi type problems: new results via a geometric approach

Abstract: We use a Poincare type formula and level set analysis to detect one-dimensional symmetry of stable solutions of possibly degenerate or singular elliptic equation of the form div a(|∇u(x)|)∇u(x) + f(u(x)) = 0 . Our setting is very general and, as particular cases, we obtain new proofs of a conjecture of De Giorgi for phase transitions in R and R and of the Bernstein problem on the flatness of minimal area graphs in R. A one-dimensional symmetry result in the half-space is also obtained as a byproduct of our analysis. Our approach is also flexible to non-elliptic operators: as an application, we prove one-dimensional symmetry for 1-Laplacian type operators.

Put‐call symmetry: extensions and applications

Abstract: Classic put-call symmetry relates the prices of puts and calls at strikes on opposite sides of the forward price. We extend put-call symmetry in several directions. Relaxing the assumptions, we generalize to unified local/stochastic volatility models and time-changed Levy processes, under a symmetry condition. Further relaxing the assumptions, we generalize to various asymmetric dynamics. Extending the conclusions, we take an arbitrarily given payoff of European style or single/double/sequential barrier style, and we construct a conjugate European-style claim of equal value, and thereby a semistatic hedge of the given payoff.

Alignment of 3D models

Abstract: In this paper we present a new method for alignment of 3D models. This approach is based on two types of symmetries of the models: the reflective symmetry and the local translational symmetry along a direction. Inspired by the work on the principal component analysis (PCA), we select the best optimal alignment axes within the PCA-axes, the plane reflection symmetry being used as a selection criterion. This pre-processing transforms the alignment problem into an indexing scheme based on the number of the retained PCA-axes. In order to capture the local translational symmetry of a shape along a direction, we introduce a new measure we call the local translational invariance cost (LTIC). The mirror planes of a model are also used to reduce the number of candidate coordinate frames when looking for the one which corresponds to the user's perception. Experimental results show that the proposed method finds the rotation that best aligns a 3D mesh.

The effect of convolving families of L-functions on the underlying group symmetries

Abstract: Let {FN} and {GM} be families of primitive automorphic L-functions for GLn(AQ) and GLm(AQ), respectively, such that, as N,M → ∞, the statistical behavior (1-level density) of the low-lying zeros of L-functions in FN (resp., GM ) agrees with that of the eigenvalues near 1 of matrices in G1 (resp., G2) as the size of the matrices tend to infinity, where each Gi is one of the classical compact groups (unitary U, symplectic Sp, or orthogonal O, SO(even), SO(odd)). Assuming that the convolved families of L-functions FN × GM are automorphic, we study their 1-level density. (We also study convolved families of the form f ×GM for a fixed f .) Under natural assumptions on the families (which hold in many cases) we can associate to each family L of L-functions a symmetry constant cL equal to 0 (resp., 1 or −1) if the corresponding low-lying zero statistics agree with those of the unitary (resp., symplectic or orthogonal) group. Our main result is that cF×G = cF · cG : the symmetry type of the convolved family is the product of the symmetry types of the two families. A similar statement holds for the convolved families f × GM . We provide examples built from Dirichlet L-functions and holomorphic modular forms and their symmetric powers. An interesting special case is to convolve two families of elliptic curves with positive rank. In this case the symmetry group of the convolution is independent of the ranks, in accordance with the general principle of multiplicativity of the symmetry constants (but the ranks persist, before taking the limit N, M →∞, as lower-order terms).

Minimizing Neumann fundamental tones of triangles: an optimal Poincare inequality

Abstract: The first nonzero eigenvalue of the Neumann Laplacian is shown to be minimal for the degenerate acute isosceles triangle, among all triangles of given diameter. Hence an optimal Poincar\'{e} inequality for triangles is derived. The proof relies on symmetry of the Neumann fundamental mode for isosceles triangles with aperture less than $\pi/3$. Antisymmetry is proved for apertures greater than $\pi/3$.

Model of leptons from $SO(3) \to A_4$

Abstract: The lepton sector masses and mixing angles can be explained in models based on $A_4$ symmetry. $A_4$ is a non-Abelian discrete group. Therefore, one issue in constructing models based on it is explaining the origin of $A_4$. A plausible mechanism is that $A_4$ is an unbroken subgroup of a continuous group that is broken spontaneously. We construct a model of leptons where the $A_4$ symmetry is obtained by spontaneous symmetry breaking of SO(3).

Non-Abelian discrete flavor symmetries from T-2/Z(N) orbifolds

Abstract: In [1] it was shown how the flavor symmetry A4 (or S4) can arise if the three fermion generations are taken to live on the fixed points of a specific 2-dimensional orbifold. The flavor symmetry is a remnant of the 6-dimensional Poincare symmetry, after it is broken down to the 4-dimensional Poincare symmetry through compactification via orbifolding. This raises the question if there are further non-abelian discrete symmetries that can arise in a similar setup. To this end, we generalize the discussion by considering all possible 2-dimensional orbifolds and the flavor symmetries that arise from them. The symmetries we obtain from these orbifolds are, in addition to S4 and A4, the groups D3, D4 and D6 D3 × Z2 which are all popular groups for flavored model building.

Curved glide-reflection symmetry detection

Abstract: We generalize reflection symmetry detection to a curved glide reflection symmetry detection problem. We propose a unifying, local feature based approach for curved glide reflection symmetry detection from real, unsegmented images, where the classic reflection symmetry becomes one of four special cases. Our method detects and groups statistically dominant local reflection axes in a 3D parameter space. A curved glid reflection symmetry axis is estimated by a set of contiguous local straight reflection axes. Experimental results of the proposed algorithm on 40 real world images demonstrate promising performance.

The role of Onofri type inequalities in the symmetry properties of extremals for Caffarelli-Kohn-Nirenberg inequalities, in two space dimensions

Abstract: We first discuss a class of inequalities of Onofri type depending on a parameter, in the two-dimensional Euclidean space. The inequality holds for radial functions if the parameter is larger than - 1. Without symmetry assumption, it holds if and only if the parameter is in the interval (- 1, 0]. The inequality gives us some insight on the symmetry breaking phenomenon for the extremal functions of the Caffarelli-Kohn-Nirenberg inequality, in two space dimensions. In fact, for suitable sets of parameters (asymptotically sharp) we prove symmetry or symmetry breaking by means of a blow-up method and a careful analysis of the convergence to a solution of a Lionville equation. In this way, the Onofri inequality appears as a limit case. of the Caffarelli-Kohn-Nirenberg inequality.

Detection of symmetry and repetition in one and two objects. Structures versus strategies.

Abstract: Symmetry is usually easier to detect within a single object than in two objects (one-object advantage), while the reverse is true for repetition (two-objects advantage). This interaction between regularity and number of objects could reflect an intrinsic property of encoding spatial relations within and across objects or it could reflect a matching strategy. To test this, regularities between two contours (belonging to a single object or two objects) had to be detected in two experiments. Projected three-dimensional (3-D) objects rotated in depth were used to disambiguate figure-ground segmentation and to make matching based on simple translations of the two-dimensional (2-D) contours unlikely. Experiment 1 showed the expected interaction between regularity and number of objects. Experiment 2 used two-objects displays only and prevented a matching strategy by also switching the positions of the two objects. Nevertheless, symmetry was never detected more easily than repetition in these two-objects displays. We conclude that structural coding, not matching strategies, underlies the one-object advantage for symmetry and the two-objects advantage for repetition.

Matrix Extension with Symmetry and Applications to Symmetric Orthonormal Complex M -wavelets

Abstract: Matrix extension with symmetry is to find a unitary square matrix P of 2π-periodic trigonometric polynomials with symmetry such that the first row of P is a given row vector p of 2π-periodic trigonometric polynomials with symmetry satisfying \(\mathbf {p}\overline{\mathbf{p}}^{T}=1\) . Matrix extension plays a fundamental role in many areas such as electronic engineering, system sciences, wavelet analysis, and applied mathematics. In this paper, we shall solve matrix extension with symmetry by developing a step-by-step simple algorithm to derive a desired square matrix P from a given row vector p of 2π-periodic trigonometric polynomials with complex coefficients and symmetry. As an application of our algorithm for matrix extension with symmetry, for any dilation factor M, we shall present two families of compactly supported symmetric orthonormal complex M-wavelets with arbitrarily high vanishing moments. Wavelets in the first family have the shortest possible supports with respect to their orders of vanishing moments; their existence relies on the establishment of nonnegativity on the real line of certain associated polynomials. Wavelets in the second family have increasing orders of linear-phase moments and vanishing moments, which are desirable properties in numerical algorithms.

Symmetry in chaos : a search for pattern in mathematics, art, and nature

Abstract: 1. Introduction to symmetry and chaos 2. Planar symmetries 3. Patterns everywhere 4. Chaos and symmetry creation 5. Symmetric icons 6. Quilts 7. Symmetric fractals Appendix A. Picture parameters Appendix B. Icon mappings Appendix C. Planar lattices Bibliography Index.

A generalized quantum nonlinear oscillator

Abstract: We examine various generalizations, e.g. exactly solvable, quasi-exactly solvable and non-Hermitian variants, of a quantum nonlinear oscillator. For all these cases, the same mass function has been used and it has also been shown that the new exactly solvable potentials possess shape invariance symmetry. The solutions are obtained in terms of classical orthogonal polynomials.

A PDE viewpoint on basic properties of coordination algorithms with symmetries

Abstract: Several recent control applications consider the coordination of subsystems through local interaction. Often the interaction has a symmetry in state space, e.g. invariance with respect to a uniform translation of all subsystem values. The present paper shows that in presence of such symmetry, fundamental properties can be highlighted by viewing the distributed system as the discrete approximation of a partial differential equation. An important fact is that the symmetry on the state space differs from the popular spatial invariance property, which is not necessary for the present results. The relevance of the viewpoint is illustrated on two examples: (i) ill-conditioning of interaction matrices in coordination/consensus problems and (ii) the string instability issue.

Projective equivalence of Einstein spaces in general relativity

Abstract: There has been some recent interest in the relation between two spacetimes which have the same geodesic paths, that is, spacetimes which are projectively equivalent (sometimes called geodesically equivalent). This paper presents a short and accessible proof of the theorem that if two spacetimes have the same geodesic paths and one of them is an Einstein space then (either each is of constant curvature or) their Levi-Civita connections are identical. It also clarifies the relationship between their associated metrics. The results are extended to include the signatures (+ + + +) and (− − + +), and some examples and discussion are given in the case of dimension n > 4. Some remarks are also made which show how these results may be useful in the study of projective symmetry.

Elliptic PDEs with fibered nonlinearities

Abstract: In ℝ m ×ℝ n−m , endowed with coordinates x=(x′,x″), we consider bounded solutions of the PDE $$\Delta u(x)=f(u(x))\chi(x').$$ We prove a geometric inequality, from which a symmetry result follows.

Geometry and Symmetries in Coordination Control

Abstract: The present dissertation studies specific issues related to the coordination of a set of “agents” evolving on a nonlinear manifold, more particularly a homogeneous manifold or a Lie group. The viewpoint is somewhere between control algorithm design and system analysis, as algorithms are derived from simple principles — often retrieving existing models — to highlight specific behaviors. With a fair amount of approximation, the objective of the dissertation can be summarized by the following question: Given a swarm of identical agents evolving on a nonlinear, nonconvex configuration space with high symmetry, how can you define specific collective behavior, and how can you design individual agent control laws to get a collective behavior, without introducing hierarchy nor external reference points that would break the symmetry of the configuration space? Maintaining the basic symmetries of the coordination problem lies at the heart of the contributions. The main focus is on the global geometric invariance of the configuration space. This contrasts with most existing work on coordination, where either the agents evolve on vector spaces — which, to some extent, can cover local behavior on manifolds — or coordination is coupled to external reference tracking such that the reference can serve as a beacon around which the geometry is distorted towards vector space-like properties. A second, more standard symmetry is to treat all agents identically. Another basic ingredient of the coordination problem that has important implications in this dissertation is the reduced agent interconnectivity: each agent only gets information from a limited set of other agents, which can be varying. In order to focus on issues related to geometry / symmetry and reduced interconnectivity, individual agent dynamics are drastically simplified to simple integrators. This is justified at a “planning” level. Making the step towards realistic dynamics is illustrated for the specific case of rigid body attitude synchronization. The main contributions of this dissertation are I. an extensive study of synchronization on the circle, (a) highlighting difficulties encountered for coordination and (b) proposing simple strategies to overcome these difficulties; II. (a) a geometric definition and related control law for “consensus” configurations on compact homogeneous manifolds, of which synchronization — all agents at the same point — is a special case, and (b) control laws to (almost) globally reach synchronization and “balancing”, its opposite, under general interconnectivity conditions; III. several propositions for rigid body attitude synchronization under mechanical dynamics; IV. a geometric framework for “coordinated motion” on Lie groups, (a) giving a geometric definition of coordinated motion and investigating its implications, and (b) providing systematic methods to design control laws for coordinated motion. Examples treated for illustration of the theoretical concepts are the circle S1 (sometimes the sphere Sn), the rotation group SO(n), the rigid-body motion groups SE(2) and SE(3) and the Grassmann manifolds Grass(p, n). The developments in this dissertation remain at a rather theoretical level; potential applications are briefly discussed.

Deriving projective hyperspace from harmonic

Abstract: We derive actions for projective N=2 superspace ('hyperspace') from those for harmonic hyperspace, including that for non-Abelian Yang-Mills theory (a new result). The method uses Wick rotation of the sphere from complex conjugate coordinates to real, null ones, which can be treated as independent. The result can be considered 'holographic' in that the dimension of the internal (R symmetry) space is reduced from 2 to 1, by solving equations of motion or gauge conditions for dependence on the other coordinate. The auxiliary nature of the redundant dimension makes the hypergraph rules and evaluation almost identical.

A geometric newton method for oja's vector field

Abstract: Newton's method for solving the matrix equation runs up against the fact that its zeros are not isolated. This is due to a symmetry of F by the action of the orthogonal group. We show how differential-geometric techniques can be exploited to remove this symmetry and obtain a “geometric” Newton algorithm that finds the zeros of F. The geometric Newton method does not suffer from the degeneracy issue that stands in the way of the original Newton method.

Prediction of human eye fixations using symmetry

Abstract: Humans are very sensitive to symmetry in visual patterns. Reaction time experiments show that symmetry is detected and recognized very rapidly. 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. In this paper, we discuss local symmetry as a measure of saliency. We propose a number of symmetry models and perform 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, Koch and Niebur (1998). The results show that the symmetry models better match the human data than the contrast model, which indicates that symmetry can be regarded as a salient feature.

A Real-Time Road Sign Detection Using Bilateral Chinese Transform

Abstract: We present a real-time approach for circular and polygonal road signs detection in still images, regardless of their pose and orientation. Object detection is done using a pairwise gradient-based symmetry transform able to detect circles and polygons indistinctly. This symmetry transform of gradient orientation, the so-called Bilateral Chinese Transform BCT, decomposes an object into a set of parallel contours with opposite gradients, and models this gradient field symmetry using an accumulation of radial symmetry evidences. On a test database of 89 images 640x480 containing 92 road signs, 79 are correctly detected (86%) with 25 false positives using the BCT approach in about 30 ms/image.

Projective reduction of the discrete Painlev\'e system of type $(A_2+A_1)^{(1)}$

Abstract: We consider the q-Painlev\'e III equation arising from the birational representation of the affine Weyl group of type $(A_2 + A_1)^{(1)}$. We study the reduction of the q-Painlev\'e III equation to the q-Painlev\'e II equation from the viewpoint of affine Weyl group symmetry. In particular, the mechanism of apparent inconsistency between the hypergeometric solutions to both equations is clarified by using factorization of difference operators and the $\tau$ functions.

The stationary Navier–Stokes equation on the whole plane with external force with antisymmetry

Abstract: This paper is concerned with the stationary Navier–Stokes equation in the two-dimensional whole plane with external force given by a potential with some symmetry, and gives a condition on the potential sufficient for the existence of a solution of the problem above. This paper also proves the uniqueness of the solution small in appropriate function spaces.

Twisted quantum fields on moyal and wick–voros planes are inequivalent

Abstract: The Moyal and Wick–Voros planes $\mathcal{A}_{\theta}^{\mathcal{M}},V$ are *-isomorphic. On each of these planes the Poincare group acts as a Hopf algebra symmetry if its coproducts are deformed by twist factors $F_{\theta}^{\mathcal{M},V}$. We show that the *-isomorphism ${\rm T}$: $\mathcal{A}_{\theta}^{\mathcal{M}}to\mathcal{A}_{\theta}$ also does not map the corresponding twists of the Poincare group algebra. The quantum field theories on these planes with twisted Poincare–Hopf symmetries are thus inequivalent. We explicitly verify this result by showing that a nontrivial dependence on the noncommutative parameter is present for the Wick–Voros plane in a self-energy diagram whereas it is known to be absent on the Moyal plane (in the absence of gauge fields).1–3 Our results differ from those of Ref. 4 because of differences in the treatments of quantum field theories.

On implementing symmetry detection

Abstract: Automatic symmetry detection has received a significant amount of interest, which has resulted in a large number of proposed methods. This paper reports on our experiences while implementing the approach of Puget (CP2005, LNCS, vol. 3709, pp. 475---489. Springer, 2005). In particular, it proposes a modification to the approach to deal with general expressions, discusses the insights gained, and gives the results of an experimental evaluation of the accuracy and efficiency of the approach.

Efficient image matching

Abstract: A system described herein includes a receiver component that receives a first image and a symmetry signature generator component that generates a first global symmetry signature for the image, wherein the global symmetry signature is representative of symmetry existent in the first image. The system also includes a comparer component that compares the first global symmetry signature with a second global symmetry signature that corresponds to a second image, wherein the second global symmetry signature is representative of symmetry existent in the second image. The system additionally includes an output component that outputs an indication of similarity between the first image and the second image based at least in part upon the comparison undertaken by the comparer component.