Showing papers on "Symmetrization published in 2006"

Partial and approximate symmetry detection for 3D geometry

Abstract: "Symmetry is a complexity-reducing concept [...]; seek it every-where." - Alan J. PerlisMany natural and man-made objects exhibit significant symmetries or contain repeated substructures. This paper presents a new algorithm that processes geometric models and efficiently discovers and extracts a compact representation of their Euclidean symmetries. These symmetries can be partial, approximate, or both. The method is based on matching simple local shape signatures in pairs and using these matches to accumulate evidence for symmetries in an appropriate transformation space. A clustering stage extracts potential significant symmetries of the object, followed by a verification step. Based on a statistical sampling analysis, we provide theoretical guarantees on the success rate of our algorithm. The extracted symmetry graph representation captures important high-level information about the structure of a geometric model which in turn enables a large set of further processing operations, including shape compression, segmentation, consistent editing, symmetrization, indexing for retrieval, etc.

A symmetric theory of electrons and positrons

Abstract: It is shown that it is possible to achieve complete formal symmetrization in the electron and proton quantum theory by means of a new quantization process. The meaning of Dirac equations is somewhat modified and there is no longer any reason to speak of negative-energy states nor to assume, for any other types of particles, especially neutral ones, the existence of antiparticles, corresponding to the “holes” of negative energy.

Symmetry-adapted Localized Wannier Functions Suitable for Periodic Local Correlation Methods

Abstract: A scheme for the a posteriori symmetrization of a set of localized Wannier functions is presented and illustrated with some examples. The method has been implemented in the periodic code CRYSTAL using the LCAO approach and it is shown that they can be useful in the computational implementation of periodic local correlation methods. The resulting functions feature very accurate symmetry correspondences without a significant loss of the original localization properties.

Transitions between symmetric and asymmetric solitons in dual-core systems with cubic-quintic nonlinearity

Abstract: It is well known that a symmetric soliton in coupled nonlinear Schroedinger (NLS) equations with the cubic nonlinearity loses its stability with the increase of its energy, featuring a transition into an asymmetric soliton via a subcritical bifurcation. A similar phenomenon was found in a dual-core system with quadratic nonlinearity, and in linearly coupled fiber Bragg gratings, with a difference that the symmetry-breaking bifurcation is supercritical in those cases. We aim to study transitions between symmetric and asymmetric solitons in dual-core systems with saturable nonlinearity. We demonstrate that a basic model of this type, viz., a pair of linearly coupled NLS equations with the intra-core cubic-quintic (CQ) nonlinearity, features a bifurcation loop: a symmetric soliton loses its stability via a supercritical bifurcation, which is followed, at a larger value of the energy, by a reverse bifurcation that restores the stability of the symmetric soliton. If the linear-coupling constant is small enough, the second bifurcation is subcritical, and there is a broad interval of energies in which the system is bistable, with coexisting stable symmetric and asymmetric solitons. At larger values of the coupling constant, the reverse bifurcation is supercritical, and the bifurcation loop disappears if the linear coupling is very strong. Collisions between moving solitons are studied too. Symmetric solitons always collide elastically, while collisions between asymmetric solitons turns them into breathers, that subsequently undergo dynamical symmetrization.

Symmetrization inequalities and Sobolev embeddings

Abstract: We prove new extended forms of the Polya-Szego symmetrization principle. As a consequence new sharp embedding theorems for generalized Besov spaces are proved, including a sharpening of the limiting cases of the classical Sobolev embedding theorem. In particular, a surprising self-improving property of certain Sobolev embeddings is uncovered.

The random paving property for uniformly bounded matrices

Abstract: This note presents a new proof of an important result due to Bourgain and Tzafriri that provides a partial solution to the Kadison--Singer problem. The result shows that every unit-norm matrix whose entries are relatively small in comparison with its dimension can be paved by a partition of constant size. That is, the coordinates can be partitioned into a constant number of blocks so that the restriction of the matrix to each block of coordinates has norm less than one half. The original proof of Bourgain and Tzafriri involves a long, delicate calculation. The new proof relies on the systematic use of symmetrization and (noncommutative) Khintchine inequalities to estimate the norms of some random matrices.

Approximation of symmetrizations and symmetry of critical points

Abstract: We give a sufficient condition in order that a sequence of cap or Steiner symmetrizations or of polarizations approximates some fixed cap or Steiner symmetrization. This condition is used to obtain the almost sure convergence for random sequences of symmetrization taken in an appropriate set. The results are applicable to the symmetrization of sets. An application is given to the study of the symmetry of critical points obtained by minimax methods based on the Krasnosel'skiĭ genus.

Schur-convexity of the complete symmetric function

Abstract: This paper investigates Schur-convexity of the complete symmetric function cr(x) = ∑ i1+...+in=r x i1 1 ...x in n and the function φr(x) = cr(x) cr−1(x) , where i1, ..., in are non-negative integers and r 1. Some inequalities, including Ky Fan type inequality, are established by use of the theory of majorization. It is also concerned with an open problem proposed by Menon [1]. Mathematics subject classification (2000): 05E05, 26E60, 26D20.

Symmetrization results for classes of nonlinear elliptic equations with q-growth in the gradient

Abstract: In this paper we study the Dirichlet problem for a class of nonlinear elliptic equations in the form A ( u ) = H ( x , u , Du ) + g ( x , u ) , where the principal term is a Leray–Lions operator defined on W 0 1 , p ( Ω ) . Comparison results are obtained between the rearrangement of a solution u of Dirichlet problem quoted above and the rearrangement of the solution of a problem whose data are radially symmetric.

A weighted symmetrization for nonlinear elliptic and parabolic equations in inhomogeneous media

Abstract: In the theory of elliptic equations, the technique of Schwarz symmetrization is one of the tools used to obtain a priori bounds for classical and weak solutions in terms of general information on the data. A basic result says that, in the absence of the zero-order term, the symmetric rearrangement of the solution u of an elliptic equation, that we write u ⁄ , can be compared pointwise with the solution of the symmetrized problem. The main question we address here is the modiflcation of the method to take into account degenerate equations posed in inhomogeneous media. Moreover, the equations we want to deal with involve weights that make them non-divergent, at least when written in terms of the natural variables. We flnd comparison results covering the elliptic case and the corresponding evolution models of parabolic type, with attention to equations of porous medium type. More speciflcally, we obtain a priori bounds and decay estimates for wide classes of solutions of those equations.

Minimal rearrangements, strict convexity and critical points

Abstract: An asymmetry estimate for extremal functions in Polya–Szego type inequalities for Dirichlet integrals is established in presence of critical points and nonstrictly convex integrands.

A blowup analysis of the mean field equation for arbitrarily signed vortices

Abstract: We study the noncompact solution sequences to the mean field equation for arbitrarily signed vortices and observe the quantization of the mass of concentration, using the rescaling argument.

On the isoperimetric problem in Euclidean space with density

Abstract: We study the isoperimetric problem for Euclidean space endowed with a continuous density. In dimension one, we characterize isoperimetric regions for a unimodal density. In higher dimensions, we prove existence results and we derive stability conditions, which lead to the conjecture that for a radial log-convex density, balls about the origin are isoperimetric regions. Finally, we prove this conjecture and the uniqueness of minimizers for the density $\exp (|x|^2)$ by using symmetrization techniques.

Symmetric energy-momentum tensor in Maxwell, Yang-Mills, and Proca theories obtained using only Noether's theorem

Abstract: The symmetric and gauge-invariant energy-momentum tensors for source-free Maxwell and Yang-Mills theories are obtained by means of translations in spacetime via a systematic implementation of Noether's theorem. For the source-free neutral Proca field, the same procedure yields also the symmetric energy-momentum tensor. In all cases, the key point to get the right expressions for the energy-momentum tensors is the appropriate handling of their equations of motion and the Bianchi identities. It must be stressed that these results are obtained without using Belinfante's symmetrization techniques which are usually employed to this end.

On symmetric functionals of the gradient having symmetric equidistributed minimizers

Abstract: Within a general family of functionals, not necessarily of integral type, depending on the modulus of the gradient, we characterize those possessing only spherically symmetric minimizers in classes of Sobolev functions with level sets of prescribed Lebesgue measures.

FEM Computation of Magnetic Fields in Anisotropic Magnetic Materials

Abstract: The magnetic fields in nonlinear anisotropic magnetic materials were analyzed by using the Finite Element Method (FEM). The measured data was directly used in the computation without a complicateded smoothing. The resultant asymmetric linear equations were solved by using the ILUBiCGStab method without symmetrization or the ICCG method with symmetrization. The magnetic flux distributions in a ring core model showed the characteristic patterns according to the non-oriented, grain-oriented and doubly-oriented magnetic properties. The good convergence of the Newton-Raphson nonlinear iteration was attained by the iterative solvers without special techniques for the smoothing.

Hyperspherical harmonic formalism for tetraquarks

Abstract: We present a generalization of the hyperspherical harmonic formalism to study systems made of quarks and antiquarks of the same flavor. This generalization is based on the symmetrization of the $N-$body wave function with respect to the symmetric group using the Barnea and Novoselsky algorithm. Our analysis shows that four-quark systems with non-exotic $2^{++}$ quantum numbers may be bound independently of the quark mass. $0^{+-}$ and $1^{+-}$ states become attractive only for larger quarks masses.

Tail Bounds for the Suprema of Empirical Processes over Unbounded Classes of Functions

Abstract: So far the study of exponential bounds of an empirical process has been restricted to a bounded index class of functions. The case of an unbounded index class of functions is now studied on the basis of a new symmetrization idea and a new method of truncating the original probability space; the exponential bounds of the tail probabilities for the supremum of the empirical process over an unbounded class of functions are obtained. The exponential bounds can be used to establish laws of the logarithm for the empirical processes over unbounded classes of functions.

On the Volume Formulas of Cones and Orthogonal Multi-cones in S n (1) and H n (—1)

Abstract: In the study of n-dimensional spherical or hyperbolic geometry, n≥ 3, the volume of various objects such as simplexes, convex polytopes, etc. often becomes rather difficult to deal with. In this paper, we use the method of infinitesimal symmetrization to provide a systematic way of obtaining volume formulas of cones and orthogonal multiple cones in S n (1) and H n (—1).

Inclusion of inversion symmetry in centroid molecular dynamics: A possible avenue to recover quantum coherence

Abstract: Inversion symmetry is included in the operator formulation of the centroid molecular dynamics (CMD). This work involves the development of a symmetry-adapted CMD (SA-CMD), here particularly for symmetrization and antisymmetrization projections. A symmetry-adapted quasidensity operator, as defined by Blinov and Roy [J. Chem. Phys. 115, 7822 (2001)], is employed to obtain the centroid representation of quantum mechanical operators. Numerical examples are given for a single particle confined to one-dimensional symmetric quartic and symmetric double-well potentials. Two SA-CMD simulations are performed separately for both projections, and centroid position autocorrelation functions are obtained. For each projection, the quality of the approximation as well as the accuracy are similar to those of regular CMD. It is shown that individual trajectories from two separate SA-CMD simulations can be properly combined to recover trajectories for Boltzmann statistics. Position autocorrelation functions are compared to the exact quantum mechanical ones. This explicit account of inversion symmetry provides a qualitative improvement on the conventional CMD approach and allows the recovery of some quantum coherence.

A minimal area problem for nonvanishing functions

Abstract: The minimal area covered by the image of the unit disk is found for non- vanishing univalent functions normalized by the conditions f (0) = 1, f(0) = α .T wo different approaches are discussed, each of which contributes to the complete solution of the problem. The first approach reduces the problem, via symmetrization, to the class of typically real functions, where the well-known integral representation can be employed to obtain the solution upon ap rioriknowledge of the extremal function. The second approach, requiring smoothness assumptions, leads, via some variational formulas, to a boundary value problem for analytic functions, which admits an ex- plicit solution.

Symmetrization of brace algebras

Abstract: We show that the symmetrization of a brace algebra structure yields the structure of a symmetric brace algebra. We also show that the symmetrization of the natural brace structure on L k≥1 Hom(V ⊗k, V ) coincides with the natural symmetric brace structure on L k≥1 Hom(V ⊗k, V )as, the space of antisymmetric maps V ⊗K → V .

Geometric symmetry in the quadratic fisher discriminant operating on image pixels

Abstract: This correspondence examines the design of Quadratic Fisher Discriminants (QFDs) that operate directly on image pixels, when image ensembles are taken to comprise all rotated and reflected versions of distinct sample images. A procedure based on group theory is devised to identify and discard QFD coefficients made redundant by symmetry, for arbitrary sampling lattices. This procedure introduces the concept of a degeneracy matrix. Tensor representations are established for the square lattice point group (8-fold symmetry) and hexagonal lattice point group (12-fold symmetry). The analysis is largely applicable to the symmetrization of any quadratic filter, and generalizes to higher order polynomial (Volterra) filters. Experiments on square lattice sampled synthetic aperture radar (SAR) imagery verify that symmetrization of QFDs can improve their generalization and discrimination ability

The number-phase wigner function in the extended fock space

Abstract: It is shown that the number-phase (or rotational) Wigner function is obtained from the Weyl symmetrization rule for the correspondence between classical functions and quantum operators. In spite of the complicated form of the commutator for the number and phase operators, the Weyl symmetrization rule is similar to that in the case of the position and momentum operators. In addition, it is found that the ordering of the number and phase operators also has the same structure as that for the position-momentum pair.

Denotation and application for d_j-Map of symmetric function.

Abstract: Symmetry is a significant property of a logic function.However,the conventional methods of detecting symmetry of a logic function are too complicated if the function is expanded to CRM in OR-COINCIDENCE algebraic system.According to the property of symmetric function,the denotation about d_j map of partially symmetric function and totally symmetric function was presented based on the K-map of symmetric function and b_j map of symmetric function.Symmetry of the function can be detected by means of d_j map of symmetric function,and some examples were given.The method has several advantages that the logic design based on the symmetry of logic function is simpler and more effective as compared with conventional design.