Showing papers on "Antisymmetric relation published in 2011"

Stability of solitons in parity-time-symmetric couplers

Abstract: Families of analytical solutions are found for symmetric and antisymmetric solitons in a dual-core system with Kerr nonlinearity and parity-time (PT)-balanced gain and loss. The crucial issue is stability of the solitons. A stability region is obtained in an analytical form, and verified by simulations, for the PT-symmetric solitons. For the antisymmetric ones, the stability border is found in a numerical form. Moving solitons of both types collide elastically. The two soliton species merge into one in the “supersymmetric” case, with equal coefficients of gain, loss, and intercore coupling. These solitons feature a subexponential instability, which may be suppressed by periodic switching (“management”).

Stability of solitons in PT-symmetric couplers

Abstract: Families of analytical solutions are found for symmetric and antisymmetric solitons in the dual-core system with the Kerr nonlinearity and PT-balanced gain and loss. The crucial issue is stability of the solitons. A stability region is obtained in an analytical form, and verified by simulations, for the PT-symmetric solitons. For the antisymmetric ones, the stability border is found in a numerical form. Moving solitons of both types collide elastically. The two soliton species merge into one in the "supersymmetric" case, with equal coefficients of the gain, loss and inter-core coupling. These solitons feature a subexponential instability, which may be suppressed by periodic switching ("management").

Superpolynomials for toric knots from evolution induced by cut-and-join operators

Abstract: The colored HOMFLY polynomials, which describe Wilson loop averages in Chern-Simons theory, possess an especially simple representation for torus knots, which begins from quantum R-matrix and ends up with a trivially-looking split W representation familiar from character calculus applications to matrix models and Hurwitz theory. Substitution of MacDonald polynomials for characters in these formulas provides a very simple description of "superpolynomials", much simpler than the recently studied alternative which deforms relation to the WZNW theory and explicitly involves the Littlewood-Richardson coefficients. A lot of explicit expressions are presented for different representations (Young diagrams), many of them new. In particular, we provide the superpolynomial P_[1]^[m,km\pm 1] for arbitrary m and k. The procedure is not restricted to the fundamental (all antisymmetric) representations and the torus knots, still in these cases some subtleties persist.

Focusing of the lowest antisymmetric Lamb wave in a gradient-index phononic crystal plate

Abstract: In this letter, we numerically demonstrate focusing of the lowest antisymmetric Lamb wave in a gradient-index phononic crystal (PC) silicon plate and its application as a beam-width compressor for compressing Lamb wave into a stubbed phononic tungsten/silicon plate waveguide. The results show that beam width of the lowest antisymmetric Lamb wave in the PC thin plate can be compressed efficiently and fitted into tungsten/silicon PC plate waveguide over a wide range of frequency.

On the Efficiency of Algorithms for Solving Hartree-Fock and Kohn-Sham Response Equations.

Abstract: The response equations as occurring in the Hartree-Fock, multiconfigurational self-consistent field, and Kohn-Sham density functional theory have identical matrix structures. The algorithms that are used for solving these equations are discussed, and new algorithms are proposed where trial vectors are split into symmetric and antisymmetric components. Numerical examples are given to compare the performance of the algorithms. The calculations show that the standard response equation for frequencies smaller than the highest occupied molecular orbital-lowest unoccupied molecular orbital gap is best solved using the preconditioned conjugate gradient or conjugate residual algorithms where trial vectors are split into symmetric and antisymmetric components. For larger frequencies in the standard response equation as well as in the damped response equation in general, the preconditioned iterative subspace approach with symmetrized trial vectors should be used. For the response eigenvalue equation, the Davidson algorithm with either paired or symmetrized trial vectors constitutes equally good options.

A ten-dimensional action for non-geometric fluxes

Abstract: The NSNS Lagrangian of ten-dimensional supergravity is rewritten via a change of field variables inspired by Generalized Complex Geometry. We obtain a new metric and dilaton, together with an antisymmetric bivector field which leads to a ten-dimensional version of the non-geometric Q-flux. Given the involved global aspects of non-geometric situations, we prescribe to use this new Lagrangian, whose associated action is well-defined in some examples investigated here. This allows us to perform a standard dimensional reduction and to recover the usual contribution of the Q-flux to the four-dimensional scalar potential. An extension of this work to include the R-flux is discussed. The paper also contains a brief review on non-geometry.

Spontaneous symmetry breaking in Schrödinger lattices with two nonlinear sites

Abstract: We introduce discrete systems in the form of straight (infinite) and ring-shaped chains, with two symmetrically placed nonlinear sites. The systems can be implemented in nonlinear optics (as waveguiding arrays) and Bose-Einstein condensates (by means of an optical lattice). A full set of exact analytical solutions for symmetric, asymmetric, and antisymmetric localized modes is found, and their stability is investigated in a numerical form. The symmetry-breaking bifurcation, through which the asymmetric modes emerge from the symmetric ones, is found to be of the subcritical type. It is transformed into a supercritical bifurcation if the nonlinearity is localized in relatively broad domains around two central sites, and also in the ring of a small size, i.e., in effectively nonlocal settings. The family of antisymmetric modes does not undergo bifurcations and features both stable and unstable portions. The evolution of unstable localized modes is investigated by means of direct simulations. In particular, unstable asymmetric states, which exist in the case of the subcritical bifurcation, give rise to breathers oscillating between the nonlinear sites, thus restoring an effective dynamical symmetry between them.

Dispersion relation for surface plasmon polaritons in metal/nonlinear-dielectric/metal slot waveguides

Abstract: We present the first (to our knowledge) exact dispersion relation for the transverse-magnetic surface plasmon polariton (SPP) modes of a plasmonic slot waveguide, which is formed by a nonlinear Kerr medium sandwiched between two metallic slabs The obtained relation is then simplified to the case of small field intensities, while retaining nonlinear terms, to derive approximate dispersion equations for the symmetric and antisymmetric SPP modes

Generalized Nonlinear Complementary Attitude Filter

Abstract: A = matrices a = true value ~ A = Â T A ; left multiply error â = estimated value a = measured value ~ a = error between estimated and true values hai = time-averaged expectation value A;B , fA;Bg = commutator or anticommutator of A and B A = equilibrium point AI = CAC ; matrix in inertial frame Ps A , Pa A = symmetric or antisymmetric projection of A x = vectors x = matrix equivalent form of cross product xI = Cx; vector in inertial frame

High-sensitivity charge detection using antisymmetric vibration in coupled micromechanical oscillators

Abstract: High-sensitivity charge detection using antisymmetric vibration in two coupled GaAs oscillators is demonstrated. The antisymmetric mode under in-phase simultaneous driving of the two oscillators disappears with perfect frequency tuning. The piezoelectric stress induced by a small gate-voltage modulation breaks the balance of the two oscillators, leading to the re-emergence of the antisymmetric mode. Measurement of the amplitude change enables detection of the applied voltage or, equivalently, added charges. In contrast to the frequency-shift detection using a single oscillator, our method allows a large readout up to the strongly driven nonlinear response regime, providing the high room-temperature sensitivity of 147 e/Hz0.5.

Symmetry breaking of solitons in two-component Gross-Pitaevskii equations

Abstract: We revisit the problem of the spontaneous symmetry breaking (SSB) of solitons in two-component linearly coupled nonlinear systems, adding the nonlinear interaction between the components. With this feature, the system may be realized in new physical settings, in terms of optics and the Bose-Einstein condensate (BEC). SSB bifurcation points are found analytically, for both symmetric and antisymmetric solitons (the symmetry between the two components is meant here). Asymmetric solitons, generated by the bifurcations, are described by means of the variational approximation (VA) and numerical methods, demonstrating good accuracy of the variational results. In the space of the self-phase-modulation (SPM) parameter and soliton's norm, a border separating stable symmetric and asymmetric solitons is identified. The nonlinear coupling may change the character of the SSB bifurcation, from subcritical to supercritical. Collisions between moving asymmetric and symmetric solitons are investigated too. Antisymmetric solitons are destabilized by a supercritical bifurcation, which gives rise to self-confined modes featuring Josephson oscillations, instead of stationary states with broken antisymmetry. An additional instability against delocalized perturbations is also found for the antisymmetric solitons.

Multistable dissipative structures pinned to dual hot spots

Abstract: We analyze the formation of one-dimensional localized patterns in a nonlinear dissipative medium including a set of two narrow ``hot spots'' (HSs), which carry the linear gain, local potential, cubic self-interaction, and cubic loss, while the linear loss acts in the host medium. This system can be realized as a spatial-domain one in optics and also in Bose-Einstein condensates of quasiparticles in solid-state settings. Recently, exact solutions were found for localized modes pinned to the single HS represented by the \ensuremath{\delta} function. The present paper reports analytical and numerical solutions for coexisting two- and multipeak modes, which may be symmetric or antisymmetric with respect to the underlying HS pair. Stability of the modes is explored through simulations of their perturbed evolution. The sign of the cubic nonlinearity plays a crucial role: in the case of the self-focusing, only the fundamental symmetric and antisymmetric modes, with two local peaks tacked to the HSs, and no additional peaks between them, may be stable. In this case, all the higher-order multipeak modes, being unstable, evolve into the fundamental ones. Stability regions for the fundamental modes are reported. A more interesting situation is found in the case of the self-defocusing cubic nonlinearity, with the HS pair giving rise to a multistability, with up to eight coexisting stable multipeak patterns, symmetric and antisymmetric ones. The system without the self-interaction, the nonlinearity being represented only by the local cubic loss, is investigated too. This case is similar to those with the self-focusing or defocusing nonlinearity, if the linear potential of the HS is, respectively, attractive or repulsive. An additional feature of the former setting is the coexistence of the stable fundamental modes with robust breathers.

Schunck classes of soluble Leibniz algebras

Abstract: I set out the theory of Schunck classes and projectors for soluble Leibniz algebras, parallel to that for Lie algebras. Primitive Leibniz algebras come in pairs, one (Lie) symmetric, the other antisymmetric. A Schunck formation containing one member of a pair also contains the other. Projectors for a Schunck formation are intravariant.

Time evolution of decay of two identical quantum particles

Abstract: An analytical solution for the time evolution of decay of two identical noninteracting quantum particles seated initially within a potential of finite range is derived using the formalism of resonant states. It is shown that the wave function, and hence also the survival and nonescape probabilities, for factorized symmetric and entangled symmetric or antisymmetric initial states evolve in a distinctive form along the exponentially decaying and nonexponential regimes. Our findings show the influence of the Pauli exclusion principle on decay. We exemplify our results by solving exactly the $s$-wave $\ensuremath{\delta}$ shell potential model.

Polyakov loop of antisymmetric representations as a quantum impurity model

Abstract: The Polyakov loop of an operator in the anti-symmetric representation inN = 4 SYM theory on spacial R 3 is calculated, to leading order in 1=N and at large ’t Hooft coupling, by solving the saddle point equations of the corresponding quantum impurity model. Agreement is found with previous results from the supergravity dual, which is given by a D5-brane in an asymptotically AdS5 S 5 black brane background. It is shown that the azimuth angle, at which the dual D5brane wraps the S 5 , is related to the spectral asymmetry angle in the spectral density associated with the Green’s function of the impurity fermions. Much of the calculation also applies to the Polyakov loop on spacial S 3 or H 3 .

Towards extending the Chern-Simons couplings at order O ( α ′ 2 )

Abstract: Using the compatibility of the anomalous Chern-Simons couplings on D p -branes with the linear T-duality and with the antisymmetric B-field gauge transformations, some couplings have been recently found for C (p−3) at order O(α′2). We examine these couplings with the S-matrix element of one RR and two antisymmetric B-field vertex operators. We find that the S-matrix element reproduces these couplings as well as some other couplings. Each of them is invariant under the linear T-duality and the B-field gauge transformations.

Energy loss of an infinitely massive half-Bogomol'nyi-Prasad-Sommerfeld particle by radiation to all orders in 1/N.

Abstract: We use the AdS/CFT correspondence to compute the energy radiated by an infinitely massive half-Bogomol'nyi-Prasad-Sommerfeld particle charged under N=4 super Yang-Mills theory, transforming in the symmetric or antisymmetric representation of the gauge group, and moving in the vacuum, to all orders in 1/N and for large 't Hooft coupling. For the antisymmetric case we consider D5-branes reaching the boundary of five-dimensional anti-de Sitter space (AdS(5)) at arbitrary timelike trajectories, while for the symmetric case, we consider a D3-brane in AdS(5) that reaches the boundary at a hyperbola. We compare our results to the one obtained for the fundamental representation, deduced by considering a string in AdS(5).

Methyl groups at dielectric and metal surfaces studied by sum-frequency generation in co- and counter-propagating configurations

Abstract: The optimum experimental geometry for visible-infrared sum-frequency generation experiments depends rather sensitively on the molecules adsorbed at the surface, their orientation, and the nature of the adjacent bulk media. We consider the commonly encountered case of methyl groups situated at air–water, air–gold, and polymer–water interfaces. We provide expressions that may be used to determine the optimal visible and IR beam incident angles, considering the symmetric and antisymmetric modes separately and then together. The analysis is carried out for co-propagating (collinear and non-collinear geometries) and counter-propagating configurations. We first consider that one or more vibrational modes are of interest, and the goal is to study them quantitatively under a single polarization scheme; our results enable the user to set the beam angles for such an experiment. In the second case, molecular orientation information is desired, and so the calibrated response is required in all accessible polarization...

Large rank Wilson loops in N=2 superconformal QCD at strong coupling

Abstract: We compute the expectation values of circular Wilson loops in large representations at strong coupling, in the large-N limit of the N=2 superconformal theory with SU(N) gauge group and 2N hypermultiplets. Employing Pestun's matrix integral, we focus attention on symmetric and antisymmetric representations with ranks of order N. We find that large rank antisymmetric loops are independent of the coupling at strong 't Hooft coupling while symmetric Wilson loops grow exponentially with it. Symmetric loops display a non-analyticity as a function of the rank, characterized by the splitting of a single matrix model eigenvalue from the continuum, bearing close resemblance to Bose-Einstein condensation in an ideal gas. We discuss implications of these for a putative large-N string dual. The method of calculation we adopt makes explicit the connection to Fermi and Bose gas descriptions and also suggests a tantalizing connection of the above system to a multichannel Kondo model.

Hydroelastic waves on fluid sheets

Abstract: Nonlinear travelling waves on a two-dimensional inviscid fluid sheet are investigated when the sheet is bounded above and below by two thin elastic plates. Symmetric and antisymmetric solution branches are identified, together with a pair of bifurcation branches. It is shown that far along the branches the solutions approach limiting configurations that correspond to static solutions valid in the absence of fluid forcing.

Multi-stable dissipative structures pinned to dual hot spots

Abstract: We analyze the formation of one-dimensional localized patterns in a nonlinear dissipative medium including a set of two narrow "hot spots" (HSs), which carry the linear gain, local potential, cubic self-interaction, and cubic loss, while the linear loss acts in the host medium. This system can be realized, as a spatial-domain one, in optics, and also in Bose-Einstein condensates of quasi-particles in solid-state settings. Recently, exact solutions were found for localized modes pinned to the single HS represented by the delta-function. The present paper reports analytical and numerical solutions for coexisting two- and multi-peak modes, which may be symmetric or antisymmetric with respect to the underlying HS pair. Stability of the modes is explored through simulations of their perturbed evolution. The sign of the cubic nonlinearity plays a crucial role: in the case of the self-focusing, only the fundamental symmetric and antisymmetric modes, with two local peaks tacked to the HSs, and no additional peaks between them, may be stable. In this case, all the higher-order multi-peak modes, being unstable, evolve into the fundamental ones. Stability regions for the fundamental modes are reported. A more interesting situation is found in the case of the self-defocusing cubic nonlinearity, with the HS pair giving rise to a multi-stability, with up to eight coexisting stable multi-peak patterns, symmetric and antisymmetric ones. The system without the self-interaction, the nonlinearity 2 being represented only by the local cubic loss, is investigated too. This case is similar to those with the self-focusing or defocusing nonlinearity, if the linear potential of the HS is, respectively, attractive or repulsive. An additional feature of the former setting is the coexistence of the stable fundamental modes with robust breathers.

Angular Energy Quantization for Linear Elliptic Systems with Antisymmetric Potentials and Applications

Abstract: In the present work we establish a quantization result for the angular part of the energy of solu- tions to elliptic linear systems of Schrodinger type with antisymmetric potentials in two dimension. This quantization is a consequence of uniform Lorentz-Wente type estimates in degenerating annuli. We derive from this angular quantization the full energy quantization for general critical points to functionals which are conformally invariant or also for pseudo-holomorphic curves on degenerating Riemann surfaces.

Non-dissipative electromagnetic medium with a double light cone

Abstract: We study Maxwell's equations on a 4-manifold where the electromagnetic medium is modelled by an antisymmetric (2, 2)-tensor kappa with real coefficients In this setting the Fresnel surface is a fourth order polynomial surface in each cotangent space that acts as a generalisation of the light cone determined by a Lorentz metric; the Fresnel surface parameterises electromagnetic wave-speeds as a function of direction The contribution of this paper is the complete pointwise description of all electromagnetic medium tensors that satisfy the following conditions: (i) kappa is invertible, (ii) kappa is skewon-free, (iii) kappa is birefringent, that is, the Fresnel surface of kappa is the union of two distinct light cones We show that there are only three classes of mediums with these properties Moreover, we give explicit expressions in local coordinates for each class

Approximations of arbitrary binary relations by partial orders: classical and rough set models

Abstract: The problem of approximating an arbitrary binary relation by a partial order is formally defined and analysed. Five different partial order approximations of an arbitrary binary relation are provided and their relationships analysed. Both the classical relational algebra model and a model based on the Rough Set paradigm are discussed.

Vortex shedding patterns, their competition, and chaos in flow past inline oscillating rectangular cylinders

Abstract: The flow past inline oscillating rectangular cylinders is studied numerically at a Reynolds number representative of two-dimensional flow. A symmetric mode, known as S-II, consisting of a pair of oppositely signed vortices on each side, observed recently in experiments, is obtained computationally. A new symmetric mode, named here as S-III, is also found. At low oscillation amplitudes, the vortex shedding pattern transitions from antisymmetric to symmetric smoothly via a regime of intermediate phase. At higher amplitudes, this intermediate regime is chaotic. The finding of chaos extends and complements the recent work of Perdikaris et al. [Phys. Fluids 21(10), 101705 (2009)]. Moreover, it shows that the chaos results from a competition between antisymmetric and symmetric shedding modes. For smaller amplitude oscillations, rectangular cylinders rather than square are seen to facilitate these observations. A global, and very reliable, measure is used to establish the existence of chaos.

Towards extending the Chern-Simons couplings at order $O(\alpha'^2)$

Abstract: Using the compatibility of the anomalous Chern-Simons couplings on D$_p$-branes with the linear T-duality and with the antisymmetric B-field gauge transformations, some couplings have been recently found for $C^{(p-3)}$ at order $O(\alpha'^2)$. We examine these couplings with the S-matrix element of one RR and two antisymmetric B-field vertex operators. We find that the S-matrix element reproduces these couplings as well as some other couplings. Each of them is invariant under the linear T-duality and the B-field gauge transformations.

Spontaneous symmetry breaking of photonic and matter waves in two-dimensional pseudopotentials

Abstract: We introduce the two-dimensional Gross–Pitaevskii/nonlinear-Schrodinger (GP/NLS) equation with the self-focusing nonlinearity confined to two identical circles, separated or overlapped. The model can be realised in terms of Bose–Einstein condensates (BECs) and photonic-crystal fibers. Following the recent analysis of the spontaneous symmetry breaking (SSB) of localized modes trapped in 1D and 2D double-well nonlinear potentials (also known as pseudopotentials), we aim to find 2D solitons in the two-circle setting, using numerical methods and the variational approximation (VA). Well-separated circles support stable symmetric and antisymmetric solitons. The decrease of separation L between the circles leads to destabilisation of the solitons. The symmetric modes undergo two SSB transitions. First, they are transformed into weakly asymmetric breathers, which is followed by a transition to single-peak modes trapped in one circle. The antisymmetric solitons perform a direct transition to the single-peak mode. ...