Affine Symmetry Research Articles

Switching rule design for affine switched systems using a max-type composition rule

This paper presents conditions for designing a switching rule that drives the state of the switched dynamic system to a desired equilibrium point. The proposed method deals with the class of switched systems where each subsystem has an affine vector field and considers a switching rule using ‘max’ composition. The results guarantee global asymptotic stability of the tracking error dynamics even if sliding mode occurs at any switching surface of the system. In addition, the method does not require a Hurwitz convex combination of the dynamic matrices of the subsystems. Two numerical examples are used to illustrate the results.

Viruses and fullerenes – symmetry as a common thread?

The principle of affine symmetry is applied here to the nested fullerene cages (carbon onions) that arise in the context of carbon chemistry. Previous work on affine extensions of the icosahedral group has revealed a new organizational principle in virus structure and assembly. This group-theoretic framework is adapted here to the physical requirements dictated by carbon chemistry, and it is shown that mathematical models for carbon onions can be derived within this affine symmetry approach. This suggests the applicability of affine symmetry in a wider context in nature, as well as offering a novel perspective on the geometric principles underpinning carbon chemistry.

Affine operations plus symmetry yield perception of metric shape with large perspective changes (≥45°): Data and model.

G. P. Bingham and M. Lind (2008, Large continuous perspective transformations are necessary and sufficient for accurate perception of metric shape, Perception & Psychophysics, Vol. 70, pp. 524-540) showed that observers could perceive metric shape, given perspective changes ≥ 45° relative to a principal axis of elliptical cylinders. In this article, we tested (a) arbitrary perspective changes of 45°, (b) whether perception gradually improves with more perspective change, (c) speed of rotation, (d) whether this works with other shapes (asymmetric polyhedrons), (e) different slants, and (f) perspective changes >45°. Experiment 1 compared 45° perspective change away from, versus centered on, a principal axis. Observers adjusted an ellipse to match the cross-section of an elliptical cylinder viewed in a stereo-motion display. Experiment 2 tested whether performance would improve gradually with increases in perspective change, or suddenly with a 45° change. We also tested speed of rotation. Experiment 3 tested (a) asymmetric polyhedrons, (b) perspective change beyond 45°, and (c) the effect of slant. The results showed (a) a particular perspective was not required, (b) judgments only improved with ≥ 45° change, (c) speed was not relevant, (d) it worked with asymmetric polyhedrons, (e) slant was not relevant, and (f) judgments remained accurate beyond 45° of change. A model shows how affine operations, together with a symmetry yielded by 45° perspective change, bootstrap perception of metric shape.

Emergent Gravity, Violated Relativity and Dark Matter

The nonlinear affine Goldstone model of the emergent gravity, built on the nonlinearly realized/ hidden affine symmetry, is concisely revisited. Beyond General Relativity, the explicit violation of general invariance/relativity, under preserving general covariance, is exposed. Dependent on a nondynamical affine connection, a generally covariant second-order effective Lagrangian for metric gravity is worked out, with the general relativity violation and the gravitational dark matter serving as the signatures of emergence.

An infinite surface with the lattice property I: Veech groups and coding geodesics

We study the symmetries and geodesics of an infinite translation surface which arises as a limit of translation surfaces built from regular polygons, studied by Veech. We find the affine symmetry group of this infinite translation surface, and we show that this surface admits a deformation into other surfaces with topologically equivalent affine symmetries. The geodesics on these new surfaces are combinatorially the same as the geodesics on the

Submaximal metric projective and metric affine structures

We prove that the next possible dimension after the maximal n2+2n for the Lie algebra of local projective symmetries of a metric on a manifold of dimension n>1 is n2−3n+5 if the signature is Riemannian or n=2, n2−3n+6 if the signature is Lorentzian and n>2, and n2−3n+8 elsewise. We also prove that the Lie algebra of local affine symmetries of a metric has the same submaximal dimensions (after the maximal n2+n) unless the signature is Riemannian and n=3,4, in which case the submaximal dimension is n2−3n+6.

Affine infinitesimal generators of semigroups on the polydisk

This paper concerns affine infinitesimal generators of continuous semigroups of holomorphic functions on the polydisk. An easy-to-check necessary and sufficiently condition for an affine vector field to be an infinitesimal generator is given.

Constructing curves and triangular patches by Beta functions

A class of new basis functions for curve and triangular patch modeling is constructed by means of Beta functions. Based on these basis functions, a new scheme of generating curves and triangular patches is proposed. First we demonstrate that these basis functions have similar properties as those of the Bernstein–Bézier basis functions, such as non-negativity, partition of unity and others. Thus, these basis functions give rise to curve and triangular patch representations with affine invariance, convex hull, symmetry and endpoint interpolation, as well as an evaluation algorithm, which is similar to the de Casteljau’s algorithm for Bézier curves and surfaces. In addition, these basis functions have a shape parameter. The shape of the curve or triangular patch can be modified by changing the value of the shape parameter under the same control polygon or control net. The modeling examples illustrate the validity of the new methods.

Anomalous dimensions in deformed WZW models on supergroups

Abstract We investigate a class of current-current, Gross-Neveu like, perturbations of WZW models in which the full left-right affine symmetry is broken to the diagonal global algebra only. Our analysis focuses on those supergroups for which such a perturbation preserves conformal invariance. A detailed calculation of the 2-point functions of affine primary operators to 3-loops is presented. Furthermore, we derive an exact formula for the anomalous dimensions of a large subset of fields to all orders in perturbation theory. Possible applications of our results, including the study of non-perturbative dualities, are outlined.

ON THE DESCRIPTION OF SURFACE OPERATORS IN ${\mathcal N} = 2^*$ SYM

Alday and Tachikawa [Lett. Math. Phys.94, 87 (2010)] observed that the Nekrasov partition function of [Formula: see text] superconformal gauge theories in the presence of fundamental surface operators can be associated to conformal blocks of a 2D CFT with affine sl(2) symmetry. This can be interpreted as the insertion of a fundamental surface operator changing the conformal symmetry from the Virasoro symmetry discovered in Ref. 2 to the affine Kac–Moody symmetry. A natural question arises as to how such a 2D CFT description can be extended to the case of non-fundamental surface operators. Motivated by this question, we review the results [Y. Hikida and V. Schomerus, JHEP0710, 064 (2007); S. Ribault, JHEP0805, 073 (2008)] and put them together to suggest a way to address the problem: It follows from this analysis that the expectation value of a non-fundamental surface operator in the SU(2) [Formula: see text] super Yang–Mills (YM) theory would be in correspondence with the expectation value of a single vertex operator in a two-dimensional CFT with reduced affine symmetry and whose central charge is parametrized by the integer number that labels the type of singularity of the surface operator.

THE CLASSICAL EQUIVALENCE OF MONODROMY MATRICES IN SQUASHED SIGMA MODEL

We show Yangian and quantum affine symmetries are realized in two-dimensional non-linear sigma models with squashed spheres target spaces. According to these symmetries, there are two descriptions for the classical dynamics. In each description, a Lax pair and the corresponding monodromy matrix are constructed. Both Lax pairs reproduce the same equations of motion and hence they are equivalent. We show the gauge-equivalence by constructing a map between monodromy matrices with the relation of spectral parameters.

On the semiclassical three-point function inAdS3

We reconsider the problem of determining the semiclassical three-point function in the Euclidean ${\mathrm{AdS}}_{3}$ model. Exploiting the affine symmetry of the model, we use solutions of the classical Knizhnik-Zamolodchikov equation to compute the saddle point of the action in the presence of three vertex operators. This alternative derivation reproduces the ``heavy charge'' classical limit of the quantum three-point correlator. It is different from the recently proposed expression obtained by the generalized Pohlmeyer reduction in ${\mathrm{AdS}}_{2}$.

Conformal and affine Hamiltonian dynamics of general relativity

The Hamiltonian approach to the General Relativity is formulated as a joint nonlinear realization of conformal and affine symmetries by means of the Dirac scalar dilaton and the Maurer-Cartan forms. The dominance of the Casimir vacuum energy of physical fields provides a good description of the type Ia supernova luminosity distance--redshift relation. Introducing the uncertainty principle at the Planck's epoch within our model, we obtain the hierarchy of the Universe energy scales, which is supported by the observational data. We found that the invariance of the Maurer-Cartan forms with respect to the general coordinate transformation yields a single-component strong gravitational waves. The Hamiltonian dynamics of the model describes the effect of an intensive vacuum creation of gravitons and the minimal coupling scalar (Higgs) bosons in the Early Universe.

Polynomial processes and their applications to mathematical finance

We introduce a class of Markov processes, called m-polynomial, for which the calculation of (mixed) moments up to order m only requires the computation of matrix exponentials. This class contains affine processes, processes with quadratic diffusion coefficients, as well as Levy-driven SDEs with affine vector fields. Thus, many popular models such as exponential Levy models or affine models are covered by this setting. The applications range from statistical GMM estimation procedures to new techniques for option pricing and hedging. For instance, the efficient and easy computation of moments can be used for variance reduction techniques in Monte Carlo methods.

Holomorphic affine vector fields on weil bundles

We obtain necessary and sufficient conditions for a holomorphic vector field to be affine for a holomorphic linear connection defined on aWeil bundle. We also prove that the Lie algebra over R of holomorphic affine vector fields for the natural prolongation of a linear connection from the base to theWeil bundle is isomorphic to the tensor product of theWeil algebra by the Lie algebra of affine vector fields on the base.

The quantum affine origin of the AdS/CFT secret symmetry

We find a new quantum affine symmetry of the S-matrix of the one-dimensional Hubbard chain. We show that this symmetry originates from the quantum affine superalgebra , and in the rational limit exactly reproduces the secret symmetry of the AdS/CFT worldsheet S-matrix.

Media with integrable constraints

The application of the affine symmetry to the construction of models of anisotropic media with constraints is carried out. The theory is based on the classification of simple medium models by non-sensitivity groups, i.e. by continuum subgroups of the general group of linear three-dimensional transformations of Lagrangian variables, and on the calculation of their invariants from Lagrangian components of the metric tensor (Golubiatnikov, 1978,Golubiatnikov, 2003). Plane problems with two homogeneous non-stationary constraints are efficiently solved. Some other problems of continuum mechanics are discussed.

Corrections to “Switching Rule Design for Switched Dynamic Systems With Affine Vector Fields” [Sep 09 2215-2222

In this note, we present corrections to our previous paper "Switching rule design for switched dynamic systems with affine vector fields" [1], [2]. The correction is necessary to guarantee the stability of the system under sliding motion, which is not ensured from [1], [2] due to an error in the expressions (11) of the original paper [1]. To correct the results it is necessary to add a new condition to the results of [1], [2]. If there is no sliding motion or if the sliding motion dynamics, in the sense of Filippov, satisfies a quadratic stability condition, the additional condition is not necessary. The corrections in this note are restricted to the case of two operation modes.

Succinct Representation of Codes with Applications to Testing

Motivated by questions in property testing, we search for linear error-correcting codes that have the “single local orbit” property, i.e., they are specified by a single local constraint and its translations under the symmetry group of the code. We show that the dual of every “sparse” binary code whose coordinates are indexed by elements of $\mathbb{F}_{2^n}$ for prime $n$ and whose symmetry group includes the group of nonsingular affine transformations of $\mathbb{F}_{2^n}$ has the single local orbit property. (A code is said to be sparse if it contains polynomially many codewords in its block length.) In particular this class includes the dual-BCH codes for whose duals (i.e., for BCH codes) simple bases were not known. Our result gives the first short ($O(n)$-bit, as opposed to the natural $\exp(n)$-bit) description of a low-weight basis for BCH codes. The interest in the single local orbit property comes from the recent result of Kaufman and Sudan (STOC 2008) that shows that the duals of codes that have the single local orbit property under the affine symmetry group are locally testable. When combined with our main result, this shows that all sparse affine-invariant codes over the coordinates $\mathbb{F}_{2^n}$ for prime $n$ are locally testable. If, in addition to $n$ being prime, $2^n-1$ does not have large divisors, then we get that every sparse cyclic-invariant code also has the single local orbit. In particular this implies that BCH codes of such length are generated by a single low-weight codeword and its cyclic shifts.

Universe as a representation of affine and conformal symmetries

The evolution of the Universe is considered by means of a nonlinear realization of affine and conformal symmetries via Maurer-Cartan forms. Conformal symmetry is realized by the geometry of similarity with the Dirac scalar dilaton. We provide preliminary quantitative evidence that the zeroth harmonic of the Dirac scalar dilaton may lead to observationally viable cosmology, where the type la supernova luminosity distances-redshift relation can be explained by vacuum dilaton dark energy. The diffeo-invariance of spin connection coefficients of the affine formulation leaves only one degree of freedom of strong gravitation waves. We discuss that the dark matter effect in spiral galaxies can be described by the gravitation waves expressed through the spin connection coefficients of the affine formulation. We show that, the evolution equations of the affine gravitons with respect to the dilaton zeroth mode coincide with the equations of “squeezed oscillator”. The list of theoretical and observational arguments is given in favor of that the origin of the Universe can be described by quantum vacuum creation of these squeezed oscillators.