Recursions of Symmetry Orbits and Reduction without Reduction

We consider a four-dimensional PDE possessing partner symmetries mainly on the example of complex Monge-Amp\`ere equation (CMA). We use simultaneously two pairs of symmetries related by a recursion relation, which are mutually complex conjugate for CMA. For both pairs of partner symmetries, using Lie equations, we introduce explicitly group parameters as additional variables, replacing symmetry characteristics and their complex conjugates by derivatives of the unknown with respect to group parameters. We study the resulting system of six equations in the eight-dimensional space, that includes CMA, four equations of the recursion between partner symmetries and one integrability condition of this system. We use point symmetries of this extended system for performing its symmetry reduction with respect to group parameters that facilitates solving the extended system. This procedure does not imply a reduction in the number of physical variables and hence we end up with orbits of non-invariant solutions of CMA, generated by one partner symmetry, not used in the reduction. These solutions are determined by six linear equations with constant coefficients in the five-dimensional space which are obtained by a three-dimensional Legendre transformation of the reduced extended system. We present algebraic and exponential examples of such solutions that govern Legendre-transformed Ricci-flat K\"ahler metrics with no Killing vectors. A similar procedure is briefly outlined for Husain equation

On a class of second-order PDEs admitting partner symmetries

Recently we have demonstrated how to use partner symmetries for obtaining noninvariant solutions of heavenly equations of Plebanski that govern heavenly gravitational metrics. In this paper, we present a class of scalar second-order PDEs with four variables, that possess partner symmetries and contain only second derivatives of the unknown. We present a general form of such a PDE together with recursion relations between partner symmetries. This general PDE is transformed to several simplest canonical forms containing the two heavenly equations of Plebanski among them and two other nonlinear equations which we call mixed heavenly equation and asymmetric heavenly equation. On an example of the mixed heavenly equation, we show how to use partner symmetries for obtaining noninvariant solutions of PDEs by a lift from invariant solutions. Finally, we present Ricci-flat self-dual metrics governed by solutions of the mixed heavenly equation and its Legendre transform.Comment: LaTeX2e, 26 pages. The contents change: Exact noninvariant solutions of the Legendre transformed mixed heavenly equation and Ricci-flat metrics governed by solutions of this equation are added. Eq. (6.10) on p. 14 is correcte

Multi-Hamiltonian structure of Plebanski's second heavenly equation

We show that Plebanski's second heavenly equation, when written as a first-order nonlinear evolutionary system, admits multi-Hamiltonian structure. Therefore by Magri's theorem it is a completely integrable system. Thus it is an example of a completely integrable system in four dimensions

Exact Solvability of Superintegrable Systems

It is shown that all four superintegrable quantum systems on the Euclidean plane possess the same underlying hidden algebra $sl(3)$. The gauge-rotated Hamiltonians, as well as their integrals of motion, once rewritten in appropriate coordinates, preserve a flag of polynomials. This flag corresponds to highest-weight finite-dimensional representations of the $sl(3)$-algebra, realized by first order differential operators.Comment: 14 pages, AMS LaTe

Differences in the production and perception of communicative kinematics in autism

In human communication, social intentions and meaning are often revealed in the way we move. In this study, we investigate the flexibility of human communication in terms of kinematic modulation in a clinical population, namely, autistic individuals. The aim of this study was twofold: to assess (a) whether communicatively relevant kinematic features of gestures differ between autistic and neurotypical individuals, and (b) if autistic individuals use communicative kinematic modulation to support gesture recognition. We tested autistic and neurotypical individuals on a silent gesture production task and a gesture comprehension task. We measured movement during the gesture production task using a Kinect motion tracking device in order to determine if autistic individuals differed from neurotypical individuals in their gesture kinematics. For the gesture comprehension task, we assessed whether autistic individuals used communicatively relevant kinematic cues to support recognition. This was done by using stick-light figures as stimuli and testing for a correlation between the kinematics of these videos and recognition performance. We found that (a) silent gestures produced by autistic and neurotypical individuals differ in communicatively relevant kinematic features, such as the number of meaningful holds between movements, and (b) while autistic individuals are overall unimpaired at recognizing gestures, they processed repetition and complexity, measured as the amount of submovements perceived, differently than neurotypicals do. These findings highlight how subtle aspects of neurotypical behavior can be experienced differently by autistic individuals. They further demonstrate the relationship between movement kinematics and social interaction in high-functioning autistic individuals

Differences in functional brain organization during gesture recognition between autistic and neurotypical individuals

Persons with and without autism process sensory information differently. Differences in sensory processing are directly relevant to social functioning and communicative abilities, which are known to be hampered in persons with autism. We collected functional magnetic resonance imaging (fMRI) data from 25 autistic individuals and 25 neurotypical individuals while they performed a silent gesture recognition task. We exploited brain network topology, a holistic quantification of how networks within the brain are organized to provide new insights into how visual communicative signals are processed in autistic and neurotypical individuals. Performing graph theoretical analysis, we calculated two network properties of the action observation network: local efficiency, as a measure of network segregation, and global efficiency, as a measure of network integration. We found that persons with autism and neurotypical persons differ in how the action observation network is organized. Persons with autism utilize a more clustered, local-processing-oriented network configuration (i.e., higher local efficiency), rather than the more integrative network organization seen in neurotypicals (i.e., higher global efficiency). These results shed new light on the complex interplay between social and sensory processing in autism

Anti-self-dual Riemannian metrics without Killing vectors, can they be realized on K3?

Explicit Riemannian metrics with Euclidean signature and anti-self dual curvature that do not admit any Killing vectors are presented. The metric and the Riemann curvature scalars are homogenous functions of degree zero in a single real potential and its derivatives. The solution for the potential is a sum of exponential functions which suggests that for the choice of a suitable domain of coordinates and parameters it can be the metric on a compact manifold. Then, by the theorem of Hitchin, it could be a class of metrics on $K3$, or on surfaces whose universal covering is $K3$.Comment: Misprints in eqs.(9-11) corrected. Submitted to Classical and Quantum Gravit

Solutions of the sDiff(2)Toda equation with SU(2) Symmetry

We present the general solution to the Plebanski equation for an H-space that admits Killing vectors for an entire SU(2) of symmetries, which is therefore also the general solution of the sDiff(2)Toda equation that allows these symmetries. Desiring these solutions as a bridge toward the future for yet more general solutions of the sDiff(2)Toda equation, we generalize the earlier work of Olivier, on the Atiyah-Hitchin metric, and re-formulate work of Babich and Korotkin, and Tod, on the Bianchi IX approach to a metric with an SU(2) of symmetries. We also give careful delineations of the conformal transformations required to ensure that a metric of Bianchi IX type has zero Ricci tensor, so that it is a self-dual, vacuum solution of the complex-valued version of Einstein's equations, as appropriate for the original Plebanski equation.Comment: 27 page