Generalized Lenard Chains, Separation of Variables and Superintegrability

We show that the notion of generalized Lenard chains naturally allows formulation of the theory of multi-separable and superintegrable systems in the context of bi-Hamiltonian geometry. We prove that the existence of generalized Lenard chains generated by a Hamiltonian function defined on a four-dimensional \omega N manifold guarantees the separation of variables. As an application, we construct such chains for the H\'enon-Heiles systems and for the classical Smorodinsky-Winternitz systems. New bi-Hamiltonian structures for the Kepler potential are found.Comment: 14 pages Revte

The Stabilized Poincare-Heisenberg algebra: a Clifford algebra viewpoint

The stabilized Poincare-Heisenberg algebra (SPHA) is the Lie algebra of quantum relativistic kinematics generated by fifteen generators. It is obtained from imposing stability conditions after attempting to combine the Lie algebras of quantum mechanics and relativity which by themselves are stable, however not when combined. In this paper we show how the sixteen dimensional Clifford algebra CL(1,3) can be used to generate the SPHA. The Clifford algebra path to the SPHA avoids the traditional stability considerations, relying instead on the fact that CL(1,3) is a semi-simple algebra and therefore stable. It is therefore conceptually easier and more straightforward to work with a Clifford algebra. The Clifford algebra path suggests the next evolutionary step toward a theory of physics at the interface of GR and QM might be to depart from working in space-time and instead to work in space-time-momentum.Comment: 14 page

Efficient indexing of necklaces and irreducible polynomials over finite fields

We study the problem of indexing irreducible polynomials over finite fields, and give the first efficient algorithm for this problem. Specifically, we show the existence of poly(n, log q)-size circuits that compute a bijection between {1, ... , |S|} and the set S of all irreducible, monic, univariate polynomials of degree n over a finite field F_q. This has applications in pseudorandomness, and answers an open question of Alon, Goldreich, H{\aa}stad and Peralta[AGHP]. Our approach uses a connection between irreducible polynomials and necklaces ( equivalence classes of strings under cyclic rotation). Along the way, we give the first efficient algorithm for indexing necklaces of a given length over a given alphabet, which may be of independent interest

Equivalence problem for the orthogonal webs on the sphere

We solve the equivalence problem for the orthogonally separable webs on the three-sphere under the action of the isometry group. This continues a classical project initiated by Olevsky in which he solved the corresponding canonical forms problem. The solution to the equivalence problem together with the results by Olevsky forms a complete solution to the problem of orthogonal separation of variables to the Hamilton-Jacobi equation defined on the three-sphere via orthogonal separation of variables. It is based on invariant properties of the characteristic Killing two-tensors in addition to properties of the corresponding algebraic curvature tensor and the associated Ricci tensor. The result is illustrated by a non-trivial application to a natural Hamiltonian defined on the three-sphere.Comment: 32 page

Does cultural background influence the intellectual performance of children from immigrant groups?

This paper addresses both the construct validity and the criterion-related validity of the "Revisie Amsterdamse Kinder Intelligentie Test" (RAKIT), which is a cognitive ability test developed for primary school children. The present study compared immigrant primary school children (N = 559) and Dutch children (N = 604). The mean scores of Surinamese/Netherlands Antillean, Moroccan, and Turkish children differed from each other and were lower than those of the Dutch children. Comparison of the test dimensions showed that group differences with respect to the construct validity were small. We found some item bias, but the combined effects on the sum score were not large. The estimate of general intelligence (g) as computed with the RAKIT showed strong predictive validity for most school subjects and standardized achievement tests. Although some criteria revealed significant prediction bias, the effects were very small. Most of the analyses we performed on differences in test scores and differences in criterion scores supported Spearman's hypothesis that g is the predominant factor determining the size of the differences between two groups. The conclusion that the RAKIT can be used for the assessment of groups from various backgrounds seems warranted

Counting Complex Disordered States by Efficient Pattern Matching: Chromatic Polynomials and Potts Partition Functions

Counting problems, determining the number of possible states of a large system under certain constraints, play an important role in many areas of science. They naturally arise for complex disordered systems in physics and chemistry, in mathematical graph theory, and in computer science. Counting problems, however, are among the hardest problems to access computationally. Here, we suggest a novel method to access a benchmark counting problem, finding chromatic polynomials of graphs. We develop a vertex-oriented symbolic pattern matching algorithm that exploits the equivalence between the chromatic polynomial and the zero-temperature partition function of the Potts antiferromagnet on the same graph. Implementing this bottom-up algorithm using appropriate computer algebra, the new method outperforms standard top-down methods by several orders of magnitude, already for moderately sized graphs. As a first application, we compute chromatic polynomials of samples of the simple cubic lattice, for the first time computationally accessing three-dimensional lattices of physical relevance. The method offers straightforward generalizations to several other counting problems.Comment: 7 pages, 4 figure

Supersymmetric Deformations of Maximally Supersymmetric Gauge Theories

We study supersymmetric and super Poincar\'e invariant deformations of ten-dimensional super Yang-Mills theory and of its dimensional reductions. We describe all infinitesimal super Poincar\'e invariant deformations of equations of motion of ten-dimensional super Yang-Mills theory and its reduction to a point; we discuss the extension of them to formal deformations. Our methods are based on homological algebra, in particular, on the theory of L-infinity and A-infinity algebras. The exposition of this theory as well as of some basic facts about Lie algebra homology and Hochschild homology is given in appendices.Comment: New results added. 111 page

Fact or Factitious? A Psychobiological Study of Authentic and Simulated Dissociative Identity States

BACKGROUND: Dissociative identity disorder (DID) is a disputed psychiatric disorder. Research findings and clinical observations suggest that DID involves an authentic mental disorder related to factors such as traumatization and disrupted attachment. A competing view indicates that DID is due to fantasy proneness, suggestibility, suggestion, and role-playing. Here we examine whether dissociative identity state-dependent psychobiological features in DID can be induced in high or low fantasy prone individuals by instructed and motivated role-playing, and suggestion. METHODOLOGY/PRINCIPAL FINDINGS: DID patients, high fantasy prone and low fantasy prone controls were studied in two different types of identity states (neutral and trauma-related) in an autobiographical memory script-driven (neutral or trauma-related) imagery paradigm. The controls were instructed to enact the two DID identity states. Twenty-nine subjects participated in the study: 11 patients with DID, 10 high fantasy prone DID simulating controls, and 8 low fantasy prone DID simulating controls. Autonomic and subjective reactions were obtained. Differences in psychophysiological and neural activation patterns were found between the DID patients and both high and low fantasy prone controls. That is, the identity states in DID were not convincingly enacted by DID simulating controls. Thus, important differences regarding regional cerebral bloodflow and psychophysiological responses for different types of identity states in patients with DID were upheld after controlling for DID simulation. CONCLUSIONS/SIGNIFICANCE: The findings are at odds with the idea that differences among different types of dissociative identity states in DID can be explained by high fantasy proneness, motivated role-enactment, and suggestion. They indicate that DID does not have a sociocultural (e.g., iatrogenic) origin

Poisson structures for reduced non-holonomic systems

Borisov, Mamaev and Kilin have recently found certain Poisson structures with respect to which the reduced and rescaled systems of certain non-holonomic problems, involving rolling bodies without slipping, become Hamiltonian, the Hamiltonian function being the reduced energy. We study further the algebraic origin of these Poisson structures, showing that they are of rank two and therefore the mentioned rescaling is not necessary. We show that they are determined, up to a non-vanishing factor function, by the existence of a system of first-order differential equations providing two integrals of motion. We generalize the form of that Poisson structures and extend their domain of definition. We apply the theory to the rolling disk, the Routh's sphere, the ball rolling on a surface of revolution, and its special case of a ball rolling inside a cylinder.Comment: 22 page