On Galilean connections and the first jet bundle

We see how the first jet bundle of curves into affine space can be realized as a homogeneous space of the Galilean group. Cartan connections with this model are precisely the geometric structure of second-order ordinary differential equations under time-preserving transformations — sometimes called KCC-theory. With certain regularity conditions, we show that any such Cartan connection induces “laboratory” coordinate systems, and the geodesic equations in this coordinates form a system of second-order ordinary differential equations. We then show the converse — the “fundamental theorem” — that given such a coordinate system, and a system of second order ordinary differential equations, there exists regular Cartan connections yielding these, and such connections are completely determined by their torsion.     

On quantum mechanics and generalized Clifford algebras

In this note we give elementary examples of the naturalness of generalized Clifford algebras appearance, in some particular quantum mechanical models. First Weyl’s program [1] for quantum kinematics for the case of simplest Galois fieldsZ _n is realized in terms of generalized Clifford algebras. Dynamics might then be introduced, following the ideas of Hanney and Berry [2], as shown in [3]. Second the coherent state picture of the finite dimensional “Z _n — Quantum Mechanics” is presented. In the last part the known coherent states ofq-deformed quantum oscillators (q≡ω) are explicitly shown in the generalized Grassman algebras and the generalized Clifford algebras settings. Presented atThe Polish-Mexican Seminar, Kazimierz Dolny, August 1998 — Poland. 176     

The quantum euler class and the quantum cohomology of the Grassmannians

The Poincaré duality of classical cohomology and the extension of this duality to quantum cohomology endows these rings with the structure of a Frobenius algebra. Any such algebra possesses a canonical “characteristic element;” in the classical case this is the Euler class, and in the quantum case this is a deformation of the classical Euler class which we call the “quantum Euler class.” We prove that the characteristic element of a Frobenius algebraA is a unit if and only ifA is semisimple, and then apply this result to the cases of the quantum cohomology of the finite complex Grassmannians, and to the quantum cohomology of hypersurfaces. In addition we show that, in the case of the Grassmannians, the [quantum] Euler class equals, as [quantum] cohomology element and up to sign, the determinant of the Hessian of the [quantum] Landau-Ginzbug potential.     

Covariant and dynamical reduction for principal bundle field theories

Reduction for field theories with symmetry can be done either covariantly—that is, on spacetime—or dynamically—that is, after spacetime is split into space and time. The purpose of this article is to show that these two reduction procedures are, in an appropriate sense, equivalent for a class of field theories whose fields take values in a principal bundle. One can think of this class of field theories as including examples such as a “sea of rigid bodies” with and appropriate interbody coupling potential.     

Parton version of the quantum model with memory and the Zeno effect

A parton version of the model with memory is proposed to describe quantum objects. These objects are assumed to consist of valence subpartons and “sea subpartons,” which are carriers of the particle and wave properties, respectively. Subpartons are similar, but not identical, to quantum chromodynamic partons. An interpretation of the quantum Zeno effect is given in the framework of the model. This effect is shown to be caused by a specific interaction of the quantum atomic system with classical fields. This interaction breaks down the coherence of “sea subpartons” localized at different energy levels of the atom, which can be considered as the physical reason behind the wave-function collapse. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 110, No. 2, pp. 298–307, February, 1997.     

Weyl algebras over quantum groups

The term “Weyl algebras” is proposed for differential algebras associated with dual pairs of Hopf algebras. The principle of complete reducibility for the category of “admissible” modules over Weyl algebras is proved. Comodule structures that connect Weyl algebras with the Drinfeld quantum double are investigated. Translated from Teoreticheskaya i Matematicheskaya Fizika. Vol. 118, No. 2, pp. 190–204, February, 1999.     

Langlands duality for representations of quantum groups

We establish a correspondence (or duality) between the characters and the crystal bases of finite-dimensional representations of quantum groups associated to Langlands dual semi-simple Lie algebras. This duality may also be stated purely in terms of semi-simple Lie algebras. To explain this duality, we introduce an “interpolating quantum group” depending on two parameters which interpolates between a quantum group and its Langlands dual. We construct examples of its representations, depending on two parameters, which interpolate between representations of two Langlands dual quantum groups.     

Approximation of impulse controls

The application of impulse controls—delta-function and its higher derivatives—essentially improves our ability to control various systems. However, the delta-function and its derivatives are “idealizations.” Controls applied to model and control real systems are finite (although possibly quite large in magnitude). In this article we consider bounded approximations of impulse controls—so-called fast controls—and examine methods of their construction.     

Why vagueness is a mystery

This paper considers two “mysteries” having to do with vagueness. The first pertains to existence. An argument is presented for the following conclusion: there are possible cases in which ‘There exists something that is F’ is of indeterminate truth-value and with respect to which it is not assertable that there are borderline-cases of “being F.” It is contended that we have no conception of vagueness that makes this result intelligible. The second mystery has to do with “ordinary” vague predicates, such as ‘tall’. An argument is presented for the conclusion that although there are people who are “tall to degree 1”—definitely tall, tall without qualification—, no greatest lower bound can be assigned to the set of numbers n such that a man who is n centimeters tall is tall to degree 1. But, since this set is bounded from below, this result seems to contradict a well-known property of the real numbers.     

On the quiver-theoretical quantum Yang–Baxter equation

Quivers over a fixed base set form a monoidal category with tensor product given by pullback. The quantum Yang–Baxter equation, or more properly the braid equation, is investigated in this setting. A solution of the braid equation in this category is called a “solution” for short. Results of Etingof–Schedler–Soloviev, Lu–Yan–Zhu and Takeuchi on the set-theoretical quantum Yang–Baxter equation are generalized to the context of quivers, with groupoids playing the role of groups. The notion of “braided groupoid” is introduced. Braided groupoids are solutions and are characterized in terms of bijective 1-cocycles. The structure groupoid of a non-degenerate solution is defined; it is shown that it is a braided groupoid. The reduced structure groupoid of a non-degenerate solution is also defined. Non-degenerate solutions are classified in terms of representations of matched pairs of groupoids. By linearization we construct star-triangular face models and realize them as modules over quasitriangular quantum groupoids introduced in papers by M. Aguiar, S. Natale and the author.     

“Locus” and “Trace” in Cabrigéomètre: relationships between geometric and functional aspects in a study of transformations

The present text describes and characterises the tools “Locus” and “Trace” of Cabri-géomètre II, in relations to a study of geometric transformation, more precisely, the passage from the notion of transformation of figures to the notion of applications¹ that map points on the plane onto the plane itself. In particular it discusses how the conception of image of a figure under a transformation can evolve—through interaction in a “milieu” organised around Cabri-géomètre—such that students move from views of figure-images as undecomposible entities to see them as sets of image-points. Moreover, the study allowed the identification that the notion of trajectory (in a dynamic interpretation) has an important role in this conceptually difficult passage and that dynamic geometry environment renovate this notion.     

Number theoretical peculiarities in the dimension theory of dynamical systems

We show that dimensional theoretical properties of dynamical systems can considerably change because of number theoretical peculiarities of some parameter values. Supported by “DFG-Schwerpunktprogramm — Dynamik: Analysis, effiziente Simulation und Ergodentheorie”. We refer to the book of Falconer [6] for an introduction to dimension theory and recommend the book of Pesin [17] for the dimension theory of dynamical systems.     

Aging of spherical spin glasses

Sompolinski and Zippelius (1981) propose the study of dynamical systems whose invariant measures are the Gibbs measures for (hard to analyze) statistical physics models of interest. In the course of doing so, physicists often report of an “aging” phenomenon. For example, aging is expected to happen for the Sherrington-Kirkpatrick model, a disordered mean-field model with a very complex phase transition in equilibrium at low temperature. We shall study the Langevin dynamics for a simplified spherical version of this model. The induced rotational symmetry of the spherical model reduces the dynamics in question to an N-dimensional coupled system of Ornstein-Uhlenbeck processes whose random drift parameters are the eigenvalues of certain random matrices. We obtain the limiting dynamics for N approaching infinity and by analyzing its long time behavior, explain what is aging (mathematically speaking), what causes this phenomenon, and what is its relationship with the phase transition of the corresponding equilibrium invariant measures. Received: 8 July 1999 / Revised version: 2 June 2000 / Published online: 6 April 2001     

Comparison of two kinds of Hankel type operators

We compare two kinds of Hankel type operators, the “BJP-operators” introduced by Boman—Janson—Peetre and the “Z-operators” considered by Zhang. We show that every BJP-operator comes as an orthogonal sum ofZ-operators.     

Separation of variables in the quantum integrable models related to the Yangiany[sl(3)]

In the absence of a precise definition of the quantum integrability, the separability of variables can serve as its practical substitute. For any quantum integrable model generated by the Yangian Y[sl(3)], the canonical coordinates and the conjugate operators are constructed which satisfy the “quantum characteristic equation” (the quantum counterpart of the spectral algebraic curve for the L-operator). The coordinates constructed provide a local separation of variables. Conditions are listed which are necessary for the separation of variables to take place. Bibliography: 17 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 205, 1993, pp. 166–178. Translated by E. K. Sklyanin.     

Quantum dynamics as a stochastic process in phase space

It is shown that quantum dynamics is equivalent to a stochastic process in phase space. The process is described by normalized but not necessarily positive probability distributions (“pseudoprobabilities”). Evolution of the simultaneous probability distribution of momentum and coordinate exactly conincides with the dynamics of the Wigner function of a quantum system. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 110, No. 3, pp. 454–458, March, 1997.     

A quantum analog of the Poincare-Birkhoff-Witt theorem

We reduce the basis construction problem for character Hopf algebras to a study of special elements, called “super-letters,” which are defined by Shirshov standard words. It is shown that character Hopf algebras having not more than finitely many “hard” super-letters share some of the properties of universal envelopings of finite-dimensional lie algebras. The background for our proofs is the construction of a filtration such that the associated graded algebra is obtained by iterating the skew polynomials construction, possibly followed with factorization. Supported by the National Society of Researchers, México (SNI, exp. 18740, 1997–2000) Translated fromAlgebra i Logika, Vol. 38, No. 4, pp. 476–507, July–August, 1999.     

Interaction of Two Charges in a Uniform Magnetic Field: II. Spatial Problem

The interaction of two charges moving in ℝ³ in a magnetic field B can be formulated as a Hamiltonian system with six degrees of freedom. Assuming that the magnetic field is uniform and the interaction potential has rotation symmetry, we reduce this system to one with three degrees of freedom. For special values of the conserved quantities, choices of parameters or restriction to the coplanar case, we obtain systems with two degrees of freedom. Specialising to the case of Coulomb interaction, these reductions enable us to obtain many qualitative features of the dynamics. For charges of the same sign, the gyrohelices either “bounce-back”, “pass-through”, or exceptionally converge to coplanar solutions. For charges of opposite signs, we decompose the state space into “free” and “trapped” parts with transitions only when the particles are coplanar. A scattering map is defined for those trajectories that come from and go to infinite separation along the field direction. It determines the asymptotic parallel velocities, guiding centre field lines, magnetic moments and gyrophases for large positive time from those for large negative time. In regimes where gyrophase averaging is appropriate, the scattering map has a simple form, conserving the magnetic moments and parallel kinetic energies (in a frame moving along the field with the centre of mass) and rotating or translating the guiding centre field lines. When the gyrofrequencies are in low-order resonance, however, gyrophase averaging is not justified and transfer of perpendicular kinetic energy is shown to occur. In the extreme case of equal gyrofrequencies, an additional integral helps us to analyse further and prove that there is typically also transfer between perpendicular and parallel kinetic energy.