A WKB treatment of diffusion in a multidimensional bistable potential

We extend to the case of a finite set of stochastic variables whose distributionP obeys a nonlinear Fokker-Planck equation our previous treatment of diffusion in a bistable potentialU, in the limit of small, constant diffusion coefficient. This is done with the help of an extended WKB approximation due to Gervais and Sakita. The treatment is valid if there exists a well-defined most probable path connecting the minima ofU, and if the valley ofU along that path has a slowly varying width, and weak curvature and twisting. We find that: (i) the final approach to equilibrium is governed by Eyring's generalization of the Kramers high-viscosity rate, which we rederive; (ii) for intermediate times, if the initial distribution is concentrated in the region of instability (close vicinity of the saddle point ofU),P has, along the most probable path, the behavior described by Suzuki's scaling statement for a one-dimensional system. In a second part of this time domain,P enters the diffusive regions around the minima ofU and relaxes toward local longitudinal equilibrium on a time comparable with Suzuki's time scale. The time for relaxation toward transverse local equilibrium may, depending on the initial conditions, compete with these longitudinal times.We dedicate this work to our colleague, Yuri Orlov.     

Spectral Measures of Small Index Principal Graphs

The principal graph X of a subfactor with finite Jones index is one of the important algebraic invariants of the subfactor. If Δ is the adjacency matrix of X we consider the equation Δ = U + U ⁻¹. When X has square norm ≤ 4 the spectral measure of U can be averaged by using the map u→ u ⁻¹, and we get a probability measure on the unit circle which does not depend on U. We find explicit formulae for this measure for the principal graphs of subfactors with index ≤ 4, the (extended) Coxeter-Dynkin graphs of type A, D and E. The moment generating function of is closely related to Jones’ Θ-series.D.B. was supported by NSF under Grant No. DMS-0301173.     

Polynomial Asymptotic Representation of Subharmonic Functions in a Half-Plane

Let u(z) be a subharmonic function in a half-plane such that its Riesz measure is concentrated on the finite system of rays.In the paper the connection between the behavior of u(z) and the distribution of its measure (including boundary measure) is investigated in terms of polynomial asymptotic representations.     

The jordanian deformation of Su(2) and clebsch-gordan coefficients

Representation theory for the Jordanian quantum algebraU _h (sl(2)) is developed using a nonlinear relation between its generators and those of sl(2). Closed form expressions are given for the action of the generators ofU _h (sl(2)) on the basis vectors of finite dimensional irreducible representations. In the tensor product of two such representations, a new basis is constructed on which the generators ofU _h (sl(2)) have a simple action. Using this basis, a general formula is obtained for the Clebsch-Gordan coefficients ofU _h (sl(2)). Some remarkable properties of these Clebsch-Gordan coefficients are derived. Presented at the 6th International Colloquium on Quantum Groups: “Quantum Groups and Integrable Systems”, Prague, 19–21 June 1997.     

The convergence of the slaving principle in a simplified model

We prove the convergence of the slaving principle in a model consisting of two nonlinear equations with two variables,s andu, which represent in the linear regime the stable mode and unstable mode, respectively.We show explicitly how the stable modes becomes increasingly dependent on the unstable modeu and approaches a definite power series ofu regardless of the initial condition fors. This power series is called slaving function and is shown to be absolutely and uniformly convergent on a closed disc, which contains the point describing the asymptotic behavior of the system. For some finite time, we show that the approximation involved in the substitution of the slaving function for the original stable modes decreases exponentially with time.     

New approach in theory of Clebsch-Gordan coefficients foru(n) andU q(u(n))

A new method for calculation of Clebsch-Gordan coefficients (CGCs) of the Lie algebrau(n) and its quantum analogU _q(u(n)) is developed. The method is based on the projection operator method in combination with the Wigner-Racah calculus for the subalgebrau(n−1) (U _q(u(n−1))). The key formulas of the method are couplings of the tensor and projection operators and also a tensor form of the projection operator ofu(n) andU _q(u(n)). It is shown that theU _q(u(n)) CGCs can be presented in terms of theU _q(u)(n−1)) q−9j-symbols. Presented at the 9th International Colloquium: “Quantum Groups and Integrable Systems”, Prague, 22–24 June 2000. Supported by Russian Foundation for Fundamental Research, grant 99-01-01163. Supported in part by the U.S. National Science Foundation under Grant PHY-9970769 and Cooperative Agreement EPS-9720652 that includes matching from the Louisiana Board of Regents Support Fund.     

On the microcanonical ensemble for a spin system interacting with one-mode electromagnetic field

A system described by the Dicke Hamiltonian is considered. It is known that for the canonical density operator describing such a system, the internal energyU is strictly negative. On the other hand, the microcanonical density operator can be defined for an arbitrary value ofU. The thermodynamic limit for the microcanonical density operator is investigated. In the caseU<0 the obtained results are completely equivalent to those of the canonical density operator. However, in the caseU>0 it turns out that the photon density is non-zero, and that the entropy and the mean spin are independent ofU. Moreover, forU>0 the coupling constant of the Hamiltonian does not appear in any thermodynamic formula after the thermodynamic limit is performed.     

Mappin class group actions on quantum doubles

We study representations of the mapping class group of the punctured torus on the double of a finite dimensional possibly non-semisimple Hopf algebra that arise in the construction of universal, extended topological field theories. We discuss how for doubles the degeneracy problem of TQFT's is circumvented. We find compact formulae for theS ^±1-matrices using the canonical, non-degenerate forms of Hopf algebras and the bicrossed structure of doubles rather than monodromy matrices. A rigorous proof of the modular relations and the computation of the projective phases is supplied using Radford's relations between the canonical forms and the moduli of integrals. We analyze the projectiveSL(2, Z)-action on the center ofU _q(sl₂) forq anl=2m+1^st root of unity. It appears that the 3m+1-dimensional representation decomposes into anm+1-dimensional finite representation and a2m-dimensional, irreducible representation. The latter is the tensor product of the two dimensional, standard representation ofSL(2, Z) and the finite,m-dimensional representation, obtained from the truncated TQFT of the semisimplified representation category ofU _q(sl₂).     

Vortex rings moving into turbulence

We report on an experimental study of turbulent vortex rings injected with velocity U _v0 into a grid-generated turbulent flow (with RMS streamwise velocity u _*) and followed relative to the mean flow. The initial Reynolds number of the vortices varies from 4500 to 11,500. The turbulence was characterised by an intensity I_t =u _*/U _v0, which varied over the range 0<I_t <0.03. A mathematical model based on a stochastic model of the vortex core is developed to explain and interpret the results. The vortex radius grows diffusively in time with the rate of increase of the square of the vortex radius increasing linearly with I_t . As the vortices grow, they slow down sufficiently rapidly in a manner that they penetrate a finite distance into the turbulence. The vortex velocity, averaged over many experiments, showed an initial t ^?1 decay, consistent with Maxworthy’s experiments. The analysis and experiments show that such vortices ultimately only move a finite distance from their point of generation and this distance varies inversely with I_t .     

On Gell-Mann's λ-matrices,d- andf-tensors,octets, and parametrizations ofS U (3)

The algebra ofS U (3) is developed on the basis of the matrices _i ofGell-Mann, and identities involving the tensorsd _{i j k} andf _{i j k} occurring in their multiplication law are derived. Octets and the tensor analysis of the adjoint groupS U (3)/Z(3) ofS U (3) are discussed. Various explicit parametrizations ofS U (3) are presented as generalizations of familiarS U (2) results.     

Speed and shape of solitary waves in relativistic warm plasma

Large-amplitude solitary waves are investigated in a relativistic plasma with finite ion-temperature. The mass of electron is also considered. The Sagdeev’s pseudopotential is determined in terms ofu, the ion speed. It is found that there exists a critical value ofu ₀, the value ofu at which (u′)²=0, beyond which the solitary waves cease to exist. The critical value also depends on the parameters likeν, the soliton velocity;μ, the electronion mass ratio orσ, the temperature ratio of ion to electron. This result reproduces our previous result [Czech. J. Phys., Vol. 54 (2004), No. 4, 489–496] when the ion temperature is neglected.     

Discrete series of unitary irreducible representations of the Uq(u(3, 1)) and Uq(u(n, 1)) noncompact quantum algebras

The structure of all discrete series of unitary irreducible representations of the U _q (u(3, 1)) and U _q (u(n, 1)) noncompact quantum algebras are investigated with the aid of extremal projection operators and the q-analog of the Mickelsson-Zhelobenko algebra Z(g, g′) _q . The orthonormal basis constructed in the infinite-dimensional space of irreducible representations of the U _q (u(n, 1)) ⊇ U _q (u(n)) algebra is the q-analog of the Gelfand-Graev basis in the space of the corresponding irreducible representations of the u(n, 1) ⊇ u(n) classical algebra.     

Stochastic operators,information, and entropy

For a stochastic operatorU on andL ₁-space, i.eU is linear, positive, and norm preserving on the positive cone ofL ₁, it is shown thatU decreases relative information between two nonnegativeL ₁-functions. Furthermore it is shown that the following properties ofU are closely related:U is energy decreasing (energy preserving),U isH-decreasing, whereH is Boltzmann'sH-functional, and the Maxwell distributions are fixed points ofU.     

Projective indecomposable modules over the small quantum osp(1|2)

In this paper, we construct the finite dimensional Hopf superalgebra u _q (osp(1|2)) arising from U _q(osp(1|2)) when q is a root of unity and describe the projective objects and the irreducible morphisms in a category of Z-graded u _q (osp(l|2))-modules.     

Integration theory of observables

Representations of abstract observables on a generalised logic are given in terms of bounded vector-valued Borel measures on the real line whose ranges are in the dual spaceX ^* of the Banach space of statesX. Each bounded observable is furthermore represented by an elementu ^* ofX ^* such that for any proper statep X, u ^* (p) is the expectation value ofu when the system is in the statep.     

Completely analytical interactions: Constructive description

An interactionU is called a completely analytical (CA) interaction, if it satisfies one of 12 given conditions formulated in terms of analyticity properties of the partition functions Z_v(u), or correlation decay, or truncated correlation bounds, or asymptotic behavior of ln Z_v(u), v→∞. The 12 conditions are presented, together with part of the proof of their equivalence. The main result of the paper is that each condition is constructive in the following sense: instead of checking it in all finite volumesv??^v, it is enough to consider only (a finite amount of) volumes with restricted size. In particular, the partition functions Z _v (u+?) for the complex perturbationsu+? ofu do not vanish for all V? ^v and all ? with ∥?∥<?, provided this is true only forv with diam v?C(?) and ∥?∥<?′ (but with?<?′).     

The localb-completeness of space-times

It is shown that any point of space-time has a neighbourhoodU such that theb-boundary ofU coincides with/U.     

Centre and representations of small quantum superalgebras at roots of unity

When the deformation parameter is a root of unity, the centre of a quantum group can be described by a set of generators and non trivial relations. In the case ofU _q (sl(N)), these relations simply derive from the expressions of the deformed Casimir operators. In the case ofU _q (osp(1|2)), the relation is simple if we use an operator which anticommutes with the fermionic generators and whose square is the quadratic Casimir. This operator also simplifies the classification of finite dimensional irreducible representations. In the case ofU _q (sl(1|2)), the relations derive from the (infinite set of) standard Casimir operators.Presented at the 5th International Colloquium on Quantum Groups: Quantum Groups and Integrable Systems, Prague, 20–22 June 1996.     

On the irreducible representations of groups that contain a subgroup of index 2

The objects under consideration are a groupG containing a subgroupN of index 2 and an irreducible multiplier representationU ofG by semiunitary (=unitary or antiunitary) operators on a complex Hilbert space of arbitrary dimension. It is assumed thatU(g) is unitary for allg belonging toN. Then the following assertion is proved. The representation ofN that is obtained by restrictingU toN is either irreducible or an orthogonal sum of two irreducible representations.     

Properties of narrowU(3.1) based on theM-diquonium 651-1651-1651-1interpretation

We study the properties ofU(3.1) assuming that theU is anM-diquoniumsq \(\bar q\) ²(q = u ord) state. It is shown that the annihilation decay which becomes the most important for usual diquonia is forbidden forU. We show there exist various reasons which makeU narrow. NearU(3.1) we expect other narrow diquonia. We also compute the electromagnetic mass splitting and find thatU ^?? is the heaviest andU ⁰ is the lightest.