String Thermalization in Static Spacetimes

We study the evolution, the transverse spreadingand the subsequent thermalization of string states inthe Weyl static axisymmetric spacetime. This possessesa singular event horizon on the symmetry axis and a naked singularity along the otherdirections. The branching diffusion process of stringbits approaching the singular black-hole horizonprovides the notion of the temperature that iscalculated for this process. We find that the solution of theFokker-Planck equation in the phase space of thetransverse variables of the string, can be factored asa product of two thermal distributions, provided that the classical conjugate variables satisfy theuncertainty principle. We comment on the possiblephysical significance of this result.     

Unimodularity criteria for Poisson structures on foliated manifolds

We study the behavior of the modular class of an orientable Poisson manifold and formulate some unimodularity criteria in the semilocal context, around a (singular) symplectic leaf. Our results generalize some known unimodularity criteria for regular Poisson manifolds related to the notion of the Reeb class. In particular, we show that the unimodularity of the transverse Poisson structure of the leaf is a necessary condition for the semilocal unimodular property. Our main tool is an explicit formula for a bigraded decomposition of modular vector fields of a coupling Poisson structure on a foliated manifold. Moreover, we also exploit the notion of the modular class of a Poisson foliation and its relationship with the Reeb class.     

Picard-Fuchs Equation for Glueball Superfield for the SO(N) Gauge Group

In this paper, by using the factorization equation of the N = 2 supersymmetric gauge theory, we study N = 1 theory in Argyres-Douglas points. We suppose that all monopoles become massive. We derive general Picard-Fuchs equations for glueball superfields. These equations are hypergeometric equations and have regular singular points corresponding to Argyres-Douglas points. Furthermore, we obtain the solution of these differential equations.     

Perturbations des superalgèbres de Lie

We develop the notion of perturbations of Lie Superalgebras in terms of infinitesimals. We solve the classical equation of deformation, by using perturbations. We are interested in the rigid Lie Superalgebras and we give such a Lie superalgebra, in all dimensions.     

Interaction of regular and chaotic states

Modelling the chaotic states in terms of the Gaussian Orthogonal Ensemble of random matrices (GOE), we investigate the interaction of the GOE with regular bound states. The eigenvalues of the latter may or may not be embedded in the GOE spectrum. We derive a generalized form of the Pastur equation for the average Green’s function. We use that equation to study the average and the variance of the shift of the regular states, their spreading width, and the deformation of the GOE spectrum non-perturbatively. We compare our results with various perturbative approaches.     

Singular Hartree equation in fractional perturbed Sobolev spaces

We establish the local and global theory for the Cauchy problem of the singular Hartree equation in three dimensions, that is, the modification of the non-linear Schrödinger equation with Hartree non-linearity, where the linear part is now given by the Hamiltonian of point interaction. The latter is a singular, self-adjoint perturbation of the free Laplacian, modelling a contact interaction at a fixed point. The resulting non-linear equation is the typical effective equation for the dynamics of condensed Bose gases with fixed point-like impurities. We control the local solution theory in the perturbed Sobolev spaces of fractional order between the mass space and the operator domain. We then control the global solution theory both in the mass and in the energy space.     

A new approach to the Schrödinger equation with rational potentials

A new analytic theory is established for the Schrödinger equation with a rational potential, including a complete classification of the regular eigenfunctions into three different types, an exact method of obtaining wavefunctions, an explicit formulation of the spectral equation (3 x 3 determinant) etc. All representations are exhibited in a unifying way via function-theoretic methods and therefore given in explicit form, in contrast to the prevailing discussion appealing to perturbation or variation methods or continued-fraction techniques. The irregular eigenfunctions at infinity can be obtained analogously and will be discussed separately as another solvable case for singular potentials.     

广义色散Camassa-Holm模型的奇异孤子

引进一类广义色散Camassa-Holm模型，对其做奇异性分析.通过改进的WTC-Kruskal算法，证明该模型在Painlevé意义下可积，得到了它的一组Painlevé-Bcklund系统和Bcklund变换.应用Maple进行代数运算，得到了丰富的规则(regular)孤子和一类奇异(singular)孤子，扭结(kink)孤子,紧孤子(compacton)和反紧孤子(anti-compacton).特别地，推导出一类在扭结孤子的中间区域包含有一列周期尖点(cuspon)波的奇异结构.在这些规则的孤子系统的基础上，对可积广义系统应用Bcklund变换，得到三类奇异孤子，分别是具有驼峰结构的周期爆破波，具有爆破波结构的扭结孤子和紧孤子. 关键词： 广义Camassa-Holm 模型 周期尖点波 紧孤子 周期爆破波     

Static,Spherically Symmetric Solutions of Yang-Mills-Dilaton Theory

Static, spherically symmetric solutions of the Yang-Mills-Dilaton theory are studied. It is shown that these solutions fall into three different classes. The generic solutions are singular. Besides there is a discrete set of globally regular solutions further distinguished by the number of nodes of their Yang-Mills potential. The third class consists of oscillating solutions playing the role of limits of regular solutions, when the number of nodes tends to infinity. We show that all three sets of solutions are non-empty. Furthermore we give asymptotic formulae for the parameters of regular solutions and confront them with numerical results.     

Poisson-Nernst-Planck Equations for Simulating Biomolecular Diffusion-Reaction Processes I: Finite Element Solutions

In this paper we developed accurate finite element methods for solving 3-D Poisson-Nernst-Planck (PNP) equations with singular permanent charges for electrodiffusion in solvated biomolecular systems. The electrostatic Poisson equation was defined in the biomolecules and in the solvent, while the Nernst-Planck equation was defined only in the solvent. We applied a stable regularization scheme to remove the singular component of the electrostatic potential induced by the permanent charges inside biomolecules, and formulated regular, well-posed PNP equations. An inexact-Newton method was used to solve the coupled nonlinear elliptic equations for the steady problems; while an Adams-Bashforth-Crank-Nicolson method was devised for time integration for the unsteady electrodiffusion. We numerically investigated the conditioning of the stiffness matrices for the finite element approximations of the two formulations of the Nernst-Planck equation, and theoretically proved that the transformed formulation is always associated with an ill-conditioned stiffness matrix. We also studied the electroneutrality of the solution and its relation with the boundary conditions on the molecular surface, and concluded that a large net charge concentration is always present near the molecular surface due to the presence of multiple species of charged particles in the solution. The numerical methods are shown to be accurate and stable by various test problems, and are applicable to real large-scale biophysical electrodiffusion problems.     

Lösung singulärer Streugleichungen mittels Borel-Summation

In the case of zero energy scattering it is shown how to reconstruct the regular solution of a singular scattering equation from an asymptotic (divergent) power series expansion.     

Very weak solutions of wave equation for Landau Hamiltonian with irregular electromagnetic field

In this paper, we study the Cauchy problem for the Landau Hamiltonian wave equation, with time-dependent irregular (distributional) electromagnetic field and similarly irregular velocity. For such equations, we describe the notion of a ‘very weak solution’ adapted to the type of solutions that exist for regular coefficients. The construction is based on considering Friedrichs-type mollifier of the coefficients and corresponding classical solutions, and their quantitative behaviour in the regularising parameter. We show that even for distributional coefficients, the Cauchy problem does have a very weak solution, and that this notion leads to classical or distributional-type solutions under conditions when such solutions also exist.     

One-dimensional mappings of the eigenelements of the Schrodinger problem

An approach is proposed for analyzing the inverse spectral problems for the Schrodinger equation based on writing the equation for the analog of the number-of-quanta operator for a harmonic oscillator. This equation makes it possible to determine not only the one-dimensional mapping of the energy eigenvalues but also the linear equation for the point spectrum shift operator of the Schrodinger problem. The solvability conditions of the latter lead to a nonlinear equation that determines the class of allowable potentials. Two classes of potentials regular in R(1) and symmetrical are isolated on the basis of the proposed approach. The first of these leads to equidistant spectra with a gap of arbitrary size and location. The spectrum of the second potential class is a combination of three rigorously equidistant spectra with ground states that are shifted by an arbitrary amount. Generalizations to the case of essentially nonequidistant spectra are shown to be possible.     

The geometry of the hyperbolic system for an anisotropic perfectly elastic medium

We evaluate the fundamental solution of the hyperbolic system describing the generation and propagation of elastic waves in an anisotropic solid by studying the homology of the algebraic hypersurface defined by the characteristic equation, also known as the slowness surface. Our starting point is the Herglotz-Petrovsky-Leray integral representation of the fundamental solution. We find an explicit decomposition of the latter solution into integrals over vanishing cycles associated with the isolated singularities on the slowness surface. As is well known in the theory of isolated singularities, integrals over vanishing cycles satisfy a system of differential equations known as Picard-Fuchs equations. Such equations are linear and can have at most regular singular points. We discuss a method to obtain these equations explicitly. Subsequently, we use the monodromy properties around the regular singular points to find the asymptotic behavior according to the different types of singularities that may appear on a wave front in three dimensions. This is a method alternative to the one that arises in the Maslov theory of oscillating integrals. Our work sheds new light on how to compute and classify the Cagniard-De Hoop contour in the complex radial horizontal slowness plane; this contour is used in numerical integration schemes to obtain the full time behaviour of the fundamental solution for a given direction of propagation.     

A smoothing technique for discrete delta functions with application to immersed boundary method in moving boundary simulations

The effects of complex boundary conditions on flows are represented by a volume force in the immersed boundary methods. The problem with this representation is that the volume force exhibits non-physical oscillations in moving boundary simulations. A smoothing technique for discrete delta functions has been developed in this paper to suppress the non-physical oscillations in the volume forces. We have found that the non-physical oscillations are mainly due to the fact that the derivatives of the regular discrete delta functions do not satisfy certain moment conditions. It has been shown that the smoothed discrete delta functions constructed in this paper have one-order higher derivative than the regular ones. Moreover, not only the smoothed discrete delta functions satisfy the first two discrete moment conditions, but also their derivatives satisfy one-order higher moment condition than the regular ones. The smoothed discrete delta functions are tested by three test cases: a one-dimensional heat equation with a moving singular force, a two-dimensional flow past an oscillating cylinder, and the vortex-induced vibration of a cylinder. The numerical examples in these cases demonstrate that the smoothed discrete delta functions can effectively suppress the non-physical oscillations in the volume forces and improve the accuracy of the immersed boundary method with direct forcing in moving boundary simulations.     

Numerical treatment of acoustic problems with boundary singularities by the singular boundary method

The singular boundary method (SBM) is a novel boundary-type meshless method based on the fundamental solution of the given governing equation. The SBM employs the origin intensity factors to circumvent the singularities resulting from the fundamental solutions. In this paper, we investigate the acoustic problems with boundary singularities using the SBM. This is achieved by combining the SBM with the singularity subtraction techniques where the solution is decomposed into the singular solution and the regular solution. The singular solution is derived analytically which satisfies the governing equation and the corresponding boundary conditions containing the singularities. Then the regular solution is obtained by the SBM. Numerical examples show the excellent performance of the proposed technique.     

Noncommutative baby Skyrmions

We subject the baby Skyrme model to a Moyal deformation, for unitary or Grassmannian target spaces and without a potential term. In the Abelian case, the radial BPS configurations of the ordinary noncommutative sigma model also solve the baby Skyrme equation of motion. This gives a class of exact analytic noncommutative baby Skyrmions, which have a singular commutative limit but are stable against scaling due to the noncommutativity. We compute their energies, investigate their stability and determine the asymptotic two-Skyrmion interaction.     

A kinetic theory of dense fluids

A new kinetic equation is developed which incorporates the desirable features of the Enskog, the Rice-Allnatt, and the Prigogine-Nicolis-Misguich kinetic theories of dense fluids. Advantages of the present theory over the latter three theories are (1) it yields the correct local equilibrium hydrodynamic equations, (2) unlike the Rice-Allnatt theory, it determines the singlet and doublet distribution functions from the same equation, and (3) unlike the Prigogine-Nicolis-Misguich theory, it predicts the kinetic and kinetic-potential transport coefficients. The kinetic equation is solved by the Chapman-Enskog method and the coefficients of shear viscosity, bulk viscosity, thermal conductivity, and self-diffusion are obtained. The predicted bulk viscosity and thermal conductivity coefficients are singular at the critical point, while the shear viscosity and self-diffusion coefficients are not.     

Schrödinger Operators with Random Sparse Potentials. Existence of Wave Operators

We consider the connection problem for the Heun differential equation, which is a Fuchsian differential equation that has four regular singular points. We consider the case in which the parameters in this equation satisfy a certain set of conditions coming from the eigenvalue problem of the non-commutative harmonic oscillators. As an application, we describe eigenvalues with multiplicities greater than 1 and the corresponding odd eigenfunctions of the non-commutative harmonic oscillators. The existence of a rational or a certain algebraic solution of the Heun equation implies that the corresponding eigenvalues has multiplicities greater than 1.The research of the author is supported in part by a Grant-in-Aid for Scientific Research (B) (No. 15340005) from the Ministry of Education, Culture, Sports, Science and Technology.Mathematics Subject classifications (2000). primary, 34M35, secondary, 33E20.This revised version was published online in March 2005 with corrections to the cover date.