Rows of optical vortices from elliptically perturbing a high-order beam

An optical vortex (phase singularity) with a high topological strength resides on the axis of a high-order light beam. The breakup of this vortex under elliptic perturbation into a straight row of unit-strength vortices is described. This behavior is studied in helical Ince-Gauss beams and astigmatic, generalized Hermite-Laguerre-Gauss beams, which are perturbations of Laguerre-Gauss beams. Approximations of these beams are derived for small perturbations, in which a neighborhood of the axis can be approximated by a polynomial in the complex plane: a Chebyshev polynomial for Ince-Gauss beams, and a Hermite polynomial for astigmatic beams.     

Singularities in Einstein–conformally coupled Higgs cosmological models

The dynamics of Einstein–conformally coupled Higgs field (EccH) system is investigated near the initial singularities in the presence of Friedman–Robertson–Walker symmetries. We solve the field equations asymptotically up to fourth order near the singularities analytically, and determine the solutions numerically as well. We found all the asymptotic, power series singular solutions, which are (1) solutions with a scalar polynomial curvature singularity but the Higgs field is bounded (‘Small Bang’), or (2) solutions with a Milne type singularity with bounded spacetime curvature and Higgs field, or (3) solutions with a scalar polynomial curvature singularity and diverging Higgs field (‘Big Bang’). Thus, in the present EccH model there is a new kind of physical spacetime singularity (‘Small Bang’). We also show that, in a neighbourhood of the singularity in these solutions, the Higgs sector does not have any symmetry breaking instantaneous vacuum state, and hence then the Brout–Englert–Higgs mechanism does not work. The large scale behaviour of the solutions is investigated numerically as well. In particular, the numerical calculations indicate that there are singular solutions that cannot be approximated by power series.     

On the classification of quasihomogeneous functions

We give a criterion for the existence of a non-degenerated quasihomogeneous polynomial in a configuration, i.e. in the space of polynomials with a fixed set of weights, and clarify the relation of this criterion to the necessary condition derived from the formula for the Poincaré polynomial. We further prove finiteness of the number of configurations for a given value of the singularity index. For the value 3 of this index, which is of particular interest in string theory, a constructive version of this proof implies an algorithm for the calculation of all non-degenerate configurations.     

Slow Light Via a Tapered Grating: Transfer Matrix Approach

Reflection properties of a nonuniform apodized Bragg grating connecting a homogeneous medium with a (half-infinite) uniform grating are investigated for system parameters such that the group velocity in the uniform grating is low. In particular a smooth polynomial transition of the index profile to the uniform region is considered. Exact coupled mode equations for local Bloch amplitudes are set up in a transfer matrix formalism. For a linear taper the coupling matrix appears to be singular for parameter values near the edge of the photonic band gap. The polynomial profile removes the singularity in this coupling term.     

Chiral rings,singularity theory and electric-magnetic duality

We study in detail the space of perturbations of a pair of dual N = 1 supersymmetric theories based on an SU(N_c) gauge theory with an adjoint X and fundamentals with a superpotential which is polynomial in X. The equivalence between them depends on non-trivial facts about polynomial equations, i.e. singularity theory. The classical chiral rings of the two theories are different. Quantum mechanically there are new relations in the chiral rings which ensure their equivalence. Duality interchanges “trivial” classical relations in one theory with quantum relations in the other and vice versa. We also speculate about the behavior of the theory without the superpotential.     

A scalar polynomial singularity without an event horizon

It is shown that the solution of the field equations for a static spherically symmetric scalar field has a scalar polynomial singularity and no event horizon. The solution does not develop from nonsingular data on any Cauchy surface. The possible existence of a universal scalar field, the conformal diagram and geodesies of the solution, and the energy and momentum of the field present are discussed.     

基于双树复小波变换的非平稳时间序列去趋势波动分析方法

多重分形去趋势波动分析是研究非平稳时间序列非均匀性和奇异性的有效工具, 针对该方法中趋势项难以确定的问题, 提出一种基于双树复小波变换的方法, 实现了非平稳信号的多重分形自适应去趋势波动分析. 利用双树复小波变换提取信号的多尺度趋势和波动信息, 通过小波系数的希尔伯特变换确定每个时间尺度不重叠子区间的长度, 使多重分形分析具有信号自适应性及较高的计算效率. 以具有解析形式分形特征的倍增级联信号和分数布朗运动时间序列为例验证本文方法的有效性, 所得结果与解析解相吻合. 与传统的多项式去趋势多重分形方法相比, 本文方法根据信号自身特点自适应地确定信号的趋势和不重叠等长度子区间长度, 所得结果更加精确. 对倍增级联信号时间序列取不同的长度, 验证了算法的稳定性. 分别与基于极大重叠离散小波变换和离散小波变换多重分形方法进行比较, 表明本文方法具有更精确的结果和更快的运算速度.     

Further remark concerning spherically symmetric nonstatic separable solutions to the Einstein equations in the comoving frame

We consider nonstatic spherically symmetric fluid solutions to the Einstein equations which, in the comoving frame, have metric coefficients that are separable functions of their arguments and that have an origin. Subject to the vanishing of the heat flux, we show that all such solutions with shear and non-vanishing shear viscosity have a scalar polynomial singularity at the origin if the fluid satisfies both the weak and strong energy conditions. When combined with previous results [1] we conclude that for the metric forms under consideration, the only fluid solutions to the Einstein equations with vanishing heat flux which satisfy the energy conditions and are free of singularities at the origin are the Robertson-Walker solutions.     

Discretizing families of linearizable equations

We investigate generalizations of the discrete Riccati equation as a linearizable system, to multicomponent linearizable systems. These are discretizations of nonlinear ordinary differential equations with superposition formulas. We present discrete matrix Riccati equations, projective, conformal, orthogonal and symplectic Riccati equations. Also obtained are discrete equations, based on complex orthogonal and symplectic groups, that in the continuous limit involve fourth order polynomial nonlinearities. All these equations satisfy the criterion of singularity confinement.     

半平面对称非类光电磁场产生的半平面对称时空

本文通过严格求解Einatein-Maxwell场方程,找到了由半平面对称非类光电磁场产生的半平面对称静态通解,并证明此类时空只具有(?/?x)^a,(?/?y)^a和(?/?t)^a三个Killing矢量场,本文还研究了时空奇异性,证明这类时空都是测地不完备的,且所有不完备测地线都对应着Scalar Polynomial曲率奇异性. 关键词：     

超短脉冲复宗量拉盖尔-高斯光束

从超短高斯脉冲光束出发。根据拉盖尔多项式的性质,用理论解析推导的方法。给出了一组新的超短脉冲光束的解析解,称为超短复宗量拉盖尔-高斯脉冲光束。此脉冲光束解的每个频率分量都是复宗量拉盖尔-高斯光束。时间脉冲的形状是任意的,具有相同的衍射距离参量,并且可以描述短于一个光学周期的超短脉冲。对这种超短脉冲光束及其在自由空间中的传输过程进行了较为细致的研究,分析了超短复宗量拉盖尔一高斯脉冲光束的轴上光强、光强的横向分布、衍射性质、脉冲极性反转、脉冲延迟等。讨论了引入缓变包络近似后出现的时空奇异性。     

An Explicit Family of Probability Measures for Passive Scalar Diffusion in a Random Flow

We explore the evolution of the probability density function (PDF) for an initially deterministic passive scalar diffusing in the presence of a uni-directional, white-noise Gaussian velocity field. For a spatially Gaussian initial profile we derive an exact spatio-temporal PDF for the scalar field renormalized by its spatial maximum. We use this problem as a test-bed for validating a numerical reconstruction procedure for the PDF via an inverse Laplace transform and orthogonal polynomial expansion. With the full PDF for a single Gaussian initial profile available, the orthogonal polynomial reconstruction procedure is carefully benchmarked, with special attentions to the singularities and the convergence criteria developed from the asymptotic study of the expansion coefficients, to motivate the use of different expansion schemes. Lastly, Monte-Carlo simulations stringently tested by the exact formulas for PDF’s and moments offer complete pictures of the spatio-temporal evolution of the scalar PDF’s for different initial data. Through these analyses, we identify how the random advection smooths the scalar PDF from an initial Dirac mass, to a measure with algebraic singularities at the extrema. Furthermore, the Péclet number is shown to be decisive in establishing the transition in the singularity structure of the PDF, from only one algebraic singularity at unit scalar values (small Péclet), to two algebraic singularities at both unit and zero scalar values (large Péclet).     

Neutrino exchange and asymptotic behaviour of weak interaction cross sections

It is argued that total cross sections cannot increase asymptotically as a power of energy even in the presence of massless particle exchange. Elastic νν scattering is considered as an example. The properties of s- and t-channel partial waves are analysed and it is shown that the above statement is true if the amplitude is bounded by a polynomial in s for s→∞ and t>0 and if its singularity at t=0 is dominated by two-particle exchange.     

Exponential Convergence to Equilibrium for the Homogeneous Boltzmann Equation for Hard Potentials Without Cut-Off

This paper deals with the long time behavior of solutions to the spatially homogeneous Boltzmann equation. The interactions considered are the so-called (non cut-off with a moderate angular singularity and non mollified) hard potentials. We prove an exponential in time convergence towards the equilibrium, improving results of Villani (Commun Math Phys 234(3): 455–490, 2003) where a polynomial decay to equilibrium is proven. The basis of the proof is the study of the linearized equation for which we prove a new spectral gap estimate in a \(L^1\) space with a polynomial weight by taking advantage of the theory of enlargement of the functional space for the semigroup decay developed by Gualdani et al. (http://hal.archives-ouvertes.fr/ccsd-00495786, 2013). We then get our final result by combining this new spectral gap estimate with bilinear estimates on the collisional operator that we establish.     

Orthogonal polynomial approach to discrete Lax pairs for initial boundary-value problems of the QD algorithm

Using orthogonal polynomial theory, we construct the Lax pair for the quotient-difference algorithm in the natural Rutishauser variables. We start by considering the family of orthogonal polynomials corresponding to a given linear form. Shifts on the linear form give rise to adjacent families. A compatible set of linear problems is made up from two relations connecting adjacent and original polynomials. Lax pairs for several initial boundary-value problems are derived and we recover the discrete-time Toda chain equations of Hirota and of Suris. This approach allows us to derive a Bäcklund transform that relates these two different discrete-time Toda systems. We also show that they yield the same bilinear equation up to a gauge transformation. The singularity confinement property is discussed as well.     

The Kerr space-time near the ring singularity

We investigate the geometry of the Kerr space-time near the ring singularity. A systematic study of the mathematical and physical structure of this region reveals that the singularity in the Kerr space-time is naturally understood in terms of a subset of the immersion of the set defined byr=0 (in Boyer-Lindquist coordinates) in the Kerr space-time. It is well known that the Kerr space-time is not a differentiable manifold (due to the curvature singularity) or a topological manifold, but a well defined topological space with a structure that is manifested by the constrast in taking limits along the hypersurface atr=0 and the equatorial plane which approach singularity. We find that the ring singularity is either an edge or a self-intersection of the topological space depending on which extension of the metric throughr=0 is implemented. A major result of this analysis is the extrapolation to the general accelerating case of Carter's proof that the only nonspacelike geodesics which can reach the ring singularity are restricted to the equatorial plane. For finite magnitudes of proper acceleration, it is shown that only lightlike trajectories that asymptotically approach the null generator of the ring singularity can reach it from above or below the equatorial plane.     

On the physical interpretation of the central-symmetric gravitational-field singularities

It is shown that the existence of singularity in the centrally symmetric gravitational field, which is interpreted as a surface with unusual physical properties, follows from equations for the action and the energy of a test particle not using Einstein equations and their solutions. A black hole is treated as a physical model of the singularity in question. The results are compared with the usual interpretation of the singularity in Schwarzschild solutions to Einstein equations.     

Vortex structures with complex points singularities in two-dimensional Euler equations. New exact solutions

In this work we present a new class of exact stationary solutions for two-dimensional (2D) Euler equations. Unlike already known solutions, the new ones contain complex singularities. We consider point singularities which have a vector field index greater than 1 as complex. For example, the dipole singularity is complex because its index is equal to 2. We present in explicit form a large class of exact localized stationary solutions for 2D Euler equations with a singularity whose index is equal to 3. The solutions obtained are expressed in terms of elementary functions. These solutions represent a complex singularity point surrounded by a vortex satellite structure. We also discuss the motion equation of singularities and conditions for singularity point stationarity which provide the stationarity of the complex vortex configuration.     

On the non-integrability of the spherically-symmetric nonlinear Schrödinger equation in the Langmuir collapse problem

In the present paper we give some analytic considerations of the spherically-symmetric nonlinear Schrödinger equation arising in the Langmuir collapse problem. We will make a systematic exploration of the various group symmetries of thie equation and show that the latter possesses only a three-parameter symmetry group. We will then give a variational formulation of this equation and use the three-parameter symmetry group to show that the equation in question possesses apparently only two polynomial conservation laws. Finally, we will make a study of the singularity structure of the present equation and show that it does not seem to possess the Painlevé property. The conclusion is that the spherically-symmetric nonlinear Schrödinger equation in question is apparently not integrable.