On the influence of extra dimensions on the homogeneous isotropic universe

The model of a homogeneous isotropic universe is studied in the presence of gauge fields, noninteracting dust, and two extra compact dimensions. It is found that the singular “big bang” type solution can be rejected because of the drastic growth of the radius of the universe. On the other hand, solutions without singularity can be found showing a very rapid oscillation (Planck frequency) with small amplitude around the data prescribing the present status of the universe.     

A Multi-Symplectic Compact Method for the Two-Component Camassa-Holm Equation with Singular Solutions

The two-component Camassa-Holm equation includes many intriguing phenomena. We propose a multi-symplectic compact method to solve the two-component Camassa-Holm equation. Based on its multi-symplectic formulation, the proposed method is derived by the sixth-order compact finite difference method in spatial discretization and the symplectic implicit midpoint scheme in temporal discretization. Numerical experiments finely describe the velocity and density variables in the two-component integrable system and distinctly display the evolvement of the singular solutions. Moreover, the proposed method shows good conservative properties during long-time numerical simulation.     

Spectral theory of Schrödinger operators of many-body systems with permutation and rotation symmetries

We investigate the spectra of many-body Schrödinger operators with permutation and rotation symmetry on subspaces of a given symmetry. For relatively compact two-body interactions the essential spectrum is a half line starting at the lowest two-body threshold with compatible symmetry. For dilatation analytic interactions the singular continuous spectrum is empty, and the eigenvalues accumulate at most at thresholds with compatible symmetries. For systems of atomic type there is an infinite sequence of discrete eigenvalues of each symmetry type.     

Static spherically symmetric solutions of the Einstein-Yang-Mills equations

We study the global behaviour of static, spherically symmetric solutions of the Einstein-Yang-Mills equations with gauge groupSU(2). Our analysis results in three disjoint classes of solutions with a regular origin or a horizon. The 3-spaces (t=const.) of the first, generic class are compact and singular. The second class consists of an infinite family of globally regular, resp. black hole solutions. The third type is an oscillating solution, which although regular is not asymptotically flat.This article was processed by the author using the Springer-Verlag TEX CoMaPhy macro package 1991.     

Reduced Hamiltonian for intersecting shells

The gauge usually adopted for extracting the reduced Hamiltonian of a thin spherical shell of matter in general relativity, becomes singular when dealing with two or more intersecting shells. We introduce here a more general class of gauges which is apt for dealing with intersecting shells. As an application we give the Hamiltonian treatment of two intersecting shells, both massive and massless. Such a formulation is applied to the computation of the semiclassical tunneling probability of two shells. The probability for the emission of two shells is simply the product of the separate probabilities thus showing no correlation in the emission probabilities in this model.     

Self-dual Chern–Simons Vortices on Bounded Domains

In this Letter we consider the Abelian Chern–Simons vortices on a bounded simply connected domain. We establish the existence of solutions for the self-duality equations. We prove the uniqueness of solutions when all the vortex points are equal and the domain is star-shaped. We also show the radial symmetry of solutions on balls centered at the vortex point.     

Spherical Gravitational Collapse: Tangential Pressure and Related Equations of State

We derive an equation for the acceleration of a fluid element in the spherical gravitational collapse of a bounded compact object made up of an imperfect fluid. We show that non-singular as well as singular solutions arise in the collapse of a fluid initially at rest and having only a tangential pressure. We obtain an exact solution of the Einstein equations, in the form of an infinite series, for collapse under tangential pressure with a linear equation of state. We show that if a singularity forms in the tangential pressure model, the conditions for the singularity to be naked are exactly the same as in the model of dust collapse.     

Existence and Dynamics of Bounded Traveling Wave Solutions to Getmanou Equation

In this paper, we study the existence and dynamics of bounded traveling wave solutions to Getmanou equations by using the qualitative theory of differential equations and the bifurcation method of dynamical systems. We show that the corresponding traveling wave system is a singular planar dynamical system with two singular straight lines, and obtain the bifurcations of phase portraits of the system under different parameters conditions. Through phase portraits, we show the existence and dynamics of several types of bounded traveling wave solutions including solitary wave solutions, periodic wave solutions, compactons, kink-like and antikink-like wave solutions. Moreover, the expressions of solitary wave solutions are given. Additionally, we confirm abundant dynamical behaviors of the traveling wave s olutions to the equation, which are summarized as follows: i) We confirm that two types of orbits give rise to solitary wave solutions, that is, the homoclinic orbit passing the singular point, and the composed homoclinic orbit which is comprised of two heteroclinic orbits and tangent to the singular line at the singular point of associated system. ii) We confirm that two types of orbits correspond to periodic wave solutions, that is, the periodic orbit surrounding a center, and the homoclinic orbit of associated system, which is tangent to the singular line at the singular point of associated system.     

Synthesis of hollow silver spheres using poly-(styrene-methyl acrylic acid) as templates in the presence of sodium polyacrylate

Hollow silver spheres were successfully prepared by reducing AgNO₃ with ascorbic acid and using negatively charged poly-(styrene-methyl acrylic acid) (PSA) spheres as templates in the presence of sodium polyacrylate as a stabilizer. Firstly, silver cations adsorbed on the surface of PSA spheres via electrostatic attraction between the carboxyl groups and silver cations were reduced in situ by ascorbic acid. The silver nanoparticles deposited on the surface of PSA spheres served as seeds for the further growth of silver shells. After that, extra amount of AgNO₃ and ascorbic acid solutions were added to form PSA/Ag composites with thick silver shells. In order to obtain compact silver shells, the as-prepared PSA/Ag composites were heated at 150 °C for 3 h. Then hollow silver spheres were prepared by dissolving PSA templates with tetrahydrofuran.     

Magnetic monopoles in SU(3) and SU(2) gauge theories without Higgs' scalars

We show that certain known singular solutions for pure SU(3) Yang-Mills theory carry magnetic charges with respect to both U(1) subgroups of SU(3). A topological characterisation in terms of monopoles is given to the SU(2) singular solutions of Wu and Yang.     

Dielectric permittivity of suspensions

A strict macroscopic analysis of the limiting long-wavelength permittivity of a model suspension is presented in which the suspension is considered as a finely dispersed system consisting of isotropic dielectric balls with piecewise-continuous radial permittivity profile. The analysis is performed within the framework of the notion of compact groups of inhomogeneities and the procedure of field averaging over volumes significantly exceeding the scale of these groups. The indicated value is described by the Lorentz-Lorenz formula. The effective polarizability of balls in the suspension is reconstructed from their parameters and the parameters of the medium by means of integration. The result is valid for any concentration of the balls at which the suspension remains macroscopically homogeneous and isotropic with respect to the field and for an arbitrary difference between the ball and medium permittivities.     

Convergence of the Vanishing Viscosity Approximation for Superpositions of Confined Eddies

A confined eddy is a circularly symmetric flow with vorticity of compact support and zero net circulation. Confined eddies with disjoint supports can be superimposed to generate stationary weak solutions of the two-dimensional incompressible inviscid Euler equations. In this work, we consider the unique weak solution of the two-dimensional incompressible Navier-Stokes equations having a disjoint superposition of very singular confined eddies as the initial datum. We prove the convergence of these weak solutions back to the initial configuration, as the Reynolds number goes to infinity. This implies that the stationary superposition of confined eddies with disjoint supports is the unique physically correct weak solution of the two-dimensional incompressible Euler equations.     

ON FINITE NORMAL SERIES SOLUTIONS OF THE SCHRÖDINGER EQUATION WITH IRREGULAR SINGULARITY

We compute all potentials with the following property: The one-dimensional nonrelativistic Schrödinger equation for these potentials has irregular singular points at infinity and/or zero and is solved by a finite normal series. We restrict to expansion order zero, discuss some properties of the potentials obtained and, as an application, calculate for some given potentials exact solutions and energies. The aim of this paper is to provide a tool for finding exact solutions of the Schrödinger equation for a large class of singular potentials.     

Singular Shell Embedded into a Cosmological Model

We generalize Israel's formalism to cover singular shells embedded in a non-vacuum Universe. That is, we deduce the relativistic equation of motion for a thin shell embedded in a Schwarzschild/Friedmann-Lemaître-Robertson-Walker spacetime. Also, we review the embedding of a Schwarzschild mass into a cosmological model using curvature coordinates and give solutions with (Sch/FLRW) and without the embedded mass (FLRW).     

A first approximation for quantization of singular spaces

Many mathematical models of physical phenomena that have been proposed in recent years require more general spaces than manifolds. When taking into account the symmetry group of the model, we get a reduced model on the (singular) orbit space of the symmetry group action. We investigate quantization of singular spaces obtained as leaf closure spaces of regular Riemannian foliations on compact manifolds. These contain the orbit spaces of compact group actions and orbifolds. Our method uses foliation theory as a desingularization technique for such singular spaces. A quantization procedure on the orbit space of the symmetry group–that commutes with reduction–can be obtained from constructions which combine different geometries associated with foliations and new techniques originated in Equivariant Quantization. The present paper contains the first of two steps needed to achieve these just detailed goals.     

Theory of singular vortex solutions of the nonlinear Schrödinger equation

We present a systematic study of singular vortex solutions of the critical and supercritical two-dimensional nonlinear Schrödinger equation. In particular, we study the critical power for collapse and the asymptotic blowup profile of singular vortices.     

Wave propagation characteristics of helically orthotropic cylindrical shells and resonance emergence in scattered acoustic field. Part 2. Numerical results

In the present work as the second part of the research work on wave propagation characteristics of helically orthotropic cylindrical shells, the main aim is to use the developed solution for resonance isolation and identification of an air-filled and water submerged Graphite/Epoxy cylindrical shell and quantitative sensitivity analysis of excited resonance frequencies to the perturbation in the material’s elastic constants. The physical justifications are presented for the singular features associated with the stimulated resonance frequencies according to their style of propagation and polarization, induced stress-strain fields and wave type. For evaluation purposes, the wave propagation characteristics of the anisotropic shell and the far-field form function amplitude of a limiting case are considered and good agreement with the solutions available in the literature is established.     

Singularity of Projections of 2-Dimensional Measures Invariant Under the Geodesic Flow

We show that on any compact Riemann surface with variable negative curvature there exists a measure which is invariant and ergodic under the geodesic flow and whose projection to the base manifold is 2-dimensional and singular with respect to the 2-dimensional Lebesgue measure.     

On dynamical behavior for optical solitons sustained by the perturbed Chen–Lee–Liu model

This study investigates the perturbed Chen–Lee–Liu model that represents the propagation of an optical pulse in plasma and optical fiber. The generalized exponential rational function method is used for this purpose. As a result, we obtain some non-trivial solutions such as the optical singular, periodic, hyperbolic, exponential, trigonometric soliton solutions. We aim to express the pulse propagation of the generated solutions, by taking specific values for the free parameters existed in the obtained solutions. The obtained results show that the generalized exponential rational function technique is applicable, simple and effective to get the solutions of nonlinear engineering and physical problems. Moreover, the acquired solutions display rich dynamical evolutions that are important in practical applications.     

Relativistic dynamics of cylindrical shells of counter-rotating particles

Although infinite cylinders are not astrophysical entities, it is possible to learn a great deal about the basic qualitative features of generation of gravitational waves and the behavior of the matter conforming such shells in the limits of very small radius. We study an analytical model of a relativistic cylindrical shell of counter-rotating particles using kinetic theory for the matter and the junction conditions through the shell to obtain its equation of motion. The nature of the static solutions are analyzed, both for a single shell as well as for two coaxial shells. In the latter case, we integrate numerically the time dependent equation of motion of the external shell, when we neglect the wave components of the gravitational field at the shells locations. We obtain solutions that correspond to shells that perform damped oscillations, collapse, or are locally expanding. The collapse ends (numerically) when the external shell hits the interior shell. The numerically work also shows that the radiation becomes important after the bounce of the external shell.