Symmetry and bifurcations of momentum mappings

The zero set of a momentum mapping is shown to have a singularity at each point with symmetry. The zero set is diffeomorphic to the product of a manifold and the zero set of a homogeneous quadratic function. The proof uses the Kuranishi theory of deformations. Among the applications, it is shown that the set of all solutions of the Yang-Mills equations on a Lorentz manifold has a singularity at any solution with symmetry, in the sense of a pure gauge symmetry. Similarly, the set of solutions of Einstein's equations has a singularity at any solution that has spacelike Killing fields, provided the spacetime has a compact Cauchy surface.This work was partially supported by the National Science Foundation. The second author was supported by a Killam Visiting fellowship at the University of Calgary during the completion of the paper     

Symmetries, chirp-free points, and bistability in dispersion-managed fiber lines

We show from an elementary symmetry analysis that, in dispersion-compensated systems for which a lossless model is valid, nonlinearity requires a chirp-free point at the center of symmetry (if such exists) of the map for any kind of unique periodic solution. We also present an example of a more-complex map when the periodic solution is not unique.     

层状介质中异质散射源三维定位逆问题的加权傅里叶变换研究

为了获得散射介质中异质结构信息,建立了层状均匀介质中含有一个较小的异质球模型,考虑小异质球对光(电磁波)的传播的影响是微扰的,在吸收边界条件下求得漫射方程一级微扰的基本解.同时考察漫射方程在二维傅里叶空间里形式解的特性,提出一种新颖的加权逆傅里叶变换.在加权傅里叶变换作用下,模型的表面数据在异质球位置上存在奇异性,结合数据本身的对称性,从而可以确定异质球的三维位置     

Some Schwarzschild solutions and their singularities

A number of different forms of the Schwarzschild solution are considered. The static forms all have a singularity at the Schwarzschild radius. This Schwarzschild singularity can be eliminated if one goes over to a stationary or time-dependent form of solution. However, the coordinate transformations needed for this have singularities. It is stressed that coordinate systems connected by singular transformations are not equivalent and the corresponding metrics may describe different physical situations.     

Spatial Kasner Solution and an Infinite Slab with a Constant Energy Density

We study the solutions of the Einstein equations in the presence of a thick infinite slab with constant energy density. When there is an isotropy in the plane of the slab, we find an explicit exact solution that matches with the Rindler and Weyl-Levi-Civita spacetimes outside the slab. We also show that there are solutions that can be matched with general anisotropic Kasner spacetime outside the slab. In any case, it is impossible to avoid the presence of the Kasner type singularities in contrast to the well-known case of spherical symmetry, where by matching the internal Schwarzschild solution with the external one, the singularity in the center of coordinates can be eliminated.

     

On the possibility of avoiding singularities by dilaton emission

We analyze the classical equations of supergravity theories containing a dilaton field, investigating the possibility that dilaton emission may prevent the formation of singularities. An initial cosmological singularity can be avoided in a no-scale supergravity theory if there is a nonzero charge density associated with the R symmetry current. However, this is only possible if some fields have negative metric initially, which may indicate a breakdown of the classical equations. A similar situation seems to occur with the Schwarzschild singularity.     

Comment on “Strength and genericity of singularities in Tolman–Bondi–de Sitter collapse” and a note on central singularities

It has been claimed in Phys. Lett. A 287 (2001) 53 that the Lemaitre–Tolman–Bondi–de Sitter solution always admits future-pointing radial time-like geodesics emerging from the shell-focussing singularity, regardless of the nature of the (regular) initial data. This is despite the fact that some data rule out the emergence of future pointing radial null geodesics. We correct this claim and show that, in general in spherical symmetry, the absence of radial null geodesics emerging from a central singularity is sufficient to prove that the singularity is censored.     

A detection method of symmetry restoration process of attractor merging crisis

We propose a method to detect the approach to a specific unstable symmetric mediating solution, which characterises the symmetry restoration process close to a bifurcation point of an attractor merging crisis. This method captures a temporary restoration of the symmetry, and it does not require neither the exact parameter value of the bifurcation point nor the mediating solution. We study a forced XY model as an example and show that this method figures out the singularity caused by the approach from the asymmetric side of the crisis. An analysis of the repulsively coupled Stuart Landau system suggests the feasibility of this method even when the mediating solution is a symmetric torus.     

Gravitational Collapse of Self-Similar Perfect Fluid in 2 + 1 Gravity

Perfect fluid with kinematic self-similarity is studied in 2+1 dimensional spacetimes with circular symmetry, and various exact solutions to the Einstein field equations are given. These include all the solutions of dust and stiff perfect fluid with self-similarity of the first kind, and all the solutions of perfect fluid with a linear equation of state and self-similarity of the zeroth and second kinds. It is found that some of these solutions represent gravitational collapse, and the final state of the collapse can be either a black hole or a null singularity. It is also shown that one solution can have two different kinds of kinematic self-similarity.     

Singularities in general relativity

The problem of singularities is examined from the stand-point of a local observer. A singularity is defined as a state with an infinite proper rest mass density. The approach consists of three steps: (i) The complete system of equations describing a non-symmetric motion of a perfect fluid under assumption of adiabatic thermodynamic processes and of no release of nuclear energy is reduced to six Einstein field equations and their four first integrals for six remaining unknown componentsgik. (ii) A differential relation for the behavior of the rest mass density is deduced. It shows that any inhomogeneity and anisotropy in the distribution and motion of a non-rotating ideal fluid accelerates collapse to a singularity which will be reached in a finite proper time. Collapse is also inevitable in a rotating fluid in the case of extremely high pressure when the relativistic limit of the equation of state must be applied. In the case of a lower or zero pressure the relation does not give an unambiguous answer if the matter is rotating. (iii) The influence of rotation on the motion of an incoherent matter is investigated. Some qualitative arguments are given for a possible existence of a narrow class of singularity-free solutions of Einstein equations. Assuming rotational symmetry the Einstein partial differential equations together with their first integrals are reduced to a system of simultaneous ordinary differential equations suitable for numerical integration. Without integrating this system the existence of the class of singularity-free solutions is confirmed and exactly delimited. These solutions, representing a new general relativistic effect, are, however, of no importance for the application in cosmology or astrophysics. It is proved that in all the other cases interesting from the point of view of application the occurrence of a point singularity in incoherent matter with a rotational symmetry is inevitable even if the rotation is present.Read on 15 May 1970 at the Gwatt Seminar on the Bearings of Topology upon General Relativity     

A scalar field with self-interaction leads to the absence of a singularity in cosmology

A self-consistent treatment of the Higgs conformal field with spontaneously broken symmetry and gravitational field in the semiclassical approximation is shown (by computer calculations) to lead to the absence of a singularity in the anisotropic model if the domain structure is taken into account.     

On the definition of phase transition

It is shown that there exists a phase transition associated with a singularity of the free energy for a model such that for all temperatures the equilibrium state is unique and thus stable with respect to boundary perturbations. It is also shown on this model that there exist phase transitions without symmetry breakdown, which can be related to a phase transition with symmetry breakdown on an equivalent model.     

Geometry and regularity of moving punctures

Significant advances in numerical simulations of black-hole binaries have recently been achieved using the puncture method. We examine how and why this method works by evolving a single black hole. The coordinate singularity and hence the geometry at the puncture are found to change during evolution, from representing an asymptotically flat end to being a cylinder. We construct an analytic solution for the stationary state of a black hole in spherical symmetry that matches the numerical result and demonstrates that the evolution is not dominated by artefacts at the puncture but indeed finds the analytical result.     

One-channel Roy equations revisited

The Roy equation in the single-channel case is a nonlinear, singular integral equation for the phase shift in the low-energy region. We first investigate the infinitesimal neighborhood of a given solution, and then present explicit expressions for amplitudes that satisfy the nonlinear equation exactly. These amplitudes contain free parameters that render the non-uniqueness of the solution manifest. They display, however, an unphysical singularity at the upper end of the interval considered. This singularity disappears and uniqueness is achieved if one uses analyticity properties of the amplitudes that are not encoded in the Roy equation. Received: 25 March 1999 / Published online: 15 July 1999     

Lagrange Structure and Dynamics for Solutions to the Spherically Symmetric Compressible Navier-Stokes Equations

The compressible Navier-Stokes system (CNS) with density-dependent viscosity coefficients is considered in multi-dimension, the prototype of the system is the viscous Saint-Venat model for the motion of shallow water. A spherically symmetric weak solution to the free boundary value problem for CNS with stress free boundary condition and arbitrarily large data is shown to exist globally in time with the free boundary separating fluids and vacuum and propagating at finite speed as particle path, which is continuous away from the symmetry center. Detailed regularity and Lagrangian structure of this solution have been obtained. In particular, it is shown that the particle path is uniquely defined starting from any non-vacuum region away from the symmetry center, along which vacuum states shall not form in any finite time and the initial regularities of the solution is preserved. Starting from any non-vacuum point at a later-on time, a particle path is also uniquely defined backward in time, which either reaches at some initial non-vacuum point, or stops at a small middle time and connects continuously with vacuum. In addition, the free boundary is shown to expand outward at an algebraic rate in time, and the fluid density decays and tends to zero almost everywhere away from the symmetry center as the time grows up. This finally leads to the formation of vacuum state almost everywhere as the time goes to infinity.     

Elasticity and dislocations in quasicrystals with 18-fold symmetry

Elasticity problems of quasicrystals with 18-fold rotational symmetry are studied. Constitutive equations and governing equations are obtained. For static elasticity problems, the displacement vectors in two phason fields are expressed in terms of two pairs of associated harmonic functions or two analytic functions. For dynamic problems, the displacement vectors can be represented in terms of an auxiliary function satisfying a fourth-order partial differential equation. A general solution of phasons is given by the solution of two diffusion equations. Phason elastic fields induced by a dislocation in a quasicrystal with 18-fold symmetry are determined and exhibit an inverse singularity.     

Complex random matrix models with possible applications to spin-impurity scattering in quantum Hall fluids

We study the one-point and two-point Green functions in a complex random matrix model to sub-leading orders in the large-N limit. We take this complex matrix model as a model for the two-state scattering problem, as applied to spin-dependent scattering of impurities in quantum Hall fluids. The density of state shows a singularity at the band center due to reflection symmetry. We also compute the one-point Green function for a generalized situation by putting random matrices on a lattice of arbitrary dimensions.     

Neue Lösungsformen der Boltzmann-Gleichung für monoenergetischen Neutronentransport in kugelförmiger Geometrie

In Chapter I thesingular solution of the Boltzmann equation for neutron transport in spherical geometry will be derived. The calculation will be performed in two steps. First, a partial differential equation (7) with an assumed density (6) on its right hand side will be solved. But the partial solution found in this way will generally not yield the assumed density. Therefore on has to add a suitable solution of the homogeneous differential equation (10). This addition leads to an equation of compatibility which turns out to be a Sonine integral equation (12). The second step of the calculation is the solution of this integral equation. The total solution of the Boltzmann equation will be written down in two different representations, (15) and (31), but its uniqueness has been proved. The main singularity at the center of the sphere is proportional to l/(?√1 μ²). A term log ? does not appear, but a term proportional to log [(1+μ)/(1?μ)] does which, however, loses its importance at the center of the sphere ?=0 in comparison with the main singularity. A characteristic equation needs not occur in this mathematical procedure; it may or may not be introduced. Therefore no hint at the spectrum of the Boltzmann operator in spherical geometry will be given. In Chapter II it will be shown that there exists a remarkably short integral representation of theregular solution (38) which satisfies from the first all requirements, if the validity of the characteristic equation (3) is supposed. But there are also regular solutions, given by the difference of two singular solutions, which need not satisfy a characteristic equation. In Chapter III both kinds of regular solutions in spherical geometry are given assuperpositions of solutions in plane geometry which belong to the discrete or to the continuous spectrum of the Boltzmann operator. The regular solutions are identical with the corresponding well-known series of spherical harmonics, where the supposition of a characteristic equation needs also not necessarily be made for exact solutions in the infinite space. A preliminary discussion of this problem is given in the introduction.     

An exact solution for uniformly accelerated particles in general relativity

Recently an exact solution of Einstein's empty-space equations referring to four uniformly accelerated particles was given. The relation of this to static axially symmetric metrics of the Weyl and Einstein-Rosen classes is investigated in the present paper. A physical interpretation of the singularity along half of the axis of symmetry of the uniformly accelerated metric in Weyl's form is given. An exact solution corresponding to an expanding (contracting) singular null surface is obtained by a limiting process from that for uniformly accelerated particles.     

Physical properties of an exact spherically symmetric solution with shear in general relativity

The properties of an exact spherically symmetric perfect fluid solution obtained in non-comoving coordinates are examined. This solution contains shear, and the pressure and the density are positive in the interior of the fluid. Their respective gradients with respect to comoving radial coordinate are equal and negative, and the speed of sound in this fluid is less than the speed of light in vacuum and is increasing outwards. There is a singularity at the center of the fluid since the pressure and the density become infinite there, though their ratio becomes unity. This singularity is naked, since there does not exist a trapped surface in the fluid outside this singularity. The circumference is an increasing function of radical comoving coordinate, and the mass function is positive and is increasing outwards. There are no tidal forces in radial direction, but the tidal forces normal to this direction are non-vanishing. We also give the kinematic quantities for this fluid. However, it is not possible to match this solution with an exterior vacuum Schwarzschild solution. Moreover, the dominant energy condition produces imaginary values for the sound speed.