The Extensions of Gravitational Soliton Solutions with Real Poles

We analyse vacuum gravitationalsoliton solutions with real poles in thecosmological context. It is well known that thesesolutions contain singularities on certain nullhypersurfaces. Using a Kasner seed solution, we demonstrate thatthese may contain thin sheets of null matter or may besimple coordinate singularities, and we describe anumber of possible extensions through them.     

Dynamics of anisotropic cosmological model with ultrarelativistic matter,magnetic field,and fluxes of free isotropic particles

Within the framework of general relativity a dynamics of homogeneous anistropic axially symmetric model of the Bianchi type I is considered for the case when sources of gravitational field are ultrarelativistic matter, homogeneous magnetic field, and fluxes of free particles. Qualitative analysis of the field equations on a phase plane is given. All solutions of a considered type for large values of proper time asymptotically approach the flat Friedmann model while the value of energy density of free particles approaches the double value of magnetic field energy density. Near a singular state the solution exhibits oscillating behavior with successive interchange of Kasner singularities of pancake-like and filament-like types. It is also shown that in the absence of matter a solution retains its character.     

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.     

Singularities in Gravitational Collapse with Radial Pressure

We analyze spherical dust collapse with non-vanishing radial pressure, II, and vanishing tangential stresses. Considering a barotropic equation of state, II = , we obtain an analytical solution in closed form—which is exact for = –1, 0, and approximate otherwise—near the center of symmetry (where the curvature singularity forms). We study the formation, visibility, and curvature strength of singularities in the resulting spacetime. We find that visible, Tipler strong singularities can develop from generic initial data. Radial pressure alters the spectrum of possible endstates for collapse, increasing the parameter space region that contains no visible singularities, but cannot by itself prevent the formation of visible singularities for sufficiently low values of the energy density. Known results from pressureless dust are recovered in the = 0 limit.     

Recovery of singularities for formally determined inverse problems

In this paper we considered several formally determined problems in two dimensions. There are no global identifiability results for these problems. However, we can recover an important feature of these functions, namely their singularities. More precisely, we prove that one can determine the location and strength of singularities of anL compactly supported potential by knowing the associated scattering amplitude at a fixed energy. Also we prove that one can determine the location and strength of the singularities of the sound speed of a medium by making measurements just on the boundary of the medium.Partially supported by NSF grant DMS-9123742Partially supported by NSF grant DMS-9100178     

Motion of an ideal fluid in the field of a plane gravitational wave

An exact solution of the equations of relativistic hydrodynamics is found which describes the motion of an initially uniform ideal fluid in the field of a plane gravitational wave of arbitrary amplitude and polarization. For all solutions we find that the pressure and energy density develop singularities on the singular surface, and the velocity of the fluid in the direction of propagation of the gravitational wave approaches the speed of light. In the case of the equation of state =p, the solution becomes intrinsically unstable and describes the generation of sound waves.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 11, pp. 96–99, November, 1982.     

Description of dislocation cores

In this paper we explore the possibility of describing the dislocation core of a dislocation by a distribution of infinitesimal dislocations smooth enough to remove all singularities. The results may be extended straightforwardly to the case of anisotropic elasticity and can be used to calculate accurately contrast effects. It is shown that precipitation at dislocation cores may induce a glissile to sessile transformation.Dedicated to Dr. Frantiek Kroupa in honour of his 70^th birthday.     

Einstein-Proca Model, Micro Black Holes, and Naked Singularities

The Einstein-Proca equations, describing a spin-1 massive vector field in general relativity, are studied in the static spherically-symmetric case. The Proca field equation is a highly nonlinear wave equation, but can be solved to good accuracy in perturbation theory, which should be very accurate for a wide range of mass scales. The resulting first order metric reduces to the Reissner-Nordström solution in the limit as the range parameter goes to zero. The additional terms in the g ₀₀ metric coefficient are positive, as in Reissner-Nordström, in agreement with previous numerical solutions, and hence involve naked singularities.     

On complex singularities of the 2D Euler equation at short times

We present a study of complex singularities of a two-parameter family of solutions for the two-dimensional Euler equation with periodic boundary conditions and initial conditions in the short-time asymptotic régime. As has been shown numerically in Pauls et al. [W. Pauls, T. Matsumoto, U. Frisch, J. Bec, Nature of complex singularities for the 2D Euler equation, Physica D 219 (2006) 40-59], the type of the singularities depends on the angle ? between the modes p and q. Thus, the Fourier coefficients of the solutions decrease as with the exponent α depending on ?. Here we show for the two particular cases of ? going to zero and to π that the type of the singularities can be determined very accurately, being characterised by α=5/2 and α=3 respectively. In these two cases we are also able to determine the subdominant corrections. Furthermore, we find that the geometry of the singularities in these two cases is completely different, the singular manifold being located “over” different points in the real domain.     

Relativistic jost functions and potentials

An extension of the solution of the inversion problem reproducing the potentials and wave functions from the given singularities of the Jost functions on a static limit relativistic case of theS-wave Klein-Gordon equation is investigated by way of the method elaborated by Petrá in Czech. J. Phys.B12 (1962), 67. It is shown that similarly as in the non-relativistic case, a transparent representation of the relativistic Jost solution permits to determine uniquely the potentials decreasing exponentially at large distances and leads to the dispersive Fredholm integral equations for the Jost solution components. The proposed method might be considered as an alternative to that used by Cornille in J. Math. Phys.11 (1970), 79.     

The conical point in the ferroelectric six-vertex model

We examine the last unexplored regime of the asymmetric six-vertex model: the low-temperature phase of the so-called ferroelectric model. The original publication of the exact solution by Sutherland, Yang, and Yang and various derivations and reviews published afterward do not contain many details about this regime. We study the exact solution for this model by numerical and analytical methods. In particular, we examine the behavior of the model in the vicinity of an unusual coexistence point that we call the conical point. This point corresponds to additional singularities in the free energy that were not discussed in the original solution. We show analytically that at this point many polarizations coexist, and that unusual scaling properties hold in its vicinity.     

G2-Monopoles with Singularities (Examples)

G₂-Monopoles are solutions to gauge theoretical equations on G₂-manifolds. If the G₂-manifolds under consideration are compact, then any irreducible G₂-monopole must have singularities. It is then important to understand which kind of singularities G₂-monopoles can have. We give examples (in the noncompact case) of non-Abelian monopoles with Dirac type singularities, and examples of monopoles whose singularities are not of that type. We also give an existence result for Abelian monopoles with Dirac type singularities on compact manifolds. This should be one of the building blocks in a gluing construction aimed at constructing non-Abelian ones.     

Normal-dominated singularities in static space-times

We define normal-dominated singularities of static solutions of the Einstein equations and show that a uniquely and invariantly defined structure can be assigned to these singularities. We find for the general solution that the dominant term of the Riemann tensor near the singularity is of Petrov Type N. Except for one special class of solutions, it seems that in general the shear of the null geodesics blows up at the same rate as their convergence near the singularity, in contradistinction to the elementary singularity of Newman and Posadas. We compute the structure for a variety of known static solutions as well as the stationary Kerr-Newman metrics.Supported in part by NSF Grant GP 34639 X.     

On the Singularities in the Susceptibility Expansion for the Two-Dimensional Ising Model

For temperatures below the critical temperature, the magnetic susceptibility for the two-dimensional isotropic Ising model can be expressed in terms of an infinite series of multiple integrals. With respect to a parameter related to temperature and the interaction constant, the integrals may be extended to functions analytic outside the unit circle. In a groundbreaking paper, Nickel (J Phys A 32:3889–3906, 1999) identified a class of singularities of these integrals on the unit circle. In this note we show that there are no other singularities on the unit circle.     

Another Note on Focus-Focus Singularities

We show a natural relation between the monodromy formula for focus-focus singularities of integrable Hamiltonian systems and a formula of Duistermaat–Heckman, and extend the main results of our previous note ( ¹-action, monodromy, and topological classification) to the case of degenerate focus-focus singularities. We also consider the non-Hamiltonian case, local normal forms, etc.     

Vaidya spacetime in the diagonal coordinates

We have analyzed the transformation from initial coordinates (v, r) of the Vaidya metric with light coordinate v to the most physical diagonal coordinates (t, r). An exact solution has been obtained for the corresponding metric tensor in the case of a linear dependence of the mass function of the Vaidya metric on light coordinate v. In the diagonal coordinates, a narrow region (with a width proportional to the mass growth rate of a black hole) has been detected near the visibility horizon of the Vaidya accreting black hole, in which the metric differs qualitatively from the Schwarzschild metric and cannot be represented as a small perturbation. It has been shown that, in this case, a single set of diagonal coordinates (t, r) is insufficient to cover the entire range of initial coordinates (v, r) outside the visibility horizon; at least three sets of diagonal coordinates are required, the domains of which are separated by singular surfaces on which the metric components have singularities (either g ₀₀ = 0 or g ₀₀ = ∞). The energy–momentum tensor diverges on these surfaces; however, the tidal forces turn out to be finite, which follows from an analysis of the deviation equations for geodesics. Therefore, these singular surfaces are exclusively coordinate singularities that can be referred to as false fire-walls because there are no physical singularities on them. We have also considered the transformation from the initial coordinates to other diagonal coordinates (η, y), in which the solution is obtained in explicit form, and there is no energy–momentum tensor divergence.     

Global Stringy Orbifold Cohomology,K-Theory and de Rham Theory

There are two approaches to constructing stringy multiplications for global quotients. The first one is given by first pulling back and then pushing forward. The second one is given by first pushing forward and then pulling back. The first approach has been used to define a global stringy extension of the functors K ₀ and K ^top by Jarvis–Kaufmann–Kimura, A* by Abramovich–Graber–Vistoli, and H* by Chen–Ruan and Fantechi–Göttsche. The second approach has been applied by the author in the case of cyclic twisted sector and in particular for singularities with symmetries and for symmetric products. The second type of construction has also been discussed in the de Rham setting for Abelian quotients by Chen–Hu. We give a rigorous formulation of de Rham theory for any global quotient from both points of view. We also show that the pull–push formalism has a solution by the push–pull equations in the setting case of cyclic twisted sectors. In the general, not necessarily cyclic case, we introduce ring extensions and treat all the stringy extension of the functors mentioned above also from the second point of view. A first extension provides formal sections and a second extension fractional Euler classes. The formal sections allow us to give a pull–push solution while fractional Euler classes give a trivialization of the co-cycles of the pull–push formalism. The main tool is the formula for the obstruction bundle of Jarvis–Kaufmann–Kimura. This trivialization can be interpreted as defining the physics notion of twist fields. We end with an outlook on applications to singularities with symmetries aka. orbifold Landau–Ginzburg models.     

The magnetic susceptibility of two-dimensional Ising model on a finite-size lattice

The form factor representation of the correlation function of the 2D Ising model on a cylinder is generalized to the case of arbitrary arrangement of correlating spins. The magnetic susceptibility on a lattice, one of whose dimensions (N) is finite, is calculated in both the ferromagnetic and the paramagnetic region of the parameters of the model. The structure of singularities of susceptibility in the complex temperature plane at finite values of N and the transition to the thermodynamic limit N→∞ are discussed.