An interior solution for the gamma metric

Interior solutions for a static, axially symmetric family of solutions of Einstein's equations are described. The interior solutions correspond to spatially bound matter and are properly matched to an exterior vacuum solution. The family of solutions discussed include the Schwarzschild solution as a special case. A general method is exhibited for transforming any spherically symmetric interior solution to an interior for the other members of the family of solutions. The energy density remains positive for at least a finite range of the parameter that describes the family of solutions. Two solutions are explicitly exhibited. One is transformed from the constant density Schwarzschild interior solution and one from the Adler interior solution. The first solution would be expected to be unstable under adiabatic perturbations of the matter, the second would be expected to be stable.Supported in part by The National Science Foundation under Grant No. INT 782-5663.Supported in part by Consejo Nacional de Investigaciones Cientificas y Technologicas (CONICIT), Venezuela.     

Factorization of Correlation Functions in Coset Conformal Field Theories

We use the conformal Ward identities to study the structure of correlation functions in coset conformal field theories. For a large class of primary fields of arbitrary g/h theory, a factorization anzatz is found. Corresponding correlation functions are explicitly expressed in terms of correlation functions of two independent WZNW theories for g and h.     

A class of solutions of Einstein's equations for the interior of a rigidly rotating perfect fluid

A general class of solutions of Einstein's equations for the interior of a rigidly rotating axisymmetric perfect fluid is presented, which depends on an arbitrary function. To get solutions explicitly one has to calculate two integrals involving the arbitrary function. The equipressure surfaces of all solutions of the class are spheres or planes. A family of solutions, which depend on four arbitrary real constants, is calculated explicitly. The solution of the family, which is obtained if we assign a specific value to one of its parameters, and which was found before, is futher generalized with the addition of one more parameter.     

Kink stability of isothermal spherical self-similar flow revisited

The problem of kink stability of isothermal spherical self-similar flow in newtonian gravity is revisited. Using distribution theory we first develop a general formula of perturbations, linear or non-linear, which consists of three sets of differential equations, one in each side of the sonic line and the other along it. By solving the equations along the sonic line we find explicitly the spectrum, k, of the perturbations, whereby we obtain the stability criterion for the self-similar solutions. When the solutions are smoothly across the sonic line, our results reduce to those of Ori and Piran. To show such obtained perturbations can be matched to the ones in the regions outside the sonic line, we study the linear perturbations in the external region of the sonic line (the ones in the internal region are identically zero), by taking the solutions obtained along the line as the boundary conditions. After properly imposing other boundary conditions at spatial infinity, we are able to show that linear perturbations, satisfying all the boundary conditions, exist and do not impose any additional conditions on k. As a result, the complete treatment of perturbations in the whole spacetime does not alter the spectrum obtained by considering the perturbations only along the sonic line.     

Collapsing shear-free radiating fluid spheres

We present a new class of exact solutions of relativistic field equations for a collapsing spherically symmetric shear-free isotopic fluid undergoing radial heat flow. The interior solutions are matched with Vaidya exterior metric over the boundary. Initially the interior solutions represent a static configuration of perfect fluid which then gradually starts evolving into radiating collapse.     

Symmetric solutions to the four dimensional degenerate Painlevé type equation NYA4

We have classified symmetric solutions around the origin to the four dimensional degenerate Painlevé type equation NY^A₄ with generic values of parameters. We obtained sixteen meromorphic solutions, which are transformed each other by the Bäcklund transformation. We calculated the linear monodromy for one of them, explicitly.     

On nodal solutions of the Yamabe equation on products

Let (M,g), (N,h) be closed Riemannian manifolds of constant scalar curvature. We prove the existence of nodal solutions of the Yamabe equation on the Riemannian product which depend on only one of the factors. We do this by studying the second Yamabe invariant introduced by Ammann and Humbert. We work out the case when M=S¹ explicitly showing the existence of an infinite number of solutions.     

SU(N) Yang-Mills solutions with constant field strength onT 4

We study forT ⁴ the class of solutions to the SU(N) Yang-Mills equations with constant field strength. The fluctuation spectrum is explicitly calculated in terms of generalized Riemann theta functions. We show that if these solutions are stable, they are necessarily (anti)-selfdual, in which case we verify the index theorem.     

A new class of interior solutions for a stress-energy tensor of Segré characteristic [111, 1]

The interior solutions of (the tetrad versions of) Einstein's field equations withT _AB having Segré characteristic [111, 1] (which has all four eigenvalues distinct), are investigated. For this purpose amixed method, which combines Synge'sg method andT method, is introduced. Some of the tetrad equations are solved for the metric functions while the remaining equations are used to define the corresponding components ofT _AB . As necessary conditions of the consistency of the mixed method the conservation equationsT ^AB _B =0 are explicitly verified. Several simplifications and analysis of some differential inequalities show the existence of a new class of solutions which, in addition to having Segré characteristic [111, 1], also satisfy the strong energy conditions of Hawking and Ellis.This Author is a member of the Theoretical Sciences Institute, Simon Fraser University, Burnaby, B.C., V5A 1S6, Canada.     

Computational soliton solutions to $$$$-dimensional generalised Kadomtsev–Petviashvili and $$(2+1)$$(2+1)-dimensional Gardner–Kadomtsev–Petviashvili models and their applications

In this paper, we present a formalism to generate a family of interior solutions to the Einstein–Maxwell system of equations for a spherically symmetric relativistic charged fluid sphere matched to the exterior Reissner–Nordström space–time. By reducing the Einstein–Maxwell system to a recurrence relation with variable rational coefficients, we show that it is possible to obtain closed-form solutions for a specific range of model parameters. A large class of solutions obtained previously are shown to be contained in our general class of solutions. We also analyse the physical viability of our new class of solutions.     

Behavior as t → ∞ of solutions of a problem in mathematical physics

A class of solutions, decaying as t → ∞, of a two-dimensional model problem on the oscillations of an ideal rotating fluid in some domains with angular points is constructed explicitly. The existence of solutions whose L ₂-norms decrease more rapidly than any negative power of t is established.     

Generating functions for the affine symplectic group

The generating function notion is used to give a representation of the inhomogeneous symplectic group as group of affine canonical transformations. Then the classical action for linear mechanical systems, the Hamiltonians of which belong to the algebrah sp(2n,R), is deduced; it is explicitely constructed for all the Hamiltonians belonging to some particular subalgebras ofh sp(2n,R). The metaplectic representation ofW Sp(2n,R) onL ²(R) and the solutions of the Schrödinger equation for linear systems are also obtained in terms of generating functions. The Maslov index is explicitly constructed for the quantum corresponding sets of Hamiltonians considered in the classical case.Members of the Centre National de la Recherche Scientifique (France)Recipient of aid from the Ministère de l'Education du Gouvernement du Québec     

Anisotropic static solutions in modelling highly compact bodies

Einstein field equations for static anisotropic spheres are solved and exact interior solutions obtained. This paper extends earlier treatments to include anisotropic models which accommodate a wider variety of physically viable energy densities. Two classes of solutions are possible. The first class contains the limiting caseμ,∝ r^-2 for the energy density which arises in many astrophysical applications. In the second class the singularity at the centre of the star is not present in the energy density.     

Quasi-Stationary States for Particle Systems in the Mean-Field Limit

We prove that the N particles approximation of a class of stable stationary solutions of the Vlasov equation is uniformly valid on a time scale N ^β for β>0 (explicitly given in various cases) much longer than the usual log N scale. The vortex blob method in dimension 2 is also discussed. The result applies to a class of stationary solutions more general than in a previous work.     

Exact anisotropic viscous fluid solutions of Einstein's equations

A method for obtaining anisotropic, rotationless viscous fluid matter solutions of Bianchi type I and Segré type [1, 111] with the barotropic equation of state is presented. Solutions for which the anisotropy decreases exponentially or with a power law as well as solutions with average Hubble parameterH t ^–1 are discussed. Also, a class of solutions with constant anisotropy and Bianchi type VI_h is found. The dominant energy condition holds and the transport coefficients show the right sign.     

A Class of Charged Relativistic Superdense Star Models

In this article we obtain a new class of well behaved charged solutions by using particular forms of the metric potential g ₄₄ and electric intensity, which involves a parameter K. The metric describing the superdense stars joins smoothly with the Reissner-Nordstrom metric at the pressure free boundary. This class of solutions describes well behaved charged fluid balls. The class of solutions gives range of parameter K (0.13≤K≤1.9999) for which the solution is well behaved hence, suitable for modeling of super dense star. The interior of the stars possess there energy density, pressure, pressure-density ratio and velocity of sound to be monotonically decreasing towards the pressure free interface. In view of the surface density 2×10¹⁴ gm/cm³, the maximum mass of the charged fluid balls and corresponding radius are 0.4711M _Θ and 7.0122 km. The red shift at the centre and boundary are found to be 0.1640 and 0.1100 respectively.     

A study on the determination of mechanical properties of a power law material by its indentation force–depth curve

In this study a novel optimization approach is proposed to extract mechanical properties of a power law material whose stress–strain relationship may be expressed as a power law from its given experimental indentation P–h curve. A set of equations have been established to relate the P–h curve to mechanical properties E, σ _y and n of the material. For the loading part of a P–h curve this approach is based on the assumption that the indentation response of an elastic–plastic material is a linear combination of the corresponding elastic and elastic–perfect plastic materials. For the unloading part of the P–h curve it is based on the assumption that the unloading response of the elastic–plastic material is a linear combination of the full contact straight line and the purely elastic curve. Using the proposed optimization approach it was found that the mechanical properties of an elastic–plastic material usually cannot be decided uniquely by using only a single indentation P–h curve of the material. This is because in general a few matched sets of mechanical properties were found to produce a given P–h curve. It is however possible to identify the best matched set of mechanical properties by knowing some background information of the material. If the best matched material is identified, the predictions of mechanical properties are quite accurate.     

Study of CdSe/CdS quantum dots in solutions and gels by small-angle X-ray scattering

The sizes of semiconductor nanocrystals of CdSe/CdS quantum dots (QDs) synthesized by the colloidal method were estimated using small-angle X-ray scattering. The distribution of QD nanocrystals in organic solvents of different polarities and in polymer gels and matrices is studied. Structural invariants of scattering QD particles (heterogeneities of the electron density)—namely, inertia radii and sizes, forms, and dispersive composition of particles—are determined. The contribution of scattering by QDs in solutions and gels is calculated. The effective sizes of particles and their aggregates are determined, and the parameters of the distribution over the QD sizes in organic solvents and polymer matrices are estimated. The typical distance between particles in samples is determined. The position of the maximum at the beginning of the small-angle scattering curves corresponds to the distance d _m = 2π/h ₀ between the planes (here, h ₀ is the position of the maximum on the scale h). It is 74–76.9 ? for solutions, 60 ? for gels, and 99 ? for polymer matrices with concentrations of up to 0.15% and 77 ? for those with the concentrations exceeding 0.15%, which is close to the estimation of the sizes of separate CdSe QDs that was obtained from the distribution histograms (60–80 ?). This result shows that CdSe/CdS QDs introduced in the polymer matrices disperse to form either separate particles or small aggregates and located at a distance on the order of 80 ? from each other.     

Part 1 : Topological aspects of Yang-Mills fields in curved spaces. (Exact solutions)

To explore in its full richness the topological possibilities of gauge fields one should allow for simultaneous presence of gravitational and Yang-Mills ones. Thus if the integral topological indices of the Yang-Mills field for a flat Euclidean base space is associated with the structure of the vacuum, one may ask among other questions of interest, how this spectrum might be modified when the base space itself has non trivial indices. Exact solutions of SU(2) Yang-Mills fields are presented for metrics corresponding to well-known gravitational instantons. Such selfdual solutions, with vanishing energy monien-tunl tensor T_μv for Euclidean signature of the base space, do not perturb the metric. Thus they provide solutions of the combined gravitational-Y.M. system. New topological possibilities, such as finite action SU(2) fields with fractional indices for many centre inetrics are displayed explicitly. As another type of possibility non selfdual, finite action solutions are constructed explicitly on Schwarzschild and de Sitter metrics, the solution being real in the first and complex in second case respectively. It is also shown how various meron type solutions in flat space can be derived systematically from a very simple static solution in de Sitter.