Spherical conformal models for compact stars

We consider spherical exact models for compact stars with anisotropic pressures and a conformal symmetry. The conformal symmetry condition generates an integral relationship between the gravitational potentials. We solve this condition to find a new anisotropic solution to the Einstein field equations. We demonstrate that the exact solution produces a relativistic model of a compact star. The model generates stellar radii and masses consistent with PSR J1614-2230, Vela X1, PSR J1903+327 and Cen X-3. A detailed physical examination shows that the model is regular, well behaved and stable. The mass–radius limit and the surface red shift are consistent with observational constraints.     

Shear-free spherically symmetric solutions with conformal symmetry

Dyer, McVittie and Oattes (1987) presented the field equations for shearfree perfect fluid spacetimes which are spherically symmetric and admit a conformal symmetry. Two special solutions of these equations are found. Furthermore, in the general case, one field equation is solved in terms of a Painlevé transcendent, while the remaining equation is reduced to an Emden-Fowler equation.     

Classes of exact Einstein–Maxwell solutions

We find new classes of exact solutions to the Einstein–Maxwell system of equations for a charged sphere with a particular choice of the electric field intensity and one of the gravitational potentials. The condition of pressure isotropy is reduced to a linear, second order differential equation which can be solved in general. Consequently we can find exact solutions to the Einstein–Maxwell field equations corresponding to a static spherically symmetric gravitational potential in terms of hypergeometric functions. It is possible to find exact solutions which can be written explicitly in terms of elementary functions, namely polynomials and product of polynomials and algebraic functions. Uncharged solutions are regainable with our choice of electric field intensity; in particular we generate the Einstein universe for particular parameter values.     

Generating Interior Sources for the Reissner-Nordström Metric

We study the Einstein-Maxwell equations for isotropic pressure distributions. We postulate a relationship between the electric field intensity and one of the gravitational potentials. An algorithm is developed that allows us to systematically generate new classes of exact solutions for charged relativistic stars. The solutions are expressed in terms of simple elementary functions; it is possible to parametrize the solutions so that different values of a constant allows us to tabulate the models. For a particular class it is possible to generate models without any integration. We study the qualitative features of a particular solution, and show that it is physically reasonable in the region of a spherical shell surrounding the centre.     

First integrals for charged perfect fluid distributions

We study the evolution of shear-free spherically symmetric charged fluids in general relativity. We find a new parametric class of solutions to the Einstein-Maxwell system of field equations. Our charged results are a generalisation of earlier treatments for neutral relativistic fluids. We regain the first integrals found previously for uncharged matter as a special case. In addition an explicit first integral is found which is necessarily charged.     

Homogeneous Embeddings of Cycles in Graphs

If G and H are vertex-transitive graphs, then the framing number fr(G,H) of G and H is defined as the minimum order of a graph every vertex of which belongs to an induced G and an induced H. This paper investigates fr(C _m,C _n) for m<n. We show first that fr(C _m,C _n)≥n+2 and determine when equality occurs. Thereafter we establish general lower and upper bounds which show that fr(C _m,C _n) is approximately the minimum of and n+n/m. Received: June 12, 1996 / Revised: June 2, 1997     

On the probability of Magnus Carlsen reaching 2900

How likely is it that Magnus Carlsen will achieve his goal of a 2900 Elo rating? At what level of play does Magnus have a reasonable chance of reaching the 2900 goal? These two questions are of great current interest to Magnus and the chess community. The probabilistic properties of Elo's rating system are well known, and together with a Brownian motion model of rating evolution, we use simulation-based methods to address these questions. Our model assesses that Magnus has a 4.5% chance of reaching 2900 if he continues his 2020–2022 level of play. However, this increases dramatically to $80$% chance if he can repeat his hot streak performance of 2019 which is not an easy undertaking. The probabilities are intimately related to Elo's choice $K$-factor used for grandmaster chess play. Finally, we conclude with a discussion of the policy issues involved with the choice of $K$-factor.