We discuss some aspects of recent research as well as more general issues about motivation, useful methods and open problems in the field of cosmological singularities. In particular, we review some of the approaches to the general area and include discussions of the method of asymptotic splittings, singularity and completeness theorems and the use of the Bel-Robinson energy to prove completene...

In a one parameter family of dilaton gravity theories which allow the coupling of the dilaton to gravity and to a U(1) gauge field to differ, we have found the existence of everywhere regular spacetimes describing black holes hiding expanding universes inside their horizon.

For the multivariate one-sample location model (relating to a diagonally symmetric distribution). sequential non-parametric (point as well as interval) estimators based on appropriate rank statistics are considered and their asymptotic properties studied. In this context. asymptotic risk-efficiency of the proposed estimators and asymptotic normality of the associated stopping times are establis...

the aim of this paper is to present a numerical method for singularly perturbed convection-diffusion problems with a delay. the method is a combination of the asymptotic expansion technique and the reproducing kernel method (rkm). first an asymptotic expansion for the solution of the given singularly perturbed delayed boundary value problem is constructed. then the reduced regular delayed diffe...

It is now well known that standard asymptotic inference techniques for instrumental variable estimation may perform very poorly in the presence of weak instruments. In some circumstances, standard asymptotic techniques give spuriously small standard errors, leading investigators to apparently tight confidence regions which may be very far from the true parameter of interest. While much research...

We study the rate of convergence of linear two-time-scale stochastic approximation methods. We consider two-time-scale linear iterations driven by i.i.d. noise, prove some results on their asymptotic covariance and establish asymptotic normality. The well-known result [Polyak, B. T. (1990). Automat. Remote Contr. 51 937–946; Ruppert, D. (1988). Technical Report 781, Cornell Univ.] on the optima...

We perform the asymptotic enumeration of two classes of rooted maps on orientable surfaces:m-hypermaps andm-constellations. Form = 2 they correspond respectively to maps with even face degrees and bipartite maps. We obtain explicit asymptotic formulas for the number of such maps with any finite set of allowed face degrees. Our proofs combine a bijective approach, generating series techniques re...

Global asymptotic stability conditions for discrete nonlinear scalar stochastic systems with state delay are obtained based on the convergence theorem for semimartingale inequalities, without assuming the Lipschitz conditions for nonlinear drift functions. The Lyapunov-Krasovskii and degenerate functionals techniques are used. The derived stability conditions are directly expressed in terms of ...

In this paper, the asymptotic representation of the corresponding eigenfunctions of the eigenvalues has been investigated. Furthermore, we obtain the zeros of eigenfunctions.