We consider two quantum cosmological models with a massive scalar field: an ordinary Friedmann universe and a universe containing primordial black holes. For both models we discuss the complex solutions to the Euclidean Einstein equations. Using the probability measure obtained from the Hartle-Hawking no-boundary proposal, we find that the only unsuppressed black holes start at the Planck size ...

This paper studies gravitational collapse of a complex scalar field at the threshold for black hole formation, assuming that the collapse is spherically symmetric and continuously self-similar. A new solution of the coupled Einstein-scalar field equations is derived, after a small amount of numerical work with ordinary differential equations. The universal scaling and echoing behavior discovere...

Let A belong to an automorphism group, Lie algebra or Jordan algebra of a scalar product. When A is factored, to what extent do the factors inherit structure from A? We answer this question for the principal matrix square root, the matrix sign decomposition, and the polar decomposition. For general A, we give a simple derivation and characterization of a particular generalized polar decompositi...

In this paper, we study the rate of convergence in periodic homogenization of scalar ordinary differential equations. We provide a quantitative error estimate between the solutions of a first-order ordinary differential equation with rapidly oscillating coefficients, and the solution of the limiting homogenized equation. As an application of our result, we obtain an error estimate for the solut...

This paper studies the gravitational collapse of a complex scalar Beld at the threshold for black hole formation, assuming that the collapse is spherically symmetric and continuously self-similar. A new solution of the coupled Einstein-scalar Beld equations is derived, after a small amount of numerical work with ordinary difFerential equations. The universal scaling and echoing behavior discove...

It is well known [1, 2] that the Lorentz part of any quantum (or Poisson) Poincaré group is triangular. This is in fact a general feature, which excludes the standard q-deformation from the context of inhomogeneous quantum groups [3]. In order to make the standard q-deformation compatible with inhomogeneous groups one has to consider some generalization of the notion of quantum (Poisson) group,...

Symmetries of the fundamental first integrals for scalar second-order ordinary differential equations ODEs which are linear or linearizable by point transformations have already been obtained. Firstly we show how one can determine the relationship between the symmetries and the first integrals of linear or linearizable scalar ODEs of order two. Secondly, a complete classification of point symme...

Since their Newtonian inception, differential equations have been a fundamental tool for modeling the natural world. As the name suggests, these equations involve the derivatives of dependent variables (e.g. viral load, species densities, genotypic frequencies) with respect to independent variables (e.g. time, space). When the independent variable is scalar, the differential equation is called ...