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We re-analyze the observed magnitude-redshift relation of type Ia supernovae (SNe Ia) and examine the possibility that the apparent acceleration of the cosmic expansion is not caused by dark energy but is instead a consequence of the…

Propagation of light in nonlinear materials is here studied in the regime of the geometrical optics. It is shown that a spherically symmetric medium at rest with some specific dielectric properties can be used to produce an exact analogue…

We derive the relationship of the redshift and the angular diameter distance to the average expansion rate for universes which are statistically homogeneous and isotropic and where the distribution evolves slowly, but which have otherwise…

It has been suggested recently that the appparent accelerated expansion of the universe could be explained by a bias in the SNIa measurements. Such events indeed occur mainly in overdense regions, where matter is located, and whose dynamics…

An important open question in cosmology is the degree to which the Friedmann-Lemaitre-Robertson-Walker (FLRW) solutions of Einstein's equations are able to model the large-scale behaviour of the locally inhomogeneous observable universe. We…

In this note, we propose Bernstein's problem and De Giorgi's conjecture for spatially inhomogeneous equations, as well as De Giorgi's conjecture for system of reaction-diffusion equations.

We compare relativistic approximation methods, which describe gravitational instability in the expanding universe, in a spherically symmetric model. Linear perturbation theory, second-order perturbation theory, relativistic Zel'dovich…

We challenge the widely held belief that the cosmological principle is an obvious consequence of the observed isotropy of the cosmic microwave background radiation, combined with the Copernican principle. We perform a detailed analysis of a…

The diffusion approximation of stochastic gradient descent (SGD) in current literature is only valid on a finite time interval. In this paper, we establish the uniform-in-time diffusion approximation of SGD, by only assuming that the…

We consider cosmological models in scalar tensor theories of gravity that describe an accelerating universe, and we study a family of inverse power law potentials, for which exact solutions of the Einstein equations are known. We also…

First, we consider the problem of hedging in complete binomial models. Using the discrete-time F\"ollmer-Schweizer decomposition, we demonstrate the equivalence of the backward induction and sequential regression approaches. Second, in…

For time-homogeneous stochastic differential equations (SDEs) it is enough to know that the coefficients are Lipschitz to conclude existence and uniqueness of a solution, as well as the existence of a strongly convergent numerical method…

Self-similar solutions of the coherent diffusion equation are derived and measured. The set of real similarity solutions is generalized by the introduction of a nonuniform phase surface, based on the elegant Gaussian modes of optical…

We study the bending of light for static spherically symmetric (SSS) space-times which include a dark energy contribution. Geometric dark energy models generically predict a correction to the Einstein angle written in terms of the distance…

Extending the approach developed by Ara\'ujo and Stoeger [1] and improved in Ara\'ujo {\it et al} [2], we have shown how to construct dust-filled $\Lambda \neq 0$ Friedmann-Lema\^{\i}tre-Robertson-Walker (FLRW) cosmological models from FLRW…

The classical Reifenberg's theorem says that a set which is sufficiently well approximated by planes uniformly at all scales is a topological H\"older manifold. Remarkably, this generalizes to metric spaces, where the approximation by…

We review various cosmological models with a local underdense region (local void) and the averaged models with the backreaction of inhomogeneities, which have been proposed to explain (without assuming a positive cosmological constant) the…

We present exact solutions in Einstein-aether theory in a static spherically symmetric background space with a spacelike aether field, as a difference with the usual selection of timelike aether field. We assume a coupling between the…

In a recent paper Davis et al. make the counter intuitive assertion that a galaxy held `tethered' at a fixed distance from our own could emit blueshifted light. Moreover, this effect may be derived from the simplest…

Working in a semi-classical setting, we consider solutions of the Einstein equations that exhibit light trapping in finite time according to distant observers. In spherical symmetry, we construct near-horizon quantities from the assumption…