New Planck scale physics may solve the singularity problems of classical general relativity and may lead to interesting consequences for very early Universe cosmology. Two approaches to these questions are reviewed in this article. The first is an effective action approach to including the effects of Planck scale physics in the basic framework of general relativity. It is shown that effective a...

A model for the pinching of an inviscid liquid column, originally developed by Lee, is considered. We find that Lee’s model has a singularity in the curvature at a finite thread radius, and therefore does not describe breakup. However, the observed steepening of the profile corresponds to experiments with fluids at low viscosity. The singularity has similarity form, which we compute analyticall...

We study the self-tuning of general brane junctions and brane networks on the 6-dimensional space-time. For the general brane junctions, there may exist one fine-tuning among the brane tensions. For the brane networks, similar to the 5-dimensional self-tuning brane models, the brane tensions can be set arbitrarily and there exists the singularity for the metric and bulk scalar. And if we want t...

Stability and convergence of a time-weighted discrete scheme with nonuniform time steps are established for linear reaction-subdiffusion equations. The Caupto derivative is approximated at an offset point by using quadratic polynomial interpolation. Our analysis relies on two tools: fractional Gr\"{o}nwall inequality the global consistency analysis. new makes use interpolation error formula pol...

We study singular real-analytic Levi-flat hypersurfaces in complex projective space. We give necessary and sufficient conditions for such a hypersurface to be a pullback of a real-analytic curve in C via a meromorphic function. We define the rank of a real hypersurface and study the connections between rank, degree, and the type and size of the singularity for Levi-flat hypersurfaces. Finally, ...

We study the multiplicity modulo 2 of real analytic hypersurfaces. We prove that, under some assumptions on the singularity, the multiplicity modulo 2 is preserved by subanalytic bi-Lipschitz homeomorphisms of R. In the first part of the paper, we find a subset of the tangent cone which determines the multiplicity mod 2 and prove that this subset of S is preserved by the antipodal map. The stud...

The quotient-cusp singularities are isolated complex surface singularities that are double-covered by cusp singularities. We show that the universal abelian cover of such a singularity, branched only at the singular point, is a complete intersection cusp singularity of embedding dimension 4. This supports a general conjecture that we make about the universal abelian cover of a Q-Gorenstein sing...

in this paper, we obtain a suitable mathematical model for the seismic response of dams. by using the shear beam model (sb model), we give a mathematical formulation that it is a partial differential equation and transform it to the sturm-liouville equation.

Conditions under which gravity coupled to self interacting scalar field determines singularity formation are found and discussed. It is shown that, under a suitable matching with an external space, the boundary, if collapses completely, may give rise to a naked singularity. Issues related to the strength of the singularity are discussed.

Abstract. We prove an inequality on the Kantorovich-Rubinstein distance – which can be seen as a particular case of a Wasserstein metric– between two solutions of the spatially homogeneous Boltzmann equation without angular cutoff, but with a moderate angular singularity. Our method is in the spirit of [7]. We deduce some well-posedness and stability results in the physically relevant cases of ...