Related papers: A Coordinate-Dependent Local Resolution of Singula…
The first part of my thesis lays the foundations to generalized Lorentz geometry. The basic algebraic structure of finite-dimensional modules over the ring of generalized numbers is investigated. The motivation for this part of my thesis…
Harmonic slicing has in recent years become a standard way of prescribing the lapse function in numerical simulations of general relativity. However, as was first noticed by Alcubierre (1997), numerical solutions generated using this…
We prove existence and uniqueness of solutions for an entropic version of the semi-geostrophic equations. We also establish convergence to a weak solution of the semi-geostrophic equations as the entropic parameter vanishes. Convergence is…
Singular equations with rank-deficient Jacobians arise frequently in algebraic computing applications. As shown in case studies in this paper, direct and intuitive modeling of algebraic problems often results in nonisolated singular…
Image data are often composed of two or more geometrically distinct constituents; in galaxy catalogs, for instance, one sees a mixture of pointlike structures (galaxy superclusters) and curvelike structures (filaments). It would be ideal to…
Equivariant localization expresses global invariants in terms of local invariants, and many of them appearing in equivariant index theory, (holomorphic) Morse theory, geometric quantization and supersymmetric localization can be…
We introduce a method of rigorous analysis of the location and type of complex singularities for nonlinear higher order PDEs as a function of the initial data. The method is applied to determine rigorously the asymptotic structure of…
We give a simple algorithm showing that the reduction of the multiplicity of a characteristic p>0 hypersurface singularity along a valuation is possible if there is a finite linear projection which is defectless. The method begins with the…
This article presents a comprehensive and rigorous overview of spacetime singularities within the framework of classical General Relativity. Singularities are defined through the failure of geodesic completeness, reflecting the limits of…
We prove well-posedness and regularity results for elliptic boundary value problems on certain domains with a smooth set of singular points. Our class of domains contains the class of domains with isolated oscillating conical singularities,…
We study a Lorentzian version of the well-known Calder\'{o}n problem that is concerned with determination of lower order coefficients in a wave equation on a smooth Lorentzian manifold, given the associated Dirichlet-to-Neumann map. In the…
This paper considers the problem of solving a special quartic-quadratic optimization problem with a single sphere constraint, namely, finding a global and local minimizer of…
A differential geometric approach to singular perturbation theory is presented. It is shown that singular perturbation problems such as multiple-scale and boundary layer problems can be treated more easily on a differential geometric basis.…
In this paper, we prove several generalizations and applications of a fixed point theorem. This theorem is used to prove the existence and uniqueness of solutions of the linear sparse matrix problem considered.
We make a detailed investigation of the generic properties that polynomial mappings possess. An important starting point is the work by Farnik, Jelonek and Ruas in 2019, where they prove some of those properties in the context of…
Discrete gradient methods are geometric integration techniques that can preserve the dissipative structure of gradient flows. Due to the monotonic decay of the function values, they are well suited for general convex and nonconvex…
Through an illuminating thought experiment we demonstrate that the nonsingular "continued collapse" picture of a black hole is the only consistent and physical one. We provide a class exact solutions on the boundary of the space of physical…
For a Lipschitz domain we show that solutions of certain first order systems are unique. This result is then applied to prove a crucial step for showing Korn's first inequality as well as to prove the 'infinitesimal rigid displacement lemma…
In this paper, we investigate the formation of singularity for general two dimensional and radially symmetric solutions for rotating shallow water system from different aspects. First, the formation of singularity is proved via the study…
An iterative optimization method applied to a function $f$ on $\mathbb{R}^n$ will produce a sequence of arguments $\{\mathbf{x}_k\}_{k \in \mathbb{N}}$; this sequence is often constrained such that $\{f(\mathbf{x}_k)\}_{k \in \mathbb{N}}$…