Related papers: The Cauchy problem for indefinite improper affine …
We introduce a new family of invariants of real algebraic sets defined in terms of the topology of their complexifications and compute some of these invariants for spheres. This allows us to completely classify topological isomorphism…
We consider the conformal Einstein equations for massless collisionless gas cosmologies which admit an isotropic singularity. It is shown that the Cauchy problem for these equations is well-posed with data consisting of the limiting…
We consider the Cauchy problem for incompressible viscoelastic fluids in the whole space $\mathbb{R}^d$ ($d=2,3$). By introducing a new decomposition via Helmholtz's projections, we first provide an alternative proof on the existence of…
The classical Cauchy completion of a metric space (by means of Cauchy sequences) as well as the completion of a uniform space (by means of Cauchy filters) are well-known to rely on the symmetry of the metric space or uniform space in…
The geodesic total curvature of rectifiable spherical curves is analyzed. We extend to the case of high dimension spheres the explicit formula that holds true for curves supported into the 2-sphere. For this purpose, we take advantage of…
We develop integral geometry for non-compactly causal symmetric spaces. We define a complex horospherical transform and, for some cases, identify it with a Cauchy type integral.
We consider the Cauchy problem for a second-order nonlinear evolution equation in a Hilbert space. This equation represents the abstract generalization of the Ball integro-differential equation. The general nonlinear case with respect to…
We consider the Cauchy problem for the wave equation in the whole space, R^n, with initial data which are distributions supported on finite sets. The main result is a precise description of the geometry of the sets of stationary points of…
This paper is concerned with the analysis of the Cauchy problem of a general class of two-dimensional nonlinear nonlocal wave equations governing anti-plane shear motions in nonlocal elasticity. The nonlocal nature of the problem is…
The global characteristic initial value problem for linear wave equations on globally hyperbolic Lorentzian manifolds is examined, for a class of smooth initial value hypersurfaces satisfying favourable global properties. First it is shown…
We consider a Cauchy problem for an overdetermined system of PDEs, and give necessary and sufficient conditions for solvability of this Cauchy problem for all data. As an application, we find all real tube hypersurfaces in complex space…
We show the existence of a complete, strictly locally convex hypersurface within $\mathbb{H}^{n+1}$ that adheres to a curvature equation applicable to a broad range of curvature functions. This hypersurface possesses a prescribed asymptotic…
We study rotational hypersurfaces with constant Gauss-Kronecker curvature. We solve the ODE for the generating curves of such hypersurfaces and analyze several geometric properties of such hypersurfaces. In particular, we discover a class…
We study the basic geometric properties of an indefinite locally conformal Kaehler manifold.
One of the major problems in the structural theory of skew braces consists in the classification of skew braces of finite order up to isomorphism. In this light, the open question of the existence of a Cauchy theorem for finite skew braces…
In this paper we provide a framework for the study of isoperimetric problems in finitely generated group, through a combinatorial study of universal covers of compact simplicial complexes. We show that, when estimating filling functions,…
For $S$ a closed surface of genus at least $2$, let $\mathrm{Hit}_3(S)$ be the Hitchin component of representations to $\mathrm{SL}(3,\mathbb{R}),$ equipped with the Labourie-Loftin complex structure. We construct a mapping class group…
We establish the local well-posedness for a new nonlinearly dispersive wave equation and we show that the equation has solutions that exist for indefinite times as well as solutions which blowup in finite times. Furthermore, we derive an…
We give an algebraic description of wave fronts that appear in strictly hyperbolic Cauchy problem. Concrete form of definig function of wave front issued from initial algebraic variety is obtained by the aid of the Gauss-Manin systems…
In a recent paper, it was shown that the problem of existence of a continuous map $X \to Y$ extending a given map $A \to Y$ defined on a subspace $A \subseteq X$ is undecidable, even for $Y$ an even-dimensional sphere. In the present paper,…