Related papers: P.d.e.'s which imply the Penrose conjecture
We suggest a novel extension to the Kaluza-Klein scheme that allows us to obtain consistently all SU(n) Einstein-Yang-Mills theories. This construction is based on allowing the five-dimensional spacetime to carry some non-vanishing torsion;…
The Wigner-Dyson-Gaudin-Mehta conjecture asserts that the local eigenvalue statistics of large real and complex Hermitian matrices with independent, identically distributed entries are universal in a sense that they depend only on the…
Coarse spaces are essential to ensure robustness w.r.t. the number of subdomains in two-level overlapping Schwarz methods. Robustness with respect to the coefficients of the underlying partial differential equation (PDE) can be achieved by…
In this article, under mild constraints on the sectional curvature, we exploit a divergence formula for symmetric endomorphisms to deduce a general Poincar\'e type inequality. We apply such inequality to higher-order mean curvature of…
We estabish an analog of the Cauchy-Poincare separation theorem for normal matrices in terms of majorization. Moreover, we present a solution to the inverse spectral problem (Borg-type result) for a normal matrix. Using this result we…
We consider complete asymptotically flat Riemannian manifolds that are the graphs of smooth functions over $\mathbb R^n$. By recognizing the scalar curvature of such manifolds as a divergence, we express the ADM mass as an integral of the…
Many physical systems are described by partial differential equations (PDEs). Determinism then requires the Cauchy problem to be well-posed. Even when the Cauchy problem is well-posed for generic Cauchy data, there may exist characteristic…
We derive generalizations of McShane's identity for higher ranked surface group representations by studying a family of mapping class group invariant functions introduced by Goncharov and Shen which generalize the notion of horocycle…
Solving partial differential equations (PDEs) using an annealing-based approach involves solving generalized eigenvalue problems. Discretizing a PDE yields a system of linear equations (SLE). Solving an SLE can be formulated as a general…
This article is a guide to theorems on existence and global dynamics of solutions of the Einstein equations. It draws attention to open questions in the field. The local in time Cauchy problem, which is relatively well understood, is…
In this paper we investigate the Cauchy problem of d-dimensional Euler-Poincar\'{e} equations. By choosing a class of new and special initial data, we can transform this d-dimensional Euler-Poincar\'{e} equations into the Camassa-Holm type…
The Alder-Andrews Theorem, a partition inequality generalizing Euler's partition identity, the first Rogers-Ramanujan identity, and a theorem of Schur to $d$-distinct partitions of $n$, was proved successively by Andrews in 1971, Yee in…
In this series of studies on Cauchy's function $f(z)$ ($z=x+iy$) and its integral $J[f(z)]\equiv (2\pi i)^{-1}\oint_C f(t)dt/(t-z)$ taken along a Jordan contour $C$, the aim is to investigate their comprehensive properties over the entire…
Chung-Grigor'yan-Yau's inequality describes upper bounds of eigenvalues of Laplacian in terms of subsets ("input") and their volumes. In this paper we will show that we can reduce "input" in Chung-Grigor'yan-Yau's inequality in the setting…
We give explicit formulas as well as a quadratic time algorithm to solve (so called) generalized Vandermonde's systems of p linear equations and n variables. It allows in particular to find all (so called Lagrange's) interpolation polynoms…
Discrete state spaces represent a major computational challenge to statistical inference, since the computation of normalisation constants requires summation over large or possibly infinite sets, which can be impractical. This paper…
Based on the reduction of degree in polynomial mappings and some known results in algebraic geometry, by introducing the Brouwer degree, a tool from differential topology, algebraic topology and algebraic geometry, we completely prove the…
In this paper, we consider the Cauchy problem for semi-linear wave equations with structural damping term $\nu (-\Delta)^2 u_t$, where $\nu >0$ is a constant. As being mentioned in [8,10], the linear principal part brings both the diffusion…
The Riemannian Penrose inequality is a remarkable geometric inequality between the ADM mass of an asymptotically flat manifold with non-negative scalar curvature and the area of its outermost minimal surface. A version of the Riemannian…
We propose a geometric inequality for two-dimensional spacelike surfaces in the Schwarzschild spacetime. This inequality implies the Penrose inequality for collapsing dust shells in general relativity, as proposed by Penrose and Gibbons. We…