Let (Ω, M, µ) be a measure space. Then for any two complex valued measurable functions f and g, we have the Cauchy-Schwarz inequality: 0.1 THEOREM (Cauchy-Schwarz inequality). Let (Ω, M, µ) be a measure space. Then for any two complex valued measurable functions f and g such that

A Theorem is proved which reduces the problem of completeness of orbits of Killing vector fields in maximal globally hyperbolic, say vacuum, space–times to some properties of the orbits near the Cauchy surface. In particular it is shown that all Killing orbits are complete in maximal developements of asymptotically flat Cauchy data, or of Cauchy data prescribed on a compact manifold.

Let $G$ be a compact Lie group. In this article, we investigate the Cauchy problem for nonlinear wave equation with viscoelastic damping on . More precisely, some $L^2$ -estimates solution to homogeneous damped utilizing group Fourier transform We also prove that there is no improvement of any decay rate norm $\|u(t,\,\cdot )\|_{L^2(G)}$ by further assuming $L^1(G)$ -regularity initial data. Fi...

Cauchy problems with SPDEs on the whole space are localized to Cauchy problems on a ball of radius R. This localization reduces various kinds of spatial approximation schemes to finite dimensional problems. The error is shown to be exponentially small. As an application, a numerical scheme is presented which combines the localization and the space and time discretization, and thus is fully impl...

We propose a new O(n)-space implementation of the GKO-Cauchy algorithm for the solution of linear systems with Cauchy-like matrix. Despite its slightly higher computational cost, this new algorithm makes a more efficient use of the processor cache memory. Thus, for matrices of size larger than n ≈ 500− 1000, it outperforms the existing algorithms. We present an applicative case of Cauchy-like m...

We consider a strictly hyperbolic n × n system of conservation laws in one space dimension ut + f(u)x = 0, together with Cauchy initial data u(0, x) = ū(x), that is a small BV ∩ L perturbation of fixed Riemann data (u− 0 , u 0 ). We a priori assume that the latter problem is solved by M large shocks (2 ≤ M ≤ n) of different characteristic families, each of them Majda stable and Lax compressive....

A singular Cauchy-Nicoletti problem for a system of three ordinary differential equations is considered. An approach which combines topological method of T. Ważewski and Schauder’s principle is used. Theorem concerning the existence of a solution of this problem (a graph of which lies in a given domain) is proved. Moreover, an estimation of its coordinates is obtained.

We characterize hyperbolic evolution families on a Banach space X by means of spectral properties of the induced semigroup on the spaces C 0 (IR; X) and L p 1. Exponentially dichotomic and hyperbolic evolution families Given a non{autonomous Cauchy problem _ u(t) = A(t)u(t); u(s) = x s 2 X ; t s 2 IR; (nCP) on a Banach space X with possibly unbounded operators A(t), t 2 IR, on X, the solutions ...

With a semiclassical polymerization in the loop quantum gravity (LQG), interior of Schwarzschild black holes provides captivating single-horizon regular hole spacetime. The shortage rotating models (LQG) substantially restrains progress testing LQG from observations. Motivated by this, starting with spherical as seed metric, we construct spacetime using revised Newman-Janis algorithm, namely, L...

In this paper, we consider in R the Cauchy problem for nonlinear Schrödinger equation with initial data in Sobolev space W s,p for p < 2. It is well known that this problem is ill posed. However, We show that after a linear transformation by the linear semigroup the problem becomes locally well posed in W s,p for 2n n+1 < p < 2 and s > n(1− 1 p ). Moreover, we show that in one space dimension, ...