Let b k := (C 2 A)(M b;k), then lim k!1 a k = lim k!1 b k. The proof is complete. 2 From the preceding lemmas, we obtain the following theorem. As fhg ? ; M k(m) ig is a Cauchy sequence, hg ? ; M m;1 i 0 hg ? ; M k(m) i ! 0 (as m ! 1): We obtain that lim m!1 b m = lim m!1 a k(m). The proof is complete. 2 Lemma 23 We obtain Proof It suces to prove that for a Cauchy sequence fa k g F such that li...

‎we study dual integral equations which appear in formulation of the‎ ‎potential distribution of an electrified plate with mixed boundary‎ ‎conditions‎. ‎these equations will be converted to a system of‎ ‎singular integral equations with cauchy type kernels‎. ‎using‎ ‎chebyshev polynomials‎, ‎we propose a method to approximate the‎ ‎solution of cauchy type singular integral equation which will ...

1 A Model Diffusion Problem . . . . . . . . . . . . . . . . . . . . 1 x.1 Domain . . . . . . . . . . . . . . . . . . . . . . . . . . 1 x.2 Continuous Function . . . . . . . . . . . . . . . . . . . 2 x.3 Normed Vector Space . . . . . . . . . . . . . . . . . . 2 x.4 Square Integrable Function . . . . . . . . . . . . . . . 3 x.5 Inner Product Space . . . . . . . . . . . . . . . . . . . 4 x.6 Cauch...

Recently, folk questions on the smoothability of Cauchy hypersurfaces and time functions of a globally hyperbolic spacetime M , have been solved. Here we give further results, applicable to several problems: (1) Any compact spacelike acausal submanifold H with boundary can be extended to a spacelike Cauchy hypersurface S. If H were only achronal, counterexamples to the smooth extension exist, b...

Today, great observatories around the world, devote a substantial amount of observing time to sky surveys. The resulted images are inputs of source finder modules. These modules search for the target objects and provide us with source catalogues. We sought to quantify the ability of detection tools in recovering faint galaxies regularly encountered in deep surveys. Our approach was based on com...

The main goal of this paper is to present an alternative, real variable proof of the T(1)-Theorem for the Cauchy Integral. We then prove that the estimate from below of analytic capacity in terms of total Menger curvature is a direct consequence of the T(1)-Theorem. An example shows that the L 1-BMO estimate for the Cauchy Integral does not follow from L 2 boundedness when the underlying measur...

We explore two generalizations of the Cauchy-Schwarz Bessel’s inequality and the Selberg inequality and their application to probability theory. We then give a tautological proof of the De Caen-Selberg Inequality and a proof of the second Borel-Cantelli Lemma with negative dependence. We finish with a suggestion of how linear operator theory can help us understand the tightness of many Selberg-...

We study the theory of scattering for a Schrödinger equation in an external time dependent magnetic field in the Coulomb gauge. The magnetic vector potential is assumed to satisfy decay properties that are typical of solutions of the free wave equation, and even in some cases to be actually a solution of that equation. That problem appears as an intermediate step in the theory of scattering for...