The internal structure of a charged spherical black hole is still a topic of debate. In a nonrotating but aspherical gravitational collapse to form a spherical charged black hole, the backscattered gravitational wave tails enter the black hole and are blueshifted at the Cauchy horizon. This has a catastrophic effect if combined with an outflux crossing the Cauchy horizon: a singularity develops...

Formulæ are derived for expressing Cauchy and Hilbert transforms of a function f in terms of Cauchy and Hilbert transforms of f(xr). When r is an integer, this corresponds to evaluating the Cauchy transform of f(xr) at all choices of z1/r. Related formuæ for rational r result in a reduction to a generalized Cauchy transform living on a Riemann surface, which in turn is reducible to the standard...

in this article, we apply two new fixed point theorems to investigate the existence of mild solutions for a nonlocal fractional cauchy problem with an integral initial condition in banach spaces.

A subset E of a metric space (X, d) is totally bounded if and only if any sequence of points in E has a Cauchy subsequence. We call a sequence (xn) statistically quasiCauchy if st − limn→∞ d(xn+1, xn) = 0, and lacunary statistically quasi-Cauchy if Sθ − limn→∞ d(xn+1, xn) = 0.Weprove that a subset E of ametric space is totally bounded if and only if any sequence of points in E has a subsequence...

Principal Component Analysis (PCA) aims to learn compact and informative representations for data and has wide applications in machine learning, text mining and computer vision. Classical PCA based on a Gaussian noise model is fragile to noise of large magnitude. Laplace noise assumption based PCA methods cannot deal with dense noise effectively. In this paper, we propose Cauchy Principal Compo...

sets which admit a definition of distance. Fréchet gives the following definition of a complete set (JD) : a set (D) with preassigned elements of accumulation is complete if among all possible definitions of distance in (D) for which the elements of accumulation remain unchanged there is one with respect to which (D) admits a generalization of the theorem of Cauchy regarding the convergence of ...

We study the Cauchy problem for the semilinear parabolic equations ∆u−Ru+ up − ut = 0 on Mn × (0,∞) with initial value u0 ≥ 0, where Mn is a Riemannian manifold including the ones with nonnegative Ricci curvature. In the Euclidean case and when R = 0, it is well known that 1 + 2 n is the critical exponent, i.e., if p > 1+ 2 n and u0 is smaller than a small Gaussian, then the Cauchy problem has ...

Censoring, 116 Coefficient concentration, 287 confidence, 173 uncertainty, 287 Competing risks, 371 Confidence interval, 13, 173 coverage probability, 173 unbiased, 96 uniformly most accurate, 96 Contingency tables, 274 Convergence almost certain, 120 almost sure, 120, 121 almost sure convergence of series, 164 complete, 120, 129 equivalent, 148 in distribution, 125 in law, 125 in probability, ...

we study different completeness definitions for two categories of lattice-valued cauchy spaces and the relations between these definitions. we also show the equivalence of a so-called completion axiom and the existence of a completion.

<p style='text-indent:20px;'>In this paper, we investigate existence of global-in-time strong solutions to the Cauchy problem kinetic Cucker–Smale model coupled with incompressible Navier–Stokes equations in two dimensional space. By introducing a weighted Sobolev space and using maximal regularity estimate on linear non-stationary Stokes equations, present complete analysis model, withou...