By combining with the physical concept of inscribed surface, the standard Cauchy-Born rule (CBR) is straightly extended to have a rigorous and accurate atomistic continuum theory for the monolayer crystal films. Resorting to using Tersoff-Brenner potential, the present theory to graphite sheet and single-walled carbon nanotubes (SWCNTs) is applied to evaluate the mechanical properties. The resu...

Introduction. Hamada [1] proved the following result about the propagation of singularities in the Cauchy problem for an analytic linear partial differential operator. Assume that the initial data are analytic at the point 0 except for singularities along a submanifold T of the initial surface containing 0. Let K\ • • •, K be the characteristic surfaces of the operator emanating from T. Under t...

We obtain a flagged form of the Cauchy determinant and establish a correspondence between this determinant and nonintersecting lattice paths, from which it follows that Cauchy identity on Schur functions. By choosing different origins and destinations for the lattice paths, we are led to an identity of Gessel on the Cauchy sum of Schur functions in terms of the complete symmetric functions in t...

we show that  higher derivations on a hilbert$c^{*}-$module associated with the cauchy functional equation satisfying generalized hyers--ulam stability.

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...