Approximations by Generalized Discrete Singular Operators

Here, we give the approximation properties with rates of generalized discrete versions of Picard, Gauss-Weierstrass, and Poisson-Cauchy singular operators. We cover both the unitary and non-unitary cases of the operators above. We present quantitatively the point-wise and uniform convergences of these operators to the unit operator by involving the higher modulus of smoothness of a uniformly continuous function. We also establish our results with respect to L_p norm, 1≤p\u3c∞. Additionally, we state asymptotic Voronovskaya type expansions for these operators. Moreover, we study the fractional generalized smooth discrete singular operators on the real line regarding their convergence to the unit operator with fractional rates in the uniform norm. Then, we give our results for the operators mentioned above over the real line regarding their simultaneous global smoothness preservation property with respect to L_p norm for 1≤p≤∞, by involving higher order moduli of smoothness. Here we also obtain Jackson type inequalities of simultaneous approximation which are almost sharp, containing neat constants, and they reflect the high order of differentiability of involved function. Next, we cover the approximation properties of on the general complex-valued discrete singular operators over the real line regarding their convergence to the unit operator with rates in the L_p norm for 1≤p≤∞. Finally, we establish the approximation properties of multivariate generalized discrete versions of these operators over R^N,N≥1. We give pointwise, uniform, and L_p convergence of the operators to the unit operator by involving the multivariate higher order modulus of smoothness

The space of complete embedded maximal surfaces with isolated singularities in the 3-dimensional Lorentz-Minkowski space \l^3

We show that a complete embedded maximal surface in the 3-dimensional Lorentz-Minkowski space $L^3$ with a finite number of singularities is, up to a Lorentzian isometry, an entire graph over any spacelike plane asymptotic to a vertical half catenoid or a horizontal plane and with conelike singular points. We study the space $G_n$ of entire maximal graphs over $\{x_3=0\}$ in $L^3$ with $n+1 \geq 2$ conelike singularities and vertical limit normal vector at infinity. We show that $G_n$ is a real analytic manifold of dimension $3n+4,$ and the coordinates are given by the position of the singular points in $R^3$ and the logarithmic growth at the end. We also introduce the moduli space $M_n$ of {\em marked} graphs with $n+1$ singular points (a mark in a graph is an ordering of its singularities), which is a $(n+1)$-sheeted covering of $G_n.$ We prove that identifying marked graphs differing by translations, rotations about a vertical axis, homotheties or symmetries about a horizontal plane, the corresponding quotient space $M_n$ is an analytic manifold of dimension $3n-1.$Comment: 32 pages, 4 figures, corrected typos, former Theorem 3.3 (now Theorem 2.2) modifie

Lectures on Minimal Surface Theory

An article based on a four-lecture introductory minicourse on minimal surface theory given at the 2013 summer program of the Institute for Advanced Study and the Park City Mathematics Institute.Comment: 46 pages, 6 figures. Some references added/corrected on August 2, 2014. A few minor corrections on October 16, 2015. Additional typos corrected on January 17, 201

How smooth are particle trajectories in a Λ\LambdaCDM Universe?

It is shown here that in a flat, cold dark matter (CDM) dominated Universe with positive cosmological constant ($\Lambda$), modelled in terms of a Newtonian and collisionless fluid, particle trajectories are analytical in time (representable by a convergent Taylor series) until at least a finite time after decoupling. The time variable used for this statement is the cosmic scale factor, i.e., the "$a$-time", and not the cosmic time. For this, a Lagrangian-coordinates formulation of the Euler-Poisson equations is employed, originally used by Cauchy for 3-D incompressible flow. Temporal analyticity for $\Lambda$CDM is found to be a consequence of novel explicit all-order recursion relations for the $a$-time Taylor coefficients of the Lagrangian displacement field, from which we derive the convergence of the $a$-time Taylor series. A lower bound for the $a$-time where analyticity is guaranteed and shell-crossing is ruled out is obtained, whose value depends only on $\Lambda$ and on the initial spatial smoothness of the density field. The largest time interval is achieved when $\Lambda$ vanishes, i.e., for an Einstein-de Sitter universe. Analyticity holds also if, instead of the $a$-time, one uses the linear structure growth $D$-time, but no simple recursion relations are then obtained. The analyticity result also holds when a curvature term is included in the Friedmann equation for the background, but inclusion of a radiation term arising from the primordial era spoils analyticity.Comment: 16 pages, 4 figures, published in MNRAS, this paper introduces a convergent formulation of Lagrangian perturbation theory for LCD

Numerical methods for integral equations of Mellin type

We present a survey of numerical methods (based on piecewise polynomial approximation) for integral equations of Mellin type, including examples arising in boundary integral methods for partial differential equations on polygonal domains

Some numerical applications of generalized Bernstein operators

Inthispaper,somerecentapplicationsoftheso-calledGeneralizedBernsteinpolynomialsarecollected. This polynomial sequence is constructed by means of the samples of a continuous function f on equispaced points of [0, 1] and depends on an additional parameter which can be suitable chosen in order to improve the rate of convergence to the function f , as the smoothness of f increases, overcoming the well-known low degree of approximation achieved by the classical Bernstein polynomials or by the piecewise polynomial approximation. The applications considered here deal with the numerical integration and the simultaneous approximation. Quadrature rules on equidistant nodes of [0, 1] are studied for the numerical computation of ordinary integrals in one or two dimensions, and usefully em- ployed in Nyström methods for solving Fredholm integral equations. Moreover, the simultaneous approximation of the Hilbert transform and its derivative (the Hadamard transform) is illustrated. For all the applications, some numerical details are given in addition to the error estimates, and the proposed approximation methods have been implemented providing numerical tests which confirm the theoretical estimates. Some open problems are also introduced

Constant mean curvature surfaces

In this article we survey recent developments in the theory of constant mean curvature surfaces in homogeneous 3-manifolds, as well as some related aspects on existence and descriptive results for $H$-laminations and CMC foliations of Riemannian $n$-manifolds.Comment: 102 pages, 17 figure