Real Interpolation of Compact Bilinear Operators

We establish an analog for bilinear operators of the compactness interpolation result for bounded linear operators proved by Cwikel and Cobos, Kühn and Schonbek. We work with the assumption that \(T:(A_0+A_1) \times (B_0+B_1) \longrightarrow E_0+E_1\) is bounded, but we also study the case when this does not hold. Applications are given to compactness of convolution operators and compactness of commutators of bilinear Calderón–Zygmund operators.     

Tensor Characterizations of Summing Polynomials

Operators T that belong to some summing operator ideal can be characterized by means of the continuity of an associated tensor operator \({\overline{T}}\) that is defined between tensor products of sequences spaces. In this paper, we provide a unifying treatment of these tensor product characterizations of summing operators. We work in the more general frame provided by homogeneous polynomials, where an associated “tensor” polynomial—which plays the role of \({\overline{T}}\)—, needs to be determined first. Examples of applications are shown.     

Compact Perturbations of a Strongly Positive Operator

Let X be a real Hilbert space and \(X^{*}\) its dual space. Let A be a strongly positive operator from X to \(X^{*}\). Sufficient conditions are provided to assert that a compact perturbation of A reaches a fixed value at least once and at most finitely many times. When the compact perturbation is linear, the value is reached just once. The same conclusions are obtained when operators map the space X into itself. As an application of our results two examples are given related to some types of integral equations. The proof of results is constructive and is based upon a continuation method.     

Generalized Bernstein Operators on the Classical Polynomial Spaces

We study generalizations of the classical Bernstein operators on the polynomial spaces \(\mathbb {P}_{n}[a,b]\), where instead of fixing \(\mathbf {1}\) and x, we reproduce exactly \(\mathbf {1}\) and a polynomial \(f_1\), strictly increasing on [a, b]. We prove that for sufficiently large n, there always exist generalized Bernstein operators fixing \(\mathbf {1}\) and \(f_1\). These operators are defined by non-decreasing sequences of nodes precisely when \(f_1^\prime > 0\) on (a, b), but even if \(f_1^\prime \) vanishes somewhere inside (a, b), they converge to the identity.     

Compact $$\lambda $$-translating Solitons with Boundary

A \(\lambda \)-translating soliton with density vector \(\mathbf {v}\) is a surface \(\varSigma \) in Euclidean space \(\mathbb {R}^3\) whose mean curvature H satisfies \(2H=2\lambda +\langle N,\mathbf {v}\rangle \), where N is the Gauss map of \(\varSigma \). In this article, we study the shape of a compact \(\lambda \)-translating soliton in terms of its boundary. If \(\varGamma \) is a given closed curve, we deduce under what conditions on \(\lambda \) there exists a compact \(\lambda \)-translating soliton \(\varSigma \) with boundary \(\varGamma \) and we provide estimates of the surface area depending on the height of \(\varSigma \). Finally, we study the shape of \(\varSigma \) related with the geometry of \(\varGamma \), in particular, we give conditions that assert that \(\varSigma \) inherits the symmetries of its boundary \(\varGamma \).     

Local and Global Approximation for Certain (<Emphasis Type="Italic">p</Emphasis>, <Emphasis Type="Italic">q</Emphasis>)-Durrmeyer Type Operators

The present paper pertains to special approximation properties of certain (p, q)-Durrmeyer variant. The local and global approximation results are estimated along with comparison to the optimal convergence using the technique given by King. Graphically, we illustrate the convergence of these operators for different values of the two parameters p and q using MATLAB.     

Strong regularizing effect of a gradient term in the heat equation with the Hardy potential

We deal with the following parabolic problem,

     

All harmonic 2-spheres in the unitary group, completely explicitly

We give a completely explicit formula for all harmonic maps of finite uniton number from a Riemann surface to the unitary group U(n) in any dimension, and so all harmonic maps from the 2-sphere, in terms of freely chosen meromorphic functions on the surface and their derivatives, using only combinations of projections and avoiding the usual ${\bar{\partial}}We give a completely explicit formula for all harmonic maps of finite uniton number from a Riemann surface to the unitary group U(n) in any dimension, and so all harmonic maps from the 2-sphere, in terms of freely chosen meromorphic functions on the surface and their derivatives, using only combinations of projections and avoiding the usual [`(?)]{\bar{\partial}} -problems or loop group factorizations. We interpret our constructions using Segal’s Grassmannian model, giving an explicit factorization of the algebraic loop group, and showing how to obtain harmonic maps into a Grassmannian.     

Preparation,Characterization and Application of Modified Surfaces with 3,5-Bis(3,5-dinitrobenzoyl-amino) benzoic acid

Summary: The spontaneous adsorption of the dendron 3,5-Bis (3,5-dinitrobenzoylamino) benzoic acid (D-NO₂) onto gold and carbon electrodes produced conductive surfaces with electroactive chemical functions. A comparative electrochemical behavior of both electrodes after dendron immobilization led us to conclude that the self-assembly of D-NO₂ on carbon is faster and stronger. Considering this advantage, the surface of magnetic maghemite nanoparticles (MNPs) was modified using D-NO₂. Firstly, MNPs were modified with APS as silane coupling agent and afterwards, D-NO₂ was covalently attached to the surface, achieving nitro-functionalized MNPs. Subsequently, the immobilization of these modified MNPs onto glassy carbon surfaces was explored to generate a novel platform promising for biosensors development.