We examine the relation of the set of peak points with several boundaries defined in the spectrum of a given topological algebra relative to a subspace of it. In this respect, we show that the peak points are contained in the Choquet boundary and, under suitable conditions, are dense in the Šilov boundary. Furthermore, the set of points at issue coincide with the Bishop boundary if, and only if, it constitutes a weakly boundary set. On the other hand, in appropriate topological algebras, the Bishop, Choquet and strong boundaries coincide with the peak points, so that they are dense in the Šilov boundary. Finally there are topological algebras for which all the above boundaries and points remain invariant, under restriction of the Gel'fand transform algebras to subsets of the spectra of the topological algebras involved, containing the Šilov boundary.

In the previous paper [12] we introduced the definition of the strict topology $\beta(X)$ on the measure space $M(X)$ for a locally compact Hausdorff space $X$. In this paper, we consider on $M(X)$ the topology $\beta(X)$ and we show that $\beta(X)$ is the weak topology under all left multipliers induced by a function space on $M(X)$. We then show that $\beta(X)$ can be considered as a mixed topology. This result is not only of interest in its own right, but also it paves the way to prove that $(M(X),\beta(X))$ is a Mazur space and the locally convex space $(M(S),\beta(S))$, equipped with the convolution multiplication is a complete semitopological algebra, for a wide class of locally compact semigroups $S$.

Some sufficient conditions under which the tuple of the adjoint of weighted composition operators $(C_{\omega_1,\varphi_1}^* , C_{\omega_2,\varphi_2}^*)$ on the Hilbert space $\mathcal{H}$ of analytic functions is disk-cyclic (or codisk-cyclic) were investigated.

Upper-lower and left-right approaches coincide for semi-Fredholm operators on Hilbert spaces but, in general, they are distinct for operators on a Banach space. The purpose of this paper is to investigate the difference between the upper-lower and left-right approaches for semi-Fredholm operators on a Banach space and, in particular, to verify when these approaches coincide. The program is based on the classes $\Gamma_R[\X]$ and $\Gamma_N[\X]$ of all operators on a Banach space $\X$ with complemented range and complemented kernel. It is shown that the intersection ${\Gamma_R[\X]\cap\Gamma_N[\X]}$ is algebraically and topologically large, and also that if $\Gamma_R[\X]$ and $\Gamma_N[\X]$ are either open, or closed, or if they coincide, then there is no difference between the upper-lower and left-right approaches for semi-Fredholm operators on a Banach space.

We study $f-$statistical convergence, which is a generalization of the classical statistical convergence. In terms of it, we give a characterization of completeness in a normed space. We also introduce `$f-$statistical cluster points', which is a richer concept than the classic one. Namely, each (usual) limit point of a sequence is an $f-$statistical cluster point for some $f$.

In the present work we prove the equivalence of the fractional modulus of smoothness to the realization functional and to the Peetre K-functional in weighted Lorentz spaces. Using this equivalence we obtain an improvement of the direct approximation theorem. Furthermore we prove the improved converse theorem in this space.

We consider a mathematical model describing the quasistatic frictional contact between an electro-elasto-viscoplastic body and an adhesive conductive foundation. The contact is described with a normal compliance condition with adhesion, the associated general version of Coulomb's law of dry friction in which the adhesion of contact surfaces is taken into account and a regularized electrical conductivity condition. The existence of a unique weak solution is established under smallness assumption on the surface conductance. The proof is based on arguments of time-dependent variational inequalities, differential equations and Banach fixed point theorem.

In this present work, we continue studying the Bernstein-Walsh type estimations for complex algebraic polynomials in the bounded and unbounded regions with quasiconformal boundary.

Some categorical conditions are given that are sufficient to show that an enriched monad with a strength is a double exponential monad. The conditions hold for the double power locale monad (enriched over posets) and so as an application it is shown that the double power locale monad is a double exponential monad. A benefit is that this result about the double power locale monad can be established without the need for any detailed discussion of frame presentations or topos theory.