MathSciDoc: An Archive for Mathematician ∫

We show how to construct the Perron–Bremermann function by using proper analytic disks. We apply this result to the polynomial hull of a compact set$K$defined on the boundary of the unit ball.

We show that a family $\mathcal{F}$ of analytic functions in the unit disk ${\mathbb{D}}$ all of whose zeros have multiplicity at least$k$and which satisfy a condition of the form $$f^n(z)f^{(k)}(xz)\ne1$$ for all $z\in{\mathbb{D}}$ and $f\in\mathcal{F}$ (where$n$≥3,$k$≥1 and 0<|$x$|≤1) is normal at the origin. The proof relies on a modification of Nevanlinna theory in combination with the Zalcman–Pang rescaling method. Furthermore we prove the corresponding Picard-type theorem for entire functions and some generalizations.

We consider the convolution operators in spaces of functions which are holomorphic in a bounded convex domain in ℂ^{$n$}and have a polynomial growth near its boundary. A characterization of the surjectivity of such operators on the class of all domains is given in terms of low bounds of the Laplace transformation of analytic functionals defining the operators.

Let$M$be a smooth hypersurface of constant signature in$CP$^{$n$},$n$≥3. We prove the regularity foron$M$in bidegree (0,1). As a consequence, we show that there exists no smooth hypersurface in$CP$^{$n$},$n$≥3, whose Levi form has at least two zero-eigenvalues.

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The topological center of the spectrum of the Weyl algebra$W$, i.e. the norm closure of the algebra generated by the set of functions $\{n\mapsto\lambda^{n^{i}};\lambda\in\mathbb{T}\mbox{ and }i\in\mathbb {N}\}$ , is characterized in a recent paper by$Jabbari$and$Namioka$(Ellis group and the topological center of the flow generated by the map $n\mapsto \lambda^{n^{k}}$ , to appear in$Milan J. Math.$). By the techniques essentially used in the cited paper, the topological center of the spectrum of the subalgebra$W$_{$k$}, the norm closure of the algebra generated by the set of functions $\{n\mapsto\lambda^{n^{i}};\lambda\in\mathbb{T}\mbox{ and }i=0,1,2,\ldots,k\}$ , will be characterized, for all$k$∈ℕ. Also an example of a non-minimal dynamical system, with the enveloping semigroup Σ, for which the set of all continuous elements of Σ is not equal to the topological center of Σ, is given.

Through a geometric capacitary analysis based on space dualities, this paper addresses several fundamental aspects of functional analysis and potential theory for the Morrey spaces in harmonic analysis over the Euclidean spaces.

For a positive integer$k$and a non-negative integer$t$, a class of simplicial complexes, to be denoted by$k$-CM_{$t$}, is introduced. This class generalizes two notions for simplicial complexes: being$k$-Cohen–Macaulay and$k$-Buchsbaum. In analogy with the Cohen–Macaulay and Buchsbaum complexes, we give some characterizations of CM_{$t$}(=1−CM_{$t$}) complexes, in terms of vanishing of some homologies of its links, and in terms of vanishing of some relative singular homologies of the geometric realization of the complex and its punctured space. We give a result on the behavior of the CM_{$t$}property under the operation of join of two simplicial complexes. We show that a complex is$k$-CM_{$t$}if and only if the links of its non-empty faces are$k$-CM_{$t$−1}. We prove that for an integer$s$≤$d$, the ($d$−$s$−1)-skeleton of a ($d$−1)-dimensional$k$-CM_{$t$}complex is ($k$+$s$)-CM_{$t$}. This result generalizes Hibi’s result for Cohen–Macaulay complexes and Miyazaki’s result for Buchsbaum complexes.

We introduce a technique for handling Whitney decompositions in Gaussian harmonic analysis and apply it to the study of Gaussian analogues of the classical tent spaces$T$^{1,$q$}of Coifman–Meyer–Stein.

We introduce tropical vector bundles, morphisms and rational sections of these bundles and define the pull-back of a tropical vector bundle and of a rational section along a morphism. Most of the definitions presented here for tropical vector bundles will be contained in$Torchiani, C.$,$Line Bundles on Tropical Varieties$, Diploma thesis, Technische Universität Kaiserslautern, Kaiserslautern,2010, for the case of line bundles. Afterwards we use the bounded rational sections of a tropical vector bundle to define the Chern classes of this bundle and prove some basic properties of Chern classes. Finally we give a complete classification of all vector bundles on an elliptic curve up to isomorphisms.