It is known that the derivative of a Blaschke product whose zero sequence lies in a Stolz angle belongs to all the Bergman spaces ${{A}^{P}}$ with $0<p<3/2$. The question of whether this result is best possible remained open. In this paper, for a large class of Blaschke products $B$ with zeros in a Stolz angle, we obtain a number of conditions which are equivalent to the membership of ${B}'$ in the space ${{A}^{p}}\left( p>1 \right)$. As a consequence, we prove that there exists a Blaschke product $B$ with zeros on a radius such that ${B}'\,\notin \,{{A}^{3/2}}$.

An invariant is presented which classifies, up to equivariant isomorphism, ${{C}^{*}}$-dynamical systems arising as limits from inductive systems of elementary ${{C}^{*}}$-algebras on which the Euclidean motion group acts by way of unitary representations that decompose into finite direct sums of irreducibles.

This paper studies Hausdorff–Young inequalities for certain group extensions, by use of Mackey's theory. We consider the case in which the dual action of the quotient group is free almost everywhere. This result applies in particular to yield a Hausdorff–Young inequality for nonunimodular groups.

In this article, we give an explicit formula to compute the non abelian twisted sign-determined Reidemeister torsion of the exterior of a fibered knot in terms of its monodromy. As an application, we give explicit formulae for the non abelian Reidemeister torsion of torus knots and of the figure eight knot.

A sharp upper bound on the first ${{S}^{1}}$ invariant eigenvalue of the Laplacian for ${{S}^{1}}$ invariant metrics on ${{S}^{2}}$ is used to find obstructions to the existence of ${{S}^{1}}$ equivariant isometric embeddings of such metrics in (${{\mathbb{R}}^{3}}$, can). As a corollary we prove: If the first four distinct eigenvalues have even multiplicities then the metric cannot be equivariantly, isometrically embedded in (${{\mathbb{R}}^{3}}$, can). This leads to generalizations of some classical results in the theory of surfaces.

In this note we consider $\bar{\partial }$-problem in line bundles over complex projective space $\mathbb{C}{{\mathbb{P}}^{1}}$ and prove that the equation can be solved for (0, 1) forms with compact support. As a consequence, any Cauchy-Riemann function on a compact real hypersurface in such line bundles is a jump of two holomorphic functions defined on the sides of the hypersurface. In particular, the results can be applied to $\mathbb{C}{{\mathbb{P}}^{2}}$ since by removing a point from it we get a line bundle over $\mathbb{C}{{\mathbb{P}}^{1}}$.

Free analogues of the logarithmic Sobolev inequality compare the relative free Fisher information with the relative free entropy. In the present paper such an inequality is obtained for measures on the circle. The method is based on a random matrix approximation procedure, and a large deviation result concerning the eigenvalue distribution of special unitary matrices is applied and discussed.

In this article we will show that there are infinitely many symmetric, integral 3 × 3 matrices, with zeros on the diagonal, whose eigenvalues are all integral. We will do this by proving that the rational points on a certain non-Kummer, singular $\text{K3}$ surface are dense. We will also compute the entire Néron–Severi group of this surface and find all low degree curves on it.

Continuous mappings defined from compact subsets $K$ of complex Euclidean space ${{\mathbb{C}}^{n}}$ into complex projective space ${{\mathbb{P}}^{m}}$ are approximated by rational mappings. The fundamental tool employed is homotopy theory.

This note shows that any set of cofibrations containing the standard set of generating projective cofibrations determines a cofibrantly generated proper closed model structure on the category of simplicial presheaves on a small Grothendieck site, for which the weak equivalences are the local weak equivalences in the usual sense.

We characterize the weight functions $u,v,w$ on $\left( 0,\infty \right)$ such that

$${{\left( \int\limits_{0}^{\infty }{{{f}^{*}}{{\left( t \right)}^{q}}w\left( t \right)}\,dt \right)}^{1/q}}\le C\,\,\underset{t\in \left( 0,\infty \right)}{\mathop{\sup }}\,{{f}_{u}}^{**}\left( t \right)v\left( t \right),$$

where

$${{f}_{u}}^{**}\left( t \right):={{\left( \int\limits_{0}^{t}{u\left( s \right)}\,ds \right)}^{-1}}\int\limits_{0}^{t}{{{f}^{*}}}\left( s \right)u\left( s \right)\,ds.$$

As an application we present a new simple characterization of the associate space to the space ${{\Gamma }^{\infty }}\left( v \right)$, determined by the norm

$${{\left\| f \right\|}_{\Gamma \infty \left( v \right)}}=\,\underset{t\in \left( 0,\infty \right)}{\mathop{\sup }}\,{{f}^{**}}\left( t \right)v\left( t \right),$$

where

$${{f}^{**}}\left( t \right):=\frac{1}{t}\int\limits_{0}^{t}{{{f}^{*}}}\left( s \right)\,ds.$$

In this paper we determine the structure of the full lift for the Howe correspondence of $\left( Sp\left( n \right),O\left( V \right) \right)$ for rank-one reducibilities.

We introduce roots of indecomposable modules over group algebras of finite groups, and we investigate some of their properties. This allows us to correct an error in Landrock's book which has to do with roots of simple modules.

Let $X$ be a Polish space. We will prove that

$${{\dim}_{T}}X=\inf \left\{ {{\dim}_{L}}{X}'\,:\,{X}'\,\text{is homeomorphic to }X\, \right\},$$

where ${{\dim}_{L}}\,X$ and ${{\dim}_{T}}\,X$ stand for the concentration dimension and the topological dimension of $X$, respectively.

In this paper, we consider the behavior of the commutators of convolution operators on the Triebel–Lizorkin spaces ${{\dot{F}}_{p}}^{s,q}$.

In this paper we describe six pencils of $K3$-surfaces which have large Picard number $\left( \rho =19,20 \right)$ and each contains precisely five special fibers: four have $\text{A-D-E}$ singularities and one is non-reduced. In particular, we characterize these surfaces as cyclic coverings of some $K3$-surfaces described in a recent paper by Barth and the author. In many cases, using 3-divisible sets, resp., 2-divisible sets, of rational curves and lattice theory, we describe explicitly the Picard lattices.

We give a geometric proof of classical results that characterize Pisot numbers as algebraic $\text{ }\lambda \,>1$ for which there is $x\ne 0$ with $\text{ }\lambda {{\text{ }}^{n}}x\to 0\left( \,\bmod \,\,1 \right)$ and identify such $x$ as members of $\mathbb{Z}\left[ \text{ }\lambda {{\text{ }}^{-1}} \right]\cdot$$\mathbb{Z}{{\left[ \text{ }\!\!\lambda\!\!\text{ } \right]}^{*}}$ where $\mathbb{Z}{{\left[ \text{ }\!\!\lambda\!\!\text{ } \right]}^{*}}$ is the dual module of $\mathbb{Z}\left[ \text{ }\!\!\lambda\!\!\text{ } \right]$.

Using the Polyak–Viro Gauss diagram formula for the degree-4 Vassiliev invariant, we extend some previous results on positive knots and the non-triviality of the Jones polynomial of untwisted Whitehead doubles.

A Bernstein–Walsh type inequality for ${{C}^{\infty }}$ functions of several variables is derived, which then is applied to obtain analogs and generalizations of the following classical theorems: (1) Bochnak– Siciak theorem: a ${{C}^{\infty }}$ function on ${{\mathbb{R}}^{n}}$ that is real analytic on every line is real analytic; (2) Zorn–Lelong theorem: if a double power series $F\left( x,y \right)$ converges on a set of lines of positive capacity then $F\left( x,y \right)$ is convergent; (3) Abhyankar–Moh–Sathaye theorem: the transfinite diameter of the convergence set of a divergent series is zero.

We construct vector-valued modular forms of weight 2 associated to Jacobi-like forms with respect to a symmetric tensor representation of $\Gamma$ by using the method of Kuga and Shimura as well as the correspondence between Jacobi-like forms and sequences of modular forms. As an application, we obtain vector-valued modular forms determined by theta functions and by pseudodifferential operators.