Related papers: On Convergence Sets of Power Series with Holomorph…

A family of sets has the $(p,q)$ property if among any $p$ members of the family some $q$ have a nonempty intersection. It is shown that for every $p\ge q\ge d+1$ there is a $c=c(p,q,d)<\infty$ such that for every family $\scr F$ of…

We show that if $n\geq 1$, $\Omega\subset \mathbb R^{n+1}$ is a connected domain with porous boundary, and $E\subset \partial\Omega$ is a set of finite and positive Hausdorff $H^{n}$-measure upon which the harmonic measure $\omega$ is…

Let $(\phi_t)$ be a holomorphic semigroup of the unit disc (i.e., the flow of a semicomplete holomorphic vector field) without fixed points in the unit disc and let $\Omega$ be the starlike at infinity domain image of the Koenigs function…

We show that a fairly arbitrary Frechet space topology on the space of holomorphic functions on a domain controls the topology of uniform convergence on compact sets. In fact it turns out that the result we present can be proved more simply…

Let $S\subset \mathbb{R}^d$ $(d\geq 2)$. A set $S$ is said to be $m$-point convex, if for every $m$ distinct points in $S$, at least one of the line-segments determined by them lies in $S$. We also say that $S$ has property $P_m$. Let…

In recent work by Zimmer it was proved that if $\Omega\subset\mathbb C^n$ is a bounded convex domain with $C^\infty$-smooth boundary, then $\Omega$ is strictly pseudoconvex provided that the squeezing function approaches one as one…

We say that a subset of C^n is hypoconvex if its complement is the union of complex hyperplanes. Let D be the closed unit disk in C, T the unit circle. We prove two conjectures of Helton and Marshall. (See ``Frequency domain design and…

In this paper, we extend the notions of statistically convergence of order $\beta $ and strong Ces\`{a}ro summability of order $\beta ,$ and introduce the notions $f-$statistically convergence of order $\beta $ and strong Ces\`{a}ro…

For any power series $a(t)$ with exponentially bounded nonnegative integer coefficients we suggest a simple construction of a finitely generated monomial associative algebra $R$ with Hilbert series $H(R,t)$ very close to $a(t)$. If $a(t)$…

It is shown that a formal mapping between two real-analytic hypersurfaces in complex space is convergent provided that neither hypersurface contains a nontrivial holomorphic variety. For higher codimensional generic submanifolds,…

A new convenient method of describing flat convex compact sets is proposed. It generalizes classical trigonometric functions $\sin$ and $\cos$. Apparently, this method may be very useful for explicit description of solutions of optimal…

For a finite point set $P \subset \mathbb{R}^d$, denote by $\text{diam}(P)$ the ratio of the largest to the smallest distances between pairs of points in $P$. Let $c_{d, \alpha}(n)$ be the largest integer $c$ such that any $n$-point set $P…

We give an overview of the fundamental definitions and results concerning hypersurface singularities, defined by convergent power series over an arbitrary real valued field. This approach combines, on the one hand, the classical case of…

We revisit the phenomenon where, for certain domains $D$, if the squeezing function $s_D$ extends continuously to a point $p\in \partial{D}$ with value $1$, then $\partial{D}$ is strongly pseudoconvex around $p$. In $\mathbb{C}^2$, we…

Let $\Omega$ be a compact subset of $\mathbb{C}$ and let $A$ be a unital simple, separable $C^*$-algebra with stable rank one, real rank zero and strict comparison. We show that, given a Cu-morphism $\alpha:{\rm Cu}(C(\Omega))\to {\rm…

The Thoma simplex $\Omega$ is an infinite-dimensional space, a kind of dual object to the infinite symmetric group. The z-measures are a family of probability measures on $\Omega$ depending on three continuous parameters. One of them is the…

The purpose of this article is to provide an exposition of domains of convergence of power series of several complex variables without recourse to relatively advanced notions of convexity.

Let be F a family of curves in the unit disc. We show that the set of all functions f holomorphic on the unit disc, which satisfy the following condition, is G-delta and dense in the space of all functions holomorphic on the unit disc: For…

A topological group $G$ is {\em sequentially $h$-complete} if all the continuous homomorphic images of $G$ are sequentially complete. In this paper we give necessary and sufficient conditions on a complete group for being compact, using the…

A compact subset $K$ of the complex plane $\C$ is a set of polynomial (respectively rational) approximation if $P(K)=A(K)$ (respectively $R(K)=A(K)$), where $P(K)$ (respectively $R(K)$) is the family of functions on $K$ which are uniform…