On coherent states for the simplest quantum groups

The coherent states for the simplest quantum groups (q-Heisenberg-Weyl, SU _q (2) and the discrete series of representations of SU _q (1, 1)) are introduced and their properties investigated. The corresponding analytic representations, path integrals, and q-deformation of Berezin's quantization on , a sphere, and the Lobatchevsky plane are discussed.     

Concentrations and velocity distributions of positive ions in laser ablation of copper

It is shown that the generation of secondary electrons by excited neutrals hitting a surface makes the so-called probe method unsuited for measuring the positive ion fraction in laser ablation plumes. Experiments have been performed in a modified set-up, in which the disturbance by secondary electrons is avoided. For a typical case of Cu irradiated in ultra-high vacuum by nanosecond excimer laser pulses of 3 J cm^–2 the ionized fraction is about 10^–8 at a distance of 60 cm. This number is in fair agreement with Saha-Langmuir predictions based on the assumption of local thermal equilibrium at an estimated temperature of about 3000K. Angular-resolved time-of-flight measurements show that there are three different ion velocity distributions. A slow contribution (kinetic energy 2eV) with an angular distribution peaked along the normal, and two fast, isotropic contributions (kinetic energy 20–50 eV). The fast contributions are attributed to ions involved in a Coulomb explosion.     

Continuous modified Newton’s-type method for nonlinear operator equations

A nonlinear operator equation F(x)=0, F:H→H, in a Hilbert space is considered. Continuous Newton’s-type procedures based on a construction of a dynamical system with the trajectory starting at some initial point x ₀ and becoming asymptotically close to a solution of F(x)=0 as t→+∞ are discussed. Well-posed and ill-posed problems are investigated. Received: June 29, 2001; in final form: February 26, 2002?Published online: February 20, 2003 This paper was finished when AGR was visiting Institute for Theoretical Physics, University of Giessen. The author thanks DAAD for support     

A Bitangential Interpolation Problem on the Closed Unit Ball for Multipliers of the Arveson Space

We solve a bitangential interpolation problem for contractive multipliers on the Arveson space with an arbitrary interpolating set in the closed unit ball . The solvability criterion is established in terms of positive kernels. The set of all solutions is parametrized by a Redheffer transform. Submitted: February 2, 2002.     

The dimension of the kernel in an intersection of starshaped sets

We establish the following Helly-type theorem: Let ${\cal K}$ be a family of compact sets in $\mathbb{R}^d$. If every d + 1 (not necessarily distinct) members of ${\cal K}$ intersect in a starshaped set whose kernel contains a translate of set A, then $\cap \{ K : K\; \hbox{in}\; {\cal K} \}$ also is a starshaped set whose kernel contains a translate of A. An analogous result holds when ${\cal K}$ is a finite family of closed sets in $\mathbb{R}^d$. Moreover, we have the following planar result: Define function f on $\{0, 1, 2\}$ by f(0) = f(2) = 3, f(1) = 4. Let ${\cal K}$ be a finite family of closed sets in the plane. For k = 0, 1, 2, if every f(k) (not necessarily distinct) members of ${\cal K}$ intersect in a starshaped set whose kernel has dimension at least k, then $\cap \{K : K\; \hbox{in}\; {\cal K}\}$ also is a starshaped set whose kernel has dimension at least k. The number f(k) is best in each case.Received: 4 June 2002     

Symmetric incidence groupoids

In [7] point-reflection geometries were studied which can be derived from commutative kinematic spaces without involutory elements. But the class of point-reflection geometries is larger. For example, elliptic planes with their reflections cannot be derived from commutative kinematic spaces. Here we investigate a larger class of reflection geometries.This paper was sponsored by Vigoni Program 1999.     

A posteriori error estimates for optimal control problems governed by parabolic equations

Summary. In this paper, we derive a posteriori error estimates for the finite element approximation of quadratic optimal control problem governed by linear parabolic equation. We obtain a posteriori error estimates for both the state and the control approximation. Such estimates, which are apparently not available in the literature, are an important step towards developing reliable adaptive finite element approximation schemes for the control problem. Received July 7, 2000 / Revised version received January 22, 2001 / Published online January 30, 2002 RID="*" ID="*" Supported by EPSRC research grant GR/R31980     

Hypersurfaces of the Euclidean sphere with nonnegative Ricci curvature

In this paper we prove that a compact oriented hypersurface of a Euclidean sphere with nonnegative Ricci curvature and infinite fundamental group is isometric to an H(r)-torus with constant mean curvature. Furthermore, we generalize, whithout any hypothesis about the mean curvature, a characterization of Clifford torus due to Hasanis and Vlachos. Received: 19 March 2002     

Inner Radius of Univalence for a Strongly Starlike Domain

 The inner radius of univalence of a domain D with Poincaré density ρ _D is the possible largest number σ such that the condition ∥ S _f ∥ _D  = sup _{w∈ D} ρ _D (w) ⁻²∥ S _f (z) ∥ ≤ σ implies univalence of f for a nonconstant meromorphic function f on D, where S _f is the Schwarzian derivative of f. In this note, we give a lower bound of the inner radius of univalence for strongly starlike domains of order α in terms of the order α. The author was partially supported by the Ministry of Education, Grant-in-Aid for Encouragement of Young Scientists, 11740088. A part of this work was carried out during his visit to the University of Helsinki under the exchange programme of scientists between the Academy of Finland and the JSPS. Received November 26, 2001; in revised form September 24, 2002 Published online May 9, 2003