Leggett-Williams norm-type theorems for coincidences

We study the existence of positive solutions to the operator equation Lx = Nx, where L is a linear Fredholm mapping of index zero and N is a nonlinear operator. Using the properties of cones in Banach spaces and Leray-Schauder degree for completely continuous operators, k-set contractions and condensing mappings, we obtain some refinements of the results established in [3] and [14]. Received: 9 July 2005; revised: 12 January 2006     

The effect of fluorine-based plasma treatment on morphology and chemical surface composition of biocompatible silicone elastomer

X-ray photoelectron spectroscopy (XPS) and scanning electron microscopy (SEM) have been used to investigate the effect of reactive ion etching (RIE) on poly(methylhydrogensiloxane-co-dimethylsiloxane) surface in fluorine-based plasmas. Polysiloxane layers supported on the standard silicon wafers were etched using SF₆ + O₂ or CF₄ + O₂ plasmas. SEM studies show that the polysiloxane morphology depends on plasma chemical composition strongly. Presence of a columnar layer likely covered with a fluorine rich compound was found on the elastomer surface after the CF₄ + O₂ plasma exposure. After the SF₆ + O₂ or CF₄ + O₂ plasma treatment the polysiloxane surface enriches with fluorine or with fluorine and aluminum, respectively. Different morphologies and surface chemical compositions of the silicone elastomer etched in both plasmas indicate different etching mechanisms.     

At the other side of a saddle-node

We describe phenomena occurring just before a saddle-node bifurcation for one-parameter families of interval maps. In particular, as a parameter approaches the bifurcation value, attracting periodic orbits of periodsk, k+1,k+2,k+3,... can appear. We make a detailed study of a family of cusp-shaped maps, where this phenomenon occurs in a pure form.     

On automorphisms of n-dimensional Laguerre space

We examine some properties of two types of inversions in the isotropical model of n-dimensional Laguerre space. Moreover we establish an isomorphism between the isotropical and the cylinder models of the n-dimensional Laguerre space. We prove also that every automorphism of the n-dimensional Laguerre space is a composition of a pseudodilatation and some involutive maps.     

The measure of a translated ball in uniformly convex spaces

Let (E, ¦·¦) be a uniformly convex Banach space with the modulus of uniform convexity of power type. Let be the convolution of the distribution of a random series inE with independent one-dimensional components and an arbitrary probability measure onE. Under some assumptions about the components and the smoothness of the norm we show that there exists a constant such that |{·<t}–{·+r<t}|r ^q , whereq depends on the properties of the norm. We specify it in the case ofL spaces, >1.     

Crystal structure of m-nitrobenzoic anhydride

The title compound (C₁₄H₈N₂O₇,M _r =306.2) crystallizes in the orthorhombic space group Pbca witha=6.962(1).b=24.688(1), andc=15.890(1)Å,V=2731.0 ų,D _x =1.489 g·cm^–3 forZ=8,=0.98 mm^–1,F(000)=1296,T=293 K. FinalR=0.053 for 1873 observed reflections. The structure was solved by direct methods. Approximately planar molecules lie perpendicular to the [100] direction and show partial stacking. The structure is the first example of a symmetric anhydride which does not retain the symmetry in the crystal state. The two independent nitro groups twist out of the ring planes by 10.5 and 14.8°, respectively.     

Structure of 18-cyanoprogesterone,C22H29O2N

The crystal structure of 18-cyanoprogesterone was determined by X-ray diffraction methods:P2₁2₁2₁ a=7.436(2),b=11.322(2),c=22.642(2) Å. The structure was solved usingShelx-86. Final conventionalR=0.054.R _w =0.051 for 1841 reflections. TheA ring has an intermediate sofa-half-chair conformation with asymmetry parameters C _s ^/1 =11.0, C ₂ ^3,4 =14.9. The steroid skeleton exhibits a flattening of theA ring relative to the rest of the molecule. The progesterone side chain has a typical conformation, and the C16-C17-C20-O20 torsion angle is –19.0(6)°.     

In-trees and plane embeddings of outerplanar graphs

A notion of anin-tree is introduced. It is then used to characterize and count plane embeddings of outerplanar graphs. In-trees have also been applied in the study of independent vertex covers of faces in outerplanar graphs.This research was partially supported by the grant RP.I.09, from the Institute of Informatics, University of Warsaw. Hospitality of the Institute of Datalogy, University of Copenhagen where this research has been completed is gratefully acknowledged.     

The dynamics of multidimensional cosmological models in the presence of quantum effects. I. The case FRW ×T D at high temperatures

The dynamical system approach is applied to the study of dynamics of multidimensional cosmological models with topology FRW ×T ^D (D-dimensional torus) in the presence of high-temperature quantum effects. The stability methods developed in the paper of Szydowski (Gen. Rel. Grav.,20, 221, 1988) are used in the analysis of typical states of the metric in the neighborhood of singularities and for large time values. The problems of dynamical dimensional reduction, structure of singularities, isotropization, etc., are discussed in this context.     

Plane contact problems for a periodic two-layered elastic composite

Summary This paper deals with the two-dimensional static punch problems in the presence of friction for a periodically layered half-space. Within the framework of the homogenized model of linear elasticity with microlocal parameters [8–11] the exact solutions of the considered problems are obtained. The case of the indentation of the composite body by a rigid rectangular punch has been discussed in detail.

---

Ebene Kontaktprobleme für ein periodisches elastisches Zweischichten-Komposit

Übersicht In dieser Arbeit wird das zweidimensionale statische Stempelproblem mit Reibung für einen periodisch geschichteten Halbraum betrachtet. Im Rahmen eines Homogenisierungsmodells der linearen Elastizitätstheorie mit mikropolaren Parametern werden die exakten Lösungen der betrachteten Probleme gewonnen. Im Detail wird das Eindringen eines rechteckigen starren Stempels behandelt.