Dimensional reduction for Helmholtz's equation on a bounded domain

Summary. The dimensional reduction method for solving boundary value problems of Helmholtz's equation in domain by replacing them with systems of equations in dimensional space are investigated. It is proved that the existence and uniqueness for the exact solution and the dimensionally reduced solution of the boundary value problem if the input data on the faces are in some class of functions. In addition, the difference between and in is estimated as and are fixed. Finally, some numerical experiments in a domain are given in order to compare theretical results. Received April 2, 1996 / Revised version received July 30, 1990     

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.     

On an application of Hall's representatives theorem to a finite geometry problem

It is well-known thatn points not belonging to a hyperplane determine at leastn hyperplanes. The possible configurations of hyperplanes in the case when the number of hyperplanes is equal ton are known, too. In this paper we obtain these results by means of Hall's representatives theorem. The setting is that of a finite geometry.     

Sensitivity of hydrodynamic system of the Obukhov type

The paper deals with the regime of advanced Bénard instabilities (described usually by the Lorenz system) in a more general system formulated firstly by Obukhov. It has been shown that triplet co-operation leads to characteristic courses where some new regularities occur.     

Kinetics of the photodissociation iodine laser with slow pumping

Systematic calculations of maximum inversion were performed for iodine photodissociation laser in dependence on the mixture composition. The flash-lamp radiation was modelled as emanating from an optically thin Xe plasma. Duration of the pumping pulse is about 300 s. The sensitivity of the model to the values of kinetic constants was also tested.