Sets which are almost starshaped

A nonempty setS in a real topological linear spaceL is said to be quasi-starshaped if and only if there is some pointq in clS such that the subset of points ofS visible viaS fromq is everywhwere dense inS and contains intS, and the set of all such pointsq is called the quasi-kernel ofS and denoted by qkerS. It is proved that forS connected with slncS nonempty {conv Â_z:z slncS} qkerS, where slncS denotes the set of strong local nonconvexity points ofS and Â_z={s clS:z is clearly visible froms via S}. Familiar procedures generate then the Krasnosel'skii-type characterizations for the dimension of the quasi-kernel ofS. This contributes to an open problem.Jakub Oswald gewidmet     

Spatial structure and NMR spectra of strained [2.2.2]cyclophanes

The chemical shifts and coupling constants in the NMR spectra of the title compounds are influenced by their strain and the proximity of the aromatic rings. These parameters are studied by density functional theory (DFT) calculations for 1–4 and measured for 3 and 4. We find that, in spite of the strain, the calculations reproduce the experimental values satisfactorily. This finding is of special importance for novel hypothetical molecules like hexahydrosuperphane 5 for their future identification and in searches of hydrocarbons exhibiting unusual NMR parameters. Our results demonstrate the influence of strain on the parameters studied. Most proton and carbon chemical shifts for the molecules under study having nonplanar aromatic rings differ considerably from the corresponding values for the relatively unstrained trimethylbenzenes and ethylbenzene. In addition, the calculated values of the coupling constants through three bonds in most cases do not follow Karplus relation. Copyright © 2009 John Wiley & Sons, Ltd.     

Exploration of 4,5-dimethyl-1H-imidazole N-oxide derivatives in the synthesis of new achiral and chiral ionic liquids

New 1-alkoxy-3-alkyl-4,5-dimethylimidazolium bromides were synthesized by alkylation of the corresponding 1-alkylimidazole 3-oxides, which were conveniently prepared via condensation of α-(hydroxyimino)ketones, primary aliphatic amines, and formaldehyde. By using enantiomerically pure chiral amines, optically active imidazolium salts were obtained. Treatment with sodium tetrafluoroborate in acetone yielded the corresponding imidazolium tetrafluoroborates. All these compounds, with only one exception, were obtained as oils, which are considered as potential ionic liquids and ‘chiral ionic liquids’. The reduction of the chiral or non-chiral 1-alkylimidazole 3-oxides with Raney-Ni, followed by alkylation with alkyl bromides and subsequent ion exchange to tetrafluoroborates, gave the corresponding 1,3-dialkylimidazolium salts, most of them showing properties of ionic liquids. The alkylation of 1-butyl-4,5-dimethylimidazole 3-oxide and the corresponding imidazole, respectively, with 1,3-dibromopropane led to the first bis-imidazolium dibromides and bis-tetrafluoroborates.     

Thue type problems for graphs, points, and numbers

A sequence S=s₁s₂…s_n is said to be nonrepetitive if no two adjacent blocks of S are the same. A celebrated 1906 theorem of Thue asserts that there are arbitrarily long nonrepetitive sequences over the set {0,1,2}. This result is the starting point of Combinatorics on Words—a wide area with many deep results, sophisticated methods, important applications and intriguing open problems.The main purpose of this survey is to present a range of new directions relating Thue sequences more closely to Graph Theory, Combinatorial Geometry, and Number Theory. For instance, one may consider graph colorings avoiding repetitions on paths, or colorings of points in the plane avoiding repetitions on straight lines. Besides presenting a variety of new challenges we also recall some older problems of this area.     

Tolerance averaging of heat conduction in transversally graded laminates

It is considered a heat conduction in a layer made of two conductors distributed in the form of laminas with varied thicknesses. Macroscopic (averaged) properties of the layer are continuously “transversally” graded across its thickness (TGL layer), cf. Fig. 1. The aim of the paper is to present and apply an averaged model of the heat conduction, obtained within the tolerance averaging technique, discussed in the book edited by Woźniak et al. (Thermomechanics of microheterogeneous solids and structures. Tolerance averaging approach, Łódź, Wydawnictwo Politechniki Łódzkiej, 2008). It is shown that the proposed model describes the microstructural effect on the heat conduction of the TGL layer. Moreover, results obtained within this model are compared to results by the higher order theory (cf. Aboudi et al., Composites B, 30:777–832, 1999).     

Mathematical and physical aspect of the solution of the Cauchy problem for a system of equations describing a nonlocal model of propagation of heat with finite speed

In our paper we present a new system of equations describing a nonlocal model of propagation of heat with finite speed in three-dimensional space. Such a system of equations is described by a system of integral – differential equations. At first using the modiffied Cagniard de Hoop method, we construct the fundamental solution of this system of equations. On the basis of the constructed fundamental solution we obtain the explicite formulate of the solution of the Cauchy problem for this system of equations and applying the method of Sobolev and Biesov spaces, we get L^p – L^q time decay estimate for the solution of the Cauchy problem. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Eigenmobilities in background electrolytes for capillary zone electrophoresis. I. System eigenpeaks and resonance in systems with strong electrolytes

A background electrolyte system for capillary zone electrophoresis which is composed of three strong univalent ionic constituents is investigated. The ion 1 is considered as a counter-ion and two ions, 2 and 3, are considered as co-ions in relation to the analyte ion 4. We investigate the linearized model of electromigration in such a system and calculate the eigenvalues of a corresponding matrix. The model is formulated in such a way that the eigenvalues of the system are certain mobilities, which we call eigenmobilites, which characterize specific features of the electrophoretic migration. One of the eigenmobilities is the system eigenmobility u_S causing the rise of the system peak, called here the system eigenpeak. A situation when the analyte has the same mobility as the system eigenmobility, u₄=u_S, is analyzed in detail. We show that it leads to the resonance—the mutual jump in the concentration profile of both co-ions, 2 and 3, has a shape of the spatial derivation of the originally sampled analyte profile and, moreover, it grows linearly with time. After a sufficiently long time it can be “amplified” to any value. The resonance has then a great impact on signals of indirect detection methods, like indirect UV detection or conductivity detection. In the framework of the linearized model the relative velocity slope S_X, a measure of electromigration dispersion, is expressed as S_X=F(u₁+u₄)(u₂−u₄)(u₃−u₄)/[u₄(u_S−u₄)], where u_i is the mobility of the ith ion and F is the Faraday constant. As in practice the concentration of the analyte is not infinitely small and has a certain finite value, the analyte will be at the resonance severely dispersed to a much broader spatial interval. When a specific detector is used, the signal of such an analyte can apparently be missed without any notice.     

Legendrian Subvarieties of Projective Space

I prove that every smooth Legendrian variety generated by quadrics is a homogeneous variety and, further, I give a list of all such Legendrian varieties. Another result is that no complete intersection is a Legendrian variety.Mathematics Subject Classification (2000): Primary: 14M99, secondary: 17B10, 22E46 *The author was partially supported by Marie Curie Programme.