Solution of the problem of combinatorial characterization of the dimension of the kernel of a starshaped set

It is proved for a nonempty proper subset S of a real topological linear space L that kerS ={convA_z: z bdryS} and for a close connected nonconvex subset S of L that kerS ={convA_z: z slncS}, where bdryS and slncS denote the sets of boundary points and strong local nonconvexity points of S, respectively, and A_z = {s S: z is clearly visible from s via S}.This extends previous results and, combined with standard techniques, yields among others two Krasnosel'skii-type characterizations for the dimension of kerS in R^d in case of a nonempty proper set S with bdryS bounded and a closed connected nonconvex set S with lncS bounded.The assumption of boundedness of S turns out to be irrelevant in these criteria.Herrn Professor Dr. Wilhelm Stoll gewidmetResearch partially supported by the grant PB 2 1140 91 01     

Two-point Padé approximants for formal Stieltjes series

We prove that certain two-point Padé approximants occupying the diagonal of the Padé table form monotone sequences of lower and upper bounds uniformly converging to a Stieltjes function. The results can be applied to the theory of inhomogeneous media for the calculation of the bounds on the effective transport coefficients of heterogeneous materials.     

Fuzzy quantum logics and infinite-valued Łukasiewicz logic

A new, physically more plausible definition of a fuzzy quantum logic is proposed. It is shown that this definition coincides with the previously studied definition of a fuzzy quantum logic; therefore it defines objects which are traditional quantum logics with ordering sets of states. The new definition is expressed exclusively in terms of fuzzy set operations which are generated by connectives of multiple-valued logic studied by ukasiewicz at the beginning of the 20th century. Therefore, the logic of quantum mechanics is recognized as a version of infinite-valued ukasiewicz logic.     

A new heuristic algorithm solving the linear ordering problem

The linear ordering problem is an NP-hard combinatorial problem with a large number of applications. Contrary to another very popular problem from the same category, the traveling salesman problem, relatively little space in the literature has been devoted to the linear ordering problem so far. This is particularly true for the question of developing good heuristic algorithms solving this problem.In the paper we propose a new heuristic algorithm solving the linear ordering problem. In this algorithm we made use of the sorting through insertion pattern as well as of the operation of permutation reversal. The surprisingly positive effect of the reversal operation, justified in part theoretically and confirmed in computational examples, seems to be the result of a unique property of the problem, called in the paper the symmetry of the linear ordering problem. This property consists in the fact that if a given permutation is an optimal solution of the problem with the criterion function being maximized, then the reversed permutation is a solution of the problem with the same criterion function being minimized.     

Sensitive spectrophotometric determination of tin with Pyrogallol Red and cetyldimethylbenzylammonium ions

Summary Tin(IV), in the presence of cetyldimethylbenzylammonium chloride (CDBA), forms with Pyrogallol Red (PR) a complex SnPR-CDBA. This complex floates at the interface between hexane and aqueous medium. It dissolves in acetone and can be advantageously used for the determination of tin. The molar absorptivity is 6.5×10⁴ l·mole^–1·cm^–1 at 480 nm. The method is selective when citric acid is used as masking agent. Phosphate, fluoride, EDTA, nitrite and sulphide interfere.

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Zusammenfassung Zinn(IV) bildet mit Pyrogallolrot (PR) in Gegenwart von Cetyldimethylbenzylammoniumchlorid (CDBA) einen Komplex. Dieser wird flotiert, in Aceton gelöst und zur Bestimmung von Zinn verwendet. Der molare Absorptionskoeffizient bei =480 nm beträgt =6,5·10⁴ l·mol^–1 cm^–1. Die Methode ist selektiv, wenn Zitrate zur Maskierung verwendet werden. Es stören Phosphat, Fluorid, ÄDTA, Nitrit und Sulfid.

     

Uniqueness and the global Markov property for Euclidean fields: The case of general exponential interaction

The uniqueness and the global Markov property for the regular Gibbs measure corresponding to the interaction $$U_\Lambda (\varphi ): = \lambda \int\limits_\Lambda {d_2 x\int {d\varrho (\alpha ):e^{\alpha \varphi } :_0 (x)} } $$ [forλ>0,d?(α) a probability measure with support in \(( - 2\sqrt {\pi ,} 2\sqrt \pi )\) ] is proved.     

Energetics of proton transfer between carbon atoms (H3CH ? CH3)−

Ab initio calculations were carried out to study the potential energy surface of (H₃C? H? CH₃)^?. The 6–31G* basis set is supplemented by a set of diffuse p functions on both C and H (with a range of exponents for the latter). The binding energy of CH₄ and CH₃^? to form the (H₃CH? CH₃)^? complex is about 2 kcal/mol, much smaller than for comparable ionic H-bonded systems involving O or N atoms. Nearly half of this interaction energy is due to correlation effects, computed at second and third orders of Møller-Plesset perturbation theory. Correlation is also responsible for substantial reductions in the energy barrier to proton transfer within the complex. This barrier is computed to be 13?15 kcal/mol at the MP3 level, depending upon the exponent used for the H p functions.     

Kinetics and mechanism of the cationic polymerization of tetrahydrofuran in solution. I. THF–CCI4 system

Polymerization of THF in CCl₄ solvent was initiated with 1,3-dioxolan-2-ylium eations with AsF₆^?, PF₆^?, and SbF₆^? anions as well as with esters of fluorosulfonic and trifluoromethanesulfonic acids. With these esters polymerization proceeds with a marked acceleration period, due to slow initiation. The corresponding rate constants of initiation and their dependence on the polarity of the THF/CCl₄ mixture were determined. The rate constant of propagation on the macroion-pairs (k_p^±) of the polytetrahydrofurylium cation with AsF₆^?, PF₆^?, and SbF₆^? and CF₃SO₃^?, anions was found to be independent in CCl₄ solvent on the anion structure and given by the expression: k_p^± = 2.93 × 10^?2 exp {?4.7 × 10³/T} at [THF]₀ = 8.0M. This constant depends on the polarity of the polymerization mixture, and at 25°C for the THF-CCl₄ system, k_p^± = 1.78 × 10^?2 exp {?4.9/D}; thus, in CCl₄ at [THF]₀ = 8.0M, and at 25° k_p^± = 4.0 × 10^?21/mole-sec. In the polymerization with derivatives of CF₃SO₃H (able to form the corresponding macroester) the overall polymerization rate is much lower than that with complex anions because of the reversible conversion of the macroion-pairs into the macroester (internal return). The macroester is much less reactive than the macroionpair (10²–10³ times) in the monomer addition reaction. At [THF]₀ = 8.0M and at 25°C, 96.5% of the growing species exists in the macroester form. Polymerization of THF initiated with derivatives of CF₃SO₃H is a subject of a strong special salt-effect. At a sufficiently high ratio of [AgSbF₆] to [I]₀, where the initiator I is C₂H₅OSO₂CF₃, the overall polymerization rate is equal to that observed for the polymerization of THF on the macroion-pairs, since the internal return within the triflate ion-pair (the macroester formation) is eliminated and polymerization proceeds on the macroion-pairs with SbF₆- anions exclusively.