The probability thatn random points in a triangle are in convex position

We show thatn random points chosen independently and uniformly from a triangle are in convex position with probability $$\frac{{2^n (3n - 3)!}}{{((n - 1)!)^3 (2n)!}}$$ .     

Lines,line-point incidences and crossing families in dense sets

LetP be a set ofn points in the plane. We say thatP isdense if the ratio between the maximum and the minimum distance inP is of order . A setC of line segments in the plane is calleda crossing family if the relative interiors of any two line segments ofC intersect. Vertices of line segments of a crossing familyC are calledvertices of C. It is known that for any setP ofn points in general position in the plane there is a crossing family of size with vertices inP. In this paper we show that ifP is dense then there is a crossing family of almost linear size with vertices inP.The above result is related to well-known results of Beck and of Szemerédi and Trotter. Beck proved that any setP ofn points in the plane, not most of them on a line, determines at least (n ²) different line. Szemerédi and Trotter proved that ifP is a set ofn points and is a set ofm lines then there are at mostO(m ^2/3 n ^2/3 +m+n) incidences between points ofP and lines of . We study whether or not the bounds shown by Beck and by Szemerédi and Trotter hold for any dense setP even if the notion of incidence is extended so that a point is considered to be incident to a linel if it lies in a small neighborhood ofl. In the first case we get very close to the conjectured bound (n ²). In the second case we obtain a bound of order .The work on this paper was supported by Czech Republic grant GAR 201/94/2167, by Charles University grants No. 351 and 361, by Deutsche Forschungsgemeinschaft, grant We 1265/2-1, and by DIMACS.     

Grain boundary segregation diagrams of α-iron

Based on thermodynamic analysis of interfacial segregation, the segregation enthalpy H ^o of a solute I in a given matrix was found to depend linearly on two mutually independent terms reflecting the type of interface and the solid solubility limit X _infI ^sup* at temperature T and can be written as In this equation, the structural dependence of interfacial segregation is contained in H ^*() which corresponds to the extrapolated segregation enthalpy of a solute with unlimited solubility in the matrix. The product [Tln(X _infI ^sup* )] is essentially constant with temperature, and can therefore be obtained from data for maximum solid solubility, [Tln(X _infI ^sup* )]_max. The parameter v>0 represents the relationship between the activity a _infI ^sup* of a solute at the bulk solid solubility limit in a given matrix and X _infI ^sup* , a _infI ^sup* =(X _infI ^sup* ) ^v , and is characteristic for the matrix. Using recent experimental data for silicon, phosphorus, and carbon segregation at well-characterized grain boundaries in oriented bicrystals of -iron, the averaged value was determined. Values of H ^*() range from -8 kJ/mol (general grain boundaries) up to +8 kJ/mol (special grain boundaries). These values are discussed and used for a more precise and generalized construction of grain boundary segregation diagrams of -iron.     

Extremal problems for certain analytic (Nevanlinna) functions

We obtain sharp bounds on some basic functionals defined on the sets of all analytic functions having the representations \(f\left( z \right) \equiv \int\limits_{ - 1}^1 {\frac{{d\mu \left( t \right)}}{{z - t}}} \) and \(\varphi \left( z \right) \equiv \int\limits_{ - 1}^1 {\frac{{z\mu \left( t \right)}}{{1 - tz}}} \) ; respectively. Here μ is a probability measure.     

Facile one-pot synthesis of BINOL- and H8-BINOL-based aryl phosphites and their use in palladium catalysed asymmetric allylation

Novel P-monodentate aryl phosphite ligands have been synthesised in one step from (R)-BINOL, (R)-H₈-BINOL and (R)-H₈-3,3′-dibromo-BINOL. With the new aryl phosphites, up to 86% ee was observed in the asymmetric Pd-catalysed amination of 1,3-diphenyl-2-propenyl acetate with sodium diformylamide. In the enantioselective alkylation of 1,3-diphenyl-2-propenyl acetate with dimethyl malonate, up to 97% enantioselectivity was achieved.     

Beitrag zur Stickstoffschnellbestimmung nach Merz

Zusammenfassung Einige Änderungen an der Apparatur zur Stickstoffschnellbestimmung nachMerz wurden vorgenommen. Dadurch wird erreicht, daß zwischen den einzelnen Verbrennungen Sauerstoff das Verbrennungsrohr durchspült und vor dem Reduktionsrohr nach außen tritt. Der Verbrennungsraum ist daher bereits zu Beginn jeder Analyse mit Sauerstoff gefüllt, wodurch günstigere Bedingungen für die schlagartige Verbrennung erzielt werden. Der Umbau der von der Firma Heraeus gelieferten Apparatur und die Herstellung eines Oxydationskatalysators (Kobaltoxid-Wolframoxid) werden beschrieben.

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Contribution to the rapid nitrogen determination according to merz

Summary The apparatus for rapid nitrogen determination according toMerz has been modified. Between each two combustions, oxygen is flushed through the combustion tube and is vented before the reduction tube. Therefore, at the beginning of each analysis, the combustion chamber is filled with oxygen, providing a more favorable atmosphere for the rapid ignition. The alterations in the Hereaus apparatus and the preparation of an oxidation catalyst (cobalt oxide and tungsten oxide) are described.

     

Concrete quantum logics with covering properties

LetL be a concrete (=set-representable) quantum logic. Letn be a natural number (or, more generally, a cardinal). We say thatL admits intrinsic coverings of the ordern, and writeL C _n , if for any pairA, BL we can find a collection {C _i iI}, where cardI<n andC _i L for anyiI, such thatA B= _il C _i . Thus, in a certain sense, ifLC _n , then the rate of noncompatibility of an arbitrary pairA,BL is less than a given numbern. In this paper we first consider general and combinatorial properties of logics ofC _n and exhibit typical examples. In particular, for a givenn we construct examples ofL C _n+1\C _n . Further, we discuss the relation of the classesC _n to other classes of logics important within the quantum theories (e.g., we discover the interesting relation to the class of logics which have an abundance of Jauch-Piron states). We then consider conditions on which a class of concrete logics reduce to Boolean algebras. We conclude with some open questions.     

The sorption of some radiocations on microporous titanosilicate ETS-10

The microporous titanosilicate ETS-10 synthesized from gel with following molar composition: 1.0 Na₂O: 1.49 SiO₂ : 0.2 TiO₂ : 0.6 KF : 1.28 HCl : 39.5 H₂O was subjected to sorption of radioactive cations ¹¹⁵Cd²⁺, ²⁰⁴Hg²⁺, ⁶⁰Co²⁺ and ¹³⁷Cs⁺ (M) from aqueous solution, in the absence of ionic competition. The uptake of these cations on the ETS-10 was compared by means of the distribution coefficient (K_d) versus contact time and sorption capacity (R) at equilibrium. The FT-IR spectra of M-ETS-10 sorption products exhibit a modification of the absorption band, principally at 381 cm^-1.     

Zwitterionic ion chromatography with carboxybetaine surfactant-coated particle packed and monolithic type columns

Both particle packed (25 cm x 0.46 cm I.D. SUPELCOSIL 5 microm C18) and monolithic type (10 cm x 0.46 cm I.D. Merck Chromolith Performance C18) reversed-phase substrates were dynamically coated with a carboxybetaine type zwitterionic surfactant ((dodecyldimethyl-amino) acetic acid) and investigated as stationary phases for use in zwitterionic ion chromatography (ZIC). Investigations into eluent concentration and pH were carried out using KCl eluents containing 0.2 mM of the carboxybetaine surfactant to stabilise the column coatings. It was found that eluent concentration decreased anion retention whilst simultaneously increasing peak efficiencies, which may be due to the dissociation of intra- and inter-molecular salts of the carboxybetaine surfactant under higher ionic strength conditions. The Effect of eluent pH was an increase in anion retention with decreased eluent pH due to the increased protonation of the weak acid terminal group of the carboxybetaine, causing both a relative increase in the positive charge of the stationary phase and less repulsion of the anions by the dissociated weak acid group. The carboxybetaine-coated monolithic phase was applied to rapid anion separations using elevated flow rates and flow rate gradients.     

Unexpected formation of dinaphthoaza-17-crown-5 ether containing γ-aminopiperidine subunit

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