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.     

Ion-chromatographic separation of metal cations in the presence of a-hydroxyisobutyric acid

Separations of metal cations on a column packed with the strongly acidic cation exchanger Separon SGX CX were investigated in the presence of -hydroxyisobutyric acid (HIBA) in the mobile phase. A retention model based on the general theory of side equilibria was elaborated and relations describing dependences of capacity factors of analytes on the compositon of the mobile phase were derived. Effects of HIBA concentration and pH of the mobile phase on the analyte retention were studied in detail. Stability constants of divalent metal cations (Cd²⁺, Co²⁺, Mn²⁺, Ni²⁺ and Zn²⁺) with HIBA were calculated from the experimental dependences of the reciprocal values of capacity factors on the ligand concentration.     

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.     

Impact of adsorption isotherm parameters on the performance of enantioseparation using simulated moving bed chromatography

Often there are several chromatographic systems, i.e., combinations of mobile and stationary phases, available to solve a certain separation problem. Essential differences of these chromatographic systems are the separation factors and the efficiencies. For preparative applications in addition also the column saturation capacities and solubility limits are of importance. The impact of all these parameters appears to be rather well understood for conventional overloaded elution chromatography using a single column. In the last years the continuous simulated moving bed (SMB) process was increasingly used as a powerful alternative to batch elution since increased productivities and reduced solvent consumptions could be realised. However, the selection of suitable chromatographic systems is more sophisticated for this process. In this paper five different chromatographic systems capable of separating the enantiomers of mandelic acid are compared based on the achievable productivities using SMB chromatography. For these five systems the adsorption isotherms have been determined experimentally. Subsequently, an analysis of the SMB process was performed numerically using a well-established model.     

2,8'-disubstituted-1,1'-binaphthyls: a new pattern in chiral ligands

The title binaphthyls 19 and 26, which are the positional isomers of 2-methoxy-2'-(diphenylphosphino)-1,1'-binaphthyl (MOP, 19) and 2-amino-2'-hydroxy-1,1'-binaphthyl (NOBIN, 26), have been synthesized by Suzuki coupling as the key step (10 + 15-->18), followed by functional group transformations, involving C-P and C-N bond formation (18-->19 and 18-->23). Racemic intermediate 22 was resolved by co-crystallization with N-benzylcinchonidinium chloride and the absolute configuration determined by X-ray crystallography. These novel binaphthyls are configurationally stable and, as such, potentially usable as chiral ligands in asymmetric reactions. Michael addition of the glycine-derived enolate 40 to methyl acrylate, carried out in the presence of (R)-(-)-27 as the chiral phase-transfer catalyst, afforded L-glutamic acid (S)-(+)-43 of 92% ee (after hydrolysis of the primary product).     

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.     

Synthesis and modeling study of (2S,5R,6R)- and (2S,5R,6S)-6-hydroxy- 8-(1-decynyl)-benzolactam-V8 as protein kinase C modulators

[structures: see text] Both (2S,5R,6R)- and (2S,5R,6S)-6-hydroxy-8-(1-decynyl)benzolactam-V8 were designed and synthesized as PKC modulators. Biological assays reveal the (6R)-ligand to be 20-fold more potent than its (6S)-counterpart in binding to PKC alpha.     

Photophysical processes and the photodissociation of chemical bonds in polyatomic molecules

The main concepts of the nature of electronically excited states in polyatomic molecules and the intramolecular and intermolecular processes of their evolution are reported. The dependence of the probabilities of these processes on the electronic structure of the molecule is considered. Possible mechanisms of the dissociation of electronically excited molecules with bond cleavage are discussed, and the theoretical results of this consideration are given. The experimental data obtained by the authors are interpreted. In this case, attention is focused on C-H-bond photodissociation processes in a condensed phase, which are the best studied processes. The dissociation of other bonds is briefly discussed.     

Total synthesis of polycavernoside A, a lethal toxin of the red alga Polycavernosa tsudai

[structure: see text] Two approaches to the synthesis of the aglycon 120 of polycavernoside A (1) were developed, only one of which was completed. The successful "second-generation" route assembled the aglycon seco acids 102 and 106 via Nozaki-Hiyama-Kishi coupling of aldehyde 70, prepared from methyl (S)-3-hydroxy-2-methylpropionate (72) and (S)-pantolactone (73), with vinyl bromide 71. The latter was obtained from a sequence which commenced from the silyl ether 24 of 3-hydroxypropionaldehyde and entailed cyclization of (Z)-zeta-hydroxy-alpha,beta-unsaturated ester 82. Regioselective Yamaguchi lactonization of trihydroxycarboxylic acids 102 and 106 and subsequent functional-group adjustments led to macrolactone 120, to which the fucopyranosylxylopyranoside moiety was attached. Stille coupling of the glycosidated aglycon 128 with dienylstannane 129 furnished polycavernoside A in a synthesis for which the longest linear sequence was 25 steps. The overall yield to lactone 120 was 4.7%.