Approximation of convex bodies by simplices

This paper deals with the following kind of approximation of a convex bodyQ in Euclidean space E ⁿ by simplices: which is the smallest positive numberh _S(Q) such thatS ₁ Q S ₂ for a simplexS ₁ and its homothetic copyS ₂ of ratioh _S(Q). It is shown that ifS ₀ is a simplex of maximal volume contained inQ, then a homothetic copy ofS ₀ of ratio 13/3 containsQ.     

Finely holomorphic functions and quasi-analytic classes

We study the set of functions in quasi-analytic classes and the set of finely holomorphic functions. We show that no one of these two sets is contained in the other.LetI denote the set of complex functionsf: for which there exists a quasi-analytic classC{M _n} containingf. Let denote the set of complex functionsf: for which there exist a fine domainU containing the real line and a function finely holomorphic onU satisfyingf(x)= (x) for allx . The power of unique continuation is incomparable in these two cases (I\ is non-empty, \I is non-empty).Research supported by the grant No. 201/93/2174 of Czech Grant Agency and by the grant No. 354 of Charles University.     

Energy-level statistics of model quantum systems: Universality and scaling in a lattice-point problem

We investigate the statistics of the numberN(R, S) of lattice pointsnZ ², in an annular domain (R, w)=(R+w)A\RA, whereR, w>0. HereA is a fixed convex set with smooth boundary andw is chosen so that the area of (R, w) isS. The statistics comes fromR being taken as random (with a smooth density) in some interval [c ₁ T,c ₂,T],c ₂>c ₁>0. We find that in the limitT the variance and distribution of N=N(R; S)–S depend strongly on howS grows withT. There is a saturation regimeS/T, asT, in which the fluctuations in N coming from the two boundaries of are independent. Then there is a scaling regime,S/Tz, 0<z<, in which the distribution depends onz in an almost periodic way going to a Gaussian asz0. The variance in this limit approachesz for genericA, but can be larger for degenerate cases. The former behavior is what one would expect from the Poisson limit of a distribution for annuli of finite area.     

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.