Low temperature absorption spectra of TcO 4 − and ReO 4 − in KClO4

The low temperature spectra of TcO ₄ ^– and ReO ₄ ^– both show two band systems with pronounced vibrational structures. The bands are identified as¹ A ₁¹ T ₂ transitions. No other bands are observed with certainty. It seems likely that the KClO₄ crystals contain KReO₄ crystallites. They are therefore not pure mixed crystals. It is concluded that the virtual orbital (2e) used in the construction of the low lying states resembles an atomic nd orbital more and more when going from n=3, Mn to n=5, Re.

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Zusammenfassung Die Tieftemperaturspektren von TcO ₄ ^– und ReO ₄ ^– zeigen beide zwei Bandensysteme mit ausgeprägten Schwingungsstrukturen. Die Banden werden als ¹ A ₁¹ T ₂-Übergänge identifiziert. Keine anderen Banden werden mit Sicherheit beobachtet. Es scheint wahrscheinlich, daß die KClO₄-Kristalle KReO₄-Kristallite enthalten und deswegen keine reinen Mischkristalle sind. Es wird geschlossen, daß das virtuelle Orbital (2e), welches zur Konstruktion der tiefliegenden Zustände gebraucht wird, in der Reihe n=3, Mn bis n=5, Re immer weitgehender einem nd-Atomorbital ähnelt.

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Nachwuchsstipendiat, Schweizerischer Nationalfonds.     

Neue Näherungsgleichung zur einheitlichen Berechnung von Wärmeübertragern

Zusammenfassung Es wird eine für alle Stromführungen einheitliche Näherungsgleichung mit drei oder vier anpaßbaren Parametern zur Berechnung des Korrekturfaktors für die mittlere logarithmische Temperaturdifferenz angegeben. Die anpaßbaren Parameter wurden für etwa 50 verschiedene Stromführungen durch Ausgleichsrechnung bestimmt. Die Genauigkeit der Gleichung ist für die Berechnung im praktisch wichtigen Bereich mehr als ausreichend.

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New approximate equation for uniform heat exchanger design

An approximate equation with three or four empirical parameters for the uniform calculation of the LMTD-correction factor of all heat exchanger configurations is proposed. The empirical parameters have been determined for about 50 different flow configurations using least squares estimation. The accuracy of the equation is more than sufficient for practical design purposes.

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Formelzeichen A Übertragungsfläche - a, b, c, d Parameter der Näherungsgleichung - Wärmekapazitätsstrom - F Korrekturfaktor für die logarithmische mittlere Temperaturdifferenz - k Wärmedurchgangskoeffizient - m, n Zahl der Durchgänge oder Einzelapparate - NTU Anzahl der Übertragungseinheiten (number of transfer units); NTU_i=kA/ _i - P dimensionslose Temperaturänderung - R Wärmekapazitätsstromverhältnis;R ₁=₁/₂;R ₂=₂/₁ - relativer Fehler - Mittelwert von NTU₁ und NTU₂ Indizes 1, 2 Stoffstrom 1, 2 - G Gegenstrom - s Schätzwert Herrn Prof. Dr.-Ing. E.h. K. Stephan zum 65. Geburtstag gewidmet.     

Distribution of energy levels of quantum free particle on the Liouville surface and trace formulae

We consider the Weyl asymptotic formula

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.