The complexity of many cells in arrangements of planes and related problems

We consider several problems involving points and planes in three dimensions. Our main results are: (i) The maximum number of faces boundingm distinct cells in an arrangement ofn planes isO(m ^2/3 n logn +n ²); we can calculatem such cells specified by a point in each, in worst-case timeO(m ^2/3 n log³ n+n ² logn). (ii) The maximum number of incidences betweenn planes andm vertices of their arrangement isO(m ^2/3 n logn+n ²), but this number is onlyO(m ^3/5– n ^4/5+2 +m+n logm), for any>0, for any collection of points no three of which are collinear. (iii) For an arbitrary collection ofm points, we can calculate the number of incidences between them andn planes by a randomized algorithm whose expected time complexity isO((m ^3/4– n ^3/4+3 +m) log² n+n logn logm) for any>0. (iv) Givenm points andn planes, we can find the plane lying immediately below each point in randomized expected timeO([m ^3/4– n ^3/4+3 +m] log² n+n logn logm) for any>0. (v) The maximum number of facets (i.e., (d–1)-dimensional faces) boundingm distinct cells in an arrangement ofn hyperplanes ind dimensions,d>3, isO(m ^2/3 n ^d/3 logn+n ^d–1). This is also an upper bound for the number of incidences betweenn hyperplanes ind dimensions andm vertices of their arrangement. The combinatorial bounds in (i) and (v) and the general bound in (ii) are almost tight.Work on this paper by the first author has been supported by Amoco Fnd. Fac. Dev. Comput. Sci. 1-6-44862 and by NSF Grant CCR-8714565. Work by the third author has been supported by Office of Naval Research Grant N00014-87-K-0129, by National Science Foundation Grant DCR-82-20085, by grants from the Digital Equipment Corporation, and the IBM Corporation, and by a research grant from the NCRD—the Israeli National Council for Research and Development. An abstract of this paper has appeared in theProceedings of the 13th International Mathematical Programming Symposium, Tokyo, 1988, p. 147.     

Implicitly representing arrangements of lines or segments

Anarrangement ofn lines (or line segments) in the plane is the partition of the plane defined by these objects. Such an arrangement consists ofO(n ²) regions, calledfaces. In this paper we study the problem of calculating and storing arrangementsimplicitly, using subquadratic space and preprocessing, so that, given any query pointp, we can calculate efficiently the face containingp. First, we consider the case of lines and show that with (n) space¹ and (n ^3/2) preprocessing time, we can answer face queries in (n)+O(K) time, whereK is the output size. (The query time is achieved with high probability.) In the process, we solve three interesting subproblems: (1) given a set ofn points, find a straight-edge spanning tree of these points such that any line intersects only a few edges of the tree, (2) given a simple polygonal path , form a data structure from which we can find the convex hull of any subpath of quickly, and (3) given a set of points, organize them so that the convex hull of their subset lying above a query line can be found quickly. Second, using random sampling, we give a tradeoff between increasing space and decreasing query time. Third, we extend our structure to report faces in an arrangement of line segments in (n ^1/3)+O(K) time, given(n ^4/3) space and (n ^5/3) preprocessing time. Lastly, we note that our techniques allow us to computem faces in an arrangement ofn lines in time (m ^2/3 n ^2/3+n), which is nearly optimal.The first author is pleased to acknowledge the support of Amoco Fnd. Fac. Dev. Comput. Sci. 1-6-44862 and National Science Foundation Grant CCR-8714565. Work on this paper by the fifth author has been supported by Office of Naval Research Grant N00014-87-K-0129, by National Science Foundation Grant NSF-DCR-83-20085, by grants from the Digital Equipment Corporation, and the IBM Corporation, and by a research grant from the NCRD—the Israeli National Council for Research and Development. The sixth author was supported in part by a National Science Foundation Graduate Fellowship. This work was begun while the non-DEC authors were visiting at the DEC Systems Research Center.     

Block renormalization group for Euclidean Fermions

Block renormalization group transformations (RGT) for lattice and continuum Euclidean Fermions in d dimensions are developed using Fermionic integrals with exponential and -function weight functions. For the free field the sequence of actionsD _k generated by the RGT from D, the Dirac operator, are shown to have exponential decay; uniform ink, after rescaling to the unit lattice. It is shown that the two-point functionD ^–1 admits a simple telescopic sum decomposition into fluctuation two-point functions which for the exponential weight RGT have exponential decay. Contrary to RG intuition the sequence of rescaled actions corresponding to the -function RGT do not have uniform exponential decay and we give examples of initial actions in one dimension where this phenomena occurs for the exponenential weight RGT also.     

The dynamics of multidimensional cosmological models in the presence of quantum effects. I. The case FRW ×T D at high temperatures

The dynamical system approach is applied to the study of dynamics of multidimensional cosmological models with topology FRW ×T ^D (D-dimensional torus) in the presence of high-temperature quantum effects. The stability methods developed in the paper of Szydowski (Gen. Rel. Grav.,20, 221, 1988) are used in the analysis of typical states of the metric in the neighborhood of singularities and for large time values. The problems of dynamical dimensional reduction, structure of singularities, isotropization, etc., are discussed in this context.     

On an application of Hall's representatives theorem to a finite geometry problem

It is well-known thatn points not belonging to a hyperplane determine at leastn hyperplanes. The possible configurations of hyperplanes in the case when the number of hyperplanes is equal ton are known, too. In this paper we obtain these results by means of Hall's representatives theorem. The setting is that of a finite geometry.     

Forward-backward multiplicity correlations in σ+ p,K + p andpp collisions at 250 GeV/c

Forward-backward multiplicity correlations in σ⁺,K ⁺ p andpp collisions at 250 GeV/c ( \(\sqrt s \) =22 GeV) are given for all charges and for the different charge combinations. The correlations are found to be caused predominantly by centrally produced particles. It is demonstrated that this result is an agreement with observations at the ISR and the CERNp \(\bar p\) -Collider. The results are compared to expectations from LUND, DPM and FRITIOF Monte Carlo models and a geometrical picture relating correlations in hadron-hadron collisions toe ⁺ e ^? data in terms of impact parameters is tested.     

A model of adsorption at liquid-solid interface involving association in the bulk phase

A model of monolayer adsorption of binary liquid mixtures on homogeneous and heterogeneous solid surfaces involving association of one component in the bulk phase is discussed. Suitable model calculations, illustrating association and heterogeneity effects, have been performed according to an equation derived for adsorption excess. This equation has been examined by using the experimental data of adsorption of alcohols from benzene andn-heptane on silica gel.

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Adsorptionsmodell für die Grenzfläche Feststoff-Flüssigkeit unter Berücksichtigung der Assoziation in der Flüssigkeitsphase

Zusammenfassung Es wird ein Adsorptionsmodell binärer, flüssiger Mischungen an homogenen und heterogenen Oberflächen von Feststoffen unter Beachtung der Assoziation eines der Bestandteile in der Flüssigkeitsphase diskutiert. Mit der aus dem Oberflächenüberschuß abgeleiteten Gleichung wurden entsprechende Modellberechnungen durchgeführt, die die mit Assoziation und Heterogenität verbundenen Effekte illustrieren. Die Gleichung wurde für die experimentellen Daten der Alkoholadsorption aus Benzol undn-Heptan an Kieselgel überprüft.

     

DTA determination of the anhydrite content in rock salts

The DTA method was used to determine the phase diagram of the NaCl-CaSO₄ system over the range 0.55 wt.% of CaSO₄ in air atmosphere. The effects of SiO₂, CaCO₃ and Fe₂O₃ on the melting temperatures of pure NaCl and the eutectic were determined. On the basis of the above temperatures, a method of quantitative determination of the anhydrite content in rock salts has been developed.

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Zusammenfassung DTA wurde benutzt, um in Luft im Bereich 0 bis 55,0 Gew.% CaSO₄ das Phasendiagramm des Systemes NaCl-CaSO₄ zu bestimmen. Es wurde auch der Einfluß von SiO₂, CaCO₃ und Fe₂O₃ auf die Schmelztemperatur von reinem NaCl bzw. des Eutektikums bestimmt. Auf der Grundlage obiger Temperaturen wurde eine Methode zur quantitativen Bestimmung des Anhydritgehaltes in Steinsalzen entwickelt.

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- 0–55,0 . , . , ( ) .

     

Protolytic reactions of 3,3'-dimethylnaphthidine and 3,3'-dimethoxybenzidine

The dissociation constants of diprotonated 3,3'-dimethylnaphthidine (DMN) and 3,3'-dimethoxybenzidine (DMB) have been determined spectrophotometrically. They are: pK(a1) = 2.62 +/- 0.03, pK(a2) = 3.33 +/- 0.09 for DMN: pK(a1) = 2.83 +/- 0.07, pK(a2) = 4.05 +/- 0.12 for DMB. The molar absorptivities (l.mole(-1).cm(-1)) of all forms of the indicators have been also determined: epsilon(B) = 1.68 x 10(4), epsilon(BH(+)) = 9.34 x 10(3), epsilon(BH(2+)(2)) = 1.80 x 10(3) at 300 nm for DMB; epsilon(B) = 7.33 x 10(3), epsilon(BH(+)) = 3.73 x 10(3), epsilon(BH(2+)(2)) = 0 at 330 nm for DMN.