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.     

Functional analysis and boundary-value problems: An introductory treatment

Book Review

Functional analysis and boundary-value problems: An introductory treatmentB. Dayanand Reddy: Pitman Monographs and Survey in Pure and Applied Mathematics, Longman Scientific and Technical, 1986, 333 pp., £45.00 net     

Kaon production in\bar pp interactions at c.m. energies from 200 to 900 GeV

A detailed analysis ofK _s ⁰ production in \(\bar pp\) |<2.5 the average transverse momentum is found to be 0.53±0.07 GeV/c at 200 GeV and 0.62±0.08 GeV/c at 900 GeV, which is an increase with respect to data at c.m. energies below 60 GeV. TheK _s ⁰ production cross sections in inelastic collisions are 29±4 mb at 200 GeV and 63±6 mb at 900 GeV, showing an increase compared to lower energy data. The central kaon density is found to increase as a logarithmic function of energy. At 900 GeV, where statistics are sufficient to allow one to draw conclusions, the average transverse momentum is higher in events with large charged multiplicity than in events with low multiplicity.     

Phase space dependence of the multiplicity distribution in π+ andpp collisions at 250 GeV/c

The charged particle multiplicity distribution has been studied for non-single-diffractive π⁺ p andpp collisions at \(\sqrt s = 22\) GeV, for full phase space as well as for intervals in rapidity, azimuthal angle and transverse momentum. In general, the multiplicity distribution is well described by a negative binomial. From comparison of the distribution for negative or positive particles to that of all charged particles, cascading is favoured as an interpretation over stimulated emission. Interesting consequences follow from a comparison of our results to those at collider energies and toe ⁺ e ^? data at comparable energy. Furthermore, evidence is given that the multiplicity distribution is not exactly of negative binomial type in every (connected or disconnected) phase space region.     

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.     

New books

Few-Body Systems — News Section

New books     

A validated spectrofluorometric method for the determination of celecoxib in capsules

A rapid, selective, sensitive and simple fluorescence method was developed for the direct determination of celecoxib in capsules. The capsules were emptied, pulverized and dissolved in either ethanol or acetonitrile, sonicated and filtered. Direct fluorescence emission was measured at 355±5 nm (exciting at 272 nm). The method was fully validated and the recoveries were excellent, even in presence of excipients.     

Effect of Oxalic Acid on Stabilization and Property of Sb-Doped Tin Oxide Sol

Zeta potential studies show that the ATO nanoparticle surface is positively charged in pH range from 2 to 5 without oxalic acid. The addition of oxalic acid brings a charge reversal on the surface of ATO particle in a wide pH range. The interaction of oxalic acid with surface of ATO nanoparticle was studied by FTIR and indicates that the most probable mechanism is the formation of Sn-O-C bonds via the condensation reaction between the oxalic acid and surface hydroxyl groups. TEM, X-ray diffraction and UV-Vis-Near IR spectra were used to characterize the nanocrystalline ATO sol and thin gel film.     

Metal components analysis of metallothionein-III in the brain sections of metallothionein-I and metallothionein-II null mice exposed to mercury vapor with HPLC/ICP-MS

Mercury vapor is effectively absorbed via inhalation and easily passes through the blood–brain barrier; therefore, mercury poisoning with primarily central nervous system symptoms occurs. Metallothionein (MT) is a cysteine-rich metal-binding protein and plays a protective role in heavy-metal poisoning and it is associated with the metabolism of trace elements. Two MT isoforms, MT-I and MT-II, are expressed coordinately in all mammalian tissues, whereas MT-III is a brain-specific member of the MT family. MT-III binds zinc and copper physiologically and is seemed to have important neurophysiological and neuromodulatory functions. The MT functions and metal components of MTs in the brain after mercury vapor exposure are of much interest; however, until now they have not been fully examined. In this study, the influences of the lack of MT-I and MT-II on mercury accumulation in the brain and the changes of zinc and copper concentrations and metal components of MTs were examined after mercury vapor exposure by using MT-I, II null mice and 129/Sv (wild-type) mice as experimental animals. MT-I, II null mice and wild-type mice were exposed to mercury vapor or an air stream for 2 h and were killed 24 h later. The brain was dissected into the cerebral cortex, the cerebellum, and the hippocampus. The concentrations of mercury in each brain section were determined by cold vapor atomic absorption spectrometry. The concentrations of mercury, copper, and zinc in each brain section were determined by inductively coupled plasma mass spectrometry (ICP-MS). The mercury accumulated in brains after mercury vapor exposure for MT-I, II null mice and wild-type mice. The mercury levels of MT-I, II null mice in each brain section were significantly higher than those of wild-type mice after mercury vapor exposure. A significant change of zinc concentrations with the following mercury vapor exposure for MT-I, II null mice was observed only in the cerebellum analyzed by two-way analysis of variance. As for zinc, the copper concentrations only changed significantly in the cerebellum. Metal components of metal-binding proteins of soluble fractions in the brain sections were analyzed by size-exclusion high-performance liquid chromatography (HPLC) connected with ICP-MS. From the results of HPLC/ICP-MS analyses, it was concluded that the mercury components of MT-III and high molecular weight metal-binding proteins in the cerebellum of MT-I, II null mice were much higher than those of wild-type mice. It was suggested that MT-III is associated with the storage of mercury in conditions lacking MT-I, and MT-II. It was also suggested that the physiological role of MT-III and some kind of high molecular weight proteins might be impaired by exposure to mercury vapor and lack of MT-I and MT-II.