Algorithms for ham-sandwich cuts

Given disjoint setsP ₁,P ₂, ...,P _d inR ^d withn points in total, ahamsandwich cut is a hyperplane that simultaneously bisects theP _i . We present algorithms for finding ham-sandwich cuts in every dimensiond>1. Whend=2, the algorithm is optimal, having complexityO(n). For dimensiond>2, the bound on the running time is proportional to the worst-case time needed for constructing a level in an arrangement ofn hyperplanes in dimensiond−1. This, in turn, is related to the number ofk-sets inR ^d−1 . With the current estimates, we get complexity close toO(n ^3/2 ) ford=3, roughlyO(n ^8/3 ) ford=4, andO(n ^d−1−a(d) ) for somea(d)>0 (going to zero asd increases) for largerd. We also give a linear-time algorithm for ham-sandwich cuts inR ³ when the three sets are suitably separated. A preliminary version of the results of this paper appeared in [16] and [17]. Part of this research by J. Matoušek was done while he was visiting the School of Mathematics, Georgia Institute of Technology, Atlanta, and part of his work on this paper was supported by a Humboldt Research Fellowship. W. Steiger expresses gratitude to the NSF DIMACS Center at Rutgers, and his research was supported in part by NSF Grants CCR-8902522 and CCR-9111491.     

Rank-one completions of partial matrices and completely rank-nonincreasing linear functionals

We obtain necessary and sufficient conditions for the existence and the uniqueness of rank-one completions of a partial matrix, and we verify a conjecture of Hadwin and Larson concerning the nature of completely rank-nonincreasing linear functionals defined on pattern subspaces.

     

Investigation of the degradation of commercial polyoxymethylene copolymer in water service applications

An investigation has been carried out on the influence of degradation on various properties of commercial polyoxymethylene (POM) copolymer fittings in chlorinated cold water service lines. Scanning electron microscope (SEM) was used to document interior bulk morphological changes and typical defects. The various types of irregular cracks and pits were observed at the innerface of the samples. DSC and intrinsic viscosity determination were carried out for investigation of crystallinity and molecular weight changes. It was seen that the crystallinity of the POM increases at a much higher rate at the inner surface layers than the outer layer in the used samples. The FT-IR spectra of the new and used samples indicate the extent of degradation of the samples.     

Minimal regular polygons serving as universal covers in R 2

A universal cover is a set K with the property that each set of unit diameter is a subset of a congruent copy of K. It is shown that the smallest regular n-gon, for fixed n 4, which serves as an universal cover in R ² is the smallest regular n-gon covering a Reuleaux triangle of unit width.     

Depolymerization and chemical modification of lignite coal byPseudomonas cepacia strain DLC-07

Applied Biochemistry and Biotechnology - Coal metabolizingPseudomonas cepacia DLC-07 was isolated from soil and shown to use soluble lignite coal as a carbon/energy source.Pseudomonas modified coal...     

Synthesen von Heterocyclen, 130. Mitt.: Über das Glomerin und das Arborin

Zusammenfassung Es werden einstufige Synthesen desGlomerins (2) [1,2-Dimethyl-4(1H)-chinazolinon] und desArborins (3) [1-Methyl-2-benzyl-4(1H)-chinazolinon] bekanntgegeben.

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Syntheses of heterocycles, CXXX: Glomerine and arborine

A one step synthesis ofglomerine (2) [1.2-dimethyl-4(1H)-quinazolinone] andarborine (3) [1-methyl-2-benzyl-4(1H)-quinazolinone] is described.

     

Synthesis, structure, and alpha-elimination chemistry of hafnocene triarylstannyl complexes

New hafnocene triarylstannyl complexes were prepared and were shown to undergo clean thermal decompositions via alpha-aryl-elimination to produce the corresponding stannylene and a hafnocene aryl complex. The rate of the decomposition is highly dependent on the nature of the ancillary ligand, with the stabilities of the CpCp*Hf(SnPh(3))X compounds following the order X = NMe(2) > Np (alpha-agostic) > OMe > Cl > Me. Mechanistic information suggests that alpha-aryl-elimination may be viewed as a concerted process involving nucleophilic attack of the migrating aryl group onto the electrophilic metal center.     

Synthesis and characterization of scandium silyl complexes of the type Cp*2ScSiHRR'. sigma-Bond metathesis reactions and catalytic dehydrogenative silation of hydrocarbons

The scandium dihydrosilyl complexes Cp*(2)ScSiH(2)R (R = Mes (4), Trip (5), SiPh(3) (6), Si(SiMe(3))(3) (7); Mes = 2,4,6-Me(3)C(6)H(2), Trip = 2,4,6-(i)()Pr(3)C(6)H(2)) and Cp*(2)ScSiH(SiMe(3))(2) (8) were synthesized by addition of the appropriate hydrosilane to Cp*(2)ScMe (1). Studies of these complexes in the context of hydrocarbon activation led to discovery of catalytic processes for the dehydrogenative silation of hydrocarbons (including methane, isobutene and cyclopropane) with Ph(2)SiH(2) via sigma-bond metathesis.