The overlay of lower envelopes and its applications

Let and be two collections of a total ofn (possibly partially defined) bivariate, algebraic functions of constant maximum degree. The minimization diagrams of are the planar maps obtained by the xy-projections of the lower envelopes of , respectively. We show that the combinatiorial complexity of the overlay of the minimization diagrams of and of is O(n^2+ɛ), for any ɛ>0. This result has several applications: (i) a near-quadratic upper bound on the complexity of the region in 3-space enclosed between the lower envelope of one such collection of functions and the upper envelope of another collection; (ii) an efficient and simple divide-and-conquer algorithm for constructing lower envelopes in three dimensions; and (iii) a near-quadratic upper bound on the complexity of the space of all plane transversals of a collection of simply shaped convex sets in three dimensions. Work on this paper by the first author was supported by National Science Foundation Grant CCR-93-01259 and an NYI award. Work on this paper by the second author was supported by the Netherlands’ Organization for Scientific Research (NWO) and partially supported by ESPRIT Basic Research Action No. 6546 (project PROMotion). His current address is Department of Computer Science, Postech, San 31, Hyoja-Dong, Pohang, Kyungbuk 790–784, South Korea. Email: otfried@vision.postech.ac.kr. Work on this paper by the third author was supported by NSF Grant CCR-91-22103, by a Max-Planck Research Award, and by grants from the U.S.-Israeli Binational Science Foundation the Fund for Basic Research administered by the Israeli Academy of Sciences, and the G.I.F., the German-Israeli Foundation for Scientific Research and Development.     

Preparation and Characterisation of Meloxicam Hydroxy Propyl <Emphasis Type="Italic">β</Emphasis>-Cyclodextrin Inclusion Complex

Meloxicam is a non steroidal anti inflammatory drug, used in the treatment of rheumatoid and osteoarthiritis. It is practically insoluble in water and its prolonged use is associated with the incidence of side effects like gastro intestinal perforations, ulcerations and bleeding. Therefore, an attempt has been made to improve the aqueous solubility of the drug by making an inclusion complex using hydroxy propyl β cyclodextrin(HPβ-CD). The complexes were prepared by physical mixture and freeze drying method. The different methods employed for evaluation such as DSC, XRD, SEM and FT-IR studies indicated complete formation of the complex by freeze drying method in a molar ratio of 1:2. The prepared complexes showed improved in-vitro dissolution profile as compared to the pure drug.     

The molecular structure of 4a, 5, 8, 8a-tetrahydro-11, 14-dimethoxy-7-methyl-4a-(3-methyl-2-butenyl)-5, 8a-o-benzeno-1, 4-naphthoquinone

A novel natural product, microphyllone, has been isolated from Ehretia microphylla² together with baurenol and ursolic acid. Spectroscopic techniques, derivative formation and finally X-ray diffraction have been utilized in the structure elucidation of microphyllone. The structure appears to be 4a, 5, 8,8a - tetrahydro -11,14- dihydroxy-7-methyl-4a-(3-methyl-2-butenyl)-5, 8a-o-benzeno-1, 4-naphthoquinone.     

N,N-bis[2-oxo-3-oxazolidinyl]phosphorodiamidic chloride: a novel coupling reagent in the synthesis of oligodeoxyribonucleotides

Use of a novel coupling reagent, N,N-bis[2-oxo-3-oxazolidinyl]phosphorodiamidic chloride (BOPDC) in the synthesis of oligodeoxyribonucleotides is described. This reagent allows the synthesis of phosphotriesters in high yields (70–80%) without detectable side reactions. Synthesis of a hexanucleotide, d(A-A-C-C-C-G) is presented as an example.