Bounds for arrays of dots with distinct slopes or lengths

Ann×m sonar sequence is a subset of then×m grid with exactly one point in each column, such that the vectors determined by them are all distinct. We show that for fixedn the maximalm for which a sonar sequence exists satisfiesn–Cn ^11/20<m<n+4n ^2/3 for alln andm>n+c logn log logn for infinitely manyn.Another problem concerns the maximal numberD of points that can be selected from then×m grid so that all the vectors have slopes. We proven ^1/2Dn ^4/5 Supported by Hungarian National Foundation for Scientific Research, Grant No. 1901Research conducted by Herbert Taylor was sponsored in part by the Office of Naval Research under ONR Contract No. N00014-90-J-1341.     

Tilivalline,a new pyrrolo[2, 1-c][1,4] benzodiazepine metabolite from klebsiella

From Klebsiella pneumoniae (+)(11S, 11aS) - 1,2,3,10,11,11a - hexahydro - 9 - hydroxy - 11 - (3′ - indolyl) - 5H - pyrrolo[2,1-c][1,4]benzodiazepin - 5 - one (1) has been isolated for which the name tilivalline is suggested. Structure elucidation and synthesis are reported.     

Modal interference of X-intersecting waveguide switches at large branching angles

The X-intersecting waveguide switch with large branching angles (=2–12°) is analysed using coupled mode theory. The results strongly highlight the importance of modal interference and are in good agreement with measured data.     

Combinatorial complexity bounds for arrangements of curves and spheres

We present upper and lower bounds for extremal problems defined for arrangements of lines, circles, spheres, and alike. For example, we prove that the maximum number of edges boundingm cells in an arrangement ofn lines is (m ^2/3 n ^2/3 +n), and that it isO(m ^2/3 n ^2/3 (n) +n) forn unit-circles, where(n) (and later(m, n)) is a function that depends on the inverse of Ackermann's function and grows extremely slowly. If we replace unit-circles by circles of arbitrary radii the upper bound goes up toO(m ^3/5 n ^4/5 (n) +n). The same bounds (without the(n)-terms) hold for the maximum sum of degrees ofm vertices. In the case of vertex degrees in arrangements of lines and of unit-circles our bounds match previous results, but our proofs are considerably simpler than the previous ones. The maximum sum of degrees ofm vertices in an arrangement ofn spheres in three dimensions isO(m ^4/7 n ^9/7 (m, n) +n ²), in general, andO(m ^3/4 n ^3/4 (m, n) +n) if no three spheres intersect in a common circle. The latter bound implies that the maximum number of unit-distances amongm points in three dimensions isO(m ^3/2 (m)) which improves the best previous upper bound on this problem. Applications of our results to other distance problems are also given.The research of the second author was supported by the National Science Foundation under Grant CCR-8714565. Work by the fourth author has been supported by Office of Naval Research Grant N00014-87-K-0129, by National Science Foundation Grant No. 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. A preliminary version of this paper has appeared in theProceedings of the 29th IEEE Symposium on Foundations of Computer Science, 1988.     

The\bar \partial -operator on algebraic curves

For a singular algebraic curve we show that all closed extensions of are Fredholm, and we give a general index formula. In particular, we prove a modified version of a conjecture due to MacPherson.     

Bis(alk-4-enyl)zink-verbindungen durch addition von alk-2-enylzink an olefine

The dialk-2-enylzinc compounds I–IV add to ethylene, oct-1-ene and 3,3-dimethylcyclopropene to give the corresponding di-alk-4-enylzinc compounds VI–IX (88–98% yield), XV and XVI (90–94% yield), XVII (7%) and XXIIIa–XXVIa (91–98% yield). Additions to XIII are regioselective with Zn → C(1) while stereoselective cis-addition to XXII is observed.     

Synthesen und Strukturen funktionell substituierter Ferrocene

Syntheses and Structures of functional substituted Ferrocenes Substituted ferrocenes have been prepared using carbonic acid chloride of ferrocene. New pathways have been found reacting the acid chloride with sodium hexamethyldisilazane and tris(trimethylsilyl)phosphane, respectively. The compounds FecC(OSiMe₃)N(SiMe₃) (Fec = C₅H₅FeC₅H₄) 1 and FecC(OSiMe₃)P(SiMe₃) 2 have been obtained and characterized by X-ray single crystal structure analysis. Experiments for the preparation of FecCP 3 are reported.     

Syntheses and structure of the first eight-membered fluoro and chloro hafnium siloxane complexes

The eight-membered chloro hafnium siloxane complex [Cl₂HfOtBu₂SiO(THF)₂]₂ 2 has been prepared by the reaction of hafnium tetrachloride and tBu₂Si(OSnMe₃)₂ 1 . The X-ray single crystal structure of [Cl₂HfOtBu₂SiO(THF)₂]₂ is reported. The reaction of 2 with Me₃SnF in a molar ratio of 1:4 leads to the corresponding fluoro complex 3 . The reaction of [(C₅Me₅)HfCl₃] with the lithiated silanediol tBu₂Si(OLi)₂ yields [(C₅Me₅)ClHfOtBu₂SiO]₂ 4 .     

Cationic Graft Copolymers as Carriers for Delivery of Antisense‐Oligonucleotides

Self‐assembling systems based on ionic complexes of DNA fragments (36 base pairs), bcl‐2 antisense oligonucleotides (octadecamer), oligophosphates (25 phosphate groups) or acrylic oligomers (18 groups of phosphonic acid) with poly(L ‐lysine) (PLL) ( = 130 000 and 88 000) grafted with short poly[N‐(2‐hydroxypropyl)methacrylamide] (PHPMA) chains ( = 4 300 or 8 600) were studied by static and dynamic light scattering methods as systems suitable for gene therapy applications. The graft copolymers (GPLLs) with shorter PHPMA grafts ( = 4 300) provide polyelectrolyte complexes (PECs) with smaller and R_H than the corresponding GPLLs with longer grafts ( = 8 600) and the same content of PLL. The lowest aggregation number of 2 was observed for PECs prepared from the GPLL with short grafts and 40 wt.‐% of PLL. The complexes of oligonucleotides and DNA fragments with GPLLs showed quite similar behavior to that with oligophosphates and acrylic oligomer. The complexes prepared from GPLLs containing 40 wt.‐% of PLL and at excess of oligophosphate were stable for at least 48 h under physiological conditions (0.15 M NaCl) and in bovine serum albumin solutions (1 mg · mL^?1). Additionally, polyanion exchange reactions of the PECs in contact with poly(styrenesulfonate) and DNA were studied in 0.15 M NaCl solutions. The oligophosphates in complexes were at least partially substituted with high‐molecular‐weight polyanions. The structure of the initial PECs dominated the PEC structure after the exchange reaction.

The dependence of the molecular weight (a) and the hydrodynamic radius R_H (b) of complexes of the oligophosphate (OPP) and four graft copolymers (GPLLi, i = 0–3) on the mixing ratio X.