WCl6/LiAlH4 Promoted transformation of imines into secondary and tertiary amines

The reaction of WCl₆/LiAlH₄ with imines, R′NCHR, gave tertiary amines, R′N(CH₂R)₂, and secondary amines, R′NHCHRCH₂R. Isotope labeling experiments revealed that the reaction involved two types of azatungstenacyclobutanes, $\text{WNR′CHRC}$HR and $\text{WCHRNR′C}$HR, produced from the reaction of an alkylidene tungsten intermediate with the imine CN double bond. Formation of these metallacyclobutanes is highly dependent on the solvent used.     

Lower bounds for combinatorial problems on graphs

Nontrivial lower bounds are given for the computation time of various combinatorial problems on graphs under a linear or algebraic decision tree model. An $\text{Ω(n}{}^3\text{log}\text{n)}$ bound is obtained for a “travelling salesman problem” on a weighted complete graph of n vertices. Moreover it is shown that the same bound is valid for the “subgraph detection problem” with respect to property P if P is hereditary and determined by components. Thus an $\text{Ω(n}{}^3\text{log}\text{n)}$ bound is established in a unified way for a rather large class of problems.     

A linear 5-coloring algorithm of planar graphs

A simple linear algorithm is presented for coloring planar graphs with at most five colors. The algorithm employs a recursive reduction of a graph involving the deletion of a vertex of degree 6 or less possibly together with the identification of its several neighbors.     

Hermite Constants of Division Algebras

 We introduce a constant attached to a central division algebra D over a number field which is a generalization of the Hermite–Rankin constant. Geometrically, equals the maximum of minimal twisted heights of rational points of a generalized Brauer–Severi variety. We will deduce a duality result, an analog of Rankin’s inequality and an upper estimate for . Received 23 October 2000; in revised form 8 September 2001     

Finding the connected components and a maximum clique of an intersection graph of rectangles in the plane

An intersection graph of rectangles in the (x, y)-plane with sides parallel to the axes is obtained by representing each rectangle by a vertex and connecting two vertices by an edge if and only if the corresponding rectangles intersect. This paper describes algorithms for two problems on intersection graphs of rectangles in the plane. One is an O(n log n) algorithm for finding the connected components of an intersection graph of n rectangles. This algorithm is optimal to within a constant factor. The other is an O(n log n) algorithm for finding a maximum clique of such a graph. It seems interesting that the maximum clique problem is polynomially solvable, because other related problems, such as the maximum stable set problem and the minimum clique cover problem, are known to be NP-complete for intersection graphs of rectangles. Furthermore, we briefly show that the k-colorability problem on intersection graphs of rectangles is NP-complete.     

The gonality of smooth curves with plane models

The Hausdorff measure with fractional index is used in order to define a functional on measurable sets of the plane. A fractal set, constructed using the well-known Von Koch set, is involved in the definition. This functional is proved to arise as the limit of a sequence of classical functionals defined on sets of finite perimeter. Thus it is shown that a natural extension of the ordinary functionals of the calculus of variations leads both to fractal sets and to the fractional Hausdorff measure.     

A theorem on paths in planar graphs

C. Thomassen extended Tutte's theorem on cycles in planar graphs in the paper “A Theorem on Paths in Planar Graphs”. This note corrects a flaw in his proof.     

Polarography in polymer science

A review of polarography in polymer science is presented.