Step Find a point Pin the plane w%ch is in &he In&$x of Cl-l(s). At worst, this can be done in clfl sQp9 by te dting 3 element subsets of S for collineti@, discarding middle p&n& of collinear rets ar5b bq@rig when the fust noncollinear set (if there i# or&j. &y X, y arrcf z, is found. P can be chosen to I&E the centroid oC the triangle formed by X, y and z. Sfq 2: Express each si E S in polar c...

A tandem catalytic reaction has been developed as part of a process to discover tungsten-based olefin metathesis catalysts that have a strong preference for terminal olefins over cis or trans internal isomers in olefin metathesis. This tandem isomerization/terminal olefin metathesis reaction (ISOMET) converts Cn trans internal olefins into C2n−2 cis olefins and ethylene. This reaction is made p...

In this talk I will review the concept of the biembedding of two latin squares. Of particular interest will be the regular biemedding of two isomorphic copies of the latin square corresponding to the cyclic group of order n, denoted Cn. Grannell and Griggs have shown that, for all n, a regular biembedding exists, and in addition, that the automorphism group of the regular biembedding has order ...

The crossing number, cr(G), of a graph G is the least number of crossing points in any drawing of G in the plane. According to the Crossing Lemma of Ajtai, Chvátal, Newborn, Szemerédi [ACNS82] and Leighton [L83], the crossing number of any graph with n vertices and e > 4n edges is at least constant times e/n. Apart from the value of the constant, this bound cannot be improved. We establish some...

The lowest-lying 1(01) <-- 0(00) transition of para-H(2)D(+) and 1(11) <-- 0(00) of ortho-D(2)H(+) has been detected by the enhancement of the D/H isotope exchange reaction in collisions with p-H2 upon rotational excitation. These are the first pure rotational spectra of molecular ions by action spectroscopy. For this purpose, a cryogenic multipole ion trap has been combined with narrow-band tu...