Intersections and Unions of Orthogonal Polygons Starshaped Via Staircase n-Paths

For n ≥ 1, define p (n) to be the smallest natural number r for which the following is true: For any finite family of simply connected orthogonal polygons in the plane and points x and y in , if every r (not necessarily distinct) members of contain a common staircase n-path from x to y, then contains such a path. We show that p(1) = 1 and p(n) = 2 (n − 1) for n ≥ 2. The numbers p(n) yield an improved Helly theorem for intersections of sets starshaped via staircase n-paths. Moreover, we establish the following dual result for unions of these sets: Let be any finite family of orthogonal polygons in the plane, with simply connected. If every three (not necessarily distinct) members of have a union which is starshaped via staircase n-paths, then T is starshaped via staircase (n + 1)-paths. The number n + 1 in the theorem is best for every n ≥ 2.     

A Krasnosel’skii theorem for orthogonal polygons starshaped via staircase (n + 1)-paths

Let S be a simply connected orthogonal polygon in the plane, and let n be fixed, n ≥ 1. If every two points of S are visible via staircase n-paths from a common point of S, then S is starshaped via staircase (n + 1)-paths. Moreover, the associated staircase (n + 1)-kernel is staircase (n + 1)-convex. The number two is best possible, and the number n + 1 is best possible for n ≥ 2.     

Primitive radon partitions for cyclic polytopes

It is proved that the setA∪B is a primitive Radon partition in the cyclic polytopeC(n,d) if and only if thed+2 points ofA andB alternate along the moment curve.     

Low-temperature catalytic decomposition of N2O on platinum and bismuth-modified platinum: identification of active sites

Classification of the active surface sites of platinum catalysts responsible for low temperature N2O decomposition, in terms of steps, kinks and terraces, has been achieved by controlled addition of bismuth to as-received platinum/graphite catalysts.     

Essential and inessential points of local nonconvexity

LetS be a closed connected subset of a Hausdorff linear topological space,Q the points of local nonconvexity ofS, E the essential members ofQ, N the inessential. IfS～Q is connected, then the following are true: Theorem 1.If Q ?Øis countable, then S is planar. Theorem 2.If Q is finite and nonempty, then cardE≧cardN+1. Theorem 3.If SυR ² and N is infinite, then E is infinite.     

Active-site coating for molecular discrimination in heterogeneous catalysis

A coherent layer of zeolite A has been applied to a SiO2-supported Pt-Fe oxidation catalyst, using total surface charge-reversal. The zeolite has tracked the metals into the pore structure of the support, covering the active sites. The zeolite channel size is large enough to allow access of CO and O2 to the active sites, and to allow CO2 to emerge, but it excludes larger molecules. The presence of the zeolite membrane transforms the supported Pt-Fe into a highly specific catalyst, which can discriminate between CO and butane, even after the macroscopic catalyst particles are crushed.     

The mechanism of 7,8-diaminopelargonate synthase; the role of S-adenosylmethionine as the amino donor

The product of the first transamination step in the reaction catalysed by diaminopelargonate (DAPA) synthase has been shown to be 4-(S-adenosyl)-2-oxobutanoate, which has been trapped as the corresponding alcohol.     

Helly-type theorems for appropriate colorings of visibility sets

Summary. For positive integers q and n, think of P as the vertex set of a (qn + r)-gon, 0 ￡ r ￡ q - 1 0 \leq r \leq q - 1 . For 1 ￡ i ￡ qn + r 1 \leq i \leq qn + r , define V(i) to be a set of q consecutive points of P, starting at p(i), and let S be a subset of {V(i) : 1 ￡ i ￡ qn + r } \lbrace V(i) : 1 \leq i \leq qn + r \rbrace . A q-coloring of P = P(q) such that each member of S contains all q colors is called appropriate for S, and when 1 ￡ j ￡ q 1 \leq j \leq q , the definition may be extended to suitable subsets P(j) of P. If for every 1 ￡ j ￡ q 1 \leq j \leq q and every corresponding P(j), P(j) has a j-coloring appropriate for S, then we say P = P(q) has all colorings appropriate for S. With this terminology, the following Helly-type result is established: Set P = P(q) has all colorings appropriate for S if and only if for every (2n + 1)-member subset T of S, P has all colorings appropriate for T. The number 2n + 1 is best possible for every r 3 1 r \geq 1 . Intermediate results for q-colorings are obtained as well.