An example concerning unions of two starshaped sets in the plane

LetS be a closed, simply connected subset of the plane,J a line segment (or a one-pointed set),J⊂S. If for every three points ofS there is a point ofJ seeing at least two of these points viaS, thenS is a union of two starshaped sets. IfJ⊈S or ifS is not simply connected, the result fails.     

Increasing unions of starlike sets

Let S be an arbitrary set in R². Every three points of cl S are clearly visible via S from a common point of cl S if and only if each bounded subset of S may be extended to a starlike set in S. When this occurs, set S is expressible as an increasing union of starlike sets.     

A Helly-type theorem for intersections of orthogonally starshaped sets in \mathbb R^d

Let $\mathcal K$ be a finite family of orthogonal polytopes in $\mathbb R^d$ such that, for every nonempty subfamily $\mathcal K^\prime $ of $\mathcal K, \cap \{K : K$ in $\mathcal K^\prime \}$ , if nonempty, is a finite union of boxes whose intersection graph is a tree. Assume that every $d + 1$ (not necessarily distinct) members of $\mathcal K$ meet in a (nonempty) staircase starshaped set. Then $S \equiv \cap \{ K : K$ in $\mathcal K\}$ is nonempty and staircase starshaped.     

Principal ideals in the semigroup of binary relations on a finite set: What happens when one element is added to the set

The principal ideals in the semigroup of binary relations on a finite set X form a partially ordered set under inclusion. When the cardinality of the set X is increased by one, some of the covering relations present in the original set will be preserved and some will not. This article describes the difference.     

Steric effects in 31P NMR spectra: ‘Gamma’ shielding in aliphatic phosphorus compounds

An examination of published ³¹P NMR spectral data for aliphatic phosphorus compounds has revealed that chain lengthening and branching effects on the chemical shift can be interpreted in terms similar to those used for ¹³C and ¹⁵N shifts. For six families of phosphorus compounds, a β-carbon substituent was shown to deshield phosphorus, while a γ-carbon caused shielding. The effects are additive, and good agreement was obtained between ³¹P shifts calculated with the appropriate constants and the experimental values. Shielding by γ-carbon is indicative of the operation of a steric influence on ³¹P chemical shifts, not heretofore articulated. The γ-effect is also useful in explaining the unusually large shielding found in six-membered cyclic phosphines.     

The reaction of the hydroxyl radical with acetylene

The reaction of OH with acetylene was studied in a discharge flow system at room temperature. OH was generated by the reaction of atomic hydrogen with NO₂ and was monitored throughout the reaction using ESR spectroscopy. Mass-spectrometric analysis of the reaction products yielded the following results: (1) less than 3 molecules of OH were consumed, and less than 2 molecules of H₂O were formed for every molecule of acetylene that reacted; (2) CO was identified as the major carbon-containing product; (3) NO, formed in the generation of OH, reacted with a reaction intermediate to give among other products N₂O. These observations placed severe limitations on the choice of a reaction mechanism. A mechanism containing the reaction OH + C₂H₂ → HC₂O + H₂ better accounted for the experimental results than one involving the abstraction reaction OH + C₂H₂ → C₂H + H₂O. The rate constant for the initial reaction was measured as 1.9 ± 0.6 × 10^?13 cm³ molecule^?1 sec^?1.     

Leading large order asymptotics for (φ4)2 perturbation theory

Communications in Mathematical Physics - We develop a rigorous semiclassical expansion to compute the radius of convergence of the Borel transform for the pressure in (φ4)2 field theory. This...     

A decomposition theorem form-convex sets

LetS be a closedm-convex subset of the plane,m≧2,Q the set of points of local nonconvexity ofS, with convQ ⊆S. If there is some pointp in [(bdryS) ∩ (kerS)] ∼Q, thenS is a union ofm−1 closed convex sets. The result is best possible for everym.