The Representation Polyhedron of a Semiorder

Let a finite semiorder, or unit interval order, be given. When suitably defined, its numerical representations are the solutions of a system of linear inequalities. They thus form a convex polyhedron. We show that the facets of the representation polyhedron correspond to the noses and hollows of the semiorder. Our main result is to prove that the system defining the polyhedron is totally dual integral. As a consequence, the coordinates of the vertices and the components of the extreme rays of the polyhedron are all integral multiples of a common value. Total dual integrality is in turn derived from a particular property of the oriented cycles in the directed graph of noses and hollows of a strictly upper diagonal step tableau. Our approach delivers also a new proof for the existence of the minimal representation of a semiorder, a concept originally discovered by Pirlot (Theory Decis 28:109–141, 1990). Finding combinatorial interpretations of the vertices and extreme rays of the representation polyhedron is left for future work.     

Improved Molecular Weight Control in Ring‐Opening Metathesis Polymerization (ROMP) Reactions with Ru‐Based Olefin Metathesis Catalysts Using N Donors and Acid: A Kinetic and Mechanistic Investigation

The effect of the addition of H₃PO₄ on the ROMP activity of cyclooctene (COE) with first‐ [Cl₂(PCy₃)₂Ru?CHPh] and second‐generation [(H₂IMes)Cl₂(PCy₃)Ru?CHPh] Grubbs’ catalysts 1 and 4 (Cy=cyclohexyl, Ph=phenyl, Mes=2,4,6‐trimethylphenyl (mesityl)), their inhibited mixtures with 1‐methylimidazole (MIM), as well as their isolated bis‐N,N′‐dimethylaminopyridine (DMAP) derivatives [Cl₂(PCy₃)(DMAP)₂Ru?CHPh)] ( 5 b ) and [Cl₂(H₂IMes)(DMAP)₂Ru?CHPh] ( 7 b ) (DMAP=dimethylaminopyridine), a novel catalyst, has been investigated. The studies include the determination of their initiation rates, as well as a determination of the molecular weights and molecular weight distributions of the polymers obtained with these catalysts and catalyst mixtures from the exo‐7‐oxanorbornene derivative 11 . The structure of catalyst 7 b was confirmed by means of X‐ray diffraction. All N‐donor‐bearing catalysts or N‐donor‐containing catalyst mixtures not only exhibited elevated activity in the presence of acid, but also increased initiation rates. Using the reversible inhibition/activation protocol with MIM and H₃PO₄ enabled us to conduct controlled ROMP with catalyst 4 producing the isolated exo‐7‐oxanorbornene‐based polymer 12 with predetermined molecular weights and narrow molecular weight distributions. This effect was based on fast and efficient catalyst initiation in contrast to the parent catalyst 4 . Hexacoordinate complex 5 b also experienced a dramatic increase in initiation rates upon acid‐addition and the ROMP reactions became well‐controlled in contrast to the acid‐free reaction. In contrast, complex 7 b performs well‐controlled ROMP in the absence of acid, whereas the polymerization of the same monomer becomes less controlled in the presence of H₃PO₄. The closer evaluation of catalysts 5 b and 7 b demonstrated that their initiation rates exhibit a linear dependency on the substrate concentration in contrast to catalysts 1 and 4 . As a consequence, their initiation rates are determined by an associative step, not a dissociative step as seen for catalysts 1 and 4 . A feasible associative metathesis initiation mechanism is proposed.     

Olefin metathesis catalysts bearing a pH-responsive NHC ligand: a feasible approach to catalyst separation from RCM products

Two novel ruthenium-based olefin metathesis catalysts, H(2)ITap(PCy(3))Cl(2)Ru[double bond, length as m-dash]CH-Ph and H(2)ITapCl(2)Ru[double bond, length as m-dash]CH-(C(6)H(4)-O-iPr) (H(2)ITap = 1,3-bis(2',6'-dimethyl-4'-dimethylaminophenyl)-4,5-dihydroimidazol-2-ylidene), were synthesized bearing a pH-responsive NHC ligand with two aromatic NMe(2) groups. The crystal structures of complexes and were determined via X-ray crystallography. Both catalysts perform ring opening metathesis polymerization (ROMP) of cyclooctene (COE) at faster rates than their commercially available counterparts H(2)IMes(PCy(3))Cl(2)Ru[double bond, length as m-dash]CH-Ph and H(2)IMesCl(2)Ru[double bond, length as m-dash]CH-(C(6)H(4)-O-iPr) (H(2)IMes = 1,3-bis(2',4',6'-trimethylphenyl)-4,5-dihydroimidazol-2-ylidene) and perform at similar rates during ring closing metathesis (RCM) of diethyldiallylmalonate (DEDAM). Upon addition of 2 equiv. of HCl, catalyst is converted into a mixture of several mono and diprotonated Ru-carbene species 12' which are soluble in methanol but degrade within a few hours at room temperature. Catalyst can be protonated with 2 equiv. of HCl and the resulting complex is moderately water-soluble. The complex is stable in aqueous solution in air for >4 h, but over prolonged periods of time shows degradation in acidic media due to hydrolysis of the NHC-Ru bond. Catalysts and perform RCM of diallylmalonic acid in acidic protic media with only moderate activity at 50 degrees C and do not produce polymer in the ROMP of cationic 7-oxanorbornene derivative under the same conditions. Catalyst was used for Ru-seperation studies when RCM of DEDAM or 3,3-diallypentadione (DAP) was conducted in low-polar organic solution and the Ru-species was subsequently precipitated by addition of strong acid. The Ru-species were removed by (1) filtration and (2) filtration and subsequent extraction with water. The residual Ru-levels could be reduced to as far as 11 ppm (method 2) and 24 ppm (method 1) without the use of chromatography or other scavenging methods.     

Linker‐Free,Silica‐Bound Olefin‐Metathesis Catalysts: Applications in Heterogeneous Catalysis

A set of heterogenized olefin‐metathesis catalysts, which consisted of Ru complexes with the H₂ITap ligand (1,3‐bis(2′,6′‐dimethyl‐4′dimethyl aminophenyl)‐4,5‐dihydroimidazol‐2‐ylidene) that had been adsorbed onto a silica support, has been prepared. These complexes showed strong binding to the solid support without the need for tethering groups on the complex or functionalized silica. The catalysts were tested in the ring‐opening–ring‐closing‐metathesis (RO‐RCM) of cyclooctene (COE) and the self‐metathesis of methyl oleate under continuous‐flow conditions. The best complexes showed a TON>4000, which surpasses the previously reported materials that were either based on the Grubbs–Hoveyda II complex on silica or on the classical heterogeneous Re₂O₇/B₂O₃ catalyst.     

Imaging cytoplasmic cAMP in mouse brainstem neurons

Background  

cAMP is an ubiquitous second messenger mediating various neuronal functions, often as a consequence of increased intracellular Ca²⁺ levels. While imaging of calcium is commonly used in neuroscience applications, probing for cAMP levels has not yet been performed in living vertebrate neuronal tissue before.     

Simple Inductive Proofs of the Fishburn and Mirkin Theorem and the Scott–Suppes Theorem

This paper presents new proofs of two classic characterization theorems for families of ordered sets. The first is that any finite poset with no restriction isomorphic to has an interval representation. The second is that any finite poset with no restriction isomorphic to or to has a unit interval representation. Both proofs are straightforward and inductive.     

Stable isotopic studies of earthworm feeding ecology in tropical ecosystems of Puerto Rico

Feeding strategies of earthworms and their influence on soil processes are often inferred from morphological, behavioral and physiological traits. We used (13)C and (15)N natural abundance in earthworms, soils and plants to explore patterns of resource utilization by different species of earthworms in three tropical ecosystems in Puerto Rico. In a high altitude dwarf forest, native earthworms Trigaster longissimus and Estherella sp. showed less (15)N enrichment ((15)N = 3-6 per thousand) than exotic Pontoscolex corethrurus ((15)N =7-9 per thousand) indicating different food sources or stronger isotopic discrimination by the latter. Conversely, in a lower altitude tabonuco forest, Estherella sp. and P. corethrurus overlapped completely in (15)N enrichment ((15)N = 6-9 per thousand), suggesting the potential for interspecific competition for N resources. A tabonuco forest converted to pasture contained only P. corethrurus which were less enriched in (15)N than those in the forest sites, but more highly enriched in (13)C suggesting assimilation of C from the predominant C(4) grass. These results support the utility of stable isotopes to delineate resource partitioning and potential competitive interactions among earthworm species. Copyright 1999 John Wiley & Sons, Ltd.     

A Finite Non-Unit Free Triangle Order

The notion of a free triangle representation of a partially ordered set was first introduced by Josh Laison [4] as a generalization of the ideas of interval and trapezoid representations. A free triangle representation assigns a triangle to each element of a partially ordered set, with all triangles having one vertex on each of two parallel baselines and a third ‘free’ vertex between the two baselines. In a previous paper [1] we presented an example of an infinite non-unit free triangle order. In this paper we use some of the same ideas to construct an example of a finite, albeit more complicated, non-unit free triangle order.The majority of the content in this paper (theorems, proofs, etc.) was prepared before Ken's untimely death in March of 2005.