Near-perfect token distribution

Suppose that n tokens are arbitrarily placed on the n nodes of a graph. At each parallel step one token may be moved from each node to an adjacent node. An algorithm for the near-perfect token distribution problem redistributes the tokens in a finite number of steps, so that, at the end, no more than O(1) tokens reside at each node. (In perfect distribution, at the end, exactly one token resides at each node.) In this paper we present a simple algorithm that works for all extrovert graphs, a new property which we define and study. In terms of connectivity requirements, extrovert graphs are roughly in-between expanders and compressors. Our results lead to an optimal solution for the near-perfect token distribution problem on almost all cubic graphs. The new solution is conceptually simpler than previous algorithms, and applies to graphs of minimum possible degree. © 1994 John Wiley & Sons, Inc.     

Evaluation of nitrogen supplements for bioconversion of municipal solid waste to lactic acid

Applied Biochemistry and Biotechnology - Previous studies conducted by the authors showed that the ingredients of Elliker broth could be used as fermentation adjuncts for the bioconversion of...     

Completeness and robustness properties of min‐wise independent permutations * †

We provide several new results related to the concept of min‐wise independence. Our main result is that any randomized sampling scheme for the relative intersection of sets based on testing equality of samples yields an equivalent min‐wise independent family. Thus, in a certain sense, min‐wise independent families are “complete” for this type of estimation. We also discuss the notion of robustness, a concept extending min‐wise independence to allow more efficient use of it in practice. A surprising result arising from our consideration of robustness is that under a random permutation from a min‐wise independent family, any element of a fixed set has an equal chance to get any rank in the image of the set, not only the minimum as required by definition. © 2001 John Wiley & Sons, Inc. Random Struct. Alg., 18: 18–30, 2001     

A highly functionalized ferrocenyl­pyrazolo­[2,3‐a]­pyridine

The crystal structure of [2‐(4‐bromo­phenyl)‐4‐cyano‐5‐ferrocenyl­pyrazolo­[2,3‐a]­pyridin‐7‐yl]­aceto­nitrile, C₂₆H₁₇N₄FeBr or [Fe(C₅H₅)(C₂₁H₁₂BrN₄)], shows that the pyrazolo­pyridine ring system (PP), the bromo­phenyl ring (BP) and the cyclo­penta­diene ring (Cp) are nearly planar. The PP ring system is twisted out of the plane of the BP and Cp rings by about 20°.     

Cinnamoyl shikonin

The title compound, 1-(5,8-di­hydro-1,4-di­hydroxy-5,8-dioxo-2-naphthyl)-4-methyl­pent-3-en-1-yl cinnamate, C₂₅H₂₂O₆, crystallizes in space group P2₁. The phenyl ring of the cinnamate is anti to the carbonyl group of the same moiety [C—C—C—C = −175.6 (2)°] and is nearly parallel to the naphthyl ring system. Two six-membered rings formed by intramolecular hydrogen bonds, with O—H⃛O distances of 2.587 (2) and 2.589 (2) Å, occur on either side of the fused ring system, creating a tetracyclic pyrene-shaped system. The phenyl ring forms an intermolecular stack with the benzo­quinone ring, as a result of aromatic π–π interactions.     

Dilemmas in Crystallization. The ‘Unusual' Behavior of trans‐1,5‐Dichloro‐9,10‐diethynyl‐9,10‐dihydroanthracene‐9,10‐diol and the More ‘Normal' Behavior of Its Pseudopolymorphs with Dipolar Aprotic Solvents

The compound trans‐1,5‐dichloro‐9,10‐diethynyl‐9,10‐dihydroanthracene‐9,10‐diol (DDDA) has an inversion center as the only molecular symmetry element and yet does not occupy an inversion center in the centrosymmetric space group that it adopts in the crystal structure. The reason for this very unusual occurrence is the crowded environment of the H‐bond donors and acceptors that leads to less than optimal H‐bonding. A centrosymmetric supramolecular synthon constituted with four Cl‐atoms in a planar array occupies an i site in the crystal, and this appears to provide a satisfactory alternative packing. Based on the hypothesis that H‐bonding is less than optimal in the crystal structure of DDDA, pseudopolymorphs were prepared with strongly H‐bond‐accepting solvents. The crystal structures of five of these solvates are described, wherein the DDDA molecule is able to occupy an i site and form strong and linear O? H ???O H‐bonds with the solvent molecules. Competition experiments show that a smaller solvent molecule with a greater H‐bond‐accepting ability is included more readily and that the H‐bonds formed are correspondingly better.     

Orthogonal Polynomials and Second-order Differential Equations

We show that a sequence of polynomials can be eigenfunctionsof a second-order differential operator only under severe restrictions.     

The r-Stirling numbers

The r-Stirling numbers of the first and second kind count restricted permutations and respectively restricted partitions, the restriction being that the first r elements must be in distinct cycles and respectively distinct subsets. The combinatorial and algebraic properties of these numbers, which in most cases generalize similar properties of the regular Stirling numbers, are explored starting from the above definition.