Palladium-catalyzed selective acyloxylation using sodium perborate as oxidant

Sodium perborate (SPB), a principal component of washing powders, was employed as an inexpensive and eco-friendly oxidant in the palladium-catalyzed C-H acyloxylation of alkenes in excellent regio- and stereochemistry. The reactions used anhydrides as acyloxy sources. The method applies to both terminal and internal alkenes, and even benzylic C-H oxidation.     

Upper tails for counting objects in randomly induced subhypergraphs and rooted random graphs

General upper tail estimates are given for counting edges in a random induced subhypergraph of a fixed hypergraph ℋ, with an easy proof by estimating the moments. As an application we consider the numbers of arithmetic progressions and Schur triples in random subsets of integers. In the second part of the paper we return to the subgraph counts in random graphs and provide upper tail estimates in the rooted case.     

Some Remarks on the Combinatorics of ISn

\noindent We describe the asymptotic behavior of the cardinalities of the finite symmetric inverse semigroup IS_n and its endomorphism semigroup. This is applied to show that the ratio |IS_n|/|End(IS_n)| is asymptotically 0, answering a question of Schein and Teclezghi. We also apply our results to compute the distributions of elements from IS_n with respect to certain combinatorial properties, and to compute the generating functions for |IS_n| and for the number of nilpotent elements in IS_n.     

Perfect matchings in random s-uniform hypergraphs

Let E = {X₁, X₂…, X_m} where the X_i ? V for 1 ≤ i ≤ m are distinct. The hypergraph G = (V, E) is said to be s-uniform if |X₁| = s for 1 ≤ i ≤ m. A set of edges M = {X_i : i ? I } is a perfect matching if (i) i ≠ j ? I implies X_i ∩ X_i = 0, and (ii) ∪_i?I X_i = V. In this article we consider the question of whether a random s-uniform hypergraph contains a perfect matching. Let s ≥ 3 be fixed and m/n^4/3 → ∞. We show that an s-uniform hypergraph with m edges chosen uniformly from [74] contains a perfect matching with high probability. This improves an earlier result of Schmidt and Shamir who showed that m/n^3/2 → ∞ suffices. © 1995 John Wiley & Sons, Inc.     

The asymptotic distributions of incomplete U-statistics

Summary An incomplete U-statistic is obtained by sampling the terms of an U-statistic. This paper derives the asymptotic distribution (if the variance is finite). Depending on the number of sampled terms, the resulting distribution is either the same as for the U-statistic, a normal distribution, or something intermediate. Also the case of a non-random sampling of the terms is treated. As an example, a non-parametric test of the independence of two circular random variables is studied. The results are generalized to generalized U-statistics.     

The Cut Metric,Random Graphs,and Branching Processes

In this paper we study the component structure of random graphs with independence between the edges. Under mild assumptions, we determine whether there is a giant component, and find its asymptotic size when it exists. We assume that the sequence of matrices of edge probabilities converges to an appropriate limit object (a kernel), but only in a very weak sense, namely in the cut metric. Our results thus generalize previous results on the phase transition in the already very general inhomogeneous random graph model introduced by the present authors in Random Struct. Algorithms 31:3–122 (2007), as well as related results of Bollobás, Borgs, Chayes and Riordan (Ann. Probab. 38:150–183, 2010), all of which involve considerably stronger assumptions. We also prove corresponding results for random hypergraphs; these generalize our results on the phase transition in inhomogeneous random graphs with clustering (Random Struct. Algorithms, 2010, to appear).     

Sesqui-type branching processes

We consider branching processes consisting of particles (individuals) of two types (type $L$ and type $S$) in which only particles of type $L$ have offspring, proving estimates for the survival probability and the (tail of) the distribution of the total number of particles. Such processes are in some sense closer to single- than to multi-type branching processes. Nonetheless, the second, barren, type complicates the analysis significantly. The results proved here (about point and survival probabilities) are a key ingredient in the analysis of bounded-size Achlioptas processes in a recent paper by the last two authors.     

One-step purification of 3,4-dihydroxyphenyllactic acid, salvianolic acid B, and protocatechualdehyde from Salvia miltiorrhiza Bunge by isocratic stepwise hydrogen bond adsorption chromatography on cross-linked 12% agarose

Three major active components of the traditional Chinese medicinal herb Salvia miltiorrhiza Bunge, 3,4-dihydroxyphenyllactic acid, salvianolic acid B, and protocatechualdehyde, are separated and purified from a crude water extract in one step by isocratic hydrogen bond adsorption chromatography on cross-linked 12% agarose (Superose 12 HR 10/30). Separation is achieved by stepwise elution with mobile phases composed of mixtures of ethanol and acetic acid: 0-50 mL, 5% ethanol, 5% acetic acid; 50-100 mL, 20% ethanol, 20% acetic acid; and 100-200 mL, 30% ethanol, 30% acetic acid. The 3,4-dihydroxyphenyllactic acid is obtained with a purity of 97.3% and with a recovery of 88.1%. The corresponding figures for protocatechualdehyde are a purity of 99.4% with a recovery of 90.7%, and for salvianolic acid B a purity of 90.4% with a recovery of 50.3%, respectively. At a sample load of 40 mg crude extract dissolved in 0.5 mL mobile phase (corresponding to a load of 1.6 mg/mL gel), a 3,4-dihydroxyphenyllactic acid purity of approximately 94% with a recovery of 80.2% is obtained.