One-step purification of epigallocatechin gallate from crude green tea extracts by mixed-mode adsorption chromatography on highly cross-linked agarose media

(-)-Epigallocatechin gallate (EGCG) was purified in one step from a green tea polyphenol (GTP) crude extract by adsorption chromatography on a Superose 12 HR 10/30 column. The mobile phase used was a mixture of acetonitrile and water with an optimum mobile phase compositions regarding purity, recovery and separation time of 78/22 (v/v). Maximum practical sample loading was 100 mg GTP per run (corresponding to 4.2 mg/ml Superose). An EGCG purity of 99% with recoveries in the range 60-65% was achieved in one step directly from the crude GTP extract. Full column regeneration was obtained using solvents in the following order: 0.5 M NaOH, distilled water and 30% acetic acid.     

Precise Logarithmic Asymptotics for the Right Tails of Some Limit Random Variables for Random Trees

For certain random variables that arise as limits of functionals of random finite trees, we obtain precise asymptotics for the logarithm of the right-hand tail. Our results are based on the facts (i) that the random variables we study can be represented as functionals of a Brownian excursion and (ii) that a large deviation principle with good rate function is known explicitly for Brownian excursion. Examples include limit distributions of the total path length and of the Wiener index in conditioned Galton-Watson trees (also known as simply generated trees). In the case of Wiener index (where we recover results proved by Svante Janson and Philippe Chassaing by a different method) and for some other examples, a key constant is expressed as the solution to a certain optimization problem, but the constant’s precise value remains unknown. Research supported by NSF grants DMS-0104167 and DMS-0406104 and by The Johns Hopkins University’s Acheson J. Duncan Fund for the Advancement of Research in Statistics.     

Sorting Using Complete Subintervals and the Maximum Number of Runs in a Randomly Evolving Sequence

We study the space requirements of a sorting algorithm where only items that at the end will be adjacent are kept together. This is equivalent to the following combinatorial problem: Consider a string of fixed length n that starts as a string of 0’s, and then evolves by changing each 0 to 1, with the n changes done in random order. What is the maximal number of runs of 1’s? We give asymptotic results for the distribution and mean. It turns out that, as in many problems involving a maximum, the maximum is asymptotically normal, with fluctuations of order n ^1/2, and to the first order well approximated by the number of runs at the instance when the expectation is maximized, in this case when half the elements have changed to 1; there is also a second order term of order n ^1/3. We also treat some variations, including priority queues and sock-sorting. The proofs use methods originally developed for random graphs.     

A new approach to the giant component problem

We study the largest component of a random (multi)graph on n vertices with a given degree sequence. We let n → ∞. Then, under some regularity conditions on the degree sequences, we give conditions on the asymptotic shape of the degree sequence that imply that with high probability all the components are small, and other conditions that imply that with high probability there is a giant component and the sizes of its vertex and edge sets satisfy a law of large numbers; under suitable assumptions these are the only two possibilities. In particular, we recover the results by Molloy and Reed on the size of the largest component in a random graph with a given degree sequence. We further obtain a new sharp result for the giant component just above the threshold, generalizing the case of G(n,p) with np = 1 + ω(n)n^?1/3, where ω(n) → ∞ arbitrarily slowly. Our method is based on the properties of empirical distributions of independent random variables, and leads to simple proofs. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2009     

Electrophilic trifluoromethylation by copper-catalyzed addition of CF3-transfer reagents to alkenes and alkynes

Regio- and stereoselective Cu-catalyzed addition of the above hypervalent iodine reagent to alkynes and alkenes was achieved. In the presence of CuI, the reaction is suitable to perform trifluoromethyl-benzoyloxylation and trifluoromethyl-halogenation of alkenes and alkynes. Electron-donating substituents accelerate the process, and alkenes react faster than alkynes emphasizing the electrophilic character of the addition reaction.     

Liquid-liquid/solid three-phase high-speed counter-current chromatography, a new technique for separation of polyphenols from Geranium wilfordii Maxim

High-speed counter-current chromatography using a new liquid-liquid/solid three-phase system was used for the separation of the polyphenols corilagin and geraniin from a crude extract of Geranium wilfordii Maxim in one step. The optimized three-phase system was composed of n-hexane/ethyl acetate/methanol/acetic acid/water and to which was added 10-μm average diameter microspheres of cross-linked 12% agarose at the ratio of 0.2:10:2:1:5 and 0.1 g/mL, respectively. The purities of geraniin and corilagin were 82 and 90%, which were determined by HPLC at 280 nm. A 14.5 and 7 mg of geraniin and corilagin were purified from 160 mg crude extract with the yields of 70 and 78%, respectively.     

Big Hankel operators of higher weight

Known results concerning the smoothness and boundedness of «big» Hankel operators (Hankel operators in the sense of Axler) are generalized to the case of higher weight (in the sense of representation theory). The key result is a certain estimate for thes-numbers of a particular such operator, involving a combinatorial sum.     

Preservation of convexity of solutions to parabolic equations

In the present paper we find necessary and sufficient conditions on the coefficients of a parabolic equation for convexity to be preserved. A parabolic equation is said to preserve convexity if given a convex initial condition, any solution of moderate growth remains a convex function of the spatial variables for each fixed time.     

Rademacher chaos: tail estimates versus limit theorems

We study Rademacher chaos indexed by a sparse set which has a fractional combinatorial dimension. We obtain tail estimates for finite sums and a normal limit theorem as the size tends to infinity. The tails for finite sums may be much larger than the tails of the limit.