An Asymptotic Ratio in the Complete Binary Tree

Let T _n be the complete binary tree of height n considered as the Hasse-diagram of a poset with its root 1 _n as the maximum element. For a rooted tree T, define two functions counting the embeddings of T into T _n as follows A(n;T)=|{S T _n  : 1 _n ∈S, S≅T}|, and B(n;T)=|{S T _n :1 _n ∉S, S≅T}|. In this paper we investigate the asymptotic behavior of the ratio A(n;T)/B(n;T), and we show that lim  _n→∞[A(n;T)/B(n;T)]=2^ℓ;−1−1, for any tree T with ℓ leaves. This revised version was published online in June 2006 with corrections to the Cover Date.     

A numerical model of multimode fiber PON frequency response

Frequency response of passive optical network (PON) based on multimode fibers is investigated. The network comprises fibers, connectors and splitters/couplers. It is shown that due to mode filtering at splitters, the frequency response is different for different network nodes in otherwise symmetrical network.     

Normality of orbit closures for Dynkin quivers¶of type ?n

It is known that the orbit closures for the representations of the equioriented Dynkin quivers ? _n are normal and Cohen–Macaulay varieties with rational singularities. In the paper we prove the same for the Dynkin quivers ? _n with arbitrary orientation. Received: 25 October 2000 / Revised version: 28 February 2001     

Multivariate approximation in the worst case setting over reproducing kernel Hilbert spaces

We study the worst case setting for approximation of d variate functions from a general reproducing kernel Hilbert space with the error measured in the L_∞ norm. We mainly consider algorithms that use n arbitrary continuous linear functionals. We look for algorithms with the minimal worst case errors and for their rates of convergence as n goes to infinity. Algorithms using n function values will be analyzed in a forthcoming paper.We show that the L_∞ approximation problem in the worst case setting is related to the weighted L₂ approximation problem in the average case setting with respect to a zero-mean Gaussian stochastic process whose covariance function is the same as the reproducing kernel of the Hilbert space. This relation enables us to find optimal algorithms and their rates of convergence for the weighted Korobov space with an arbitrary smoothness parameter α>1, and for the weighted Sobolev space whose reproducing kernel corresponds to the Wiener sheet measure. The optimal convergence rates are n^-(α-1)/2 and n^-1/2, respectively.We also study tractability of L_∞ approximation for the absolute and normalized error criteria, i.e., how the minimal worst case errors depend on the number of variables, d, especially when d is arbitrarily large. We provide necessary and sufficient conditions on tractability of L_∞ approximation in terms of tractability conditions of the weighted L₂ approximation in the average case setting. In particular, tractability holds in weighted Korobov and Sobolev spaces only for weights tending sufficiently fast to zero and does not hold for the classical unweighted spaces.     

A 2-approximation algorithm for interval data minmax regret sequencing problems with the total flow time criterion

In this paper we discuss a minmax regret version of the single-machine scheduling problem with the total flow time criterion. Uncertain processing times are modeled by closed intervals. We show that if the deterministic problem is polynomially solvable, then its minmax regret version is approximable within 2.     

Solvent extraction of Cu,Mo, V,and U from leach solutions of copper ore and flotation tailings

A micro-cloud point extraction method was discussed for preconcentration and spectrophotometric quantification of U(VI). The method depends on complex formation between U(VI) and 2-(4-sulphophenyloazo)-1,8-dihydroxy-3,6-naphtalenedisulphonic acid (SPADNS) at pH 7.0 and subsequent extraction of the complex in a mixed surfactant medium (cethyltrimethyl ammonium bromide and Triton X-114). The separation was carried out in the presence of 1% Na₂SO₄ at room temperature. The calibration curve was linear up to 3000 µg L^?1. The enrichment factor, detection limit and precision were 16.0, 1.05 µg L^?1, and 2.3%, respectively. The method was employed for the determination of U(VI) in real samples with different matrices.     

Circular logic: nonribosomal peptide-like macrocyclization with a ribosomal peptide catalyst

A protease from ribosomal peptide biosynthesis macrocyclizes diverse substrates, including those resembling nonribosomal peptide and hybrid polyketide-peptide products. The proposed mechanism is analogous to thioesterase-catalyzed chemistry, but the substrates are amide bonds rather than thioesters.     

Mononuclear and dinuclear chiral vanadium(V) complexes with tridentate Schiff bases derived from R(−)-1,2-diaminopropane: Synthesis,structure, characterization and catalytic properties

A series of vanadium(V) complexes with unsymmetrical tridentate Schiff base ligands, obtained by the single condensation of R(−)-1,2-diaminopropane with salicylaldehyde and its derivatives, 2-hydroxy-1-naphthaldehyde, 2-hydroxyacetophenone, 1-hydroxy-2-acetonaphthone and 2-hydroxybenzophenone, were prepared. The complexes were characterized by elemental analysis and by their IR, CD, UV–Vis, 1D (¹H, ⁵¹V) and 2D (COSY, NOESY, gHSQC) NMR spectra. Crystal structures of the mononuclear complex {R(−)-2-amino-1-N-[(2′-oxido-κO-4′,6′-dimethoxyphenyl)methylene]aminopropane-κ²N}dioxidovanadium(V), VO₂(C₁₂H₁₇N₂O₃), 4, and of the dinuclear complex, di-μ-oxido-bis({R(−)-2-[1-(2-aminopropylimino)ethyl]-4-methylphenolato-κ³N,N′,O}oxidovanadium(V)), V₂O₄(C₁₁H₁₅N₂O)₂, 5, have been obtained by X-ray diffraction studies. The structure of 4 was revealed to be a distorted trigonal–bipyramidal coordination geometry, rarely encountered in VO₂(tridentate Schiff base) complexes. Complexes 2 and 3 have the ability to catalyze the oxidation of prochiral sulfide substrates PhSR (R = Me, Bz) utilizing hydrogen peroxide or cumene hydroperoxide (CHPO) as the oxidant.