Factorizing the complete graph into factors with large star number

The graph G has star number n if any n vertices of G belong to a subgraph which is a star. Let f(n, k) be the smallest number m such that the complete graph on m vertices can be factorized into k factors with star number n. In the present paper we prove that $\text{c}{}_1{}^{\mathrm{n}}\text{k ≤ f(n, k) < c}{}_2{}^{\mathrm{n}}\text{k}$.     

The Birth of the Infinite Cluster:¶Finite-Size Scaling in Percolation

We address the question of finite-size scaling in percolation by studying bond percolation in a finite box of side length n, both in two and in higher dimensions. In dimension d= 2, we obtain a complete characterization of finite-size scaling. In dimensions d>2, we establish the same results under a set of hypotheses related to so-called scaling and hyperscaling postulates which are widely believed to hold up to d= 6. As a function of the size of the box, we determine the scaling window in which the system behaves critically. We characterize criticality in terms of the scaling of the sizes of the largest clusters in the box: incipient infinite clusters which give rise to the infinite cluster. Within the scaling window, we show that the size of the largest cluster behaves like n ^d π _n , where π _n is the probability at criticality that the origin is connected to the boundary of a box of radius n. We also show that, inside the window, there are typically many clusters of scale n ^d π _n , and hence that “the” incipient infinite cluster is not unique. Below the window, we show that the size of the largest cluster scales like ξ ^d π_ξ log(n/ξ), where ξ is the correlation length, and again, there are many clusters of this scale. Above the window, we show that the size of the largest cluster scales like n ^d P _∞, where P _∞ is the infinite cluster density, and that there is only one cluster of this scale. Our results are finite-dimensional analogues of results on the dominant component of the Erdős–Rényi mean-field random graph model. Received: 6 December 2000 / Accepted: 25 May 2001     

Fast,Facile and Solvent-Free Dry-Melt Synthesis of Oxovanadium(IV) Complexes: Simple Design with High Potency towards Cancerous Cells

A range of oxobis(phenyl-1,3-butanedione) vanadium(IV) complexes have been successfully synthesized from cheap starting materials and a simple and solvent-free one-pot dry-melt reaction. This direct, straightforward, fast and alternative approach to inorganic synthesis has the potential for a wide range of applications. Analytical studies confirm their successful synthesis, purity and solid-state coordination, and we report the use of such complexes as potential drug candidates for the treatment of cancer. After a 24 hour incubation of A549 lung carcinoma cells with the compounds, they reveal cytotoxicity values elevenfold greater than cisplatin and remain non-toxic towards normal cell types. Additionally, the complexes are stable over a range of physiological pH values and show the potential for interactions with bovine serum albumin.     

Synthesis of Polymers Containing Potassium Acyltrifluoroborates (KATs) and Post-polymerization Ligation and Conjugation

We report the synthesis of monomers for atom-transfer radical polymerization (ATRP) and a reversible addition-fragmentation chain transfer (RAFT) agent bearing trifluoroborate iminiums (TIMs), which are quantitatively converted into potassium acyltrifluoroborates (KATs) after polymerization. The resulting KAT-containing polymers are suitable for rapid amide-forming ligations for both post-polymerization modification and polymer conjugation. The polymer conjugation occurs rapidly, even under dilute (micromolar) aqueous conditions at ambient temperatures, thereby enabling the synthesis of a variety of linear and star-shaped block copolymers. In addition, we applied post-polymerization modification to the covalent linking of a photocaged cyclic antibiotic (gramicidin S) to the side chains of the KAT-containing copolymer. Cellular assays revealed that the polymer–antibiotic conjugate is biocompatible and provides efficient light-controlled release of the antibiotic on demand.     

Efficient algorithms for finding minimum spanning trees in undirected and directed graphs

Recently, Fredman and Tarjan invented a new, especially efficient form of heap (priority queue). Their data structure, theFibonacci heap (or F-heap) supports arbitrary deletion inO(logn) amortized time and other heap operations inO(1) amortized time. In this paper we use F-heaps to obtain fast algorithms for finding minimum spanning trees in undirected and directed graphs. For an undirected graph containingn vertices andm edges, our minimum spanning tree algorithm runs inO(m logβ (m, n)) time, improved fromO(mβ(m, n)) time, whereβ(m, n)=min {i|log⁽ⁱ⁾ n ≦m/n}. Our minimum spanning tree algorithm for directed graphs runs inO(n logn + m) time, improved fromO(n log n +m log log log_(m/n+2) n). Both algorithms can be extended to allow a degree constraint at one vertex. Research supported in part by National Science Foundation Grant MCS-8302648. Research supported in part by National Science Foundation Grant MCS-8303139. Research supported in part by National Science Foundation Grant MCS-8300984 and a United States Army Research Office Program Fellowship, DAAG29-83-GO020.     

Synthesis and reactivity of bis(imido) uranium(VI) cyclopentadienyl complexes

The bis(imido) uranium(VI)-C(5)H(5) and -C(5)Me(5) complexes (C(5)H(5))(2)U(N(t)Bu)(2), (C(5)Me(5))(2)U(N(t)Bu)(2), (C(5)H(5))U(N(t)Bu)(2)(I)(dmpe), and (C(5)H(5))(2)U(N(t)Bu)(2)(dmpe) can be synthesized from reactions between U(N(t)Bu)(2)(I)(2)(L)(x) (L=THF, x=2; L=dmpe, x=1) and Na(C(5)R(5)) (R=H, Me); these complexes represent the first structurally characterized C(5)H(5)-compounds of uranium(VI) and they further highlight the differences between UO(2)(2+) and the bis(imido) fragment.     

The neomycin biosynthetic gene cluster of Streptomyces fradiae NCIMB 8233: genetic and biochemical evidence for the roles of two glycosyltransferases and a deacetylase

An efficient protocol has been developed for the genetic manipulation of Streptomyces fradiae NCIMB 8233, which produces the 2-deoxystreptamine (2-DOS)-containing aminoglycoside antibiotic neomycin. This has allowed the in vivo analysis of the respective roles of the glycosyltransferases Neo8 and Neo15, and of the deacetylase Neo16 in neomycin biosynthesis. Specific deletion of each of the neo8, neo15 and neo16 genes confirmed that they are all essential for neomycin biosynthesis. The pattern of metabolites produced by feeding putative pathway intermediates to these mutants provided unambiguous support for a scheme in which Neo8 and Neo15, whose three-dimensional structures are predicted to be highly similar, have distinct roles: Neo8 catalyses transfer of N-acetylglucosamine to 2-DOS early in the pathway, while Neo15 catalyses transfer of the same aminosugar to ribostamycin later in the pathway. The in vitro substrate specificity of Neo15, purified from recombinant Escherichia coli, was fully consistent with these findings. The in vitro activity of Neo16, the only deacetylase so far recognised in the neo gene cluster, showed that it is capable of acting in tandem with both Neo8 and Neo15 as previously proposed. However, the deacetylation of N-acetylglucosaminylribostamycin was still observed in a strain deleted of the neo16 gene and fed with suitable pathway precursors, providing evidence for the existence of a second enzyme in S. fradiae with this activity.     

Formation of the serine octamer: Ion evaporation or charge residue

The mechanism of formation for clusters of serine generated by electrospray ionization is hypothesized to play a critical role in determining their ultimate properties. Under carefully manipulated electrospray source conditions, two distinct and well-separated distributions of clusters can be observed. The characteristics of the two cluster populations are consistent with different formation mechanisms, namely ion evaporation and charge residue. Upon further inspection, it is proposed that the magic number intensity, homochiral selectivity, and unique formation of the serine octamer are best explained within the context of the ion evaporation mechanism. As a consequence, solution phase properties of the octamer become important, particularly in relation to interface effects present on the surface of the charged droplet. In contrast, other clusters of serine, including the B form of the octamer, are probably generated by the charge residue mechanism and may have no connection to condensed phase phenomena.     

Cholesterol surrogates incorporating a benzophenone as part of the sterol tetracycle

Photoactivatable analogues 4-6 of cholesterol (1), having their cross-linking site in the ring D sterol region, have been synthesized starting from bromotetralone 14 via enantioselective Robinson annulation to enone 13 and Suzuki carbonylative coupling to the appropriate phenylboronic acid. Each of 4-6 was shown to substitute successfully for 1 in an assay of apo A-I-induced cellular cholesterol efflux, indicating that these analogues equilibrated with 1 in all major cellular pools.     

Thermophysical Properties of High-Temperature Reacting Mixtures of Carbon and Water in the Range 400–30,000 K and 0.1–10 atm. Part 1: Equilibrium Composition and Thermodynamic Properties

This paper is devoted to the calculation of the chemical equilibrium composition and thermodynamic properties of reacting mixtures of carbon and water at high temperature. Equilibrium particle concentrations and thermodynamic properties including mass density, molar weight, entropy, enthalpy and specific heat at constant pressure, sonic velocity, and heat capacity ratio are determined by the method of Gibbs free energy minimization, using species data from standard thermodynamic tables. The calculations, which assume local thermodynamic equilibrium, are performed in the temperature range from 400 to 30,000 K for pressures of 0.10, 1.0, 3.0, 5.0 and 10.0 atm. The properties of the reacting mixture are affected by the possible occurrence of solid carbon formation at low temperature, and therefore attention is paid to the influence of the carbon phase transition by comparing the results obtained with and without considering solid carbon formation. The results presented here clarify some basic chemical process and are reliable reference data for use in the simulation of plasmas in reacting carbon and water mixtures together with the need of transport coefficients computation.