Upper bounds on the general covering number Cλ(v,k, t,m)

A collection of k‐subsets (called blocks) of a v‐set X (v) = {1, 2,…, v} (with elements called points) is called a t‐(v, k, m, λ) covering if for every m‐subset M of X (v) there is a subcollection of with such that every block K ∈ has at least t points in common with M. It is required that v ≥ k ≥ t and v ≥ m ≥ t. The minimum number of blocks in a t‐(v, k, m, λ) covering is denoted by C_λ(v, k, t, m). We present some constructions producing the best known upper bounds on C_λ(v, k, t, m) for k = 6, a parameter of interest to lottery players. © 2004 Wiley Periodicals, Inc.     

Thermodynamic theory of incompressible hydrodynamics

The grand potential for open systems describes thermodynamics of fluid flows at low Mach numbers. A new system of reduced equations for the grand potential and the fluid momentum is derived from the compressible Navier-Stokes equations. The incompressible Navier-Stokes equations are the quasistationary solution to the new system. It is argued that the grand canonical ensemble is the unifying concept for the derivation of models and numerical methods for incompressible fluids, illustrated here with a simulation of a minimal Boltzmann model in a microflow setup.     

Balanced incomplete block designs with block size 8

The necessary conditions for the existence of a balanced incomplete block design on υ ≥ k points, with index λ and block size k, are that: For k = 8, these conditions are known to be sufficient when λ = 1, with 38 possible exceptions, the largest of which is υ = 3,753. For these 38 values of υ, we show (υ, 8, λ ) BIBDs exist whenever λ > 1 for all but five possible values of υ, the largest of which is υ = 1,177, and these five υ's are the only values for which more than one value of λ is open. For λ>1, we show the necessary conditions are sufficient with the definite exception of two further values of υ, and the possible exception of 7 further values of υ, the largest of which is υ=589. In particular, we show the necessary conditions are sufficient for all λ> 5 and for λ = 4 when υ ≠ 22. We also look at (8, λ) GDDs of type 7^m. Our grouplet divisible design construction is also refined, and we construct and exploit α ‐ frames in constructing several other BIBDs. In addition, we give a PBD basis result for {n: n ≡ 0, 1; mod 8, n ≥ 8}, and construct a few new TDs with index > 1. © 2001 John Wiley & Sons, Inc. J Combin Designs 9: 233–268, 2001     

The preparata codes and a class of 4-designs

An extension theorem for t-designs is proved. As an application, a class of 4-(4^m + 1,5,2) designs is constructed by extending designs related to the 3-designs formed by the minimum weight vectors in the Preparata code of length n = 4^m, m ≥ 2. © 1994 John Wiley & Sons, Inc.     

Synthesis and Characterization of Poly(Oxyethylene)S with (2-Benzoxazolon-3-yl)Acetyl End-Groups. Complex Formation with Polycarboxylic Acids

Abstract

Chemical modification of poly(oxyethylene)s (MW from 600 Da to 3000 Da), leading to the obtaining of poly(oxyethylene)s with (2-benzoxazolon-3-yl)acetyl endgroups, is described. The corresponding diesters were synthesized in high yields-70—84% and degree of esterification-82—92% under mild conditions in the presence of dicyclohexylcarbodiimide and 4-dimethylaminopyridine as catalyst. The diesters were characterized by ¹H-NMR, ¹³C-NMR and IR spectra, and SEC analysis. The relatively narrow molecular weight distribution was retained in the process of chemical modification. The complex formation between polyacrylic or polymeth-acrylic acids and (2-benzoxazolon-3-yl)acetyl derivatives of poly -(oxyethylene)s was studied in dilute aqueous solutions by potentio-metric titration and viscometry. The presence of (2-benzoxazolon-3-yl)acetyl end groups attached to the poly(oxyethylene)s chains results in a significant decrease of the critical poly(oxyethylene) chain length necessary for the formation of stable polymer-polymer complexes.     

Multifunctional Hybrid Materials From Poly(3‐Hydroxybutyrate), TiO2 Nanoparticles,and Chitosan Oligomers by Combining Electrospinning/Electrospraying and Impregnation

Several types of fibrous material containing poly(3‐hydroxybutyrate) (PHB), nanosized TiO₂‐anatase (nanoTiO₂), and chitosan oligomers are prepared by combining the electrospinning, electrospraying, and impregnation techniques. Simultaneous electrospinning/electrospraying provides uniform distribution of electrosprayed nanoTiO₂ along the PHB fibers and throughout the mat. Hybrid materials of different design manifest excellent photocatalytic activity, even after repeated use. They exhibit high bactericidal activity against Escherichia coli. In addition, the fibrous scaffolds are compatible with human mesenchymal stem cells and provide a favorable environment for their development.

     

Discotic liquid crystals of cucurbit[7]uril (CB[7])-functionalised Laponite clays

ABSTRACT

Here we demonstrate that in the presence of the small, barrel-shaped molecule cucurbit[7]uril (CB[7]) Laponite clay particles undergo a phase transition into a birefringent gel with lamellar structure and a fluid, Laponite-poor aqueous solution. We employ various characterisation methods and polarised light microscopy to study this transition and the system's structural and optical properties. Furthermore, when dried completely we obtain a layered, transparent composite film, still retaining some liquid-crystalline structure that can be rehydrated reversibly. We argue that this new system may be a model system for anisotropic patchy particles.     

Quadratic corrections to three‐point functions

Following the recent progress on the calculation of three‐point correlators with two “heavy” (with large quantum numbers) and one “light” states at strong coupling, we compute the logarithmic divergent terms of leading bosonic quantum corrections to correlation functions with “heavy” operators corresponding to simple string solutions in AdS₅ × S⁵. The “light” operator is chosen to be the dilaton. An important relation connecting the corrections to both the dimensions of “heavy” states, and the structure constants is recovered.