Upper Bounds for Erdös–Hajnal Coefficients of Tournaments

A celebrated unresolved conjecture of Erdös and Hajnal (see Discrete Appl Math 25 (1989), 37–52) states that for every undirected graph H, there exists , such that every graph on n vertices which does not contain H as an induced subgraph contains either a clique or an independent set of size at least . In (Combinatorica (2001), 155–170), Alon et al. proved that this conjecture was equivalent to a similar conjecture about tournaments. In the directed version of the conjecture cliques and stable sets are replaced by transitive subtournaments. For a fixed undirected graph H, define to be the supremum of all ε for which the following holds: for some n₀ and every every undirected graph with vertices not containing H as an induced subgraph has a clique or independent set of size at least . The analogous definition holds if H is a tournament. We call the Erdös–Hajnal coefficient of H. The Erdös–Hajnal conjecture is true if and only if for every H. We prove in this article that:

• the Erdös–Hajnal coefficient of every graph H is at most ,
• there exists such that the Erdös–Hajnal coefficient of almost every tournament T on k vertices is at most , i.e. the proportion of tournaments on k vertices with the coefficient exceeding goes to 0 as k goes to infinity.

     

On quadruples of Griffiths points

Tabov (Math Mag 68:61–64, 1995) has proved the following theorem: if points A ₁, A ₂, A ₃, A ₄ are on a circle and a line l passes through the centre of the circle, then four Griffiths points G ₁, G ₂, G ₃, G ₄ corresponding to pairs (Δ _i ,l) are on a line (Δ _i denotes the triangle A _j A _k A _l , j,k,l ≠ i). In this paper we present a strong generalisation of the result of Tabov. An analogous property for four arbitrary points A ₁, A ₂, A ₃, A ₄, is proved, with the help of the computer program “Mathematica”.     

Chain Geometry Determined by the Affine Group

Chain geometry associated with an affine group and with a linear group is studied. In particular, closely related to the respective chain geometries affine partial linear spaces and generalizations of sliced spaces are defined. The automorphisms of thus obtained structures are determined.     

On ω-categorical,generically stable groups and rings

We prove that every ω-categorical, generically stable group is nilpotent-by-finite and that every ω-categorical, generically stable ring is nilpotent-by-finite.     

An alternative for polarized antiproton beams

Presently the most popular way to prepare high quality polarized antiproton beams is the so called spin filter method. The feasibility of the method has been proven for a proton beam and measurements of the spin dependent interaction of antiprotons have been proposed by the PAX collaboration. Another well known source for polarized antiprotons is the $\bar{\Lambda}$ decay which was used at FERMILAB in the only experiment performed so far with polarized antiprotons. An alternative approach for polarized antiproton beams may be the production process itself. If the produced antiprotons show polarization it would be rather simple to handle a polarized antiproton beam in the existing antiproton collector and cooler at CERN just like in the unpolarized case.     

The use of the Hurst exponent to investigate the global maximum of the Warsaw Stock Exchange WIG20 index

The WIG20 index–the index of the 20 biggest companies traded on the Warsaw Stock Exchange–reached the global maximum on 29th October 2007. I have used the local DFA (Detrended Functional Analysis) to obtain the Hurst exponent (diffusion exponent) and investigate the signature of anti-correlation of share price evolution around the maximum. The analysis was applied to the share price evolution for variable DFA parameters. For many values of parameters, the evidence of anti-correlation near the WIG20 maximum was pointed out.     

Chiral J-aggregates of atropo-enantiomeric perylene bisimides and their self-sorting behavior

Herein we report on structural, morphological, and optical properties of homochiral and heterochiral J-aggregates that were created by nucleation-elongation assembly of atropo-enantiomerically pure and racemic perylene bisimides (PBIs), respectively. Our detailed studies with conformationally stable biphenoxy-bridged chiral PBIs by UV/Vis absorption, circular dichroism (CD) spectroscopy, and atomic force microscopy (AFM) revealed structurally as well as spectroscopically quite different kinds of J-aggregates for enantiomerically pure and racemic PBIs. AFM investigations showed that enantiopure PBIs form helical nanowires of unique diameter and large length-to-width ratio by self-recognition, while racemic PBIs provide irregular-sized particles by self-discrimination of the enantiomers at the stage of nucleation. Steady-state fluorescence spectroscopy studies revealed that the photoluminescence efficiency of homochiral J-aggregated nanowires (47±3%) is significantly higher than that of heterochiral J-aggregated particle-like aggregates (12±3%), which is explained in terms of highly ordered molecular stacking in one-dimensional nanowires of homochiral J-aggregates. Our present results demonstrate the high impact of homochirality on the construction of well-defined nanostructures with unique optical properties.     

Uptake of Well‐Defined,Highly Glycosylated,Pentafluorostyrene‐Based Polymers and Nanoparticles by Human Hepatocellular Carcinoma Cells

Chain length, size, composition, surface charge, and other properties of polymeric materials affect their recognition and uptake by cells and must be optimized to deliver polymers selectively to their target. However, it is often not possible to precisely modify selected properties without changing other parameters. To overcome these difficulties, well‐defined poly(pentafluorostyrene)‐based polymers are prepared that can be grafted via thiol/para‐fluorine “click” reaction with 1‐thio‐β‐D ‐glucose and 1‐thio‐β‐D ‐galactose. Fluorescence microscopy and flow cytometry show that nanoparticles are taken up by HepG2 cells to a higher degree than the respective water‐soluble polymers, and that internalization of both galactosylated homo‐ and nanoprecipitated block copolymers is enhanced.

     

High-pressure accelerated asymmetric organocatalytic Friedel-Crafts alkylation of indoles with enones: application to quaternary stereogenic centers construction

An organocatalytic Friedel-Crafts alkylation of indoles with α,β-unsaturated ketones was found to be efficiently accelerated under high-pressure conditions with a low loading of chiral primary amine salts with good yield and enantioselectivity up to 90%. This approach also allows, for the first time, selected indole derivatives containing quaternary stereogenic centers to be obtained from prochiral β,β-disubstituted enones with an enantioselectivity up to 80%.     

A volume microstrip RF coil for MRI microscopy

Quantitative magnetic resonance imaging (MRI) studies of small samples such as a single cell or cell clusters require application of radiofrequency (RF) coils that provide homogenous B₁ field distribution and high signal-to-noise ratio (SNR).We present a novel design of an MRI RF volume microcoil based on a microstrip structure. The coil consists of two parallel microstrip elements conducting RF currents in the opposite directions, thus creating homogenous RF field within the space between the microstrips. The construction of the microcoil is simple, efficient and cost-effective.Theoretical calculations and finite element method simulations were used to optimize the coil geometry to achieve optimal B₁ and SNR distributions within the sample and predict parameters of the coil. The theoretical calculations were confirmed with MR images of a 1-mm-diameter capillary and a plant obtained with the double microstrip RF microcoil at 11.7 T. The in-plane resolution of MR images was 24 μm×24 μm.