The number of K m,m -free graphs

A graph is called H-free if it contains no copy of H. Denote by f _n (H) the number of (labeled) H-free graphs on n vertices. Erdős conjectured that f _n (H) ≤ 2^{(1+o(1))ex(n,H)}. This was first shown to be true for cliques; then, Erdős, Frankl, and R?dl proved it for all graphs H with χ(H)≥3. For most bipartite H, the question is still wide open, and even the correct order of magnitude of log₂ f _n (H) is not known. We prove that f _n (K _m,m ) ≤ 2 ^O (n ^2−1/m ) for every m, extending the result of Kleitman and Winston and answering a question of Erdős. This bound is asymptotically sharp for m∈{2,3}, and possibly for all other values of m, for which the order of ex(n,K _m,m ) is conjectured to be Θ(n ^2−1/m ). Our method also yields a bound on the number of K _m,m -free graphs with fixed order and size, extending the result of Füredi. Using this bound, we prove a relaxed version of a conjecture due to Haxell, Kohayakawa, and Łuczak and show that almost all K _3,3-free graphs of order n have more than 1/20·ex(n,K _3,3) edges.     

Automated on-line dispersive liquid–liquid microextraction based on a sequential injection system

A novel approach for sequential injection-dispersive liquid–liquid microextraction (SI-DLLME) has been suggested. The method is based on the aspiration and mixing of a sample and all required aqueous reagents in the holding coil of an SIA system, delivering it into a conical tube and adding in a mixture of extraction solvent, auxiliary solvent and disperser solvent at high flow rate, resulting in the formation of a cloudy state and the extraction of an analyte. The mixture of extraction and auxiliary solvent is immiscible with water and has a density significantly higher than that of water; consequently, the resulting fine droplets in the mixture, which contain the extracted analyte, are self-sedimented in a short time at the bottom of conical tube. Thus, no centrifugation and no use of a microcolumn are required for separation of the extraction phase. Afterwards, the extracted analyte is aspirated and transferred to a micro-volume Z-flow cell, and the absorbance is measured.The performance of the suggested approach is demonstrated by the SI-DLLME of thiocyanate ions in the form of ion associate with dimethylindocarbocyanine reagent, followed by spectrophotometric detection. A mixture of amyl acetate (as extraction solvent), tetrachloromethane (as auxiliary solvent) and acetonitrile (as disperser solvent) was selected for the DLLME procedure. The appropriate experimental conditions for conventional DLLME and automated SI-DLLME were investigated. The analytical performance of both these procedures was compared. The absorbance of the colored extracts at wavelength 555 nm obeys Beer's law in the range of 3.13–28.2 for conventional DLLME and 0.29–5.81 mg L^? 1 of SCN for SI-DLLME, and the limit of detection, calculated from a blank test based on 3 s, is 0.110 for conventional DLLME and 0.017 mg L^? 1 for SI-DLLME.     

Lie derived length and involutions in group algebras

Let $G$ be a group such that the set of $p$-elements of $G$ forms a finite nonabelian subgroup, where $p$ is an odd prime, and let $F$ be a field of characteristic $p$. In this paper we prove that the lower bound of the Lie derived length of the group algebra $FG$ given by Shalev in [11] is also a lower bound for the Lie derived length of the set of symmetric elements of $FG$ for every involution which is linear extension of an involutive anti-automorphism of $G$. Furthermore, we provide counterexamples to the interesting cases which are not covered by the main theorem.     

Anisotropic turbulent spectra in the terrestrial magnetosheath as seen by the cluster spacecraft

Here we report the first three-dimensional spatial spectrum of the low frequency magnetic turbulence obtained from the four Cluster spacecraft in the terrestrial magnetosheath close to the magnetopause. We show that the turbulence is compressible and dominated by mirror structures, its energy is injected at a large scale kp approximately 0.3 (l approximately 2000 km) via a mirror instability well predicted by linear theory, and cascades nonlinearly and unexpectedly up to kp approximately 3.5 (l approximately 150 km), revealing a new power law in the inertial range not predicted by any turbulence theory, and its strong anisotropy is controlled by the static magnetic field and the magnetopause normal.