Effects of substituents on activation parameters in SN2 reactions at aliphatic carbon in solution

Changes of the activation parameters in aliphatic S_N2 reactions with anionic and neutral nucleophiles in various solvents, ΔH^≠ and ΔS^≠, were correlated with σ constants of the substituents. The resultant δΔH^≠ and δΔS^≠ reaction constants are linearly related for variations of substituents at the substrate, leaving group and nucleophile. Correlations of δΔH^≠ versus δΔS^≠ allow the estimation of the contribution of changes of the internal enthalpy, δΔH, to the enthalpy reaction constant, δΔH^≠, which gives a single linear dependence on the Hammett ρ reaction constants. The deviations from the dependence of δΔH versus ρ can be interpreted in terms of changes in the transition state structure in S_N2 reactions. The results obtained show that the substituent effects on the charge development in the transition state are governed by the magnitude of δΔH. Copyright © 2009 John Wiley & Sons, Ltd.     

Explicit solutions of generalized Cauchy-Riemann systems using the transplant operator

In Kravchenko (2008) [8] it was shown that the tool introduced there and called the transplant operator transforms solutions of one Vekua equation into solutions of another Vekua equation, related to the first via a Schrödinger equation. In this paper we prove a fundamental property of this operator: it preserves the order of zeros and poles of generalized analytic functions and transforms formal powers of the first Vekua equation into formal powers of the same order for the second Vekua equation. This property allows us to obtain positive formal powers and a generating sequence of a “complicated” Vekua equation from positive formal powers and a generating sequence of a “simpler” Vekua equation. Similar results are obtained regarding the construction of Cauchy kernels. Elliptic and hyperbolic pseudoanalytic function theories are considered and examples are given to illustrate the procedure.     

Low-energy electron beam source

Experimental results of research into a ferroelectric-plasma-source-assisted hollow anode (HA) discharge as a source of low-energy electron beams are presented. To generate electron beams, the HA auto-bias negative voltage was achieved by the discharge current flowing through the resistor connecting the HA and the grounded electrode. It is shown that this method allows reliable and reproducible generation of low-energy electron beams with electron energy of several hundreds of eV, electron current density up to several A/cm² and pulse duration of several tens of microseconds.     

Recoil leader formation and development

The existing interpretation in the lightning literature, based on field measurements, defines recoil leaders as negative leaders. However recoil leaders are floating conductors, and, based on this physical assumption, they should be defined as bipolar and bidirectional leaders. This physics-based assumption has never previously been verified experimentally. Such verification, reported in this paper, has been obtained from observations of branched upward positive leaders from a tall tower using a high-speed video system synchronized with electric and magnetic field change and luminosity measurements on the ground. The analysis of these observations clearly reveals the nature of recoil and dart leaders as bidirectional and bipolar electrodeless discharges that develop from a small region along a path of the decayed channels of a previous positive leader, or a positively charged return stroke of negative CG flashes.     

The chromatic number of random Borsuk graphs

We study a model of random graph where vertices are n i.i.d. uniform random points on the unit sphere S^d in , and a pair of vertices is connected if the Euclidean distance between them is at least 2??. We are interested in the chromatic number of this graph as n tends to infinity. It is not too hard to see that if ?>0 is small and fixed, then the chromatic number is d+2 with high probability. We show that this holds even if ?→0 slowly enough. We quantify the rate at which ? can tend to zero and still have the same chromatic number. The proof depends on combining topological methods (namely the Lyusternik–Schnirelman–Borsuk theorem) with geometric probability arguments. The rate we obtain is best possible, up to a constant factor—if ?→0 faster than this, we show that the graph is (d+1)‐colorable with high probability.25     

The Bergman kernel for the Vekua equation

The paper extends some well-known results for analytic functions onto solutions of the Vekua equation $\partial \overline{{}_z}W=aW+b\overline{W}$ regarding the existence and construction of the Bergman kernel and of the corresponding Bergman projection operator.