Physico-chemical factors governing partition behaviour of solutes and particles in aqueous polymeric biphasic systems. III. Features of solutes and biological particles detected by the partition technique

The partition behaviour of human serum albumin and oxyhaemoglobin and several amino acids and small peptides was studied in the aqueous Ficoll-dextran biphasic system as a function of the ionic composition and pH. The partition coefficients of the solutes were expressed in terms of the equivalent number of CH2 groups, nCH2, and the equivalent number of carboxyl groups, m. The physical meaning of these two parameters and of the relationships found between them and pH for the proteins examined are discussed. A correlation was established between the difference in the relative hydrophobicities of the individual phases of various water-organic solvent systems and the interfacial tension, gamma 12, of the systems. It is argued that a relation of a similar type exists for the aqueous polymeric biphasic systems. The possibility of estimating the relative intensity of Van der Waals and hydration interactions of a solute and particle surface by examination of their partitioning in a biphasic system calibrated for the hydrophobic and hydration properties of the phases is discussed.     

Nonattacking Queens in a Rectangular Strip

The function that counts the number of ways to place nonattacking identical chess or fairy chess pieces in a rectangular strip of fixed height and variable width, as a function of the width, is a piecewise polynomial which is eventually a polynomial and whose behavior can be described in some detail. We deduce this by converting the problem to one of counting lattice points outside an affinographic hyperplane arrangement, which Forge and Zaslavsky solved by means of weighted integral gain graphs. We extend their work by developing both generating functions and a detailed analysis of deletion and contraction for weighted integral gain graphs. For chess pieces we find the asymptotic probability that a random configuration is nonattacking, and we obtain exact counts of nonattacking configurations of small numbers of queens, bishops, knights, and nightriders.     

High-power terahertz-range planar gyrotrons with transverse energy extraction

To increase the output power of terahertz gyrotrons to several hundred kilowatts, we suggest using a planar geometry of interaction space with a sheet electron beam and transverse energy extraction. An advantage of this scheme in comparison with conventional cylindrical geometry is the possibility to ensure effective mode selection over the open transverse coordinate in combination with radiation outcoupling that leads to a substantial reduction of Ohmic losses. Similar to unstable resonators in optics for further growth of the radiation power it is beneficial to introduce waveguide tapering.     

Six signed Petersen graphs,and their automorphisms

Up to switching isomorphism, there are six ways to put signs on the edges of the Petersen graph. We prove this by computing switching invariants, especially frustration indices and frustration numbers, switching automorphism groups, chromatic numbers, and numbers of proper 1-colorations, thereby illustrating some of the ideas and methods of signed graph theory. We also calculate automorphism groups and clusterability indices, which are not invariant under switching. In the process, we develop new properties of signed graphs, especially of their switching automorphism groups.     

The Largest Parity Demigenus of a Simple Graph

A graphΓisparity embeddedin a surface if a closed path in the graph is orientation preserving or reversing according as its length is even or odd. Theparity demigenusofΓis the minimum of 2−χ(S) (whereχis Euler characteristic) over all surfacesSin whichΓcan be parity embedded. We calculate the maximum parity demigenus over all loopless graphs of ordern. As a corollary we strengthen the calculation by Jungerman, Stahl, and White of the genus ofK_n,nbsp;nwith a perfect matching removed. We conclude by discussing numerous related problems.     

Partitioning of alkali halides in aqueous dextran-ficoll two-phase system

Partitioning of alkali halides MeX in aqueous dextran-Ficoll two-phase system of fixed polymer and ionic composition was studied. It is found that NaCl and LiX concentrate in the Ficoll-rich phase, while all KX, RbX, and CsX concentrate in the dextran-rich phase. Partitioning of KX, RbX, and CsX in the presence of a large excess of NaCl is found to depend on the type of the halide anion X^–.     

Solvent properties governing protein partitioning in polymer/polymer aqueous two-phase systems

Distribution coefficients of various proteins were measured in aqueous Dextran-Ficoll, Dextran-PES, and Ficoll-PES two-phase systems, containing 0.15M NaCl in 0.01 M phosphate buffer, pH 7.4. The acquired data were combined with data for the same proteins in different systems reported previously and known solvatochromic solvent properties of the systems to characterize the protein-solvent interactions. The relative susceptibilities of proteins to solvent dipolarity/polarizability, solvent hydrogen bond acidity, solvent hydrogen bond basicity, and solvent ability to participate in ion-ion and ion-dipole interactions were characterized. These parameters, which are representative of solute-solvent interactions, adequately described the partitioning of the proteins in each system. It was found that the relative susceptibilities of proteins to solvent dipolarity/polarizability are interrelated with their relative susceptibilities to solvent hydrogen bond acidity and solvent hydrogen bond basicity similarly to those established previously for small nonionic organic compounds.     

On examples in mathematical thinking and learning

The importance of examples and exemplification in mathematical thinking, learning and teaching, is well recognized by mathematicians, epistemologists and mathematics educators. In the collection of papers on these topics presented in this issue, we aim to contribute to the debate on this theme, proposing original studies carried out from different approaches and perspectives, and linked to other relevant topics within mathematics education.     

Stimulated Cherenkov radiation of a relativistic electron beam moving over a periodically corrugated surface (quasi-optical theory)

The propagation of waves over periodically corrugated surfaces and their excitation by relativistic electron beams are investigated within the framework of a quasi-optical approach. The dispersion equation is derived for normal waves under the assumption of a small (in the scale of the period and wavelength) corrugation depth, based on which two limiting cases are identified. In the first limiting case, the wave frequency is far from the Bragg resonance, and the propagation of waves can be described in terms of the impedance approximation, in which the fundamental spatial harmonic slows down. In the second limiting case realized at frequencies close to the Bragg resonance, the field is represented as two counterpropagating quasi-optical wave beams coupled on a corrugated surface and forming a normal surface wave. When interacting with an electron beam, convective instability, which can be used to realize amplifier regimes, corresponds to the first case, and absolute one, which is applied in surface-wave oscillators, corresponds to the second case. The developed theory is used to determine basic characteristics of amplifier and oscillator schemes: the growth rates, the energy exchange efficiency, and the formation of a self-consistent spatial structure of the radiated field. The practical realization of relativistic submillimeter amplifiers and surface-wave oscillators is shown to hold promise.