General theorem for the existence of iterative roots of homeomorphisms with periodic points

We apply Zdun’s factorization theorem (see Zdun (2008) [3]) to give the conditions for the existence and the form of continuous and orientation-preserving iterative roots of homeomorphisms of the circle with a rational rotation number. Our theorem generalizes the previous results given by Jarczyk (2003) in [2], Zdun (2008) in [3] and Solarz (2003, 2009) in [4] and [5].     

Stereoselective separations of chiral phosphinic acid pseudodipeptides by CEC using silica monoliths modified with an anion-exchange-type chiral selector

A nonaqueous CEC method for the simultaneous separation of the four stereoisomers of the N-benzyloxycarbonyl phosphinic pseudodipeptide methyl ester benzyloxycarbonyl-homophenylalanine Z-hPhepsi(PO2HCH2)Phe-OCH3 as well as of the corresponding N-2,4-dinitrophenyl (DNP)-derivative with free C-terminal carboxylic group DNP-hPhepsi(PO2HCH2)Phe-OH was developed. For this purpose, a monolithic silica capillary column modified with a cinchona alkaloid-derived anion-exchange-type chiral selector, namely O-9-(tert-butylcarbamoyl)quinidine (tBuCQD) was prepared. The mobile phase composition (ACN/methanol ratio, counterion type) was thoroughly optimized to end up with baseline resolution of all four stereoisomers with critical resolution of as high as about 2. The CEC method proved to be superior over the corresponding HPLC separations primarily due to significantly enhanced plate numbers (between 200,000 and 600,000 m(-1) in CEC). Diastereoselectivity contributions arising from electrophoretic mobility differences of the diastereomers facilitated the separation of the later eluted diastereomeric peak pair (peaks III and IV), but had a negative influence on the selectivity of the earlier eluted diastereomeric peak pair (peaks I and II). The stereoselective CEC assay allowed the assessment of the stereoisomeric purity of the individual isomers which were obtained by preparative HPLC on a CHIRALPAK QD-AX column that is based on the same tBuCQD selector. The present study demonstrates that there exist problems which are hard to solve by HPLC, yet can be conveniently solved by CEC. Moreover, it was intended to prove by this practical application that CEC with monolithic columns is robust enough to be used for solving real-life problems.     

Theoretical investigations of the chromatographic separation of interacting enantiomers

Separation of a pair of enantiomers by liquid chromatography is modeled using the equilibrium dispersive (ED) model of chromatography. It is assumed that the chiral stationary phase used for the separation consists of two types of adsorption sites, including chiral selectors linked to the surface and nonselective centers belonging to the achiral matrix. Additionally, intermolecular interactions between adsorbed enantiomers are taken into account. The corresponding equilibrium adsorption isotherms of the enantiomers are derived by means of the mean field approximation (MFA) and used as input data for the ED model. Special attention is paid to the influence of the lateral interactions on the effectiveness of the enantiomer separation. In particular, we examine the effect of the interactions on the shape and relative position of the chromatographic peaks associated with the enantiomers. Furthermore, the influence of the spacer length, which modifies screening of the lateral interactions, on the adsorption process is studied. The obtained results suggest that the lateral interactions combined with the screening effect may cause serious changes in the separation, depending on the nature (attraction or repulsion) and strength of the interactions as well as on the spacer length.     

Implementation of a symplectic multiple-time-step molecular dynamics algorithm, based on the united-residue mesoscopic potential energy function

A symplectic multiple-time-step (MTS) algorithm has been developed for the united-residue (UNRES) force field. In this algorithm, the slow-varying forces (which contain most of the long-range interactions and are, therefore, expensive to compute) are integrated with a larger time step, termed the basic time step, and the fast-varying forces are integrated with a shorter time step, which is an integral fraction of the basic time step. Based on the split operator formalism, the equations of motion were derived. Separation of the fast- and slow-varying forces leads to stable molecular dynamics with longer time steps. The algorithms were tested with the Ala(10) polypeptide chain and two versions of the UNRES force field: the current one in which the energy components accounting for the energetics of side-chain rotamers (U(rot)) can lead to numerically unstable forces and a modified one in which the the present U(rot) was replaced by a numerically stable expression which, at present, is parametrized only for polyalanine chains. With the modified UNRES potential, stable trajectories were obtained even when extending the basic time step to 15 fs and, with the original UNRES potentials, the basic time step is 1 fs. An adaptive multiple-time-step (A-MTS) algorithm is proposed to handle instabilities in the forces; in this method, the number of substeps in the basic time step varies depending on the change of the magnitude of the acceleration. With this algorithm, the basic time step is 1 fs but the number of substeps and, consequently, the computational cost are reduced with respect to the MTS algorithm. The use of the UNRES mesoscopic energy function and the algorithms derived in this work enables one to increase the simulation time period by several orders of magnitude compared to conventional atomic-resolution molecular dynamics approaches and, consequently, such an approach appears applicable to simulating protein-folding pathways, protein functional dynamics in a real molecular environment, and dynamical molecular recognition processes.     

Watched guards in art galleries

The art gallery problem asks how many guards are sufficient to see every point of the interior of a polygon. A set of guards is called watched if each guard itself is seen by at least one of its colleagues. In 1994, Hernández-Pe?alver wrote that watched guards always suffice to guard any polygon with n vertices. However in 2001, Michael and Pinciu, and independently Żyliński, presented a class of polygons that required more than watched guards – which disproved the Hernández-Pe?alver’s result – and they established a new tight bound for watched guards: . Combinatorial bounds for watched guards in orthogonal polygons were independently given by Hernández-Pe?alver , and by Michael and Pinciu, who proved the -bound to be tight. In this paper, tight bounds for polygons of miscellaneous shapes are presented: watched guards for monotone and spiral polygons, and vertex watched guards for star-shaped polygons.     

Determination of 210Po and uranium in high salinity water samples

A method for the determination of uranium and ²¹⁰Po in high salinity water samples has been elaborated. Both radionuclides are preconcentrated from 0.5 dm³ saline media by co-precipitation with hydrated manganese dioxide, followed by dissolution of the precipitate in 200 mL of 1 M HCl. Uranium isotopes ²³⁵U and ²³⁸U can be directly determined by ICP MS method with a detection limit of 0.01 ppb for ²³⁸U. Prior to a selective determination of ²¹⁰Po, the majority of other naturally occurring α-emitting radionuclides (uranium, thorium and protactinium) can be stripped from this solution by their extraction with a 50% solution of HDEHP in toluene. Finally, ²¹⁰Po is simply separated by direct transfer to an extractive scintillator containing 5% of trioctylphosphine oxide in Ultima Gold F cocktail and determined by an α/β separation liquid scintillation technique with detection limit below 0.1 mBq/dm³.     

Microelectrode array fabricated in low temperature cofired ceramic (LTCC) technology

This work presents the development of a novel construction of an integrated microelectrode array. The device was fabricated on a ceramic support, with the use of low temperature cofired ceramics technology. Model potassium-selective membranes were applied on the surface of PdAg/AgCl electrodes formed on the ceramic substrate. The obtained microsensors exhibited very good repeatability, reproducibility, and sensitivity. The array of microelectrodes covered with polymeric layers of various selectivities was applied as an electronic tongue to differentiate between various diet supplements.