Ergodic properties of Markov maps inR d

Summary For domainsIR ^d (bounded or not) the notion of a Markov map fromI into itself is developed. It is shown that under a condition of Rényi type and the assumption that the map is Markov, any probability density tends inL ¹-norm to a unique invariant measure under the action of the Perron-Frobenius operatorP . The smoothness and ergodic properties of that invariant measure are studied. The paper generalizes results of Lasota and Yorke from dimension one to higher dimension.     

The influence of reverse energy transport on emission anisotropy in two-component viscous solutions

The effect of nonradiative reverse energy transport (NRET) in two donor-acceptor systems was studied experimentally. It was found that the NRET occurring in system I; rhodamine 6G (donor) and rhodamine B (acceptor), considerably lowers the emission anisotropy at medium and high concentrations. These results qualitatively confirm the predictions of the approximate theoretical approach of L. Kulak and C. Bojarski (see the preceding paper). In system II; rhodamine 6G (donor) and Nile Blue (acceptor), for which the NRET process does not occur, a good agreement with no-back-transport theory was obtained.     

Invariant measures ofC 2-diffeomorphisms of Markov type inR d

A class of uniformly expanding, piecewiseC ²-diffeomorphisms from domainsIR ^d (bounded or not) into themselves is considered. It is shown that the number of the extreme points of Fix (P )={gG:Pg=g} whereP is the Frobenius-Perron operator associated with andG={gL ¹: g0 g=1}, can be determined in an effective way. Moreover, it is shown that the sequence {P ^j g} is convergent inL ¹ for anygG, and in the topology of uniform convergence for anygG(1). The limit is a linear projectionR inL ¹ (defined by (3.1)) which mapsG onto Fix (P ) (see Th. 3.1).Dedicated to professor A. Lasota on the occasion of his 60th birthday     

The classification of differential structures on quantum 2-spheres

Exterior algebras of differential forms on quantum 2-spheresS _qc ² ,q[–1, 1]/{0},c[0, ] (c=0 forq=±1), are classified. In the definition of exterior algebras we assume the invariance w.r.t. the action of the quantumSU(2) group and dimensionality conditions (which imply that we deal with two-dimensional manifolds). The exterior algebras exist only forc=0 and are unique in that case. The corresponding generalized directional derivatives are provided.     

Non-smoothness of event horizons of Robinson-Trautman black holes

It is shown that generic small data Robinson-Trautman space-times cannot beC ¹²³ extended beyond the r=2m Schwarzschild-like event horizon. This implies that an observer living in such a space-time can determine by local measurements whether or not he has crossed the event-horizon of the black-hole.     

Suppression of deleterious effects of free silanols in liquid chromatography by imidazolium tetrafluoroborate ionic liquids

Silica-based stationary phases are commonly used in liquid chromatography, but their surface acidity causes known problems, especially when separating basic compounds. Deleterious effects of free silanols are not fully removed by standard prevention procedures consisting in adding alkylamines or other amino quenchers to the eluents. We found that ionic liquids of the imidazolium tetrafluoroborate class, added to mobile phases at concentrations of 0.5-1.5% (v/v), blocked silanols and provided excellent thin-layer chromatographic separations of strongly basic drugs which were otherwise not eluted, even with neat acetonitrile as the mobile phase. The silanol suppressing potency of imidazolium tetrafluoroborates was demonstrated to markedly exceed that of the standard mobile phase additives, like triethylamine, dimethyloctylamine and ammonia. The proposed new mobile phase additives were also demonstrated to provide reliable lipophilicity parameters of base drug analytes as determined by gradient mode of high-performance liquid chromatography. By applying the readily available and environmentally friendly imidazolium tetrafluoroborate ionic liquids, simple and efficient means of improvement of liquid chromatographic analysis of organic bases were elaborated.     

Diastereoselective assembling of 21-C-alkylated nickel(II) complexes of inverted porphyrin on a platinum(II) template

Reaction of 21-C-methyl and 21-C-benzyl nickel(II) complexes of inverted meso-tetratolylporphyrin with platinum(II) dichloride or its bis(benzonitrile) complex yields a chloroplatinum(II) species containing two nickel(II) carbaporphyrinoids in a cis arrangement. One of the carbaporphyrinoids coordinates to the platinum ion with the external nitrogen while the other is bound with the external nitrogen and one ortho-carbon of the adjacent meso-aryl ring. The reaction is highly chemoselective. (1)H and (13)C NMR experiments in solution show the diastereoselectivity of the reaction. Single-crystal X-ray data confirm the presence of the diastereomer with opposite configurations of the Ni(II)-coordinated carbons in the subunits of the dimer. Cyclovoltammetric measurements reveal an anodic shift of the nickel(II) oxidation potentials of dimers with respect to those of the parent monomers and two different reduction couples. Reaction of unsubstituted inverted porphyrin with Pt(PhCN)(2)Cl(2) in chlorobenzene yields a monomeric platinum(II) complex of inverted porphyrin. This complex displays a markedly upfield (195)Pt NMR shift compared to tetraphenylporphyrinatoplatinum(II). Under strongly basic conditions deprotonation of the external nitrogen of inverted porphyrin and both electrochemical and chemical oxidation of platinum(II) center are observed.     

The Rothe method for the Mckendrick-von Foerster equation

We present the Rothe method for the McKendrick-von Foerster equation with initial and boundary conditions. This method is well known as an abstract Euler scheme in extensive literature, e.g. K. Rektorys, The Method of Discretization in Time and Partial Differential Equations, Reidel, Dordrecht, 1982. Various Banach spaces are exploited, the most popular being the space of bounded and continuous functions. We prove the boundedness of approximate solutions and stability of the Rothe method in L ^∞ and L ¹ norms. Proofs of these results are based on comparison inequalities. Our theory is illustrated by numerical experiments. Our research is motivated by certain models of mathematical biology.     

Novel (R)-(+)-limonene-derived ligands: synthesis and application in asymmetric transfer hydrogenations

(R)-(+)-Limonene was transformed into mono-N-tosylated-1,2-diamine derivatives using an N-tosylaziridination procedure followed by sodium azide treatment and reduction on Pd/C. The ligands obtained proved effective in the asymmetric transfer hydrogenation protocol on aromatic ketones.     

Luminescent cellulose fibers activated by Eu3+-doped nanoparticles

UV- active cellulose fibers were obtained by dry-wet method spinning an 8?% by weight α-cellulose solution in N-methylomorpholine-N-oxide (NMMO) modified by europium-doped gadolinium oxyfluoride Gd₄O₃F₆:Eu³⁺ containing 5?mol (%) of the dopant. Photoluminescent nanoparticles were introduced in the in powder form into a polymer matrix during the process of cellulose dissolution in NMMO. The dependencies of emission intensity on excitation energy and the concentration of Gd₄O₃F₆:Eu³⁺ nanoparticles in the final cellulosic products were examined by photoluminescence spectroscopy (excitation and emission). The fiber structure was studied by X-ray powder diffraction analysis. The size and dispersity of the nanoparticles in the polymer matrix were evaluated using scanning electron microscopy and X-ray microanalysis. The influence of different concentration particles (in the range from 0.5 to 5?% by weight) on the mechanical properties of the fibers, such as tenacity and elongation at break, were determined.