On topological properties of the Hartman-Mycielski functor

We investigate some topological properties of a normal functorH introduced earlier by Radul which is some functorial compactification of the Hartman-Mycielski construction HM. We prove that the pair (H X, HMY) is homeomorphic to the pair (Q, σ) for each nondegenerated metrizable compactumX and each denseσ-compact subsetY.     

Hyperspaces of Normed Linear Spaces with the Attouch–Wets Topology

Let Cld _AW (X) be the hyperspace of nonempty closed subsets of a normed linear space X with the Attouch–Wets topology. It is shown that the space Cld _AW (X) and its various subspaces are AR's. Moreover, if X is an infinite-dimensional Banach space with weight w(X) then Cld _AW (X) is homeomorphic to a Hilbert space with weight 2 ^w(X).     

Division and <Emphasis Type="Italic">k</Emphasis>-th root theorems for <Emphasis Type="Italic">Q</Emphasis>-manifolds

We prove that a locally compact ANR-space X is a Q-manifold if and only if it has the Disjoint Disk Property (DDP), all points of X are homological Z _∞-points and X has the countable-dimensional approximation property (cd-AP), which means that each map f: K → X of a compact polyhedron can be approximated by a map with the countable-dimensional image. As an application we prove that a space X with DDP and cd-AP is a Q-manifold if some finite power of X is a Q-manifold. If some finite power of a space X with cd-AP is a Q-manifold, then X ² and X × [0, 1] are Q-manifolds as well. We construct a countable family χ of spaces with DDP and cd-AP such that no space X ∈ χ is homeomorphic to the Hilbert cube Q whereas the product X × Y of any different spaces X, Y ∈ χ is homeomorphic to Q. We also show that no uncountable family χ with such properties exists. This work was supported by the Slovenian-Ukrainian (Grant No. SLO-UKR 04-06/07)     

Noncommutative differential geometry related to the Young-Baxter equation

An analogue of the differential calculus associated with a unitary solution of the quantum Young-Baxter equation is constructed. An example of a ring sheaf is considered in which local solutions of the Young-Baxter quantum equation are defined but there is no global section. Bibliography: 13 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 199, 1992, pp. 51–70. Translated by I. A. Izergina.     

On universal spaces and absorbing sets related to a transfinite extension of covering dimension

R. Pol has shown that for every countable ordinal number α there exists a universal space for separable metrizable spaces X with trindX?α. W. Olszewski has shown that for every countable limit ordinal number λ there is no universal space for separable metrizable space with trIndX?λ. T. Radul and M. Zarichnyi have proved that for every countable limit ordinal number there is no universal space for separable metrizable spaces with dimWX?α where dimW is a transfinite extension of covering dimension introduced by P. Borst. We prove the same result for another transfinite extension dimC of the covering dimension.As an application, we show that there is no absorbing sets (in the sense of Bestvina and Mogilski) for the classes of spaces X with dimCX?α belonging to some absolute Borel class.     

1, 5-Rearrangement of enol acylates of aryl 1H.-1, 2, 4-triazol-1-ylmethyl ketones

The rearrangement of enol acylates of aryl 1H-1, 2, 4-triazol-1-ylmethyl ketones at 140–150°C in acetic anhydride is studied. The migration of the acyl group to the C(5) atom of the heterocycle is found to be intramolecular. The characteristics of the original and final products are presented. X-ray structural studies of the enol acetates of 2, 4-dichlorophenyl 1H-1, 2, 4-triazol-1-ylmethyl ketone and 2, 4-dichlorophenyl 5-acetyl-1H-1, 2, 4-triazol-1-ylmethyl ketone are carried out.Translated fromIzvestiya Akademii Nauk. Seriya Khimicheskaya, No. 3, pp. 560–564, March, 1993.     

Origin of visible-light-driven photocatalysis: A comparative study on N/F-doped and N-F-codoped TiO2 powders by means of experimental characterizations and theoretical calculations

An overall comparative study was carried out on N-doped, F-doped, and N-F-codoped TiO₂ powders (NTO, FTO, NFTO) synthesized by spray pyrolysis in order to elucidate the origin of their visible-light-driven photocatalysis. The comparisons in their experimentally obtained characteristics were based on the analysis of XPS, UV-Vis, PL, NH₃-TPD and ESR spectra. The comparisons in their theoretically predicted properties were based on the analysis of the calculated electronic structures. As the results, N-doping into TiO₂ resulted in not only the improvement in visible-light absorption but also the creation of surface oxygen vacancies. F-doping produced several beneficial effects including the creation of surface oxygen vacancies, the enhancement of surface acidity and the increase of Ti³⁺ ions. Doped N atoms formed a localized energy state above the valence band of TiO₂, whereas doped F atoms themselves had no influence on the band structure. The photocatalytic tests indicated that the NFTO demonstrated the highest visible-light activity for decompositions of both acetaldehyde and trichloroethylene. This high activity was ascribed to a synergetic consequence of several beneficial effects induced by the N-F-codoping.     

FORMATION OF SINGLE AND DOUBLE STRAND BREAKS IN DNA ULTRAVIOLET IRRADIATED AT HIGH INTENSITY

The induction of single-strand breaks (SSB) by two quantum processes in DNA is well established. We now report that biphotonic processes result in double-strand breaks (DSB) as well. pUC19 and bacteriophage M13 RF DNA were irradiated using an excimer laser (248 nm) at intensities of 10(7), 10(9), 10(10) and 10(11) W/m2 and doses up to 30 kJ/m2. The proportion of DNA as supercoil, open circular, linear and short fragments was determined by gel electrophoresis. Linear molecules were noted at fluences where supercoiled DNA was still present. The random occurrence of independent SSB in proximity to each other on opposite strands (producing linear DNA) implies introduction of numerous SSB per molecule in the sample. If so, supercoiled DNA that has sustained no SSB should not be observed. A model accounting for the amounts of supercoiled, open circular, linear and shorter fragments of DNA due to SSB, DSB and Scissions (opposition of two independently occurring SSB producing an apparent DSB) was developed, our experimental data and those of others were fit to the model, and quantum yields determined for SSB and DSB formation at each intensity. Results showed that high intensity laser radiation caused an increase in the quantum yields for both SSB and DSB formation. The mechanism of DSB formation is unknown, and may be due to simultaneous cleavage of both strands in one biphotonic event or the biased introduction of an SSB opposite a preexisting SSB, requiring two biphotonic events.     

Instability of the finite‐difference split‐step method applied to the generalized nonlinear Schrödinger equation. III. external potential and oscillating pulse solutions

This is the final part of a series of articles where we have studied numerical instability (NI) of localized solutions of the generalized nonlinear Schrödinger equation (gNLS). It extends our earlier studies of this topic in two ways. First, it examines differences in the development of the NI between the case of the purely cubic NLS and the case where the gNLS has an external bounded potential. Second, it investigates how the NI is affected by the oscillatory dynamics of the simulated pulse. The latter situation is common when the initial condition is not an exact stationary soliton. We have found that in this case, the NI may remain weak when the time step exceeds the threshold quite significantly. This means that the corresponding numerical solution, while formally numerically unstable, can remain sufficiently accurate over long times, because the numerical noise will stay small. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 633–650, 2017