Blow–Up Profiles in One–Dimensional. Semilinear Parabolic Problems

Partially supporte by CICYT Research Grant PB86–0112–00202 and EEC Contrast SCI–0019–c.

We consider the Cauchy Problem where p > 1 and U$sub:o$esub:(x) is continuous, nonnegative and bounded. Let u(x.t) be the solution of (1). (2). and assume that u blows up at t=T . and We then show that the blow–up set is discrete. Moreover, if x=0 is a blow–up point, one of the two following possibilities occurs. Either There exist c > 0 and an even number m such that     

On-Line Dimension of Semi-Orders

We analyze the on-line dimension of partially ordered sets as a value of a two-person game between Algorithm and Spoiler. The game is played in rounds. Spoiler presents an on-line order of width at most w, one point at a time. Algorithm maintains its realizer, i.e., the set of d linear extensions which intersect to the presented order. Algorithm may not change the ordering of the previously introduced elements in the existing linear extensions. The value of the game val(w) is the least d such that Algorithm has a strategy against Spoiler presenting any order of width at most w. For interval orders Hopkins showed that val $(w) \leqslant 4w-4$ . We analyze the on-line dimension of semi-orders i.e., interval orders admitting a unit-length representation. For up-growing semi-orders of width w we prove a matching lower and upper bound of w. In the general (not necessarily up-growing) case we provide an upper bound of 2w.     

Triangle-Free Geometric Intersection Graphs with Large Chromatic Number

Several classical constructions illustrate the fact that the chromatic number of a graph may be arbitrarily large compared to its clique number. However, until very recently no such construction was known for intersection graphs of geometric objects in the plane. We provide a general construction that for any arc-connected compact set $X$ X in $\mathbb{R }^2$ R 2 that is not an axis-aligned rectangle and for any positive integer $k$ k produces a family $\mathcal{F }$ F of sets, each obtained by an independent horizontal and vertical scaling and translation of $X$ X , such that no three sets in $\mathcal{F }$ F pairwise intersect and $\chi (\mathcal{F })>k$ χ ( F ) > k . This provides a negative answer to a question of Gyárfás and Lehel for L-shapes. With extra conditions we also show how to construct a triangle-free family of homothetic (uniformly scaled) copies of a set with arbitrarily large chromatic number. This applies to many common shapes, like circles, square boundaries or equilateral L-shapes. Additionally, we reveal a surprising connection between coloring geometric objects in the plane and on-line coloring of intervals on the line.     

Neutron scattering studies of solid-state polymorphism in dimethyl butanol glass formers

Inelastic incoherent neutron scattering (IINS) and neutron powder diffraction (NPD) studies have been performed for two dimethyl 1-butanols and two dimethyl 2-butanols with CH₃ side molecular groups. Low-temperature vibrational density of states confirmed solid-state polymorphism detected by calorimetric methods, i.e., existence of crystalline and ODIC phases for all isomers, orientationally disordered glass for 2,2-DM 1-B and 3,3-DM 2-B, and glass of isotropic phase for 3,3-DM 1-B. Difference in vibrational density of states between glass and the ordered crystal has shown the so-called Boson peak. Influence of the OH group position in the molecules on their vibrational dynamics up to 50 meV is discussed.     

Reactions of nitroxides,part 7: Synthesis of novel nitroxide selenoureas

The reactions of 4‐isoselenocyanato‐2,2,6,6‐tetramethylpiperidine‐1‐oxyl with selected amines and lower alcohols give the corresponding novel selenoureas and selenocarbamates, all bearing the nitroxyl moiety. Some of the synthesized selenoureas and selenocarbamates show moderate‐to‐good activity against pathogenic fungi. © 2008 Wiley Periodicals, Inc. Heteroatom Chem 19:549–556, 2008; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/hc.20454     

Self-similar asymptotic behavior for the solutions of a linear coagulation equation

In this paper we consider the long-time asymptotics of a linear version of the Smoluchowski equation which describes the evolution of a tagged particle moving in a random distribution of fixed particles. The volumes v of these particles are independently distributed according to a probability distribution which decays asymptotically as a power law $v^{?\sigma }$. The validity of the equation has been rigorously proved in [22] taking as a starting point a particle model and for values of the exponent $\sigma >3$, but the model can be expected to be valid, on heuristic grounds, for $\sigma >\frac{5}{3}$. The resulting equation is a non-local linear degenerate parabolic equation. The solutions of this equation display a rich structure of different asymptotic behaviors according to the different values of the exponent σ. Here we show that for $\frac{5}{3}<\sigma <2$ the linear Smoluchowski equation is well-posed and that there exists a unique self-similar profile which is asymptotically stable.     

Multi-electrode Harmonized Kingdon Traps

Based on the analysis of the results of the study of various designs of multi-electrode harmonized Kingdon traps, we propose a new type of trap with two merged internal electrodes that has the ability to capture and accumulate ions formed inside. We also investigated the influence of inaccuracies in the manufacture of the electrodes on the field inside such trap. The four-electrode trap, which actually degenerates into a two-electrode device with traces of two other electrodes present at the ends of the internal electrodes (their splitting) has been found as the less sensitive to inaccuracies caused by manufacturing and cutting the ends of trap electrodes. We show that a mass spectrometer with a relatively high resolving power can be created on the basis of such a trap. The creation of the traps requires the manufacture of complex electrodes with demanded accuracy of their surfaces. This becomes possible with the advent of 3D printers.

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Graphical Abstract Multi electrode harmonize Kingdon trap with fused elctrode pares

     

Influence of constant magnetic field on electrodeposition of metals,alloys, conductive polymers,and organic reactions

The paper presents and summarizes some research on constant magnetic field effects in chemistry. Metals and alloys electrodeposited under constant magnetic field have greater thickness and smoother surface with finest grains. Metallic materials deposited under the influence of uniform magnetic field may have stronger corrosion resistance, than those obtained without the presence of magnetic field. Constant magnetic field also causes an increase of the electropolymerization rate and yield of some organic reactions. Our research also shows that the presence of constant magnetic field affects the electrodeposition process of alloys and their morphology to a great extent. The effects of magnetic field on metals, alloys, composites, polymers and other materials are due to the Lorentz force and the magnetohydrodynamic effect. It is possible that the further development of magnetoelectrodeposition will allow for using the constant magnetic field to improve the properties of metal coatings, alloys, polymers, and other materials in the industry.