Conciliating the nonadditive entropy approach and the fractional model formulation when describing subdiffusion

We consider here two different models describing subdiffusion. One of them is derived from Continuous Time Random Walk formalism and utilizes a subdiffusion equation with a fractional time derivative. The second model is based on Sharma-Mittal nonadditive entropy formalism where the subdiffusive process is described by a nonlinear equation with ordinary derivatives. Using these two models we describe the process of a substance released from a thick membrane and we find functions which determine the time evolution of the amount of substance remaining inside this membrane. We then find ‘the agreement conditions’ under which these two models provide the same relation defining subdiffusion and give the same function characterizing the process of the released substance. These agreement conditions enable us to determine the relation between the parameters occuring in both models.     

A wavelet basis within Bethe ansatz

We describe here an example of a dynamical system which preserves the distance between constituent particles. Namely, we consider the one-dimensional finite magnetic Heisenberg ring of an even numberN of nodes with the spin 1/2, solved exactly by the well-known Bethe ansatz (XXX model). It is described in terms of a number of spin deviations from the ferromagnetic saturation state (a system of Bethe pseudoparticles) which undergo processes of mutual solitonic scattering when moving on the ring. We point out that, when restricting to (i) the subspace of quantum states with two spin deviations and (ii) to the boundary of the Brillouin zone, Bethe pseudoparticles perfectly avoid any mutual scattering. Therefore, each state with a fixed distance between pseudoparticles (the so called basis of wavelets) is already an eigenstate of the initial Heisenberg Hamiltonian. Presented by T. Lulek at the DI-CRM Workshop held in Prague, 18–21 June 2000. The authors are very much indebted to Professors J. Patera and J. Tolar for their invitation to DI-CRM Workshop on Mathematical Physics, and for their kind hospitality during the Workshop.     

The weak lower semicontinuity of the Kobayashi distance and its applications

In this paper we prove the lower semicontinuity with respect to the weak topology of the Kobayashi distance in a bounded, convex and open subset of a reflexive Banach space. We apply this result to the Denjoy-Wolff theorem for condensing mappings in the unit open ball in a strictly convex reflexive Banach space with the Kadec-Klee property. Received June 30, 1998 / in final form December 20, 1999 / Published online July 20, 2000     

Preparation, Structure, and Thermal Decomposition of Co(II) Complexeswith 2,3- and 2,5-Dichlorobenzoic Acid and Imidazole

Summary.  The reaction products of Co(II)-2,3- and -2,5-dichlorobenzoate with imidazole (1, 2; CoL ₂⋯2imdċ2H₂O, L=C₇H₃Cl₂O, imd=imidazole) were characterized by their spectroscopic and thermochemical properties. The compounds crystallize in the monoclonic system with space group = P2₁/c, a=13.848(3), b=12.841(3) ?, c= 7.064(2) ?, β=98.12 °, V=1243.5(4) ?³, Z=2 for 1 and space group =P2₁/n, a=13.293(3), b= 6.964(2), c=13.800(3) ?, β=108.92(3) °, V=1208.6(4) ?³, Z=2 for 2. The complexes lose their crystal water in one step at 333 K and subsequently decompose to CoO with intermediate formation of Co₃O₄. Received August 9, 1999. Accepted (revised) February 11, 2000     

Reaction of diethyl thiocyanatomethylphosphonate with aldehydes as a route to diethyl Z‐1‐alkenylphosphonates

Diastereoselective synthesis of diethyl Z‐1‐alkenylphosphonates from easily available diethyl thiocyanatomethylphosphonate and aromatic aldehydes has been developed. Olefination of the aldehydes occurs under mild conditions and affords the title compounds with moderate yields. A plausible mechanism of the above‐mentioned reaction is also discussed. © 2007 Wiley Periodicals, Inc. Heteroatom Chem 18:732–739, 2007; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/hc.20371     

On the Development of the VIPAR Polymer Gel Dosimeter for Three-Dimensional Dose Measurements

Summary: The new polymer gel dosimeter, based on the modification of the VIPAR gel composition, is described for the purpose of radiation dose distribution measurement in radiotherapy. It features increased concentration of the two VIPAR substrates: N-vinylpyrrolidone (8%) and gelatine (7.5%) (N,N′-methylenebisacrylamide was maintained at 4%), and the addition of copper sulphate (0.0008%) and ascorbic acid (0.007%) in order to facilitate the preparation through elimination of the need for deoxygenation of the gel. Following the exposure to ionizing radiation, polymerisation and cross-linking of the new gel monomers occurs retaining the spatial distribution of absorbed dose and causing opacity of the gel. Quantitative parameters of the new gel dose response were studied using magnetic resonance imaging to relate polymerisation induced physicochemical changes of the gel to dose. The dose threshold is found significantly lower than that of the original VIPAR gel. The linear part of measured spin-spin relaxation rate R₂(D) ( = 1/T₂(D)) reaches up to 35 Gy. Its slope and an intercept are slightly higher relative to the original VIPAR. The efficiency of the new polymer gel-magnetic resonance imaging dosimeter was also tested for dose verification of a 3D dose distribution planned by a commercially available treatment planning software (Eclipse External Beam v.6.5) and delivered by a 6 MV medical linear accelerator. The new polymer gel is proposed to be called, VIPAR^nd (after VIPAR-normoxic-double).     

On the Statistical Discrepancy and Affinity of Priority Vector Heuristics in Pairwise-Comparison-Based Methods

There are numerous priority deriving methods (PDMs) for pairwise-comparison-based (PCB) problems. They are often examined within the Analytic Hierarchy Process (AHP), which applies the Principal Right Eigenvalue Method (PREV) in the process of prioritizing alternatives. It is known that when decision makers (DMs) are consistent with their preferences when making evaluations concerning various decision options, all available PDMs result in the same priority vector (PV). However, when the evaluations of DMs are inconsistent and their preferences concerning alternative solutions to a particular problem are not transitive (cardinally), the outcomes are often different. This research study examines selected PDMs in relation to their ranking credibility, which is assessed by relevant statistical measures. These measures determine the approximation quality of the selected PDMs. The examined estimates refer to the inconsistency of various Pairwise Comparison Matrices (PCMs)—i.e., W = (w_ij), w_ij > 0, where i, j = 1,…, n—which are obtained during the pairwise comparison simulation process examined with the application of Wolfram’s Mathematica Software. Thus, theoretical considerations are accompanied by Monte Carlo simulations that apply various scenarios for the PCM perturbation process and are designed for hypothetical three-level AHP frameworks. The examination results show the similarities and discrepancies among the examined PDMs from the perspective of their quality, which enriches the state of knowledge about the examined PCB prioritization methodology and provides further prospective opportunities.     

Photochemical reduction of iron trichloride in ethyl acetate: synthesis,Mössbauer spectra and the crystal structure at 80 K of hexakis(ethyl acetate)iron(II) bis-tetrachloroironate(III)

FeCl₃ in ethyl acetate under the influence of sunlight, undergoes partial reduction yields the [Fe(CH₃CO₂Et)₆](FeCl₄)₂ salt. The Mössbauer spectra showed that the iron atoms are at +2 and +3 oxidation states. The crystal structure determined by X-ray diffraction methods at 80 K and refined by full-matrix least-squares techniques toR=0.028 for 2410 independent non-zero reflections is in good agreement with the Mössbauer results. The [Fe(CH₃CO₂Et)₆]²⁺ cations occupy centers of symmetry and the Fe²⁺ ions are octahedrally coordinated by six carbonyl oxygen atoms of six ethyl acetate molecules.