Characterizing the number of coloured m-ary partitions modulo m,with and without gaps

In a pair of recent papers, Andrews, Fraenkel and Sellers provide a complete characterization for the number of $m$-ary partitions modulo $m$, with and without gaps. In this paper we extend these results to the case of coloured $m$-ary partitions, with and without gaps. Our method of proof is different, giving explicit expansions for the generating functions modulo $m$.     

On profinite groups with word values covered by nilpotent subgroups

Let N stand for the class of nilpotent groups or one of its well-known generalizations. For a multilinear commutator word w and a profinite group G we show that w(G) is finite-by-N if and only if the set of wvalues in G is covered by countably many finite-by-N subgroups. Earlier this was known only in the case where w = x or w = [x, y].     

Asymptotics of solutions to the periodic problem for the nonlinear damped wave equation

We study large time asymptotic behavior of solutions to the periodic problem for the nonlinear damped wave equation

Spreading of Gas from a Line Source in the Flow over Two-dimensional Hill

Flow around the two-dimensional hill is of great interest in engineering application, namely in transport and dispersion of pollutants in atmosphere. A case of polynomial shaped 2D hill with the line source of tracer gas has been studied in detail. Experimental study of both, the time-averaged velocity field and the binary-mixture concentration field has been done. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On a Congruence Involving Generalized Fibonomial Coefficients

Let (F_n)_n≥0 be the Fibonacci sequence. For 1 ≤ k ≤ m, the Fibonomial coefficient is defined as

$${\left[ {\begin{array}{*{20}{c}} n \\ k \end{array}} \right]_F} = \frac{{{F_{n - k + 1}} \cdots {F_{n - 1}}{F_n}}}{{{F_1} \cdots {F_k}}}$$

. In 2013, Marques, Sellers and Trojovský proved that if p is a prime number such that p ≡ ±1 (mod 5), then p?\({\left[ {\begin{array}{*{20}{c}} {{p^{a + 1}}} \\ {{p^a}} \end{array}} \right]_F}\) for all integers a ≥ 1. In 2010, in particular, Kilic generalized the Fibonomial coefficients for

$${\left[ {\begin{array}{*{20}{c}} n \\ k \end{array}} \right]_{F,m}} = \frac{{{F_{\left( {n - k + 1} \right)m}} \cdots {F_{\left( {n - 1} \right)m}}{F_{nm}}}}{{{F_m} \cdots {F_{km}}}}$$

. In this note, we generalize Marques, Sellers and Trojovský result to prove, in particular, that if p ≡ ±1 (mod 5), then \({\left[ {\begin{array}{*{20}{c}} {{p^{a + 1}}} \\ {{p^a}} \end{array}} \right]_{F,m}} \equiv 1\) (mod p), for all a ≥ 0 and m ≥ 1.

     

Tetramethyl(perfluoroalkyl)cyclopentadienyl rhodium(I) complexes with ethylene or diene ligands. Crystal structure of [(η-C5Me4C6F13)Rh(CO)2]

Tetramethyl(perfluoroalkyl)cyclopentadienyl rhodium(I) complexes with ethylene or diene (norbornadiene, cycloocta-1,5-diene, 2,3-dimethylbuta-1,3-diene, cyclohexa-1,3-diene) ligands were obtained by reduction of tetramethyl(perfluoroalkyl)rhodium(III) dichloro dimers by zinc in THF or by propan-2-ol/sodium carbonate in the presence of the ligands. Reduction in the presence of cycloocta-1,3-diene gave a different product, an η³-cyclooctenyl complex, which was not reduced further. During the reduction in the presence of ethylene, a new tetramethyl(perfluoroalkyl)-η⁴-cyclopentadiene complex was observed by NMR. This compound, formed by hydrogen transfer from the metal to the ligand, is probably in an equilibrium with the parent hydridocyclopentadienyl complex. Crystal and molecular structure of dicarbonyltetramethyl(perfluorohexyl)cyclopentadienylrhodium(I) complex was determined by X-ray diffraction. The structure shows a moderate ring slippage of the rhodium atom which was not observed in the only other known structure of a complex with the same ligand, the rhodium(III) dichloro dimer.     

Tungsten carbide precursors as an example for influence of a binder on the particle formation in the nanosecond laser ablation of powdered materials

A study of LA-ICP-MS analysis of pressed powdered tungsten carbide precursors was performed to show the advantages and problems of nanosecond laser ablation of matrix-unified samples. Five samples with different compositions were pressed into pellets both with silver powder as a binder serving to keep the matrix unified, and without any binder. The laser ablation was performed by nanosecond Nd:YAG laser working at 213 nm.The particle formation during ablation of both sets of pellets was studied using an optical aerosol spectrometer allowing the measurement of particle concentration in two size ranges (10-250 nm and 0.25-17 μm) and particle size distribution in the range of 0.25-17 μm. Additionally, the structure of the laser-generated particles was studied after their collection on a filter using a scanning electron microscope (SEM) and the particle chemical composition was determined by an energy dispersive X-ray spectroscope (EDS).The matrix effect was proved to be reduced using the same silver powdered binder for pellet preparation in the case of the laser ablation of powdered materials.The LA-ICP-MS signal dependence on the element content present in the material showed an improved correlation for Co, Ti, Ta and Nb of the matrix-unified samples compared to the non-matrix-unified pellets. In the case of W, the ICP-MS signal of matrix-unified pellets was influenced by the changes in the particle formation.     

Stereoselective hetero-Diels-Alder reaction of 3-nitro-2-trihalomethyl-2H-chromenes with 2,3-dihydrofuran and ethyl vinyl ether under solvent-free conditions

3-Nitro-2-trifluoro(trichloro)methyl-2H-chromenes undergo heterodiene cycloaddition to 2,3-dihydrofuran and ethyl vinyl ether under solvent-free conditions producing novel cyclic nitronates with high stereoselectivity and in good yields. 3,6-Dinitro-2-trifluoromethyl-2H-chromene reacts with two molecules of ethyl vinyl ether to give the tandem [4+2]/[3+2] cycloaddition adduct in 48% yield. The stereochemistry of the products was established based on 2D COSY, NOESY, HSQC, and HMBC experiments and an X-ray diffraction study.     

Electronic properties of N(5)-ethyl flavinium ion

We investigated the electronic properties of N(5)-ethyl flavinium perchlorate (Et-Fl(+)) and compared them to those of its parent compound, 3-methyllumiflavin (Fl). Absorption and fluorescence spectra of Fl and Et-Fl(+) exhibit similar spectral features, but the absorption energy of Et-Fl(+) is substantially lower than that of Fl. We calculated the absorption signatures of Fl and Et-Fl(+) using time-dependent density functional theory (TD-DFT) methods and found that the main absorption bands of Fl and Et-Fl(+) are (π,π*) transitions for the S(1) and S(3) excited states. Furthermore, calculations predict that the S(2) state has (n,π*) character. Using cyclic voltammetry and a simplistic consideration of the orbital energies, we compared the HOMO/LUMO energies of Fl and Et-Fl(+). We found that both HOMO and LUMO orbitals of Et-Fl(+) are stabilized relative to those in Fl, although the stabilization of the LUMO level was more pronounced. Visible and mid-IR pump-probe experiments demonstrate that Et-Fl(+) exhibits a shorter excited-state lifetime (590 ps) relative to that of Fl (several nanoseconds), possibly due to faster thermal deactivation in Et-Fl(+), as dictated by the energy gap law. Furthermore, we observed a fast (23-30 ps) S(2) → S(0) internal conversion in transient absorption spectra of both Fl and Et-Fl(+) in experiments that utilized pump excitations with higher energy.