Coloured Permutations Containing and Avoiding Certain Patterns

Let be the set of all coloured permutations on the symbols 1, 2, . . . , n with colours 1, 2, . . . , r, which is the analogous of the symmetric group when r = 1, and the hyperoctahedral group when r = 2. Let be a subset of d colours; we define to be the set of all coloured permutations . We prove that the number of -avoiding coloured permutations in . We then prove that for any , the number of coloured permutations in which avoid all patterns in except for and contain exactly once equals . Finally, for any , this number equals . These results generalize recent results due to Mansour, Mansour and West, and Simion.AMS Subject Classification: 05A05, 05A15.     

Glyoxal sample preparation for high-performance liquid chromatographic detection of 2,4-dinitro-phenylhydrazone derivative: Suppression of polymerization and mono-derivative formation by using methanol medium

Some difficulties on a sampling of gaseous glyoxal using DNPH-silica cartridge were discussed, and an alternative sampling procedure was proposed. When glyoxal was sampled using the cartridge, it partially formed mono-hydrazone with various degrees, whereas it was quantitatively converted into its bis-hydrazone when sample gas was directly bubbled into a DNPH acidic solution. Additionally, glyoxal polymerized on the inner wall of the system during trapping to the silica cartridge. We found that the polymerization of glyoxal was effectively suppressed by dissolving glyoxal into methanol under reduced pressure; glyoxal in methanol readily reacted with DNPH in a hydrochloric acid solution and gave bis-hydrazone derivative, without any interference of methanol and any formation of mono-hydrazone derivative. When glyoxal was sampled by the proposed method, 102% of recovery was obtained, whereas it was reduced to 92.7% when the trapped glyoxal was dissolved into methanol under atmosphere.     

Approximation by polynomials with bounded coefficients

Let θ be a real number satisfying 1<θ<2, and let A(θ) be the set of polynomials with coefficients in {0,1}, evaluated at θ. Using a result of Bugeaud, we prove by elementary methods that θ is a Pisot number when the set (A(θ)−A(θ)−A(θ)) is discrete; the problem whether Pisot numbers are the only numbers θ such that 0 is not a limit point of (A(θ)−A(θ)) is still unsolved. We also determine the three greatest limit points of the quantities , where C(θ) is the set of polynomials with coefficients in {−1,1}, evaluated at θ, and we find in particular infinitely many Perron numbers θ such that the sets C(θ) are discrete.     

On the genus distributions of wheels and of related graphs

We are concerned with families of graphs in which there is a single root-vertex ofunbounded valence, and in which, however, there is a uniform upper bound for the valences of all the other vertices. Using a result of Zagier, we obtain formulas and recursions for the genus distributions of several such families, including the wheel graphs. We show that the region distribution of a wheel graph is approximately proportional to the sequence of Stirling numbers of the first kind. Stahl has previously obtained such a result for embeddings in surfaces whose genus is relatively near to the maximum genus. Here, we generalize Stahl’s result to the entire genus distributions of wheels. Moreover, we derive the genus distributions for four other graph families that have some similarities to wheels.     

Combining ligand-based and structure-based drug design approaches to study the structure-activity relationships of a β-carboline derivative series

In this study, we investigated the structure-activity relationships of a series of β-carboline alkaloid derivatives using the 2D-QSAR and molecular docking, in order to identify the mode of interaction between β-carboline derivatives and the PLK1 kinase, and determine their key substituents responsible for the cytotoxic activity. The obtained QSAR models using multiple linear regression (MLR) and partial least squares (PLS) methods showed a high correlation between the experimental activity and the predicted one by PLS (R²_PLS?=?0.82, q²?=?0.72) and MLR (R²_MLR?=?0.82, q²?=?0.72). An external dataset was used to test the extrapolation power of the models which resulted in an R²_PLS (EV)?=?0.76; RMSE?=?0.39. The 2D-QSAR analysis reveals that lipophilicity plays an important role in the cytotoxic activity of this group of β-carboline derivatives. Indeed, the molecular docking study into the active site of the polo-like kinase (PLK1) revealed that the most active ligand 57 shows higher binding energy and interacts, especially by H-bonds and hydrophobic interactions, with the active site of the PLK1 kinase. Consequently, the results obtained from the 2D-QSAR and docking studies provided a useful tool to design new and potent β-carboline derivatives as cytotoxic agents.     

Kernel method and linear recurrence system

Based on the kernel method, we present systematic methods to solve equation systems on generating functions of two variables. Using these methods, we get the generating functions for the number of permutations which avoid 1234 and 12k(k-1)…3$12k(k-1)\dots 3$ and permutations which avoid 1243 and 12…k$12\dots k$.     

On Λ2-strong convergence of numerical sequences and Fourier series

We introduce the Λ²-strong convergence of numerical sequences and with it we generalize the concept of Λ-strong convergence of the results published by F. Móricz [2].