Ballot matrix as Catalan matrix power and related identities

We use an analytical approach to find the kth power of the Catalan matrix. Precisely, it is proven that the power of the Catalan matrix is a lower triangular Toeplitz matrix which contains the well-known ballot numbers. A result from [H. S. Wilf, Generatingfunctionology, Academic Press, New York, 1990, Free download available from http://www.math.upenn.edu/~wilf/Downld.html.], related to the generating function for Catalan numbers, is extended to the negative integers. Three interesting representations for Catalan numbers by means of the binomial coefficients and the hypergeometric functions are obtained using relations between Catalan matrix powers.     

On the divisor function and the Riemann zeta-function in short intervals

We obtain, for T ^ε ≤U=U(T)≤T ^1/2−ε , asymptotic formulas for

On the ordering of trees by the Laplacian coefficients

We generalize the results from [X.-D. Zhang, X.-P. Lv, Y.-H. Chen, Ordering trees by the Laplacian coefficients, Linear Algebra Appl. (2009), doi:10.1016/j.laa.2009.04.018] on the partial ordering of trees with given diameter. For two n-vertex trees T₁ and T₂, if c_k(T₁)c_k(T₂) holds for all Laplacian coefficients c_k,k=0,1,…,n, we say that T₁ is dominated by T₂ and write T_1cT₂. We proved that among n vertex trees with fixed diameter d, the caterpillar C_n,d has minimal Laplacian coefficients c_k,k=0,1,…,n. The number of incomparable pairs of trees on 18 vertices is presented, as well as infinite families of examples for two other partial orderings of trees, recently proposed by Mohar. For every integer n, we construct a chain of n-vertex trees of length , such that T₀S_n,T_mP_n and T_icT_i+1 for all i=0,1,…,m-1. In addition, the characterization of the partial ordering of starlike trees is established by the majorization inequalities of the pendent path lengths. We determine the relations among the extremal trees with fixed maximum degree, and with perfect matching and further support the Laplacian coefficients as a measure of branching.     

Extremal 1-codes in distance-regular graphs of diameter 3

We study 1-codes in distance-regular graphs of diameter 3 that achieve three different bounds. We show that the intersection array of a distance-regular graph containing such a code has the form $${\{a(p+1), cp, a+1; 1, c, a p\}\quad{\rm or}\quad\{a(p+1), (a+1)p,c; 1, c, a p\}}$$ for c =? c ₂,?a =? a ₃ and ${p = p_{33}^3}$ . These two families contain 10?+?15 known feasible intersection arrays out of which four are uniquely realized by the Sylvester graph, the Hamming graph H(3, 3), the Doro graph and the Johnson graph J(9, 3), but not all members of these two families are feasible. We construct four new one-parameter infinite subfamilies of feasible intersection arrays, two of which have a nontrivial vanishing Krein parameter: $${\{(2r^2-1)(2r+1), 4r(r^2-1), 2r^2; 1, 2(r^2-1), r(4r^2-2)\}}$$ and $${\{2r^2(2r+1), (2r-1)(2r^2+r+1), 2r^2; 1, 2r^2, r(4r^2-1)\}}$$ for r > 1 (the second family actually generalizes to a two-parameter family with the same property). Using this information we calculate some triple intersection numbers for these two families to show that they must contain the desired code. Finally, we use some additional combinatorial arguments to prove nonexistence of distance-regular graphs with such intersection arrays.     

An Information-Theoretic Bound on p-Values for Detecting Communities Shared between Weighted Labeled Graphs

Extraction of subsets of highly connected nodes (“communities” or modules) is a standard step in the analysis of complex social and biological networks. We here consider the problem of finding a relatively small set of nodes in two labeled weighted graphs that is highly connected in both. While many scoring functions and algorithms tackle the problem, the typically high computational cost of permutation testing required to establish the p-value for the observed pattern presents a major practical obstacle. To address this problem, we here extend the recently proposed CTD (“Connect the Dots”) approach to establish information-theoretic upper bounds on the p-values and lower bounds on the size and connectedness of communities that are detectable. This is an innovation on the applicability of CTD, broadening its use to pairs of graphs.     

Antifungal Activity of Lavandula angustifolia Essential Oil against Candida albicans: Time-Kill Study on Pediatric Sputum Isolates

The aim of our study was to determine the susceptibility of 15 Candida albicans sputum isolates on fluconazole and caspofungin, as well as the antifungal potential of Lavandula angustifolia essential oil (LAEO). The commercial LAEO was analyzed using gas chromatography-mass spectrometry. The antifungal activity was evaluated using EUCAST protocol. A killing assay was performed to evaluate kinetics of 2% LAEO within 30 min treatment. LAEO with major constituents’ linalool (33.4%) and linalyl acetate (30.5%) effective inhibited grows of C. albicans in concentration range 0.5–2%. Fluconazole activity was noted in 67% of the isolates with MICs in range 0.06–1 µg/mL. Surprisingly, 40% of isolates were non-wild-type (non-WT), while MICs for WT ranged between 0.125–0.25 µg/mL. There were no significant differences in the LAEO MICs among fluconazole-resistant and fluconazole-susceptible sputum strains (p = 0.31) and neither among caspofungin non-WT and WT isolates (p = 0.79). The 2% LAEO rapidly achieved 50% growth reduction in all tested strains between 0.2 and 3.5 min. Within 30 min, the same LAEO concentration exhibited a 99.9% reduction in 27% isolates. This study demonstrated that 2% solution of LAEO showed a significant antifungal activity which is equally effective against fluconazole and caspofungin susceptible and less-susceptible strains.     

A detailed experimental and computational study of Cd complexes with pyridyl-based thiazolyl hydrazones

Interest in Cd complexes has been growing in recent years. Cd complexes are considered a potential solution in the search for novel antibiotics that can fight antimicrobial resistance. In addition, Cd complexes draw attention to material chemistry. The main objective of this work was to prepare the first Cd(II) complexes with anionic forms of pyridine-based thiazolyl hydrazone (THs) ligands HLS² [(E)-4-(4-methoxyphenyl)-2-(2-[pyridine-2-ylmethylene]hydrazinyl)thiazole] and HLS³ [(E)-2-(2-[pyridine-2-ylmethylene]hydrazinyl)-4-(p-tolyl)thiazole] and perform their structural and spectroscopic characterization, as well as stability in solution and upon heating. Studies related to their biological activities and possible electrochromic applications are also being conducted. Complexes [Cd(HLS²)₂] ( 1 ) and [Cd(HLS³)₂] ( 2 ) have been characterized by a single-crystal X-ray diffraction and computational analysis of intermolecular interactions responsible for their solid-state structures was performed. Thermal stability of 1 and 2 in the solid-state was analyzed by TGA/MS, where as their solution stability was determined by the spectrophotometric titration method. Electrochemical and in situ UV–Vis spectroelectrochemical analyses of 1 and 2 were carried out to determine redox mechanisms and the influence of the substituents and electrolytes on their redox responses. The antioxidant capacity of both complexes was tested in antioxidant assays, while their antimicrobial activity was tested against five Gram-positive and four Gram-negative bacteria, as well as against three fungi. The obtained results indicate their potent antioxidant capacity. The antimicrobial activity of investigated compounds on almost all tested bacterial strains was stronger than that of the standard antibiotic erythromycin. The results of docking studies indicate that the minor groove DNA is the possible biological target of 1 and 2 .     

Galvanic displacement of Co with Rh boosts hydrogen and oxygen evolution reactions in alkaline media

Journal of Solid State Electrochemistry - The growing energy crisis put an emphasis on the development of novel efficient energy conversion and storage systems. Here we show that surface...