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.

     

Specific ion effects at protein surfaces: a molecular dynamics study of bovine pancreatic trypsin inhibitor and horseradish peroxidase in selected salt solutions

The distribution of sodium, choline, sulfate, and chloride ions around two proteins, horseradish peroxidase (HRP) and bovine pancreatic trypsin inhibitor (BPTI), is investigated by means of molecular dynamics simulations with the aim to elucidate ion adsorption at the protein surface. Although the two proteins under investigation are very different from each other, the ion distributions around them are remarkably similar. Sulfate is always strongly attached to the proteins, choline shows a significant, but unspecific, propensity for the protein surfaces, and sodium ions have a weak surface affinity, while chloride has virtually no preference for the protein surface. In mixtures of all four ion species in protein solutions, the resulting distributions are almost a superposition of the distributions of sodium sulfate and choline chloride, except that sodium partially replaces choline close to the proteins. The present simulations support a picture of ions interacting with individual ionic and polar amino acid groups rather than with an averaged protein surface. The results thus show how subtle the so-called Hofmeister and electroselectivity effects are in salt solution of proteins, making all simplified interaction models questionable.     

How the stabilization of INK4 tumor suppressor 3D structure evaluated by quantum chemical and molecular mechanics calculations corresponds well with experimental results: interplay of association enthalpy, entropy, and solvation effects

The folding free energy of the INK4c tumor suppressor core, consisting of 10 helices, was determined as the sum of gas-phase interaction enthalpy, gas-phase interaction entropy, and dehydration and hydration free energy. The interaction energy and the hydration free energy were determined using the nonempirical density functional theory (DFT) method, augmented by a dispersion-energy correction term, the semiempirical density-functional tight-binding method covering the dispersion energy, and the density functional theory/conductor-like screening model (DFT/COSMO) procedure, whereas the interaction entropy was calculated with the empirical Cornell et al. force field. Alternatively, all contributions were evaluated consistently using empirical methods. All the values of the interaction energy of helix pairs are stabilizing, and the dominant stabilizing terms stem from the London dispersion energy and, in the case of charged systems, the electrostatic energy. The stabilization energy of the core, determined as the difference of the energy of the core and 10 separate helices, amounts to approximately 450 kcal/mol. Systematically, the difference in the hydration free energy of a helix pair and its separate components is smaller in magnitude than the interaction energy, and it is negative for some pairs while positive for others. The average total free energy of a core formation amounts to -29.6 kcal/mol (yielded by scaled quantum-chemical methods) and +13.9 kcal/mol (resulting from empirical methods). These values are considerably smaller than their single components, which are dominated by the interaction energy. The computationally predicted interval encloses the experimental value of the folding free energy (-2.8 kcal/mol).     

Disaster preparedness in humanitarian logistics: A collaborative approach for resource management in floods

The logistical deployment of resources to provide relief to disaster victims and the appropriate planning of these activities are critical to reduce the suffering caused. Disaster management attracts many organisations working alongside each other and sharing resources to cope with an emergency. Consequently, successful operations rely heavily on the collaboration of different organisations. Despite this, there is little research considering the appropriate management of resources from multiple organisations, and none optimising the number of actors required to avoid shortages or convergence.This research introduces a disaster preparedness system based on a combination of multi-objective optimisation and geographical information systems to aid multi-organisational decision-making. A cartographic model is used to avoid the selection of floodable facilities, informing a bi-objective optimisation model used to determine the location of emergency facilities, stock prepositioning, resource allocation and relief distribution, along with the number of actors required to perform these activities.The real conditions of the flood of 2013 in Acapulco, Mexico, provided evidence of the inability of any single organisation to cope with the situation independently. Moreover, data collected showed the unavailability of enough resources to manage a disaster of that magnitude at the time. The results highlighted that the number of government organisations deployed to handle the situation was excessive, leading to high cost without achieving the best possible level of satisfaction. The system proposed showed the potential to achieve better performance in terms of cost and level of service than the approach currently employed by the authorities.     

Two-dimensional and serial column reversed-phase separation of phenolic antioxidants on octadecyl-, polyethyleneglycol-, and pentafluorophenylpropyl-silica columns

The separation selectivity of octadecyl-silica (C18) and of bonded pentafluorophenylpropyl-silica (F5) and PEG-silica columns was compared for natural phenolic antioxidants. The separation selectivities for phenolic antioxidants on C18 and F5 columns are strongly correlated, but low selectivity correlation indicating strong differences in the retention mechanism was observed between the C18 and PEG columns. Hence, the combination of a C18 and a PEG column is useful for separation of phenolic antioxidants that are not fully separated on single columns. Two-dimensional comprehensive liquid chromatography using a short PEG-silica column in the first dimension and a conventional C18-silica in the second dimension has the advantage of on-column focusing of the fractions transferred onto the C18 column in the second dimension, as a weaker mobile phase is used in the first dimension than in the second dimension. However, a stop-flow set-up in the first dimension system is necessary after the transfer of each fraction to the second dimension. Peak capacity is considerably larger but the separation time is much longer than with serially coupled PEG and C18 columns, which were employed for separation of beer and hop extract samples in connection with coulometric detection.     

Diastereotopos-differentiating C-H activation reactions at methylene groups

The activation of C-H bonds has become a widely used method which allows for the direct transformation of C-H bonds into synthetically more valuable C-C and C-X bonds in a selective manner. This critical review aims to summarize and to highlight a specific subgroup of these transformations, C-H activation reactions of chiral substrates bearing diastereotopic hydrogen atoms at methylene groups (95 references).     

Surface self-diffusion of organic molecules adsorbed in porous silicon

The pulsed field gradient nuclear magnetic resonance method has been employed to probe self-diffusion of organic guest molecules adsorbed in porous silicon with a 3.6 nm pore size. The molecular self-diffusion coefficient and intrapore adsorption were simultaneously measured as a function of the external vapor pressure. The latter was varied in a broad range to provide pore loading from less than monolayer surface coverage to full pore saturation. The measured diffusivities are found to be well-correlated with the adsorption isotherms. At low molecular concentrations in the pores, corresponding to surface coverages of less than one monolayer, the self-diffusion coefficient strongly increases with increasing concentration. This observation is attributed to the occurrence of activated diffusion on a heterogeneous surface. Additional experiments in a broad temperature range and using binary mixtures confirm this hypothesis.     

Effect of glass composition on activation energy of viscosity in glass-melting-temperature range

In the high-temperature range, where the viscosity (η) of molten glass is < 10³ Pa s, the activation energy (B) is virtually independent of temperature (T). Moreover, the coefficient A in the Arrhenius relationship, ln(η) = A + B/T, is nearly independent of melt composition. Hence, the viscosity–composition relationship for η < 10³ Pa s is defined by B as a function of composition. Using a database encompassing over 1300 compositions of high-level waste glasses with nearly 7000 viscosity data, we developed mathematical models for B(x), where x is the composition vector in terms of mass fractions of components. In this paper, we present 13 versions of B(x) as first- and second-order polynomials with coefficients for 15 to 39 components, including Others, a component that sums constituents having little effect on viscosity.     

A practical asymmetric conjugate addition to cyclic enones with chiral bifunctional Ru amido catalysts

A practical asymmetric C-C bond formation to synthetically useful β-chiral cyclic ketones (>99% ee) using bifunctional chiral amido Ru catalysts under an S/C = 1000, the highest ratio achieved so far in the literature for this class of reactions, is described. The catalytic reactivity decreases in the order of Ru(Msdpen)(hmb) > Ru(Pfbsdpen)(hmb) > Ru(Tsdpen)(hmb) > Ru(PMsdpen)(hmb), where Ru(Pfbsdpen)(hmb) is a newly developed chiral bifunctional catalyst. Complex Ru(Msdpen)(hmb) was identified as the best in terms of reactivity and enantioface selectivity, whereas Ru(PMsdpen)(hmb) gave unsatisfactory results. An importance of the NH proton in the bifunctional catalyst for determining the enantioselectivity has been experimentally demonstrated. Valuable information for the reaction mechanism was accumulated.