The Role of Hydrogen Bonds in Interactions between [PdCl4]2− Dianions in Crystal

[PdCl₄]²⁻ dianions are oriented within a crystal in such a way that a Cl of one unit approaches the Pd of another from directly above. Quantum calculations find this interaction to be highly repulsive with a large positive interaction energy. The placement of neutral ligands in their vicinity reduces the repulsion, but the interaction remains highly endothermic. When the ligands acquire a unit positive charge, the electrostatic component and the full interaction energy become quite negative, signalling an exothermic association. Raising the charge on these counterions to +2 has little further stabilizing effect, and in fact reduces the electrostatic attraction. The ability of the counterions to promote the interaction is attributed in part to the H-bonds which they form with both dianions, acting as a sort of glue.     

1H, 13C, 15N NMR and X-Ray Diffractometry in Structural Studies of Macrocyclic Lactams Containing Pyridine Moiety

Abstract

We examined complexing sites of the Pb²⁺ complex of the macrocyclic lactam 1 using ¹⁵N NMR and other spectroscopies and we have found that the amide groups undergo conformational changes to allow the complexation process to proceed via the pyridine nitrogen atom and carbonyl and ethereal oxygen atoms. X-Ray analysis of compound 1 was carried out successfully. Space group I4₁/a, a=28.332(4)Å, b=28.332(4)Å, c=10.7379(4)Å, Z=16, V=8619.3(18)ų, D_c=1.197gcm^?3, R1=0.0479 (based on 2510 reflections I>2[sgrave](I). It shows presence of intramolecular hydrogen bonds, which are broken during the complexation. Molecules form supramolecular tetragonal assemblies in the crystal, which form channels the walls of which are 7.42 Å apart.     

Estimating the domain of attraction of a stable equilibrium state for certain delayed differential systems of order two

Summary We discuss the influence exerted on the stability of an equilibrium state of a second order differential-difference system by the delays existing in the coupling terms of the two system equations. On certain hypotheses an asymptotic stability condition is determined which is independent of the magnitude of delays but dependent on the magnitude of coupling. A simple method for constructing regions belonging to the domain of attraction of the considered equilibrium state is developed. Several alternative constructions and applications are discussed in detail.

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Sommario Viene discussa l'influenza che sulla stabilità ai uno stato di equilibrio di un sistema differenziale alle differenze di secondo ordine banno i ritardi incorporati nei termini di accoppiamento fra le due equazioni del sistema. Si mostra come sotto opportune ipotesi si possa determinare una condizione di stabilità asintotica che non dipende dall'entità dei ritardi, mentre dipende da quella degli accoppiamenti, e si propone un metodo semplice e flessibile per costruire regioni contenute nel dominio di attrazione dello stato di equilibrio considerato. Si discutono in dettaglio numerose costruzioni alternative ed applicazioni.

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On leave of absence from CYFRONET, Institute of Nuclear Research, Swierk k/Otwocha, Poland.     

Graph coloring and Graham’s greatest common divisor problem

In this paper we introduce and study two graph coloring problems and relate them to some deep number-theoretic problems. For a fixed positive integer $k$ consider a graph $B_k$ whose vertex set is the set of all positive integers with two vertices $a,b$ joined by an edge whenever the two numbers $a∕gcd(a,b)$ and $b∕gcd(a,b)$ are both at most $k$. We conjecture that the chromatic number of every such graph $B_k$ is equal to $k$. This would generalize the greatest common divisor problem of Graham from 1970; in graph-theoretic terminology it states that the clique number of $B_k$ is equal to $k$. Our conjecture is connected to integer lattice tilings and partial Latin squares completions.     

Two pentadehydropeptides with different configurations of the ΔPhe residues

Comparison of the crystal structures of two pentadehydropeptides containing ΔPhe residues, namely (Z,Z)‐N‐(tert‐butoxycarbonyl)glycyl‐α,β‐phenylalanylglycyl‐α,β‐phenylalanylglycine (or Boc⁰–Gly¹–Δ^ZPhe²–Gly³–Δ^ZPhe⁴–Gly⁵–OH) methanol solvate, C₂₉H₃₃N₅O₈·CH₄O, (I), and (E,E)‐N‐(tert‐butoxycarbonyl)glycyl‐α,β‐phenylalanylglycyl‐α,β‐phenylalanylglycine (or Boc⁰–Gly¹–Δ^EPhe²–Gly³–Δ^EPhe⁴–Gly⁵–OH), C₂₉H₃₃N₅O₈, (II), indicates that the Δ^ZPhe residue is a more effective inducer of folded structures than the Δ^EPhe residue. The values of the torsion angles ϕ and ψ show the presence of two type‐III′β‐turns at the Δ^ZPhe residues and one type‐II β‐turn at the Δ^EPhe residue. All amino acids are linked trans to each other in both peptides. β‐Turns present in the peptides are stabilized by intramolecular 4→1 hydrogen bonds. Molecules in both structures form two‐dimensional hydrogen‐bond networks parallel to the (100) plane.     

Stabilization of solutions of dissipative Hamiltonian systems

We study the stabilization of solutions of damped Hamiltonian systems. We give sufficient conditions for convergence of these solutions, decay estimate and examples of applications.     

A Hamiltonian formulation of the Einstein-Cartan-Sciama-Kibble theory of gravity

We postulate the energy-momentum functionE for the ECSK theory of gravity and formulate the functional Hamiltonian equation in terms of the energy-momentum functionE and the symplectic 2-form . The system of partial differential equations which follows from the functional Hamilton equation is equivalent to the system of variational equations of the ECSK theory. The Hamiltonian method gives rise to a natural division of these equations into 10 constraint equations and the set of dynamical equations. We discuss the geometric sense of the constraint equations and their relations to the initial value problem.     

Complete 1H and 13C signal assignment of prenol‐10 with 3D NMR spectroscopy

The complete assignment of ¹H and ¹³C chemical shifts of natural abundance prenol‐10 is reported for the first time. It was achieved using 3D NMR experiments, which were based on random sampling of the evolution time space followed by multidimensional Fourier transform. This approach makes it possible to acquire 3D NMR spectra in a reasonable time and preserves high resolution in indirectly detected dimensions. It is shown that the interpretation of 3D COSY–HMBC and 3D TOCSY–HSQC spectra is crucial in the structural analysis of prenol‐10. Copyright © 2009 John Wiley & Sons, Ltd.