Dynamic Supramolecular Polymers Based on Benzene‐1,3,5‐tricarboxamides: The Influence of Amide Connectivity on Aggregate Stability and Amplification of Chirality

N‐Centred benzene‐1,3,5‐tricarboxamides (N‐BTAs) composed of chiral and achiral alkyl substituents were synthesised and their solid‐state behaviour and self‐assembly in dilute alkane solutions were investigated. A combination of differential scanning calorimetry (DSC), polarisation optical microscopy (POM) and X‐ray diffraction revealed that the chiral N‐BTA derivatives with branched 3,7‐dimethyloctanoyl chains were liquid crystalline and the mesophase was assigned as Col_ho. In contrast, N‐BTA derivatives with linear tetradecanoyl or octanoyl chains lacked a mesophase and were obtained as crystalline compounds. Variable‐temperature infrared spectroscopy showed the presence of threefold, intermolecular hydrogen bonding between neighbouring molecules in the mesophase of the chiral N‐BTAs. In the crystalline state at room temperature a more complicated packing between the molecules was observed. Ultraviolet and circular dichroism spectroscopy on dilute solutions of N‐BTAs revealed a cooperative self‐assembly behaviour of the N‐BTA molecules into supramolecular polymers with preferred helicity when chiral alkyl chains were present. Both the sergeants‐and‐soldiers as well as the majority‐rules principles were operative in stacks of N‐BTAs. In fact, the self‐assembly of N‐BTAs resembles closely that of their carbonyl (C?O)‐centred counterparts, with the exception that aggregation is weaker and amplification of chirality is less pronounced. The differences in the self‐assembly of N‐ and C?O‐BTAs were analysed by density functional theory (DFT) calculations. These reveal a substantially lower interaction energy between the monomeric units in the supramolecular polymers of N‐BTAs. The lower interaction energy is due to the higher energy penalty for rotation around the Ph? NH bond compared to the Ph? CO bond and the diminished magnitude of dipole–dipole interactions. Finally, we observed that mixed stacks are formed in dilute solution when mixing N‐BTAs and C?O BTAs.     

Coordination properties of ionic liquid-mediated chromium(II) and copper(II) chlorides and their complexes with glucose

The structural and coordination properties of complexes formed upon the interaction of copper(II) and chromium(II) chlorides with dialkylimidazolium chloride (RMIm(+)Cl(-)) ionic liquids and glucose are studied by a combination of density functional theory (DFT) calculations and X-ray absorption spectroscopy (XAS). In the absence of the carbohydrate substrate, isolated mononuclear four-coordinated MeCl(4)(2-) species (Me = Cu, Cr) dominate in the ionic liquid solution. The organic part of the ionic liquid does not directly interact with the metal centers. The interactions between the RMIm(+) cations and the anionic metal chloride complexes are limited to hydrogen bonding with the basic Cl(-) ligands and the overall electrostatic stabilization of the anionic metal complexes. Exchange of Cl(-) ligands by a hydroxyl group of glucose is only favorable for CrCl(4)(2-). For Cu(2+) complexes, the formation of hydrogen bonded complexes between CuCl(4)(2-) and glucose is preferred. No preference for the coordination of metal chloride species to specific hydroxyl group of the carbohydrate is found. The formation of binuclear metal chloride complexes is also considered. The reactivity and selectivity patterns of the Lewis acid catalyzed reactions of glucose are discussed in the framework of the obtained results.     

Approximability results for the resource-constrained project scheduling problem with a single type of resources

In this paper, we consider the well-known resource-constrained project scheduling problem. We give some arguments that already a special case of this problem with a single type of resources is not approximable in polynomial time with an approximation ratio bounded by a constant. We prove that there exist instances for which the optimal makespan values for the non-preemptive and the preemptive problems have a ratio of O(logn), where n is the number of jobs. This means that there exist instances for which the lower bound of Mingozzi et al. has a bad relative error of O(logn), and the calculation of this bound is an NP-hard problem. In addition, we give a proof that there exists a type of instances for which known approximation algorithms with polynomial time complexity have an approximation ratio of at least equal to $O(\sqrt{n})$ , and known lower bounds have a relative error of at least equal to O(logn). This type of instances corresponds to the single machine parallel-batch scheduling problem 1|p?batch,b=∞|C _max.     

Resonances for Dirac operators on the half-line

We consider the 1D Dirac operator on the half-line with compactly supported potentials. We study resonances as the poles of scattering matrix or equivalently as the zeros of modified Fredholm determinant. We obtain the following properties of the resonances: (1) asymptotics of counting function, (2) estimates on the resonances and the forbidden domain.     

Schrödinger operators on periodic discrete graphs

We consider Schrödinger operators with periodic potentials on periodic discrete graphs. The spectrum of the Schrödinger operator consists of an absolutely continuous part (a union of a finite number of non-degenerated bands) plus a finite number of flat bands, i.e., eigenvalues of infinite multiplicity. We obtain estimates of the Lebesgue measure of the spectrum in terms of geometric parameters of the graph and show that they become identities for some class of graphs. Moreover, we obtain stability estimates and show the existence and positions of large number of flat bands for specific graphs. The proof is based on the Floquet theory and the precise representation of fiber Schrödinger operators, constructed in the paper.