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.     

Speeding up branch and bound algorithms for solving the maximum clique problem

In this paper we consider two branch and bound algorithms for the maximum clique problem which demonstrate the best performance on DIMACS instances among the existing methods. These algorithms are MCS algorithm by Tomita et al. (2010) and MAXSAT algorithm by Li and Quan (2010a, b). We suggest a general approach which allows us to speed up considerably these branch and bound algorithms on hard instances. The idea is to apply a powerful heuristic for obtaining an initial solution of high quality. This solution is then used to prune branches in the main branch and bound algorithm. For this purpose we apply ILS heuristic by Andrade et al. (J Heuristics 18(4):525–547, 2012). The best results are obtained for p_hat1000-3 instance and gen instances with up to 11,000 times speedup.     

Little-Preemptive Scheduling on Unrelated Processors

We consider the preemptive scheduling of n independent jobs on m unrelated machines to minimize the makespan. Preemptive schedules with at most 2m–3 preemptions are built, which are optimal when the maximal job processing time is no more than the optimal schedule makespan. We further restrict the maximal job processing time and obtain optimal schedules with at most m–1 preemptions. This is better than the earlier known best bound of 4m ²–5m+2 on the total number of preemptions. Without the restriction on the maximal job processing time, our (2m–3)-preemptive schedules have a makespan which is no more than either of the following two magnitudes: (a) the maximum between the longest job processing time and the optimal preemptive makespan, and (b) the optimal nonpreemptive makespan. Our (m–1)-preemptive schedules might be at most twice worse than an optimal one.     

Upper semicontinuity of automorphism groups

The second author was partially supported by NSF#DMS-9101826     

Crystal Structures of Hydrazinium(II) Salts of [SbF6]– and [Sb2F11]–

N₂H₆(Sb₂F₁₁)₂ was synthesized by the reaction of N₂H₆F₂ with excess of SbF₅ in anhydrous hydrogen fluoride (aHF). It crystallizes in the triclinic space group P$\bar{1}$ (No. 2) with a = 6.6467(3) Å, b = 8.3039(4) Å, c = 8.3600(5) Å, α = 76.394(5) ^o, β = 70.161(5) ^o, γ = 70.797(5) ^o, V = 405.90(4) ų at 150 K, Z = 2. When it is redissolved in aHF, it solvolysis with the release of SbF₅ yielding N₂H₆(SbF₆)₂ which crystallizes in the monoclinic C2/c space group (No. 15) with a = 7.3805(3) Å, b = 12.3248(5) Å, c = 10.4992(4) Å, β = 92.218(4) ^o, V = 954.33(7) ų at 150 K, and Z = 8. No other phases were observed in crystallization products when different molar ratios of N₂H₆F₂/SbF₅ (1:1,2:3,1:3) in aHF were used as starting materials.     

Non-Pincer-Type Manganese Complexes as Efficient Catalysts for the Hydrogenation of Esters

Catalytic hydrogenation of carboxylic acid esters is essential for the green production of pharmaceuticals, fragrances, and fine chemicals. Herein, we report the efficient hydrogenation of esters with manganese catalysts based on simple bidentate aminophosphine ligands. Monoligated Mn PN complexes are particularly active for the conversion of esters into the corresponding alcohols at Mn concentrations as low as 0.2 mol % in the presence of sub-stoichiometric amounts of KO^tBu base.