On generalized Nash equilibrium problems with linear coupling constraints and mixed-integer variables

ABSTRACT

We define and discuss different enumerative methods to compute solutions of generalized Nash equilibrium problems with linear coupling constraints and mixed-integer variables. We propose both branch-and-bound methods based on merit functions for the mixed-integer game, and branch-and-prune methods that exploit the concept of dominance to make effective cuts. We show that under mild assumptions the equilibrium set of the game is finite and we define an enumerative method to compute the whole of it. We show that our branch-and-prune method can be suitably modified in order to make a general equilibrium selection over the solution set of the mixed-integer game. We define an application in economics that can be modelled as a Nash game with linear coupling constraints and mixed-integer variables, and we adapt the branch-and-prune method to efficiently solve it.     

Selective MAO-B inhibitors: a lesson from natural products

Monoamine oxidases (MAOs) are mitochondrial bound enzymes, which catalyze the oxidative deamination of monoamine neurotransmitters. Inside the brain, MAOs are present in two isoforms: MAO-A and MAO-B. The activity of MAO-B is generally higher in patients affected by neurodegenerative diseases like Alzheimer’s and Parkinson’s. Therefore, the search for potent and selective MAO-B inhibitors is still a challenge for medicinal chemists. Nature has always been a source of inspiration for the discovery of new lead compounds. Moreover, natural medicine is a major component in all traditional medicine systems. In this review, we present the latest discoveries in the search for selective MAO-B inhibitors from natural sources. For clarity, compounds have been classified on the basis of structural analogy or source: flavonoids, xanthones, tannins, proanthocyanidins, iridoid glucosides, curcumin, alkaloids, cannabinoids, and natural sources extracts. MAO inhibition values reported in the text are not always consistent due to the high variability of MAO sources (bovine, pig, rat brain or liver, and human) and to the heterogeneity of the experimental protocols used.     

Efficient implementation of the analytic second derivatives of Hartree–Fock and hybrid DFT energies: a detailed analysis of different approximations

In this paper, various implementations of the analytic Hartree–Fock and hybrid density functional energy second derivatives are studied. An approximation-free four-centre implementation is presented, and its accuracy is rigorously analysed in terms of self-consistent field (SCF), coupled-perturbed SCF (CP-SCF) convergence and prescreening criteria. The CP-SCF residual norm convergence threshold turns out to be the most important of these. Final choices of convergence thresholds are made such that an accuracy of the vibrational frequencies of better than 5 cm^?1 compared to the numerical noise-free results is obtained, even for the highly sensitive low frequencies (<100–200 cm^?1). The effects of the choice of numerical grid for density functional exchange–correlation integrations are studied and various weight derivative schemes are analysed in detail. In the second step of the work, approximations are introduced in order to speed up the computation without compromising its accuracy. To this end, the accuracy and efficiency of the resolution of identity approximation for the Coulomb terms and the semi-numerical chain of spheres approximation to the exchange terms are carefully analysed. It is shown that the largest performance improvements are realised if either Hartree–Fock exchange is absent (pure density functionals) and otherwise, if the exchange terms in the CP-SCF step of the calculation are approximated by the COSX method in conjunction with a small integration grid. Default values for all the involved truncation parameters are suggested. For vancomycine (176 atoms and 3593 basis functions), the RIJCOSX Hessian calculation with the B3LYP functional and the def2-TZVP basis set takes ～3 days using 16 Intel® Xeon® 2.60GHz processors with the COSX algorithm having a net parallelisation scaling of 11.9 which is at least ～20 times faster than the calculation without the RIJCOSX approximation.     

Hexamethylphosphoramide Compounds: II. The Structure of a Complex Containing Two Different Coordination Polyhedra for Two Independent Neodymium(III) Ions

Abstract

The crystal structure of the complex of formula {|Nd(NCS)₃ (HMPA)₃‖ |Nd(NCS)₃ (HMPA)₄|} was determined by three-dimensional X-ray diffraction methods and refined anisotropically to a R=0.040. The compound crystallizes in the trigonal system, space group R3 (No 146), with a=19.947(3), b=19.947(3), c=20.106(3) Å, α=β=90, γ=120°, V=6928(4) ų, M=1891.4, Z=3, D_c=1.360 g cm^?3, λ(MoKα)=0.71073 Å, μ=1.4 cm^?1, F (000)=2922.01. There are two independent Nd³⁺ ions located in the three fold axis. One of them, located at the origin (000) is coordinated to nitrogen atoms of three symmetry related NCS^? anions which are below the (x y) plane (negative z) and also to the oxygen atoms of three symmetry related HMPA groups above that plane. The coordination polyhedron is a slightly distorted octahedron. The other Nd³⁺ ion is located at (00,0.5025(1)). It is coordinated to the nitrogen atoms of three symmetry related NCS^? anions above the (xy 1/2) plane and to oxygen atoms of three symmetry related HMPA groups below that plane. Another HMPA group has the O and P atoms located on the three-fold axis, above the (xy 1/2) plane. The coordination number is in this case 7 and the polyhedron is a capped trigonal antiprism. The mean distances are: Nd-N=2.44 Å, Nd-0=2.35 Å and Nd-N=2.52 Å, Nd-0=2.36 Å for the octahedron and antiprism configurations, respectively. (CNPq, FAPESP, FINEP)     

Site-Specific Differences in Proteasome-Dependent Degradation of Monoubiquitinated α-Synuclein

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