Multiple critical points for non-differentiable parametrized functionals and applications to differential inclusions

In this paper we deal with a class of non-differentiable functionals defined on a real reflexive Banach space X and depending on a real parameter of the form ${\mathcal{E}_\lambda(u)=L(u)-(J_1\circ T)(u)-\lambda (J_2\circ S)(u)}$ , where ${L:X \rightarrow \mathbb R}$ is a sequentially weakly lower semicontinuous C ¹ functional, ${J_1:Y\rightarrow\mathbb R, J_2:Z\rightarrow \mathbb R}$ (Y, Z Banach spaces) are two locally Lipschitz functionals, T : X → Y, S : X → Z are linear and compact operators and λ > 0 is a real parameter. We prove that this kind of functionals posses at least three nonsmooth critical points for each λ > 0 and there exists λ* > 0 such that the functional ${\mathcal{E}_{\lambda^\ast}}$ possesses at least four nonsmooth critical points. As an application, we study a nonhomogeneous differential inclusion involving the p(x)-Laplace operator whose weak solutions are exactly the nonsmooth critical points of some “energy functional” which satisfies the conditions required in our main result.     

Disjoint Compatible Geometric Matchings

We prove that for every set of n pairwise disjoint line segments in the plane in general position, where n is even, there is another set of n segments such that the 2n segments form pairwise disjoint simple polygons in the plane. This settles in the affirmative the Disjoint Compatible Matching Conjecture by Aichholzer et al. (Comput. Geom. 42:617–626, 2009). The key tool in our proof is a novel subdivision of the free space around n disjoint line segments into at most n+1 convex cells such that the dual graph of the subdivision contains two edge-disjoint spanning trees.     

Infinitely many reducts of homogeneous structures

It is shown that the countably infinite dimensional pointed vector space (the vector space equipped with a constant) over a finite field has infinitely many first order definable reducts. This implies that the countable homogeneous Boolean-algebra has infinitely many reducts.     

Theoretical study on the structure and stability of some unusual boron-nitrogen helices

Ab initio self-consistent field molecular orbital and density functional theory calculations have been performed on a series of helical structures comprised of boron-nitrogen analogues of extended helicenes, with helically arranged N fused benzene rings, and alternating N benzene units fused to N - 1 cyclobutadiene rings as reference structures. The electronic structure and stability of boron-nitrogen analogues of angular [N]helicenes, [N]phenylenes (N = 5, 6, 7, 12), and [N]methylenylnaphthalenes (N = 6) were investigated at the HF/6-31G(d) and the B3LYP/6-31G(d) levels of theory. The presence of an even number N of rings in the boron-nitrogen [N]helicenes leads to the possibility of angular isomers. Electron density contours were calculated in order to interpret the existing bonding patterns. These structures may provide supramolecular building blocks and macromolecular "springs" with unusual electronic properties.     

Impact of ligand protonation on virtual screening against beta-secretase (BACE1)

Structure-based virtual screens were carried out against beta-secretase (BACE1) to investigate the impact of ligand protonation on screening efficacy. A comparative evaluation of the performance and its dependence on ligand protonation states docking by Surflex, eHiTS, GOLD, and FlexX-Pharm was performed. Virtual screening performed by FlexX-Pharm (EF(1%)=69) and Surflex (EF(1%)=58) provided the best efficiency. Screening protocols by FlexX-Pharm and GOLD were affected by ligand protonation, while performance of Surflex did not depend on ligand protonation.     

Stereoselective Reduction of `Capsanthol‐3′‐ones' (=3,6′‐Dihydroxy‐β,κ‐caroten‐3′‐ones) by Complex Hydrides

The (3R,5′R,6′R)‐ and (3R,5′R,6′S)‐capsanthol‐3′‐one (=3,6′‐dihydroxy‐β,κ‐caroten‐3′‐one; 4 and 5 , resp.) were reduced by different complex metal hydrides containing organic ligands. The ratio of the thus obtained diastereoisomeric (3′S)‐capsanthols 2 and 3 or (3′R)‐capsanthols 6 and 7 , respectively, was investigated. Four complex hydrides showed remarkable stereoselectivity and produced the (3′R,6′S)‐capsanthol ( 6 ) in 80 – 100% (see Table 1). The starting materials and the products were characterized by UV/VIS, CD, ¹H‐ and ¹³C‐NMR, and mass spectra.