Some alternating sums of Lucas numbers

We consider alternating sums of squares of odd and even terms of the Lucas sequence and alternating sums of their products. These alternating sums have nice representations as products of appropriate Fibonacci and Lucas numbers.     

Quantum Interference, Quantum Theory of Measurement, and (In)completeness of Quantum Mechanics

The new techniques and ideas in quantum interferometry with neutrons, photons, atoms, electrons, and Bose condensates that fluorished in the last two decades have influenced in a decisive way the thinking and the research in the foundations and interpretation of quantum mechanics. The controversies existing among different schools on the reality of matter waves of quantum theory, the postulates of quantum measurement theory, and the (in)completeness of quantum mechanics have to be approached now in a new way. Our argumentation follows the spirit of the Paris school.     

On Properties of Rectangular Hyperbolas

This paper improves an old theorem about a rectangular hyperbola : its centre lies on the pedal circle of any point on with respect to any triangle inscribed in . We also prove that an analogous result holds for Cevian circles. These results are used to obtain new characterisations of the Feuerbach, Jarabek, and Kiepert hyperbolas of a triangle.     

Compatible statistical interpretation of a wave packet

A compatible statistical interpretation of a wave packet is proposed. De Broglian probabilities which unite wave and particle features of quantons are evaluated for free wave packets and Jor a superposition of wave packets. The obtained expressions provide a very plausible and physically appealing explanation of coherence in apparently incoherent beams and of the characteristic modulation of the momentum distribution, found recently in neutron interferometry combined with spectral filtering. Certain conclusions about dualism and objectivity in quantum domain are also derived.     

Synthesis,structure and magnetism of a novel Cu4^ⅡTi5^Ⅳ heterometallic cluster

A nonanuclear Cu4^ⅡTi5^Ⅳheterometallic cluster,[Ti5 Cu4 O6(ba)(16)](1,Hba=benzoic acid)was synthesized in one-pot reaction under the solvothermal condition.The metallic skeleton 1 contains a Ti5 core constructed from two vertex-shared Ti3 triangles and four separated Cu atoms outside which are connected together byμ3-O^2- ions.Total 16 ba ligands adoptμ2-η^1:η^1 coordination mode to protect the overall hetero metallic co re.Due to the unique d-d transitions of Cu^Ⅱion,the reflectance spectrum of 1 displays broad and strong absorption towards visible light extending to the near-infrared region.Moreover,1 shows almost purely paramagnetic behavior with the presence of weak antiferromagnetic interactions at low temperatures.     

Interplay between the structural and magnetic probes in the elucidation of the structure of a novel 2D layered [V4O4(OH)2(O2CC6H4CO2)4]·DMF

The title compound has been synthesized under solvothermal conditions by reacting vanadium(V) oxytriisopropoxide with terephthalic acid in N,N-dimethylformamide. A combination of synchrotron powder diffraction, infrared spectroscopy, scanning and transmission electron microscopy, and thermal and chemical analysis elucidated the chemical, structural and microstructural features of a new 2D layered inorganic-organic framework. Due to the low-crystallinity of the final material, its crystal structure has been solved from synchrotron X-ray powder diffraction data using a direct space global optimization technique and subsequent constraint Rietveld refinement. [V(4)O(4)(OH)(2)(O(2)CC(6)H(4)CO(2))(4)]·DMF crystallizes in the monoclinic system (space group P2/m (No. 10)); cell parameters: a = 20.923(4) ?, b = 5.963(4) ?, c = 20.425(1) ?, β = 123.70(6)°, V = 2120.1(9) ?(3), Z = 2. The overall structure can be described as an array of parallel 2D layers running along [-101] direction, consisting of two types of vanadium oxidation states and coordination polyhedra: face-shared trigonal prisms (V(4+)) and distorted corner-shared square pyramids (V(5+)). Both configurations form independent parallel chains oriented along the 2-fold symmetry crystallographic b-axis mutually interlinked with terephthalate ligands in a monodentate mode perpendicular to it. The morphology of the compound exhibits long nanofibers, with the growth direction along the layered [-101] axis. The magnetic susceptibility measurements show that the magnetic properties of [V(4)O(4)(OH)(2)(O(2)CC(6)H(4)CO(2))(4)]·DMF can be described by a linear antiferromagnetic chain model, with the isotropic exchange interaction of J = -75 K between the nearest V(4+) neighbours of S = 1/2.     

Cis-trans isomeric and polymorphic effects on the magnetic properties of water-bridged copper coordination chains

The synthesis and magnetic characterization of vanillin-based Cu(II) mononuclear complexes of formula [Cu(van)(2)(H(2)O)(2)](H(2)O)(x) (van = vanillinate; x = 0, compound 1; x = 2, compounds 2 and 3) were performed. Despite the presence of very similar [Cu(van)(2)(H(2)O)(2)] moieties, the crystal structures exhibit distinct Cu···Cu contacts and display three different through-H-bond exchange-coupling pathways. As a result of the relative positions of the water molecules, the experimental (MAGSUS) exchange-coupling constants are dissimilar, i.e., J(1) = -3.0 cm(-1) (the data have been fitted to the Bleaney-Bowers equation considering a dimer; 2J = -6.0 cm(-1)), J(2) = -4.0 cm(-1) (the data have been fitted to the Bonner-Fischer equation for a chain of monomeric copper(II) units), whereas compound 3 is paramagnetic. Subsequently, the theoretical density functional theory (DFT) and wave function theory-based (DDCI) calculations were carried out to better understand the role of the water molecule as a mediator of the magnetic coupling. The use of localized orbitals allows one to elucidate the role of the H-bonds in generating exchange interactions. Since the exchange-coupling constants are strongly dependent on the mechanisms selectively introduced, the role of the H-bond is demonstrated.