Coupled intervals in the discrete calculus of variations: necessity and sufficiency

In this work we study nonnegativity and positivity of a discrete quadratic functional with separately varying endpoints. We introduce a notion of an interval coupled with 0, and hence, extend the notion of conjugate interval to 0 from the case of fixed to variable endpoint(s). We show that the nonnegativity of the discrete quadratic functional is equivalent to each of the following conditions: The nonexistence of intervals coupled with 0, the existence of a solution to Riccati matrix equation and its boundary conditions. Natural strengthening of each of these conditions yields a characterization of the positivity of the discrete quadratic functional. Since the quadratic functional under consideration could be a second variation of a discrete calculus of variations problem with varying endpoints, we apply our results to obtain necessary and sufficient optimality conditions for such problems. This paper generalizes our recent work in [R. Hilscher, V. Zeidan, Comput. Math. Appl., to appear], where the right endpoint is fixed.     

Electronic structure and vibrational spectra of cis-diammine(orotato)platinum(II), a potential cisplatin analogue: DFT and experimental study

Orotic acid (vitamin B₁₃) is a key intermediate in biosynthesis of the pyrimidine nucleotides in living organisms, moreover, it may serve as the biological carrier for some metal ions. cis-Diammine(orotato)platinum(II), cis-[Pt(C₅H₂N₂O₄)(NH₃)₂] can be considered as a new potential cisplatin analogue. The FT-Raman and FT-IR spectra of the title complex are reported, for the first time. The molecular structure, vibrational frequencies, and the theoretical infrared and Raman intensities have been calculated by the density functional mPW1PW91 method. The detailed vibrational assignment has been made on the basis of the calculated potential energy distribution. The theoretically predicted IR and Raman spectra show very good agreement with experiment. Natural bond orbital (NBO) analyses were performed for cisplatin, carboplatin and the title complex. The results provided new data on the nature of platinum–ligand bonding in these compounds. Strong intramolecular hydrogen bond between the orotate ligand and the coordinated ammonia group stabilizes the structure of the platinum(II) complex. Thus, it is suggested that the orotate ligand in the title complex is more inert to the substitution reactions than the chloride ligands in cisplatin.     

Pressure-temperature phase diagrams of smectogenic 4'-alkyl-4-cyanobiphenyls (9CB, 10CB, 11CB and 12CB)

The pressure-temperature ( p - T ) phase diagrams for four smectogenic members of the 4'-alkyl-4-cyanobiphenyl homologous series ( n CB, n =9, 10, 11 and 12) over the temperature range 320-410 K and pressure range 0.1-300 MPa (3 kbar) were constructed using DTA. At 1 atm 9CB exhibits nematic and smectic A d phases, while the other members show only the smectic A d phase. However, at elevated pressures the clearing line splits in the case of 10CB and 11CB which indicates the induction of a nematic phase. It was found that the triple point, where the isotropic, nematic and smectic phases coexist, is strongly shifted to higher pressures with increasing chain length. This was interpreted as being caused by a loss of the rod-like shape of the molecules containing longer alkyl tails which explore a range of conformations. The slope of the clearing line, d T /d p , depends strongly on the length of the alkyl chain for the n CB series, but does not show a step-wise change between the nematogenic and smectogenic members.     

Partitions of a Polytope and Mappings of a Point Set to Facets

Three theorems of this paper generalize previous results of the author on conjectures of A. Bezdek and V.V. Proizvolov. They show the existence of mappings from a given point set to the set of facets of a given polytope that satistfy some special conditions. Developing the same technique, some results on convex polytope partitions are presented, two of them dealing with partitions with prescribed measures of parts. Then we prove a corollary on the existence of a possibly nonconvex polytope with a given set of vertices, containing given points in its interior. We also consider problems of the following type: find an assignment of vectors from a given set to the parts of a given convex partition of ℝⁿ so that the shifts of the parts by their corresponding vectors either do not intersect by interior points or cover ℝⁿ     

The high‐pressure form of cadmium vanadate,CdV2O6

The crystal structure of a new high‐pressure modification of cadmium divanadium hexaoxide, CdV₂O₆, was refined from X‐ray single‐crystal data. It contains zigzag chains of edge‐sharing VO₆ octahedra. Octahedra in adjacent chains share corners and form corrugated layers. Octahedrally coordinated Cd atoms, which lie on twofold axes, are situated between the layers. The columbite‐like structure results in a strong distortion of the CdO₆ octahedra which may be stabilized only at high pressure.     

Aluminum(III) complexes containing O,O chelating ligand

The stoichiometric reactions of trimethylaluminum with 2,6‐(MeOCH₂)₂C₆H₃OH (LH) revealed compounds L₃Al ( 1 ) and L₂AlMe ( 2 ). On the other hand reaction of 1 equiv. of LH with trimethylaluminum did not lead to the formation of complex LAlMe₂ ( 3 ), rather 2 together with Me₃Al were observed as a result of a disproportionation of 3 . Compounds 1 and 2 were characterized by elemental analysis, ¹H and ¹³C NMR spectroscopy and in the case of 1 by X‐ray diffraction. Derivative 2 underwent transmetalation with Ph₃SnOH, giving LSnPh₃ ( 4 ) as the result of a migration of ligand L from the aluminum to the tin atom. The identity of 4 was established by elemental analysis, ¹H, ¹³C and ¹¹⁹Sn NMR spectroscopy and ¹H, ¹¹⁹Sn HMBC experiments. The system 2 and B(C₆F₅)₃ in a 1:1 molar ratio was shown to be active in the polymerization of propylene oxide and ε‐caprolactone. Copyright © 2007 John Wiley & Sons, Ltd.     

Definable sets and expansions of models of Peano arithmetic

We consider expansions of models of Peano arithmetic to models ofA ₂ ^s -¦ ₁ ¹ + ₁ ¹ –AC which consist of families of sets definable by nonstandard formulas.