Kinetics of solid-state synthesis and properties of lead-containing niobate pyrochlores

The main features of the synthesis of (Pb_1-x Sr_x)₂Nb₂O₇ and (Pb_1-x Ba_x)₂Nb₂O₇ pyrochlore solid solutions were determined using isothermal thermogravimetry in tandem with formal-kinetic analysis. The stability and electron-transport properties of the triple niobates were studied, which allowed us to test them as electrode-active materials for membranes in ion-selective electrodes (ISEs).     

Synthesis and structural characterization of two novel heterometallic clusters: [Rh4Pt2(CO)11(dppm)2] and [Ru2Rh2Pt2(CO)12(dppm)2

Two novel heterometallic octahedral clusters [Rh(4)Pt(2)(CO)(11)(dppm)(2)](1) and [Ru(2)Rh(2)Pt(2)(CO)(12)(dppm)(2)](2) were synthesized by the reaction of [Rh(2)Pt(2)(CO)(6)(dppm)(2)] with [Rh(6)(CO)(14)(NCMe)(2)] and Ru(3)(CO)(12), respectively. Solid state structures of 1 and 2 have been established by a single crystal X-ray diffraction study. Two dppm ligands in 1 are bonded to one platinum and three rhodium atoms, which form an equatorial plane of the Rh(4)Pt(2) octahedron. Two rhodium and two platinum atoms bound to the diphosphine ligands in 2 are nonplanar to give an octahedral C2 symmetric Ru(2)Rh(2)Pt(2)(dppm)2 framework. The (31)P NMR investigation of and (1D, (31)P COSY, (31)P-[(103)Rh] HMQC) and simulation of 1D spectral patterns showed that in both clusters the structures of the M(6)(PP)(2) fragments found in the solid state are maintained in solution.     

A V. A. Markov type inequality in Lp

It is proved that for any algebraic polynomial P of degree at most n we have for 1 p ≤ + t8, x ≥ 1 the inequality For p ≥1 and x ≥ 1 we construct a polynomial P_* of degree n for which      

Order of a Function on the Bruschlinsky Group

For an arbitrary function F defined on the group of homotopy classes of mappings of a finite polyhedron X to the circle and taking values in an Abelian group Q, the notion of order is defined. The order ord F is compared with the algebraic degree of F. It is proved that ord Fdeg F and deg Fdim X ord F. The inequality ord Fdeg F is proved in the case where Q is torsion-free or ord F1. Bibliography: 2 titles.     

Application of HMQC spectroscopy to the observation of the metal resonance in polymetallic compounds: the effect of multiple-metal spin transitions

Multiple-metal spin transitions which distort the HMQC spectra of rhodium carbonyl clusters are discussed. These effects are seen whenever the detector nucleus, e.g. 13C or 31P, couples to more than one metal spin and are not restricted to detector ligands occupying edge- or face-bridging sites. These effects are illustrated in, but not limited to, the 13C-{103Rh} and 31P-{103Rh} HMQC spectra of [Rh6(CO)15L], (where L = P(4-F-C6H4)3), [Rh4(CO)11{P(OPh)3}], [Rh6C(CO)15]2- and [Rh2(carboxylate)2PPh3]. The effect is to modulate the intensity and position of the correlations in the metal dimension; cross peaks are displaced from the true chemical shift, additional cross peaks are seen and the intensity of the coherences varies as a function of the preparation delay, d2, and coupling constant, and may go to zero at the conventional value of 1/(2J). Analyses of the relevant spin systems are given together with experimental strategies to overcome these effects.     

On asymptotics of the uniform norm of polynomials with zeros at the roots of unity

The work is related to the problem by P. Erds about the estimation of the numbers

Quadratic Property of the Rational Semicharacteristic

Let . Assume that V is a manifold, is the set of germs of n-dimensional oriented submanifolds of V, and is the ₂-module of all ₂-valued functions on E _n (V). If is an oriented submanifold, let be the indicator function of the set of germs of X. It is proved that there exists a quadratic map such that for any compact oriented submanifold one has the relation , where is the (rational) semicharacteristic of , i.e., the residue class defined by the formula

Homotopy invariants of mappings to the circle

Homotopy classes of mappings of a space X to the circle T form an Abelian group B(X) (the Bruschlinsky group). If a: X → T is a continuous mapping, then [a] denotes the homotopy class of a, and I _r (a): (X × T) _r → \mathbbZ \mathbb{Z} is the indicator function of the rth Cartesian power of the graph of a. Let C be an Abelian group and let f: B(X) → C be a mapping. By definition, f has order not greater than r if the correspondence I _r (a) → f([a]) extends to a (partly defined) homomorphism from the Abelian group of Z-valued functions on (X × T) ^r to C. It is proved that the order of f equals the algebraic degree of f. (A mapping between Abelian groups has degree at most r if all of its finite differences of order r +1 vanish.) Bibliography: 2 titles.     

Equilibrium structure and properties of metallic sodium calculated from first principles by the pseudopotential method

The form factor, characteristic function, total energy, interatomic pair potential, and electron and phonon spectra of metallic Na are calculated from first principles by the pseudopotential method, taking exchange-correlation effects, orthogonalizational holes, and energy shifts of the internal levels into account. Good agreement is obtained between these values calculated in the second order of perturbation theory and experimental values.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 39–46, August, 1979.