On linear congruence relations for Kubota-Leopoldt 2-adic L-functions

Our purpose in the paper is to find the most general linear congruence relation of the Hardy-Williams type for linear combinations of special values of Kubota-Leopoldt 2-adic L-functions L₂(k,χω^1−k) with k running over any finite subset of not necessarily consisting of consecutive integers (see Acta Arith. 47 (1986) 263; Publ. Math. Fac. Sci. Besançon, Théorie des Nombres, 1995/1996; Publ. Math. Debrecen 56 (2000) 677 and cf. Mathematics and Its Applications, Vol. 511, Kluwer Academic Publishers, Dordrecht, Boston, London, 2000). If k runs over finite subsets of consisting of consecutive integers see Compositio Math. 111 (1998) 289; Publ. Math. Debrecen 56 (2000) 677; Hardy and Williams, 1986; Compositio Math. 75 (1990) 271; Acta Arith. 71 (1995) 273; 52 (1989) 147; J. Number Theory 34 (1990) 362. In order to obtain the most general congruences of this type we make use of divisibility properties of the generalized Vandermonde determinants obtained in Spie? et al. (Divisibility properties of generalized Vandermonde and Cauchy determinants, Preprint 627, Institute of Mathematics, Polish Academy of Sciences, Warsaw, 2002). This allows us to simplify our main Theorem 2 and obtain Theorem 3 where the most general form of the linear congruence relation is given.     

Discrete dynamical equations in Minkowski space

Discrete difference equations in Minkowski space are obtained and the discrete Minkowski force is shown to be a four-vector. A transformation from a discrete dynamical equation in Minkowski space to a Lorentz-invariant difference equation in one-dimensional space is given.     

On the universality of Einstein equations

It is proved that a Lagrangian field theory based on a linear connection in space-time is equivalent to Einstein's general relativity interacting with additional matter fields.     

The localization of energy in gauge field theories and in linear gravitation

It is shown how the energy-positivity criterion enables us to localize the energy in various field theories. For this purpose the role of surface integrals in a canonical formalism is investigated. The same techniques are applied to linearized gravity, where the mixed Cauchy-boundary value problem in a finite volume is analyzed. Unconstrained degrees of freedom and boundary data which have to be controlled are found. This paper is part of a program to analyze the possibility of localization of gravitational energy in complete General Relativity.     

A numerical model of multimode fiber PON frequency response

Frequency response of passive optical network (PON) based on multimode fibers is investigated. The network comprises fibers, connectors and splitters/couplers. It is shown that due to mode filtering at splitters, the frequency response is different for different network nodes in otherwise symmetrical network.     

A novel catalytic adsorptive stripping voltammetric method for the determination of germanium ultratraces in the presence of chloranilic acid and the V(IV)·HEDTA complex

A novel, sensitive catalytic adsorptive stripping voltammetric procedure which can be used to determine trace amounts of germanium is described. The method is based on the interfacial accumulation of the complex formed by Ge(IV) and the product of the reduction of chloranilic acid on the hanging mercury drop electrode or the renewable silver amalgam film electrode, and its subsequent reduction from the adsorbed state followed by the catalytic action of the V(IV)·HEDTA complex. The presence of V(IV)·HEDTA greatly enhances the adsorptive stripping response of Ge. The reduction of the Ge(IV) in the presence of chloranilic acid and V(IV)·HEDTA was investigated in detail and the effects of pH, electrolyte composition, and instrumental parameters were studied. Under optimal conditions, the catalytic peak current of germanium exhibited good linearity for Ge(IV) concentrations in the range of 0.75–60 nM (for 60 s of accumulation at −0.1 V, r² = 0.995) and a low limit of detection (LOD = 0.085 nM). The procedure was successfully applied to determine Ge in water samples.

     

Structures and energies of seleno derivatives of biuret.Ab Initio comparative studies of diselenobiuret, selenobiuret, and selenothiobiuret

Ab initio studies (LCAO-MO method) on conformers of three seleno derivatives of the biuret molecules diselenobiuret [I], selenobiuret [II], and selenothiobiuret [III] were carried out at the Hartree-Fock (HF) and MP2 levels. The molecular geometries of these species were fully optimized at the HF level and characterized by analysis of the harmonic vibrational frequencies using a split-valence triple-zeta basis set augmented by a set ofd polarization functions on heavy atoms andp polarization functions on hydrogen atoms [TZP(d, p)]. The total energies of the HF-optimized structures were calculated at the MP2 (frozen core) level using a larger TZP (2df, 2pd) basis set. The potential energy searches revealed a total of 11 minimum-energy conformers (assigned astrans-trans, trans-cis, cis-trans, andcis-cis) and seven transition-state species for the title molecules. The two predicted conformers for diselenobiuret (Ia=trans-trans andIc=cis-cis) are characterized byC ₂ and the third byC _s symmetry. For selenothiobiuret two forms (IIIa=trans-trans andIIId=cis-cis) possessC ₁ and two (IIIb=trans-cis andIIIc=cis-trans) possessC _s symmetries, respectively. For selenobiuret, four formsIIa=trans-trans (C₁),IIb=trans-cis (C _s),IIc=cis-trans (C ₁), andIId=cis-cis (C₁), were obtained as a result of gradient optimization. Comparison of the relative energies for the considered species indicated that thecis-trans forms are the most stable conformations for all three systems at both the HF and MP2 levels of theory.