A Lower Bound for the Height of a Rational Function at S-unit Points

Let a,b be given, multiplicatively independent positive integers and let >0. In a recent paper jointly with Y. Bugeaud we proved the upper bound exp(n) for g.c.d.(aⁿ–1, bⁿ–1); shortly afterwards we generalized this to the estimate g.c.d.(u–1,v–1)<>u,v) for multiplicatively independent S-units u,vZ. In a subsequent analysis of those results it turned out that a perhaps better formulation of them may be obtained in terms of the language of heights of algebraic numbers. In fact, the purposes of the present paper are: to generalize the upper bound for the g.c.d. to pairs of rational functions other than {u–1,v–1} and to extend the results to the realm of algebraic numbers, giving at the same time a new formulation of the bounds in terms of height functions and algebraic subgroups of G_m².     

Synthesis and crystal structure of bis(dimethylthioacetonedicarboxylate)nickel(II)

Summary The title compound has been synthesized and characterized. The crystal structure has been determined by single-crystal x-ray analysis from diffractometer data, and refined to R= 0.063 for 1889 observed reflections. Crystals are monoclinic, space group P2₁/n, witha=21.97(2),b=8.21(1),c=10.44(1) A, and =101.87(3), and Z=4. Both ligands are coordinated to nickel through sulphur and one oxygen atom to realize thecis form of the complex in the square planar coordination geometry. Bond distances are: Ni-S, 2.14(1), Ni-O, 1.87(1) and 1.90(1) Å. Each nickel atom also forms two close contacts with oxygen atoms from adjacent molecules.     

Nature of point defects on SiO2/Mo(112) thin films and their interaction with Au atoms

We have studied by means of periodic DFT calculations the structure and properties of point defects at the surface of ultrathin silica films epitaxially grown on Mo(112) and their interaction with adsorbed Au atoms. For comparison, the same defects have been generated on an unsupported silica film with the same structure. Four defects have been considered: nonbridging oxygen (NBO, [triple bond]Si-O(*)), Si dangling bond (E' center, [triple bond]Si(*)), oxygen vacancy (V(O), [triple bond]Si-Si[triple bond]), and peroxo group ([triple bond]Si-O-O-Si[triple bond]), but only the NBO and the V(O) centers are likely to form on the SiO(2)/Mo(112) films under normal experimental conditions. The [triple bond]Si-O(*) center captures one electron from Mo forming a silanolate group, [triple bond]Si-O(-), sign of a direct interaction with the metal substrate. Apart from the peroxo group, which is unreactive, the other defects bind strongly the Au atom forming stable surface complexes, but their behavior may differ from that of the same centers generated on an unsupported silica film. This is true in particular for the two most likely defects considered, the nonbridging oxygen, [triple bond]Si-O(*), and the oxygen vacancy, [triple bond]Si-Si[triple bond].     

Storage of localized structure matrices in nematic liquid crystals

We show experimentally that large matrices of localized structures can be stored as elementary pixels in a nematic liquid crystal cell. Based on optical feedback with phase modulated input beam, our system allows us to store, erase, and actualize the localized structures in the matrix.     

Autocorrelation function formulations and the turbulence dissipation rate: Application to dispersion models

The classical statistical diffusion theory and the binomial autocorrelation function are used to obtain a new formulation for the turbulence dissipation rate ε. The approach employs the Maclaurin series expansion of a logarithm function contained in the dispersion parameter formulation. The numerical coefficient of this new relation for ε is 100% larger than the numerical coefficient of the classical relation derived from the exponential autocorrelation function. A similar approach shows that the dispersion parameter obtained from the even exponential autocorrelation function does not result in a relation for ε and, therefore, is not suitable for application in dispersion models. In addition, a statistical comparison to experimental ground-level concentration data demonstrates that this newly derived relation for ε as well as other formulations for the turbulence dissipation rate are suitable for application in Lagrangian stochastic dispersion models. Therefore, the analysis shows that there is an uncertainty regarding the turbulence dissipation rate function form and the autocorrelation function form.     

On the operations of sequences in rings and binomial type sequences

Given a commutative ring with identity R, many different and interesting operations can be defined over the set \(H_R\) of sequences of elements in R. These operations can also give \(H_R\) the structure of a ring. We study some of these operations, focusing on the binomial convolution product and the operation induced by the composition of exponential generating functions. We provide new relations between these operations and their invertible elements. We also study automorphisms of the Hurwitz series ring, highlighting that some well-known transforms of sequences (such as the Stirling transform) are special cases of these automorphisms. Moreover, we introduce a novel isomorphism between \(H_R\) equipped with the componentwise sum and the set of the sequences starting with 1 equipped with the binomial convolution product. Finally, thanks to this isomorphism, we find a new method for characterizing and generating all the binomial type sequences.

     

On a characterization of orthogonality with respect to particular sequences of random variables in <Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>

This note deals with the orthogonality between sequences of random variables. The main idea of the note is to apply the results on equidistant systems of points in a Hilbert space to the case of the space L ²(Ω, F, ℙ) of real square integrable random variables. The main result gives a necessary and sufficient condition for a particular sequence of random variables (elements of which are taken from sets of equidistant elements of L ²(Ω, F, ℙ) to be orthogonal to some other sequence in L ²(Ω, F, ℙ). The result obtained is interesting from the point of view of the time series analysis, since it can be applied to a class of sequences random variables that exhibit a monotonically increasing variance. An application to ergodic theorem is also provided.     

Carnot efficiency: Why?

In 1824, Carnot proposed a cycle operating on reversibility principles. He proved that there exists an upper limit of the efficiency of this cycle and this limit is also the upper limit for any real process. The irreversibility related to the finite-time and the finite-size constraints are fundamental for the optimization of any real thermodynamic system. It has been pointed out how fundamental is the interaction between any open system and its surroundings. The meaning of the Carnot efficiency is that even in the ideal condition, when there is no dissipation, there exists something that does not allow the system to convert all the energies absorbed in work. The aim of this paper is to show why this happens, starting from a variational approach of thermodynamics.