There are infinitely many limit points of the fractional parts of powers

Suppose that α > 1 is an algebraic number and ξ > 0 is a real number. We prove that the sequence of fractional partsξα ⁿ , n = 1, 2, 3, …, has infinitely many limit points except when α is a PV-number and ξ ∈ ℚ(α). For ξ = 1 and α being a rational non-integer number, this result was proved by Vijayaraghavan.     

Banach-Saks operators on spaces of continuous functions

Summary Let K be a compact Hausdorff space and let E be a Banach space. We denote by C(K, E) the Banach space of all E-valued continuous functions defined on K, endowed with the supremum norm. We study in this paper Banach-Saks operators defined on C(K, E) spaces. We characterize these operators for a large class of compacts K (the scattered ones), or for a large class of Banach spaces E (the superreflexive ones). We also show by some examples that our theorems can not be extended directly.Partially supported by C.A.I.C.Y.T. grant 0338-84. The author wishes to thank Professor F.Bombal for his encouragement.     

Characterization of Titania/Silica Gel by Means of Low-Pressure Nitrogen Adsorption

Adsorbents synthesized by grafting of titania onto mesoporous silica gel surfaces at different temperatures were studied by means of nitrogen adsorption–desorption and water desorption. The pore size distribution f(R_p) of titania/silica gel depends on the titania concentration (C_TiO2) and the temperature of titania synthesis. Nonuniformity of TiO₂ phase is maximal at a low C_TiO2 value (3.2 wt.% anatase deposited at 473 K), and two peaks of the fractal dimension distribution f(D) are observed at such a concentration of titania, but at larger C_TiO2 values, only one f(D) peak is seen. More ordered filling of pores and adsorption sites by nitrogen, reflecting in the shape of adsorption energy distributions f(E) at different pressures of adsorbate, is observed for adsorbent with titania (rutile+anatase) grafted on silica gel at a higher temperature (673 K).     

Structural-chemical disorder of manganese dioxides 1. Influence on surface properties at the solid-electrolyte interface

Relationships between lattice parameters of manganese dioxides and their surface properties at the solid-aqueous solution interface were investigated. The studied series ranged from ramsdellite to pyrolusite and encompassed disordered MD samples. The structural model used takes into account structural defects: Pr (rate of pyrolusite intergrowth) and Tw (rate of microtwinning). Water adsorption isotherms showed that the cross sectional area of water molecules adsorbed in the first monolayer is positively correlated to Pr. Titration of the surface charge of the MD series evidenced a positive linear relationship between the PZC and Pr (Pr=0, Tw=0, PZC=1 for ramsdellite; Pr=1, Tw=0, PZC=7.3 for pyrolusite; gamma-MD with intermediate values of Pr (0.2 to 0.45) have increasing PZC values). The rate of microtwinning appeared as a secondary factor for the increase of the PZC. The above correlations are explained by the chemical defects at the origin of the structural disorder, respectively Mn(3+)/Mn4+ substitution for Pr and Mn vacancies for Tw, which result in proton affinity and thus in increased PZC. The experimental results are compared with data collected in the literature for manganese dioxides as well as for dioxides of transition elements with tetragonal structure.     

Aromaticity of annulenes and annulene ions with 4v+2 electrons from the viewpoint of the theory of Hartree-Fock instabilities

The connection between the aromaticity of annulenes and annulene ions with 4v+2 -electrons and the stability of the closed-shell restricted Hartree-Fock (RHF) solutions for these systems is discussed in the framework of the PPP-approximation. It is shown that the tendency towards an uniform electron density distribution in aromatic cycles is paralleled by the stability properties of the corresponding closed-shell RHF solutions. The stability investigations are demonstrated to provide realistic estimates of the critical ring sizes at which Hückel's 4v+2 rule breaks down.     

Hexacyanometalate molecular chemistry: di-, tri-, tetra-, hexa- and heptanuclear heterobimetallic complexes; control of nuclearity and structural anisotropy

Following a bottom-up approach to nanomaterials, we present a rational synthetic route to high-spin and anisotropic molecules based on hexacyanometalate [M(CN)(6)](3-) cores. Part 1 of this series was devoted to isotropic heptanuclear clusters; herein, we discuss the nuclearity and the structural anisotropy of nickel(II) derivatives. By changing either the stoichiometry, the nature of the terminal ligand, or the counterion, it is possible to tune the nuclearity of the polynuclear compounds and therefore to control the structural anisotropy. We present the synthesis and the characterisation by mass spectrometry, X-ray crystallography and magnetic susceptibility of bi-, tri-, tetra-, hexa- and heptanuclear species [M(CN)(n)(CN-M'L)(6-n)](m+) (with n=0-5; M=Cr(III), Co(III), M'=Ni(II); L=pentadentate ligand). Thus, with M=Cr(III), d(3), S=3/2, a dinuclear complex [Cr(III)(CN)(5)(CN-NiL(n))](9+), (L(n)=polydentate ligand) was built and characterised, showing a spin ground state, S(G)=5/2, with a ferromagnetic interaction J(Cr,Cu)=+18.5 cm(-1). With M=Co(III) (d(6), S=0) were built di-, tri-, tetra-, hexa and hepanuclear CoNi species: CoNi, CoNi(2), CoNi(3), CoNi(5) and CoNi(6). By a first approximation, they behave as one, two, three, five and six isolated nickel(II) complexes, respectively, but more accurate studies allow us to evaluate the weak antiferromagnetic coupling constant between two next-nearest neighbours M'-Co-M'.     

HIGH RESOLUTION GAS ADSORPTION STUDY ON ILLITES PERMUTED WITH VARIOUS CATIONS ASSESSMENT OF SURFACE ENERGETIC PROPERTIES

ABSTRACT

Ar and N₂ high resolution adsorption volumetry coupled to the Derivative Isotherm Simulation method was applied on nine homoionic illite samples (Li ?, Na ? K ? Rb ? Cs ? Mg ? Ca ? Sr ? and Ba ? illites). By comparing the adsorption isotherms it appears that the nature of the surface cation strongly influences the adsorption mechanisms.

The lateral and basal surface areas can only be derived from argon adsorption on monovalent samples. The argon adsorption energies varv with the nature of the surface compensating cation. Both cation size and electronegativity are involved. These interactions can, in first approach, be expressed in terms of surface tensionsγ_sv ^LW. Specific interactions appear when nitrogen is used revealing high energy adsorption sites located on basal faces. These sites could be related to some polar properties of the surface.