A dimension gap for continued fractions with independent digits

Kinney and Pitcher (1966) determined the dimension of measures on [0, 1] which make the digits in the continued fraction expansion i.i.d. variables. From their formula it is not clear that these dimensions are less than 1, but this follows from the thermodynamic formalism for the Gauss map developed by Walters (1978). We prove that, in fact, these dimensions are bounded by 1−10⁻⁷. More generally, we considerf-expansions with a corresponding absolutely continuous measureμ under which the digits form a stationary process. Denote byE _δ the set of reals where the asymptotic frequency of some digit in thef-expansion differs by at leastδ from the frequency prescribed byμ. ThenE _δ has Hausdorff dimension less than 1 for anyδ>0.     

X-ray crystal structure* and kinetics of oxidation of 1,4,8,11-tetraazaundecanenickel(II), [NiII(2,3,2-tet)]2+ by peroxodisulphate: Formation of a stable non-macrocyclic nickel(III) polyamine

While macrocyclic ligands are well-known to stabilize tervalent nickel, it is very unusual to form stable nickel(III)-amine species in aqueous solution. We describe the kinetics of oxidation of 1,4,8,11-tetraazaundecanenickel(II), [Ni^II(2,3,2-tet)]²⁺ by peroxodisulphate in aqueous solution to the tervalent state. The reaction follows a second order rate law, first order in [Ni^II(2,3,2-tet)]²⁺ and first order in peroxodisulphate. The activation enthalpy and entropy are 55 ± 3 kJ mol^–1 and –38 ± 10 J K^–1 mol^–1 respectively. The X-ray crystal structure of the [Ni^II(2,3,2-tet)]²⁺ cation is also described. The title compound crystallizes in the monoclinic space group P2(1)/n. The cation has a slightly distorted square planar structure, with the bite angle of the 5-membered chelate rings being slightly less than 90°, and that of the 6-membered chelate ring being correspondingly greater than 90°.     

Mean topological dimension

In this paper we present some results and applications of a new invariant for dynamical systems that can be viewed as a dynamical analogue of topological dimension. This invariant has been introduced by M. Gromov, and enables one to assign a meaningful quantity to dynamical systems of infinite topological dimension and entropy. We also develop an alternative approach that is metric dependent and is intimately related to topological entropy.     

MOLECULAR AND SUPRAMOLECULAR ORDERING IN CONFINED ENVIRONMENTS

Crystal and phase morphologies and structures determined by self-organization of crystalline-amorphous diblockcopolymers, crystallization of the crystallizable blocks, and vitrification of the amorphous blocks are reviewed through asystematic study on a series of poly(ethylene oxide)-b-polystyrene (PEO-b-PS) diblock copolymers. On the base ofcompetitions among these three processes, molecular and supramolecular ordering in confined environments can beinvestigated. In a concentration-fluctuation-induced disordered (D_(CF)) diblock copolymer, the competition between crystalli-zation of the PEO blocks and vitrification of the PS blocks is momtored by time-resolved simultaneous small angle X-rayscattering (SAXS) and wide angle X-ray diffraction (WAXD) techniques. In the case of T_c相似文献   

Controlling the Spontaneous Precipitation of Silver Nanoparticles in Sol-Gel Materials

The mechanisms responsible for spontaneous silver precipitation in silver-doped sol-gel materials are identified. The chemistry of the solvent phase is found to be the critical factor in controlling this phenomenon. The addition of HCl as catalyst leads to the formation of AgCl and subsequent formation of silver upon light exposure. Another factor leading to silver precipitation is the reducing capability of methanol radicals. Silver precipitation is inhibited by simply washing out the pore solvents by a solvent exchange method.     

Electrospinning of Hyaluronic Acid (HA) and HA/Gelatin Blends

Summary: In the present study, electrospinning of hyaluronic acid (HA) and hyaluronic acid/gelatin (HA‐GE) blends in N,N‐dimethylformamide (DMF)/water‐mixed solvents have been investigated. When the volume ratio of DMF to water was in the range of 1.5–0.5, HA solutions could be electrospun into fibrous membranes successfully. The average diameter of HA fibers was about 200 nm. The HA‐GE composite nanofibrous membranes with varied HA/GE weight ratio in the range of 100/20–100/100 have also been successfully fabricated. The average diameter of HA‐GE fibers was in the range of 190–500 nm. The decrease in surface tension could promote fiber formation. Thus, an introduction of DMF that could decrease the surface tension distinctively, without significant change or increase in viscosity of the solution, could bypass the use of blowing‐assisted electrospinning. Our postulated picture is that the lower surface tension could help the ejection of stream with relatively high viscosity and reduce or prevent the droplet formation during the spinning process.

HA/GE (100/80) nanofibrous membrane produced by electrospinning.     

A Rapid Process for Producing Cellulose Multi‐Filament Fibers from a NaOH/Thiourea Solvent System

Summary: Cellulose was dissolved rapidly in 9.5 wt.‐% NaOH and 4.5 wt.‐% thiourea aqueous solution pre‐cooled to −5 °C to prepare a transparent solution. Novel cellulose multi‐filament fibers were spun successfully, for the first time, from the cellulose dope on an extended laboratory scale. The results from ¹³C NMR, scanning electron microscopy and wide angle X‐ray diffraction (WAXD) patterns indicated that the fibers exhibited cellulose II character and possessed a circular cross‐section and smooth surface. The tensile strength of the novel fibers reached 1.9–2.2 cN · dtex⁻¹. 2D WAXD and SAXS patterns revealed that, with a drawing progress, the orientation factor increased and mechanical properties were improved.

SEM micrographs of the novel multi‐filament fibers spun from cellulose solution in a NaOH/thiourea aqueous system pre‐cooled to −5 °C on an extended laboratory scale.     

Synthesis and Properties of Oligo-/Polymers Based on a “Flexible” Swivel-Cruciform Structure

We have introduced the concept of the so-called swivel type cruciforms as a highly flexible tool to tune the molecular properties from amorphous to highly crystalline features. Simple alkoxy substituted binaphthyl building blocks have been introduced randomly into poly(9,9-dioctylfluorene) (PFO) thereby suppressing the formation of side-chain crystallization. On the other hand the all aryl penta-thiophene based cruciform exhibited strong π–π interaction between the arms and gave one of the highest values reported to date for wet processed organic field effect transistors (OFETs) utilising oligothiophenes.     

How Carbon Nanotube Crushing can Improve Flame Retardant Behaviour in Polymer Nanocomposites?

Nanocomposites based on an ethylene–vinyl acetate copolymer (27 wt.‐% vinyl acetate) and multiwall carbon nanotubes (MWNTs) have been prepared by melt blending and their thermal degradation and flame retardant properties have been evaluated. Special attention has been paid to the influence of the nanotube nature on the flammability properties and more particularly on the time to ignition (TTI) as measured by cone calorimetry. It has been shown that there is a strong influence of the nature of carbon nanotubes on the fire behaviour of the composites, especially previous MWNTs crushing proved to substantially delay the TTI while maintaining much reduced heat release rate (HRR). Such a remarkable behaviour might be explained by the chemical reactivity of radical species present at the surface/extremities of crushed MWNTs during the combustion process.