Prevalence of Multifractal Functions in S<Superscript>v</Superscript> Spaces

Spaces called S^v were introduced by Jaffard [16] as spaces of functions characterized by the number ≃ 2^ν(α)j of their wavelet coefficients having a size ≳ 2^−αj at scale j . They are Polish vector spaces for a natural distance. In those spaces we show that multifractal functions are prevalent (an infinite-dimensional “almost-every”). Their spectrum of singularities can be computed from ν, which justifies a new multifractal formalism, not limited to concave spectra.     

The Acetolysis of exo- and endo-5,6-Dimethylidene-2-Norbornyl p-Bromobenzenesulfonates and of their Optically Active and Deuterium-Labelled Derivatives

The buffered (AcOK) acetolyses of exo, (11) and endo-5, 6-dimethylidene-2-norbornyl brosylate (12) yielded exo5, 6-dimethylidene-2-norbornyl (16) and (3-methylidene-2-nortricyclyl)methyl acetates (18) . Endo-5, 6-dimethylidene-2-norbornyl (17) and 2-methylidene-3-tricyclo [3.2.1.0^3,6]octyl acetates (20) could not be detected. The titrimetric rate constants of the acetolysis of 11 (k_t(exo)=4.49 ± 0.02) · 10^?5 s^?1 at 25°, ΔH^≠=23.6 ±0.7 kcal mol^?1, ΔS^≠=0.7 ±2 calmol^?1 K^?1 and 12 (k_t(endo)=1.9 ±0.08) · 10^?9 s^?1 at 25°, ΔH^≠=27 ±1 kcal mol^?1, ΔS^≠=-8 ±2.5 calmol^?1 K^?1) were measured and compared with the polarimetric rate constants (k_α/k_(exo)=6.8 at 25°,(k_α/k_(exo)=1.0 at 121°) of the buffered acetolyses of the optically active brosylates (+)- 11 and (+)- 12 . Neither a common-ion (KOBs) nor a special ion effect (LiClO₄) on k_t(endo) could be detected, although external return might well intervene as some exo-5,6-dimethylidene-2-norbornyl tosylate (21) was formed upon solvolysis in the presence of KOTs. Acetolysis of (+)- 11 yielded completely racemized products, whereas (+)- 12 led to incomplete racemization. The buffered acetolysis of exo-(3exo-D)-5,6-dimethylidene-2-norbornyl brosylate (24) furnished (3exo-D)-( 26 :37.5%), exo-(7syn-D)-5,6-dimethylidene-2-norbornyl brosylate (27 : 37.5%) and [(5anti-D)-3-methylidene-2-nortricyclyl]methyl acetates (28 : 25 %). The acetolysis of endo-(2exo-D)-5,6-dimethylidene-2-norbornyl brosylate (25) yielded (2endo-D)-( 29 : 54%), exo-(1-D)-5,6-dimethylidene-2-norbornyl ( 30 : 36%) and [(6-D)-3-methylidene-2-nortricyclyl]methyl acetates ( 31 : 10%). Product analysis and deuterium label distribution was established by a combination of GC., ¹H-NMR., ²H-{¹H}-NMR. and MS. techniques. The results are rationalized by invoking anchimerically assisted ionization of the exo-brosylate 11 to symmetrical ion-pairs (cyclopropylcarbinyl cation intermediates) which undergo internal (and probably also external) return. Acetolysis of the endo-brosylate 12 is not anchimerically assisted and leads initially to non-symmetrical ion pairs. These evolve to symmetrical ion pair intermediates or, to a minor extent, are intercepted by solvent.     

The equality relations in scientific computing

The equality relation (more generally, the ordering relations) in floating point arithmetic is the exact translation of the mathematical equality relation. Because of the propagation of round-off errors, the floating point arithmetic is not the exact representation of the theoretical arithmetic which is continuous on the real numbers.This leads to some incoherence when the equality concept is used in floating point arithmetic. A well known example is the detection of a zero element in the pivoting column and equation when applying Gaussian elimination, which is almost impossible in floating point arithmetic.We shall begin by showing the inadequacy of the equality relation used in floating point arithmetic (we will call it floating point equality), and then introduce two new concepts: stochastic numbers and the equality relation between such numbers which will be called the stochastic equality. We will show how these concepts allow to recover the coherence between the arithmetic operators and the ordering relations that was missing in floating point computations.     

Double dynamers: molecular and supramolecular double dynamic polymers

The formation of double dynamers, polymers that are dynamic on both the molecular and supramolecular levels, was achieved in solution and corroborated by electron microscopy; NMR and mass spectrometry studies confirm their double molecular and supramolecular nature.     

Supramolecular polymers generated from heterocomplementary monomers linked through multiple hydrogen-bonding arrays--formation, characterization, and properties

Supramolecular polymers are described that are derived from the association of two homoditopic heterocomplementary monomers through sextuple hydrogen-bonding arrays. They form fibers and a variety of different materials depending on the conditions. The strong affinity of the DAD-DAD (D=donor, A=acceptor) hydrogen-bonding sites for double-faced cyanuric acid type wedges drives the supramolecular polymeric assembly in apolar and chlorinated organic solvents. The marked influence of stoichiometry, as well as end-capping and cross-linking agents upon fiber formation is revealed in solution and by electron microscopy (EM). The results further contribute to the development of a supramolecular polymer chemistry that comprises reversible polymers formed through recognition-controlled noncovalent connections between the molecular components. Such materials are, by nature, dynamic and present adaptive character in view of their ability to respond to external stimuli.     

Monitoring the differential ordering of enantiomers included into cyclodextrins through deuterium NMR in lyotropic liquid crystals

Deuterium NMR in an aqueous non-chiral liquid crystal allows the discrimination of enantiomers through their ordering inside beta-cyclodextrins.     

Lyapunov Characteristic Exponents for smooth dynamical systems and for hamiltonian systems; A method for computing all of them. Part 2: Numerical application

Summary The present paper, together with the previous one (Part 1: Theory, published in this journal) is intended to give an explicit method for computing all Lyapunov Characteristic Exponents of a dynamical system. After the general theory on such exponents developed in the first part, in the present paper the computational method is described (Chapter A) and some numerical examples for mappings on manifolds and for Hamiltonian systems are given (Chapter B).

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Sommario Questo articolo, insieme con il precedente (Parte 1: Teoria, pubblicato in questa stessa rivista) è inteso a fornire un metodo esplicito per il calcolo di tutti gli esponenti caratteristici di Lvapunov per un sistema dinamico. Dopo la teoria generale su tali esponenti sviluppata nella prima parte, qui si illustra il metodo di calcolo (Capitolo A) e si danno esempi numerici per applicazioni di varietà in sè e per sistemi Hamiltoniani (Capitolo B).