Uniform and Lipschitz homotopy classes of maps

If is a compact connected polyhedron, we associate with each uniform homotopy class of uniformly continuous mappings from the real line into an element of where is the space of uniformly continuous functions from to and is the subspace of bounded uniformly continuous functions. This map from uniform homotopy classes of functions to is surjective. If is the -dimensional torus, it is bijective, while if is a compact orientable surface of genus 1$">, it is not injective.

In higher dimensions we have to consider smooth Lipschitz homotopy classes of smooth Lipschitz maps from suitable Riemannian manifolds to compact smooth manifolds With each such Lipschitz homotopy class we associate an element of where is the dimension of is the space of bounded continuous functions from the positive real axis to and is the set of all such that

     

Growth and characterization of Nd-doped SBN single crystal fibers

The growth from ceramic pedestals of single crystal fibers of strontium barium niobate (SBN), doped with different Nd concentrations, was studied. High-quality fibers were grown for Nd doping levels of up to 4.2 at.%, and doping levels as high as 9.9 at.% were reached. No signs of core effects were observed, even for the highest concentration samples. The emission spectra of Nd³⁺ ions in the fibers did not differ from those obtained from bulk SBN crystals, showing very broad bands due to the typical disorder of the crystallographic tungsten-bronze structure. Fluorescence quenching was clearly observed for Nd doping levels above 2 at.%. PACS 81.10.Fq; 78.55.-m     

Smooth Linearization for a Saddle on Banach Spaces

As a continuation of a previous work on linearization of class C¹ of diffeomorphisms and flows in infinite dimensions near a fixed point, in this work we deal with the case of a saddle point with some non-resonance restrictions for the linear part. Our result can be seen as an extension of results by Hartman [Boletin de la Sociedad Matematica Mexicana 5(2), 220–241 (1960)] and Aronson, Belitskii and Zhuzhoma [Introduction to the Qualitative Theory of Dynamical systems on surfaces, AMS Transl. Math. Monog. vol.153, pp. 268–277 (1996)] in dimension two. We also present an application to a system of nonlinear wave equations.AMS Subject Classifications: Primary: 35B05, 34G20. Secondary: 35B40, 34D05.Dedicated to Professor Shui-Nee Chow on the occasion of his 60th birthday     

Phase transitions in unstable cancer cell populations

The dynamics of cancer evolution is studied by means of a simple quasispecies model involving cells displaying high levels of genetic instability. Both continuous, mean-field and discrete, bit-string models are analysed. The string model is simulated on a single-peak landscape. It is shown that a phase transition exists at high levels of genetic instability, thus separating two phases of slow and rapid growth. The results suggest that, under a conserved level of genetic instability the cancer cell population will be close to the threshold level. Implications for therapy are outlined.Received: 22 April 2003, Published online: 22 September 2003PACS: 87.10.+e Biological physics: General theory and mathematical aspects - 87.23.Kg Dynamics of evolution - 87.23.-n Ecology and evolution - 89.75.Fb Structures and organization in complex systems     

Anomalous behavior of ideal Fermi gas below two dimensions

Normal behavior of the thermodynamic properties of a Fermi gas in d > 2 dimensions, integer or not, means monotonically increasing or decreasing of its specific heat, chemical potential or isothermal sound velocity, all as functions of temperature. However, for 0 < d < 2 dimensions these properties develop a "hump" (or "trough") which increases (or deepens) as d? 0 d\rightarrow 0 . Though not the phase transition signaled by the sharp features ("cusp" or "jump") in those properties for the ideal Bose gas in d > 2 (known as the Bose-Einstein condensation), it is nevertheless an intriguing structural anomaly which we exhibit in detail.     

Superdiffusion and encounter rates in diluted,low dimensional worlds

Rate limitation due to encounters is fundamental to many ecological interactions. Since encounter rate governs reaction rates, and thus, dynamics of systems, it deserves systematic study. In classical population biology, ecological dynamics rely on the assumption of perfectly mixed interacting entities (e.g., individuals, populations, etc.) in a spaceless world. The so-called mean field assumption assumes that encounter rates are driven exclusively by changes in the density of the interacting entities and not on how they are distributed or move in space. Therefore, the mean field assumption does not give any insight into relevant spatiotemporal statistical properties produced by the trajectories of moving entities through space. In the present study, we develop spatially explicit simulations of random walking particles (i.e., Lévy walkers) to evaluate encounter rate constraints beyond the mean field assumption. We show that encounter rate fluctuations are driven not only by physical aspects such as the size or the velocity of the interacting particles, but also by different motion patterns. In particular, superdiffusion phenomena might be relevant at low densities and/or low spatial dimensionality. Finally, we discuss potential adaptive responses of living organisms that may allow individuals to control how they diffuse through space and/or the spatial dimensions employed in the exploration process.     

New ru(II) complexes containing oxazoline ligands as epoxidation catalysts. Influence of the substituents on the catalytic performance

The synthesis of a family of new Ru complexes containing the facial tridentate ligand with general formula [Ru(II)(T)(D)(X)](n+) (T = trispyrazolylmethane (tpm); D = ((4S,4'S)-(-)-4,4',5,5'-tetrahydro-4,4'-bis(1-methylethyl)-2,2'-bioxazole) (iPr-box-C) or N-(1-hydroxy-3-methylbutan-(2S)-(-)-2-yl)-(4S)-(-)-4-isopropyl-4,5-dihydrooxazole-2-carbimidate (iPr-box-O); X = Cl, H(2)O) has been described. All complexes have been spectroscopically characterized in solution through (1)H NMR and UV-vis techniques, and the redox properties of complexes have also been studied by means of cyclic voltammetry (CV). Furthermore, the chloro complexes presented here have been characterized in the solid state through monocrystal X-ray diffraction analysis. The oxazolinic iPr-box-C ligand undergoes a Ru-assisted hydrolysis reaction generating the corresponding amidate anionic ligand iPr-box-O, that keeps coordinated to the Ru metal center and that produces a strong σ-donation effect over it. The reactivity of the Ru-OH(2) complexes described in this paper together with other similar ones, previously synthesized by us, has been tested with regard to the epoxidation of different olefins. Complexes [Ru(II)(R-box-C)(tpm)OH(2)](BF(4))(2), R = Bz, 3'c/iPr, 3c, show high stereoselectivity in the epoxidation of cis-β-methylstyrene, with the exclusive formation of the cis-epoxide. However, there is a significant difference in regioselectivity between the two catalysts in the epoxidation of 4-vinylcyclohexene; complex 3'c leads to the regioselective oxidation at the ring alkene position, whereas complex 3c leads to the oxidation at the terminal position. Computational calculations indicate only small energy differences between the two possible products of 4-vinylcyclohexene epoxidation, but the energy barriers for the interaction of the catalytic systems with the alkene groups of 4-vinylcyclohexene agree with the reactivity differences found for the two catalysts having isopropyl or benzyl as substituent of the oxazole ligand. Computed local Fukui functions help to explain the observed reactivity trends.     

Gold(I)‐Catalyzed N‐Desulfonylative Amination versus N‐to‐O 1,5‐Sulfonyl Migration: A Versatile Approach to 1‐Azabicycloalkanes

Valuable 1‐azabicycloalkane derivatives have been synthesized through a novel gold(I)‐catalyzed desulfonylative cyclization strategy. An ammoniumation reaction of ynones substituted at the 1‐position with an N‐sulfonyl azacycle took place in the presence of a gold cation by intramolecular cyclization of the disubstituted sulfonamide moiety onto the triple bond. Depending on the size of the heterocyclic ring and substitution of the substrates, two unprecedented forms of nucleophilic attack on the sulfonyl group were exploited, that is, a N‐desulfonylation in the presence of an external protic O nucleophile (37–87 %, 10 examples) and a unique N‐to‐O 1,5‐sulfonyl migration (60–98 %, 9 examples).     

Diels–Alder and Retro‐Diels–Alder Cycloadditions of (1,2,3,4,5‐Pentamethyl)cyclopentadiene to La@C2v‐C82: Regioselectivity and Product Stability

One of the most important reactions in fullerene chemistry is the Diels–Alder (DA) reaction. In two previous experimental studies, the DA cycloaddition reactions of cyclopentadiene (Cp) and 1,2,3,4,5‐pentamethylcyclopentadiene (Cp*) with La@C_2v‐C₈₂ were investigated. The attack of Cp was proposed to occur on bond 19 , whereas that of Cp* was confirmed by X‐ray analysis to be over bond o . Moreover, the stabilities of the Cp and Cp* adducts were found to be significantly different, that is, the decomposition of La@C_2v‐C₈₂Cp was one order of magnitude faster than that of La@C_2v‐C₈₂Cp*. Herein, we computationally analyze these DA cycloadditions with two main goals: First, to compute the thermodynamics and kinetics of the cycloadditions of Cp and Cp* to different bonds of La@C_2v‐C₈₂ to assess and compare the regioselectivity of these two reactions. Second, to understand the origin of the different thermal stabilities of the La@C₈₂Cp and La@C₈₂Cp* adducts. Our results show that the regioselectivity of the two DA cycloadditions is the same, with preferred attack on bond o . This result corrects the previous assumption of the regioselectivity of the Cp attack that was made based only on the shape of the La@C₈₂ singly occupied molecular orbital. In addition, we show that the higher stability of the La@C₈₂Cp* adduct is not due to the electronic effects of the methyl groups on the Cp ring, as previously suggested, but to higher long‐range dispersion interactions in the Cp* case, which enhance the stabilization of the reactant complex, transition state, and products with respect to the separated reactants. This stabilization for the La@C₈₂Cp* case decreases the Gibbs reaction energy, thus allowing competition between the direct and retro reactions and making dissociation more difficult.