Potential reduction method for harmonically convex programming

In this paper, we introduce a potential reduction method for harmonically convex programming. We show that, if the objective function and them constraint functions are allk-harmonically convex in the feasible set, then the number of iterations needed to find an -optimal solution is bounded by a polynomial inm, k, and log(1/). The method requires either the optimal objective value of the problem or an upper bound of the harmonic constantk as a working parameter. Moreover, we discuss the relation between the harmonic convexity condition used in this paper and some other convexity and smoothness conditions used in the literature.The authors like to thank Dr. Hans Nieuwenhuis for carefully reading this paper and the anonymous referees for the worthy suggestions.     

Multiple separatrix crossing in multi-degree-of-freedom Hamiltonian flows

Summary We study separatrix crossing in near-integrablek-degree-of-freedom Hamiltonian flows, 2 <k < , whose unperturbed phase portraits contain separatrices inn degrees of freedom, 1 <n <k. Each of the unperturbed separatrices can be recast as a codimension-one separatrix in the 2k-dimensional phase space, and the collection of these separatrices takes on a variety of geometrical possibilities in the reduced representation of a Poincaré section on the energy surface. In general 0 l n of the separatrices will be available to the Poincaré section, and each separatrix may be completely isolated from all other separatrices or intersect transversely with one or more of the other available separatrices. For completely isolated separatrices, transitions across broken separatrices are described for each separatrix by the single-separatrix crossing theory of Wiggins, as modified by Beigie. For intersecting separatrices, a possible violation of a normal hyperbolicity condition complicates the analysis by preventing the use of a persistence and smoothness theory for compact normally hyperbolic invariant manifolds and their local stable and unstable manifolds. For certain classes of multi-degree-of-freedom flows, however, a local persistence and smoothness result is straightforward, and we study the global implications of such a local result. In particular, we find codimension-one partial barriers and turnstile boundaries associated with each partially destroyed separatrix. From the collection of partial barriers and turnstiles follows a rich phase space partitioning and transport formalism to describe the dynamics amongst the various degrees of freedom. A generalization of Wiggins' higher-dimensional Melnikov theory to codimension-one surfaces in the multi-separatrix case allows one to uncover invariant manifold geometry. In the context of this perturbative analysis and detailed numerical computations, we study invariant manifold geometry, phase space partitioning, and phase space transport, with particular attention payed to the role of a vanishing frequency in the limit approaching the intersection of the partially destroyed separatrices. The class of flows under consideration includes flows of basic physical relevance, such as those describing scattering phenomena. The analysis is illustrated in the context of a detailed study of a 3-degree-of-freedom scattering problem.     

Superexponential stability of KAM tori

We study the dynamics in the neighborhood of an invariant torus of a nearly integrable system. We provide an upper bound to the diffusion speed, which turns out to be of superexponentially small size exp[-exp(1/)], being the distance from the invariant torus. We also discuss the connection of this result with the existence of many invariant tori close to the considered one.     

Meromorphic singular foliations on complex projective surfaces

Here we show that many numerical computations and bounds on the degrees of the algebraic leaves for singular meromorphic foliations on ² may be extended to large classes of foliations and complex projective surfaces.The author was partially supported by MURST and GNSAGA of CNR (Italy)     

Optimal life strategies in organisms exposed to recurrent critical events

The optimal partition of energy between survival and reproduction is considered for a population subject to recurrent and potentially lethal critical events. The best strategy is found by maximizing fitness, a functional derived from the Lotka equation. The dynamics is governed by a second-order, age-varying, nonlinear system. The energy storage and the probability of survival are the state variables, while the amounts of energy placed into and withdrawn from the storage are the controls. The optimal life strategy is shown to be as follows: build up the storage at the very beginning of life, and then progressively deplete it to resist the critical events.This work was partially supported by MURST Project Ricerche sui Fondamenti della Conservazione della Natura. The authors would like to thank C. Ricci for suggesting the problem and C. Matessi for helpful discussion.     

An example of applications of the elementary catastrophe theory in Hamiltonian systems

The convection in atmosphere discussed in ref. [1] is rigorously treated by considering the variation of environmental temperature with the height. This represents an example of applications of the elementary catastrophe theory in Hamiltonian systems.     

Synthesis of new functional derivatives of pyrazolidine from 1-acetyl-3-methyl-5-nitromethyl-2-phenylpyrazolidine

3-Nitromethylpyrazolidine reacts with derivatives of unsaturated acids and reducing agents with the formation of polyfunctional compounds of the pyrazolidine nucleus. __________ Translated from Khimiya Geterotsiklicheskikh Soedinenii, No. 9, pp. 1365–1369, September, 2006.     

Notiz zur Darstellung von 4-Hydroxy-6-methyl-5,6-dihydro-2H-pyran-2-on

Meldrum's acid (2) is acylated by diketen affording the acylMeldrum's acid3. In aqueous bicarbonate its 3-oxogroup is selectively reduced by sodium tetrahydroborate giving the alcohol4, which readily undergoes cyclization in refluxing dioxane leading to the title pyrone7. Under identical conditions3 produces the pyrone6 with the pyrone carboxylic acid5 as intermediate.

     

Direct Electrochemistry of Multi‐Copper Oxidases at Carbon Nanotubes Noncovalently Functionalized with Cellulose Derivatives

This study describes the direct electron transfer of multi‐copper oxidases, i.e., laccase (from Trametes versicolor) and bilirubin oxidase (BOD, from Myrothecium verrucaria) at multiwalled carbon nanotubes (MWNTs) noncovalently functionalized with biopolymers of cellulose derivatives, i.e., hydroxyethyl cellulose (HEC), methyl cellulose (MC), and carboxymethyl cellulose (CMC). The functionalization of the MWNTs with the cellulose derivatives is found to substantially solubilize the MWNTs into aqueous media and to avoid their aggregation on electrode surface. Under anaerobic conditions, the redox properties of laccase and BOD are difficult to be defined with cyclic voltammetry at either laccase/MWNT‐modified or BOD/MWNT‐modified electrodes. The direct electron transfer properties of laccase and BOD are thus studied in terms of the bioelectrocatalytic activities of the laccase/MWNT‐modified and BOD/MWNT‐modified electrodes toward the reduction of oxygen and found to be facilitated at the functionalized MWNTs. The possible application of the laccase‐catalyzed O₂ reduction at the laccase/MWNT‐modified electrode is illustrated by constructing a CNT‐based ascorbate/O₂ biofuel cell with the MWNT‐modified electrode as the anode for the oxidation of ascorbate biofuel.     

Effect of conformational control of chiral oxazaborolidine by π-π stacking interaction of a pentafluorophenyl group toward asymmetric borane reduction

A pentafluorophenyl group can act as a stereo-controlling group in oxazaborolidine-catalyzed asymmetric borane reduction through intramolecular π-π stacking interaction with a phenyl group. The intramolecular π-π interaction in oxazaborolidine bearing pentafluorophenyl group is confirmed by calculations and ¹H NMR study. The interaction affects the enantioselectivity of the asymmetric reduction of acetophenone while the extent is small.