On eigenfunctions of nonlinear heat generation

Summary We consider the equation u+ expu=0, >0,u(boundary)0 in the formv= exp (K,v), whereK ^–1=–. We give bounds on for the latter equation to be solvable by the contraction mapping principle, and estimate theL ² norm of the solution so obtained. We also give a bound on for the topological index of the solution to be non-zero and apply Krasnoselskii's results to the least squares method of approximating the solution.

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Sommario Consideriamo l'equazione u+ expu=0, >0,u(frontiera)=0 nella formav= exp (Kv), doveK ^–1=–. In questo lavoro diamo limitazioni per per cui la seconda equazione e risolubile col metodo delle contrazioni, e diamo una stima della norma inL ² della soluzione cosi ottenuta. Diamo anche una limitazione per per cui l'indice topologico della soluzione diventa non zero, e applichiamo i risultati di Krasnoselskii al metodo dei minimi quadrati per approssimare la soluzione.

     

Dications of benzylidenefluorene and diphenylmethylidene fluorene: the relationship between magnetic and energetic measures of antiaromaticity

Oxidation of m- and p-substituted benzylidene fluorenes to antiaromatic dications was attempted by electrochemical and chemical means. Electrochemical oxidation to dications was successful for benzylidene fluorenes with p-methoxy, p-methyl, p-fluoro, and unsubstituted phenyl rings in the 3-position; attempts to oxidize the m-substituted derivatives via electrochemistry were unsuccessful. Chemical oxidation with SbF(5)/SO(2)ClF gave the dication of 9-[(4-methoxyphenyl)methylene]-9H-fluorene cleanly; oxidation of all other substituted benzylidene fluorenes resulted in mixtures of products. The excellent linear relationship between the chemical shifts calculated by the GIAO method and the experimental shifts for the p-methoxy-substituted benzylidene fluorene dication suggests that the calculations satisfactorily reflect the magnetic properties of this dication and potentially those of the other dications studied. The redox potentials from electrochemical oxidation, a measure of the stability of the dications, showed a good linear relationship with another measure of stability, the calculated difference in energy between each dication and its neutral precursor. The dications of benzylidene fluorenes were less stable than the dications of diphenylmethylidene fluorenes; within each type of compound, dications with p-substituted phenyl rings were more stable than dications with m-substituted phenyl rings and dications with phenyl rings substituted with electron-donating groups were more stable than dications with phenyl rings substituted with electron-withdrawing groups. The antiaromaticity of the fluorenyl system was assessed through the nucleus-independent chemical shift (NICS) that was also calculated by the GIAO method. The plot of the NICS values per square area versus the calculated energy difference for the dications showed a moderate degree of linearity; the plot of NICS values per square area versus the oxidation potentials was less linear. Thus, a suggestive, but not conclusive, relationship between magnetic and energetic measures of antiaromaticity was observed.     

Topology control in the mobile ad hoc networks in order to intensify energy conservation

Although studied for years, due to their dynamic nature, research in the field of mobile ad hoc networks (MANETs) has remained a vast area of interest. Since once distributed, there will be less to no plausibility of recharge, energy conservation has become one of the pressing concerns regarding this particular type of network. In fact, one of the main obligations of designers is to make efficient use of these scarce resources. There has been tremendous work done in different layers of protocol stack in order to intensify energy conservation. To date, numerous topology control algorithms have been proposed, however, only a few have used meta-heuristics such as genetic algorithms, neural networks and/or learning automata to overcome this issue. On the other hand, since nodes are mobile and thus in a different spatial position, as time varies, we can expect that by regulating time intervals between topology controls, one may prolong the network’s lifetime. The main initiative of this paper is to intensify energy conservation in a mobile ad hoc network by using weighted and learning automata based algorithms. The learning automata, regulates time intervals between which the topology controls are done. The represented learning automata based algorithm uses its learning ability to find appropriate time-intervals so that the nodes would regulate the energy needed in order to exchange the information to their neighbors, accordingly. Moreover, at first we have represented two weighted based algorithms which extend two prominent protocols, namely K-Neigh and LMST. Then these algorithms are combined with a learning based algorithm which regulates time intervals between which the topology controls are done. In comparison with approaches that are based on periodic topology controls, proposed approach shows enhanced results. On the other hand, considering the learning ability of the learning automata based algorithms, composition of the aforementioned algorithms has been proven to be enhanced, in the respect of energy consumed per data transmitted, over those compared with.     

Entropy and escape of mass for SL<Subscript>3</Subscript>(ℤ)\ SL<Subscript>3</Subscript>(ℝ)

We study the relation between measure theoretic entropy and escape of mass for the case of a singular diagonal flow on the moduli space of three-dimensional unimodular lattices.     

Total weak unimodularity: testing and applications

The concept of total weak unimodularity for an integral matrix is introduced. Connections are shown to existing and well studied ideas such as TDI and integer rounding. Algorithms are provided for testing total weak unimodularity and producing integral solutions to inequality systems when they exist under the assumption of total weak unimodularity.     

Expedient synthesis of bisabolenol stink bug pheromones via stereodefined cyclohex-2-enones

We recently synthesized all stereoisomers of 1,10-bisaboladien-3-ol and 10,11-epoxy-1-bisabolen-3-ol, including three stink bug pheromones, via a rhodium-catalyzed asymmetric addition of trimethylaluminum to diastereomeric mixtures of cyclohex-2-enones. However, yields of trans isomers were low, and scaling reactions using expensive catalysts were cumbersome. Now we describe a new synthesis of bisabolenol stink bug pheromones via (S)- and (R)-4-((R)-6-methylhept-5-en-2-yl)cyclohex-2-enones prepared by enantioselective Michael additions of methyl vinyl ketone to (S)- and (R)-citronellals and lithium hydroxide monohydrate-catalyzed stereoselective cyclizations of intermediate ketoaldehydes. Addition of methyllithium to these enones provided cis- and trans-1,10-bisaboladien-3-ols, which were separated by chromatography on silica and further converted to 10,11-epoxy-1-bisabolen-3-ols. Thus, we developed more convenient syntheses of pheromones of the rice stink bug, the harlequin bug, and brown marmorated stink bug.     

Synthesis of a new fluorinated fluorescent β-cyclodextrin sensor

The synthesis of a new fluorescent sensor incorporating a fluorinated indolizine unit bound to 6-amino-β-cyclodextrin by two different synthetic ways is described. Its sensing ability toward adamantanol has been demonstrated.     

A note on Galerkin's method for nonlinear equations

Aequationes mathematicae -