Computing Zeta Functions of Kummer Curves via Multiplicative Characters

We present a practical polynomial-time algorithm for computing the zeta function of a Kummer curve over a finite field of small characteristic. Such algorithms have recently been obtained using a method of Kedlaya based upon Monsky–Washnitzer cohomology, and are of interest in cryptography. We take a different approach. The problem is reduced to that of computing the L-function of a multiplicative character sum. This latter task is achieved via a cohomological formula based upon the work of Dwork and Reich. We show, however, that our method and that of Kedlaya are very closely related.Dedicated to the memory of Gian-Carlo Rota     

Pd-catalyzed one-pot coupling of allylamines, activated alkenes, and unsaturated halides (or triflate): an atom efficient synthesis of highly functionalized pyrrolidines

A Pd-catalyzed three-component assembling of highly functionalized 4-benzyl-(and allyl-)pyrrolidines was achieved based on a combination of allyl amines, gem-diactivated alkenes, and unsaturated halides (or triflate).     

Critical Point Theorems and Applications to Differential Equations

This paper contains a generalization of the well-known Palais-Smale and Cerami compactness conditions. The compactness condition introduced is used to prove some general existence theorems for critical points. Some applications are given to differential equations.     

The study of the influence of uniaxial stress on impurity complexes in silicon

The influence of external uniaxial stress on the different indium-donor complexes in silicon has been studied using the perturbed γ –γ angular correlation (PAC) method. Such effect of an applied stress is detected by means of the probe atoms situated at different complexes in the sample. The current results showed that the responses of the probes in an extrinsic silicon samples are found to be dissimilar for the same value of stress. Such change in the local environments of the probe atoms could be associated with the various strain field created by the implantations of varied size of impurities. The phosphorous implantation in silicon has even lead to the complete absence of observable effect of the applied stress suggesting significant lose of the elasticity of the sample.     

Effects of aldehyde or dipyrromethane substituents on the reaction course leading to meso-substituted porphyrins

To better understand the effects of diverse substituents on reactions leading to porphyrins, pyrrole+aldehyde condensations and related reactions of dipyrromethanes were examined. The course of pyrrole+aldehyde condensations was investigated by monitoring the yield of porphyrin (by UV-Vis spectroscopy), reaction of aldehyde (by TLC), and changes in the composition of oligomers (by laser desorption mass spectrometry). Reaction reversibility was examined via exchange experiments. Reversibility of reactions leading to porphyrin was further probed with studies of dipyrromethanes. The reaction course was found to depend on the nature of the substituent and the acid catalyst. Alkyl or electron-donating substituents displayed levels of reversibility (exchange/scrambling) on par or greater than that of the phenyl substituent, whereas electron-withdrawing or sterically bulky substituents exhibited little to no reversibility. The results obtained provide insight into the electronic and steric effects of different substituents and should facilitate the design of synthetic plans for preparing porphyrinic macrocycles.     

Synthesis and conformational investigation of tetrapeptide analogues of the fragment B23-B26 of insulin

Tetrapeptides containing one of a set of four different α,α-dialkyl glycines at the C-terminus were synthesized by conventional methods in solution and their conformational behavior investigated by ¹H NMR spectroscopy in connection with molecular mechanics calculations. The results were consistent with conformations stabilized by a γ-turn in the case of compounds with alkyl groups larger than methyl, while the corresponding Aib derivative did not exhibit intramolecular hydrogen bonding.     

Synthesis of bioactive indolocarbazoles: synthesis, nucleophilic ring-opening and chiral base desymmetrisation of a cyclic sulfate intermediate

A number of new functionalised bridged indolocarbazole systems have been prepared by ring-opening reactions of a key cyclic sulfate intermediate, prepared from the corresponding diol by the action of sulfuryl diimidazole and DBU. The same cyclic sulfate also undergoes an unprecedented asymmetric rearrangement to a chiral ketone, on treatment with a chiral lithium amide base.     

Fractional Derivatives Applied to Phase-Space Reconstructions

The concept and application of phase-space reconstructions are reviewed. Fractional derivatives are then proposed for the purpose of reconstructing dynamics from a single observed time history. A procedure is presented in which the fractional derivatives of time series data are obtained in the frequency domain. The method is applied to the Lorenz system. The ability of the method to unfold the data is assessed by the method of global false nearest neighbors. The reconstructed data is used to compute recurrences and correlation dimensions. The reconstruction is compared to the commonly used method of delays in order to assess the choice of reconstruction parameters, and also the quality of results.