Solutions of cubic equations in quadratic fields

Let K be any quadratic field with O-K its ring of integers. We study the solutions of cubic equations, which represent elliptic curves defined over Q, in quadratic fields and prove some interesting results regarding the solutions by using elementary tools. As an application we consider the Diophantine equation r + s + t = rst = 1 in O-K. This Diophantine equation gives an elliptic curve defined over Q with finite Mordell-Weil group. Using our study of the solutions of cubic equations in quadratic fields we present a simple proof of the fact that except for the ring of integers of Q(i) and Q(root 2), this Diophantine equation is not solvable in the ring of integers of any other quadratic fields, which is already proved in [4].

Functional Significance of an Evolutionarily Conserved Alanine (GCA) Resume Codon in tmRNA in Escherichia coli

Occasionally, ribosomes stall on mRNAs prior to the completion of the polypeptide chain. In Escherichia coli and other eubacteria, tmRNA-mediated trans-translation is a major mechanism that recycles the stalled ribosomes. The tmRNA possesses a tRNA-like domain and a short mRNA region encoding a short peptide (ANDENYALAA in E. coli) followed by a termination codon. The first amino acid (Ala) of this peptide encoded by the resume codon (GCN) is highly conserved in tmRNAs in different species. However, reasons for the high evolutionary conservation of the resume codon identity have remained unclear. In this study, we show that changing the E. coli tmRNA resume codon to other efficiently translatable codons retains efficient functioning of the tmRNA. However, when the resume codon was replaced with the low-usage codons, its function was adversely affected. Interestingly, expression of tRNAs decoding the low-usage codon from plasmid-borne gene copies restored efficient utilization of tmRNA. We discuss why in E. coli, the GCA (Ala) is one of the best codons and why all codons in the short mRNA of the tmRNA are decoded by the abundant tRNAs.

Multiple scales without center manifold reductions for delay differential equations near Hopf bifurcations

We study small perturbations of three linear Delay Differential Equations (DDEs) close to Hopf bifurcation points. In analytical treatments of such equations, many authors recommend a center manifold reduction as a first step. We demonstrate that the method of multiple scales, on simply discarding the infinitely many exponentially decaying components of the complementary solutions obtained at each stage of the approximation, can bypass the explicit center manifold calculation. Analytical approximations obtained for the DDEs studied closely match numerical solutions.

Manipulation of the Hydration Levels in Minerals of Sodium Cadmium Bisulfate toward the Design of Functional Materials

Several variants of hydrated sodium cadmium bisulfate, Na(2)Cd(2)(SO(4))(3) center dot 3H(2)O, Na(2)Cd(SO(4))(2) center dot 2H(2)O, and Na(2)Cd(SO(4))(2) center dot 4H(2)O have been synthesized, and their thermal properties followed by phase transitions have been invesigated. The formation of these phases depends on the stochiometry and the time taken for crystallization from water. Na(2)Cd(2)(SO(4))(3)center dot 3H(2)O, which crystallizes in the trigonal system, space group P3c, is grown from the aqueous solution in about four weeks. The krohnkite type mineral Na(2)Cd(SO(4))(2) center dot 2H(2)O and the mineral astrakhanite, also known as blodite, Na(2)Cd (SO(4))(2)center dot 4H(2)O, crystallize concomittantly in about 24 weeks. Both these minerals belong to the monoclinic system(space group P2(1)/c). Na(2)Cd(2)(SO(4))(3)center dot 3H(2)O loses water completely when heated to 250 degrees C and transforms to a dehydrated phase (cubic system, space group I (4) over bar 3d) whose structure has been established using ab initio powder diffration techniques. Na(2)Cd(SO(4))(2)center dot 2H(2)O transforms to alpha-Na(2)Cd(SO(4))(2) (space group C2/c) on heating to 150 degrees C which is a known high ionic conductor and remains intact over prolonged periods of exposure to moisture (over six months). However, when alpha-Na(2)Cd(SO(4))(2) is heated to 570 degrees C followed by sudden quenching in liquid nitrogen beta-Na(2)Cd(SO(4))(2) (P2(1)/c) is formed. beta-Na(2)Cd(SO(4))(2) takes up water from the atmosphere and gets converted completely to the krohnkite type mineral in about four weeks. Further, beta-Na(2)Cd(SO(4))(2) has a conductivity behavior comparable to the a-form up to 280 degrees C, the temperature required for the transformation of the beta- to alpha-form. These experiments demonstrate the possibility of utilizing the abundantly available mineral sources as precursors to design materials with special properties.

Surfactant-mediated Synthesis of Functional Metal Oxide Nanostructures via Microwave Irradiation-assisted Chemical Synthesis

Nanostructured materials have attracted considerable interest in recent years due to their properties which differ strongly from their bulk phase and potential applications in nanoscale electronic and optoelectronic devices. Metal oxide nanostructures can be synthesized by variety of different synthesis techniques developed in recent years such as thermal decomposition, sol-gel technique, chemical coprecipitation, hydrothermal process, solvothermal process, spray pyrolysis, polyol process etc. All the above processes go through a tedious synthesis procedure followed by prolonged heat treatment at elevated temperature and are time consuming. In the present work we describe a rapid microwave irradiation-assisted chemical synthesis technique for the growth of nanoparticles, nanorods, and nanotubes of a variety of metal oxides in the presence of an appropriate surfactant, without the use of any templates The method is simple, inexpensive, and helps one to prepare nanostructures in a very simple way, and in a very short time, measured in minutes. The synthesis procedure employs high quality metalorganic complexes (typically -diketonates) featuring a direct metal-to-oxygen bond in its molecular structure. The complex is dissolved in a suitable solvent, often with a surfactant added, and the solution then subjected to microwave irradiation in a domestic microwave oven operating at 2.45 GHz frequency with power varying from 160-800 W, from a few seconds to a few minutes, leading to the formation of corresponding metal oxides. This method has been used successfully to synthesize nanostructures of a variety of binary and ternary metal oxides such as ZnO, CdO, Fe2O3, CuO, Ga2O3, Gd2O3, ZnFe2O4, etc. There is an observed variation in the morphology of the nanostructures with the change of different parameters such as microwave power, irradiation time, appropriate solvent, surfactant type and concentration. Cationic, anionic, nonionic and polymeric surfactants have been used to generate a variety of nanostructures. Even so, to remove the surfactant, there is either no need of heat treatment or a very brief exposure to heat suffices, to yield highly pure and crystalline oxide materials as prepared. By adducting the metal complexes, the shape of the nanostructures can be controlled further. In this manner, very well formed, single-crystalline, hexagonal nanorods and nanotubes of ZnO have been formed. Adducting the zinc complex leads to the formation of tapered ZnO nanorods with a very fine tip, suitable for electron emission applications. Particle size and their monodispersity can be controlled by a suitable choice of a precursor complex, the surfactant, and its concentration. The resulting metal oxide nanostructures have been characterized by X-ray diffraction, field emission scanning electron microscopy, transmission electron microscopy, FTIR spectroscopy, photoluminescence, and electron emission measurements.

Pseudodynamic systems approach based on a quadratic approximation of update equations for diffuse optical tomography

We explore a pseudodynamic form of the quadratic parameter update equation for diffuse optical tomographic reconstruction from noisy data. A few explicit and implicit strategies for obtaining the parameter updates via a semianalytical integration of the pseudodynamic equations are proposed. Despite the ill-posedness of the inverse problem associated with diffuse optical tomography, adoption of the quadratic update scheme combined with the pseudotime integration appears not only to yield higher convergence, but also a muted sensitivity to the regularization parameters, which include the pseudotime step size for integration. These observations are validated through reconstructions with both numerically generated and experimentally acquired data. (C) 2011 Optical Society of America

Network Rhythms Influence the Relationship between Spike-Triggered Local Field Potential and Functional Connectivity

Characterizing the functional connectivity between neurons is key for understanding brain function. We recorded spikes and local field potentials (LFPs) from multielectrode arrays implanted in monkey visual cortex to test the hypotheses that spikes generated outward-traveling LFP waves and the strength of functional connectivity depended on stimulus contrast, as described recently. These hypotheses were proposed based on the observation that the latency of the peak negativity of the spike-triggered LFP average (STA) increased with distance between the spike and LFP electrodes, and the magnitude of the STA negativity and the distance over which it was observed decreased with increasing stimulus contrast. Detailed analysis of the shape of the STA, however, revealed contributions from two distinct sources-a transient negativity in the LFP locked to the spike (similar to 0 ms) that attenuated rapidly with distance, and a low-frequency rhythm with peak negativity similar to 25 ms after the spike that attenuated slowly with distance. The overall negative peak of the LFP, which combined both these components, shifted from similar to 0 to similar to 25 ms going from electrodes near the spike to electrodes far from the spike, giving an impression of a traveling wave, although the shift was fully explained by changing contributions from the two fixed components. The low-frequency rhythm was attenuated during stimulus presentations, decreasing the overall magnitude of the STA. These results highlight the importance of accounting for the network activity while using STAs to determine functional connectivity.

Structural basis for the functional and inhibitory mechanisms of beta-hydroxyacyl-acyl carrier protein dehydratase (FabZ) of Plasmodium falciparum

The beta-hydroxyacyl-acyl carrier protein dehydratase of Plasmodium falciparum (PfFabZ) catalyzes the third and important reaction of the fatty acid elongation cycle. The crystal structure of PfFabZ is available in hexameric (active) and dimeric (inactive) forms. However, PfFabZ has not been crystallized with any bound inhibitors until now. We have designed a new condition to crystallize PfFabZ with its inhibitors bound in the active site, and determined the crystal structures of four of these complexes. This is the first report on any FabZ enzyme with active site inhibitors that interact directly with the catalytic residues. Inhibitor binding not only stabilized the substrate binding loop but also revealed that the substrate binding tunnel has an overall shape of ``U''. In the crystal structures, residue Phe169 located in the middle of the tunnel was found to be in two different conformations, open and closed. Thus, Phe169, merely by changing its side chain conformation, appears to be controlling the length of the tunnel to make it suitable for accommodating longer substrates. The volume of the substrate binding tunnel is determined by the sequence as well as by the conformation of the substrate binding loop region and varies between organisms for accommodating fatty acids of different chain lengths. This report on the crystal structures of the complexes of PfFabZ provides the structural basis of the inhibitory mechanism of the enzyme that could be used to improve the potency of inhibitors against an important component of fatty acid synthesis common to many infectious organisms. (C) 2011 Elsevier Inc. All rights reserved.

A Multidimensional Kinetic Upwind Method for Euler Equations

The study of directional derivative lead to the development of a rotationally invariant kinetic upwind method (KUMARI)3 which avoids dimension by dimension splitting. The method is upwind and rotationally invariant and hence truly multidimensional or multidirectional upwind scheme. The extension of KUMARI to second order is as well presented.