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].

Synthesis of metastable, wurzite-based zinc oxide-cobalt(II) oxide solid solutions by spray pyrolysis

Aqueous solutions of acetates and nitrates of zinc and cobalt have been spray decomposed to study the production of extended solid solutions in the ZnO-CoO system. Examination of the products of a variety of synthesis conditions indicates that up to 70% CoO may be retained in the solid solution in the wurzite phase, even though a comparison of the equilibrium solubility in the phase diagram might be expected to favor the formation of a rock-salt-based solid solution.

Synthesis of single-walled carbon nanotubes using binary (Fe, Co, Ni) alloy nanoparticles prepared in situ by the reduction of oxide solid solutions

Passing a H-2-CH4 mixture over oxide spinels containing two transition elements as in Mg0.8MyMz'Al2O4 (M, M' = Fe, Co or Ni, y + z = 0.2) at 1070 degrees C produces small alloy nanoparticles which enable the formation of carbon nanotubes. Surface area measurements are found to be useful for assessing the yield and quality of the nanotubes. Good-quality single-walled nanotubes (SWNTs) have been obtained in high yields with the FeCo alloy nanoparticles, as evidenced by transmission electron microscope images and surface area measurements. The diameter of the SWNTs is in the 0.8-5 nm range, and the multiwalled nanotubes, found occasionally, possess very few graphite layers. (C) 1999 Elsevier Science B.V. All rights reserved.

Low-pressure MOCVD of Al2O3 films using aluminium acetylacetonate as precursor: nucleation and growth

A study of the deposition of aluminium oxide films by low-pressure metalorganic chemical vapour deposition from the complex aluminium acetylacetonate, in the absence of an oxidant gas, has been carried out. Depositions on to Si(100), stainless steel, and TiN-coated cemented carbide are found to be smooth, shiny, and blackish. SIMS, XPS and TEM analyses reveal that films deposited at temperatures as low as 600 degreesC contain small crystallites Of kappa-Al2O3, embedded in an amorphous matrix rich in graphitic carbon. Optical and scanning electron microscopy reveal a surface morphology made up of spherulites that suggests that film growth might involve a melting process. A nucleation and growth mechanism, involving the congruent melting clusters of precursor molecules on the hot substrate surface, is therefore invoked to explain these observations. An effort has been made experimentally to verify this proposed mechanism. (C) 2002 Elsevier Science B.V. All rights reserved.

Microwave heating in multiphase systems: evaluation of series solutions

The 1D electric field and heat-conduction equations are solved for a slab where the dielectric properties vary spatially in the sample. Series solutions to the electric field are obtained for systems where the spatial variation in the dielectric properties can be expressed as polynomials. The series solution is used to obtain electric-field distributions for a binary oil-water system where the dielectric properties are assumed to vary linearly within the sample. Using the finite-element method temperature distributions are computed in a three-phase oil, water and rock system where the dielectric properties vary due to the changing oil saturation in the rock. Temperature distributions predicted using a linear variation in the dielectric properties are compared with those obtained using the exact nonlinear variation.

Novel phase-transition behavior near liquid/liquid critical points of aqueous solutions: Formation of a third phase at the interface

We report a novel phase behavior in aqueous solutions of simple organic solutes near their liquid/liquid critical points, where a solid-like third phase appears at the liquid/liquid interface. The phenomenon has been found in three different laboratories. It appears in many aqueous systems of organic solutes and becomes enhanced upon the addition of salt to these solutions.

Analytical solutions for heat transfer during cyclic melting and freezing of a phase change material used in electronic or electrical packaging

In this paper we develop an analytical heat transfer model, which is capable of analyzing cyclic melting and solidification processes of a phase change material used in the context of electronics cooling systems. The model is essentially based on conduction heat transfer, with treatments for convection and radiation embedded inside. The whole solution domain is first divided into two main sub-domains, namely, the melting sub-domain and the solidification sub-domain. Each sub-domain is then analyzed for a number of temporal regimes. Accordingly, analytical solutions for temperature distribution within each subdomain are formulated either using a semi-infinity consideration, or employing a method of quasi-steady state, depending on the applicability. The solution modules are subsequently united, leading to a closed-form solution for the entire problem. The analytical solutions are then compared with experimental and numerical solutions for a benchmark problem quoted in the literature, and excellent agreements can be observed.

Minimum error solutions of the Boltzmann equation for shock structure

‘Best’ solutions for the shock-structure problem are obtained by solving the Boltzmann equation for a rigid sphere gas by applying minimum error criteria on the Mott-Smith ansatz. The use of two such criteria minimizing respectively the local and total errors, as well as independent computations of the remaining error, establish the high accuracy of the solutions, although it is shown that the Mott-Smith distribution is not an exact solution of the Boltzmann equation even at infinite Mach number. The minimum local error method is found to be particularly simple and efficient. Adopting the present solutions as the standard of comparison, it is found that the widely used v2x-moment solutions can be as much as a third in error, but that results based on Rosen's method provide good approximations. Finally, it is shown that if the Maxwell mean free path on the hot side of the shock is chosen as the scaling length, the value of the density-slope shock thickness is relatively insensitive to the intermolecular potential. A comparison is made on this basis of present results with experiment, and very satisfactory quantitative agreement is obtained.

Effect of Nature of the Precursor on Crystallinity and Microstructure of MOCVD-Grown ZrO2 Thin Films

In the present work, we report the deposition of zirconia thin films on Si(100) at various substrate temperatures by low-pressure metalorganic chemical vapor deposition (MOCVD). Three different zirconium complexes, viz., tetrakis(2,4-pentadionato)zirconium(IV), [Zr(pd)4], tetrakis(2,2,6,6-tetramethyl-3,5-heptadionato)zirconium(IV), [Zr(thd)4], and tetrakis(t-butyl-3-oxo-butanoato)zirconium(IV), [Zr(tbob)4] are used as precursors. The relationship between the molecular structures of the precursors and their thermal properties, as examined by TG/DTA is presented. The films deposited using these precursors have distinctly different morphology, though all of them are of the cubic phase. The films grown from Zr(thd)4 are well crystallized, showing faceted growth at 575°C, whereas the films grown from Zr(pd)4 and Zr(tbob)4 are not well crystallized, and display cracks. These differences in the observed microstructure may be attributed to the different chemical decomposition pathways of the precursors during the film growth, which influence the nucleation and the growth processes. This is also evidenced by the different kinetics of growth from these three precursors under otherwise identical CVD conditions. The details of thin film deposition, and film microstructure analysis by XRD and SEM is presented. The dielectric behavior of the films deposited from different precursors, as studied by C-V measurements, are compared.

Analytical solutions for protective filters based on soil-retention and permeability criteria with respect to the phenomenon of soil boiling

For the successful performance of a granular filter medium, existing design guidelines, which are based on the particle size distribution (PSD) characteristics of the base soil and filter medium, require two contradictory conditions to be satisfied, viz., soil retention and permeability. In spite of the wider applicability of these guidelines, it is well recognized that (i) they are applicable to a particular range of soils tested in the laboratory, (ii) the design procedures do not include performance-based selection criteria, and (iii) there are no means to establish the sensitivity of the important variables influencing performance. In the present work, analytical solutions are developed to obtain a factor of safety with respect to soil-retention and permeability criteria for a base soil - filter medium system subjected to a soil boiling condition. The proposed analytical solutions take into consideration relevant geotechnical properties such as void ratio, permeability, dry unit weight, effective friction angle, shape and size of soil particles, seepage discharge, and existing hydraulic gradient. The solution is validated through example applications and experimental results, and it is established that it can be used successfully in the selection as well as design of granular filters and can be applied to all types of base soils.

Microwave ECR Plasma Assisted MOCVD of Y(2)O(3) Thin Films Using Y(tod)(3) Precursor and Their Characterization

Yttrium oxide (Y(2)O(3)) thin films were deposited by microwave electron cyclotron resonance (ECR) plasma assisted metal organic chemical vapour deposition (MOCVD) process using indigenously developed metal organic precursors Yttrium 2,7,7-trimethyl-3,5-octanedionates, commonly known as Y(tod)(3) which were synthesized by an ultrasound method. A series of thin films were deposited by varying the oxygen flow rate from 1-9 sccm, keeping all other parameters constant. The deposited coatings were characterized by X-ray photoelectron spectroscopy, glancing angle X-ray diffraction and infrared spectroscopy. Thickness and roughness for the films were measured by stylus profilometry. Optical properties of the coatings were studied by the spectroscopic ellipsometry. Hardness and elastic modulus of the films were measured by nanoindentation technique. Being that microwave ECR CVD process is operating-pressure-sensitive, optimum oxygen activity is very essential for a fixed flow rate of precursor, in order to get a single phase cubic yttrium oxide in the films. To the best of our knowledge, this is the first effort that describes the use of Y(tod)(3) precursor for deposition of Y(2)O(3) films using plasma assisted CVD process.

Beta-structure of polypeptides in non-aqueous solutions. I. Spectral characteristics of the polypeptide backbone

The optical rotatory features of the beta-structure of the polypeptides in non-aqueous solutions and films cast from these solutions have been investigated. The beta-structure of poly-S-benzyl-L-cysteine, poly-S-carbobenzoxy-L-cysteine and poly-S-benzyl-L-cysteine, poly-S-carbobenzoxy-L-cysteine and poly-O-carbo-bands of their films. The optical rotatory dispersion (ORD) and circular dichroism (CD) spectra of these polypeptides are found to be very similar in both film and solution. In solvents promoting the beta-structure, the polypeptides are characterized by CD troughs in the n-pi* transition region of the peptide chromophore. The ORD spectra are found to be positive in sign throughout the visible and accessible ultraviolet regions and are interpreted in terms of the possible existence of a relatively much larger positive pi-pi* CD bands as compared with the negative n-pi* band. The rotatory data obtained in the non-aqueous solution are compared with those obtained for other poly peptides in aqueous solutions, with respect to the type and extent of beta-structure present.

Beta-structure of polypeptides in non-aqueous solutions. II. Side-chain orientation

The specific side-chain orientations of the phenyl group in the polypeptides poly-S-benzyl-L-cysteine, poly-S-carbobenzoxy-L-cysteine and poly-O-carbobenzoxy-L-serine in the beta-structure have been studied by spectral measurements in solutions. All the three polypeptides exhibit aromatic CD bands, indicating the asymmetric placement of the side-chain phenyl rings when the polypeptide backbone takes up the antiparallel beta-structure. Supporting evidence for this is derived from n.m.r. spectra of the polypeptides, which show upfield shift of the phenyl protons due to the stacking of the aromatic rings. Molecular model building studies reveal the stacking of alternate phenyl groups along the polypeptide chain.

Closed form solutions for element equilibrium and flexibility matrices of eight node rectangular plate bending element using integrated force method

Closed form solutions for equilibrium and flexibility matrices of the Mindlin-Reissner theory based eight-node rectangular plate bending element (MRP8) using integrated Force Method (IFM) are presented in this paper. Though these closed form solutions of equilibrium and flexibility matrices are applicable to plate bending problems with square/rectangular boundaries, they reduce the computational time significantly and give more exact solutions. Presented closed form solutions are validated by solving large number of standard square/rectangular plate bending benchmark problems for deflections and moments and the results are compared with those of similar displacement-based eight-node quadrilateral plate bending elements available in the literature. The results are also compared with the exact solutions.