On the Caffarelli‐Kohn‐Nirenberg inequalities: Sharp constants,existence (and nonexistence), and symmetry of extremal functions

Consider the following inequalities due to Caffarelli, Kohn, and Nirenberg [6] where, for N ≥ 3, −∞ < a < (N − 2)/2, a ≤ b ≤ a + 1, and p = 2N/(N − 2 + 2(b − a)). We shall answer some fundamental questions concerning these inequalities such as the best embedding constants, the existence and nonexistence of extremal functions, and their qualitative properties. While the case a ≥ 0 has been studied extensively and a complete solution is known, little has been known for the case a < 0. Our results for the case a < 0 reveal some new phenomena which are in striking contrast with those for the case a ≥ 0. Results for N = 1 and N = 2 are also given. © 2001 John Wiley & Sons, Inc.     

Stochastic simulations based on mixed finite elements for solute transport in groundwater

An accurate mixed finite element method to solve both flow and transport is developed for stochastic simulations of transport in saturated aquifers characterized by random log-hydraulic conductivity fields. The main advantage of the mixed finite element is that it is local mass conservative. Unlike in stochastic finite element methods, this approach yields concentration fields and concentration moments for samples of the random field. In this way, it will be possible to analyze the behavior of different ensemble average observables of the transport process as well as the behavior of their fluctuations. Results of the stochastic simulations described here can be used to assess the reliability for real cases of the ensemble average quantities provided by stochastic modeling of transport in groundwater. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical analyis of turbulent flow in a Coanda ejector

The task of this study is to investigate the influence of various geometric parameters and pressure ratios on the Coanda ejector performance. For numerically investigations we use an implicit formulation of the compressible Reynolds-average Navier-Stokes equations (RANS) for axisymmetric flow with a shear stress transport k − ω (SST model) turbulence model. The numerically results was obtained for a total pressure range 1-5 Bars, imposed at the reservoir inlet. The effect of various factors, such as, the pressure ratio, primary nozzle and ejector configurations on the system performance has been evaluated based on defined performance parameters. The numerical results have been compared with theoretical and experimental results for a given Coanda ejector configuration. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Supramolecular Structure - A Key Parameter for Cellulose Biodegradation

Summary: Three different cellulosic substrata, like microcrystalline cellulose, cotton cellulose and spruce dissolving pulp, were chosen for biodegradation. The kinetics of the enzymatic hydrolysis of these celluloses by Trichoderma reesei, has been investigated. The experiments proved the fact that both the morphological structure and the crystalline one are crucial to the process and the ratio of the reactions. In addition, in order to obtain the most accessible cellulose substratum it was studied the biodegradation of cellulose allomorphs of spruce dissolving pulp. The insoluble cellulose fraction remaining after enzymatic hydrolysis was examined by X-ray diffraction method and it was established the degree of crystallinity and the average crystallite size. The enzymatic degradation is also proved by the decrease in the degree of polymerization of hydrolyzed samples.     

Linearization of the MS response function: case study for metformin assay in plasma samples for bioequivalence purposes

Calibration data of LC-MS/MS rarely fit the pure least square regression model, especially for large concentration intervals. The response function of the MS instrument is corrected by weighted regression models or logarithms. The choice of a response linearization method is based on results produced through back-interpolation of experimental data and/or evaluation of correlation coefficients. Two bioequivalence studies carried out for pharmaceutical formulations containing metformin gave us the opportunity to appreciate the impact of the MS response linearization method (logarithm and 1/x weighted linear regression) on method quality characteristics. The sample preparation was based on protein precipitation with acetonitrile. Chromatographic separation was achieved on a Zorbax CN column (mobile phase acetonitrile and aqueous 10 m m ammonium acetate solution, pH 3.5). Tandem MS detection was performed on a triple quadrupole spectrometer equipped with an electrospray source, operated under positive-ion mode. The method was validated and used for evaluation of the bioequivalence of formulations containing 500 and 1000 mg metformin. The 500 mg metformin study used logarithms for linearization of the detector response, while the 1000 mg metformin study was based on 1/x linear weighted regression. Data resulting from validations and studies completion were compared with evaluate the impact of the response linearization on the method quality characteristics.     

Influence of the heat treatment conditions on the formation of CuFe2O4 from mechanical milled precursors oxides

Stoichiometric mixture of CuO and ??-Fe₂O₃ milled in air up to 30?h was subjected to different heat treatments. The evolution of the heat treated milled powders was investigated by X-ray diffraction (XRD). The CuFe₂O₄ was partially obtained by milling, the material consisting in a mixture of phases. By applying different heat treatments in air and in vacuum, for 2?C6?h, in 500?C800?°C temperature range the phases composition of the milled samples is changed. A heat treatment at 500?°C in vacuum favours the formation of delafossite (CuFeO₂) and tenorite (CuO) phases. If the same heat treatment is made in air, the CuFe₂O₄ phase formation with a cubic structure is favoured. Differential scanning calorimetry (DSC) investigation realised in Ar atmosphere revealed two large exothermic peaks. The first one is associated with the formation of the delafossite and tenorite phases and the second one with the formation of CuFe₂O₄. The XRD patterns of the samples subjected to the DSC measurements present maxima corresponding to the delafossite and cuprospinel (CuFe₂O₄) phases. For the heat treatment at 600?°C in air the phases present in the sample are the same as for the annealing performed at 500?°C: CuFe₂O₄, ??-Fe₂O₃ and CuO. The heat treatment in air at 800?°C leads to the complete reaction between the different phases and the formation of CuFe₂O₄ phase in whole the sample volume. The CuFe₂O₄ ferrite crystallises after this heat treatment in two crystal systems: cubic and tetragonal.     

A peridynamic formulation for transient heat conduction in bodies with evolving discontinuities

We introduce a multidimensional peridynamic formulation for transient heat-transfer. The model does not contain spatial derivatives and uses instead an integral over a region around a material point. By construction, the formulation converges to the classical heat transfer equations in the limit of the horizon (the nonlocal region around a point) going to zero. The new model, however, is suitable for modeling, for example, heat flow in bodies with evolving discontinuities such as growing insulated cracks. We introduce the peridynamic heat flux which exists even at sharp corners or when the isotherms are not smooth surfaces. The peridynamic heat flux coincides with the classical one in simple cases and, in general, it converges to it in the limit of the peridynamic horizon going to zero. We solve test problems and compare results with analytical solutions of the classical model or with other numerical solutions. Convergence to the classical solutions is seen in the limit of the horizon going to zero. We then solve the problem of transient heat flow in a plate in which insulated cracks grow and intersect thus changing the heat flow patterns. We also model heat transfer in a fiber-reinforced composite and observe transient but steep thermal gradients at the interfaces between the highly conductive fibers and the low conductivity matrix. Such thermal gradients can lead to delamination cracks in composites from thermal fatigue. The formulation may be used to, for example, evaluate effective thermal conductivities in bodies with an evolving distribution of insulating or permeable, possibly intersecting, cracks of arbitrary shapes.