Radial orthogonality and Lebesgue constants on the disk

In polynomial interpolation, the choice of the polynomial basis and the location of the interpolation points play an important role numerically, even more so in the multivariate case. We explore the concept of spherical orthogonality for multivariate polynomials in more detail on the disk. We focus on two items: on the one hand the construction of a fully orthogonal cartesian basis for the space of multivariate polynomials starting from this sequence of spherical orthogonal polynomials, and on the other hand the connection between these orthogonal polynomials and the Lebesgue constant in multivariate polynomial interpolation on the disk. We point out the many links of the two topics under discussion with the existing literature. The new results are illustrated with an example of polynomial interpolation and approximation on the unit disk. The numerical example is also compared with the popular radial basis function interpolation.     

Temperature dependency of thermal conductivity of solid phases for fatty acids

Thermal conductivity variations with temperature of solid phases for lauric acid (LA), myristic acid (MA), pivalic acid (PA), and stearic acid (SA) have been measured with radial heat-flow method. Temperature dependencies of the thermal conductivity for same organic materials have been obtained by linear regression analysis. From graphs of thermal conductivity versus temperature, the thermal conductivity of solid phase at their melting temperature and temperature coefficients of thermal conductivity for LA, MA, PA, and SA have been found to be 0.37, 0.39, 0.23, and 0.35 W K^?1 m^?1 and 0.00935, 0.00446, 0.01095, and 0.00295 K^?1, respectively. The ratios of thermal conductivity of liquid phase to thermal conductivity of solid phase for LA, MA, PA, and SA have also been measured to be 0.52, 0.48, 0.25, and 0.59, respectively, with a Bridgman-type directional solidification apparatus.     

Scale-up synthesis of zinc borate from the reaction of zinc oxide and boric acid in aqueous medium

Synthesis of zinc borate was conducted in a laboratory and a pilot scale batch reactor to see the influence of process variables on the reaction parameters and the final product, 2ZnO·3B₂O₃·3.5H₂O. Effects of stirring speed, presence of baffles, amount of seed, particle size and purity of zinc oxide, and mole ratio of H₃BO₃:ZnO on the zinc borate formation reaction were examined at a constant temperature of 85 °C in a laboratory (4 L) and a pilot scale (85 L) reactor. Products obtained from the reaction in both reactors were characterized by chemical analysis, X-ray diffraction, particle size distribution analysis, thermal gravimetric analysis and scanning electron microscopy. The kinetic data for the zinc borate production reaction was fit by using the logistic model. The results revealed that the specific reaction rate, a model parameter, decreases with increase in particle size of zinc oxide and the presence of baffles, but increases with increase in stirring speed and purity of zinc oxide; however, it is unaffected with the changes in the amount of seed and reactants ratio. The reaction completion time is unaffected by scaling-up.     

N-methylpyrrolidine as an effective organocatalyst for the regioselective synthesis of 3-hydroxy-3,5/6-di-aryl-1H-imidazo[1,2-a]imidazol-2(3H)-ones

N-Methylpyrrolidine catalyzed, concise and attractive synthesis of a new class of 3-hydroxy-3,5/6-di-aryl-1H-imidazo[1,2-a]imidazol-2(3H)-ones was attained with impressive yields, in the presence of EtOH as a solvent, by means of a convenient and elegant condensation reaction between different aryl glyoxal monohydrates and guanidine hydrochloride under reflux conditions. Some specific merits of the current procedure, including encompasses low operating cost, availability of the starting substrates, reasonable reaction times, high reaction yield, operational simplicity, cleaner reaction profile, no harmful by-products, and the isolated product is in pure form. Structures of all the freshly synthesized products have been deduced by their FT-IR, ¹H-NMR, ¹³C-NMR, Mass spectrometry data and microanalysis.

Graphical abstract

     

Strength and ultrasonic properties of cemented paste backfill

This paper presents the strength (UCS) and ultrasonic pulse velocity (UPV) properties of cemented paste backfill (CPB) produced from two different mill tailings (Tailings T1 and T2). A total of 240 CPB samples with diameter × height of 5 × 10 cm and 10 × 20 cm prepared at different binder dosages (5–7 wt.%) and water-to-cement ratios (3.97–5.10) were subjected to the UPV and UCS tests at 7, 14, 28 and 56-days of curing periods. UCS and UPV of CPB samples increased with increasing the binder dosage and reducing the w/c ratio irrespective of the sample size and tailings type. CPB samples with a diameter × height of 5 × 10 cm were observed to produce consistently higher (up to 1.69-fold) UCSs than those of 10 × 20 cm CPB samples at all binder dosages and w/c ratios. However, at the corresponding binder dosages and w/c ratios, the maximum variation of UPV between the CPB samples of 5 × 10 cm and 10 × 20 cm was only 7.45%. Using the method of least squares regression, the UCS values were correlated with the UPV values for CPB samples of 10 × 20 cm in size. A linear relation with a high correlation coefficient appeared to exist between the UCS and UPV for CPB samples. These findings suggest that the UPV is essentially independent of the sample size. In this regard, the UPV test can be suitably exploited for the rapid estimation of the strength and quality of CPB samples even using small samples with concomitant benefits of reducing sample size.     

Surface pencil with a common line of curvature in Minkowski 3-space

In this paper, we analyze the problem of constructing a surface pencil from a given spacelie （timelike） line of curvature. Using the Frenet frame of the given line of curvature in Minkowski 3-space, we express the surface pencil as a linear combination of this frame and derive the necessary and sufficient conditions for the coefficients to satisfy the line of curvature requirement. We illustrate this method by presenting some examples.     

An experimental study on optimization of $${\mathbf{SiO}}_{2}$$ SiO 2 /water nanofluid flows in circular minichannels

Journal of Thermal Analysis and Calorimetry - An experimental research was performed to examine the effect of hydrodynamically and thermally developing distilled water-based SiO $$_2$$ nanofluid...