In this paper, the boundedness of Calderón-Zygmund operators is obtained on Morrey-Herz spaces with variable exponents \(MK_{q(\cdot),p(\cdot)}^{\alpha(\cdot),\lambda}(\mathbb{R}^{n})\) and several norm inequalities for the commutator generated by Calderó-Zygmund operators, BMO function and Lipschitz function are given.

The present study is based on the nonlinear analysis of unsteady magnetohydrodynamics squeezing flow and heat transfer of a third grade fluid between two parallel disks embedded in a porous medium under the influences of thermal radiation and temperature jump boundary conditions are studied using Chebyshev spectral collocation method. The results of the non-convectional numerical solutions verified with the results of numerical solutions using fifth-order Runge-Kutta Fehlberg-shooting method and also the results of homotopy analysis method as presented in literature. The parametric studies from the series solutions show that for a suction parameter greater than zero, the radial velocity of the lower disc increases while that of the upper disc decreases as a result of a corresponding increase in the viscosity of the fluid from the lower squeezing disc to the upper disc. An increasing magnetic field parameter, the radial velocity of the lower disc decreases while that of the upper disc increases. As the third-grade fluid parameter increases, there is a reduction in the fluid viscosity thereby increasing resistance between the fluid molecules. There is a recorded decrease in the fluid temperature profile as the Prandtl number increases due to decrease in the thermal diffusivity of the third-grade fluid. The results in this work can be used to advance the analysis and study of the behaviour of third grade fluid flow and heat transfer processes such as found in coal slurries, polymer solutions, textiles, ceramics, catalytic reactors, oil recovery applications etc.

We present, in a way quite accessible to undergraduate and graduate students, some basic and important facts about conics: parabola, ellipse and hyperbola. For each conic, we start by its definition, then consider tangent line and obtain an elementary proof of the reflexion property. We study intersection of tangents. We obtain the orthopic set for orthogonal tangents: the directrix for parabola and the Monge’s circle for ellipse and hyperbola. For ellipse and hyperbola we also consider intersection of tangents for parallel rays at points of intersection with the conic. Those analysis lead to geometric methods to draw conics. Finally we get the directrices for ellipse and hyperbola by considering intersections of tangents at endpoints of a secant passing through a focus.