We compare the best approximations of holomorphic functions in the Hardy space \(H^1\) by algebraic polynomials and trigonometric polynomials. Particulary, we establish a class of functions \(f\in H^1\) for which the best trigonometric approximation do not coincide with the best algebraic approximation.

This paper studies the global dynamics for discrete-time HIV infection models. The models integrate both long-lived chronically infected and short-lived infected cells. The HIV-susceptible incidence rate is taken as bilinear, saturation and general function. We discretize the continuous-time models by using nonstandard finite difference scheme. The positivity and boundedness of solutions are established. The basic reproduction number is derived. By using Lyapunov method, we prove the global stability of the models. Numerical simulations are presented to illustrate our theoretical results.

This paper studies the contact CR-warped product submanifolds of a generalized Sasakian space form admitting a nearly Sasakian structure. Some Characterization of the existence of these warped product submanifolds are also obtained. We illustrate that the warping function is a harmonic function under certain conditions. Moreover, a sharp estimate for the squared norm of the second fundamental form is investigated, and the equality case is also discussed. The results obtained in this paper generalize the results that have appeared in [I. Hasegawa, I. Mihai, Geom. Dedicata, \({\bf 102}\) (2003), 143--150], [I. Mihai, Geom. Dedicata, \({\bf 109}\) (2004), 165--173}], and [M. Atceken, Hacet. J. Math. Stat., \({\bf 44}\) (2015), 23--32].

An optimized two-step hybrid block method for the numerical solution of third-order initial value problems is presented. The method takes into regard three hybrid points which are selected suitably to optimize the local truncation errors of the main formulas for the block. The method is zero-stable and consistent with sixth algebraic order. Some numerical examples are debated to demonstrate the efficiency and the accuracy of the proposed method.

There are many systems that can handle a mix of series-parallel or parallel-series systems. Here, a new three-parameter distribution motivated mainly by dealing with series-parallel or parallel-series systems is introduced. Moments, conditional moments, mean deviations, moment generating function, quantile, Lorenz, and Bonferroni curves of the new distribution including are presented. Entropy measures are given and estimation of its parameters is studied. Two real data applications are described to show its superior performance versus some known lifetime models.

In this paper, we establish the existence of positive traveling waves solutions for the third order differential equation \(u_{t}+\alpha u_{xx}+\beta u_{xxx}+\left(f\left(x,u(x)\right)\right)_{x}=0\), where \(t,x\in\bf R\), \(f\) is a non-negative continuous function with some properties. The result is a consequence of the characterization of the travelling wave solutions as fixed points of some functional, defined using the Green's function associated to the linear problem, and the Krasnosel'skii fixed point theorem on cone expansion and compression of norm type.