The best uniform polynomial approximation of the checkmark function f(x)=|x−α| is considered, as α varies in (−1,1). For each fixed degree n, minimax error En(α) shown to be piecewise analytic α. In addition, feature n−1 linear decreasing/increasing sections, called V-shapes. points alternation set are proven and monotone increasing their dynamics completely characterized. We also prove a conje...

a class of new methods based on a septic non-polynomial spline function for the numerical solution one-dimensional bratu's problem are presented. the local truncation errors and the methods of order 2th, 4th, 6th, 8th, 10th, and 12th, are obtained. the inverse of some band matrixes are obtained which are required in proving the convergence analysis of the presented method. associated bound...

We study the computational complexity of the distance function associated with a polynomial-time computable two-dimensional domains, in the context of the Turing machine-based complexity theory of real functions. It is proved that the distance function is not necessarily computable even if a two-dimensional domain is polynomial-time recognizable. On the other hand, if both the domain and its co...

we consider the number of zeros of the integral $i(h) = oint_{gamma_h} omega$ of real polynomial form $omega$ of degree not greater than $n$ over a family of vanishing cycles on curves $gamma_h:$ $y^2+3x^2-x^6=h$, where the integral is considered as a function of the parameter $h$. we prove that the number of zeros of $i(h)$, for $0 < h < 2$, is bounded above by $2[frac{n-1}{2}]+1$.