We begin with the following question: given a closed disc D b C and a complexvalued function F ∈ C(D), is the uniform algebra on D generated by z and F equal to C(D) ? When F ∈ C1(D), this question is complicated by the presence of points in the surface S := graphD(F ) that have complex tangents. Such points are called CR singularities. Let p ∈ S be a CR singularity at which the order of contac...

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 boundary f...

2 Preliminaries 2 2.1 Homomorphism numbers and densities . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Local convergence of a graph sequence . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3 Chromatic polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.4 Subtree counts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.5 Weight...