‎This paper deals with the singular Sturm-Liouville expressions‎ ‎$ ‎ell y =‎ -‎y''+q(x)y=lambda y‎ ‎$‎ ‎on a finite interval‎, ‎where the potential function $q$ is real and‎ ‎has a singularity inside the interval‎. ‎Using the asymptotic estimates of a‎ ‎spectral fundamental system of solutions of Sturm-Liouville‎ ‎equation‎, ‎the asymptotic form of the solution of the‎ ‎equation (0.1) and the ...

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...

Abstract Conditional on the extended Riemann hypothesis, we show that with high probability, characteristic polynomial of a random symmetric $\{\pm 1\}$ -matrix is irreducible. This addresses question raised by Eberhard in recent work. The main innovation our work establishing sharp estimates regarding rank distribution -matrices over $\mathbb{F}_p$ for primes $2 < p \leq \exp(O(n^{1/4}))$ ....

Inverse polynomial images of [−1, 1], which consists of two Jordan arcs, are characterised by an explicit polynomial equation for the four endpoints of the arcs. MSC: 41A10; 30C10

A polynomial map F = (P,Q) ∈ Z[x, y]2 with Jacobian JF := PxQy −PyQx ≡ 1 has polynomial inverse of integer coefficients if the complex plane curve P = 0 has infinitely many integer points. 2000 Mathematical Subject Classification: 14R15.