The Graves Theorem Revisited II: Robust Convergence of the Newton Method

AMS subject classification: 65J15, 47H04, 90C30.

A solution of the trigonometric moment problem via Tagamlitzki's "Theorem of the Cones"

In 1952 Y. Tagamlitzki gave an elegant proof of the classical Bochner’s theorem on the positively definite functions. Unfortunately, he never published his proof. In this paper we consider a related but simpler problem, the trigonometric moment problem, by using Tagamlitzki’s approach.

On a Theorem by Van Vleck Regarding Sturm Sequences

In 1900 E. B. Van Vleck proposed a very efficient method to compute the Sturm sequence of a polynomial p (x) ∈ Z[x] by triangularizing one of Sylvester’s matrices of p (x) and its derivative p′(x). That method works fine only for the case of complete sequences provided no pivots take place. In 1917, A. J. Pell and R. L. Gordon pointed out this “weakness” in Van Vleck’s theorem, rectified it but did not extend his method, so that it also works in the cases of: (a) complete Sturm sequences with pivot, and (b) incomplete Sturm sequences. Despite its importance, the Pell-Gordon Theorem for polynomials in Q[x] has been totally forgotten and, to our knowledge, it is referenced by us for the first time in the literature. In this paper we go over Van Vleck’s theorem and method, modify slightly the formula of the Pell-Gordon Theorem and present a general triangularization method, called the VanVleck-Pell-Gordon method, that correctly computes in Z[x] polynomial Sturm sequences, both complete and incomplete.

Cayley-Hamilton Theorem for Matrices over an Arbitrary Ring

2000 Mathematics Subject Classification: 15A15, 15A24, 15A33, 16S50.

A Characterization Theorem for the K-functional Associated with the Algebraic Version of Trigonometric Jackson Integrals

2000 Mathematics Subject Classification: 41A25, 41A36.

Complex Analogues of the Rolle's Theorem

2000 Mathematics Subject Classification: 30C10.

Another Generalization of Arhangel'skii's Theorem

MSC 2010: 54A25, 54A35.

Extention of Apolarity and Grace Theorem

MSC 2010: 30C10

Growth Theorem and the Radius of Starlikeness of Close-to-Spirallike Functions

AMS Subj. Classification: 30C45

On a Generalization of a Theorem due to Larcombe on the Sum of a 3F2 Series

MSC 2010: 33C20

On Sandwich Theorem of Analytic Functions Involving Integral Operator

2000 Mathematics Subject Classification: 30C45

The Filippov-Wazewski relaxation theorem revisited

The converse statement of the Filippov-Wazewski relaxation theorem is proven, more precisely, two differential inclusions have the same closure of their solution sets if and only if the right-hand sides have the same convex hull. The idea of the proof is examining the contingent derivatives to the attainable sets.