Szegő’s theorem for matrix orthogonal polynomials

We extend some classical theorems in the theory of orthogonal polynomials on the unit circle to the matrix case. In particular, we prove a matrix analogue of Szeg?’s theorem. As a by-product, we also obtain an elementary proof of the distance formula by Helson and Lowdenslager.     

An elementary proof of Tverberg’s theorem

A new proof is suggested for Tverberg’s familiar theorem saying that an arbitrary set of q: (d + 1)(p − 1) + 1 points in ℝ ^d can be split into p parts such that their convex hulls have a nonempty intersection. Bibliography: 9 titles.     

Implementation of Pellet’s theorem

Pellet’s theorem determines when the zeros of a polynomial can be separated into two regions, based on the presence or absence of positive roots of an auxiliary polynomial, but does not provide a method to verify its conditions or to compute the roots of the auxiliary polynomial when they exist. We derive an explicit condition for these roots to exist and, when they do, propose efficient ways to compute them. A similar auxiliary polynomial appears for the generalized Pellet theorem for matrix polynomials and it can be treated in the same way.     

An analytical proof of Ogawa’s determinantal theorem

In this note, we propose a short proof, essentially based on differential calculus, of the Ogawa’s determinantal theorem, which itself is an improvement of another similar theorem.     

An alternative proof of Lickorish–Wallace theorem

We introduce the concept of s-distance of an unstabilized Heegaard splitting. We prove if a 3-manifold admits an unstabilized genus g Heegaard splitting with s-distance m  , then surgery on some (m−1)$(m-1)$ components link may produce a 3-manifold which admits a stabilized genus g Heegaard splitting. We also give an alternative proof of the fundamental theorem of surgery theory, which states that every closed orientable 3-manifold is obtained by surgery on some link in 3-sphere.     

An alternative proof of Painlevé's theorem

In this article we show some aspects of analytical and numerical solution of the n-body problem, which arises from the classical Newtonian model for gravitation attraction. We prove the non-existence of stationary solutions and give an alternative proof for Painlevé's theorem.     

A direct proof of Gromov’s theorem

A new proof of a theorem by Gromov is given: for any positive C and any integer n greater than 1, there exists a function Δ _C,n (δ) such that if the Gromov–Hausdorff distance between two complete Riemannian n-manifolds V and W is at most δ, their sectional curvaturcs |K _σ | do not exceed C, and their injectivity radii are at least 1/C, than the Lipschitz distance between V and W is less than Δ _C,n (δ), and Δ _C,n (δ) → 0 as δ → 0. Bibliography: 6 titles.     

A short proof of Levinson’s theorem

We give a short proof of Levinson’s result that over 1/3 of the zeros of the Riemann zeta function are on the critical line.     

A topological proof of Sklar’s theorem

We present a proof of Sklar’s Theorem that uses topological arguments, namely compactness (under the weak topology) of the class of copulas and some density properties of the class of distribution functions.     

New proof of a Calabi’s theorem

A Calabi’s theorem says that on a compact Riemann surface, an extremal metric is a constant scalar curvature metric. In this paper, we use a new method to prove this theorem. Then we give an interesting corollary.     

A new proof of Stetsenko’s theorem

A short proof of V.A. Stetsenko’s theorem on weakly read-many Boolean functions is given, which is based on the technique of representing read-once functions in the form of trees.     

Yet another proof of Marstrand’s theorem

In a paper from 1954 Marstrand proved that if K ⊂ ℝ² is a Borel set with Hausdorff dimension greater than 1, then its one-dimensional projection has positive Lebesgue measure for almost-all directions. In this article, we give a combinatorial proof of this theorem, extending the techniques developed in our previous paper [9].     

A proof of dilworth’s decomposition theorem for partially ordered sets

A short proof of the following theorem is given: LetP be a finite partially ordered set. If the maximal number of elements in an independent subset ofP isk, thenP is the union ofk chains.     

A proof of zelevinski’s structure theorem for psh algebras 1

We give a new proof of the structure theorem for PSH—algebras, and of a formula for primitives     

The matrix version of Hamburger’s theorem

Mathematical Notes - Necessary and sufficient conditions for the Stieltjes moment problem to have a unique solution and for the Hamburger moment problem with the same moments to have infinitely...