Related papers: From Rolle's theorem to the Sturm-Hurwitz theorem
In this note we give two proofs of Brooks' Theorem. The first is obtained by modifying an earlier proof and the second by combining two earlier proofs. We believe these proofs are easier to teach in Computer Science courses.
Simple proofs of the Hermite-Biehler and Routh-Hurwitz theorems are presented. The total nonnegativity of the Hurwitz matrix of a stable real polynomial follows as an immediate corollary.
Various generating functions of simple Hurwitz numbers of the projective line are known to satisfy many properties. They include a heat equation, the Eynard-Orantin topological recursion, an infinite-order differential equation called a…
Arguably the simplest variation of this style of proof as we avoid reducing to the cubic case entirely.
We give a new proof of Tietze Theorem on the convergence of infinite semi-regular continued fractions.
I present a simple, elementary proof of Morley's theorem, highlighting the naturalness of this theorem.
We derive the Gallai-Edmonds Structure Theorem from Hall's Theorem.
The Riemann-Roch theorem is of utmost importance in the algebraic geometric theory of compact Riemann surfaces. It tells us how many linearly independent meromorphic functions there are having certain restrictions on their poles. The aim of…
We give a short and relatively elementary proof of the Hilton-Milner Theorem.
We prove an infinitary version of the Brauer-Schur theorem.
A very simple but useful almost sure convergence theorem of probability is given.
Here we simplify the proof of the de Rham theorem for Schwartz functions on affine Nash manifolds and generalize the result to the case of non affine Nash manifolds.
The classical Sturm-Hurwitz-Kellogg theorem asserts that a function, orthogonal to an n-dimensional Chebyshev system on a circle, has at least n+1 sign changes. We prove the converse: given an n-dimensional Chebyshev system on a circle and…
We show that Thompson's $A\times B$-Lemma can be obtained as a consequence of the Brauer pair version of Brauer's Third Main Theorem.
In this expository article we provide an elegant proof of the one-sided Ingham-Karamata Tauberian theorem. As an application, we present a short deduction of the prime number theorem.
We provide several general versions of Littlewood's Tauberian theorem. These versions are applicable to Laplace transforms of Schwartz distributions. We apply these Tauberian results to deduce a number of Tauberian theorems for power series…
We give a new simpler proof of a theorem of Jayne and Rogers.
This note offers an elementary proof of the Siegel-Walfisz theorem for primes in arithmetic progressions.
We present an elementary proof of the fundamental theorem of algebra, following Cauchy's version but avoiding his use of circular functions. It is written in the same spirit as Littlewood's proof of 1941, but reduces it to more elementary…
In this note we give a detailed proof of a theorem of Aubin.