Related papers: From Rolle's theorem to the Sturm-Hurwitz theorem
In measure theory, Steinhaus theorem is a result that deals with a property of the difference between two sets of positive measure. We give a simple elementary proof of the result.
We prove the Lefschetz hyperplane section theorem using a simpler machinery by making the observation that we can compose the Lefschetz Pencil with a Real Morse function to get a map from the variety to $\mathbb{R}$ which is "close" to…
Consider a smooth map from a neighborhood of the origin in a real vector space to a neighborhood of the origin in a Euclidean space. Suppose that this map takes all germs of lines passing through the origin to germs of Euclidean circles, or…
We prove a theorem on the existence of global surfaces of section with prescribed spanning orbits and homology class. This result is a modification and a refinement of a result due to Fried, recast in terms of invariant measures instead of…
Kolmogorov's invariant torus theorem is proved using a simple fixed point theorem.
A proof based on the Chern-Gauss-Bonnet Theorem is given to Hopf Theorem concerning the degree of the Gauss map of a hypersurface in $\mathbb{R}^n$.
We show that Isserlis' theorem follows as a corollary to the invariant tensor theorem for isotropic tensors.
We present an easy proof of Polya's theorem on random walks: with the probability one a random walk on the two-dimensional lattice returns to the starting point.
We give an alternative proof of the Hurwitz existence problem for branched covers of $\mathbb{P}^1$ in the case where the number of ramification points equals the number of branch points, that is, where all the ramification profiles are of…
We give an undergraduate short and simple proof for Zariski's lemma.
In this note we extend a theorem from [13] about uniform circle random coverings
In this paper we revisit several recent results on monotone and strictly monotone Hurwitz numbers, providing new proofs. In particular, we use various versions of these numbers to discuss methods of derivation of quantum spectral curves…
In this very short note we slightly generalize some relations for one-part double Hurwitz numbers from math.AG/0209282.
We give a categorical account of Arrow's theorem, a seminal result in social choice theory.
L. Weinstein's brilliant short proof of de Branges's Theorem is made even shorter by using computer algebra.
We give a geometric proof of the Routh's theorem for tetrahedra.
In this paper, we give a new proof of an arithmetic analogue of the Riemann-Roch Theorem, due originally to Serge Lang. Lang's result was first proved using the lattice point geometry of Minkowski. By contrast, our proof is completely…
Many proofs of the Fundamental Theorem of Algebra, including various proofs based on the theory of analytic functions of a complex variable, are known. To the best of our knowledge, this proof is different from the existing ones.
We establish the connection between Sturm-Liouville equations on time scales and Sturm--Liouville equations with measure-valued coefficients. Based on this connection we generalize several results for Sturm-Liouville equations on time…
We show that Sturm's classical separation theorem on the interlacing of the zeros of linearly independent solutions of real second order two-term ordinary differential equations necessarily fails in the presence of a unique turning point in…