Related papers: From Rolle's theorem to the Sturm-Hurwitz theorem

We derive a Sturm-Liouville system of equations for the exact calculation of the survival probability in first-passage time problems. This system is the one associated with the Wiener-Hopf integral equation obtained from the theory of…

Hurwitz numbers are the Laurent coefficients of an elliptic function $\wp(u)$ of cyclotomic type, and they are natural generalization of the Bernoulli numbers. This paper gives new generalization of Bernoulli and Hurwitz numbers for higher…

I expound here in a more detailed way a proof of an important Serini's theorem, which I have already sketched in a previous Note. Two related questions are briefly discussed.

We give a new proof of a theorem of B.M. Bredihin which was originally proved by extending Linnik's solution, via his dispersion method, of a problem of Hardy and Littlewood.

We show that the homology of modules for Hurwitz spaces stabilizes and compute its stable value. As one consequence, we compute the moments of Selmer groups in quadratic twist families of abelian varieties over suitably large function…

In this paper, we propose a generalization of a congruence due to Carlitz.

Lov\'asz gave a short proof of Brooks' theorem by coloring greedily in a good order. We give a different short proof by reducing to the cubic case. Then we show how to extend the result to (online) list coloring via the Kernel Lemma.

We give a short proof of the Buchstaber-Rees theorem concerning symmetric powers. The proof is based on the notion of a formal characteristic function of a linear map of algebras.

We give a short proof of Waldhausen's homeomorphism theorem for orientable Haken 3-manifolds.

The original proof of the Sharkovsky theorem is presented in full detail. The proof should be accessible to readers with basic Real Analysis background. Although nowadays there are several alternative proofs of this classical result, we…

In this Note, we start off with the primary representation of e and from there present an elementary short proof for the Wallis formula for $\pi$.

We present an astonishingly simple and elegant proof of the celebrated Basel problem.

We prove a simultaneous generalization of the classical Riemann-Hurwitz and Plucker formulas, addressing the total inflection of a morphism from a (smooth, projective) curve to an arbitrary (smooth, projective) higher-dimensional variety.…

We obtain some results related to Romanoff's theorem.

A direct relation between the enumeration of ordinary maps and that of fully simple maps first appeared in the work of the first and last authors. The relation is via monotone Hurwitz numbers and was originally proved using Weingarten…

We obtain a system of relations between linear Hodge integrals. As an application, we show that its first non-trivial relation implies the Witten's Conjecture/Kontsevich Theorem.

A very short proof of Kneser's theorem via transversal is given.

The direct and inverse spectral problems are solved for a wide subclass of the class of Schwarz matrices. A connection between the Schwarz matrices and the so-called generalized Hurwitz polynomials is found. The known results due to H. Wall…

We present a proof system for a multimodal logic, based on our previous work on a multimodal Martin-Loef type theory. The specification of modes, modalities, and implications between them is given as a mode theory, i.e. a small 2-category.…

We give a thorough study of Hurwitz stacks both in Galois and non galois case. The construction is applied to revisit somme classical examples, the stack of stable curves equipped with a level structure, and the stacks of tamely ramified…