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A very short proof of the Fej\'er-Riesz lemma is presented in the matrix case

I present a simple derivation of the de Gennes narrowing phenomenon.

We obtain another proof of Hermite's integral for the Hurwitz zeta function.

We provide a simple proof of Kamp's theorem.

We provide a proof of the Borwein Conjecture using analytic methods.

We present a short new proof of Cobham's theorem without using Kronecker's approximation theorem, making it suitable for generalization beyond automatic sequences.

We prove an infinitary version of the Brauer-Schur theorem.

We improve the previuosly known bound for some vertex Folkman numbers.

We give a new proof of the butterfly theorem, based on the use of several expressions involving the scale factor between the two wings.

A generalization of an inequality from IMO is proven.

The converse of Fortin's Lemma in Banach spaces is established in this Note.

In this article, we prove a weighted version of Saitoh's conjecture. As an application, we prove a weighted version of Saitoh's conjecture for higher derivatives.

In comparison with the previous version of this paper, the Introduction is slightly changed and some minor typos are deleted. All results are unchanged.

We present a new proof to a general result due to Kestelman. Our proof differs completely from the other proofs we know and we hope that readers will find it clearer. We also include a quite exhaustive bibliographical analysis on related…

The article presents the proof of Casas-Alvero conjecture.

We consider the function $G(n)=\frac{\sigma(n)}{n\log\log n}$ (where $\sigma(n)=\sum_{d|n}d$) and set an imposed condition on its argument $n$, the fulfillment of which is sufficient for the existence of a prime $p$, at which $G(np)>G(n)$.…

In this paper, we prove a conjecture of Schnell in the surface case.

We give an elementary proof to Hasse theorem.

The Cram\'er-Granville conjecture is an upper bound on prime gaps, $g_n = p_{n+1} - p_n < \cCramer \, \log^2 p_n$ for some constant $\cCramer \geq 1$. Using a formula of Selberg, we first prove the weaker summed version: $\sum_{n=1}^N g_n <…

We show that it is consistent that the Borel Conjecture and the dual Borel Conjecture hold simultaneously.