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We provide elementary proof of several congruences involving single sum and multisums of binomial coefficients.

We give a short proof of a theorem of J.-E. Pin (theorem 1.1 below), which can be found in his thesis. The part of the proof which is my own (not Pin's) is a complete replacement of the same part in an earlier version of this paper.

Our goal in the present paper is to give a new ergodic proof of a well-known Veech's result, build upon our previous works.

We present a short and self-contained proof of the choosability version of Brooks' theorem.

This note presents an especially short and direct variant of Hermite's proof of the simple continued fraction expansion e = [2,1,2,1,1,4,1,1,6,...] and explains some of the motivation behind it.

In this article, we prove a decomposition theorem on differential polynomials of theta functions of high level.

We present a new, elementary, dynamical proof of the prime number theorem.

In this note, we provide a short proof of Feige's conjecture for identically distributed random variables.

Standard proofs of Lusin's theorem, using simple functions, are sometimes quite elaborate. Here, we give a one-sentence proof of Lusin's theorem. We do not believe our approach, by way of inverse images, is new. However, this particular…

Simple and shorter proofs of two Dirac-type theorems involving connectivity are presented.

The purpose of this note is to give the full and self-contained proof of Shchepin's result on a spectral representation of retracts of cubes.

This is an elementary geometrical proof of Birkhoff theorem. It is hardly important, but the pictures behind are quite nice.

We survey the classical results of the Dirichlet Approximation Theorem.

We present a simpler way than usual to deduce the completeness theorem for the second-oder classical logic from the first-order one. We also extend our method to the case of second-order intuitionistic logic.

We give a new proof of Lucas' Theorem in elementary number theory.

Proofs that a smooth morphism is flat available in the literature are long and difficult. We give a short proof of this fact.

This note is purely expository. The statement of the Gauss theorem on the constructibility of regular polygons by means of compass and ruler is simple and well-known. However, its proofs given in most textbooks rely upon much unmotivated…

The aim of this note is to give a quick algebraic proof of (the combinatorial part of) the classification theorem for compact real surfaces, whose classical proofs (as in the Massey book and in the Conway ZIP proof) are based on surgery…

This note addresses the motivic nature of some classical cohomological results due to Lefschetz, namely the primitive decomposition (for the cohomology of smooth projective varieties), and, secondly, the splitting of the cohomology of a…

We give a pen and paper and (comparatively) much simpler proof to verify of the Four Colour Theorem.