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We give a short proof of a slightly weaker version of the multilinear Kakeya inequality proven by Bennett, Carbery, and Tao.
A short and almost elementary proof of the Boros-F\"uredi-B\'ar\'any-Pach-Gromov theorem on the multiplicity of covering by simplices in $\mathbb R^d$ is given.
We give here a new proof of a Tauberian Theorem of complex Laplace transform using the Theory of measure and theory of function with bounded variations. However we deduce the simple proof of Prime Number Theorem.
We give an alternative proof of a recent result by T.D. Browning and A. Haynes (arXiv:1204.6374v1) on multiplicative inverses in sequences of intervals and improve this result under additional conditions on the spacing of these intervals.
In this short note we establish new refinements of multidimensional Szemeredi and polynomial van der Waerden theorems along the shifted primes.
In this short note, we revisit Zeilberger's proof of the classical matrix-tree theorem and give a unified concise proof of variants of this theorem, some known and some new.
We exhibit how the Rasiowa-Sikorski Lemma simplifies, in a sense, proofs of results that make use of the technique known as back-and-forth, often resulting in not very illustrative arguments. The first two sections seek to show one simple…
In arXiv:0810.2076 we presented a conjecture generalizing the Cauchy formula for Macdonald polynomials. This conjecture encodes the mixed Hodge polynomials of the representation varieties of Riemann surfaces with semi-simple conjugacy…
We provide elementary proof of several congruences involving single sum and multisums of binomial coefficients.
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.
This is an exposition, in 12 pages including all prerequisites and a generalization, of Karamata's little known elementary proof of the Landau-Ingham Tauberian theorem, a result in real analysis from which the Prime Number Theorem follows…
First proved my Donald Martin in 1975, the result of Borel determinacy has been the subject of multiple revised proofs. Following Martin's book [1], we present a recent streamlined proof which implements ideas of Martin, Moschovakis, and…
In this paper, we consider pairs of a prime and a prime power with a fixed difference. We prove an average result on the distribution of such pairs. This is a partial improvement of the result of Bauer (1998).
In this note we generalise a method of Perott to give new proofs that there are infinitely many prime numbers.
We give a proof of a result of Bonet, Engli\v{s} and Taskinen filling in several details and correcting some flaws.
We give a new simpler proof of a theorem of Jayne and Rogers.
We provide a short proof of the theorem that every real multivariate polynomial has a symmetric determinantal representation, which was first proved in J. W. Helton, S. A. McCullough, and V. Vinnikov, Noncommutative convexity arises from…
Capelli's and Turnbull's classical identities are given elegant combinatorial proofs.
We obtain a lower bound for a number of primes in tuples. As applications, we obtain a lower bound for the Romanoff type representation functions.
We give a proof of the o-minimal version of the Whitney Extension Theorem simplified as compared to the original ones. A new simplifying ingredient is a definable variant of Urysohn's lemma for class $\mathcal{C}^q$ (see Section 3).