Related papers: The last proof of extreme value theorem and interm…
We state and prove a new closure theorem closely related to the classical closure theorems of Poncelet and Steiner. Along the way, we establish a number of theorems concerning conic sections.
The paper proves that all power series over a maximal ordered Cauchy complete non-Archimedean field satisfy the intermediate value theorem on every closed interval. Hensel's Lemma for restricted power series is the main tool of the proof.
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…
We prove a limit theorem on the convergence of the distributions of the scaled last exit time over a slowly moving nonlinear boundary for a class of Gaussian stationary processes. The limit is a double exponential (Gumbel) distribution.
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 illustrate the concept of mathematical proof.
We prove that there are infinitely many integers $n$ such that $n$ and $n+1$ have the same number of distinct prime divisors.
We prove martingale-ergodic and ergodic-martingale theorems for vector valued Bochner integrable functions. We obtain dominant and maximal inequalities. We also prove weighted and multiparameter martingale-ergodic and ergodic martingale…
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.
We give simple proofs, under minimal hypotheses, of the Weak Law of Large Numbers and the Central Limit Theorem for independent identically distributed random variables. These proofs use only the elementary calculus, together with the most…
We prove the Avrunin-Scott theorem for quantum complete intersections; the rank variety of a module is isomorphic to its support variety.
The purpose of this note is to present my understanding of Tim Austin's proof of the multiple ergodic theorem for commuting transformations, emphasizing on the use of joinings, extensions and factors. The existence of a sated extension,…
We prove a combination theorem for PD(n)-pairs.
Konig's theorem states that the covering number and the matching number of a bipartite graph are equal. We prove a generalisation of this result, in which each point in one side of the graph is replaced by a subtree of a given tree. The…
In this article we show that some recent results on the existence of best proximity points can be obtained from the same result in fixed point theory.
The article presents the proof of Casas-Alvero conjecture.
We take the perspective of an advanced high school student trying to understand the proof of Fermat's Last Theorem for the first time. We collect definitions and statements needed to summarise how Fermat's Last Theorem was first proved in…
We generalize intermediate value Theorem to metric space,and make use of it to discuss existence of classic solution of the Boussinesq equation.
A proof of Sendov's conjecture is given.
We prove a central limit theorem for the entropic transportation cost between subgaussian probability measures, centered at the population cost. This is the first result which allows for asymptotically valid inference for entropic optimal…