Related papers: A Visual Proof that $\pi^e < e^\pi$

In this paper, we study the (normalized) Ricci flow on surfaces with conical singularities. Long time existence is proved for cone angle smaller than $2\pi$. In this case, convergence results are obtained if the Euler number is nonpositive.

A. Capelli gave a necessary and sufficient condition for the reducibility of $x^n-a$ over $\Q$. In this article, we are providing an alternate elementary proof for the same.

In this article, we present four issues and provide a creative and concise proof for each of them. The four issues are: 1- Inequality $\frac{1}{\sqrt{n\pi+\frac{\pi}{2}}}<\frac{\binom{2n}{n}}{2^{2n}}<\frac{1}{\sqrt{n\pi}}$ 2- A special case…

We present a new geometric proof of Pieri's formula, exhibiting an explicit chain of rational equivalences from a suitable sum of distinct Schubert varieties to the intersection of a Schubert variety with a special Schubert variety. The…

We prove a result on lower bounds in large dimensions.

We prove that given $\lambda \in \mathbb{R}$ such that $0 < \lambda < 1$, then $\pi(x + x^\lambda) - \pi(x) \sim \displaystyle \frac{x^\lambda}{\log(x)}$. This solves a long-standing problem concerning the existence of primes in short…

In this short exposition we provide a simplified proof of Buser's result for Cheeger's isoperimetric constant.

For a set of primes $\pi$, denote by $E_\pi$ the class of finite groups containing a Hall $\pi$-subgroup. We establish that $E_{\pi_1}\cap E_{\pi_2}$ is contained in $E_{\pi_1\cap\pi_2}$. As a corollary, we prove that if $\pi$ is a set of…

In this paper we establish that the well-known Arithmetic System is consistent in the traditional sense. The proof is done within this Arithmetic System.

The aim of this note is to show that Poincar\'e inequalities imply corresponding weighted versions in a quite general setting. Fractional Poincar\'e inequalities are considered, too. The proof is short and does not involve covering…

We motivate and then prove a generalized pythagorean theorem for parallelepipeds in Euclidean space.

One of the most famous results in Complex Analysis is the Little Picard Theorem, that characterizes the image set of an arbitrary entire function. Specifically, the theorem states that this image set is either the whole complex plane or the…

We present a short and elementary proof of isometric uniqueness of the Gurarii space.

In the paper, we give the proof of the polar decomposition of the Wiener measure according to the orbits of the group of diffeomorphisms.

In this work, we prove the irrationality of $\pi$ based on the nested radicals with roots of $2$ of kind $c_k = \sqrt{2 + c_{k - 1}}$ and $c_0 = 0$. Sample computations showing how the rational approximation tends to $\pi$ with increasing…

Let $E$ be an elliptic curve of rank $\text{rk}(E) \geq 1$, and let $P \in E(\mathbb{Q})$ be a point of infinite order. The number of elliptic primes $p \leq x$ for which $\langle P\rangle=E(\mathbb{F}_p)$ is expected to be…

The paper presents a counterexample to the Hodge conjecture.

A behavior of one class of mappings with finite distortion at a neighborhood of the origin is investigated. There is proved a lower estimate of distortion of a distance under mappings mentioned above.

We construct examples that show the difference in behavior or $\pi$-maximal and $\pi$-submaximal subgroups under group homomorphisms.

Gradients of the perimeter and area of a polygon have straightforward geometric interpretations. The use of optimality conditions for constrained problems and basic ideas in triangle geometry show that polygons with prescribed area…