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

In this short paper Euler gives a highly convergent series for arctan and thus pi, which converges much faster than the Leibniz series for arctan.

Here we give further evidences to support our scaling relation described in our previous paper [cond-mat/0006459, Phys. Rev. Lett. Vol.85, pp.1238 (2000)].

In 2001 W. Gosper introduced a constant Pi_{q} and conjectured without proofs many intriguing identities on this constant. In this paper we establish some modular equations of degrees 3 and 5. From these modular equations we confirm two…

We characterise the form of all simple, finite graphs for which the girth of the graph is equal to the circumference of the graph. We apply this to prove a bound on the number of edges in such a graph.

For mappings in metric spaces satisfying one inequality with respect to modulus of families of curves, there is proved a lightness of preimage under the mapping. It is proved that, the mappings, satisfying estimate mentioned above, are…

We give an infinite number of proofs of Pythagoras theorem.Some can be classified as `self-similar proofs'.

This short, expository note proves the existence of the maximal quotient of a variety by free rational curves.

In this note I go through the `proof' of frequentistic confidence intervals and show what it logically implies concerning the value of a physical quantity given an experimental observation (nothing).

We give a remarkably elementary proof of the Brouwer fixed point theorem. The proof is verifiable for most of the mathematicians.

For an elliptic curve $E$ defined over the field $\mathbb{C}$ of complex numbers, we classify all translates of elliptic curves in $E^3$ such that the $x$-coordinates satisfy a linear equation. This classification enables us to establish a…

This paper provides a new simple proof of Hesse's theorem in projective geometry for any dimension.

In this survey we report a general and systematic approach to study $[\varphi,\vec{e}_{3}]$-minimal surfaces in $\mathbb{R}^{3}$ from a geometric viewpoint and show some fundamental results obtained in the recent development of this theory.

Let $\Pi$ be a convex decomposition of a set $P$ of $n\geq 3$ points in general position in the plane. If $\Pi$ consists of more than one polygon, then either $\Pi$ contains a deletable edge or $\Pi$ contains a contractible edge.

We prove the Invariant Subspace Conjecture for separable Hilbert spaces.

In this article, we develop nested representations for cosine and inverse cosine functions, which is a generalization of Vi\`{e}te's formula for $\pi$. We explore a natural inverse relationship between these representations and develop…

We provide an alternative unified approach for proving the Pythagorean theorem (in dimension $2$ and higher), the law of sines and the law of cosines, based on the concept of shape derivative. The idea behind the proofs is very simple: we…

In this note, we extend the connection between the hydrogen atom and $\pi$ to the number $e$ via the Lerch's transcendent.

We give a sufficient condition for a Borel subset $E\subset X$ of a complete metric space with $\mathcal{H}^n(E)<\infty$ to be $n$-rectifiable. This condition involves a decomposition of $E$ into rectifiable curves known as an Alberti…

A novel approach to an old symmetry problem is developed. A new proof is given for the following symmetry problem, studied earlier.

We establish two versions of Vizing's theorem for Borel multi-graphs whose vertex degrees and edge multiplicities are uniformly bounded by respectively $\Delta$ and $\pi$. The ``approximate'' version states that, for any Borel probability…