Related papers: A Visual Proof that $\pi^e < e^\pi$
In this paper we prove the transcendence of $\pi$ using Hilbert's method. We also prove that all points constructible with compass and straightedge have algebraic coordinates. Thus we give a self-contained proof that squaring the circle is…
The better title is "Yet another FALSE proof of the 4-colour theorem." Please consider all versions of this paper as historical material on the way to a non-computer proof of the 4-colour theorem. Interpreted as proofs, all versions are…
We give a pen and paper and (comparatively) much simpler proof to verify of the Four Colour Theorem.
In this paper we give another proof of the Chudnovsky formula for calculating $\pi$ - a proof in detail with means of basic complex analysis. With the exception of the tenth chapter, the proof is self-contained, with proofs provided for all…
We use a variant of Salikhov's ingenious proof that the irrationality measure of $\pi$ is at most $7.606308\dots$ to prove that, in fact, it is at most $7.103205334137\dots$. Accompanying Maple package: While this article has a fully…
We illustrate the concept of mathematical proof.
Proofs that a smooth morphism is flat available in the literature are long and difficult. We give a short proof of this fact.
We give a unified direct proof of the lifting of PIE limits to the 2-category of algebras and (pseudo) morphisms, which specifies precisely which of the projections of the lifted limit are strict and detect strictness. In the literature,…
We use the classical definitions (i) $\pi$ is the ratio of area to the square of the radius of a circle; (ii) $\pi$ is the ratio of circumference to the diameter of a circle, to prove $\pi$'s existence within the purview of Euclidean…
We analyse two translations from the synchronous into the asynchronous $\pi$-calculus, both without choice, that are often quoted as standard examples of valid encodings, showing that the asynchronous $\pi$-calculus is just as expressive as…
This is an exposition of Gauss's proof of Descartes's rule of signs.
This document presents a simpler proof showcasing the NP-hardness of Familial Graph Compression.
We find an elementary proof for Voiculescu's theorem on the polar decomposition of circular variables.
The aim of this work is to improve Wilker inequalities near the origin and {\pi}/2.
We consider the set E of curves with positive algebraic curvature, whose extremities and tangents in their extremities are given. For each of the curves of E, we define the minimum of the radius of curvature. We first prove that there…
The aim of this short note is to present an elementary, self-contained, and direct proof for the classical Lebesgue decomposition theorem.
Let V be an n-dimensional vector space over a finite field F_q. We consider on V the $\pi$-metric recently introduced by K. Feng, L. Xu and F. J. Hickernell. In this short note we give a complete description of the group of symmetries of V…
A space $X$ is said to be $\pi$-metrizable if it has a $\sigma$-discrete $\pi$-base. In this paper, we mainly give affirmative answers for two questions about $\pi$-metrizable spaces. The main results are that: (1) A space $X$ is…
We present a new topological proof of the infinitude of prime numbers with a new topology. Furthermore, in this topology, we characterize the infinitude of any non-empty subset of prime numbers.
In this paper, we prove some foundational results on the deformation theory of E-infinity ring spectra.