Related papers: Pythagorean theorem from Heron's formula: Another …
We present an elementary proof of the cross theorem in the case of Reinhardt domains. The results illustrates the well-known interrelations between the holomorphic geometry of a Reinhardt domain and the convex geometry of its logarithmic…
We use Taylor's formula with Lagrange remainder to make a modern adaptation of Poisson's proof of a version of the fundamental theorem of calculus in the case when the integral is defined by Euler sums, that is Riemann sums with left (or…
We give an elementary proof of Kelley's theorem based on a minimax argument. Some applications to related problems are also developed.
New formulas for the construction of Pythagorean triples and generalizations to equations of higher powers. Application of formulas to some problems, in particular Fermat's equation with n=4.
We prove descent theorems for semiorthogonal decompositions using techniques from derived algebraic geometry. Our methods allow us to capture more general filtrations of derived categories and even marked filtrations, where one descends not…
It is shown that the Poincar\'e-Birkhoff fixed point theorem may be proven by extending the geometric approach originally devised by Henri Poincar\'e himself, along with several results from elementary differential topology. Beginning with…
Pisier's inequality is central in the study of normed spaces and has important applications in geometry. We provide an elementary proof of this inequality, which avoids some non-constructive steps from previous proofs. Our goal is to make…
We give an elementary, self-contained and quick proof of Belyi's theorem. As a by-product of our proof we obtain an explicit bound for the degree of the defining number field of a Belyi surface.
The Pythagorean Theorem is one of the oldest, more famous and more useful theorems of Mathematics, and possibly the one that has had the most impact in the evolution of this and other sciences. In this article, we look at it from different…
In this article, we give another visual proof of $\pi^e < e^\pi$.
A natural extension of Heron's 2000 year old formula for the area of a triangle to the volume of a tetrahedron is presented. This gives the fourth power of the volume as a polynomial in six simple rational functions of the areas of its four…
This paper gives a concise proof of the Jordan curve theorem on discrete surfaces. We also embed the discrete surface in the 2D plane to prove the original version of the Jordan curve theorem. This paper is a simple version of L. Chen, Note…
Hindman's Theorem is a prototypical example of a combinatorial theorem with a proof that uses the topology of the ultrafilters. We show how the methods of this proof, including topological arguments about ultrafilters, can be translated…
Here is present short proofing of Jordan's theorem about dividing of flat on two disjoint subsets by one closed curve.
Each triangle has three exterior or external circles tangential to the three straight lines containing the three sides of the triangle.Among the preliminaries in this paper, is deriving formulas for the radii of the three exterior circles…
In this article we show how Gr\"un's results in group theory can be used for studying the structure of class groups in normal extensions.
Eberhard-type theorems are statements about the realizability of a polytope (or more general polyhedral maps) given the valency of its vertices and sizes of its polygonal faces up to a linear linear degree of freedom. We present new…
We give an alternative, more geometric, proof of the well-known Joyal-Tierney Theorem in locale theory by utilizing Priestley duality for frames.
We give a proof of some small weight and level cases of Serre's conjecture.
This is a partial account of the fascinating history of Distance Geometry. We make no claim to completeness, but we do promise a dazzling display of beautiful, elementary mathematics. We prove Heron's formula, Cauchy's theorem on the…