相关论文: Six mathematical gems from the history of Distance…

In this paper we consider partial metric spaces in the sense of O'Neill. We introduce the notions of strong partial metric spaces and Cauchy functions. We prove a fixed point theorem for such spaces and functions that improves Matthews'…

Pythagoras' theorem, the area of a triangle as one half the base times the height, and Heron's formula are amongst the most important and useful results of ancient Greek geometry. Here we look at all three in a new and improved light, using…

We indicate that Heron's formula (which relates the square of the area of a triangle to a quartic function of its edge lengths) can be interpreted as a scissors congruence in 4-dimensional space. In the process of demonstrating this, we…

In this paper we combine the notions of partial metric spaces with negative distances, $G_p$-metric spaces and n-metric spaces together into one structure called the partial n-metric spaces. These are generalizations of all the said…

The distance on a set is a comparative function. The smaller the distance between two elements of that set, the closer, or more similar, those elements are. Fr\'echet axiomatized the distance into what is today known as a metric. In this…

The classical Reifenberg's theorem says that a set which is sufficiently well approximated by planes uniformly at all scales is a topological H\"older manifold. Remarkably, this generalizes to metric spaces, where the approximation by…

Let $(X, d)$ be a compact metric space and let $\mathcal{M}(X)$ denote the space of all finite signed Borel measures on $X$. Define $I \colon \mathcal{M}(X) \to \R$ by \[ I(\mu) = \int_X \int_X d(x,y) d\mu(x) d\mu(y), \] and set $M(X) =…

We re-derive Thales, Pythagoras, Apollonius, Stewart, Heron, al Kashi, de Gua, Terquem, Ptolemy, Brahmagupta and Euler's theorems as well as the inscribed angle theorem, the law of sines, the circumradius, inradius and some angle bisector…

In this work, using a new geometrical approach we study to the existence of the fixed-point of mappings that independence of the smoothness, and also of their single-values or multi-values. This work proved the theorems that generalize in…

Let $M$ be a compact $d$-dimensional Riemannian manifold without a boundary. Given $E \subset M$, let $\Delta_{\rho}(E)=\{\rho(x,y): x,y \in E \}$, where $\rho$ is the Riemannian metric on $M$. Let $\Delta_{\rho}^x$ denote the pinned…

We solve a very classical problem motivated by considerations in quantum gravity: providing a description of the geometry of a Euclidean tetrahedron from the initial data of the areas of the faces and the areas of the medial parallelograms…

The study of comparison theorems in geometry has a rich history. In this paper, we establish a comparison theorem for polyhedra in 3-manifolds with nonnegative scalar curvature, answering affirmatively a dihedral rigidity conjecture by…

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…

We review Euler's work on spherical geometry. After an introduction concerning the general place that trigonometric formulae occupy in geometry, we start by the two memoirs of Euler on spherical trigonometry, in which he establishes the…

Let $\Gamma$ denote a distance-regular graph with diameter $D\geq 3$ and Bose-Mesner algebra $M$. For $\theta\in C\cup \infty$ we define a 1 dimensional subspace of $M$ which we call $M(\theta)$. If $\theta\in C$ then $M(\theta)$ consists…

In 1979, Schoen and Yau proved their famous Positive Mass Theorem which is a combination of a comparison theorem: {\em a three dimensional asymptotically flat Riemannian manifold with nonnegative scalar curvature has nonnegative ADM mass},…

We define and study several new interleaving distances for persistent cohomology which take into account the algebraic structures of the cohomology of a space, for instance the cup product or the action of the Steenrod algebra. In…

Divided into three parts, the first marks out enormous geometric issues with the notion of quasi-freenss of an algebra and seeks to replace this notion of formal smoothness with an approximation by means of a minimal unital commutative…

We study the interplay between geometry and partial differential equations. We show how the fundamental ideas we use require the ability to correctly calculate the dimensions of spaces associated to the varieties of zeros of the symbols of…

The class of Schoenberg transformations, embedding Euclidean distances into higher dimensional Euclidean spaces, is presented, and derived from theorems on positive definite and conditionally negative definite matrices. Original results on…