Related papers: Two spaces looking for a geometer

Vector fields in the expanding Universe are considered within the multidimensional theory of General Relativity. Vector fields in general relativity form a three-parametric variety. Our consideration includes the fields with a nonzero…

Implicit locus equations in GeoGebra allow the user to do experiments with generalization of the concept of ellipses, namely with $n$-ellipses. By experimenting we obtain a geometric object that is very similar to a set of two circles.

We show how the Riemannian distance on $\mathbb{S}^n_{++}$, the cone of $n\times n$ real symmetric or complex Hermitian positive definite matrices, may be used to naturally define a distance between two such matrices of different…

Infinite graphs are finitary in the sense that their points are connected via finite paths. So what would an infinitary generalization of finite graphs look like? Usually this question is answered with the aid of topology, e.g. in the case…

(from the talk:) I shall here speak on gravity in (1+1)-dimensional space-time --- lineal gravity. The purpose of studying lower dimensional theories, and specifically lower dimensional gravity, is to gain insight into difficult…

We consider spacetime to be a 4-dimensional differentiable manifold that can be split locally into time and space. No metric, no linear connection are assumed. Matter is described by classical fields/fluids. We distinguish electrically…

Giorgio Parisi's recent speculations on the concept of continuous dimension (cond-mat/0207334) are compared with Von Neumann's serious work.

We consider four dimensional spaces of neutral signature and give examples of universal spaces of Walker type. These spaces have no analogue in other signatures in four dimensions and provide with a new class of spaces being universal.

This paper aims to discuss two issues that can have a significant impact on the foundations of the theory of gravitation: (1). The existence of relativity of space-time geometry with respect to the properties of used reference frame, which…

We characterize bijections on matrix spaces (operator algebras) preserving full rank (invertibility) of differences of matrix (operator) pairs in both directions.

Starting from a suggestion of Einstein on the construction of the concept of space, we elaborate an intrinsic method to obtain space and time transformations between two inertial spaces of reference, mathematically modeled as affine…

Hilbert space combines the properties of two fundamentally different types of mathematical spaces: vector space and metric space. While the vector-space aspects of Hilbert space, such as formation of linear combinations of state vectors,…

The communications and interrelations between different locations on the Earth's surface have far-reaching implications for both social and natural systems. Effective spatial analytics ideally require a spatial representation, where…

Recently, a geometrical characterization of vector spaces served to generalize them into a new class of algebras. Instead of the algebraic properties of the underlying fields, we generalized the recently discovered property of such spaces…

Prime numbers are one of the most intriguing figures in mathematics. Despite centuries of research, many questions remain still unsolved. In recent years, computer simulations are playing a fundamental role in the study of an immense…

The belief that three dimensional space is infinite and flat in the absence of matter is a canon of physics that has been in place since the time of Newton. The assumption that space is flat at infinity has guided several modern physical…

These informal notes were prepared in connection with a lecture at a high school mathematics tournament, and provide an overview of some examples of metric spaces and a few of their basic properties.

We describe the geomety of a set of scalar fields coupled to gravity. We consider the formalism of a differential Z_2-graded algebra of $2\times 2$ matrices whose elements are differential forms on space-time. The connection and the…

We study the space geometry of a rotating disk both from a theoretical and operational approach, in particular we give a precise definition of the space of the disk, which is not clearly defined in the literature. To this end we define an…

In this paper, we introduce the nonstandard vector space in which the concept of additive inverse element will not be taken into account. We also consider a metric defined on this nonstandard vector space. Under these settings, the…