Related papers: Grzegorczyk and Whitehead points: the story contin…

We construct a category, $\Omega$, of which the objects are pointed categories and the arrows are pointed correspondences. The notion of a "spec datum" is introduced, as a certain relation between categories, of which one has been given a…

Two questions are investigated by looking successively at classical mechanics, special relativity, and relativistic gravity: first, how is space related with spacetime? The proposed answer is that each given reference fluid, that is a…

For the past two decades, Einstein's Hole Argument (which deals with the apparent indeterminateness of general relativity due to the general covariance of the field equations) and its resolution in terms of Leibniz equivalence (the…

A pedagogical but concise overview of Riemannian geometry is provided, in the context of usage in physics. The emphasis is on defining and visualizing concepts and relationships between them, as well as listing common confusions,…

This paper explores relations between two separate worlds. They are the algebraic geometry of Alexander Grothendieck and the automorphic representation theory of Robert Langlands. The relation between them would be a very broad example of…

The purpose of this contribution is to give a coherent account of a particular narrative which links locales, geometric theories, sheaf semantics and constructive commutative algebra. We are hoping to convey a firm grasp of three ideas: (1)…

The Banach contraction principle is the most celebrated fixed point theorem, it has been generalized in various directions. In this paper, inspired by the concept of $(\phi, F)-$contraction in metric spaces, introduced by Wardowski. We…

Fixed point theorems are one of the many tools used to prove existence and uniqueness of differential equations. When the data involved contains products of distributions, some of these tools may not be useful. Thus rises the necessity to…

A natural one-to-one correspondence between projective spaces, defined by an axiom system published by O. Veblen and J. W. Young in 1908, and projective join spaces, defined by an axiom system published by M. Pieri in 1899, is presented. A…

One of the most distinguished features of our algebraic geometrical, pencil concept of space-time is the fact that spatial dimensions and time stand, as far as their intrinsic structure is concerned, on completely different footings: the…

We develop a unified theory of Eulerian spaces by combining the combinatorial theory of infinite, locally finite Eulerian graphs as introduced by Diestel and K\"uhn with the topological theory of Eulerian continua defined as irreducible…

Coarse median spaces simultaneously generalise the classes of hyperbolic spaces and median algebras, and arise naturally in the study of the mapping class groups and many other contexts. One issue with their definition as originally…

We prove new theorems which are higher-dimensional generalizations of the classical theorems of Siegel on integral points on affine curves and of Picard on holomorphic maps from $\mathbb{C}$ to affine curves. These include results on…

This paper presents a generalized version of a theorem of Grzegorek and Labuda in category bases and also endeavours to establish a variant formulation of the same in Marczewski structures.

In this paper, we propose a generalized notion of a distance function, called a $g$-metric. The $g$-metric with degree $n$ is a distance of $n+1$ points, generalizing the ordinary distance between two points and $G$-metric between three…

The concept of breakdown point was introduced by Hampel [Ph.D. dissertation (1968), Univ. California, Berkeley; Ann. Math. Statist. 42 (1971) 1887-1896] and developed further by, among others, Huber [Robust Statistics (1981). Wiley, New…

This paper examines the history of the development of dimensional analysis, from Euclidean geometry to hyperspace. This is largely a secondary source work intended to summarize, as briefly as possible, with enough mathematical rigor, the…

We propose to apply the idea of analytical continuation in the complex domain to the problem of geodesic completeness. We shall analyse rather in detail the cases of analytical warped products of real lines, these ones in parallel with…

Using only the principle of relativity and Euclidean geometry we show in this pedagogical article that the square of proper time or length in a two-dimensional spacetime diagram is proportional to the Euclidean area of the corresponding…

We consider more general framework than the corresponding one considered in our previous work on the Hodge-Iwasawa theory. In our current consideration we consider the corresponding more general base spaces, namely the analytic adic spaces…