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Analogous to the famous Euler angle parametrization in three-dimensional Euclidean space, a reflection-free Lorentz transformation in (2+1)-dimensional Minkowski space can be decomposed into three simple parts. Applying this decomposition…
The proposal of this work is to provide an answer to the following question: is it possible to treat the metric of space-time - that in General Relativity (GR) describes the gravitational interaction - as an effective geometry? In other…
We prove that any strongly regular Weingarten surface in Euclidean space carries locally geometric principal parameters. The basic theorem states that any strongly regular Weingarten surface is determined up to a motion by its structural…
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…
This article provides a simple geometric interpretation of the quadratic formula. The geometry helps to demystify the formula's complex appearance and casts it into a much simpler existence, thus potentially benefits early algebra students.
We discuss a version of the fundamental theorem of calculus in several variables and some applications, of potential interest as a teaching material in undergraduate courses.
Historically, there have been many attempts to produce an appropriate mathematical formalism for modeling the nature of physical space, such as Euclid's geometry, Descartes' system of Cartesian coordinates, the Argand plane, Hamilton's…
This paper proposes a totally constructive approach for the proof of Hilbert's theorem on ternary quartic forms. The main contribution is the ladder technique, with which the Hilbert's theorem is proved vividly.
In this thesis, we attempt to gain a more complete insight into Double Layer Theories in Weyl Gravity. In order to do this, we first establish the premise of Weyls Theory, including its provenance, development and flaws. This is all…
Mathematical proof is undoubtedly the cornerstone of mathematics. The emergence, in the last years, of computing and reasoning tools, in particular automated geometry theorem provers, has enriched our experience with mathematics immensely.…
We introduce the Primary Gasing Triangle, a right triangle with a hypotenuse of 1 unit, to define the primary trigonometric functions: sine and cosine. This triangle serves as the foundational element in a new approach to learning…
This text is aimed at undergraduates, or anyone else who enjoys thinking about shapes and numbers. The goal is to encourage the student to think deeply about seemingly simple things. The main objects of study are lines, squares, and the…
Graphs are a central object of study in various scientific fields, such as discrete mathematics, theoretical computer science and network science. These graphs are typically studied using combinatorial, algebraic or probabilistic methods,…
Over the last two years, the canonical approach to quantum gravity based on connections and triads has been put on a firm mathematical footing through the development and application of a new functional calculus on the space of gauge…
In this paper, we intend to revisit Theorem 2 of [3] formulating it in a way that, weakening the hypotheses and, at the same time, highlighting the richer conclusion allowed by the proof, it can potentially be applicable to a broader range…
We show a possibility to apply certain philosophical concepts to the analysis of concrete mathematical structures. Such application gives a clear justification of topological and geometric properties of considered mathematical objects.
The aim of this text is to provide an elementary and self-contained exposition of Gromov's argument on topological overlap (the presentation is based on Gromov's work, as well as two follow-up papers of Matousek and Wagner, and of…
A rigorous microscopic theory for the description of quantum-transport phenomena in systems with open boundaries is proposed. We shall show that the application of the conventional Wigner-function formalism to this problem leads to…
This document contains a description of physics entirely based on a geometric presentation: all of the theory is described giving only a pseudo-riemannian manifold (M, g) of dimension n > 5 for which the g tensor is, in studied domains,…
We try to give a pedagogical introduction to Connes' derivation of the standard model of electro-magnetic, weak and strong forces from gravity.