Related papers: Newton Revisited: An excursion in Euclidean geomet…
During the process of teaching the concept of derivative, it is common and natural to refer to geometric interpretations, such as the use of the tangent line and the maximum and minimum points of a function, to illustrate the scope of the…
This chapter reviews some past and recent developments in shape comparison and analysis of curves based on the computation of intrinsic Riemannian metrics on the space of curves modulo shape-preserving transformations. We summarize the…
Starting from first principles and general assumptions Newton's law of gravitation is shown to arise naturally and unavoidably in a theory in which space is emergent through a holographic scenario. Gravity is explained as an entropic force…
Some translations into non-euclidean geometry of classical theorems of planar projective geometry are explored. The existence of some common triangle centers is dedeuced from theorems of Pascal and Chasles. Desargues' Theorem allows to…
A class of two dimensional field theories, based on (generically degenerate) Poisson structures and generalizing gravity-Yang-Mills systems, is presented. Locally, the solutions of the classical equations of motion are given. A general…
There are many problems and configurations in Euclidean geometry that were never extended to the framework of (normed or) finite dimensional real Banach spaces, although their original versions are inspiring for this type of generalization,…
We develop an inversive geometry for anisotropic quadradic spaces, in analogy with the classical inversive geometry of a Euclidean plane.
This contribution analyses the classical laws of motion by means of an approach relating time and entropy. We argue that adopting the notion of change of states as opposed to the usual derivation of Newton's laws in terms of fields a…
Systems of Newton equations of the form $\ddot{q}=-{1/2}A^{-1}(q)\nabla k$ with an integral of motion quadratic in velocities are studied. These equations generalize the potential case (when A=I, the identity matrix) and they admit a…
We obtain a new form for the action of a nonrelativistic particle coupled to Newtonian gravity. The result is different from that existing in the literature which, as shown here, is riddled with problems and inconsistencies. The present…
An earlier paper [1] presented a gravity theory based on the optics of de Broglie waves rather than curved space-time. While the universe's geometry is flat, it agrees with the standard tests of general relativity. A second paper [2] showed…
As scientific knowledge expands, science education may not always keep pace with the latest advancements in astrophysics. A solid scientific education is crucial for preparing students for 21st-century challenges. However, science education…
We introduce new results about the shape derivatives of scalar- and vector-valued functions, extending the results from (Dogan-Nochetto 2012) to more general surface energies. They consider surface energies defined as integrals over…
This is a pedagogical paper which explains some ideas in cosmology at a level accessible to undergraduate students. It does not use general relativity, but uses the ideas of Newtonian cosmology worked out by Milne and McCrea. The…
Recent general-relativistic extensions of Newtonian rotation laws for self-gravitating stationary fluids allow one to rederive, in the first post-Newtonian approximation, the well known geometric dragging of frames, and two new weak-field…
The notion of geometric version of an infinitely divisible law is introduced. Concepts parallel to attraction and partial attraction are developed and studied in the setup of geometric summing of random variables.
Algorithms for the computation of geodesics on an ellipsoid of revolution are given. These provide accurate, robust, and fast solutions to the direct and inverse geodesic problems and they allow differential and integral properties of…
This chapter explores the notion of "dimension" of a set. Various power laws by which an Euclidean space can be characterized are used to define dimensions, which then explore different aspects of the set. Also discussed are the…
1. Introduction - 2. Astrophysics and Cosmology - 3. Neutrino Oscillations - 4. Higgs and New Physics Searches - 5. Flavour Physics and CP Violation - 6. QCD - 7. Heavy Ion Collisions - 8. Outlook
Cosmology is a field of physics in which the use of General Relativity theory is indispensable. However, a cosmology based on Newtonian gravity theory for gravity is possible in certain circumstances. The applicability of Newtonian theory…