Related papers: Solid angles in perspective
Quantum field theory in curved space-times is a well developed area in mathematical physics which has had important phenomenological applications to the very early universe. However, it is not commonly appreciated that on time dependent…
We review the concept of a graded bundle as a natural generalisation of a vector bundle. Such geometries are particularly nice examples of more general graded manifolds. With hindsight there are many examples of graded bundles that appear…
We make a systematic study of the focal surface of a congruence of lines in the projective space. Using differential techniques together with techniques from intersection theory, we reobtain in particular all the invariants of the focal…
It is customary to employ a semi-spherical scale model to describe the apparent path of the Sun across the sky, whether it be its diurnal motion or its variation throughout the year. A flat surface and three bent semi-rigid wires…
Cultural Astronomy is interdisciplinary connecting the arts, humanities, social & physical sciences. Data collection methods and theories are used from many disciplines and meld with methods and theories within cultural astronomy. The…
These notes are designed for those who either plan to work in differential geometry, or at least want to have a good reason not to do it. We discuss smooth curves and surfaces -- the main gate to differential geometry. We focus on the…
If all but two vertices of a triangulated sphere have degrees divisible by $k$, then the exceptional vertices are not adjacent. This theorem is proved for $k=2$ with the help of the coloring monodromy. For $k = 3, 4, 5$ colorings by the…
The interior structure of arbitrary sets of quaternion units is analyzed using general methods of the theory of matrices. It is shown that the units are composed of quadratic combinations of fundamental objects having a dual mathematical…
An overview is provided of the singularity theorems in cosmological contexts at a level suitable for advanced graduate students. The necessary background from tensor and causal geometry to understand the theorems is supplied, the…
Learning to use math in science is a non-trivial task. It involves many different skills (not usually taught in a math class) that help blend physical knowledge with mathematical symbology. One of these is the idea of quantification: that…
Stability plays a central role in arithmetic. In this article, we explain some basic ideas and present certain constructions for such studies. There are two aspects: namely, general Class Field Theories for Riemann surfaces using…
The unit Euclidean distance degree and the generic Euclidean distance degree are two well-studied invariants of projective varieties. These quantities measure the algebraic complexity of nearest-point problems on a variety, and in many…
This paper addresses the numerical computation of critical angles between two convex cones in finite-dimensional Euclidean spaces. We present a novel approach to computing these critical angles by reducing the problem to finding stationary…
Consider a spherically symmetric spacelike slice through a spherically symmetric spacetime. One can derive a universal bound for the optical scalars on any such slice. The only requirement is that the matter sources satisfy the dominant…
Symmetries concerning the ordinary coordinate spacetime and internal spacetime are discussed. A possible unification model of electroweak, strong and gravitational interactions is briefly described.
This is an attempt to present axioms for Euclidean geometry, aiming at the following goals: to work with geometric notions (thus not merely identify points with pairs of numbers, giving a special status to a particular coordinate system);…
This text is a support for different courses of the master of Mechanics of the University Paris-Saclay. The content of this text is an introduction, for graduate students, to tensor algebra and analysis. Far from being exhaustive, the text…
The study of the Universe and all its contents, the physical phenomena and the evolution of the sky objects has always attracted the Humanity. The Astronomy begins to try to explain these things, others sciences too. Therefore, we think a…
These lecture notes provide an informal introduction to the theory of nonnegative polynomials and sums of squares. We highlight the history and some recent developments, especially the new connections with classical (complex) algebraic…
This book is a textbook for the course of foundations of geometry. It is addressed to mathematics students in Universities and to High School students for deeper learning the elementary geometry. It can also be used in mathematics coteries…