Related papers: Theorems of Euclidean Geometry through Calculus
We find relations between quantities defining geometry and quantities defining the length of a curve in geometries underlying Electromagnetism and unified model of Electromagnetism and Gravitation. We show that the length of a vector…
This is the first paper in a series of eight where in the first three we develop a systematic approach to the geometric algebras of multivectors and extensors, followed by five papers where those algebraic concepts are used in a novel…
The main aim of this paper is to establish the Reshetnyak's theorem for quasiregualr values from generalized $n$-manifold with suitable controlled geometry to Euclidean space $\mathbb{R}^{n}.$ This generalizes a previous result due to…
Euclidean geometry is among the earliest forms of mathematical thinking. While the geometric primitives underlying its constructions, such as perfect lines and circles, do not often occur in the natural world, humans rarely struggle to…
A geometric approach to formulate the uncertainty principle between quantum observables acting on an $N$-dimensional Hilbert space is proposed. We consider the fidelity between a density operator associated with a quantum system and a…
The algebra dual to Woronowicz's deformation of the 2-\-di\-men\-sion\-al Euclidean group is constructed. The same algebra is obtained from $SU_{q}(2)$ via contraction on both the group and algebra levels.
A rigorous mathematical theory of dimensional analysis, systematically accounting for the use of physical quantities in science and engineering, perhaps surprisingly, was not developed until relatively recently. We claim that this has…
Master character of the multidimensional homogeneous Euler equation is discussed. It is shown that under restrictions to the lower dimensions certain subclasses of its solutions provide us with the solutions of various hydrodynamic type…
In Euclidean geometry, all metric notions (arc length for curves, the first fundamental form for surfaces, etc.) are derived from the Euclidean inner product on tangent vectors, and this inner product is preserved by the full symmetry group…
We elaborate on some important ideas contained in Lobachevsky's Pangeometry and in some of his other memoirs. The ideas include the following: (1) The trigonometric formulae, which express the dependence between angles and edges of…
Geometric Algebra and Calculus are mathematical languages encoding fundamental geometric relations that theories of physics seem to respect. We propose criteria given which statistics of expressions in geometric algebra are computable in…
In this paper, we give a survey of various sphere theorems in geometry. These include the topological sphere theorem of Berger and Klingenberg as well as the differentiable version obtained by the authors. These theorems employ a variety of…
We develope a difference calculus analogous to the differential geometry by translating the forms and exterior derivatives to similar expressions with difference operators, and apply the results to fields theory on the lattice [Ref. 1]. Our…
It is shown that the generalized geometries may be obtained as a deformation of the proper Euclidean geometry. Algorithm of construction of any proposition S of the proper Euclidean geometry E may be described in terms of the Euclidean…
The goal of this paper is to give a purely geometric proof of a theorem by Branko Gr\"unbaum concerning configuration of triangles coming from the classical Napoleon's theorem in planar Euclidean geometry.
``Can number and geometric spaces be reconstructed from their symmetries?'' This question, which is at the heart of anabelian geometry, a theory built on the collaborative efforts of an international community in many variants and with the…
We propose an Euclidean geometric representation for the classical detection theory. The proposed representation is so generic that can be employed to almost all communication problems. The hypotheses and observations are mapped into R^N in…
The second Poincar\'e kinematical group serves as one of new ones in addition to the known possible kinematics. The geometries with the second Poincar\'e symmetry is presented and their properties are analyzed. On the geometries, the new…
The purpose of this book is to give an exposition of geometry, from a point of view which complements Klein's Erlangen program. The emphasis is on extending the classical Euclidean geometry to the finite case, but it goes beyond that. After…
The basic laws of geometrical optics can be deduced from energy-momentum conservation for electromagnetic waves, without other wave concepts. However, the concept of quanta is required; it arises naturally, hence such a hypothesis could…