Figure 1: Fundamental region R of the Modular Group, given by {z ∈ H : |z| > 1,− 2 ≤ Re(z) ≤ 1 2} The Klein Quartic curve is famous for its symmetries and many visualizations are created to capture these in lower dimensions. In fact, it has the largest possible automorphism group for its genus, the simple group of order 168. Yet, interest in this algebraic curve started with knowledge of the ex...

Contents 1. Introduction 1 2. Structure algebra 4 2.1. Definition of a structure algebra 4 2.2. Semisimple structure algebras 7 2.3. Structure algebra of a finite group 11 3. Cuts of stratified surfaces 15 3.1. Stratified surfaces 15 3.2. Cut systems 17 3.3. Basic and simple surfaces 21 3.4. Neighboring complete cut systems 25 4. Two-dimensional topological field theory 29 4.1. Definition of Kl...

We give an explicit construction of a positive-definite invariant inner-product for the Klein-Gordon fields, thus solving the old problem of the probability interpretation of Klein-Gordon fields without having to restrict to the subspaces of the positivefrequency solutions. Our method has a much wider domain of application and may be used to obtain the most general invariant inner-product on th...

The traditional view on extra-dimensions is the Kaluza-Klein picture: the matter fields live in compact (usually flat) extra-dimensions, and their Fourier expansion along the extra-coordinates lead to an infinite collection of so-called Kaluza-Klein modes, which can be interpreted as four-dimensional fields. Their mass spectrum is very specific since it is discrete with a mass gap of the order ...

We explore the obsevability of extra dimensions through 5-dimensional squashed Kaluza-Klein black holes residing in the Kaluza-Klein spacetime. With the expectation that the Hawking radiation reflects the 5-dimensional nature of the squashed horizon, we study the Hawking radiation of a scalar field in the squashed black hole background. As a result, we show that the luminosity of Hawking radiat...

The method of computational homology is used to analyze natural image 8 × 8 and 9 × 9-patches locally. Our experimental results show that there exist subspaces of the spaces of 8× 8 and 9× 9-patches that are topologically equivalent to a circle and a Klein bottle respectively. These extend the results of the paper ”on the local behavior of spaces of natural images.” To the larger patches. The K...

We consider the supervised classification problem of machine learning in Cayley-Klein projective geometries: We show how to learn a curved Mahalanobis metric distance corresponding to either the hyperbolic geometry or the elliptic geometry using the Large Margin Nearest Neighbor (LMNN) framework. We report on our experimental results, and further consider the case of learning a mixed curved Mah...

Having shown elsewhere [1] that the Klein paradox for the Klein-Gordon (KG) equation of spin-zero particle manifests exactly the same kind of wave propagation and negative refraction phenomena, which also exist in the scattering of TM (transverse-magnetic) –polarized electromagnetic (EM) wave incident on a left handed medium (LHM), we show in this paper that it is possible to simulate the Klein...

It is well known that the world lines of charged particles in the presence of gravitational and electromagnetic fields can be viewed as geodesics in the 5D space-time first proposed by T. Kaluza [1]. Kaluza’s treatment was later improved by Klein in 1926 [2], in a quantum theoretical approach, giving rise to the so-called Kaluza-Klein theory [4]. This formalism also makes possible to propose, b...

Spherically symmetric, vacuum solutions in 5D and 7D Kaluza-Klein theory are obtained. These solutions are flux tubes with constant cross-sectional size, located between (+) and (-) Kaluza-Klein “electrical” and “magnetic” charges disposed respectively at r = ±∞ and filled with constant Kaluza-Klein “electrical” and “magnetic” fields. These objects are surprisingly similar to the flux tubes whi...