Lobachevsky Geometry Research Articles

Уравнения Максвелла в пространстве Лобачевского и моделирование среды со специальными свойствами

Lobachevsky geometry simulates a medium with special constitutive relations Di = ϵ0ϵikEk, Bi = μ0μikHk, where two matrices coincide: ϵik(x) = μik(x). The situation is specified in quasi-Cartesian coordinates (x, y, z) in Lobachevsky space, they are appropriate for modeling a medium nonuniform along the axis z. Exact solutions of the Maxwell equations in complex form of Majorana-Oppenheimer have been constructed. The problem reduces to a second-order differential equation for a certain primary function which can be associated with the one-dimensional Schrödinger problem for a particle in external potential field U(z) = U0e2z. In the frames of the quantum mechanics, Lobachevsky geometry acts as an effective potential barrier with reflection coefficient R = 1; in electrodynamic context, this geometry simulates a medium that effectively acts as an ideal mirror distributed in space. Penetration of the electromagnetic field into the effective medium along the axis z depends on the parameters of an electromagnetic waves ω, k2 1 + k2 2 and the curvature radius ρ of the used Lobachevsky model. The generalized quasi-plane wave solutions f(t, x, y, z) = E + iB and the relevant system of equations are transformed into the real form, which permit us to relate geometry characteristics with expressions for effective tensors of electric and magnetic permittivities.

Learning trajectory of non-Euclidean geometry through ethnomathematics learning approaches to improve spatial ability

Non-Euclidean geometry is an abstract subject and difficult to learn, but mandatory for students. The ethnomathematics approach as a learning approach to improve students’ spatial abilities. The aim of this research is to discover new elements of the spatial abilities of non-Euclidean geometry; determine the relationship between spatial abilities for Euclid, Lobachevsky, and Riemann geometry. This study used the micro genetic method with a 2×2 factorial experimental research design. The sample of this research is 100 students of mathematics education. There are three valid and reliable research instruments through expert trials and field trials. Data collection was carried out in two ways, namely tests and observations. Quantitative data were analyzed through ANCOVA, and observational data were analyzed through the percentage of implementation of the learning trajectory stages. The result is that the spatial ability of students who are given the ethnomathematics learning approach is higher than students who are given the conventional learning approach for Lobachevsky geometry material after controlling for the effect of Euclidean geometry spatial ability. Also, the same thing happened for the spatial abilities of Riemann geometry students. The learning trajectory is conveying learning objectives (learning objective); providing ethnomathematics-based visual problems; students do exploration; students make conclusions and summaries of exploration results; and ends with students sharing conclusions/summaries about concepts and principles in geometric systems. It was concluded that learning non-Euclid geometry through learning paths with an ethnomathematics approach had a positive impact on increasing students’ spatial abilities.

The Lobachevsky geometry method in the relativistic kinematics of particle collisions: special frame of reference

The use of the geometry of the Lobachevsky momentum space in the relativistic kinematics of particle collisions is demonstrated by the example of the problem of a special reference system. That system complements the geometric image of the process of elastic scattering of two particles of unequal masses. The speed of a special reference system relative to the center of mass and the angle of scattering of particles in it are determined. The conditions for the existence of such a reference system are analyzed. It is shown that in the case of a process with equal masses, the point corresponding to such a system goes into the ideal region of the extended Lobachevsky space - beyond the cone, and the lines intersecting in it become diverging lines in the sense of Lobachevsky geometry. In this case, the angle between the divergent straight lines (geodesics) of the geometric image is purely imaginary and connected to the minimum length of the segment perpendicular to the diverging straight lines (geodesics).

Lobachevsky Geometry and Stellar Parallaxes

Lobachevsky Geometry and Stellar Parallaxes

The understanding of the triangle in Lobachevski geometry through local culture

A triangle is an abstract object in Lobachevsky’s Geometry. The local content is needed as a starting point for learning geometry. This study aims to describe student’s understanding of triangles in Lobachevsky Geometry through local culture. This is an exploratory study. The focus is on analyzing the needs of student’s ability to understand the properties of Lobachevsky’s geometry. The subjects of this study were mathematics education students in Bengkulu, Indonesia. In-depth interviews are based on assignments about triangular problems. The problem is taken from the culture of Bengkulu society about “glutinous rice cake”. Data were analyzed through a collection of interviews and other physical activity. The results of this study are the subjects can connect the elements together to form a coherent unity based on observations of sticky rice cake. They produced a new product by organizing several elements into different forms or patterns from before, namely triangles with a large number of angles of less than 180. The conclusion of this study is that student’s understanding of triangles in Lobachevsky Geometry has reached the level of creating a new proposition, namely the number of angles in triangles in Lobachevski Geometry is less than 180.

Прямоугольники в геометрии Лобачевского

If we replace the V Postulate of Euclid with the Postulate of Lobachevsky, then we can get rectangles with more than 4 angles. The article describes the methodology for constructing such rectangles.

Lobachevsky space in analysis of relativistic nuclear interactions. New phenomenon—directed nuclear radiation

The relativistic nature of phenomena is illustrated in terms of Lobachevsky geometry. Lobachevsky space is used for description of particle production in relativistic nuclear physics on the basis of experimental data obtained at bubble chambers in π-C, p-C, C-C, n-p reactions in an energy range from units to tens of GeV . The new phenomenon—directed nuclear radiation—is discussed.

The ideas of A.M. Baldin on overcoming the reductionism principle in relativistic physics

The reductionism principle and the role of the Standard Model in the general paradigm of modern physics are discussed. The structure of the laws of nature is considered based on the idea of symmetry. The criteria of applicability of variables used for description of relativistic nuclear collisions and the ideas of A. M. Baldin on the notion “elementary particle” are discussed. Particle production is considered using the main parameter of the Lobachevsky geometry, the angle of parallelism.

Topologically embedded helicoidal pseudospherical cylinders**Authors partially supported by MIUR (Italy) under the PRIN project Varietà reali e complesse: geometria, topologia e analisi armonica; and by the GNSAGA of INDAM. The present research has also been partially supported by MIUR grant “Dipartimenti di Eccellenza” 2018-2022.

The class of traveling wave solutions of the sine-Gordon equation is known to be in 1–1 correspondence with the class of (necessarily singular) pseudospherical surfaces in Euclidean space with screw-motion symmetry: the pseudospherical helicoids. We explicitly describe all pseudospherical helicoids in terms of elliptic functions. This solves a problem posed by Popov (2014 Lobachevsky Geometry and Modern Nonlinear Problems (Berlin: Springer)). As an application, countably many continuous families of topologically embedded pseudospherical helicoids are constructed. A (singular) pseudospherical helicoid is proved to be either a dense subset of a region bounded by two coaxial cylinders, a topologically immersed cylinder with helical self-intersections, or a topologically embedded cylinder with helical singularities, called for short a pseudospherical twisted column. Pseudospherical twisted columns are characterized by four phenomenological invariants: the helicity , the parity , the wave number , and the aspect ratio , up to translations along the screw axis. A systematic procedure for explicitly determining all pseudospherical twisted columns from the invariants is provided.

G complex mass theory

Zhou Yi said, “Wuji (No have) born Tai Chi (have), Tai Chi born two state of affairs”. Tai Chi is the smallest unit of substance. The two state of affairs are real (G particle) and imaginary (field). The real state is the particle state (g particle) formed by the accumulation of Tai Chi. The imaginary state is the field (wave) state formed by the evacuation of Tai Chi. The motion of g particles in complex space-time to form quarks and leptons, Neutrinos are magnetic monopole. The field state and the particle state are both opposite and unified, and they transform each other. The Chinese diagram of the universe best defines the qualitative description of this relationship. The complex in mathematics can quantitatively describe the contradiction-the insides of things are opposite and unified in mutual conversion. All the elementary particles (fermion and bosons) are different motions of the sample particle in complex space-time. The real part of physical quantity reflects the Feynman geometry of the particle (fermion) that is macroscopic and non-linear; the imaginary part represents the Lobachevsky geometry that is microscopic, linear and fluctuating (bosons). The real part of electromagnetic force satisfies Coulomb law, and its imaginary part meets Lorentz law. The real part of mass force satisfies Newton’s law of universal gravitation, and its imaginary part meets Coriolis law. The real part of energy constitutes the macroscopical mass-energy equation of Einstein, and the imaginary part forms the Planck equation of microscopical quantum. The rest mass of electron and the charge are not in the same space, but satisfy the relationship between real and imaginary parts. The theory can explain the tests by Hao Ji, Jinsong Feng, Qi Liu, et al. It also shows that the charges with the same nature repulse each other, but attract each other with the different natures; on the contrary, the masses with the same nature attract each other. The theory can fundamentally solve the divergence in quantum field.

Lobachevsky geometry in TTW and PW systems

We review the classical properties of Tremblay-Turbiner-Winternitz and Post-Wintenitz systems and their relation with N-dimensional rational Calogero model with oscillator and Coulomb potentials, paying special attention to their hidden symmetries. Then we show that combining the radial coordinate and momentum in a single complex coordinate in proper way, we get an elegant description for the hidden and dynamical symmetries in these systems related with action-angle variables.

Lobachevsky crystallography made real through carbon pseudospheres

We realize Lobachevsky geometry in a simulation lab, by producing a carbon-based energetically stable molecular structure, arranged in the shape of a Beltrami pseudosphere. We find that this structure: (i) corresponds to a non-Euclidean crystallographic group, namely a loxodromic subgroup of ; (ii) has an unavoidable singular boundary, that we fully take into account. Our approach, substantiated by extensive numerical simulations of Beltrami pseudospheres of different size, might be applied to other surfaces of constant negative Gaussian curvature, and points to a general procedure to generate them. Our results also pave the way to test certain scenarios of the physics of curved spacetimes owing to graphene’s unique properties.

Quantum field theory in curved graphene spacetimes, Lobachevsky geometry, Weyl symmetry, Hawking effect, and all that

The solutions of many issues, of the ongoing efforts to make deformed graphene a tabletop quantum field theory in curved spacetimes, are presented. A detailed explanation of the special features of curved spacetimes, originating from embedding portions of the Lobachevsky plane into $\mathbf{R}^3$, is given, and the special role of coordinates for the physical realizations in graphene, is explicitly shown, in general, and for various examples. The Rindler spacetime is reobtained, with new important differences with respect to earlier results. The de Sitter spacetime naturally emerges, for the first time, paving the way to future applications in cosmology. The role of the BTZ black hole is also briefly addressed. The singular boundary of the pseudospheres, "Hilbert horizon", is seen to be closely related to event horizon of the Rindler, de Sitter, and BTZ kind. This gives new, and stronger, arguments for the Hawking phenomenon to take place. An important geometric parameter, $c$, overlooked in earlier work, takes here its place for physical applications, and it is shown to be related to graphene's lattice spacing, $\ell$. It is shown that all surfaces of constant negative curvature, ${\cal K} = -r^{-2}$, are unified, in the limit $c/r \to 0$, where they are locally applicable to the Beltrami pseudosphere. This, and $c = \ell$, allow us a) to have a phenomenological control on the reaching of the horizon; b) to use spacetimes different than Rindler for the Hawking phenomenon; c) to approach the generic surface of the family. An improved expression for the thermal LDOS is obtained. A non-thermal term for the total LDOS is found. It takes into account: a) the peculiarities of the graphene-based Rindler spacetime; b) the finiteness of a laboratory surface; c) the optimal use of the Minkowski quantum vacuum, through the choice of this Minkowski-static boundary.

A hyperbolic variant of the Nelder–Mead simplex method in low dimensions

Abstract The Nelder-Mead simplex method is a widespread applied numerical optimization method with a vast number of practical applications, but very few mathematically proven convergence properties. The original formulation of the algorithm is stated in Rn using terms of Euclidean geometry. In this paper we introduce the idea of a hyperbolic variant of this algorithm using the Poincaré disk model of the Bolyai- Lobachevsky geometry. We present a few basic properties of this method and we also give a Matlab implementation in 2 and 3 dimensions

Lobachevsky geometry of (super)conformal mechanics

We give a simple geometric explanation for the similarity transformation mapping one-dimensional conformal mechanics to free-particle system. Namely, we show that this transformation corresponds to the inversion of the Klein model of Lobachevsky space (non-compact complex projective plane) C P ˜ 1 . We also extend this picture to the N = 2 k superconformal mechanics described in terms of Lobachevsky superspace C P ˜ 1 | k .

TOPOLOGICALLY MASSIVE GAUGE THEORY: A LORENTZIAN SOLUTION

We obtain a Lorentzian solution for the topologically massive non-Abelian gauge theory on AdS space [Formula: see text] by means of an SU (1, 1) gauge transformation of the previously found Abelian solution. There exists a natural scale of length which is determined by the inverse topological mass ν ~ ng2. In the topologically massive electrodynamics the field strength locally determines the gauge potential up to a closed 1-form via the (anti-)self-duality equation. We introduce a transformation of the gauge potential using the dual field strength which can be identified with an Abelian gauge transformation. Then we present map [Formula: see text] including the topological mass which is the Lorentzian analog of the Hopf map. This map yields a global decomposition of [Formula: see text] as a trivial [Formula: see text] bundle over the upper portion of the pseudosphere [Formula: see text] which is the Hyperboloid model for the Lobachevski geometry. This leads to a reduction of the Abelian field equation onto [Formula: see text] using a global section of the solution on [Formula: see text]. Then we discuss the integration of the field equation using the Archimedes map [Formula: see text]. We also present a brief discussion of the holonomy of the gauge potential and the dual field strength on [Formula: see text].

Pseudospherical surfaces and some problems of mathematical physics

In the paper, some aspects of the interrelation of Lobachevsky geometry and nonlinear differential equations are discussed.

Observer’s mathematics – mathematics of relativity

Observer dependent ascending chain of embedded sets of decimal fractions and their Cartesian products is considered. For every set, arithmetic operations are defined (these operations locally coincide with standard operations), which transform every set into a local ring. The basic problems of Algebra, Geometry, Topology, Logic, and Functional Analysis are considered for this chain. Definition of Dimension of these sets is introduced. In particular, the dimension of each of these sets is greater than or equal to seven. Euclidean, Lobachevsky, and Riemannian Geometries become particular cases of the developed Geometry, although many others are possible. For example, we proved that two lines in a plane may intersect in 0 (without being parallel in the usual sense), 1, 2, 10, or even 100 points. Two of the classical Geometries depend on a particular neighborhood of a given line. For example, Euclidean Geometry works in sufficiently small neighborhood of the given line, but when we enlarge the neighborhood, Lobachevsky Geometry takes over. Developed Topology gives birth to Time, and Time becomes a function of Space. Also, the Axiom of Choice becomes invalid in the new model of Mathematics. The application of the new model to Einstein’s special physical theory of relativity is considered. The existence of Time and Space quantums is proved. We also construct a new system of coordinate transformations that substitute Lorenz transformations. We also consider the application of the new model to data-mining.

Location–Scale Depth

This article introduces a halfspace depth in the location–scale model that is along the lines of the general theory given by Mizera, based on the idea of Rousseeuw and Hubert, and is complemented by a new likelihood-based principle for designing criterial functions. The most tractable version of the proposed depth—the Student depth—turns out to be nothing but the bivariate halfspace depth interpreted in the Poincaré plane model of the Lobachevski geometry. This fact implies many fortuitous theoretical and computational properties, in particular equivariance with respect to the Möbius group and favorable time complexities of algorithms. It also opens a way to introduce some other depth notions in the location–scale context, for instance, location–scale simplicial depth. A maximum depth estimator of location and scale—the Student median—is introduced. Possible applications of the proposed concepts are investigated on data examples.

Non-euclidean geometrical aspects of the schur and levinson-szego algorithms

In this paper, we address non-Euclidean geometrical aspects of the Schur and Levinson-Szego algorithms. We first show that the Lobachevski geometry is, by construction, one natural geometrical environment of these algorithms, since they necessarily make use of automorphisms of the unit disk. We next consider the algorithms in the particular context of their application to linear prediction. Then the Schur (resp., Levinson-Szego) algorithm receives a direct (resp., polar) spherical trigonometry (ST) interpretation, which is a new feature of the classical duality of both algorithms.