pseudo-Euclidean Research Articles

AI-Powered Approaches for Hypersurface Reconstruction in Multidimensional Spaces

The present article explores the possibilities of using artificial neural networks to solve problems related to reconstructing complex geometric surfaces in Euclidean and pseudo-Euclidean spaces, examining various approaches and techniques for training the networks. The main focus is on the possibility of training a set of neural networks with information about the available surface points, which can then be used to predict and complete missing parts. A method is proposed for using separate neural networks that reconstruct surfaces in different spatial directions, employing various types of architectures, such as multilayer perceptrons, recursive networks, and feedforward networks. Experimental results show that artificial neural networks can successfully approximate both smooth surfaces and those containing singular points. The article presents the results with the smallest error, showcasing networks of different types, along with a technique for reconstructing geographic relief. A comparison is made between the results achieved by neural networks and those obtained using traditional surface approximation methods such as Bézier curves, k-nearest neighbors, principal component analysis, Markov random fields, conditional random fields, and convolutional neural networks.

Maximum acceleration and quantum clock: on the existence of a new universal constant

In the pseudo-Euclidean Minkowski space, the four-dimensional volume element is invariant under Lorentz transformations. By hypothesising that in this space there is a minimum volume, it is possible to demonstrate the existence of a maximum acceleration. The volume element cannot be derived from the theory and must be obtained through direct measurement, thus it assumes the role of a bona fide universal constant. Two different estimates of the elementary volume are given, which differ by several orders of magnitude: the first is obtained in a pseudo-Euclidean space for particles with mass, and the second represents an absolute minimum volume, independent of the mass.

Geometrical Analysis of Spacelike and Timelike Rectifying Curves and Their Applications

In the light of great importance of curves and their frames in many different branches of science, especially differential geometry as well as geometric properties and their uses in various fields, we are interested here to study a special kind of curves called rectifying curves. We consider some characterizations of a non-lightlike curve has a spacelike or timelike rectifying plane in pseudo-Euclidean space E13. Then, we demonstrate that the proportion of curvatures of any spacelike or timelike rectifying curve is a non-constant linear function of the arc length parameter s. Finally, we defray a computational example to support our main findings.

Superposition Formulae for the Geometric Bäcklund Transformations of the Hyperbolic and Elliptic Sine-Gordon and Sinh-Gordon Equations

We provide superposition formulae for the six cases of Bäcklund transformations corresponding to space-like and time-like surfaces in the 3-dimensional pseudo-Euclidean space. In each case, the surfaces have constant negative or positive Gaussian curvature and they correspond to solutions of one of the following equations: the sine-Gordon, the sinh-Gordon, the elliptic sine-Gordon and the elliptic sinh-Gordon equation. The superposition formulae provide infinitely many solutions algebraically after the first integration of the Bäcklund transformation. Such transformations and the corresponding superposition formulae provide solutions of the same hyperbolic equation, while they show an unusual property for the elliptic equations. The Bäcklund transformation alternates solutions of the elliptic sinh-Gordon equation with those of the elliptic sine-Gordon equation and the superposition formulae provide solutions of the same elliptic equation. Explicit examples and illustrations are given.

Embedding Dimensions of Matrices Whose Entries are Indefinite Distances in the Pseudo-Euclidean Space

Embedding Dimensions of Matrices Whose Entries are Indefinite Distances in the Pseudo-Euclidean Space

Features of the geometry of the five-dimensional pseudo-Euclidean space of index two

The article is devoted to the study of the geometry of subspaces of a five-dimensional pseudo-Euclidean space. This space is attractive because all kinds of semi-Euclidean, semi-pseudo-Euclidean, hyperbolic three-dimensional spaces with projective metrics are realized in its subspaces. In the sphere of the imaginary radius of space, de Sitter space is realized. Here there is a space with projective metrics in the sense of Cayley-Klein. It is a three-dimensional space with a metric that preserves space on itself when mapped linearly. The corresponding linear transformation is called the motion of this space. An interpretation of de Sitter space in a four-dimensional pseudo-Euclidean space is proved. Studies have confirmed that in subspaces of space , in addition to elliptic spaces, there is a geometry of three-dimensional spaces with projective metrics. De Sitter space of the second kind is also realized in the sphere of imaginary radius. De Sitter space is a geodesic mapping in four-dimensional Minkowski space.

Движение гиростата под действием следящего момента силы

The gyrostat moves around a fixed point under the forces influence in pseudo-Euclidean space with signature (3,1). The resulting moment of the external forces system and the gyrostatic moment are specified relative to the coordinate orthoreper, invariably associated with the gyrostat carrier, and are constant. Three types of gyrostat motion are given, in each of which the resulting force vector-moment is collinear to of the main axis of the gyrostat inertia, and the gyrostatic moment is orthogonal to this axis. The vector manifold of the gyrostat angular velocities and the field of its phase trajectories in the state space are found with the parametric integration method using. Integration of the gyrostat motion equations is carried out in quadratures and elliptic functions of time.

A classification of triharmonic hypersurfaces in a pseudo-Riemannian space form

The purpose of this article is to investigate triharmonic hypersurfaces Mtn in a pseudo-Riemannian space form Nt1n+1(c). Assuming that the shape operator is diagonalizable, we show that every CMC triharmonic hypersurface either has constant scalar curvature or it is minimal if cε>0 and every CMC triharmonic hypersurface is minimal if cε<0, where ε is the inner product of the unit normal vector of Mtn. Furthermore, we prove that any CMC triharmonic hypersurface with diagonalizable shape operator in a pseudo-Euclidean space Et1n+1 is actually minimal provided that zero is a principal curvature of multiplicity at most one.

A Hypersurfaces of Revolution Family in the Five-Dimensional Pseudo-Euclidean Space E25

We present a family of hypersurfaces of revolution distinguished by four parameters in the five-dimensional pseudo-Euclidean space E25. The matrices corresponding to the fundamental form, Gauss map, and shape operator of this family are computed. By utilizing the Cayley–Hamilton theorem, we determine the curvatures of the specific family. Furthermore, we establish the criteria for maximality within this framework. Additionally, we reveal the relationship between the Laplace–Beltrami operator of the family and a 5×5 matrix.

Classification of all constant solutions of SU(2) Yang–Mills equations with arbitrary current in pseudo-Euclidean space ℝp,q

We present a classification and an explicit form of all constant solutions of the Yang–Mills equations with [Formula: see text] gauge symmetry for an arbitrary constant non-Abelian current in pseudo-Euclidean space [Formula: see text] of arbitrary finite dimension [Formula: see text]. Using hyperbolic singular value decomposition and two-sheeted covering of orthogonal group by spin group, we solve the nontrivial system for constant solutions of the Yang–Mills equations of [Formula: see text] cubic equations with [Formula: see text] unknowns and [Formula: see text] parameters in the general case. We present a new symmetry of this system of equations. All solutions in terms of the potential, strength and invariant of the Yang–Mills field are presented. Nonconstant solutions of the Yang–Mills equations can be considered in the form of series of perturbation theory using all constant solutions as a zeroth approximation.

Triharmonic hypersurfaces with constant mean curvature in pseudo-Riemannian space forms

In this paper, triharmonic hypersurfaces with constant mean curvature in pseudo-Riemannian space forms are studied. Under the assumption that the shape operator is diagonalizable, we first classify completely the nonminimal hypersurfaces with at most two distinct principal curvatures and give some examples of non-biharmonic triharmonic hypersurfaces. Then, we prove that the hypersurfaces with at most four distinct principal curvatures have constant scalar curvature. As a consequence, we obtain that such triharmonic hypersurfaces in pseudo-Euclidean spaces are minimal, which gives an affirmative partial solution to the generalized Chen's conjecture in [21].

Тензор кривизны n-поверхности и ее сферического образа в En+k

The main task of classical multidimensional differential geometry is to study the properties of various n-surfaces. Often these studies use torsion coefficients that are defined for any n-surface with codimension k &amp;amp;gt; 1 in (n + k)-dimensional Euclidean space. For hypersurfaces, the torsion coefficients are not defined.Another important concept used to study the properties of n-surfaces is the spherical Gaussian mapping. The Gaussian mapping defined on submanifolds of Euclidean and pseudo-Euclidean spaces allows one to study the external properties of a submanifold immersed in a Euclidean or pseudo-Euclidean space. In a number of papers, the properties of the Gaussian mapping are studied, as well as the geometric characteristics of the images of submanifolds under a spherical mapping, which are submanifolds of a hypersphere or a Grassmannian.In this article, we study the local properties of the spherical image of a regular n-surface of arbitrary codimension. The spherical mapping is defined for n-surfaces with codimension greater than one in Euclidean space by means of a regular vector field. Each vector of this field at a point of the submanifold is orthogonal to the tangent space of the submanifold at the chosen point.The article uses the methods of differential and Riemannian geometry, as well as tensor analysis to study n-surfaces with a codimension greater than one. Under some additional conditions, a connection is established between the curvature tensor of a given surface and the curvature tensor of its spherical image. Under the same additional conditions, some geometric characteristics of the points of the spherical image of the original n-surface are studied.

Hypersurfaces satisfying [formula omitted] in [formula omitted

In this paper, we study hypersurfaces Mr4(r=0,1,2,3,4) satisfying △H→=λH→ (λ a constant) in the pseudo-Euclidean space Es5(s=0,1,2,3,4,5). We obtain that every such hypersurface in Es5 with diagonal shape operator has constant mean curvature, constant norm of second fundamental form and constant scalar curvature. Also, we prove that every biharmonic hypersurface in Es5 with diagonal shape operator must be minimal.

Principal angles in pseudo-euclidean spaces of index 1

In this paper, we introduce the notion of principal angles between subspaces of the same signature in a (real, complex or quaternionic) pseudo-euclidean space of index 1. We show that these determine the relative position of an important class of pairs of hyperbolic and elliptic subspaces (Theorems 3.9 and 4.10). Also, as an application, we will see that these angles can be used to study the relative position of pairs of an important class of totally geodesic submanifolds of the (real, complex or quaternionic) hyperbolic space (Theorem 6.3). As a consequence we obtain a nice interpretation of these principal angles as geometric invariants of the hyperbolic space.

Динамика перманентного движения твердого тела в псевдоевклидовом пространстве

The conditions for the existence and properties of the permanent motion of an absolutely rigid body rotating around a fixed pole are studied in the pseudo-Euclidean space. The body is affected by a force vector-moment, which is constant relative to the coordinate system, invariably associated with the body, as well as a system of gyroscopic forces with a resulting vector-moment, which is linear relative to the angular velocity of the body. The first algebraic integrals of a dynamical system are found, as well as the necessary conditions for the existence of a permanent motion of a body. The equations of the moving hodograph of the angular velocity in this motion are obtained. The structural-kinetic properties of the axes of permanent rotation and the geometry of their distribution in space relative to the body are studied.

The minimality of biharmonic hypersurfaces in pseudo-Euclidean spaces

&lt;abstract&gt;&lt;p&gt;In this paper, we investigate the minimality of biharmonic hypersurfaces with some recurrent operators in a pseudo-Euclidean space.&lt;/p&gt;&lt;/abstract&gt;

Hypersurfaces of revolution family supplying $ \Delta \mathfrak{r} = \mathcal{A}\mathfrak{r} $ in pseudo-Euclidean space $ \mathbb{E}_{3}^{7} $

&lt;abstract&gt;&lt;p&gt;In this study, we introduce a family of hypersurfaces of revolution characterized by six parameters in the seven-dimensional pseudo-Euclidean space $ {\mathbb{E}}_{3}^{7} $. These hypersurfaces exhibit intriguing geometric properties, and our aim is to analyze them in detail. To begin, we calculate the matrices corresponding to the fundamental form, Gauss map, and shape operator associated with this hypersurface family. These matrices provide essential information about the local geometry of the hypersurfaces, including their curvatures and tangent spaces. Using the Cayley-Hamilton theorem, we employ matrix algebra techniques to determine the curvatures of the hypersurfaces. This theorem allows us to express the characteristic polynomial of a matrix in terms of the matrix itself, enabling us to compute the curvatures effectively. In addition, we establish equations that describe the interrelation between the mean curvature and the Gauss-Kronecker curvature of the hypersurface family. These equations provide insights into the geometric behavior of the surfaces and offer a deeper understanding of their intrinsic properties. Furthermore, we investigate the relationship between the Laplace-Beltrami operator, a differential operator that characterizes the geometry of the hypersurfaces, and a specific $ 7\times 7 $ matrix denoted as $ \mathcal{A} $. By studying this relation, we gain further insights into the geometric structure and differential properties of the hypersurface family. Overall, our study contributes to the understanding of hypersurfaces of revolution in $ {\mathbb{ E}}_{3}^{7} $, offering mathematical insights and establishing connections between various geometric quantities and operators associated with this family.&lt;/p&gt;&lt;/abstract&gt;

Topological analysis of pseudo-Euclidean Euler top for special values of the parameters

An analogue of the Euler top is considered for a pseudo-Euclidean space is under consideration. In the cases when the geometric integral or area integral vanishes the bifurcation diagrams of the moment map are constructed and the homeomorphism class of each leaf of the Liouville foliation is determined. For each arc of the bifurcation diagram, for one of the two possible cases of the mutual arrangement of the moments of inertia, the types of singularities in the preimage of a small neighbourhood of this arc (analogues of Fomenko 3-atoms) are determined, and for nonsingular isoenergy and isointegral surfaces an invariant of rough Liouville equivalence (an analogue of a rough molecule) is constructed. The pseudo-Euclidean Euler system turns out to have noncompact noncritical bifurcations. Bibliography: 23 titles.

A family of special case sequential warped-product manifolds

&lt;abstract&gt;&lt;p&gt;We derive the general formulas for a special configuration of the sequential warped-product semi-Riemannian manifold to be Einstein, where the base-manifold is the product of two manifolds both equipped with a generic diagonal conformal metrics. Subsequently we study the case in which these two manifolds are conformal to a $ n_1 $-dimensional and $ n_2 $-dimensional pseudo-Euclidean space, respectively. For the latter case, we prove the existence of a family of solutions that are invariant under the action of a $ (n_1-1) $-dimensional group of transformations to the case of positive constant Ricci curvature ($ \lambda &amp;gt; 0 $).&lt;/p&gt;&lt;/abstract&gt;

Relational Analysis of Dirac Equation in Momentum Representation

In terms of the relational approach to space-time geometry and physical interactions, we show that the Dirac equation for a free fermion in the momentum representation can be obtained starting from a binary system of complex relations (BSCR) between elements of two abstract sets. With the derivation performed we show that the 4-dimensional pseudo-Euclidean momentum space is not needed a priori but naturally emerges from considerations of rather general character (2-spinor algebra). A bispinor wave function is constructed for a fermion with positive energy and an arbitrary distribution of momenta. Special attention is paid to physical assumptions that should be made to enable the construction.