Metrics of Homogeneous Spaces Admitting Complete Sets of the (2.1) type

A classification of spaces admitting complete sets of integrals of motion of the (2.1) type and being spatially homogeneous has been obtained. The explicit form of metrics and Killing vectors have been found for all types of such spaces. The Bianchi classification of the spaces obtained is given.     

Stäckel electrovacuum spaces with isotropic complete sets. Integration of field equations for the metric generalizing a space of type (1.1)

A system of Einstein-Maxwell equations is integrated for the metric that generalizes the metric of a space with full set of type (1.1). This is necessary for carrying out the classification of the corresponding Stäckel electrovacuum spaces. An algebraic classification of the solutions obtained is carried out.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 12, pp. 17–20, December, 1987.     

Integration of Dirac equation in Riemannian spaces with five-dimensional group of motions

In the present article a classification of Riemannian spaces with five-dimensional group of motion is described from the point of view of a solution of the Dirac equation. A class of spaces is identified in which the Dirac equation does not admit a complete separation of variables, and exact solutions of the Dirac equation are obtained in these spaces by means of the method of noncommutative integration. Omsk State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 24–28, August, 1997.     

A Geometric Approach to the Classification of the Equilibrium Shapes of Self-Gravitating Fluids

The classification of the equilibrium shapes that a self-gravitating fluid can take in a Riemannian manifold is a classical problem in Mathematical Physics. In this paper it is proved that the equilibrium shapes are isoparametric submanifolds. Some geometric properties of them are also obtained, e.g. classification and existence for some Riemannian spaces and relationship with the isoperimetric problem and the group of isometries of the manifold. Our approach to the problem is geometrical and allows to study the equilibrium shapes on general Riemannian spaces.     

Deep Learning and Entropy-Based Texture Features for Color Image Classification

In the domain of computer vision, entropy—defined as a measure of irregularity—has been proposed as an effective method for analyzing the texture of images. Several studies have shown that, with specific parameter tuning, entropy-based approaches achieve high accuracy in terms of classification results for texture images, when associated with machine learning classifiers. However, few entropy measures have been extended to studying color images. Moreover, the literature is missing comparative analyses of entropy-based and modern deep learning-based classification methods for RGB color images. In order to address this matter, we first propose a new entropy-based measure for RGB images based on a multivariate approach. This multivariate approach is a bi-dimensional extension of the methods that have been successfully applied to multivariate signals (unidimensional data). Then, we compare the classification results of this new approach with those obtained from several deep learning methods. The entropy-based method for RGB image classification that we propose leads to promising results. In future studies, the measure could be extended to study other color spaces as well.     

Integration of the einstein-dirac equations for steckel spaces of type (3.1)

We consider the problem of classification of Steckel spaces satisfying the system of self-consistent Einstein-Dirac equations with a cosmological term for the case when the spaces allow for isotropic complete sets of integrals of motion of type (3.1). The exact solution obtained contains four arbitrary functions of one variable. Tomsk State Pedagogical Institute. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 3–9, February, 1997.     

Killing-Yano Tensors and Superintegrable Systems

Killing-Yano (KY) and Killing tensors of the four types of metrics, that represent the two-dimensional spaces given in Darbouxs classification, are obtained. It is proved that all spaces admit dual manifolds and their KY tensors are calculated.     

The Vaidya problem in conformally flat Stäckel spaces of type (1.1)

The problem of classification of conformally flat radiation spaces admitting full separation of variables in the Hamilton–Jacobi equation is considered. The Vaidya problem for conformally flat St?ckel spaces of type (1.1) has been solved; the metrics and the radiation have been found in explicit form. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 12–14, January, 2009.     

A Diagram for Bianchi A-Types

A diagram for Bianchi spaces with vanishing vector of structure constants (type A in the Ellis-MacCallum classification) illustrates the relations among their different types under similarity transformations. The Ricci coefficients and the Ricci tensor are related by a Cremona transformation.     

A first approximation for quantization of singular spaces

Many mathematical models of physical phenomena that have been proposed in recent years require more general spaces than manifolds. When taking into account the symmetry group of the model, we get a reduced model on the (singular) orbit space of the symmetry group action. We investigate quantization of singular spaces obtained as leaf closure spaces of regular Riemannian foliations on compact manifolds. These contain the orbit spaces of compact group actions and orbifolds. Our method uses foliation theory as a desingularization technique for such singular spaces. A quantization procedure on the orbit space of the symmetry group–that commutes with reduction–can be obtained from constructions which combine different geometries associated with foliations and new techniques originated in Equivariant Quantization. The present paper contains the first of two steps needed to achieve these just detailed goals.     

Special geometry,cubic polynomials and homogeneous quaternionic spaces

The existing classification of homogeneous quaternionic spaces is not complete. We study these spaces in the context of certainN=2 supergravity theories, where dimensional reduction induces a mapping betweenspecial real, Kähler and quaternionic spaces. The geometry of the real spaces is encoded in cubic polynomials, those of the Kähler and quaternionic manifolds in homogeneous holomorphic functions of second degree. We classify all cubic polynomials that have an invariance group that acts transitively on the real manifold. The corresponding Kähler and quaternionic manifolds are then homogeneous. We find that they lead to a well-defined subset of the normal quaternionic spaces classified by Alekseevskiî (and the corresponding special Kähler spaces given by Cecotti), but there is a new class of rank-3 spaces of quaternionic dimension larger than 3. We also point out that some of the rank-4 Alekseevskiî spaces were not fully specified and correspond to a finite variety of inequivalent spaces. A simpler version of the equation that underlies the classification of this paper also emerges in the context ofW ₃ algebras.Communicated by K. Gawedzki     

The K-Orbits, Identities, and Classification of Four-Dimensional Homogeneous Spaces with the Group of Poincaré and de Sitter Transforms

The method of orbits traditionally applied to geometric quantization problems is used to study homogeneous spaces. Based on the proposed classification of the orbits of co-adjoint representation (K-orbits), a classification of homogeneous spaces is constructed. This classification allows one, in particular, to point out the explicit form of identities – functional relations between the transform-group generators – which are of great importance in applied problems (e.g., in the theory of separation of variables). All four-dimensional homogeneous spaces with the group of Poincaré and de Sitter transforms are classified and all independent identities on these spaces are given in explicit form.     

New approach to Einstein-Petrov type I spaces. II. A classification scheme

A classification scheme is presented which enables one to have an overall look at Einstein-Petrov type I spaces and in particular to pick out those spaces whose Petrov scalars are functionally independent. It is shown that some subclasses defined by quite severe simplifying assumptions (which lead to considerable simplification of the field equations) still retain functionally independent Petrov scalars. For these subclasses three coupled differential equations (two of which are wave equations) are calculated for the Petrov scalars.     

Classification of domains of closed operators

The structure of domain of closed operators in Hilbert space is investigated by means of sequence spaces. A complete classification leads to three classes of domains. We obtain necessary and sufficient conditions for the unitary equivalence of domains expressed by the equivalence of appropriate sequences of naturals. A connection with perturbation theory is mentioned.     

Plane Gravitational Waves in Spatially-Homogeneous Models of type-(3.1) Stäckel Spaces

Russian Physics Journal - A classification is performed of the spaces with plane-wave metrics in accordance with the presence of three-dimensional subgroups of the group of space-time motion with...     

Relativistic quantum similarities in atoms in position and momentum spaces

A study of different quantum similarity measures and their corresponding quantum similarity indices is carried out for the atoms from H to Lr (Z=1-103). Relativistic effects in both position and momentum spaces have been studied by comparing the relativistic values to the non-relativistic ones. We have used the atomic electron density in both position and momentum spaces obtained within relativistic and non-relativistic numerical-parameterized optimized effective potential approximations.     

On the use of globally symmetric pseudo-Riemannian spaces in cosmology

The Loos theory of globally symmetric manifolds as manifolds with “symmetric” multiplication is applied to the four-dimensional pseudo-Riemannian spaces of general relativity. It is shown that the classification of symmetric universes is reduced to two steps: the classification of Lie triple systems on Minkowski space which gives the possible curvature tensors and the construction of all covering spaces with the same local curvature tensors. The first step is already solved by Lister. The Loos theory can beused to construct some solutions of the second step. It turns out that the Ozsváth-Schücking universe is a special case of a certain class of symmetric spaces which can be characterized topologically. Explicit results are given for the two de-Sitter universes which are simplesymmetric spaces.     

Classification of world sheet instantons and the path integral measure in string theories

A classification of world sheet instantons propagating in some internal spaces is presented. The associated path integral measure problem or θ-angles, as discussed by Wen and Witten, and by Li, is addressed.     

Special Stäckel electrovac spacetimes

Classification of all electrovac specetimes permitting the separation of variables in the Hamilton-Jacobi equation for a charged test particle is carried out. This separation requires the existence of a complete set consisting of Killing’s vectors and tensors of a special kind. Every complete set defines its own type of metric and electromagnetic potential in the separable coordinate system. There exist seven types of separation of variables for electromagnetic spaces. For every type an additional classification is carried out by transformation of coordinates without any disturbance of the separation conditions, the gradient transformation of electromagnetic potential and the conformal-constant transformation of metric. The key step in solving the problem is the extraction of an autonomous subsystem which determines the metric from only the Einstein-Maxwell equations for every type of separation of variables. Representatives of all classes of metrics and electromagnetic potential are given for every type of separation of variables with the exception of the spaces found in the well-known work by Carter. The problem is solved in terms of metric formalism. The classes of electrovac spacetimes obtained are found to be related to Petrov’s classification.