An exact solution of a scalar-tensor theory of gravitation

A class of exact solutions of the cylindrically symmetric space-time with two degrees of freedom corresponding to the vacuum field equations of a scalar-tensor theory, formally similar to Brans-Dicke theory, is obtained. The solutions possess wave-like character.     

Exterior axisymmetric solution for a rotating charged body in the relativistic theory of gravitation

Scientific-Research Institute of Nuclear Physics at the Moscow State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 78, No. 2, pp. 305–313, February, 1989.     

Instantons and gravitation theory

Some problems related to using nonperturbative quantization methods in theories of gauge fields and gravitation are studied. The unification of interactions is considered in the context of the geometric theory of gauge fields. The notion of vacuum in the unified interaction theory and the role of instantons in the vacuum structure are considered. The relation between the definitions of instantons and the energymomentum tensor of a gauge field and also the role played by the vacuum solutions to the Einstein equations in the definition of vacuum for gauge fields are demonstrated. The Schwarzschild solution, as well as the entire class of vacuum solutions to the Einstein equations, is a gravitational instanton even though the signature of the space-time metric is hyperbolic. Gravitation, oncluding the Einstein version, is considered a special case of an interaction described by a non-Abelian gauge field. Translated from Teoreticheskaya i Matematicheskaya. Fizika. Vol. 115, No. 2, pp. 312–320, May. 1998.     

Static spherically symmetric solution for scalar and electromagnetic fields in the relativistic theory of gravitation

A general static spherically symmetric solution for free scalar and electromagnetic fields in the relativistic theory of gravitation is obtained. In the special cases of absence of the scalar, electromagnetic, or both fields, it reduces to the corresponding known solutions.Institute of Nuclear Physics at the Moscow State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 91, No. 3, pp. 418–425, June, 1992.     

Uniqueness of exterior axisymmetric solution for a rotating charged body in the relativistic theory of gravitation

Scientific-Research Institute of Nuclear Physics at the Moscow State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 79, No. 3, pp. 394–403, June, 1989.     

Causality principle in the relativistic theory of gravitation

Institute of High Energy Physics, Serpukhov. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 88, No. 3, pp. 459–466, September, 1991.     

Inequivalence of inertia of rotation and of Kerr-Newman gravitation in the relativistic theory of gravitation

Scientific-Research Institute of Nuclear Physics at the Moscow State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 80, No. 3, pp. 474–480, September, 1989.     

Phenomenological foundations of the theory of gravitation

A phenomenological expression for the vacuum amplitude of gravitons is found in the framework of Schwinger's source theory. The expression provides the basis for a formulation of the correspondence principle and solution of the energy-momentum problem in gravitation theory. Theorems on the conservation of higher moments are proved. It is shown that any theory of gravitation can always be formulated in terms of Minkowski space locally (and globally if spacetime can be covered by a single chart).Donetsk Physicotechnical Institute, Ukrainian Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 93, No. 1, pp. 154–174, October, 1992.     

Advances in metric embedding theory

Metric Embedding plays an important role in a vast range of application areas such as computer vision, computational biology, machine learning, networking, statistics, and mathematical psychology, to name a few. The mathematical theory of metric embedding is well studied in both pure and applied analysis and has more recently been a source of interest for computer scientists as well. Most of this work is focused on the development of bi-Lipschitz mappings between metric spaces. In this paper we present new concepts in metric embeddings as well as new embedding methods for metric spaces. We focus on finite metric spaces, however some of the concepts and methods are applicable in other settings as well.One of the main cornerstones in finite metric embedding theory is a celebrated theorem of Bourgain which states that every finite metric space on n points embeds in Euclidean space with distortion. Bourgain?s result is best possible when considering the worst case distortion over all pairs of points in the metric space. Yet, it is natural to ask: can an embedding do much better in terms of the average distortion? Indeed, in most practical applications of metric embedding the main criteria for the quality of an embedding is its average distortion over all pairs.In this paper we provide an embedding with constant average distortion for arbitrary metric spaces, while maintaining the same worst case bound provided by Bourgain?s theorem. In fact, our embedding possesses a much stronger property. We define the ?q-distortion of a uniformly distributed pair of points. Our embedding achieves the best possible ?q-distortion for all 1?q?∞simultaneously.The results are based on novel embedding methods which improve on previous methods in another important aspect: the dimension of the host space. The dimension of an embedding is of very high importance in particular in applications and much effort has been invested in analyzing it. However, no previous result improved the bound on the dimension which can be derived from Bourgain?s embedding. Our embedding methods achieve better dimension, and in fact, shed new light on another fundamental question in metric embedding, which is: whether the embedding dimension of a metric space is related to its intrinsic dimension? I.e., whether the dimension in which it can be embedded in some real normed space is related to the intrinsic dimension which is reflected by the inherent geometry of the space, measured by the space?s doubling dimension. The existence of such an embedding was conjectured by Assouad,⁴and was later posed as an open problem in several papers. Our embeddings give the first positive result of this type showing any finite metric space obtains a low distortion (and constant average distortion) embedding in Euclidean space in dimension proportional to its doubling dimension.Underlying our results is a novel embedding method. Probabilistic metric decomposition techniques have played a central role in the field of finite metric embedding in recent years. Here we introduce a novel notion of probabilistic metric decompositions which comes particularly natural in the context of embedding. Our new methodology provides a unified approach to all known results on embedding of arbitrary finite metric spaces. Moreover, as described above, with some additional ideas they allow to get far stronger results.The results presented in this paper⁵have been the basis for further developments both within the field of metric embedding and in other areas such as graph theory, distributed computing and algorithms. We present a comprehensive study of the notions and concepts introduced here and provide additional extensions, related results and some examples of algorithmic applications.