Separation of variables in kolmogorov's equation. II

This communication continues the investigation begun in [10] of the separation of variables (SV) in Kolmogorov's equation (KE). SV is discussed below with the help of a set of symmetry operators containing a second-order differential operator; a group of KE transformations is constructed with respect to which it is covariant.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 3, pp. 27–32, March, 1979.     

Conformal Einstein spaces in N-dimensions: Part II

This paper generalizes the main result of “Conformal Einstein spaces in N-dimensions” published in Ann. Global Anal. Geom. 20(2) (2001). We present necessary and sufficient tensorial conditions for a certain class of semi-Riemannian manifolds to be conformally related to Einstein spaces.     

Separation of variables in the diffusion equation. II

The covariant and symmetry properties of the linear diffusion equation having a scalar matrix of variable diffusion coefficients are studied. By means of differential symmetry operators of order no higher than two, a complete separation of variables is effected for the stationary and nonstationary cases.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 12, pp. 40–45, December, 1976.     

Separation of variables in the Klein-Gordon equations. II

Two kinds of external nonstationary electromagnetic fields are found containing arbitrary functions which admit of total separation of variables in the Klein-Gordon equations by using two differential symmetry operators and one second order operator. Curvilinear coordinates are presented in which the variables are divided, and equations are written down in the separated variables.Translated from Izvestiya VUZ, Fizika, No. 12, pp. 45–52, December, 1973.     

Separation of variables in 0(3)-invariant wave equations for quadrupole coordinates

Linear, 0(3) invariant equations for a multicomponent wave function, which depends on quadrupole coordinates, are examined. Vibrational, rotational, and spin parts in the Hamiltonian are separated, on which basis angular and internal variables are separated.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 80–83, July, 1982.I thank S. A. Vladimirov for formulating the problem and for useful discussions.     

Conformally Steckel metrics in Einstein spaces

Anisotropic conformally Steckel metrics that satisfy Einstein vacuum equations containing a Λ term are examined in the paper. The metrics allow the integration of the isotropic geodetic Hamilton-Jacobi equations by complete variable separation. Tomsk State University. Translated from Izeestiya Vysshikn Uchebnykh Zavedenii, Fizika, No. 10, pp. 34–43, October, 1999.     

Deformations of the embedded Einstein spaces

We discuss a method of studying the stability of solutions of Einstein's equations, which can be outlined as follows: Consider an embedding of a given Einstein spaceV ₄ into a pseudo-Euclidean spaceE _p,q ^N (N > 4,p + q =N) (p,q) describing the signature of the spaceE _p,q ^N . Then all the geometrical objects ofV ₄ can be expressed in terms of the embedding functions,Z ^A (x ⁱ ),A = 1, 2,...,N, i = 0, 1, 2, 3. Then let us deform the embedding:Z ^A Z ^A + ^A , being an infinitesimal parameter. The Einstein equations can be developed then in the powers of; we study the equations arising by requirement of the vanishing of the first- or second-order terms. Some partial results concerning the de Sitter, Einstein, and Minkowskian spaces are given.     

Highly mobile Einstein spaces in the large

We consider an Einstein spaceV of the Petrov type II or III admitting a group of motionsG of high order. First we calculate the composition law and topological structure ofG. ThenV (or its submanifolds of transitivity) is represented as the homogeneous spaceG/H ofG,H being a subgroup ofG, and the actionG onV and the topology ofV are determined. The topologies of the spacesV are as follows: ⁴ (spaceT*₂), ⁴ of ³ T¹ (spaceT ₂), ⁴ (spaceT*₃), ³ (submanifolds of transitivity in spaceT ₃).In two cases (spacesT ₂ andT ₃) we have obtained metrics free of singularities.     

Riemannian geometry in spaces with Grassman coordinates

The use of spaces containing Grassman (anticommuting) coordinates (in addition to the usual space-time coordinates) as a framework for unified gauge theories is described. The theory developed represents a local gauge-invariant extension of conventional (global) supersymmetry. Aside from containing the usual general coordinate invariance group of gravitational theory, the gauge supersymmetry group is seen to also encompass other symmetries of particle physics, e.g., electromagnetic (or Yang-Mills) invariance. The role of spontaneous symmetry breaking and the field equations unifying the Einstein, Maxwell, and Dirac interactions are discussed.Research supported in part by the National Science Foundation.Invited talk at the conference, The Riddle of Gravitation, on the Occasion of the 60th Birthday of Peter G. Bergmann, Syracuse, New York, March 1975.     

Separation of variables in Kolmogorov's equation. I

Calculations are made of the coefficients of the nonstationary Kolmogorov equation for which a solution can be found by separation of variables using a complete set of differential symmetry operators of first order; the variables are separated in accordance with this procedure. A solution is found to the equation describing the motion of a dynamical system containing a random parameter.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 16–20, August, 1977.     

Separation of variables in Kolmogorov's equation. III

We continue the investigation of the complete separation of variables (SV) in an ultraparabolic equation begun in [10, 14]. Here the classes of equivalent characteristics which allow SV are discussed in detail; the solution of the problem of Brownian motion of an oscillator with variable frequency is constructed.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 3, pp. 33–35, March, 1979.     

Algebraically special hypersurface-homogeneous Einstein spaces in general relativity

This paper illustrates the value of the Newman—Penrose complex null tetrad formalism by using it to obtain all algebraically special Einstein spaces admitting three-parameter groups of motions acting on timelike surfaces containing the repeated principal null direction. Taken together with earlier work, this enables us to give a complete list of Einstein spaces which are both algebraically special and hypersurface-homogeneous or homogeneous.     

Separation of variables in the Klein-gordon equations. III

This paper supplements [1] and [2]. All kinds of external electromagnetic fields are found which contain arbitrary functions admitting complete separation of variables in the Klein-Gordon equations by using one first and two second order differential symmetry operators. The curvilinear coordinates in which the variables separate are presented, and equations in the separated variables are written down.     

Separation of variables in the diffusion equation. I

For both stationary and nonstationary cases, all the coefficients of the linear diffusion equation with a constant diffusion tensor are enumerated for which total separation of the variables is possible by means of differential symmetry operators of a definite structure.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 11, pp. 46–53, November, 1976.     

Separation of variables in the diffusion equation. III

The Separation of variables of type N = 2 is completed, and all complete sets of type N = 1 are constructed for the diffusion equation with a scalar matrix for the leading derivatives.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 48–50, July, 1979.In conclusion, we take this opportunity to thank V. N. Shapovalov for continued interest.     

Discontinuities of solutions of the Einstein equations. II

Right zero-vectors of the characteristic matrix of the Einstein equations are constructed on isotropic cones. Relationships for the discontinuities of two derived functions of the field in the and surfaces are indicated. Quantities describing the weak discontinuities of solutions of the gravitational field equations are constructed.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 38–41, February, 1982.     

Transformation to pseudo-Cartesian coordinates in locally flat pseudo-Riemannian spaces

A tractable method is presented for obtaining transformations to pseudo-Cartesian coordinates in locally flat pseudo-Riemannian spaces. The procedure is based on the properties of parallel covector fields. As an illustration, the method is applied to obtain certain transformations that arise in the Hamilton–Jacobi theory of separation of variables.     

Quantum theory of spin-3/2 field in Einstein spaces

A quantum theory of a free massless spin-3/2 field on Einstein spaces (R _ab=g_ab) is formulated in an algebraic framework. Attention is confined to the structure of the quantum operator algebra. In particular, the issue of positivity of the anticommutator is investigated and found to depend on whether or not the space-time admits rero-frequency neutrino solutions. Using methods developed for the purpose, a class of space-time that does not admit neutrino zero modes is characterized. An appendix introduces a useful technique for obtaining an initial value formulation of spinor field equations.Supported in part by the NSF, under contract No. PHY 78-24275. Submitted to the Department of Physics, University of Chicago in partial fulfillment of the requirements for the Ph.D. degree.