Geometry of the submanifolds of SEXn. II. The generalized fundamental equations for the hypersubmanifold of SEXn

A connection which is both Einstein and semisymmetric is called an SE connection, and a generalizedn-dimensional Riemannian manifold on which the differential geometric structure is imposed byg through an SE connection is called ann-dimensional SE manifold and denoted by SEX_n. This paper is a direct continuation of earlier work. In this paper, we derive the generalized fundamental equations for the hypersubmanifold of SEX_n, including generalized Gauss formulas, generalized Weingarten equations, and generalized Gauss-Codazzi equations.     

Incompleteness of timelike submanifolds with nonvanishing second fundamental form

LetM be a properly immersed timelike hypersurface of Minkowski space and assume thatM has a strictly positive second fundamental form. If each point ofM is of diagonal type and dimM 3, then the Ricci curvature ofM is strictly positive on all (nonzero) nonspacelike vectors. ThusM satisfies both the generic and strong energy conditions and a singularity theorem forM may be established.Supported in part by a grant from the Weldon Spring fund of the University of Missouri.     

Geometry of the submanifoldsESX n . III. Parallelism inESX n and its submanifolds

A connection which is both Einstein and semisymmetric is called anES connection. And a generalizedn-dimensional Riemannian manifold on which the differential geometric structure is imposed byg through anES connection, is called ann-dimensionalES manifold and denoted byESX _n . This paper is the third part of a systematic study of the submanifoldsX _m ofESX _n . In the first part, we introduced a new concept of theC-nonholonomic frame of reference inESX _n at points ofX _m and dealt with its consequences. In the second part, the generalized fundamental equations on a hypersubmanifold ofESX _n were derived as an application of theC-nonholonomic frame of reference. The purpose of the present paper is to study parallelism inESX _n and in its submanifoldX _m , using theC-nonholonomic frame of reference and the new concept ofES _i curves.     

Lagrangian submanifolds and the Euler-Lagrange equations in higher-order mechanics

A direct construction of the Euler-Lagrange equations in higher-order mechanics as a submanifold of a higher-order tangent bundle is given, starting from the Lagrangian submanifold defined by the Lagrangian function. This construction uses higher-order tangent bundle geometry, derives the Euler-Lagrange equations as the constraint equations of a submanifold, and makes no assumptions about the regularity of the Lagrangian.     

Generalized equations of the gravitoinertial field

Generalized equations of the gravitoinertial field which contain quasi-Maxwellian field equations are obtained.     

The symmetry of the fundamental equations of the dynamics of a Hamiltonian system

This paper constructs covariant defining equations for infinitesimal operators of the Lie symmetry groups of the Hamilton-Jacobi equations, and Hamilton's and Lagrange's systems whose Hamiltonians are a quadratic function of the generalized momenta; a study is made of the relation between the groups, and in particular the relation is considered between the differential laws of conservation and the symmetry of the systems.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 12–16, February, 1977.     

Time-like minimal submanifolds as singular limits of nonlinear wave equations

We consider the sharp interface limit ?→0⁺ of the semilinear wave equation in R¹⁺ⁿ, where takes values in R^k, k=1,2, and W is a double-well potential if k=1 and vanishes on the unit circle and is positive elsewhere if k=2. For fixed ?>0 we find some special solutions, constructed around minimal surfaces in Rⁿ. In the general case, under some additional assumptions, we show that the solutions converge to a Radon measure supported on a time-like k-codimensional minimal submanifold of the Minkowski space-time. This result holds also after the appearance of singularities, and enforces the observation made by J. Neu that this semilinear equation can be regarded as an approximation of the Born-Infeld equation.     

Generalized solutions of the radiative transfer equations in a singular case

This paper is devoted to the study of the radiative transfer equations: First, we prove a global existence theorem, which allows a blow-up of the opacity _v() when 0. Thus, it extends Mercier's previous result [13]. This proof relies mainly on a nonlinear version of Hille-Yosida theorem: see Crandall-Ligett [9].Then, we prove the uniqueness of the semigroup solving (TR), and some regularity results (in the class of functions with bounded variation).Finally, we prove the convergence of some splitting algorithms associated to (TR).     

Generalized hydrodynamic equations in the hard-spheres model

Generalized linearized hydrodynamic equations in the hard-sphere model are considered on the basis of N.N. Bogolubov’s approximate approach to the analysis of collective interactions. The generalized matrix of transfer coefficients is shown not to be self-adjoint when the finiteness of the two-particle interaction region is taken into account.     

Generalized equations of the gravitoinertial field. II

A generalized Lagrangian of the gravitational and inertial fields is introduced. Modified field equations are obtained.     

Generalized Caldirola equations for the classically radiating electron

The recently proposed non-local equation of motion for the classical electron is slightly modified so that the resulting differential-difference equation becomes absolutely stable, i.e. any sort of runaway solutions are excluded, no matter which initial conditions are imposed. The electromagnetic part entering the equation of motion appears in the finite-difference form of Caldirola, whereas the non-electromagnetic part is a pure differential form. Further generalizations are discussed, thereby introducing the notion of time-like structure functions for classical charged particles with internal structure.     

The fundamental solution to equations of linear magnetic hydrodynamics in a moving medium

The fundamental solution to a system of linear differential equations of magnetic hydrodynamics in a moving medium is obtained. Using the Fourier-Laplace transform, the Green tensor function is calculated as a sum of dyadics. In this way, the integral equations of magnetic hydrodynamics can easily be derived. Particular forms of the fundamental solution that are important in applications are analyzed.