On infinitesimal conformal transformations of the tangent bundles with the synectic lift of a Riemannian metric

The purpose of the present article is to investigate some relations between the Lie algebra of the infinitesimal fibre-preserving conformal transformations of the tangent bundle of a Riemannian manifold with respect to the synectic lift of the metric tensor and the Lie algebra of infinitesimal projective transformations of the Riemannian manifold itself.     

On the elementary divisors of the Sylvester and Lyapunov maps

The paper addresses the problem of computing the elementary divisors of the tensor product of linear transformations using the analysis of the tensor products of polynomial models, as developed in Fuhrmann and Helmke [5]. We use this to study the elementary divisors of the Lyapunov and complementary Lyapunov maps.     

On Computer-Aided Solving Differential Equations and Studying Stability of Markets

For any nonholonomic manifold, i.e., a manifold with nonintegrable distribution, we define an analog of the Riemann curvature tensor and refer to Grozman's package SuperLie with the help of which the tensor had been computed in several cases. Being an analog of the usual curvature tensor, this invariant characterizes (in)stability of any nonholonomic dynamical system, in particular, of markets. Similar invariants give criteria for formal integrability of differential equations whose symmetries are induced by contact transformations similar to Goldshmidt's criteria for formal integrability of differential equations whose symmetries are induced by point transformations. As a byproduct, we obtain an approximate solution of the equation whose integrability is under study. Bibliography: 47 titles. __________ Published in Zapiski Nauchnykh Seminarov POMI, Vol. 312, 2004, pp. 165–187.     

Integral formulae for a Riemannian manifold with two orthogonal distributions

We obtain a series of new integral formulae for a distribution of arbitrary codimension (and its orthogonal complement) given on a closed Riemannian manifold, which start from the formula by Walczak (1990) and generalize ones for foliations by several authors. For foliations on space forms our formulae reduce to the classical type formulae by Brito-Langevin-Rosenberg (1981) and Brito-Naveira (2000). The integral formulae involve the conullity tensor of a distribution, and certain components of the curvature tensor. The formulae also deal with a set of arbitrary functions depending on the scalar invariants of the co-nullity tensor. For a special choice of the functions our formulae involve the Newton transformations of the co-nullity tensor.     

Anisotropic tensor spaces and functions,the mean values of tensor quantities and the limiting criteria

The concept of an anisotropic vector space with a tensor basis which is invariant under a symmetry transformations of a three-dimensional Euclidean vector space is introduced using the example of symmetric second- and fourth-rank Euclidean tensors. In addition to the traditional operation of summation, the operation of multiplication in a fixed tensor basis is introduced for the elements of this space, that is, the axioms of a ring with an identity element and zero divisors, which enable one to carry out algebraic and functional operations. The possibilities of the proposed mathematical procedure are illustrated using examples of anisotropic tensor functions of a tensor argument, by the general solution of the classical problem of calculating the mean value of the tensor of the moduli of elasticity of a single-phase grain-oriented polycrystalline material and the construction of the strength surfaces of anisotropic composite materials.     

A Contact Linearization Problem for Monge-Ampère Equations and Laplace Invariants

We solve a problem of contact linearization for non-degenerate regular Monge-Ampère equations. In order to solve the problem we construct tensor invariants of equations with respect to contact transformations and generalize the classical Laplace invariants.      

Fast evaluation of nonlinear functionals of tensor product wavelet expansions

For a nonlinear functional f, and a function u from the span of a set of tensor product interpolets, it is shown how to compute the interpolant of f (u) from the span of this set of tensor product interpolets in linear complexity, assuming that the index set has a certain multiple tree structure. Applications are found in the field of (adaptive) tensor product solution methods for semilinear operator equations by collocation methods, or after transformations between the interpolet and (bi-) orthogonal wavelet bases, by Galerkin methods.     

Tensor product of quaternion hilbert modules

One of the main problems in the theory of quaternion quantum mechanics has been the construction of a tensor product of quaternion Hilbert modules. A solution to this problem is given by studying the tensor product of quaternion algebras (over the reals) and some of its quotient modules. Real, complex, and (covariant) quaternion scalar products are found in the tensor product spaces. Annihilationcreation operators are constructed, corresponding to the second quantization of the quaternion quantum theory with Bose-Einstein or Fermi-Dirac statistics. The gauge transformations of a tensor product vector and the gauge fields are studied.On Sabbatical leave from the School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel. Work supported in part by a fellowship from the Ambrose Monell Foundation.     

On the complete decomposition of curvature tensors of Riemannian manifolds with symmetric connection

We give a complete decomposition of the space of curvature tensors with the symmetry properties as the curvature tensor associated with a symmetric connection of Riemannian manifold. We solve the problem under the action ofS0(n). The dimensions of the factors, the projections, their norms and the quadratic invariants of a curvature tensor are determined. Several applications for Riemannian manifolds with symmetric connection are given. The group of projective transformations of a Riemannian manifold and its subgroups are considered.     

TrasformazioniC-proiettive e gruppi di olonomia di varietà di quasi contatto normali

C-projective transformations on a normal almost contact manifoldsM have been defined in [11]. Among those, a particular attention is devoted toC-projective transformations which preserve the Ricci tensor and imply some interesting properties of holonomy group of special torsion-free connections ofM. Some examples of this kind of transformations are constructed. Sasakian and cosymplectic manifolds have been studied separately.     

Quadratic invariants of elastic moduli

A quadratic invariant is defined as a quadratic form in theelements of a tensor that remains invariant under a group ofcoordinate transformations. It is proved that there are sevenquadratic invariants of the 21-element elastic modulus tensorunder SO(3) and 35 under SO(2). This answers some open questionsraised by Ting and Ahmad in this journal.     

Projection operators and states in the tensor product of quaternion hilbert modules

Following the construction of tensor product spaces of quaternion Hilbert modules in our previous paper, we define the analogue of a ray (in a complex quantum mechanics) and the corresponding projection operator, and through these the notion of a state and density operators. We find that there is a one-to-one correspondence between a state and an equivalence class of vectors from the tensor product space, which gives us another method to define the gauge transformations.On sabbatical leave from the School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel. Work supported in part by a fellowship from the Ambrose Monell Foundation.     

The Algebra of the Riemann Curvature Tensor in General Relativity: Preliminaries

In a four-dimensional curved space-time it is well-known that the Riemann curvature tensor has twenty independent components; ten of these components appear in the Weyl tensor, and nine of these components appear in the Einstein curvature tensor. It is also known that there are fourteen combinations of these components which are invariant under local Lorentz transformations. In this paper, we derive explicitly closed form expressions which contain these twenty independent components in a manifest way. We also write the fourteen invariants in two ways; firstly, we write them in terms of the components; and, secondly, we write them in a covariant fashion, and we further derive the appropriate characteristic value equations and the corresponding Cayley-Hamilton equations for these invariants. We also show explicitly how all of the relevant components transform under a Lorentz transformation. We shall follow the very general and powerful methods of Sachs [1]. We shall not point out at every stage of the calculation which equations are due to Sachs, and which equations are new; this is easily ascertained. Generally speaking, however, the equations depending on the Einstein curvature tensor, and the emphasis placed on this tensor, appear to be new.     

Complex spaces with locally product metrics: general theory

Summary Locally product complex spaces are introduced with particular reference to the Calculus of Variations on complex manifolds. The goedesics of such spaces possess unusual features associated with the fact that many of the connection coefficients are tensorial in character. A restricted partial covariant derivative is introduced together with a curvature tensor. The latter is found to be invariant under a large class of gauge-like transformations. This invariance property leads naturally to the introduction of almost totally decomposable complex spaces. Necessary and sufficient conditions for a locally product complex Riemannian space to be almost totally decomposable are discussed. The counterpart of the Einstein tensor is also exhibited. On leave of absence from the Department of Mathematics, The University. Bristol. Entrata in Redazione il 30 ottobre 1968.     

Approximate decoupling of multivariate polynomials using weighted tensor decomposition

Many scientific and engineering disciplines use multivariate polynomials. Decomposing a multivariate polynomial vector function into a sandwiched structure of univariate polynomials surrounded by linear transformations can provide useful insight into the function while reducing the number of parameters. Such a decoupled representation can be realized with techniques based on tensor decomposition methods, but these techniques have only been studied in the exact case. Generalizing the existing techniques to the noisy case is an important next step for the decoupling problem. For this extension, we have considered a weight factor during the tensor decomposition process, leading to an alternating weighted least squares scheme. In addition, we applied the proposed weighted decoupling algorithm in the area of system identification, and we observed smaller model errors with the weighted decoupling algorithm than those with the unweighted decoupling algorithm.     

Relativistic kinematics of a solid medium in a three-dimensional description

We propose a three-dimensional formulation of the relativistic kinematics of a solid medium based on the geometric concept of the forms of relativistic dynamics. We introduce the concepts of instantaneous configuration of a body, the velocity field, and different measures of deformations and the deformation velocity tensor. We consider the definition of rigid motion in a given form of dynamics. We study the transformation properties of the kinematic characteristics introduced and also those of the quantities that arise under three-dimensional decomposition of the energy-momentum tensor of the medium with respect to the transformations of the Poincaré group with recomputation to a new simultaneity.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 35, 1992, pp. 173–178.     

Schwarz Problem for Multiply Connected Domains and its Application to Diffusion Around Fractals

Stationary diffusion perpendicular to unidirectional circular cylinders arranged in two-dimensional fractals set is considered. Complex potentials and the method of functional equations are used to determine the concentration field. This result is applied to the calculation of the effective diffusion tensor of fractals randomly diluted on the plane. In particular, we discuss a special fractal derived by a group of Möbius transformations and deduce a generalization of the Clausius-Mossotti approximation.     

Rank one preservers between spaces of Boolean matrices

In this paper, we characterize (i) linear transformations from one space of Boolean matrices to another that send pairs of distinct rank one elements to pairs of distinct rank one elements and (ii) surjective mappings from one space of Boolean matrices to another that send rank one matrices to rank one matrices and preserve order relation in both directions. Both results are proved in a more general setting of tensor products of two Boolean vector spaces of arbitrary dimension.     

应用于非惯性系湍流模拟的扩展内禀旋转张量

研究扩展内禀旋转张量在非惯性系湍流模拟中的作用,特别是对代数Reynolds应力湍流模式(如非线性K-ε模式)的重要性．为此,采用几个近年来发展的非线性K-ε湍流模式模拟旋转坐标系下均匀剪切湍流,并且和大涡模拟的结果进行比较．计算结果和分析表明,需要发展更先进的非线性K-ε模式从而更好地描述非惯性系下的复杂湍流．