On some problems of tensor calculus. I

Basic definitions of linear algebra and functional analysis are given. In particular, the definitions of a semigroup, group, ring, field, module, and linear space are given [1, 2, 3, 6]. A local theorem on the existence of homeomorphisms is stated. Definitions of the inner r-product, local inner product of tensors whose rank is not less than r, and of local norm of a tensor [22] are also given. Definitions are given and basic theorems and propositions are stated and proved concerning the linear dependence and independence of a system of tensors of any rank. Moreover, definitions and proofs of some theorems connected with orthogonal and biorthonormal tensor systems are given. The definition of a multiplicative basis (multibasis) is given and ways of construction bases of modules using bases of modules of smaller dimensions. In this connection, several theorems are stated and proved. Tensor modules of even orders and problems on finding eigenvalues and eigentensors of any even rank are studied in more detail than in [22]. Canonical representations of a tensor of any even rank are given. It is worth while to note that it was studied by the Soviet scientist I. N. Vekua, and an analogous problem for the elasticity modulus tensor was considered by the Polish scientist Ya. Rikhlevskii in 1983–1984.     

On symmetric configurations in some problems on extremal decomposition. II

Some problems on extremal decomposition in families of nonoverlapping domains containing systems of biangles with free vertices on a circle are considered. Simultaneously, some progress in solving the classical problem on the maximum of a well-known conformal invariant is achieved. This exhibits the role of symmetric configurations in extremal problems under consideration. Bibliography: 11 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 357, 2008, pp. 158–179.     

On the structure of some abstract differential problems II

Summary Various notions of operator homotopy are examined and results are given describing situations such that homotopy of data implies homotopy of subordinate “boundary value” problems. Research supported in part by NSF grants GP-4575 and GP-7374. Entrata in Redazione il 9 aprile 1968.     

On some properties of the multidimensional Bochner-Phillips functional calculus

We develop the multidimensional functional calculus of semigroup generators which is based on the class of Bernstein functions in several variables. We establish spectral mapping theorems, give a holomorphy condition for the semigroups generated by the operators arising in this calculus, as well as prove the moment inequality for these operators.     

Lattice tensor products. II

G. Grätzer and F. Wehrung has recently introduced the lattice tensor product, A?B, of the lattices A and B. In this note, for a finite lattice A and an arbitrary lattice B, we compute the ideal lattice of A?B, obtaining the isomorphism Id(A?B)≌A?Id B. This generalizes an earlier result of G. Grätzer and F. Wehrung proving this isomorphism for A = M_3 and B n-modular. We prove this isomorphism by utilizing the coordinatization of A?B introduced in Part I of this paper.     

On parallel solution of linear elasticity problems. Part II: Methods and some computer experiments

This is the second part of a trilogy on parallel solution of the linear elasticity problem. We consider the plain case of the problem with isotropic material, including discontinuous coefficients, and with homogeneous Dirichlet boundary condition. The discretized problem is solved by the preconditioned conjugate gradient (pcg) method. In the first part of the trilogy block‐diagonal preconditioners based on the separate displacement component part of the elasticity equations were analysed. The preconditioning systems were solved by the pcg‐method, i.e. inner iterations were performed. As preconditioner, we used modified incomplete factorization MIC(0), where possibly the element matrices were modified in order to give M‐matrices, i.e. in order to guarantee the existence of the MIC(0) factorization. In the present paper, the second part, full block incomplete factorization preconditioners are presented and analysed. In order to avoid inner/outer iterations we also study a variant of the block‐diagonal method and of the full block method, where the matrices of the inner systems are just replaced by their MIC(0)‐factors. A comparison is made between the various methods with respect to rate of convergence and work per unknown. The fastest methods are implemented by message passing utilizing the MPI system. In the third part of the trilogy, we will focus on the use of higher‐order finite elements. Copyright © 2002 John Wiley & Sons, Ltd.