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 some spectral spaces associated to tensor triangulated categories

We consider a closure operator c of finite type on the space \(SMod(\mathcal M)\) of thick \(\mathcal K\)-submodules of a triangulated category \(\mathcal M\) that is a module over a tensor triangulated category \((\mathcal K,\otimes ,1)\). Our purpose is to show that the space \(SMod^c(\mathcal M)\) of fixed points of the operator c is a spectral space that also carries the structure of a topological monoid.