On some problems of tensor calculus. II

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 [1–3, 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 the structure of some abstract differential problems. - I.

Summary Some differential type problems are formulated abstractly in terms of diagrams of Hilbert spaces and related Schwartz kernels and their structure is studied. Research supported in part by the National Science Foundation under grant GP-1448 and GP-4575.     

Automatic solution of optimal-control problems. I. simplest problem in the calculus of variations

An automatic method for obtaining the numerical solution for the simplest problem in the calculus of variations is described. The nonlinear two-point boundary-value Euler-Lagrange equation is solved using the Newton-Raphson method for obtaining successive approximations of the solution. The derivatives required for the solution of the problem are computed automatically using the table method. The user of the program need only input the integrand of the objective function in the calculus-of-variations problem and specify the boundary conditions. None of the derivatives usually associated with the Euler-Lagrange equation and the Newton-Raphson method need be calculated by hand. An example is given with numerical results. The automatic solution of the simplest problem in the calculus of variations in this paper is considered to be the first step in the automatic solution of more general optimal-control problems.     

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.     

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.     

Connections on Lagrangian submanifolds and some quasiclassical approximation problems. I

In this paper we construct a flat connection (possibly with torsion) on Lagrangian submanifolds which is connected with formulas for asymptotic wave functions generated by this submanifold. The new formalism naturally includes the Bohr-Sommerfeld condition and Maslov characteristic class and lets us establish new global formulas for asymptotic wave functions in the subtlest cases (motion in the neighborhood of a separatrix).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova, Akad. Nauk SSSR, Vol. 172, pp. 41–54, 1989.