Table of contents

An example of a Sturm-Liouville operator is presented for which the solution of the corresponding scattering problem has a resonance character.

The present paper is devoted to the construction of infinitely based varieties of associative algebras over an infinite field of arbitrary positive characteristic.

The Cauchy problem with non-negative continuous initial function for the equation

is considered for , . For generalized solutions of this problem with initial data increasing at infinity several results on their behaviour as are established.

For a non-linear eigenvalue problem similar to a linear Sturm-Liouville problem the properties of the spectrum and the eigenfunctions are analysed. The system of eigenfunctions is shown to be a Riesz basis in L₂.

The inductive limits of some families of Lie algebras are considered. Under discussion are algebras of vector fields on a manifold that preserve a volume form or a symplectic form and have supports in coordinate neighbourhoods. The family of all commutative subalgebras of the Lie algebra of the skew-Hermitian matrices of order larger than two is studied. The explicit form of the inductive limits is indicated.

Given a pair of positive integers and such that , for integer the quantity , called the partition function is considered; this by definition is equal to the cardinality of the set

The properties of and its asymptotic behaviour as are studied. A geometric approach to this problem is put forward. It is shown that

for sufficiently large , where and are positive constants depending on and , and and are characteristics of the exponential growth of the partition function. For some pair the exponents and are calculated as the logarithms of certain algebraic numbers; for other pairs the problem is reduced to finding the joint spectral radius of a suitable collection of finite-dimensional linear operators. Estimates of the growth exponents and the constants and  are obtained.

For a broad class of semilinear parabolic equations with compact attractor in a Banach space  the problem of a description of the limiting phase dynamics (the dynamics on ) of a corresponding system of ordinary differential equations in  is solved in purely topological terms. It is established that the limiting dynamics for a parabolic equation is finite-dimensional if and only if its attractor can be embedded in a sufficiently smooth finite-dimensional submanifold . Some other criteria are obtained for the finite dimensionality of the limiting dynamics: a) the vector field of the equation satisfies a Lipschitz condition on ; b) the phase semiflow extends on to a Lipschitz flow; c) the attractor has a finite-dimensional Lipschitz Cartesian structure. It is also shown that the vector field of a semilinear parabolic equation is always Holder on the attractor.

The paper contains a generalization of Calderon's theorem on the local uniqueness of the solutions of the Cauchy problem for differential equations with weighted derivatives. Anisotropic estimates of Carleman type are obtained. A class of differential equations with weighted derivatives is distinguished in which germs of solutions have unique continuation with respect to part of the variables.

This work is devoted to the construction of T-spaces with an infinite basis over a field of finite characteristic and over some other rings. Examples of T-spaces are given that are generated by polynomials in two variables or by polynomials of bounded degree in each variable.