Functional Analysis and Its Applications最新文献

The work is devoted to solving a problem of L. N. Shevrin and M. V. Sapir (Question 3.81b of the Sverdlovsk Notebook), namely, to constructing a finitely presented infinite nil-semigroup satisfying the identity (x^9 = 0). This problem is solved with the help of geometric methods of the theory of tilings and aperiodic tessellations. A semigroup of paths on a tiling, under certain conditions, inherits some properties of the tiling itself. Moreover, the defining relations in the semigroup correspond to a set of equivalent paths on the tiling.

The relationship between the geometric and the automaton approaches previously used in the construction of finitely presented objects is discussed. As noted by S. P. Novikov, the property of determinacy in the coloring of partition nodes and its extension inward is very similar to properties of a solution of a partial differential equation with a given boundary condition. The author believes that understanding this relationship between the theories of aperiodic mosaics and their arrangements and the theory of numerical methods and grids is very promising.

In this paper, we generalize the notions of quermassintegrals, harmonic quermassintegrals, and affine quermassintegrals to (p)-quermassintegrals so that the cases (p=1, -1, -n) of (p)-quermassintegrals are quermassintegrals, harmonic quermassintegrals, and affine quermassintegrals, respectively. Further, we obtain some inequalities associated with (p)-quermassintegrals, including (L_q) Brunn–Minkowski-type inequalities, a monotonic inequality, and a Bourgain–Milman-type inequality.

Large classes of nonnegative Schrödinger operators on (Bbb R^2) and (Bbb R^3) with the following properties are described:

1. The restriction of each of these operators to an appropriate unbounded set of measure zero in (Bbb R^2) (in (Bbb R^3)) is a nonnegative symmetric operator (the operator of a Dirichlet problem) with compact preresolvent;

2. Under certain additional assumptions on the potential, the Friedrichs extension of such a restriction has continuous (sometimes absolutely continuous) spectrum filling the positive semiaxis.

The obtained results give a solution of a problem by M. S. Birman.

On the basis of a new approach to the Calderón construction (X_0^{theta} X_1^{1-theta}) for ideal spaces (X_0) and (X_1) and a parameter (theta in [0,1]), final results concerning a description of multipliers spaces are obtained. In particular, it is shown that if ideal spaces (X_0) and (X_1) have the Fatou property, then (M(X_0^{theta_0} X_1^{1-theta_0},{to},X_0^{theta_1} X_1^{1-theta_1}) = M(X_1^{theta_1 - theta_0} to X_0^{theta_1 -theta_0})) for (0 <theta_0 <theta_1 <1). Due to the absence of constraints on the ideal spaces (X_0) and (X_1), the obtained results apply to a large class of ideal spaces.

We define certain subvarieties, called (theta)-Hecke correspondences, in Cartesian products of diagram automorphism fixed quiver varieties. These give us generators of diagram automorphism fixed Lie algebras.

The spectral properties of the generator of an evolution semigroup describing the dynamics of particle transport in a substance are studied. An effective estimate of the number of unstable modes is obtained, and geometric conditions for spectral stability and instability are found.

A version of Connes Integration Formula which provides concrete asymptotics of eigenvalues is given. This radically extends the class of quantum-integrable functions on compact Riemannian manifolds.

It is proved that the image of a stable germ of type (E_6^pm) of a Lagrangian map to (mathbb R^n) is homeomorphic to the germ at zero of a closed half-space.

Three-dimensional manifolds carrying the structure of (n)-valued coset topological groups originating from the Lie groups (Sp(1)) and (SO(3)) are classified.

We extend homogeneous and inhomogeneous Nash-type inequalities to abstract metric-measure spaces.