Functional Analysis and Its Applications最新文献

We propose a mathematical model for a new phenomenon: multi-dimensional hyperbolic chaos. This model is a ring chain of (Nge 2) unidirectionally coupled maps of the two-dimensional torus (mathbb{T}^2), each of which is of Arnold’s cat map type. We provide sufficient conditions (independent of (N)) under which the chain gives rise to an Anosov diffeomorphism of (mathbb{T}^{2N}) for any (Nge 2).

In this paper, we investigate the spectrum of the differential operator (T) generated by an ordinary differential expression of order (n) with (mathrm{PT})-symmertic periodic (mtimes m) matrix coefficients. We prove that if (m) and (n) are odd numbers, then the spectrum of (T) contains all the real line. Note that in standard quantum theory, observable systems must be Hermitian operators, so as to ensure that the spectrum is real. Research on (mathrm{PT})-symmetric quantum theory is based on the observation that the spectrum of a (mathrm{PT})-symmetric non-self-adjoint operator can contain real numbers. In this paper, we discover a large class of (mathrm{PT})-symmetric operators whose spectrum contains all real axes. Moreover, the proof is very short.

Generalizations of the theorems of Eberlein and Grothendieck on the precompactness of subsets of function spaces are considered: if (X) is a countably compact space and (C_p(X)) is a space of continuous functions on (X) in the topology of pointwise convergence, then any countably compact subspace of the space (C_p(X)) is precompact, that is, it has a compact closure. The paper provides an overview of the results on this topic. It is proved that if a pseudocompact (X) contains a dense Lindelöf (Sigma)-space, then pseudocompact subspaces of the space (C_p(X)) are precompact. If (X) is the product Čech complete spaces, then bounded subsets of the space (C_p(X)) are precompact. Results on the continuity of separately continuous functions are also obtained.

New families of algebras and DG algebras with two simple modules are introduced and described. Using the twisted tensor product operation, we prove that such algebras have finite global dimension, and that the resulting DG algebras are smooth. This description allows us to show that some of these DG algebras determine smooth proper noncommutative curves that provide smooth minimal noncommutative resolutions for singular rational curves.

We give a description of finite-zone (mathcal{PT})-potentials in terms of explicit theta-functional formulas.

The Poincare construction in CR geometry allows us to estimate the dimension of the stabilizer in the Lie algebra of infinitesimal holomorphic automorphisms of the germ of a CR manifold by the dimension of the stabilizer in the corresponding algebra of the model surface of this germ. We give a negative answer to the following natural question: is there an algebraic Poincare construction, i.e., is it true that the stabilizer in the Lie algebra of automorphisms of the germ of a CR manifold is isomorphic to a Lie subalgebra of the stabilizer in the algebra of its model surface? We also give a negative answer to the corresponding question for the whole automorphisms algebra.

In this paper, we demonstrate that for a locally compact Hausdorff space (S) and a decomposable Borel measure (mu), metric projectivity, injectivity, or flatness of the (C_0(S))-module (L_p(S,mu)) implies that (mu) is purely atomic with at most one atom.

The diffusion mechanism in Hamiltonian systems, close to completely integrable, is usually connected with the existence of the so-called “transition chains”. In this case slow diffusion occurs in a neighborhood of intersecting separatrices of hyperbolic periodic solutions (or, more generally, lower-dimensional invariant tori) of the perturbed system. In this note we discuss another diffusion mechanism that uses destruction of invariant tori of the unperturbed system with an almost resonant set of frequencies. We demonstrate this mechanism on a particular isoenergetically nondegenerate Hamiltonian system with three degrees of freedom. The same phenomenon also occurs for general higher-dimensional Hamiltonian systems. Drift of slow variables is shown using analysis of integrals of quasi-periodic functions of the time variable (possibly unbounded) with zero mean value. In addition, the proof uses the conditions of topological transitivity for cylindrical cascades.

We describe subgroups of elements of uniformly bounded degrees in Cremona groups of arbitrary rank. These subgroups appear naturally in (mathrm{CR})-geometry as holomorphic automorphism groups of nondegenerate homogeneous model surfaces.

We suggest a general construction of continuous Banach bundles of holomorphic function algebras on subvarieties of the closed noncommutative ball. These algebras are of the form (mathcal{A}_d/overline{I_x}), where (mathcal{A}_d) is the noncommutative disc algebra defined by G. Popescu, and (overline{I_x}) is the closure in (mathcal{A}_d) of a graded ideal (I_x) in the algebra of noncommutative polynomials, depending continuously on a point (x) of a topological space (X). Moreover, we construct bundles of Fréchet algebras (mathcal{F}_d/overline{I_x}) of holomorphic functions on subvarieties of the open noncommutative ball. The algebra (mathcal{F}_d) of free holomorphic functions on the unit ball was also introduced by G. Popescu, and (overline{I_x}) stands for the closure in (mathcal{F}_d) of a graded ideal (I_x) in the algebra of noncommutative polynomials, depending continuously on a point (xin X).