T-entropy and Variational Principle for the spectral radius of transfer and weighted shift operators

The paper deals with the variational principles for evaluation of the spectral radii of transfer and weighted shift operators associated with a dynamical system. These variational principles have been the matter of numerous investigations and the principal results have been achieved in the situation when the dynamical system is either reversible or it is a topological Markov chain. As the main summands these principles contain the integrals over invariant measures and the Kolmogorov--Sinai entropy. In the article we derive the Variational Principle for an arbitrary dynamical system. It gives the explicit description of the Legendre dual object to the spectral potential. It is shown that in general this principle contains not the Kolmogorov--Sinai entropy but a new invariant of entropy type -- the t-entropy.Comment: 51 pages, v.2: editorial correction

Development of a Calculation Methodology for the Ventilation on a Besis of a Mobile Unit

An algorithm for the analysis of safety and efficiency of the processes, which are located inside the mobile unit are developed. It follows from the calculations that the safe concentration of combustible material in the space of industrial premises is about 3.69%. Automation systems must be focused on this value. The time of occurrence of the maximum permissible concentration of pollutant was determined and amounted to 160 seconds. It is shown that the ventilation rate of 0.5 would be sufficient for functioning of the object

Noncommutative elliptic theory. Examples

We study differential operators, whose coefficients define noncommutative algebras. As algebra of coefficients, we consider crossed products, corresponding to action of a discrete group on a smooth manifold. We give index formulas for Euler, signature and Dirac operators twisted by projections over the crossed product. Index of Connes operators on the noncommutative torus is computed.Comment: 23 pages, 1 figur

Right-Side Hyperbolic Operators

In the paper a new class of linear operators was introduced: linear operator B is said to be right-side hyperbolic, if operators B−l I are right-sided invertible for any l from a neighborhood of the unit circle and moreover one can specify right-side resolvent Rr(B;l ) namely a family of right inverse to B−l I analytic in l . In the paper general form of right-side resolvents is given. We also discuss a distinguishes with the hyperbolic case

Right-Side Hyperbolic Operators

Abstract: In the paper a new class of linear operators was introduced: linear operator B is said to be right-side hyperbolic, if operators B − λ I are right-sided invertible for any λ from a neighborhood of the unit circle and moreover one can specify right-side resolvent R r (B; λ ) namely a family of right inverse to B − λ I analytic in λ . In the paper general form of right-side resolvents is given. We also discuss a distinguishes with the hyperbolic case

Uniformization and an Index Theorem for Elliptic Operators Associated with Diffeomorphisms of a Manifold

We consider the index problem for a wide class of nonlocal elliptic operators on a smooth closed manifold, namely differential operators with shifts induced by the action of an isometric diffeomorphism. The key to the solution is the method of uniformization: We assign to the nonlocal problem a pseudodifferential operator with the same index, acting in sections of an infinite-dimensional vector bundle on a compact manifold. We then determine the index in terms of topological invariants of the symbol, using the Atiyah-Singer index theorem.Comment: 16 pages, no figure

О РОСТЕ АНАЛИТИЧЕСКОЙ ФУНКЦИИ В КРУГЕ

In this article, the order of exponential of growth of analytical function ϕ on the disc is introduced, and the relation between the order of the function ϕ and its coefficients is obtained. An application of this result gives us the description of the behavior of the resolvent R(B,λ) of linear bounded operator where λ approaches the spectrum.В работе введен экспоненциальный порядок роста аналитической функции ϕ в круге и установлена связь между скоростью роста коэффициентов разложения функции и ее порядком. Дано приложение к описанию поведения нормы резольвенты R(B,λ) ограниченного линейного оператора при приближении λ к спектру

КВАЗИПЕРИОДИЧЕСКИЕ АЛГЕБРЫ, ИНВАРИАНТНЫЕ ОТНОСИТЕЛЬНО ЛИНЕЙНОГО ОТОБРАЖЕНИЯ

In this paper we consider the following problem. Let A0 be some quasi-periodic subalgebra of the algebra of almost periodic functions on Rm and a : Rm → Rm, a(x) = Mx, is a linear map defined by a square matrix of size m. There exists the smallest closed subalgebra A+, containing A0 and invariant under α, and there exists the smallest closed subalgebra A, containing A0, invariant under α and α–1. We obtain conditions on the linear map α when these subalgebras are quasi-periodic.В работе рассмотрена следующая задача. Пусть A0 есть некоторая квазипериодическая подалгебра в алгебре почти периодических функций на Rm и a : Rm → Rm, α(x) = Mx, – линейное отображение, задаваемое квадратной матрицей размера m.  Существует наименьшая замкнутая подалгебра A+ , содержащая A0 и инвариантная относительно α, и существует наименьшая замкнутая подалгебра A, содержащая A0, инвариантная относительно α и α–1. В работе получены условия на линейное отображение α, при которых эти подалгебры являются квазипериодическими