Countable System Research Articles

A modified strong dyadic integral and derivative

For a function its modified strong dyadic integral and the modified strong dyadic derivative are defined. A criterion for the existence of a modified strong dyadic integral for an integrable function is proved, and the equalities and are established under the assumption that . A countable system of eigenfunctions of the operators and is found. The linear span of this set is shown to be dense in the dyadic Hardy space , and the linear operator , , is proved to be bounded. Hence this operator can be uniquely continuously extended to and the resulting linear operator is bounded.

On the Cρ-Smoothness of an Invariant Torus of a Countable System of Difference Equations Defined on an m-Dimensional Torus

Using the method of truncation, we establish sufficient conditions of differentiability of order ρ ≥ 2 with respect to the angular variable for an invariant torus of a countable system of difference equations with deviation of the discrete argument.

Euler scheme for solutions of a countable system of stochastic differential equations

We consider a countable system of stochastic differential equation. Euler scheme for approximating these solutions is used, and the global error is estimated. Solutions are approximated by means of a process which takes values in a finite dimensional space. Finally, we expand the global error for a class of smooth functions in powers of the discretization step size.

On the Smoothness of the Invariant Torus of a Countable System of Difference Equations with Parameters

We establish sufficient conditions for the differentiability of the invariant torus of a countable system of linear difference equations defined on a finite-dimensional torus with respect to an angular variable and the parameter of the original system of equations.

On the Majorant Method for the Cauchy--Kovalevskaya Problem

We present a theory which allows us to justify the existence theorems and derive majorant estimates for solutions of the Cauchy problem for a countable system of partial differential equations. Our results also allow to estimate the maximum existence interval for a solution.

Coagulation-Fragmentation Processes

We study the well-posedness of coagulation-fragmentation models with diffusion for large systems of particles. The continuous and the discrete case are considered simultaneously. In the discrete situation we are concerned with a countable system of coupled reaction-diffusion equations, whereas the continuous case amounts to an uncountable system of such equations. These problems can be handled by interpreting them as abstract vector-valued parabolic evolution equations, where the dependent variables take values in infinite-dimensional Banach spaces. Given suitable assumptions, we prove existence and uniqueness in the class of volume preserving solutions. We also derive sufficient conditions for global existence.

The depth invariant for group actions on countable phase spaces

The construction of the depth of a countable symbolic system, which was studied earlier by the author for actions of the group\(\mathbb{Z}\), is generalized to the case of an arbitrary finitely generated Abelian group action. A new property, which is called the uniformity of the depth, is studied. The depth, which takes values in the set of at most countable ordinals, is a topological invariant of countable symbolic systems. In the paper we describe the set of possible values of the depth invariant and the method of constructing dynamical systems with an arbitrary admissible depth.

Limit theorems in the theory of multipoint boundary-value problems

We present a reduction of a countable system of differential equations with countably-point boundary conditions to the case of a finite-dimensional multipoint boundary-value problem. We separately consider the case of a linear system.

On one class of solutions of a countable quasilinear system of differential equations with slowly varying parameters

For a countable quasilinear differential system whose coefficients are represented as Fourier series with slowly varying coefficients and frequency, we present conditions under which solutions of this system have analogous structure.

A criterion of exponential dichotomy for a countable system of differential equations with quasiperiodic coefficients

We establish necesary and sufficient conditions for the exponential diehetemy with matrix projeseters of a countable system of differential equations with quasiperiodic coefficients.

A solution of a nonlinear system arising in spectral perturbation theory of nonnegative matrices

Let P and E be two n × n complex matrices such that for sufficiently small positive ε, P + εE is nonnegative and irreducible. It is known that the spectral radius of P + εE and corresponding (normalized) eigenvector have fractional power series expansions. The goal of the paper is to develop an algorithm for computing the coefficients of these expansions under two (restrictive) assumptions, namely that P has a single Jordan block corresponding to its spectral radius and that the (unique up to scalar multiples) left and right eigenvectors of P corresponding to its spectral radius, say v and w, satisfy v T Ew ≠ 0. Our approach is to consider an associated countable system of nonlinear equations and solve this system recursively. At each step, we consider the coefficients of the expansion of the spectral radius of P + εE as parameters and solve a related linear system parametrically. The next coefficient of the expansion of the spectral radius is then determined from feasibility considerations for a linear system. This solution method is novel and seems useful for computing coefficients of corresponding expansions when the two (restrictive) assumptions are relaxed. Also, interestingly, the coefficients we compute yield a preferred basis of the generalized eigenspace corresponding to the spectral radius of the unperturbed matrix P.

On the word problem and the conjugacy problem for groups of the formF/V(R)

LetF be a free group with at most countable system\(\mathfrak{x}\) of free generators, letR be its normal subgroup recursively enumerable with respect to\(\mathfrak{x}\), and let\(\mathfrak{V}\) be a variety of groups that differs from\(\mathfrak{O}\) and for which the corresponding verbal subgroupV of the free group of countable rank is recursive. It is proved that the word problem inF/V(R) is solvable if and only if this problem is solvable inF/R, and if\(|\mathfrak{x}| \geqslant 3\), then there exists anR such, that the conjugacy problem inF/R is solvable, but this problem is unsolvable inF/V(R) for any Abelian variety\(\mathfrak{V} \ne \mathfrak{C}\) (all algorithmic problems are regarded with respect to the images of\(\mathfrak{x}\) under the corresponding natural epimorphisms).

Hausdorff measure of infinitely generated self-similar sets

We analyze self-similarity with respect to infinite sets of similitudes from a measure-theoretic point of view. We extend classic results for finite systems of similitudes satisfying the open set condition to the infinite case. We adopt Vitali-type techniques to approximate overlapping self-similar sets by non-overlapping self-similar sets. As an application we show that any open and bounded set\(A \subseteq \mathbb{R}^n\) with a boundary of null Lebesgue measure always contains a self-similar set generated by a countable system of similitudes and with Lebesgue measure equal to that ofA.

Introduction of local coordinates for a countable system of differential equations in the neighborhood of an invariant manifold

We establish conditions of the existence of local coordinates for a countable system of differential equations in the neighborhood of an invariant manifold and present the form of this system in these coordinates.

Time-Space Analysis of the Cluster-Formation in Interacting Diffusions

A countable system of linearly interacting diffusions on the interval [0,1], indexed by a hierarchical group is investigated. A particular choice of the interactions guarantees that we are in the diffusive clustering regime, that is spatial clusters of components with values all close to 0 or all close to 1 grow in various different scales. We studied this phenomenon in [FG94]. In the present paper we analyze the evolution of single components and of clusters over time. First we focus on the time picture of a single component and find that components close to 0 or close to 1 at a late time have had this property for a large time of random order of magnitude, which nevertheless is small compared with the age of the system. The asymptotic distribution of the suitably scaled duration a component was close to a boundary point is calculated. Second we study the history of spatial 0- or 1-clusters by means of time scaled block averages and time-space-thinning procedures. The scaled age of a cluster is again of a random order of magnitude. Third, we construct a transformed Fisher-Wright tree, which (in the long-time limit) describes the structure of the space-time process associated with our system. All described phenomena are independent of the diffusion coefficient and occur for a large class of initial configurations (universality).

On the reducibility of countable systems of linear difference equations

We solve the problem of reducibility of a countable linear system of standard difference equations with unbounded right-hand sides by the method of construction of iterations with accelerated convergence. For systems of this type with bounded right-hand sides, this problem is reduced to a finite-dimensional case.

Pure Point Spectrum of the Laplacians on Fractal Graphs

We establish the pure point spectrum of the Laplacians on two point self-similar fractal graphs. All eigenvalues have infinite multiplicity and a countable system of orthonormal eigenfunctions with compact support is complete in the corresponding Hilbert space.

Diffusive clustering in an infinite system of hierarchically interacting diffusions

We study a countable system of interacting diffusions on the interval [0,1], indexed by a hierarchical group. A particular choice of the interaction guaranties, we are in the diffusive clustering regime. This means clusters of components with values either close to 0 or close to 1 grow on various different scales. However, single components oscillate infinitely often between values close to 0 and close to 1 in such a way that they spend fraction one of their time together and close to the boundary. The processes in the whole class considered and starting with a shift-ergodic initial law have the same qualitative properties (universality).

On decomposability of countable systems of differential equations

Conditions under which there exists a change of variables that decomposes a countable system of differential equations are established for the entire real axis and a semiaxis. Similar problems are investigated for a countable system with pulse influence.

On a countable system of convolution equations

On a countable system of convolution equations