Countable Systems Research Articles

A New Existence Result of Equilibria for Vector Equilibrium Problems

In this paper, we deal with vector equilibrium problems. We prove the nonemptiness of the solution set for this type of problems in the sequential compactness case and in the absence of convexity and lower semicontinuity assumptions. Some examples are presented and an existence result for countable systems of vector equilibrium problems is stated.

DIMENSIONS OF MEASURES, DEGREES, AND FOLDING ENTROPY IN DYNAMICS

In this survey, we present some methods in the dynamics and dimension theory for invariant measures of hyperbolic endomorphisms (smooth non-invertible maps), and for conformal iterated function systems with overlaps. For endomorphisms, we recall the notion of asymptotic degree of an equilibrium measure, which is shown to be related to the folding entropy; this degree is then applied to dimension estimates. For finite iterated function systems, we present the notion of overlap number of a measure, which is related to the folding entropy of a lift transformation, and also give some examples when it can be computed or estimated. We apply overlap numbers to prove the exact dimensionality of invariant measures, and to obtain a geometric formula for their dimension. Then, for countable conformal iterated function systems with overlaps, the projections of ergodic measures are shown to be exact dimensional, and we give a dimension formula. Relations with ergodic number theory, continued fractions, and random dynamical systems are also presented.

On a nonlocal boundary value problem for a partial integro-differential equations with degenerate kernel

On a nonlocal boundary value problem for a partial integro-differential equations with degenerate kernel\Abstracteng{In this article the problems of the unique classical solvability and theconstruction of the solution of a nonlinear boundary value problem for a fifth orderpartial integro-differential equations with degenerate kernel are studied. Dirichletboundary conditions are specified with respect to the spatial variable. So, the Fourierseries method, based on the separation of variables is used. A countable system of~thesecond order ordinary integro-differential equations with degenerate kernel is obtained.The method of degenerate kernel is applied to this countable system of ordinaryintegro-differential equations. A system of~countable systems of algebraic equations isderived. Then the countable system of nonlinear Fredholm integral equations is obtained.Iteration process of solving this integral equation is constructed. Sufficient coefficientconditions of the unique solvability of the countable system of nonlinear integralequations are established for the regular values of parameter. In proof of uniquesolvability of the obtained countable system of nonlinear integral equations the method ofsuccessive approximations in combination with the contraction mapping method is used.In the proof of the convergence of Fourier series the Cauchy--Schwarz and Besselinequalities are applied. The smoothness of solution of the boundary value problemis also proved.

Смешанная задача для многомерного интегро-дифференциального уравнения в частных производных четвертого порядка

The problems of the unique classical solvability and the construction of a solution of a multidimensional mixed problem for a homogeneous fourth order partial integro-differential equations with a degenerate kernel are studied. The multidimensional Fourier series method, based on the separation of many variables, is used. A system of countable systems of algebraic equations is derived. Iteration process of solving the problem is constructed. Sufficient coefficient conditions for the unique classical solvability of the mixed problem are established.

On the Invariant Tori of Quasilinear Countable Systems of Differential Equations Defined on Infinite–Dimensional Tori

On the Invariant Tori of Quasilinear Countable Systems of Differential Equations Defined on Infinite–Dimensional Tori

Mixed type integro-differential equation with fractional order Caputo operators and spectral parameters

The issues of unique solvability of a boundary value problem for a mixed type integro-differential equation with two Caputo time-fractional operators and spectral parameters are considered. A mixed type integro-differential equation is a partial integro-differential equation of fractional order in both positive and negative parts of multidimensional rectangular domain under consideration. The fractional Caputo operator's order is less in the positive part of the domain, than the order of Caputo operator in the negative part of the domain. Using the method of Fourier series, two systems of countable systems of ordinary fractional integro-differential equations with degenerate kernels are obtained. Further, a method of degenerate kernels is used. To determine arbitrary integration constants, a system of algebraic equations is obtained. From this system, regular and irregular values of spectral parameters are calculated. The solution of the problem under consideration is obtained in the form of Fourier series. The unique solvability of the problem for regular values of spectral parameters is proved. To prove the convergence of Fourier series, the properties of the Mittag-Leffler function, Cauchy-Schwarz inequality and Bessel inequality are used. The continuous dependence of the problem solution on a small parameter for regular values of spectral parameters is also studied. The results are formulated as a theorem.

A Concretization of an Approximation Method for Non-Affine Fractal Interpolation Functions

The present paper concretizes the models proposed by S. Ri and N. Secelean. S. Ri proposed the construction of the fractal interpolation function (FIF) considering finite systems consisting of Rakotch contractions, but produced no concretization of the model. N. Secelean considered countable systems of Banach contractions to produce the fractal interpolation function. Based on the abovementioned results, in this paper, we propose two different algorithms to produce the fractal interpolation functions both in the affine and non-affine cases. The theoretical context we were working in suppose a countable set of starting points and a countable system of Rakotch contractions. Due to the computational restrictions, the algorithms constructed in the applications have the weakness that they use a finite set of starting points and a finite system of Rakotch contractions. In this respect, the attractor obtained is a two-step approximation. The large number of points used in the computations and the graphical results lead us to the conclusion that the attractor obtained is a good approximation of the fractal interpolation function in both cases, affine and non-affine FIFs. In this way, we also provide a concretization of the scheme presented by C.M. Păcurar.

Integro-differential equation with fractional order Caputo operators and spectral parameters

The questions of unique solvability of a boundary value problem for an integro-differential equation with two Caputo fractional operators and spectral parameters are considered. The fractional Caputo operator’s order is bigger than one and smaller than two. Using the method of Fourier series, countable systems of ordinary fractional integro-differential equations with degenerate kernel is obtained. Further, is used a method of degenerate kernel. To determine arbitrary integration constants, a system of algebraic equations is obtained. From this system the regular and irregular values of the spectral parameters were calculated. The solution of the problem under consideration is obtained in the form of Fourier series. The unique solvability of the problem for regular values of spectral parameters is proved. In proofing of the convergence of Fourier series the properties of the Mittag-Leffler function, Cauchy-Schwarz inequality and Bessel inequality are used. We also studied the continuous dependence of the solution of the problem on small parameter for regular values of spectral parameters. The results obtained in this paper are formulated as a theorem.

Inverse Problem for a Mixed Type Integro-Differential Equation with Fractional Order Caputo Operators and Spectral Parameters

The questions of the one-value solvability of an inverse boundary value problem for a mixed type integro-differential equation with Caputo operators of different fractional orders and spectral parameters are considered. The mixed type integro-differential equation with respect to the main unknown function is an inhomogeneous partial integro-differential equation of fractional order in both positive and negative parts of the multidimensional rectangular domain under consideration. This mixed type of equation, with respect to redefinition functions, is a nonlinear Fredholm type integral equation. The fractional Caputo operators’ orders are smaller in the positive part of the domain than the orders of Caputo operators in the negative part of the domain under consideration. Using the method of Fourier series, two systems of countable systems of ordinary fractional integro-differential equations with degenerate kernels and different orders of integro-differentation are obtained. Furthermore, a method of degenerate kernels is used. In order to determine arbitrary integration constants, a linear system of functional algebraic equations is obtained. From the solvability condition of this system are calculated the regular and irregular values of the spectral parameters. The solution of the inverse problem under consideration is obtained in the form of Fourier series. The unique solvability of the problem for regular values of spectral parameters is proved. During the proof of the convergence of the Fourier series, certain properties of the Mittag–Leffler function of two variables, the Cauchy–Schwarz inequality and Bessel inequality, are used. We also studied the continuous dependence of the solution of the problem on small parameters for regular values of spectral parameters. The existence and uniqueness of redefined functions have been justified by solving the systems of two countable systems of nonlinear integral equations. The results are formulated as a theorem.

Complete order equivalence of spin unitaries

This paper is a study of linear spaces of matrices and linear maps on matrix algebras that arise from spin systems, or spin unitaries, which are finite sets S of selfadjoint unitary matrices such that any two unitaries in S anticommute. We are especially interested in linear isomorphisms between these linear spaces of matrices such that the matricial order within these spaces is preserved; such isomorphisms are called complete order isomorphisms, which might be viewed as weaker notion of unitary similarity. The main result of this paper shows that all m-tuples of anticommuting selfadjoint unitary matrices are equivalent in this sense, meaning that there exists a unital complete order isomorphism between the unital linear subspaces that these tuples generate. We also show that the C⁎-envelope of any operator system generated by a spin system of cardinality 2k or 2k+1 is the simple matrix algebra M2k(C). As an application of the main result, we show that the free spectrahedra determined by spin unitaries depend only upon the number of the unitaries, not upon the particular choice of unitaries, and we give a new, direct proof of the fact [13] that the spin ball Bmspin and max ball Bmmax coincide as matrix convex sets in the cases m=1,2. We also derive analogous results for countable spin systems and their C⁎-envelopes.

NONLOCAL PROBLEM FOR A MIXED TYPE FOURTH-ORDER DIFFERENTIAL EQUATION WITH HILFER FRACTIONAL OPERATOR

In this paper, we consider a non-self-adjoint boundary value problem for a fourth-order differential equation of mixed type with Hilfer operator of fractional integro-differentiation in a positive rectangular domain and with spectral parameter in a negative rectangular domain. The mixed type differential equation under consideration is a fourth order differential equation with respect to the second variable. Regarding the first variable, this equation is a fractional differential equation in the positive part of the segment, and is a second-order differential equation with spectral parameter in the negative part of this segment. A rational method of solving a nonlocal problem with respect to the Hilfer operator is proposed. Using the spectral method of separation of variables, the solution of the problem is constructed in the form of Fourier series. Theorems on the existence and uniqueness of the problem are proved for regular values of the spectral parameter. For sufficiently large positive integers in unique determination of the integration constants in solving countable systems of differential equations, the problem of small denominators arises. Therefore, to justify the unique solvability of this problem, it is necessary to show the existence of values of the spectral parameter such that the quantity we need is separated from zero for sufficiently large \(n\). For irregular values of the spectral parameter, an infinite number of solutions in the form of Fourier series are constructed. Illustrative examples are provided.

Smale endomorphisms over graph-directed Markov systems

AbstractWe study Smale skew product endomorphisms (introduced in Mihailescu and Urbański [Skew product Smale endomorphisms over countable shifts of finite type. Ergod. Th. & Dynam. Sys. doi: 10.1017/etds.2019.31. Published online June 2019]) now over countable graph-directed Markov systems, and we prove the exact dimensionality of conditional measures in fibers, and then the global exact dimensionality of the equilibrium measure itself. Our results apply to large classes of systems and have many applications. They apply, for instance, to natural extensions of graph-directed Markov systems. Another application is to skew products over parabolic systems. We also give applications in ergodic number theory, for example to the continued fraction expansion, and the backward fraction expansion. In the end we obtain a general formula for the Hausdorff (and pointwise) dimension of equilibrium measures with respect to the induced maps of natural extensions ${\mathcal{T}}_{\unicode[STIX]{x1D6FD}}$ of $\unicode[STIX]{x1D6FD}$-maps $T_{\unicode[STIX]{x1D6FD}}$, for arbitrary $\unicode[STIX]{x1D6FD}>1$.

Nonlocal Inverse Problem for a Pseudohyperbolic- Pseudoelliptic Type Integro-Differential Equations

The questions of solvability of a nonlocal inverse boundary value problem for a mixed pseudohyperbolic-pseudoelliptic integro-differential equation with spectral parameters are considered. Using the method of the Fourier series, a system of countable systems of ordinary integro-differential equations is obtained. To determine arbitrary integration constants, a system of algebraic equations is obtained. From this system regular and irregular values of the spectral parameters were calculated. The unique solvability of the inverse boundary value problem for regular values of spectral parameters is proved. For irregular values of spectral parameters is established a criterion of existence of an infinite set of solutions of the inverse boundary value problem. The results are formulated as a theorem.

Genealogical constructions of population models

Representations of population models in terms of countable systems of particles are constructed, in which each particle has a “type,” typically recording both spatial position and genetic type, and a level. For finite intensity models, the levels are distributed on $[0,\lambda]$, whereas in the infinite intensity limit $\lambda\rightarrow\infty$, at each time $t$, the joint distribution of types and levels is conditionally Poisson, with mean measure $\Xi(t)\times\ell$ where $\ell$ denotes Lebesgue measure and $\Xi(t)$ is a measure-valued population process. The time-evolution of the levels captures the genealogies of the particles in the population. Key forces of ecology and genetics can be captured within this common framework. Models covered incorporate both individual and event based births and deaths, one-for-one replacement, immigration, independent “thinning” and independent or exchangeable spatial motion and mutation of individuals. Since birth and death probabilities can depend on type, they also include natural selection. The primary goal of the paper is to present particle-with-level or lookdown constructions for each of these elements of a population model. Then the elements can be combined to specify the desired model. In particular, a nontrivial extension of the spatial $\Lambda$-Fleming–Viot process is constructed.

INVESTIGATION OF THE APPROXIMATION OF CONTINUOUS PERIODIC FUNCTIONS ON THE TORUS


 
 
 Main purpose of the present work is development of qualitative theory of difference equations in the space of bounded numeric sequences.Main result is the establishment of necessary conditions of the existence of invariant toroidal manifolds for countable systems of differential and difference equations. In order to solve this problem, observed spaces are constructed in a special way. Necessary conditions of the existence of invariant tori for countable systems of differential and difference equations are derived.A concept of a continuous periodic in each variable function with period 2Pi , values of which lie in l2 , is introduced. Spaces, in which observations are made, are constructed in a special way. A theorem on approximation of a function from the corresponding space bytrigonometric polynomials is proven.
 
 

On the Solutions of Oscillating-Type Countable Differential Systems with Slowly Varying Parameters

For a countable quasilinear system of differential equations whose coefficients have the form of absolutely and uniformly convergent Fourier series with slowly varying coefficients and frequency, we obtain conditions for the existence of a partial solution of a similar structure. As a result, we establish conditions for the possibility of complete decomposition of a countable linear homogeneous differential system with coefficients of the same structure.

Köhler theory for countable quadruple systems

From the late 1970s to the early 1980s, Köhler developed a theory for constructing finite quadruple systems with point-transitive Dihedral automorphism groups by introducing a certain algebraic graph, now widely known as the (first) Köhler graph in finite combinatorics. In this paper, we define the countable Köhler graph and discuss countable extensions of a series of Köhler's works, with emphasis on various gaps between the finite and countable cases. We show that there is a simple 2-fold quadruple system over Z with a point-transitive Dihedral automorphism group if the countable Köhler graph has a so-called [1, 2]-factor originally introduced by Kano (1986) in the study of finite graphs. We prove that a simple Dihedral $\ell$-fold quadruple system over Z exists if and only if $\ell = 2$. The paper also covers some related remarks about Hrushovski's constructions of countable projective planes.

Vectorial Ekeland variational principle for systems of equilibrium problems and its applications

For a family of vector-valued bifunctions, we introduce the notion of sequentially lower monotonity, which is strictly weaker than the lower semi-continuity of the second variables of the bifunctions. Then, we give a general version of vectorial Ekeland variational principle (briefly, denoted by EVP) for a system of equilibrium problems, where the sequentially lower monotone objective bifunction family is defined on products of sequentially lower complete spaces (concerning sequentially lower complete spaces, see Zhu et al. (2013)), and taking values in a quasi-ordered locally convex space. Besides, the perturbation consists of a subset of the ordering cone and a family {p i } i∈I of negative functions satisfying for each i ∈ I, p i (x i , y i ) = 0 if and only if x i = y i . From the general version, we can deduce several particular equilibrium versions of EVP, which can be applied to show the existence of solutions for countable systems of equilibrium problems. In particular, we obtain a general existence result of solutions for countable systems of equilibrium problems in the setting of sequentially lower complete spaces. By weakening the compactness of domains and the lower semi-continuity of objective bifunctions, we extend and improve some known existence results of solutions for countable system of equilibrium problems in the setting of complete metric spaces (or Frechet spaces). When the domains are non-compact, by using the theory of angelic spaces (see Floret (1980)), we generalize some existence results on solutions for countable systems of equilibrium problems by extending the framework from reflexive Banach spaces to the strong duals of weakly compactly generated spaces.

Limit Theorem for Countable Systems of Stochastic Differential Equations

We consider infinite systems of stochastic differential equations used to describe the motion of interacting particles in random media. It is assumed that the mass of each particle approaches zero and the density of particles infinitely increases in a proper way. It is proved that the sequence of corresponding measure-valued processes converges in distribution. We also prove the existence and uniqueness of a strong solution for the limit equation.

On the remainder of the semialgebraic Stone–Cěch compactification of a semialgebraic set

In this work we analyze some topological properties of the remainder ∂M:=βs⁎M∖M of the semialgebraic Stone–Cěch compactification βs⁎M of a semialgebraic set M⊂Rm in order to ‘distinguish’ its points from those of M. To that end we prove that the set of points of βs⁎M that admit a metrizable neighborhood in βs⁎M equals Mlc∪(Clβs⁎M(M‾≤1)∖M‾≤1) where Mlc is the largest locally compact dense subset of M and M‾≤1 is the closure in M of the set of 1-dimensional points of M. In addition, we analyze the properties of the sets ∂ˆM and ∂˜M of free maximal ideals associated with formal and semialgebraic paths. We prove that both are dense subsets of the remainder ∂M and that the differences ∂M∖∂ˆM and ∂ˆM∖∂˜M are also dense subsets of ∂M. It holds moreover that all the points of ∂ˆM have countable systems of neighborhoods in βs⁎M.