A probability algebra, which is a kind of analogue of a probability space, is designed to describe a physical system. If we have a prior distribution defined by a faithful state on this algebra and we can define the behavior of the system in terms of this state, then we can also make some prediction about its behavior. In practice, however, it is impossible to measure all observable quantities on the original algebra, so we only perform a partial measurement, that is, we measure the observable quantities on a certain subalgebra. This measurement then determines another state on this subalgebra. If the resulting state extends to the original algebra without introducing additional information about the behavior of system, then we call the resulting state on the original algebra coarse-graining. In the paper we discuss the existence of coarse-graining states, referred to as (B, omega)-coarse-graining of a state, and we also consider ultrapro ducts of sequences of probability algebras and states. We show that under the introduced definition, ultrapro ducts of sequences of probability algebras are probability algebras, and we discuss the existence of conditional mathematical expectations, coarse-grained states, and information.
We consider dynamical models with aftereffect described by functional-differential equations with fractional derivatives. These models encompass processes, in which the system state may change abruptly at certain points in time, which is interpreted as the result of impulse effects (shocks). The trajectories of such systems may have discontinuities at certain points in time, and between these points the behavior of system is described by differentiable functions, which satisfy the equation in the usual sense. We pose a general control problem for a given system. We formulate solvability conditions for this problem in the class of impulse controls, L-2-controls, and their hybrids. The proposed approach to studying systems with fractional derivatives is based on the systematic use of abstract functional-differential equation theory and offers certain advantages for studying systems and processes with aftereffects.
In this work we study the problem on small motions and normal oscillations of a homogeneous mixture of several viscous incompressible fluids. The considered model is a generalization of the well-known Navier - Stokes equations for the dynamics of a one-component incompressible viscous medium, and it involves the incompressibility and momentum equations. We prove that the corresponding initial boundary value problem is well-posed and solvable. In terms of the Stokes operator, we construct the spectrum and system of eigenelements for the problem on normal oscillations.
We study infinite order Euler operators in the space H(Omega) of all functions holomorphic on an open set Omega in C-N, with the topology of uniform convergence on compact sets in Omega. In terms of their characteristic functions, we prove necessary and sufficient conditions for the applicability of these operators to H(Omega). The special case Omega = (C\{0})(N) is considered. We study the relationship between the two representations for the Euler operator, in which the Stirling numbers of the first and second kinds play a significant role. It is expressed by means of associated functions, one of which is the sum of the Newton interpolation series. The obtained results imply that each entire function of exponential type 0 on C-N can be expanded into a multidimensional Newton interpolation series. A multidimensional version of the Wigert - Leau theorem is proved. We show that in the space H(C-N) of all integer functions in C-N, each Hadamard type operator in H(C-N) is an Euler operator, that is, each continuous linear operator in H(C-N), for which each monomial is an eigenvector.
In this paper we find a solvability criterion for the multiple interpolation problem in the preimage of a convolution operator and, consequently, a criterion for the solvability of Abel - Goncharov problem in the same space. When the kernel of the operator serves as the preimage, uniqueness of the solution to these problems holds provided the set of interpolation nodes is the uniqueness set in the kernel of the convolution operator.
We study the stability of a maximal term of a Dirichlet series with positive exponents, the sum of which is an entire function. For a class of entire Dirichlet series defined by a certain convex growth majorant, we prove a theorem on the quantitative estimate of the equivalence degree (outside of some exceptional c(q)-set) of the logarithms of maximal terms in the original series and the modified Dirichlet series. A similar problem for entire Dirichlet series of an arbitrary rapid growth, but with no quantitative estimate of the stability degree for the maximal term, was first studied by A.M. Gaisin in the late 1990s and early 2000s. He then obtained a stability criterion, which was the equivalence of the logarithms of the maximal terms of the original and modified series on the asymptotic set. This result, as well as the corresponding stability statements for Dirichlet series converging only in a certain half-plane obtained by A.M. Gaisin and T.I. Belous, found useful applications in the theory of asymptotic properties of Dirichlet series, specifically in proving Polya type identities. The formulation of the stability problem considered in this paper is relevant for its applications to the minimum modulus problem, as well as to other related problems in analysis and complex dynamics.
In this paper we study the existence of periodic solutions for a class of systems second order ordinary differential equations with a separated main nonlinear part. Taking into consideration the structure of zero set for the main nonlinear part, we find new conditions ensuring an apriori estimate for periodic solutions. Under the apriori estimate, we formulate and prove a criterion for the existence of periodic solutions under any perturbation from a given class. The proof is made by using methods for calculating mapping degree of the vector field and employing the invariance of the existence of periodic solutions under a continuous varying of the main nonlinear part.
We consider basic relationships between the initial energy and the exponent of nonlinear sources in a nonlinear coupled system of kth order wave equations with nonlinear averaged damping, and we demonstrate the global nonexistence of solutions. This approach is a variant of the method of nonlinear functional analysis with a contradiction argument, which is one of the tools for proving the blow-up of solutions for nonlinear partial differential equations. A new class of nonlinear coupled wave equations with nonlinear sources is given in high-order functional spaces.
In this paper we develop the classification method of three-dimensional integrable discrete lattices based on Darboux integrable finite-field reductions of equations with one continuous and two discrete independent variables. As an example we consider a semi-discrete Toda lattice. In the framework of the present study we obtain the Toda lattice.
We study the inverse problem on identifying two unknown potential coefficients in a system of coupled Schrodinger equations in a bounded domain in R-n subject to non-homogeneous Dirichlet and Neumann boundary conditions by using Neumann and Dirichlet boundary measurements. Under specific convexity assumptions on the geometry of domain and minimal regularity conditions on the data, we establish the Lipschitz stability of this inverse problem by employing uniqueness results as well as the observability inequality.
In this paper, we study the geometry of the Ricci tensor of a harmonic nearly trans-Sasakian manifold. On the space of associated G-structure we introduce fundamental identities of harmonic nearly trans Sasakian manifolds. We prove that Ricci-flat harmonic nearly trans Sasakian manifolds are closely cosymplectic. We obtain conditions, which ensure that harmonic nearly trans Sasakian manifolds are Einstein and eta-Einstein manifolds. We obtain identities for the Ricci tensor of harmonic nearly trans-Sasakian manifolds. We provide local characterizations for the following harmonic nearly trans-Sasakian manifolds: Einstein manifolds; manifolds, the Ricci tensor of which is parallel, eta parallel, the Codazzi tensor, the Killing tensor, and satisfies the three selected identities.
We consider an algebraic q-difference equation. We propose a sufficient condition for the existence of a formal power-logarithmic expansion in the vicinity of zero of the solution to such an equation. We apply this sufficient condition to construct the formal expansion of a solution to a certain q-difference analogue of the fifth Painleve equation for particular values of the parameters in the equations. We consider two different values of q, which lead to qualitatively different formal asymptotic expansions for the solutions.
In this work we prove several inequalities, which provide a lower bound for the norm of an elliptic operator in non-divergent form in a bounded domain with power degeneration along the entire boundary. Earlier similar operators were studied in the case when they were initially defined in divergent form or reduced to such a form. In contrast, the coefficients of the operators we study are generally non-differentiable and cannot be reduced to divergent form. Only in the final section of the paper, in order to study the solvability of the corresponding differential equations, the differentiability is assumed for the coefficients of operator, and the corresponding adjoint operator is studied. We first study degenerate elliptic operators of general form and prove an inequality for them in which the sum of norm of the action of operator and the norm of the function itself with some power weight in the space L-2 is bounded from below by the norm of function itself in a weighted Sobolev type space. We then consider the case, in which the elliptic operators are weakly positive. For such operators, we prove an inequality in which the real part of the scalar product of the action of operator and the function itself is bounded from below. In the final section we assume that weakly positive elliptic operators have strong degeneration along the entire boundary of the domain. For such operators involving a parameter lambda, we first prove an inequality in which the norm of the action of operator is bounded from below by the norm of function itself in the underlying functional space. This inequality is then proved for the adjoint operator, and as a consequence, a result on the unique solvability of the corresponding differential equation is established. The technique developed in this paper is based on the extension of some known results for elliptic operators with constant coefficients to the case of operators with degeneracy using auxiliary integral inequalities.
In this article we demonstrate new integral inequalities of the Hermite - Hadamard type for differentiable functions, which are (h, m)-convex of the second type. We show that many known results are particular cases of ours.
We study the inverse problem on determining the convolution kernel of integral term in an initial boundary value problem for a multi-dimensional time fractional diffusion-wave equation with the uniformly elliptic operator in a divergent form. Moreover, as an overdetermination condition, a single observation at the point x0 is an element of Omega of the diffusion-wave process serves, where Omega subset of R-n is a bounded domain. By the Fourier spectral method and fractional integro-differentiating technics the inverse problem is reduced to a convolution nonlinear Volterra integral equation of the second kind. The fixed point argument proves the local existence and global uniqueness results. Also the stability estimate for solution to inverse problem is obtained.
In the work we study the subspace of functions analytic in a convex domain and invariant with respect to the differentiation operator. We study the fundamental principle problem, namely, on representation of all functions in the invariant subspace by the series of exponential monomials. These exponential monomials are the eigenfunctions and generalized eigenfunctions of differentiation operator in the invariant subspace. We obtain a simple geometric criterion of fundamental principle. We also obtain a similar criterion for solvability of an interpolation problem in the spaces of entire functions of exponential type.
We consider the inverse problem on determining the kernel of an integral term in an integro-differential equation. The problem of determining the memory kernel in the wave process is reduced to a nonlinear Volterra integral equation of the first kind of convolution type, then over determination condition it brings to the Volterra integral equation of the second kind. The method of contraction maps proves the unique solvability of the problem in the space of continuous functions with weight norms, and an estimate of the conditional stability of the solution is obtained.
In the paper we obtain a series of Kolmogorov type inequalities for functions analytic in a circle of an arbitrary radius R and belonging to the Hardy space H-q,H-R (1 ( q <= infinity, R > 0). We provide some applications of these inequalities in extremal problem of best polynomial approximation.
The paper deals with the left reduced semigroup C*-algebras C-lambda*(Q) for nonabelian cancellative semigroups Q associated with finite tuples (M-1, M-2, ... , M-n) of sequences M-i of arbitrary natural numbers. Such a semigroup Q is defined to be the free product of semigroups consisting of positive numbers in the ordered groups of rationals Q(Mi )generated by the reciprocals for products of the terms in M-i. The C*-algebra C-lambda*(Q) is generated by the left regular representation of Q. It is shown that each semigroup Q is not left amenable but its full and left reduced semigroup C*-algebras are isomorphic and nuclear. We establish that every C*-algebra C-lambda*(Q) can be characterized as a universal C*- algebra defined by a countable set of isometries satisfying a countable family of polynomial relations. To prove this result, we make use of the universal property of C*-algebra C-lambda*(Q) considered as the inductive limit for the inductive sequence of Toeplitz - Cuntz algebras associated with the tuple (M-1, M-2, ... , M-n).
We consider a class SK(R) of subharmonic functions on the unbounded open semi-annulus D+(R) ={z: |z|>R, Im z>0}, which on each semi-annulus D+(R-1,R-2) = {z:R < R-1< |z| < R-2< infinity, I m z>0} possesses a positive harmonic majorant. We introduce a class JS(R) of subharmonic functions on D+(R), whose boundary values on the real boundary D+(R) are non-positive. We obtain some properties of functions in the classes SK(R) and JS(R). The class SS(R) of delta-subharmonic functions on D+(R) is defined as the difference of classes SK(R) or JS(R): delta S(R) = SK(R)-SK(R) = JS(R)-JS(R). For functions v is an element of delta S(R) we introduce a growth characteristic T-R(r, v), which differs from the characteristics used for the functions defined on the upper half-plane. It determines the growth of function in the vicinity of semi-circumference L-R = {R&(i0) : 0 <= theta <=pi }. For an arbitrary function of growth y (unbounded non-decreasing positive function defined on the real semi-axis R+ = {r : r > 0}) we define the class delta S-LR(R, gamma ) subset of delta S(R) of delta-subharmonic functions v of finite y-type on D+(R) in the vicinity of circumference L-R as T-R(r, v) <= A gamma ((B) / (r-R) ) for all R < r < 2R and some positive A and B depending on v, but independent of r. We obtain the criterions of belonging of function v to the class delta SLR(R,gamma) in terms of its Fourier coefficients.
