Doklady Mathematics最新文献

The main methods and approaches of the theory of discontinuous systems are used to construct the theory of functional differential equations with a discontinuous right-hand side. In particular, methods for describing sets of discontinuity points and sliding modes of discontinuous systems with delay are considered using a special class of invariantly differentiable functionals.

We study the topological structure of the solution set of the Cauchy problem for semilinear differential inclusions of fractional order (alpha in (1,2)) in Banach spaces. It is assumed that the linear part of the inclusions is a linear closed operator generating a strongly continuous and uniformly bounded family of cosine operator functions. The nonlinear part is represented by an upper semicontinuous multivalued operator of Carathéodory type. It is established that the solution set of the problem is an ({{R}_{delta }})-set.

Holomorphic self-maps of the unit disk with boundary fixed points are investigated. In 1982, Cowen and Pommerenke established an interesting generalization of the classical Julia–Carathéodory theorem, which allowed them to derive a sharp estimate for the derivative at the Denjoy–Wolff point on the class of functions with an arbitrary finite set of boundary fixed points. In this paper, we obtain a new generalization of the Julia–Carathéodory theorem, which contains the Cowen–Pommerenke result as a special case; moreover, it is an effective tool for solving various problems on classes of functions with fixed points.

In classical works, the Hubble constant is defined via a metric. Here we define it, as it should be, via matter, following Milne and McCrea and extending their theory of the expanding Universe to the relativistic case. This allows us to explain the accelerated expansion as a simple relativistic effect without resorting to Einstein’s lambda, dark energy, or new particles, more specifically, as an exact consequence of the classical Einstein action. The well-verified fact of accelerated expansion allows us to determine the sign of the curvature in the Friedmann model: it turns out to be negative, and we live in Lobachevsky space.

The work is devoted to the study of the combinatorial properties of determinism in a family of substitution complexes consisting of quadrilaterals glued together side-to-side. These properties are useful in the construction of algebraic structures with a finite number of defining relations. In particular, this method was used in constructing an infinite, finitely presented nilsemigroup satisfying the identity ({{x}^{9}} = 0). This construction solves the problem posed by L.N. Shevrin and M.V. Sapir. This work investigates the possibility of coloring the entire family of complexes with a finite number of colors, for which the property of weak determinism holds: if the colors of three vertices of a given quadrilateral are known, then the color of the fourth vertex is uniquely determined, except in some cases of special arrangement of the quadrilateral. Even weak determinism is sufficient to construct a finitely presented nilsemigroup; when using this construction, the proof is shortened. Determinism properties help to correctly introduce defining relations in the semigroup of paths traversing the constructed complexes. The defining relations correspond to pairs of equivalent short paths. Properties of determinism have previously been studied in the context of tiling theory; in particular, Kari and Papasoglu constructed a set of square tiles that admits only aperiodic tilings of the plane and has the property of determinism: knowing the colors of two adjacent edges uniquely determines the colors of the remaining two edges.

We give the rigorous derivation of hydrodynamic equations for an infinite harmonic crystal coupled to the Klein–Gordon field. These equations hold in the hydrodynamic limit, and they should be considered as the analog of the Euler and Navier–Stokes equations for the model under consideration.

The Vandermonde matrix is considered in an arbitrary complex Banach algebra. The accompanying Frobenius matrix is used to establish the relationship between the coefficients of an algebraic equation and the Vandermonde matrix constructed from its roots. The divided difference of arbitrary order is defined based on an invertible Vandermonde matrix. An analogue of the Hermite formula for an integral representation of the divided difference is given. An inclusion for the spectrum of the divided difference and an analogue of Dunford’s theorem on the mapping of spectra are given.

The possibility of the existence of an invariant measure with smooth density is discussed in two cases related to invariant sets, namely, at the levels of partial integrals and at the joint invariant level of two or more functions. A version of Jacobi’s last multiplier theorem is presented, which supplements similar results of S.A. Chaplygin and V.V. Kozlov. Conditions are investigated under which the invariant sets represent a two-dimensional torus with an invariant measure of smooth density defined on it. This means that Kolmogorov’s theorem is applicable, which implies that, after making an appropriate change of coordinates, the motion becomes conditionally periodic.

We give a simple proof of a result recently obtained in [12] on the completeness of modal logics with modality that corresponds to the intersection of accessibility relations in a Kripke model. Completeness is proved for logics in modal languages of two types: one has modalities ({{square }_{1}}, ldots ,{{square }_{n}}) for relations ({{R}_{1}}, ldots ,{{R}_{n}}) that satisfy a unimodal logic L and modality ({{square }_{{n + 1}}}) for the intersection ({{R}_{{n + 1}}} = {{R}_{1}} cap ldots cap {{R}_{n}}); the other language has modalities ({{square }_{i}}(i in Sigma )) for relations R_i that satisfy the logic L, and, for every nonempty subset of indices (I subseteq Sigma ), the modality ({{square }_{I}}) for the intersection (bigcapnolimits_{i in I} {{R}_{i}}). While in [12] the completeness is proved only for logics over ({mathbf{K,KD,KT,K4,S4}}), and S5, we give a “uniform” construction that enables us to obtain completeness for logics with intersection over 15 “traditional” modal logics KΛ for (Lambda subseteq { {mathbf{D,T,B,4,5}}} ). The proof method is based on unraveling a frame and then taking the Horn closure of the resulting frame.

We present new cases of integrable dynamical systems of any odd order that are homogeneous in terms of some of their variables and in which a system on the cotangent bundle of an even-dimensional manifold can be distinguished. In this case, the force field (shift generator in the system) is divided into an internal (conservative) and an external one, which has dissipation of different signs. The external field is introduced using some unimodular transformation and generalizes previously considered fields. Complete sets of both first integrals and invariant differential forms are given.