Functional differentiation of integral operators of special form and some questions of the inverse interpolation

This article is devoted to the problem of operator interpolation and functional differentiation. Some information about the variational derivatives and explicit formulas for the exact solutions of the simplest equations containing the first variational derivatives of the required functional are given. For functionals defined on sets of functions and square matrices, various interpolating polynomials of the Hermitе type with nodes of the second multiplicity, which contain the first variational derivatives of the interpolated operator, are constructed. The presented solutions of the Hermitе interpolation problems are based on the algebraic Chebyshev system of functions. For analytic functions with an argument from a set of square matrices, explicit formulas for antiderivatives of functionals are obtained. The solution of some differential equations with integral operators of a special form and the first variational derivatives is found. The problem of the inverse interpolation of functions and operators is considered. Explicit schemes for constructing inverse functions and functionals, including the case of functions of a matrix variable, obtained using certain well-known results of interpolation theory, are demonstrated. Data representation is illustrated by a number of examples.

An Analytic and Numerical Investigation of a Differential Game

In this paper we present an appropriate singular, zero-sum, linear-quadratic differential game. One of the main features of this game is that the weight matrix of the minimizer’s control cost in the cost functional is singular. Due to this singularity, the game cannot be solved either by applying the Isaacs MinMax principle, or the Bellman–Isaacs equation approach. As an application, we introduced an interception differential game with an appropriate regularized cost functional and developed an appropriate dual representation. By developing the variational derivatives of this regularized cost functional, we apply Popov’s approximation method and show how the numerical results coincide with the dual representation.

On the exact and approximate solutions of several differential equations with variational derivatives of the first and second orders

In this paper, we consider the problem of the exact and approximate solutions of certain differential equations with variational derivatives of the first and second orders. Some information about the variational derivatives and explicit formulas for the exact solutions of the simplest equations with the first variational derivatives are given. An interpolation method for solving ordinary differential equations with variational derivatives is demonstrated. The general scheme of an approximate solution of the Cauchy problem for nonlinear differential equations with variational derivatives of the first order, based on the use of the operator interpolation apparatus, is presented. The exact solution of the differential equation of the hyperbolic type with variational derivatives, similar to the classical Dalamber solution, is obtained. The Hermite interpolation problem with the conditions of coincidence in the nodes of the interpolated and interpolation functionals, as well as their variational derivatives of the first and second orders, is considered for functionals defined on the sets of differentiable functions. The found explicit representation of the solution of the given interpolation problem is based on an arbitrary Chebyshev system of functions. This solution is generalized for the case of interpolation of functionals on one out of two variables and applied to construct an approximate solution of the Cauchy problem for the differential equation of the hyperbolic type with variational derivatives. The description of the material is illustrated by numerous examples.

Towards Structuring and Updating the Typology of Semi-Presidentialism Based on the Peculiarities of Dualism, Party Affiliation and Responsibility of the Executives and the Composition of Legislatures

The article clarifies that the typology of semi-presidentialism, which outlines the real powers of political institutions and the peculiarities of relations among them in the triangle “president–prime minister/cabinet–parliament”, was initiated to place the institutions of president and prime minister in the environment of the distribution of powers and responsibilities in the executive and party composition of legislatures. Thus, it was recorded that semi-presidentialism can be represented in the form of the systems of both unified and divided government, as well as their variational derivatives. Accordingly, it was found that the functionality and dynamics of semi-presidentialism are dependent both on constitutional norms and political factors. Simultaneously, the study primarily focuses on the updated and expanded theorization of the typology of semi-presidentialism based on the peculiarities of dualism, party affiliation and responsibility of the executives and the composition of legislatures. As a result, it argues that semi-presidentialism (based on presidential party positioning against the types of cabinets and the parameters of inter-party and intra-party relations) should be typified on the fully or partly unified majority systems, fully or partly unified minority systems, divided majority systems and divided minority systems, which provide various political implications. In addition, the study shows that such a logic and construction of different types of semi-presidentialism is of utmost importance, since it allows to recognize the various effects and consequences of the analyzed system of government, including its prevalence and statistics, stability and conflicts, as well as correlations with different types of political regimes. Keywords: semi-presidentialism, system of government, dualism, party affiliation and responsibility of the executives, composition of legislature, unified and divided systems.

Differential Equation in a Banach Space Multiplicatively Perturbed by Random Noise

We consider the problem of finding the moment functions of the solution of the Cauchy problem for a first-order linear nonhomogeneous differential equation with random coefficients in a Banach space. The problem is reduced to the initial problem for a nonrandom differential equation with ordinary and variational derivatives. We obtain explicit formula for the mathematical expectation and the second-order mixed moment functions for the solution of the equation.

Variational derivatives in locally Lagrangian field theories and Noether–Bessel-Hagen currents

The variational Lie derivative of classes of forms in the Krupka’s variational sequence is defined as a variational Cartan formula at any degree, in particular for degrees lesser than the dimension of the basis manifold. As an example of application, we determine the condition for a Noether–Bessel-Hagen current, associated with a generalized symmetry, to be variationally equivalent to a Noether current for an invariant Lagrangian. We show that, if it exists, this Noether current is exact on-shell and generates a canonical conserved quantity.