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The objective of this paper is to trace the increase in the complexity of the description of classes of analytic complexity (introduced by the author in previous works) under the passage from the class Cl1 to the class Cl2. To this end, two subclasses, Cl1+ and Cl1++, of Cl2 that are not contained in Cl1 are described from the point of view of the complexity of the differential equations determining these subclasses. It turns out that Cl1+ has fairly simple defining relations, namely, two differential polynomials of differential order 5 and algebraic degree 6 (Theorem 1), while a criterion for a function to belong to Cl1++ obtained in the paper consists of one relation of order 6 and five relations of order 7, which have degree 435 (Theorem 2). The "complexity drop" phenomenon is discussed; in particular, those functions in the class Cl1+ which are contained in Cl1 are explicitly described (Theorem 3). [ABSTRACT FROM AUTHOR]

In this paper, we study scalar difference-differential equations of neutral type of general form where the σj( θ) are functions of bounded variation. For the solutions of this equation, we obtain the following estimate: where C is a constant independent of u 0( t) and the values of q and ℋ are determined by the properties of the characteristic determinant of this equation. Earlier, this estimate was proved for equations of less general form. For example, for piecewise constant functions σ j( θ) or for the case in which the function σ m( θ) has jumps at both points θ = 0 and θ = h. In the present paper, this estimate is obtained under the only condition that σ m( θ) experiences a jump at the point θ = 0; this condition is necessary for the correct solvability of the initial-value problem. [ABSTRACT FROM AUTHOR]

In this paper, the inverse spectral problem method is used to integrate the nonlinear mKdV–L equation in the class of periodic infinite-gap functions. The solvability of the Cauchy problem for an infinite system of Dubrovin differential equations in the class of times continuously differentiable periodic infinite-gap functions is proved. It is shown that the sum of a uniformly convergent function series constructed by solving the system of Dubrovin equations and by using the first trace formula satisfies the mKdV–L equations. Moreover, we prove that if the initial function is a real-valued -periodic analytic function, then the solution of the Cauchy problem for the mKdV–L equation is a real-valued analytic function in the variable as well; and if the number is a period (respectively, antiperiod) of the initial function, then the number is the period (respectively, antiperiod) in the variable of the solution of the Cauchy problem for the mKdV–L equations. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of the asymptotics of the spectrum of the boundary-value problem where ρ is the generalized derivative of the Minkowski function, i.e., ρ =?′( x) (here ?( x) is the 'question-mark function' first defined by Minkowski, who introduced this notation). For the eigenvalues of the problem, asymptotic two-sided estimates of power type are obtained. The order of the power is determined by the Hausdorff dimension of the support of the Minkowski measure d?. [ABSTRACT FROM AUTHOR]

Asymptotic solutions of the wave equation degenerating on the boundary of the domain (where the wave propagation velocity vanishes as the square root of the distance from the boundary) can be represented with the use of a modified canonical operator on a Lagrangian submanifold, invariant with respect to theHamiltonian vector field, of the nonstandard phase space constructed by the authors in earlier papers. The present paper provides simple expressions in a neighborhood of the boundary for functions represented by such a canonical operator and, in particular, for the solution of the Cauchy problem for the degenerate wave equation with initial data localized in a neighborhood of an interior point of the domain. [ABSTRACT FROM AUTHOR]

The maximally monotone operator method in real weighted Lebesgue spaces is used to study three different classes of nonlinear singular integro-differential equations with an arbitrary positive parameter. Under sufficiently clear constraints on the nonlinearity, we prove existence and uniqueness theorems for the solution covering in particular, the linear case as well. In contrast to the previous papers in which other classes of nonlinear singular integral and integro-differential equations were studied, our study is based on the inversion of the superposition operator generating the nonlinearities of the equations under consideration and the establishment of the coercitivity of the inverse operator, as well as a generalization of the well-known Schleiff inequality. [ABSTRACT FROM AUTHOR]

In the present paper, we consider estimates for the orders of zeros of polynomials in functions satisfying a system of algebraic differential equations and possessing a special D-property defined in the paper. The main result obtained in the paper consists of two theorems for the two cases in which these estimates are given. These estimates are improved versions of a similar estimate proved earlier in the case of algebraically independent functions and a single point. They are derived from a more general theorem concerning the estimates of absolute values of ideals in the ring of polynomials, and the proof of this theorem occupies the main part of the present paper. The proof is based on the theory of ideals in rings of polynomials. Such estimates may be used to prove the algebraic independence of the values of functions at algebraic points. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of weak solutions for the boundary-value problem where Ω is a bounded domain with smooth boundary in ℝ ( N ≥ 2) and Δ is a subelliptic operator of the type . We use the sub-super solution and variational methods. [ABSTRACT FROM AUTHOR]

This paper studies the relationship between the Lyapunov function of a macrosystem whose dynamics is governed by the laws of stochastic chemical kinetics and the invariant measure of this macrosystem arising at large times. A necessary and sufficient condition for the reduction of the search problem for the equilibrium of the macrosystem (the most probable macrostate of the invariant measure of this macrosystem) to an entropy-linear programming problem is given. [ABSTRACT FROM AUTHOR]

The trace-class property of Hankel operators (and their derivatives with respect to the parameter) with strongly oscillating symbol is studied. The approach used is based on Peller’s criterion for the trace-class property of Hankel operators and on the precise analysis of the arising triple integral using the saddle-point method. Apparently, the obtained results are optimal. They are used to study the Cauchy problem for the Korteweg-de Vries equation. Namely, a connection between the smoothness of the solution and the rate of decrease of the initial data at positive infinity is established. [ABSTRACT FROM AUTHOR]

The paper is devoted to the study of the solution of the Cauchy problem for a first-order differential equation in a Banach space with a small parameter on the right-hand side perturbing the equation. The coefficient of the derivative of the unknown function is a Fredholm operator with index zero and one-dimensional kernel. The case of a regular pair of operator coefficients is considered. An asymptotic expansion of the solution of the problem is constructed by using a method due to Vasil’eva, Vishik, and Lyusternik. In calculating the components of the regular and boundary-layer parts of the expansion, the cascade decomposition of the equations is used. It is proved that this expansion is asymptotic. Conditions for regular degeneration are found. The behavior of the solution as the parameter tends to zero is studied. [ABSTRACT FROM AUTHOR]

The notions of different types of boundedness in the sense of Poisson of solutions to systems of differential equations are introduced. Sufficient conditions are obtained for different types of boundedness of solutions in the sense of Poisson, which are introduced in the paper. [ABSTRACT FROM AUTHOR]

The nonexistence of a global solution of the semilinear elliptic equation Δ2u − (C/|x|4)u − |x|σ|u|q = 0 in the exterior of a ball is studied. A sufficient condition for the nonexistence of a global solution is established. The proof is based on the test function method. [ABSTRACT FROM AUTHOR]

In a multidimensional projective space, a distribution of planes is considered. Under the assumption that there is a relative invariant scoped by a subobject of a fundamental object of the first order, an internal composite equipment of the distribution is made, which is an analog of the Cartan equipment and Norder normalization of the second kind. It is proved that the composition equipment induces six bunches of group connections in the associated principal bundle which are intrinsically determined by the distribution itself. In every bundle, a unique intrinsic connection is distinguished. Analytic and geometric conditions for the coincidence of different types of connections are found. In the paper, the Cartan-Laptev method is used. All considerations are of local nature. [ABSTRACT FROM AUTHOR]

In this paper, Melikhov’s conjecture that clusters appear before passage to liquid is validated. The Poisson adiabat of the Boltzmann ideal gas is obtained as a consequence of the Bose-Einstein distribution. [ABSTRACT FROM AUTHOR]

In [1], necessary and sufficient conditions for the global asymptotic stability of the trivial solution of the Liénard equation ẍ+ f( x) ẋ+ g( x)=0, g(0)=0, were obtained under the condition In [1], the following problem was also posed: To determine whether condition (A) is a necessary condition for the global asymptotic stability of the trivial solution of the Liénard equation. The present paper answers this question, and the answer is negative, i.e., condition (A) is not a necessary condition. [ABSTRACT FROM AUTHOR]

In the paper, relations for the critical values of a mixture of new ideal gases are presented and differential equations for a mixture of real gases are written out. [ABSTRACT FROM AUTHOR]

In this paper, we compare theoretical calculations related to the thermodynamics of fluids and based on the parastatistic distribution established earlier by the author with classical experimental data. The deviation from the experiment is explained. [ABSTRACT FROM AUTHOR]

The Lambert cube Q( α, β, γ) is one of the simplest polyhedra. By definition, this is a combinatorial cube with dihedral angles α, β, and γ at three noncoplanar edges and with right angles at all other edges. The volume of the Lambert cube in hyperbolic space was obtained by R. Kellerhals (1989) in terms of the Lobachevskii function Λ( x). In the present paper, we find the volume of the Lambert cube in spherical space. It is expressed in terms of the function which can be regarded as the spherical analog of the function [ABSTRACT FROM AUTHOR]

We describe a number of properties of solutions of boundary-value problems for nonlinear ordinary differential equations of the same type as those studied by Kolmogorov, Petrovskii, and Piskunov in their well-known paper on waves described by the parabolic equation. We construct and justify the asymptotics of such solutions for large values of the modulus of the independent variable. [ABSTRACT FROM AUTHOR]

In this paper, we study the problem of the boundary accumulation of a discrete spectrum, which is essential for a boundary-value problem of fourth order arising in the theory of small transverse vibrations in an inhomogeneous viscoelastic rod (a Kelvin—Voigt body). We establish conditions for such an accumulation and its asymptotics, which are expressed in terms of the coefficients defining the problem posed by the differential expression. The results obtained are illustrated by numerical computation data. [ABSTRACT FROM AUTHOR]

In this paper, we study the relation between the notion of gauge equivalence and solutions of certain systems of nonlinear partial differential equations. This relation is based on stepwise gauge equivalence. [ABSTRACT FROM AUTHOR]

In the present paper, we prove an assertion allowing us to extend results related to the presence or absence of real zeros of functions of Mittag-Leffler type for certain values of a and µ to more extensive ranges of these parameters. We give a geometric description of the sets of pairs (a, µ) for which the function E1/a( z ; µ) has and does not have real zeros. [ABSTRACT FROM AUTHOR]

In this paper, we obtain a necessary and sufficient condition for the nontrivial solvability of homogeneous Dirichlet problems in the disk for linear equations of arbitrary even order 2 m with constant complex coefficients and homogeneous nondegenerate symbol in general position. The cases m=1, 2, 3 are studied separately. For the case m=2, we consider examples of real elliptic systems reducible to single equations with constant complex coefficients for which the homogeneous Dirichlet problem in the disk has a countable set of linearly independent polynomial solutions. [ABSTRACT FROM AUTHOR]

In this paper, sufficient conditions for the spectrum of the operator of high order to be discrete and unbounded below are obtained. [ABSTRACT FROM AUTHOR]

In this paper, we obtain identities for the squares of the components of the vector eigenfunctions of the system of Dirac equations with smooth periodic potential. [ABSTRACT FROM AUTHOR]

In this paper, we construct two polyconvolutions (generalized convolutions) with weight γ = x-ν generated by the Hankel transform possessing the factorization relations Hμ [h1] (x) = x-νHμ[f] (x) Hμ [g] (x), Hμ [h2] (x) = x-νHν[f] (x) Hμ [g] (x). Here Hμ is the Hankel transform operator of order μ. Conditions for the existence of the constructed polyconvolutions are found. On their basis, using the differential properties of the Hankel transform, we obtain two more polyconvolutions. The derived constructions allow us to solve new classes of integral and integro-differential equations and systems of equations. [ABSTRACT FROM AUTHOR]

In this paper, we obtain sufficient conditions for the existence of a unique regular solution of the boundary-value problems for operator differential equations of order $2k$ with variable coefficients. These conditions are expressed solely in terms of operator coefficients of the equations under study. [ABSTRACT FROM AUTHOR]

In the paper, the notion of F function, i.e., a partial recursive function whose resemblance type consists of one isomorphism type, is generalized. [ABSTRACT FROM AUTHOR]

Let $G$ be a group, let $A$ be an Abelian group, and let $n$ be an integer such that $n\backslash ge\; -1$. In the paper, the sets $\backslash Phi\_n(G,A)$ of functions from $G$ into $A$ of degree not greater than $n$ are studied. In essence, these sets were introduced by Logachev, Sal'nikov, and Yashchenko. We describe all cases in which any function from $G$ into $A$ is of bounded (or not necessarily bounded) finite degree. Moreover, it is shown that if $G$ is finite, then the study of the set $\backslash Phi\_n(G,A)$ is reduced to that of the set $\backslash Phi\_n(G/O^p(G),A\_p)$ for primes $p$ dividing $|G/G\text{'}|$. Here $O^p(G)$ stands for the $p$-coradical of the group $G$, $A\_p$ for the $p$-component of $A$, and $G\text{'}$ for the commutator subgroup of $G$. [ABSTRACT FROM AUTHOR]

The paper consists of two parts. The first part deals with the solvability of new boundary-value problems for the model quasihyperbolic equations (−1) D u = Au + f( x, t), where p > 1, for a self-adjoint second-order elliptic operator A. For the problems under study, the existence and uniqueness theorems are proved for regular solutions. In the second part, the results obtained in the first part are somewhat sharpened and generalized. [ABSTRACT FROM AUTHOR]

For second-order algebraic differential equations which have an explicit linear part, we describe all their possible solutions that are entire functions of finite order. [ABSTRACT FROM AUTHOR]

We introduce a number of notions related to the Lyapunov transformation of linear differential operators with unbounded operator coefficients generated by a family of evolution operators. We prove statements about similar operators related to the Lyapunov transformation and describe their spectral properties. One of the main results of the paper is a similarity theorem for a perturbed differential operator with constant operator coefficient, an operator which is the generator of a bounded group of operators. For the perturbation, we consider the operator ofmultiplication by a summable operator function. The almost periodicity (at infinity) of the solutions of the corresponding homogeneous differential equation is established. [ABSTRACT FROM AUTHOR]

In the present paper, we show that the theory of dynamic equations on time scales (with an argument with “holes,” i.e., a discontinuous argument) can significantly be simplified and generalized by using Stieltjes integration, which is inverse to the differentiation with respect to Riesz measures. [ABSTRACT FROM AUTHOR]

or the second-order differential equation ẍ + f(t) ẋ + g(t)x = 0, the method of Lyapunov functions is used to obtain sufficient conditions for the existence of homoclinic trajectories, i.e., solutions x(t), ẋ (t) satisfying the conditions limt→±∞x(t) = 0 and limt→±∞ẋ (t) = 0. The specific case in which all the solutions of this differential equation are homoclinic is considered. [ABSTRACT FROM AUTHOR]

The paper deals with the development of the theory of boundedness in part of the variables of solutions to systems of differential equations, which is in fact a version of the Lyapunov direct method in the theory of stability. [ABSTRACT FROM AUTHOR]

The present paper is devoted to the Hermitian spectral pseudoinversion and its applications to analysis, the solution and reduction of Hermitian differential-algebraic systems. New explicit formulas for the solutions of such systems and the solutions of related generalized Lyapunov equations are proposed. Attainable upper bounds for the norms of the solutions are obtained. A realization of the balanced truncation method not requiring computations involving projections onto deflating subspaces is proposed. [ABSTRACT FROM AUTHOR]

The paper deals with the problem of global asymptotic stability of the zero solution of a system of autonomous differential equations. A proposed method for studying this problem is based on the use of an auxiliary positive definite function whose derivative can be sign-alternating along solutions of the system. [ABSTRACT FROM AUTHOR]

The paper is devoted to the exposition of results announced in [1]. We construct a reduction (following an idea of S. P. Novikov) of the calculus of pseudodifferential operators on Euclidean space ℝ to a similar calculus in the space of sections of a one-dimensional fiber bundle ξ on the 2 n-dimensional torus $\mathbb{T}^{2n} $. This reduction enables us to identify the Schwartz space on ℝ with the space of smooth sections Γ( T, ξ), compare the Sobolev norms on the corresponding spaces and pseudodifferential operators in them, and describe the class of elliptic operators that reduce to Fredholm operators in Sobolev norms. Thus, for a natural class of elliptic pseudodifferential operators on a noncompact manifold of ℝ, we construct an index formula in accordance with the classical Atya-Singer formula. [ABSTRACT FROM AUTHOR]

In this paper, applications of the fixed-point index of a nonlinear operator in Menger PN-spaces are given. Theorems on the existence of fixed points of a compact continuous operator are obtained. At the same time, the existence of the solution of a class of differential equations in a complete Menger PN-space is studied, and the theorem on the existence of the solution of a class of differential equations is obtained. [ABSTRACT FROM AUTHOR]

In the present paper, we describe the structure of a strongly continuous operator semigroup T( t) (where T: ℝ → End X and X is a complex Banach space) for which Im T( t) is a finite-dimensional space for all t > 0. It is proved that such a semigroup is always the direct sum of a zero semigroup and a semigroup acting in a finite-dimensional space. As examples of applications, we discuss differential equations containing linear relations, orbits of a special form, and the possibility of embedding an operator in a C-semigroup. [ABSTRACT FROM AUTHOR]

Let a1,a2, ...,an, and λ be complex numbers, and let p1,p2, ...,pn be measurable complex-valued functions on ℝ+ (:= [0, + ∞)) such that | p 1 | + (1 + | p 2 − p 1 |) ∑ j = 2 n | p j | ∈ L loc 1 (ℝ +). A construction is proposed which makes it possible to well define the differential equation y (n) + (a 1 + p 1 (x)) y (n − 1) + (a 2 + p 2 ′ (x)) y (n − 2) + ⋯ + (a n + p n ′ (x)) y = λ y under this condition, where all derivatives are understood in the sense of distributions. This construction is used to show that the leading term of the asymptotics as x → + ∞ of a fundamental system of solutions of this equation and of their derivatives can be determined, as in the classical case, from the roots of the polynomial Q (z) = z n + a 1 z n − 1 + ⋯ + a n − λ , provided that the functions p1,p2, ...,pn satisfy certain conditions of integral decay at infinity. The case where a1 = ... = an = λ = 0 is considered separately and in more detail. [ABSTRACT FROM AUTHOR]

In this paper, we determine the coefficients of left-side linear ordinary differential equations with constant coefficients over a noncommutative Banach algebra; these equations have solutions of Euler type. [ABSTRACT FROM AUTHOR]

In this paper, we consider the stochastic analog of the Baker-Hausdorff formula for operators satisfying the canonical commutation relations. This formula can be used to obtain an exact solution to the problem of the dynamics of a quantum system interacting with the environment. [ABSTRACT FROM AUTHOR]

The paper studies the problem of uniform approximation of a continuous function on a closed interval by the class of functions with bounded second derivative. We prove an estimate of the value of best approximation of the function by this class via its second modulus of continuity. The obtained estimate is sharp for the class of continuous functions. [ABSTRACT FROM AUTHOR]

In the present paper, we describe invertible contractions of the maximal quotient generated by a differential expression with bounded operator coefficients and by a nonnegative operator function. We show that the operators inverse to such contractions are integral operators and prove a criterion for such operators to be holomorphic. Using the results obtained, we describe the generalized resolvents of symmetric quotients. [ABSTRACT FROM AUTHOR]