In this paper, we consider the necessary and sufficient conditions for the subharmonicity of functions of two variables representable as a product of two functions of one variable in the Cartesian coordinate system or in the polar coordinate system in domains on the plane. We establish a connection of such functions with functions that are convex with respect to solutions of second-order linear differential equations, that is, convex with respect to two functions.
Sharp inequalities between the best approximations of functions analytic in the unit disk are obtained using algebraic polynomials and the moduli of continuity of higher-order derivatives in the Bergman space ({{mathcal{B}}_{{2,mu }}}) Based on these inequalities, the exact values of some known (n)-widths of classes of functions analytic in the unit disk are calculated.
The problem of finding the supremums of the best simultaneous polynomial approximations of some classes of functions analytic in the unit disk and belonging to the Bergman space ({{B}_{2}}) is considered. The indicated function classes are defined by the averaged values of the (m)th-order moduli of continuity of the highest derivative bounded from above by some majorant (Phi ).
For the equation ((operatorname{sgn} y){{left| y right|}^{m}}{{u}_{{xx}}} + {{u}_{{yy}}} + {{alpha }_{0}}{{left| y right|}^{{(m - 2)/2}}}{{u}_{x}} + ({{beta }_{0}}{text{/}}y){{u}_{y}} = 0,) considered in some unbounded mixed domain, uniqueness and existence theorems are proved for a solution to the problem with the missing shift condition on the boundary characteristics and an analogue of the Frankl-type condition on the interval of degeneracy of the equation.
Let (({{x}_{n}})) be a sequence and ({ {{c}_{k}}} in {{ell }^{infty }}(mathbb{Z})) such that ({{left| {{{c}_{k}}} right|}_{{{{ell }^{infty }}}}} leqslant 1). Define (mathcal{G}({{x}_{n}}) = mathop {sup }limits_j left| {sumlimits_{k = 0}^j ,{{c}_{k}}left( {{{x}_{{{{n}_{{k + 1}}}}}} - {{x}_{{{{n}_{k}}}}}} right)} right|.) Let now ((X,beta ,mu ,tau )) be an ergodic, measure preserving dynamical system with ((X,beta ,mu )) a totally (sigma )-finite measure space. Suppose that the sequence (({{n}_{k}})) is lacunary. Then we prove the following results: 1. Define ({{phi }_{n}}(x) = frac{1}{n}{{chi }_{{[0,n]}}}(x)) on (mathbb{R}). Then there exists a constant (C > 0) such that ({{left| {mathcal{G}({{phi }_{n}} * f)} right|}_{{{{L}^{1}}(mathbb{R})}}} leqslant C{{left| f right|}_{{{{H}^{1}}(mathbb{R})}}},) for all (f in {{H}^{1}}(mathbb{R})). 2. Let ({{A}_{n}}f(x) = frac{1}{n}sumlimits_{k = 0}^{n - 1} ,f({{tau }^{k}}x),) be the usual ergodic averages in ergodic theory. Then ({{left| {mathcal{G}({{A}_{n}}f)} right|}_{{{{L}^{1}}(X)}}} leqslant C{{left| f right|}_{{{{H}^{1}}(X)}}},) for all (f in {{H}^{1}}(X)). 3. If ({{[f(x)log (x)]}^{ + }}) is integrable, then (mathcal{G}({{A}_{n}}f)) is integrable.
It is established that any effectively separable many-sorted universal algebra has an enrichment that is the only (up to isomorphism) model constructed from constants for a suitable computably enumerable set of sentences.
In the paper, a generalized Lotka–Volterra-type system with switching is considered. The conditions for the ultimate boundedness of solutions and the permanence of the system are studied. With the aid of the direct Lyapunov method, the requirements for the switching law are established to guarantee the necessary dynamics of the system. An attractive compact invariant set is constructed in the phase space of the system, and a given region of attraction for this set is provided. A distinctive feature of the work is the use of a combination of two different Lyapunov functions, each of which plays its own special role in solving the problem.
A general form of the equation of a curvilinear three-web admitting a one-parameter family of automorphisms ((AW)-webs) is found. It is proved that the trajectories of automorphisms of an (AW)-web are geodesics of its Chern connection. All (AW)-webs are found for which one of the covariant derivatives of curvature is zero.
The paper considers the problem of constructing systems of vector fields that are invariant under the action of the local Lie group of transformations. It is shown that there exists a special class of Lie groups for which this problem can be solved elementarily.
Let (f) be a locally integrable function defined on (mathbb{R}), and (({{n}_{k}})) be a lacunary sequence. Define ({{A}_{n}}f(x) = frac{1}{n}int_0^n f(x - t)dt,) and let ({{mathcal{V}}_{rho }}f(x) = {{left( {sumlimits_{k = 1}^infty {{{left| {{{A}_{{{{n}_{k}}}}}f(x) - {{A}_{{{{n}_{{k - 1}}}}}}f(x)} right|}}^{rho }}} right)}^{{1/rho }}}.) Suppose that (w in {{A}_{p}}), (1 leqslant p < infty ), and (rho geqslant 2). Then, there exists a positive constant (C) such that ({{left| {{{mathcal{V}}_{rho }}f} right|}_{{L_{w}^{1}}}} leqslant C{{left| f right|}_{{H_{w}^{1}}}}) for all (f in H_{w}^{1}(mathbb{R})).
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