Vol 40, No 9 (2019)
- Year: 2019
- Articles: 20
- URL: https://journals.rcsi.science/1995-0802/issue/view/12763
Article
Best Approximations of Solutions of Fractional-integral Equations with the Riemann-Liouville Operator
Abstract
The article is devoted to the best approximations of solutions of integral equations that are defined on the line segment and have a fractional Riemann-Liouville integral in the main part. These approximations are constructed with the “generalized” projection method using the apparatus of algebraic polynomials. At the same time, the Fredholm property of an integral equation operator in a special pair of Hölder spaces of the desired elements and right-hand sides plays an important role.
Differential Inequalities and Univalent Functions
Abstract
Let \(\mathcal{M}\) be the class of analytic functions in the unit disk \(\mathbb{D}\) with the normalization f(0) = f′(0) − 1 = 0, and satisfying the condition
Hardy Type Inequalities on Domains with Convex Complement and Uncertainty Principle of Heisenberg
Abstract
We prove new integral inequalities for real-valued test functions defined on subdomains of the Euclidean space. We assume that the complement of the subdomain is a non-empty convex set. We prove an extension of the Hadwiger theorems about approximations of convex compact sets by polytopes and obtain some generalizations and improvements of several Hardy type multidimensional inequalities. In particular, in the last section we present an improvement of a two-dimensional inequality, connected with the uncertainty principle of Heisenberg.
Projections and Traces on von Neumann Algebras
Abstract
Let P, Q be projections on a Hilbert space. We prove the equivalence of the following conditions: (i) PQ + QP ≤ 2(QPQ)p for some number 0 < p ≤ 1; (ii) PQ is paranormal; (iii) PQ is M*-paranormal; (iv) PQ = QP. This allows us to obtain the commutativity criterion for a von Neumann algebra. For a positive normal functional φ on von Neumann algebra \(\mathcal{M}\) it is proved the equivalence of the following conditions: (i) φ is tracial; (ii) φ(PQ + QP) ≤ 2φ((QPQ)p) for all projections P,Q ∈ \(\mathcal{M}\) and for some p = p(P, Q) ∈ (0,1]; (iii) φ(PQP) ≤ φ(P)1/pφ(Q)1/q for all projections P, Q ∈ \(\mathcal{M}\) and some positive numbers p = p(P, Q), q = q(P, Q) with 1/p+ 1/q = 1, p ≠ 2. Corollary: for a positive normal functional φ on \(\mathcal{M}\) the following conditions are equivalent: (i) φ is tracial; (ii) φ(A + A*) ≤ 2φ(∣A*∣) for all A ∈ \(\mathcal{M}\).
Asymptotics of Conformal Module of Nonsymmetric Doubly Connected Domain under Unbounded Stretching Along the Real Axis
Abstract
We establish an asymptotic formula describing the behavior of the conformal module of a plane doubly connected domain under its stretching along the real axis with coefficient tending to infinity. The description is given in simple geometric terms, connected with equations of the boundary curves. Therefore, in the nonsymmetric case we give an answer to a problem by Prof. M. Vourinen.
Biorthogonal Systems of Analytic Functions Generated by a Regular Triangle
Abstract
We consider the properties of biorthogonal systems induced by a convolution operator with Carleman kernel for a regular triangle. This is a perturbed singular operator with fixed singularities. We describe its set of anti-invariant points. To this end, we regularize the operator using a Carleman linear convolution shift that maps each triangle side to itself and changes its orientation, with the middle points of the sides being the fixed points of the shift. We search for a solution in the form of a Cauchy-type integral with unknown density. For this, both the theory of the Carleman boundary-value problem and the method of locally conformal gluing are used in an essential manner. We also apply the theory of elliptic functions that are generated by the corresponding doubly periodic group determined by the triangle as ‘half’ of the fundamental set. Using the method of contracting mappings in a Banach space, we study the corresponding homogeneous Fredholm integral equation of the second kind with regard to its solvability. Its fundamental system of solutions contains a single function; the fundamental system of solutions of the conjugated equation contains only the constant function. This makes it possible to use this equation for the construction of a system of biorthogonally conjugated analytic functions. More precisely, we consider a system of successive derivatives of a certain rational function determined by the Carleman kernel for the triangle and investigate the approximating properties of this system, as well as those of the corresponding biorthogonally conjugated system. This is a system of Cauchy-type integrals over the triangle boundary with a density which is invariant under the considered Carleman shift. Nontrivial decompositions of zero are obtained using the system of successive derivatives of the given rational function. The results are applied to the representation of some classes of analytical functions by means of the corresponding biorthogonal series.
On Hilbert Spaces of Entire Functions with Unconditional Bases of Reproducing Kernels
Abstract
We consider an entire function under certain conditions on the distribution of its zeros. We construct a Hilbert space of entire functions which possess unconditional basis of reproducing kernels at zeros of this function. It is proved that some known Hilbert spaces of entire functions with unconditional bases of reproducing kernels are isomorphic (as normalized spaces) to the corresponding spaces constructed by the entire functions generating the bases.
Univalent Harmonic Mappings and Lift to the Minimal Surfaces
Abstract
We construct sense-preserving univalent harmonic mappings which map the unit disk onto a domain which is convex in the horizontal direction, but with varying dilatation. Also, we obtain minimal surfaces associated with such harmonic mappings. This solves also a recent problem of Dorff and Muir. In several of the cases, we illustrate mappings together with their minimal surfaces pictorially with the help of Mathematica software.
Hohlov Effects for Pre-Schwarzian Derivatives of Functions in the Gakhov Class
Abstract
We find an example of a situation when the exit from Gakhov’s class along some parametrical family of functions is connected with boundary bifurcation of the Gakhov equation. The corresponding condition of hit in Gakhov’s class is described by the construction of the Goryainov-Hohlov type, i.e. this is a subordination condition where the majorant itself is defined by (another) subordination. Next, we introduce and study a new concept of sharpness in the conditions of belonging to Gakhov’s class in the form of subordination of pre-Schwarzian derivatives to starlike functions; this concept is based on the Novikov-Hohlov’s effect in the inverse problems for the potentials and for the analytic functions. Finally, we study the Gakhov equation for the Biernacki-Hohlov operator.
Representation of Analytic Functions by Series of Exponential Monomials in Convex Domains and Its Applications
Abstract
In this paper lower bounds for entire functions of exponential type and regular growth, zero sets of which have zero condensation indices, are obtained. In this case, the exceptional set consists of pairwise disjoint disks centered at zeroes. Sufficient conditions for radii of these circles are indicated. We also obtain a result on representation of analytic functions in the closure of a bounded convex domain (as well as analytic functions in domain and continuous up to the boundary) by series of exponential monomials. This result extends the classical result of A.F. Leont’ev to the case of multiple zero set of entire function. The obtained result is applied to the problem on distribution of singular points of a sum of series of exponential monomials at the boundary of its convergence domain.
Integral Formulas for Recovering Extremal Measures for Vector Constrained Energy Problems
Abstract
Extremal problems for vector potentials have wide applications in asymptotic analysis of Hermite-Padé approximants of analytic functions. We consider equilibrium vector logarithmic potentials with constrains on measures. We study the dependance of the supports of the equilibrium measures on their masses. We obtain the integral formulas for recovering the extremal measure of given mass from the supports of the equilibrium measures of smaller masses.
Dimensions of Solution Spaces of the Schrodinger Equation with Finite Dirichlet Integral on Non-compact Riemannian Manifolds
Abstract
Exact estimations of dimensions of spaces of bounded solutions of stationary Schrodinger equation with finite Dirichlet integral in terms of massive sets are obtained. It is proved that dimension of spaces of bounded solutions of this equation is not less than number of disjoint qD-massive subsets of manifold. This paper partly extends, the results of A.A. Grigor’yan, A.G. Losev (2017).
Inverse Nonlinear Problem of Designing Supercavitating Hydrofoils
Abstract
In the paper, we present an analytical inverse nonlinear method of designing supercavitating hydrofoils by a given velocity distribution on the foil surface with allowance for the physical realizability of the obtained flows. The role of the pressure load near the leading edge (hydrofoil nose) for obtaining one-sheeted flow domains is revealed. It is shown that even the noses of very small sizes of order 10−3–10−2 of the hydrofoil chord lead to a very significant decrease of the lift-to-drag ratio.
Multidimensional Hardy Type Inequalities with Remainders
Abstract
Hardy type inequalities with an additional nonnegative-term are established for compactly supported smooth functions on arbitrary open subsets and on convex domains of the Euclidean space. We prove Hardy-type inequalities in spatial domains with finite inner radius. Weight functions depend on the distance function to the boundary of the domain. We obtain one-dimensional L1-inequalities. In particular cases we obtained sharp constants. Also new Hardy type inequality with remainders for the Riemann-Liouville fractional integrals is proved.
Qualitative Results in the Bombieri Problem for Conformal Mappings
Abstract
Bombieri’s numbers σmn characterize a behavior of the coefficient body for the class S of all holomorphic and univalent functions f in the unit disk normalized by f(z) = z + a2z2 + …. The number σmn is the limit of ratio for Re(n − an) and Re (m − am) as f tends to the Koebe function K(z) = z(1 − z)−2. It is showed in the paper that Bombieri’s conjecture about explicit values of σmn implies a sliding regime in an associated control theory problem generated by the Loewner differential equation. We develop also an asymptotical approach in verification of necessary criteria for Bombieri’s conjecture.
On Kähler Geometry of Infinite-dimensional Complex Manifolds Diff+(S1)/S1 and Diff+(S1)/Möb(S1)
Abstract
The infinite-dimensional complex Frechet manifolds \({\cal R}: = {\rm{Dif}}{{\rm{f}}_ + }({S^1})/{S^1}\) and \({\cal S}: = {\rm{Dif}}{{\rm{f}}_ + }({S^1})/{\rm{M\ddot ob}}({S^1})\) are the quotients of the group Diff+(S1) of orientation-preserving diffeomorphisms of the unit circle S1 modulo subgroups of rotations and fractional-linear transformations respectively. These manifolds are the coadjoint orbits of the Virasoro group and the only ones having a Kähler structure. It motivates the study of their complex geometry. These manifolds are also closely related to string theory because they can be realized as the spaces of complex structures on loop spaces.
Construction of Sufficient Univalent Conditions in Domains Convex in a Sector of Directions
Abstract
We study the domains which are convex in a sector of directions. Subclasses of domains whose boundaries are quasiconformal curves are determined and some sufficient univalent conditions are constructed for functions analytic in these domains. In addition, applications of the obtained results to the strong problem of univalence in inverse boundary value problems are discussed.
Limit Points of Bernoulli Distribution Algebras Induced by Boolean Functions
Abstract
We consider Bernoulli distribution algebras, i.e. sets of distributions that are closed under transformations achieved by substituting independent random variables for arguments of Boolean functions from a given system. We establish that, unless the transforming set contains only essentially unary functions, the set of algebra limit points is either empty, single-element or no less than countable.