Vol 39, No 6 (2018)
- Year: 2018
- Articles: 18
- URL: https://journals.rcsi.science/1995-0802/issue/view/12600
Article
Paranormal Measurable Operators Affiliated with a Semifinite von Neumann Algebra
Abstract
Let M be a von Neumann algebra of operators on a Hilbert space H, τ be a faithful normal semifinite trace on M. We define two (closed in the topology of convergence in measure τ) classes P1 and P2 of τ-measurable operators and investigate their properties. The class P2 contains P1. If a τ-measurable operator T is hyponormal, then T lies in P1; if an operator T lies in Pk, then UTU* belongs to Pk for all isometries U from Mand k = 1, 2; if an operator T from P1 admits the bounded inverse T−1 then T−1 lies in P1. If a bounded operator T lies in P1 then T is normaloid, Tn belongs to P1 and a rearrangement μt(Tn) ≥ μt(T )n for all t > 0 and natural n. If a τ-measurable operator T is hyponormal and Tn is τ-compact operator for some natural number n then T is both normal and τ-compact. If an operator T lies in P1 then T 2 belongs to P1. If M= B(H) and τ = tr, then the class P1 coincides with the set of all paranormal operators onH. If a τ-measurable operator A is q-hyponormal (1 ≥ q > 0) and |A*| ≥ μ∞(A)I then Ais normal. In particular, every τ-compact q-hyponormal (or q-cohyponormal) operator is normal. Consider a τ-measurable nilpotent operator Z ≠ 0 and numbers a, b ∈ R. Then an operator Z*Z − ZZ* + aRZ + bSZ cannot be nonpositive or nonnegative. Hence a τ-measurable hyponormal operator Z ≠ 0 cannot be nilpotent.
Chernoff Equivalence for Shift Operators, Generating Coherent States in Quantum Optics
Abstract
In this paper some examples of sequences of compositions of n independent equally distributed random semigroups in the form of a shift operator generating coherent states in quantum optics have been studied. Representation of averaged coherent states in quantum optics using a procedure of averaging of random shift operators is obtained. The analytical expression for averaged value of a superposition of shift operators is presented.
Decomposition of Sum in Recurrent Formula for Exponents of the Inverse Chain Exponential
Abstract
We discuss a recurrent formula for determining the exponents of an inverse chain exponential; the formula was obtained earlier by a generalization of the original Lambert function. We describe a method of evaluation of the exponents by decomposition of the original sum into partial ones with lesser orders of summation. As an example, we evaluate by this method the sixth exponent.
On a Classic Moment Problem for Entire Functions
Abstract
The paper considers a poly-element linear equation for odd functions that are analytic on the plane with a cut along the interval [−i, i] of the imaginary axis. The results obtained here are used in the study of the lacunary Stieltjes moment problem for entire functions of exponential type.
Loewner–Kufarev Equation for a Strip with an Analogue of Hydrodynamic Normalization
Abstract
A semigroup of holomorphic self-mappings of a strip with hydrodynamic normalization is studied. A description of infinitesimal transformations of the semigroup is obtained. The differentiability with respect to the time parameter of normalized evolution families is established. An existence theorem is obtained for the evolution equation.
Meromorphization of M. I. Kinder’s Formula Via the Change of Contours
Abstract
Parametrical families of the exterior inverse boundary value problems going back to well-known R. B. Salimov’s book became a plentiful source of new statements and methods in the study of the above problems. Critical points of conformal radii acting as the free parameters of such problems show interesting interrelations between their parametrical dynamics and geometric behavior. M.I. Kinder’s formula connecting the numbers of local maxima and saddles of a conformal radius is generalized here on the case when the derivative of the mapping function has zeros and poles in the unit disk and on its boundary.
Uniform Wavelet-Approximation of Singular Integral Equation Solutions
Abstract
In this article we consider a singular integral equation of the first kind with a Cauchy kernel on a segment of the real axis, which is a mathematical model of many applied problems. It is known that such an equation is exactly solved only in rare cases, therefore, the problem of its approximate solution with obtaining uniform error estimates is very actual. This equation is considered on a pair of weighted spaces that are constrictions of the space of continuous functions. The correctness of the problem of solving this equation on a chosen pair of spaces of the desired elements and right-hand sides gives the possibility of its approximate solution with a theoretical justification. The numerical method proposed in this article is based on the approximation of the unknown function by Chebyshev wavelets of the second kind. Uniform error estimates are established depending on the structural properties of the initial data. The numerical experiment in the Wolfram Mathematica package showed a good convergence rate of the approximate solution to the exact one.
Solution of Nonhomogeneous Helmholtz Equation with Variable Coefficient Using Boundary Domain Integral Method
Abstract
The Boundary Domain Integral Method (BDIM) is applied to the solution of the nonhomogeneous Helmholtz equation with variable coefficient. The analytical formulas for the integrals over the individual boundaries and domain integrals are used to increase the accuracy of the numerical approach. Comparisons of the developed BDIM with the analytical solutions for the homogeneous Helmholtz equation with constant coefficient and the nonhomogeneous Helmholtz equation with variable coefficient are given.
Avkhadiev–Becker Type Univalence Conditions for Biharmonic Mappings
Abstract
In this paper we consider complex-valued biharmonic functions that are locally univalent. We construct families of biharmonic univalent mappings of the unite disc similar to Avkhadiev- Becker type conditions for analytic functions. Also, we investigate the case where biharmonic functions are defined on the exterior of the unit disc. In this case we obtain three analogs of Avkhadiev–Becker type conditions of univalence.
Parametric Characteristics of High-Order Tangential Loewner’s Slits
Abstract
We describe the chordal Loewner evolution with a driving function generating the slit in the upper half-plane which is high-order tangential to the real axis at 0. In this case, the slit is a joint boundary for the analytic corner and the analytic cusp. We give an asymptotic expansion of the slit and of the conformal mapping from the upper half-plane onto the upper half-plane with the slit both in the corner and the cusp. We find also a relation between harmonic measures of the two sides of the slit.
A Note about Torsional Rigidity and Euclidean Moment of Inertia of Plane Domains
Abstract
Denote by P(G) the torsional rigidity of a simply connected plane domain G, and by I2(G) the Euclidean moment of inertia of G. In 1995 F.G. Avkhadiev proved that P(G) and I2(G) are comparable quantities in sense of Pólya and Szegö. Moreover, it was shown that the ratio P(G) /I2(G) belongs to the segment [1, 64]. We investigate the following conjecture P(G) ≥ 3I2(G), where G is a simply connected domain. We prove that the conjecture is true for polygonal domains circumscribed about a circle. For convex domains we show sharp isoperimetric inequalities, which justify the conjecture, in particular, we prove that P(G) > 2I2(G). Some aspects of approximate formulas for P(G) are also discussed.
On Univalent Conditions in Domains Belonging to One of the Rahmanov Classes
Abstract
The purpose of this paper is to establish new sufficient conditions for the univalence of analytical functions. We consider the case when the functions are defined in the domains belonging to one of the Rahmanov classes. Our study is based upon an examination of the properties of some generalizations of spiral domains using the quasiconformal extension method. Possible applications of the obtained results to the solution of the strong problem of univalence in inverse boundary value problems are shown.
About the Optimal Replacement of the Lebesque Constant Fourier Operator by a Logarithmic Function
Abstract
The Lebesgue constant, corresponding to the classical Fourier operator is approaching a two-parameter family of logarithmic functions. The optimal values of these parameters are found in which the best uniform approximation of the Lebesgue constant a well-defined function of this family is achieved. We considered the case when corresponding to these functions the remaining members are strictly decreasing.
A Cutting Method with Approximation of a Constraint Region and an Epigraph for Solving Conditional Minimization Problems
Abstract
We propose a method which belongs to a class of cutting methods for solving a convex programming problem. The developed method differs from traditional algorithms of the mentioned class by the following fact. This method uses approximation of the feasible set and the epigraph of the objective function simultaneously for the solving problem. In the method cutting planes are constructed by subgradients of the objective function and functions which define the constrained set. In this case while initial approximating sets are chosen as polyhedral sets, each iteration point is found by solving a linear programming problem independently of the type of functions which define the solving problem. Moreover, unlike most the cutting methods the proposed method is characterized by possibility of updating approximating sets due to dropping accumulating constraints.
Application of Riemann–Hilbert Problem Solutions to a Study of Nonlinear Boundary Value Problems for Timoshenko Type Inhomogeneous Shells with Free Edges
Abstract
The paper deals with the study of solvability of geometrically nonlinear boundary value problem for elastic shallow isotropic inhomogeneous shells with free edges within S. P. Timoshenko shear model. The problem is reduced to one nonlinear equation whose solvability is proved with the use of contracting mappings principle.