Vol 39, No 6 (2018)

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 P_k, then UTU* belongs to P_k 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.

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.

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.

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.

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.

In this paper, we establish behavior of almost bi-cubic functions in Lipschitz spaces. Indeed, we give a set-valued and Lipschitz norm approximation of bi-cubic functional equations in Lipschitz spaces.

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.

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.

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.