2017
Том 69
№ 2

# Volume 60, № 11, 2008

Article (English)

### A property of the β-Cauchy-type integral with continuous density

Ukr. Mat. Zh. - 2008. - 60, № 11. - pp. 1443–1448

The aim of this paper is to extend a theorem from classical complex analysis proved by Davydov in 1949 to the theory of solutions of a special case of the Beltrami equation in the z-complex plane (i.e., null solutions of the differential operator $\partial_{\overline{z}} - \beta \frac{z}{\overline{z}}\partial_z,\quad 0 \leq \beta < 1$). We prove that if $\gamma$ is a rectifiable Jordan closed curve and $f$ is a continuous complex-valued function on $\gamma$ such that the integral $$\int\limits_{\gamma\setminus\{\zeta \in \gamma:\;|\zeta-t|\leq r \}} \frac{|f(\zeta) - f(t)|}{\left|\zeta - t|t/\zeta|^{\theta} \right|} \left|n(\zeta) - \beta \frac{\zeta}{\overline{\zeta}} \overline{n}(\zeta) \right|ds, \quad \theta = \frac{2\beta}{1-\beta},$$ converges uniformly on $\gamma$ as $r \rightarrow 0$, where $n(\zeta)$ is the exterior unit normal vector on $\gamma$ at a point $n(\zeta)$ and $ds$ is the arc length differential, then the $\beta$-Cauchy type integral $$\frac1{2(1 - \beta)\pi}\int\limits_{\gamma}\frac{f(\zeta)}{\zeta - z|z/\zeta|^ {\theta}} \left(n(\zeta) - \beta \frac{\zeta}{\overline{\zeta}}\overline{n}(\zeta) \right)ds,\quad z \in \gamma,$$ admits a continuous extension to $\gamma$ and a version of the Sokhotski - Plemelj formulae holds. Метою цієї статті є узагальнення теореми із класичного комплексного аналізу, що була доведена Давидовим у 1949 р., для теорії розв'язків окремого випадку рівняння Вельтрамі у $z$-комплексній площині (тобто нульових розв'язків диференціального оператора $\partial_{\overline{z}} - \beta \frac{z}{\overline{z}}\partial_z,\quad 0 \leq \beta < 1$).

Article (Russian)

### Barrier functions for one class of semilinear parabolic equations

Ukr. Mat. Zh. - 2008. - 60, № 11. - pp. 1449–1456

For a parabolic quasilinear equation with monotone convex potential, we construct superparabolic and subparabolic barrier functions by the method of decomposition.

Article (Ukrainian)

Ukr. Mat. Zh. - 2008. - 60, № 11. - pp. 1457–1469

We study Bell polynomials by using functions of triangular matrices (parapermanents and paradeterminants). Some combinatorial identities and relationships between these functions and the Stirling numbers of the first and second kinds are established.

Article (Ukrainian)

### Asymptotic normality of M-estimates in the classical nonlinear regression model

Ukr. Mat. Zh. - 2008. - 60, № 11. - pp. 1470–1488

Sufficient conditions are obtained for the asymptotic normality of M-estimates of the unknown parameters of nonlinear regression models with discrete time and independent identically distributed errors of observations.

Article (English)

### Renormalization constant for the local times of self-intersections of a diffusion process in the plane

Ukr. Mat. Zh. - 2008. - 60, № 11. - pp. 1489–1498

We study the local times of self-intersection of a diffusion process in the plane. Our main result is connected with the investigation of the asymptotic behavior of the renormalization constant of this local time.

Article (Ukrainian)

### Evolution inequalities with noncoercive w λ 0 -pseudomonotone volterra-type mappings

Ukr. Mat. Zh. - 2008. - 60, № 11. - pp. 1499 – 1519

We consider a class of differential-operator inequalities with noncoercive w λ 0 -pseudomonotone operators. The problem of the existence of solutions of the Cauchy problem for these inequalities is investigated by using the Dubinsky method. A priori estimates for these solutions and their derivatives are obtained. We give a model example that illustrates the results and generalizations obtained.

Article (English)

### Muckenhoupt–Wheeden theorem for generalized f-Riesz-type potentials

Ukr. Mat. Zh. - 2008. - 60, № 11. - pp. 1520–1528

The Muckenhoupt–Wheeden theorem is proved for one class of potentials. As a consequence, we describe the equivalent norm in the space of generalized Bessel potentials of negative order.

Article (Russian)

### Convergence of solutions of stochastic differential equations to the Arratia flow

Ukr. Mat. Zh. - 2008. - 60, № 11. - pp. 1529–1538

We consider the solution $x_{\varepsilon}$ of the equation $$dx_{\varepsilon}(u,t) = \int\limits_\mathbb{R}\varphi_{\varepsilon}(x_{\varepsilon}(u,t) - r) W(dr,dt),$$ $$x_{\varepsilon}(u,0) = u,$$ where $W$ is a Wiener sheet on $\mathbb{R} \times [0; 1].$ For the case where $\varphi_{\varepsilon}^2$ converges to $p \delta(\cdot - a_1) + q \delta(\cdot - a_2),$ i.e., where a boundary function describing the influence of a random medium is singular more than at one point, we prove that the weak convergence of $\left(x_{\varepsilon}(u_1, \cdot),...,x_{\varepsilon}(u_d, \cdot) \right)$ to $\left(X(u_1, \cdot),...,X(u_d, \cdot) \right)$ takes place as $\varepsilon\rightarrow0_+$ (here, $X$ is the Arratia flow).

Article (Ukrainian)

### Joint continuity of $K_h C$-functions with values in moore spaces

Ukr. Mat. Zh. - 2008. - 60, № 11. - pp. 1539 – 1547

We introduce a notion of a categorical cliquish mapping and prove that, for each $K_h C$-mapping $f : X \times Y \rightarrow Z$ (here, $X$ is a topological space, $Y$ is a first countable space, and $Z$ is a Moore space) with categorical cliquish horizontal $y$-sections $f_y$ , the sets $C_y (f)$ are residual $G_\delta$-sets in $X$ for each $y \in Y.$

Article (Ukrainian)

### Evaluation of the remainder term for the Thiele interpolation continued fraction

Ukr. Mat. Zh. - 2008. - 60, № 11. - pp. 1548–1554

We present an estimate of the remainder term for the Thiele interpolation continued fraction.

Article (English)

### Factorization of a convolution-type integro-differential equation on the positive half line

Ukr. Mat. Zh. - 2008. - 60, № 11. - pp. 1555–1567

Sufficient conditions for the existence of solutions are obtained for a class of convolution-type integro-differential equations on the half line. The investigation is based on the three-factor decomposition of the initial integro-differential operator.

Brief Communications (Ukrainian)

### On the application of permutation groups to some combinatorial problems

Ukr. Mat. Zh. - 2008. - 60, № 11. - pp. 1568–1571

We show that some properties of permutation groups can be applied to the construction of combinatorial objects with given properties.

Brief Communications (Russian)

### On Markov measure-valued processes in a finite space

Ukr. Mat. Zh. - 2008. - 60, № 11. - pp. 1572–1577

We consider stochastic flows with interaction in a finite phase space. The flows with variable generators generating evolutionary measure-valued processes are described. The influence of the interaction of particles on the entropy of the flow is analyzed.

Brief Communications (English)

### Fréchet-valued holomorphic functions on compact sets in (DFN)-spaces

Ukr. Mat. Zh. - 2008. - 60, № 11. - pp. 1578–1584

We establish the equivalence between the weak holomorphicity and holomorphicity of Fréchet-valued functions on compact polydisks in (DFN)-spaces. Moreover, the relations between separately holomorphic functions and holomorphic functions on compact polydisks in (DFN)-spaces are also given.