# Volume 67, № 1, 2015

### On the Square-Integrable Measure of the Divergence of Two Nuclear Estimations of the Bernoulli Regression Functions

Babilua P., Nadaraya E., Sokhadze G. A.

Ukr. Mat. Zh. - 2015. - 67, № 1. - pp. 3–18

We establish the limit distribution of the square-integrable deviation of two nonparametric nuclear-type estimations for the Bernoulli regression functions. A criterion is proposed for the verification of the hypothesis of equality of two Bernoulli regression functions. We study the problem of verification and, for some “close” alternatives, investigate the asymptotics of the power.

### New Inequalities for the $p$-Angular Distance in Normed Spaces with Applications

Ukr. Mat. Zh. - 2015. - 67, № 1. - pp. 19–31

For nonzero vectors $x$ and $y$ in the normed linear space $(X, ‖ ⋅ ‖)$, we can define the $p$-angular distance by $${\alpha}_p\left[x,y\right]:=\left\Vert {\left\Vert x\right\Vert}^{p-1}x-{\left\Vert y\right\Vert}^{p-1}y\right\Vert .$$ We show (among other results) that, for $p ≥ 2$, $$\begin{array}{l}{\alpha}_p\left[x,y\right]\le p\left\Vert y-x\right\Vert {\displaystyle \underset{0}{\overset{1}{\int }}{\left\Vert \left(1-t\right)x+ty\right\Vert}^{p-1}dt}\hfill \\ {}\kern3.36em \le p\left\Vert y-x\right\Vert \left[\frac{{\left\Vert x\right\Vert}^{p-1}+{\left\Vert y\right\Vert}^{p-1}}{2}+{\left\Vert \frac{x+y}{2}\right\Vert}^{p-1}\right]\hfill \\ {}\kern3.36em \le p\left\Vert y-x\right\Vert \frac{{\left\Vert x\right\Vert}^{p-1}+{\left\Vert y\right\Vert}^{p-1}}{2}\le p\left\Vert y-x\right\Vert {\left[ \max \left\{\left\Vert x\right\Vert, \left\Vert y\right\Vert \right\}\right]}^{p-1},\hfill \end{array}$$, for any $x, y ∈ X$. This improves a result of Maligranda from [“Simple norm inequalities,” Amer. Math. Month., 113, 256–260 (2006)] who proved the inequality between the first and last terms in the estimation presented above. The applications to functions f defined by power series in estimating a more general “distance” $‖f(‖x‖)x − f(‖y‖)y‖$ for some $x, y ∈ X$ are also presented.

### On Four-Dimensional Paracomplex Structures with Norden Metrics

Ukr. Mat. Zh. - 2015. - 67, № 1. - pp. 32-41

We study almost paracomplex structures with Norden metric on Walker 4-manifolds and try to find general solutions for the integrability of these structures on suitable local coordinates. We also discuss para-Kähler (paraholomorphic) conditions for these structures.

### Inequalities for the Fractional Derivatives of Trigonometric Polynomials in Spaces with Integral Metrics

Ukr. Mat. Zh. - 2015. - 67, № 1. - pp. 42-56

We establish necessary and sufficient conditions for the validity of Bernstein-type inequalities for the fractional derivatives of trigonometric polynomials of several variables in spaces with integral metrics. The problem of sharpness of these inequalities is investigated.

### Kontsevich Integral Invariants for Random Trajectories

Ukr. Mat. Zh. - 2015. - 67, № 1. - pp. 57–67

In connection with the investigation of the topological properties of stochastic flows, we encounter the problem of description of braids formed by several trajectories of the flow starting from different points. The complete system of invariants for braids is well known. This system is known as the system of Vasil’ev invariants and distinguishes braids to within a homotopy. We consider braids formed by the trajectories $Z_k (t) = X_k(t) + iY_k (t)$ such that $X_k, Y_k , 1 ≤ k ≤ n$, are continuous semimartingales with respect to a common filtration. For these braids, we establish a representation of the indicated invariants in the form of iterated Stratonovich integrals.

### Global Existence and Long-Term Behavior of a Nonlinear Schrödinger-Type Equation

Ukr. Mat. Zh. - 2015. - 67, № 1. - pp. 68–87

We study a global mixed problem for the nonlinear Schrödinger equation with a nonlinear term in which the coefficient is a generalized function. A global solvability theorem for the analyzed problem is proved by using the general solvability theorem from [K. N. Soltanov, *Nonlin. Anal.: Theory, Meth., Appl.*, **72**, No. 1 (2010)]. We also investigate the behavior of the solution of the problem under consideration.

### On the Solvability of a Problem Nonlocal in Time for a Semilinear Multidimensional Wave Equation

Kharibegashvili S., Midodashvili B.

Ukr. Mat. Zh. - 2015. - 67, № 1. - pp. 88–105

We study a nonlocal (in time) problem for semilinear multidimensional wave equations. The theorems on existence and uniqueness of solutions of this problem are proved.

### On the Solvability of One Class of Nonlinear Integral Equations with a Noncompact Hammerstein–Stieltjes-Type Operator on the Semiaxis

Khachatryan Kh. A., Petrosyan A.

Ukr. Mat. Zh. - 2015. - 67, № 1. - pp. 106–127

We study a class of nonlinear integral equations with a noncompact operator of the Hammerstein–Stieltjes-type on the semiaxis. The existence of positive solutions is proved in various function spaces by using the factorization methods and specially chosen successive approximations.

### Admissibility of Estimated Regression Coefficients Under Generalized Balanced Loss

Luo J., Qiu H.-B., Zhang Jiajia

Ukr. Mat. Zh. - 2015. - 67, № 1. - pp. 128-134

There are some discussions concerning the admissibility of estimated regression coefficients under the balanced loss function in the general linear model. We study this issue for the generalized linear regression model. First, we propose a generalized weighted balance loss function for the generalized linear model. For the proposed loss function, we study sufficient and necessary conditions for the admissibility of the estimated regression coefficients in two interesting linear estimation classes.

### On the Asymptotics of Some Weierstrass Functions

Kharkevych Yu. I., Korenkov M. E., Zaionts Yu.

↓ Abstract

Ukr. Mat. Zh. - 2015. - 67, № 1. - pp. 135–138

For Weierstrass functions σ(z) and ζ(z), we present the asymptotic formulas valid outside the efficiently constructed exceptional sets of discs that are much narrower than in the known asymptotic formulas.

### On the Order of Growth of the Solutions of Linear Differential Equations in the Vicinity of a Branching Point

Mokhonko A. A., Mokhonko A. Z.

↓ Abstract

Ukr. Mat. Zh. - 2015. - 67, № 1. - pp. 139-144

Assume that the coefficients and solutions of the equation $f^{(n)}+p_{n−1}(z)f^{(n−1)} +...+ p_{s+1}(z)f^{(s+1)} +...+ p_0(z)f = 0$ have a branching point at infinity (e.g., a logarithmic singularity) and that the coefficients $p_j , j = s+1, . . . ,n−1$, increase slower (in terms of the Nevanlinna characteristics) than $p_s(z)$. It is proved that this equation has at most $s$ linearly independent solutions of finite order.